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arXiv:1003.1134v2 [hepth] 25 May 2010
Preprint typeset in JHEP style  HYPER VERSION
MPP201028
PUPT2331
Fermionic Operator Mixing in Holographic pwave
Superfluids
Martin Ammon,1∗Johanna Erdmenger,1†Matthias Kaminski,2‡and Andy O’Bannon1§
1MaxPlanckInstitut f¨ ur Physik (WernerHeisenbergInstitut)
F¨ ohringer Ring 6, 80805 M¨ unchen, Germany
2Department of Physics, Princeton University
Jadwin Hall, Princeton, NJ 08544, USA
Abstract: We use gaugegravity duality to compute spectral functions of fermionic operators
in a stronglycoupled defect field theory in pwave superfluid states. The field theory is (3+1)
dimensional N = 4 supersymmetric SU(Nc) YangMills theory, in the ’t Hooft limit and with
large coupling, coupled to two massless flavors of (2+1)dimensional N = 4 supersymmetric
matter. We show that a sufficiently large chemical potential for a U(1) subgroup of the
global SU(2) isospin symmetry triggers a phase transition to a pwave superfluid state, and
in that state we compute spectral functions for the fermionic superpartners of mesons valued
in the adjoint of SU(2) isospin. In the spectral functions we see the breaking of rotational
symmetry and the emergence of a Fermi surface comprised of isolated points as we cool the
system through the superfluid phase transition. The dual gravitational description is two
coincident probe D5branes in AdS5× S5with nontrivial worldvolume SU(2) gauge fields.
We extract spectral functions from solutions of the linearized equations of motion for the D5
branes’ worldvolume fermions, which couple to one another through the worldvolume gauge
field. We develop an efficient method to compute retarded Green’s functions from a system of
coupled bulk fermions. We also perform the holographic renormalization of free bulk fermions
in any asymptotically Euclidean AdS space.
Keywords: AdS/CFT correspondence, Gauge/gravity correspondence.
∗Email address: ammon@mppmu.mpg.de
†Email address: jke@mppmu.mpg.de
‡Email address: mkaminsk@princeton.edu
§Email address: ahob@mppmu.mpg.de
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Contents
1.Introduction and Summary1
2.Holographic Fermionic Operator Mixing
2.1Review: Free Fermions
2.2 Coupled Fermions
7
7
14
3.Probe Branes and Holographic pwave Superfluids
3.1pwaves, Probe Branes, and Vector Meson Condensation
3.2Probe Dpbranes in AdS5× S5
19
19
22
4.The Worldvolume Fermions
4.1Equation of Motion I: Reduction to AdS
4.2 The Dual Operators
4.3 Equation of Motion II: Gauge Couplings
4.3.1Normal Phase
4.3.2 Superfluid Phase
24
26
30
33
35
36
5.Emergence of the pwave Fermi surface
5.1Properties of the Spectral Function
5.2Numerical Results
38
38
39
6.Conclusions49
A. Holographic Renormalization of Fermions in AdS
A.1 Solving the Equation of Motion
A.2 Determining the Counterterms
A.3 Computing Renormalized Correlators
51
52
57
61
1. Introduction and Summary
The Antide Sitter/Conformal Field Theory correspondence (AdS/CFT) [1, 2, 3], and more
generally gaugegravity duality, is a holographic duality between a weaklycoupled theory of
gravity in some spacetime and a stronglycoupled field theory living on the boundary of that
spacetime. Gaugegravity duality thus provides a powerful new tool for studying strongly
coupled, scaleinvariant field theories in states with finite charge density, and hence may be
useful in condensed matter physics, for instance in understanding lowtemperature systems
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near quantum criticality [4, 5, 6, 7]. In particular, many special properties of certain high
Tcsuperconducting materials may be due to an underlying quantum critical point [4, 5, 7].
Gaugegravity duality may provide valuable insight into the physics of such materials.
Of central importance for potential condensed matter applications is the holographic
description of a Fermi surface1[9, 10, 11, 12]. On the field theory side, the minimal ingredients
are some stronglycoupled theory with a global U(1) symmetry, in a zerotemperature state
with a finite U(1) chemical potential, and some fermionic operator charged under the U(1).
Holographic calculations of the fermionic spectral function, as a function of frequency and
momentum, reveal a pole at zero frequency but finite momentum, which defines the Fermi
momentum. The pole represents an excitation about a Fermi surface.
On the gravity side, the minimal ingredients are gravity and a U(1) gauge field, plus some
bulk Dirac fermion charged under the U(1). The bulk geometry is a ReissnerNordstr¨ om black
hole. The bulk fermion is dual to the fermionic operator, and the spectral function of the
operator is extracted from solutions of the linearized bulk equation of motion, the Dirac
equation. These Fermi liquids are, generically, not Landau Fermi liquids, although the exact
properties depend on the mass and charge of the bulk fermion.
The bulk theory can also describe a phase transition to swave superfluid states, if a scalar
charged under the U(1) is present [13, 14, 15]. On the gravity side, the ReissnerNordstr¨ om
black hole grows scalar hair at low temperature, that is, a solution with a nontrivial scalar
becomes thermodynamically preferred to ReissnerNordstr¨ om. In the dual field theory, the
thermodynamicallypreferred state includes a nonzero expectation value for a scalar operator
charged under the global U(1), which we will refer to as the operator “condensing.” The
phase transition is second order with meanfield exponents [14, 15].
Gaugegravity duality can also describe pwave superfluids, that is, superfluids in which
the condensing operator is a vector charged under the U(1), thus breaking not only the U(1)
but also rotational symmetry (to some subgroup) [16]. On the gravity side, the minimal
ingredients are gravity and nonAbelian gauge fields. The simplest case is an AdS geometry
and SU(2) gauge fields, Aa
M, with Lorentz index M and a = 1,2,3 labels the SU(2) generators
τa. Here the U(1) is a subgroup of SU(2), for example the U(1) in the τ3direction, which we
will call U(1)3. At high temperature the thermodynamically preferred geometry is Reissner
Nordstr¨ om with nonzero A3
t. At low temperature, the charged black hole grows vector hair:
the preferred solution has nontrivial A1
x. The dual field theory has three conserved currents,
Jµ
breaks SU(2) to U(1)3, and the transition occurs at large chemical potential, where the
thermodynamically preferred state has nonzero ?Jx
In bulk calculations for both the s and pwave, a major technical simplification is the so
called probe limit, in which the charge of the bulk scalar, or the SU(2) YangMills coupling, is
sent to infinity, so that the scalar or YangMills stressenergy tensor on the righthand side of
Einstein’s equation becomes negligible. The bulk calculation then reduces to solving the scalar
a, dual to the gauge fields. A chemical potential, producing a finite density ?Jt
3?, explicitly
1?.
1For an alternative approach, see ref. [8].
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or YangMills equation of motion in a fixed ReissnerNordstr¨ om background. The probe limit
is sufficient to detect the transitions and determine that they are second order. In either case,
however, if we cool the system then, as shown in refs. [15], the matter fields’ stressenergy
tensor grows and we can no longer trust the probe limit. Reaching zero temperature requires
solving the fullycoupled equations, as done in refs. [15, 17, 18].
The zerotemperature limits of the bulk hairy black hole solutions generically involve
a domain wall interpolating between two regions, one near the boundary and one deep in
the interior of the spacetime. For example, the geometry may interpolate between a near
boundary AdS space and an interior AdS space with a different radius of curvature and speed
of light [19, 20, 21, 22, 23]. In field theory language, the interior AdS space represents an
emergent conformal symmetry at low temperature and finite charge density. In other words,
the emergent AdS represents a quantum critical point.
Holographic calculations of fermionic spectral functions in zerotemperature swave su
perfluid states [24, 25, 26, 27] exhibit the socalled ‘peakdiphump’ structure [24], expected
to be relevant in highTcsuperconductors [28], as well as, for suitable mass and charge of the
bulk fermion, continuous bands of poles [26] and, for suitable coupling to the bulk scalar, a
gap, i.e. poles in the spectral function at nonzero momentum and nonzero frequency [25].
Generally, the bulk actions used in holographic constructions of superfluids and Fermi
surfaces are not derived from any particular string theory construction. In other words,
they are basically ad hoc models built from the minimal ingredients needed to capture the
essential physics. Simple models have one big advantage (besides simplicity!), namely a kind
of universality: the results may be the same for many different theories, regardless of the
details of their dynamics.
On the other hand, knowing the detailed dynamics of a specific dual theory, meaning
the fundamental fields and Lagrangian of some microscopic, weakcoupling description, also
has advantages. For example, the holographic results may tell us that a superfluid phase
transition occurs, but may not tell us why. Is a nonzero ?Jx
mechanism? If so, is the pairing mechanism the same in every dual theory? Knowing an
exact dual theory may help to answer such questions, for example by providing some weak
coupling intuition.2Finding a dual Lagrangian means “embedding” the bulk theory into a full
string or supergravity construction, built for example from Dbranes (for which we know the
worldvolume theories). String (and M) theory embeddings of holographic swave superfluids
appear in refs. [20, 30, 31].
A string theory embedding of holographic pwave superfluids, in the probe limit, appears
in refs. [32, 33, 34, 35]. Here we begin with Nc Dqbranes and Nf Dpbranes. Taking
the usual decoupling limit for the Dqbranes, which in particular means Nc→ ∞, we obtain
supergravity in the nearhorizon geometry of the Dqbranes. Nonextremal Dqbranes produce
a black hole geometry. The probe limit consists of keeping Nf fixed as Nc → ∞, such
that Nf/Nc→ 0. The dynamics of Nf coincident Dpbranes is then described by the non
1? the result of some pairing
2A good recent example is ref. [29].
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Abelian BornInfeld action (plus WessZumino terms) in the nearhorizon Dqbrane geometry.
Truncating that action to leading order in the field strength, we obtain a YangMills action
in a black hole geometry.
We then know precisely what the dual field theory is: the Dqbrane worldvolume theory,
with gauge group SU(Nc), in the largeNcand strong coupling limits, coupled to a number
Nf of fields in the fundamental representation of the gauge group, i.e. flavor fields. We
will call this the Dq/Dp theory. If the Dpbranes do not overlap with all q spatial Dqbrane
directions, then the flavor fields will only propagate along a defect. The probe limit consists
of neglecting quantum effects due to the flavor fields, such as the running of the coupling,
because these are suppressed by powers of Nf/Nc. These theories generically have bound
states similar to mesons in Quantum Chromodynamics (QCD). The U(Nf) gauge invariance
of the Dpbranes is dual to a global U(Nf) analogous to the isospin symmetry of QCD. In
such systems the pwave transition occurs when a sufficiently large isospin chemical potential
triggers vector meson condensation (as we review in section 3).
Our goal is to use such a string theory system to compute fermionic spectral functions
in the pwave phase.
We choose our Dqbranes to be D3branes. The dual field theory is then a CFT, N = 4
supersymmetric YangMills (SYM) theory with gauge group SU(Nc) in the ’t Hooft limit of
Nc→ ∞ with large ’t Hooft coupling, λ ≡ g2
is (4+1)dimensional AdS times a fivesphere, AdS5×S5, with Ncunits of RamondRamond
(RR) fiveform flux on the S5. At finite temperature AdS becomes AdSSchwarzschild.
We consider supersymetric probe Dpbranes extended along AdSP× SQ, where super
symmetry requires P − Q = 2 [36]. We focus on P ≥ 3, since only in those cases is a vector
condensate ?Jx
we will only study solutions in which all Dpbrane worldvolume scalars are trivial.
Our bulk fermions will be the Dpbranes’ worldvolume fermions. These fermions are in
a supermultiplet with the worldvolume scalars and gauge field, hence they are in the adjoint
of the worldvolume U(Nf), and couple to the gauge field via the gaugecovariant derivative.4
In other words, supersymmetry determines the charges of the fermions. For example, we will
use Nf= 2, where we find three fermions with charges +1, −1 and 0 under U(1)3.
To compute fermionic spectral functions we need the linearized equations of motion, the
Dirac equation, for these fermions. Fortunately, the fermionic part of the Dbrane action,
for Dbranes in arbitrary backgrounds (including RR fields) is known to quadratic order
[37, 38, 39]. The form of the action is determined by supersymmetry and Tduality [39], as
we review in section 4. For our Dpbranes extended along AdSP×SQ, we perform a reduction
on the SQto obtain a Dirac equation in AdSP, following ref. [40] very closely. The spectrum
of AdSPfermion masses are fixed by P, Q and the coupling to the background RR fiveform.
We emphasize a major difference between our systems and the models of refs. [9, 10, 11]:
in our embedding of the Dirac equation into string theory, the mass and charge of the fermions
Y MNc→ ∞. The nearhorizon gravity solution
1? possible.3We will study only trivial embeddings of such Dpbranes, that is,
3One exception is a D5brane along AdS2× S4, which we study in section 4.1 (but not in pwave states).
4Like all worldvolume fields, they are not charged under the diagonal U(1) ⊂ U(Nf).
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are fixed by supersymmetry and Tduality. We are not free to dial the values of the mass and
charge, unlike refs. [9, 10, 11].
Much of our analysis will be valid for any supersymmetric Dpbrane extended along
AdSP× SQ, with P ≥ 3, but one particular Dpbrane is attractive for a number of reasons,
namely the D5brane extended along AdS4× S2(P = 4 and Q = 2). From the bulk point of
view, this D5brane is the only Dpbrane with a massless worldvolume fermion, as we show
in section 4.1. That makes both our numerical analysis, and comparison to refs. [9, 11] (in
which the fermions were massless), much easier.
With two coincident P = 4, Q = 2 D5branes, the dual field theory is (3+1)dimensional
N = 4 SYM coupled to Nf = 2 massless (2+1)dimensional N = 4 supersymmetric flavor
fields. The classical Lagrangian of the theory, with couplings that preserve the SO(3,2)
conformal symmetry of the (2+1)dimensional defect, appears explicitly in refs. [41, 42].
We write the explicit form of the fermionic operators dual to the D5branes’ worldvolume
fermions in section 4.2, following refs. [40, 41] very closely. These fermionic operators are
mesinos, the supersymmetric partners of mesons.
The P = 4, Q = 2 D5brane is also attractive for potential condensed matter appli
cations.As mentioned in ref.[43], many real condensed matter systems are effectively
(2+1)dimensional degrees of freedom interacting with ambient (3+1)dimensional degrees
of freedom. The D3/D5 theory also exhibits a rich phase structure, explored in detail in
refs. [43, 44, 45, 46, 47, 48, 49, 50], including for example a BerezinskiiKosterlitzThouless
transition (with finite charge density and magnetic field for the diagonal U(1) ⊂ U(2)) [51].
Here we focus on the D3/D5 theory’s phase diagram with finite isospin chemical potential.
As always in the probe limit, we cannot access the T = 0, finite chemical potential
ground state. The P = 4, Q = 2 D5brane is again attractive, however, because we know that,
unlike many Dpbranes, with zero temperature and zero chemical potential, fully backreacted
solutions appear to preserve an AdS factor in the geometry, namely an AdS4[52, 53]. That
suggests that the field theory retains SO(3,2) conformal invariance even including quantum
effects due to the flavor, which was indeed proven in ref. [42]. Whether some scale invariance
emerges with zero temperature and finite chemical potential is unclear.
On a technical level, our goal is to solve the Dirac equation for a massless fermion in the
adjoint of SU(2) confined to an AdSPsubmanifold of (4+1)dimensional AdSSchwarzschild.
For any Dpbrane, the three worldvolume fermions decouple in the normal (nonsuperfluid)
phase, where A1
nonzero. These couplings indicate that, in the field theory, the dual fermionic operators
experience operator mixing under renormalization group flow [54, 55]. In the field theory,
the retarded Green’s function, and hence the spectral function, becomes a matrix with off
diagonal entries.
We thus develop a method to compute the retarded Green’s function for bulk fermions
coupled to one another. Our method is essentially a combination of the method of ref. [54, 55],
for coupled bosonic fields, with the method of ref. [10, 56], for free fermions. Our method
is actually very general, i.e. applicable to any system of coupled bulk fermions, not just to
xis zero, but couple to one another in the superfluid phase, where A1
xis
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fermions on the worldvolume of probe Dpbranes, and is especially convenient for numerical
analysis. We thus explain our method first, in section 2.
As an added bonus, we also perform, to our knowledge for the first time, holographic
renormalization for fermions in AdS.5More precisely, we study a single free fermion in any
space that asymptotically approaches Euclideansignature AdS and determine the countert
erms needed to render the onshell action finite without spoiling the stationarity of the action.
Our results rigorously justify many of the ad hoc prescriptions used in the literature, where
divergences of the onshell action were simply discarded.
For the P = 4, Q = 2 D5brane, using our method for coupled bulk fermions, we
numerically compute spectral functions for mesinos as we cool the system through the p
wave superfluid phase transition. Due to the operator mixing, or equivalently the coupling of
the fermions in the bulk, we see that the spectral function of even a neutral fermion develops
a nontrivial feature, a peak, as the system enters the pwave phase.
Furthermore, as we lower the temperature, the zerofrequency spectral measure6is clearly
no longer rotationally invariant, and in fact at the lowest temperatures we can reliably access
in the probe limit, the main features of the spectral measure are five largely isolated peaks
in the (kx,ky) plane, two on the kxaxis, two on the kyaxis, and one at the origin. These
results are very similar to the T = 0 results of ref. [18], where the bulk theory was gravity
and SU(2) gauge fields in (3+1)dimensions, in the T = 0 vectorhairy black hole geometry.
In that case, for a fermion in the fundamental representation of SU(2), the spectral measure
consisted of two points on the kxaxis, located symmetrically about the origin. The prediction
of ref. [18] for fermions in the adjoint representation would be three points on the kxaxis,
one at the origin and two at finite kx, positioned symmetrically about the origin. At finite
temperature we see five points emerging, but we strongly suspect that, if we could access the
T = 0 limit, we would indeed see only three points, as we discuss in section 5.
We cannot resist drawing an analogy between our system and certain experimentally
realized pwave superconductors (see also ref. [58]).7In that context, a“reduction of the
Fermi surface” to certain points in momentum space has been proposed for the ruthenate
compound Sr2RuO4[59]: the pwave state is supported by ferromagnetic fluctuations that
increase the propensity for electrons to form spin triplet Cooper pairs, with an odd (pwave)
Cooper pair wave function.8Scattering channels with momentum transfer Q = (0,0), as is the
case in a ferromagnet, should be enhanced in the system, as opposed to scattering channels
of Q = (π,π), which is the case in an antiferromagnet. Small momentum transfer is best
accomplished by a strongly peaked density of states at the Fermi level, as occurs for example
with van Hove singularities, where the density of states diverges. This lies at the heart of the
5For the holographic renormalization of fermions in Schr¨ odinger spacetime, see ref. [57].
6As mentioned above, the retarded Green’s function, and hence the spectral function, is generically a matrix.
The spectral measure is simply the trace of the spectral function.
7We thank Ronny Thomale for many useful conversations about real pwave superconductors.
8This is rather particular, bearing in mind that a large number of generic spin interactions, for example
induced by superexchange processes, favor antiferromagnetic fluctuations.
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strong suspicion that a Fermi surface localized to certain points with a high density of states
may account for a suitable setup to support pwave pairing.
The paper is organised as follows. In section 2, we describe our method for computing
retarded Green’s functions for coupled bulk fermions. In section 3 we review general features
of Dq/Dp holographic pwave superfluids and demonstrate a pwave transition using the
P = 4, Q = 2 D5brane. In section 4, we write the fermionic part of the Dpbrane action,
perform the reduction of the worldvolume Dirac equation to AdSP, and, for the P = 4,
Q = 2 D5brane, match bulk fermions to dual field theory operators. In section 5 we present
our numerical results for the fermionic retarded Green’s functions using the P = 4, Q = 2
D5brane. We conclude with suggestions for future research in section 6. The holographic
renormalization of fermions in AdS appears in the appendix.
Section 2 and the appendix are technical, and may be read independently of the rest
of the paper. Readers who only want to understand our system and our numerical results
should read at least sections 3.2, 4.3 and 5.
2. Holographic Fermionic Operator Mixing
2.1 Review: Free Fermions
We begin by studying a single free fermion in AdS space. In particular we will review how to
extract the field theory fermionic twopoint function from a solution for a bulk Dirac fermion,
following refs. [9, 11, 60].
In this section we will work mainly with Euclideansignature AdS space, with the metric
written in FeffermanGraham form,9
ds2= gABdxAdxB=du2
u2+1
u2δijdxidxj.(2.1)
The boundary is at u = 0. Notice that throughout the paper we use units in which the radius
of AdSd+1is equal to one.
We will study a bulk Dirac spinor Ψ. The Dirac action (plus boundary terms) is
S =
?
dd+1x√g?¯Ψ?DΨ − m¯ΨΨ?+ Sbdy, (2.2)
where, picking one of the spatial directions to be “time,” with corresponding γt, we define
¯Ψ = Ψ†γt. We write the AdSd+1Dirac operator ?D below. Here Sbdyincludes boundary terms
that do not affect the equation of motion.
The AdS/CFT correspondence is the statement that a theory of dynamical gravity on
AdSd+1is equivalent to a d (spacetime) dimensional CFT that “lives” on the boundary of
AdSd+1. Every bulk field is dual to some operator in the boundary CFT. The precise state
ment of the correspondence equates the onshell bulk action with the generating functional
9Capital Latin letters A,B,... will always denote all the AdSd+1 directions, including the radial direction
u, while lowercase Latin letters will denote field theory directions: i,j = 1,...d.
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of connected CFT correlation functions. The bulk field Ψ is dual to some fermionic operator
O in the dual ddimensional field theory. The onshell bulk action, S, acts as the generating
functional for correlators involving O. In other words, to compute renormalized correlators
of O, we take functional derivatives of S with respect to some source.
Generically, however, both the onshell bulk action and the CFT generating functional
diverge. On the bulk side, the divergences arise from the infinite volume of AdSd+1, i.e. they
are longdistance or infrared (IR) divergences. In the field theory the divergences are short
distance, ultraviolet (UV) divergences. To make the AdS/CFT correspondence meaningful
we must regulate and renormalize these divergences.
Holographic renormalization proceeds as follows (see ref. [61] and references therein). We
first regulate the onshell bulk action by introducing a cutoff on the integration in the radial
direction: we integrate not to u = 0 but to some u = ǫ. We then add counterterms on the
u = ǫ surface to cancel any terms that diverge as we remove the regulator by taking ǫ → 0.
Generically, the form of the counterterms is fixed by symmetries, and the coefficients of the
counterterms are adjusted to cancel the divergences. Once the counterterms are known, we
can proceed to compute functional derivatives of the onshell bulk action, always taking ǫ → 0
in the end, thus obtaining renormalized CFT correlation functions in a way that is manifestly
covariant and preserves all symmetries.
As first observed in ref. [62], when we evaluate the Dirac action on a solution, the bulk
term obviously vanishes. The nonzero contribution to the onshell action comes from Sbdy,
which involves terms localized on the u = ǫ surface. As observed in ref. [63], the form of Sbdy
is fixed by demanding a welldefined variational principle for the Dirac action. Formally, Svar
thus includes two types of terms,
Sbdy= Svar+ SCT,(2.3)
where Svarare the terms required for the variation of the action to be welldefined [63], while
SCT are the counterterms, which do not affect the variation of the action.
In the appendix we perform the holographic renormalization of the Dirac action. In
particular, we determine the counterterms in SCT. The details of holographic renormaliza
tion are wellknown for various species of bulk fields, for example for the metric [64], scalar
fields [64], and gauge fields [65]. To our knowledge, the only detailed analysis of holographic
renormalization for fermions was in the (more complicated) context of nonrelativistic gauge
gravity duality, in ref. [57]. As shown in the appendix, however, in the relativistic case the
holographic renormalization procedure for fermions very closely parallels the procedure for
scalars.
As shown in the appendix, the details of the holographic renormalization depend on the
value of m. Some values of m are special, for example when m is halfinteger (in units of
the AdSd+1 radius), counterterms logarithmic in ǫ (rather than just polynomial in ǫ) are
needed. For simplicity, in this section we will restrict to values of m that are positive and not
halfinteger. Our arguments are easy to generalize to any value of m.
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In this section we will also restrict to four and fivedimensional AdS spaces, which we
will collectively denote as AdSd+1with d = 3,4, primarily for pedagogical reasons: in these
cases the bulk Dirac spinor has four complex components, and we can write explicit 4×4 bulk
Dirac Γmatrices. Additionally, we note that AdSd+1spaces with d ≤ 4 are the cases most
relevant for condensed matter applications (as opposed to, say, AdS7). The generalization to
other dimensions is straightforward. In the appendix we work with arbitrary d.
In later sections we will be interested in computing finitetemperature, realtime correla
tion functions, in particular the retarded Green’s functions, in which case the bulk geometry
will be Lorentziansignature AdSSchwarzschild. We review the prescription for obtaining the
retarded Green’s function in such cases at the end of this subsection.
Varying the above action we obtain the bulk equation of motion, the Dirac equation,
eM
AγADMΨ − mΨ = 0,(2.4)
where eM
curvedspace covariant derivative is
A= uδM
Aare the inverse vielbeins associated with the metric in eq. (2.1).10The
DM= ∂M+1
4(ωM)AB
?γA,γB?, (2.5)
where (ωM)ABis the spin connection associated with the metric in eq. (2.1). The only
nonzero components of the spin connection are (ωi)uj=1
components of DMare
Di= ∂i+1
4
We can now simplify the Dirac equation,
uδij, so that Du= ∂uand the other
1
u
?γu,γi?. (2.6)
0 = eM
= uγM∂MΨ +1
AγADMΨ − mΨ
4γi?γu,γi?Ψ − mΨ
2γu− m
=
?
uγM∂M−d
?
Ψ.(2.7)
We will work with a single Fourier mode, so we let Ψ → eikxΨ, where, without loss of
generality, we have chosen the momentum to point in the ˆ x direction.11The Dirac equation
is then
?
We will now choose an explicit basis for the Γmatrices. We will use a basis in which all
the Γmatrices are Hermitian,
uγu∂u+ ikuγx−d
2γu− m
?
Ψ = 0.(2.8)
γu=
?
−σ3
0
0
−σ3
?
,γt=
?
σ1 0
0 σ1
?
,γx=
?
−σ2 0
0σ2
?
,(2.9)
10Recall that for inverse vielbeins, the upper index is general coordinate and the lower index is local Lorentz.
The γAobey the usual algebra {γA,γB} = 2δAB.
11In a pwave superfluid phase rotational symmetry is broken, so there, to study the most general case, we
must use a momentum with nonzero components in different directions, as we will discuss in section 4.
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where σ1, σ2and σ3are the usual Pauli matrices,
σ1=
?
0 1
1 0
?
,σ2=
?
0 −i
i 0
?
,σ3=
?
1 0
0 −1
?
. (2.10)
Next we will define two sets of projectors. The first set is
Π+=1
2(1 + γu) =
0
1
0
1
,Π−=1
2(1 − γu) =
1
0
1
0
.(2.11)
We use these to define Ψ±=1
was used for example in refs. [9, 11],
2(1 ± γu)Ψ so that γuΨ±= ±Ψ±. The second set of projectors
Π1=1
2
?1 + iγuγtγx?=
0
0
1
1
,Π2=1
2
?1 − iγuγtγx?=
1
1
0
0
. (2.12)
To make converting between Ψ±and Ψ1,2easy, we explicitly write Ψ first as Ψ++ Ψ−and
then as Ψ1+ Ψ2,
where the subscripts u and d indicate the “up” and “down” components of the effectively
twocomponent Ψ±and Ψ1,2. Identifications such as Ψ+u= Ψ2dare then obvious.
We have a choice of whether to use Ψ±or Ψ1,2, although of course, we can easily translate
between the two options using eq. (2.13). We will choose whatever is most convenient for a
given question.
For example, the projectors Π1,2commute with the operator in eq. (2.8), which tells us
that, for a free fermion, the equations for Ψ1,2decouple. That makes Ψ1,2especially attractive
for numerical analysis, hence we employ them in sections 4 and 5.12Explicitly, the equations
for Ψ1,2are
Ψ =
0
Ψ+u
0
Ψ+d
+
Ψ−u
0
Ψ−d
0
=
0
0
Ψ1u
Ψ1d
+
Ψ2u
Ψ2d
0
0
,(2.13)
?
?
u∂u−d
u∂u−d
2+ mσ3− ku
?
?
Ψ1= 0,(2.14)
2+ mσ3+ kuΨ2= 0.(2.15)
12As mentioned in footnote 11, in the pwave superfluid phase, the most general momentum has nonzero
components in multiple directions. That means Ψ1 and Ψ2 will no longer decouple because other Γmatrices,
such as γy, will appear in the equation of motion, and these do not commute with Π1,2. Nevertheless, when
studying the pwave superfluid phase we use Ψ1,2 to make the comparison with the rotationallysymmetric
case easier.
– 10 –
Page 12
On the other hand, the asymptotic behavior of Ψ is most succinctly described using Ψ±,
hence we use these frequently below, especially in the appendix. In terms of Ψ±, the equation
of motion becomes
?
?
u∂u−d
u∂u−d
2− m
?
?
Ψ++ kuσ3Ψ−= 0, (2.16)
2+ mΨ−+ kuσ3Ψ+= 0. (2.17)
These firstorder equations give rise to the secondorder equations
?
∂2
u−d
u∂u+1
u2
?
−m2± m +d2
4+d
2
?
− k2
?
Ψ±= 0. (2.18)
The leading asymptotic behaviors of Ψ±are
Ψ±= c±(k)u
d
2±m+ O
?
u
d
2+1±m?
. (2.19)
where c±(k) are spinors that obey Π±c±(k) = ±c±(k), and which may depend on k, as
indicated.
As reviewed above, to compute renormalized correlators of the dual operator O, we take
functional derivatives of S with respect to some source. We identify the source for O as the
coefficient of the dominant term in Ψ’s nearboundary expansion (the term that grows most
quickly as u → 0). From eq. (2.19), we see that the dominant term is the u
we identify c−(k) as the source for O. More formally, we equate
?
where the lefthandside is the exponential of minus the action in eq. (2.2), evaluated on
a solution and properly renormalized (hence the subscript), and the righthandside is the
generating functional of the dual field theory, with c−(k) acting as the source for the operator
O.13Upon taking minus the logarithm of both sides, we find that the onshell bulk action is
the generator of connected correlators.
For bulk bosonic fields, we must solve a straightforward Dirichlet problem: we fix the
leading asymptotic value of the field, allow the field to vary, and then impose a regularity
condition in the interior of the space to fix the entire solution. This procedure is dual to
the statement that once we choose a source, the dynamics of the theory determines the
expectation values of the dual operator.
The story for fermions is more subtle, because Ψ+(u,k) and Ψ−(u,k) are not independent
[62, 63]. Each one determines the canonical momentum associated with the other (see for
example ref. [60]). In the bulk Dirichlet problem, then, we cannot fix their asymptotic values
d
2−mterm, hence
e−Sren[c−,¯ c−]= exp
??
ddx?¯ c−O +¯ Oc−
???
,(2.20)
13As we review in the appendix, for a bulk fermion with mass m, in a standard quantization the dimension
∆ of O is ∆ =d
2+ m [62, 63]. In the appendix we also discuss the chirality of O (when d is even).
– 11 –
Page 13
c±(k) simultaneously, but can fix only one, the coefficient of the dominant term, c−(k), and
then vary the field. As shown in refs. [63], for the action to remain stationary under such
variations, we must add a boundary term to the action,
Svar=
?
ddx√γ¯Ψ+Ψ−, (2.21)
where the integration is over the u = ǫ hypersurface,√γ = ǫ−dis the square root of the
determinant of the induced metric at u = ǫ, and Ψ±are evaluated at u = ǫ.
Indeed, since the bulk action is firstorder in derivatives, the only nonzero contribution
to the onshell action comes from the boundary terms Sbdy = Svar+ SCT.
when evaluated on a solution, divergent terms appear in Svar, which are canceled by the
counterterms in SCT. Notice that, to preserve stationarity of the action, SCT must involve
only Ψ−(ǫ,k), since that is held fixed under variations. We write the counterterms explicitly
in the appendix.
The principal result of the appendix is the renormalized onshell action: we evaluate Sbdy
on a solution and take ǫ → 0 to obtain (for positive, nonhalfinteger m)
?
We can now easily compute the renormalized connected correlators of O and¯ O by taking
functional derivatives of Sren. For example, the renormalized onepoint function of¯O is
Generically,
Sren=ddx¯ c+c−, (2.22)
?¯O?
ren= −δSren
δc−
= −¯ c+. (2.23)
If we use the fact that the onshell bulk action must be Hermitian, S = S†, then we also have
Sren= S†
ren=
?
ddx [¯ c+c−]†=
?
ddx¯ c−c+,
hence we also find, as we should,
?O?ren= −δSren
δ¯ c−
= −c+. (2.24)
We can obtain twopoint functions via second functional derivatives, for example
?O¯ O?
ren= −δ2Sren
δc−δ¯ c−
= −δc+
δc−. (2.25)
The equation of motion plus some regularity condition in the interior of the spacetime will
relate c+and c−(recalling that we fix c−and vary c+). The equation is linear, hence the
relation will be linear: c+= −G(k)γtc−, for some matrix G(k) which will turn out to be the
Euclidean Green’s function. We include a factor of γtbecause, as discussed in refs. [9, 11],
– 12 –
Page 14
the Euclidean Green’s function is actually?OO†?
?O¯O?
In general, we must extract G(k)γtfrom a solution by imposing some regularity condition in
the bulk of the spacetime (in our coordinates, the u → ∞ region), which fixes c+in terms
of c−. We review that procedure for Euclidean AdSd+1in the appendix and for Lorentzian
signature AdSSchwarzschild below.
We can also reproduce the formulas used in refs. [9, 11] by switching to Ψ1,2. In that
case, the equations for Ψ1and Ψ2decouple, hence in the Green’s function the Π1and Π2
subspaces will not mix. Writing c+= −G(k)γtc−explicitly, we will have (suppressing the k
dependence of c±(k))
ren, which differs from?O¯ O?
?
renby a factor
of γt. We indeed find
ren= G(k)γt,
OO†?
ren= G(k).(2.26)
where blank entries represent zero, 12is the 2×2 identity matrix, and G11and G22represent
the components of the Green’s function in the Π1and Π2subspaces, respectively. Given a
bulk solution for Ψ, we obtain the Green’s functions simply by reading off the asymptotic
values of c+(k) and c−(k) and then constructing
G22(k) = −c+u
0
c+u
0
c+d
= −
?
G22(k)12
G11(k)12
?
0 1
1 0
0 1
1 0
c−u
0
c−d
0
= −
?
G22(k)12
G11(k)12
?
0
c−u
0
c−d
,
c−u,G11(k) = −c+d
c−d.(2.27)
Finally, we review the prescription of ref. [60] to compute the retarded twopoint function
in the finitetemperature, Lorentziansignature case. Here the geometry is AdSSchwarzschild,
with a horizon at some position uh. To obtain the retarded twopoint function, we require
that, near the horizon, the bulk solution for Ψ has the form of wave traveling into the horizon
(out of the spacetime), i.e. an ingoing wave. The asymptotic form for Ψ near the boundary
is the same as in eq. (2.19) (for positive, nonhalfinteger m). Following ref. [60], in the
regime of linear response, we have
c+(ω,k) = −iGR(ω,k)γtc−(ω,k),(2.28)
where GR(ω,k) is the retarded Green’s function. Notice that here we distinguish the frequency
ω from the momentum k, and γtis now antiHermitian,
γt=
?
iσ1 0
0 iσ1
?
.(2.29)
Eq. (2.28) is essentially just an analytic continuation from the Euclidean case: γt→ iγt. For
a free fermion, we obtain (see also eq. (A17) of ref. [11])
22(ω,k) =c+u
c−u,
GR
GR
11(ω,k) =c+d
c−d. (2.30)
– 13 –
Page 15
2.2 Coupled Fermions
We now consider multiple bulk fermions, say N of them, Ψawith a = 1,...,N, coupled to
one another. The fact that the linearized fluctuation of the Ψacouple in the bulk is dual to
the statement that the fermionic operators in the field theory mix with one another under
renormalization group flow.
We will work in Lorentzian signature, and finite temperature, so that the bulk geometry is
AdSSchwazrschild, with a horizon at some position uh. We consider fermions with quadratic
couplings of the form (with implicit summation over a,b)
S = i
?
dd+1x√g?¯Ψa?DΨa−¯ΨaΛabΨb
?+ Sbdy,(2.31)
for some matrix Λabthat need not be diagonal in either the a,b indices or the spinor indices.
As a concrete example, in later sections we will introduce a bulk SU(2) gauge field AM and
a bulk fermion valued in the adjoint of SU(2). The indices a,b are then SU(2) indices, hence
we will have three bulk fermions (for τ1, τ2, and τ3) with a coupling, coming from the gauge
covariant derivative, of the form ǫabc¯ΨaeM
either SU(2) indices or in spinor indices (because of the γA).
For the following arguments, we do not need to know any details about the equations of
motion. We will only exploit one important feature. Using the Π±projectors, we will always
obtain equations similar to eqs. (2.16) and (2.17). We will then always be able to write these
equations in the form
∇ab±Ψb±= Mac±Ψc∓,
where ∇ab±is some differential operator, involving in particular ∂u, and Mac±is a matrix
representing the couplings among not only the Ψa, which come from Λab, but also the terms
from ?DΨathat produce couplings between Ψa+and Ψa−, for example the terms proportional
to the momentum k in eqs. (2.16) and (2.17). The key feature is that only the Ψa±are on
the lefthandside, while only the Ψa∓are on the righthandside.
In practical terms, the total number of complex functions for which we must solve is
4 × N, since each Ψahas four complex components. In other words, we need to decompose
the Ψanot only into Ψa+and Ψa−, but also into the up and down components, Ψa+u, Ψa+d,
Ψa−u, and Ψa−d. When convenient, we may sometimes think of eq. (2.32) as equations
describing these 4 × N coupled functions, which we may sometimes refer to as “fields.”
Clearly, if we solve for all the Ψa, insert the solutions into the bulk action, and take
functional derivatives, we will obtain field theory retarded Green’s functions that are matrices,
GR
ab(ω,k). In principle, we may be able to diagonalize the equations of motion and obtain
decoupled equations, in which case the Green’s function will be diagonal. Given the bulk
solutions for the Ψa, we then extract the elements of GR
however, diagonalizing the equations of motion may be prohibitively difficult, i.e. practically
impossible. We can always resort to numerics to find solutions, but we will then be forced to
compute elements of the undiagonalized GR
ab(ω,k). We thus need to know what combinations
of the asymptotic values ca+and ca−give an arbitrary element GR
AγA(AM)bΨc, which is obviously not diagonal in
(2.32)
ab(ω,k) using eq. (2.30). In some cases,
ab(ω,k).
– 14 –
Page 16
We will describe a prescription to obtain the matrix GR
solutions for the Ψa. The method is a hybrid of the methods in refs. [54, 55] and refs. [10, 56].
Refs. [54, 55] described a general method to construct a retarded Green’s function for coupled
bulk scalar and gauge fields, while refs. [10, 56] described general methods for computing
Green’s functions from fermions in the bulk.
The first observation is that we can construct secondorder equations for the bulk fields,
the Ψa±, that will be similar to eq. (2.18). We actually don’t care about the exact form of
these equations. We only need to know that such equations exist. We thus have a system of
2N secondorder linear equations, for which we expect 2×2N linearlyindependent solutions.
We must therefore fix two boundary conditions for each field to specify a solution for the
entire system. Following refs. [55], we fix these boundary conditions near the horizon uh. For
example, the Ψa+uwill have the nearhorizon form
ab(ω,k), assuming we have bulk
Ψa+u= na+u(u − uh)iα+ ....(2.33)
where na+uand α are constants (independent of u) and ... represents terms that decay faster,
as u → uh, than the terms shown. The two constants na+uand α are the two degrees of
freedom we have to specify the solution. Generically, the equation of motion will only be
satisfied for two values of α, one describing an ingoing wave and the other describing an
outgoing wave. As is wellknown, to obtain the retarded Green’s function, we must use an
ingoing wave. We still need to choose the normalization na+u. As shown in refs. [9, 11],
for fermions, once we choose an ingoing wave solution, if we use the projectors Π1,2, then
when we fix the normalization of the up component Ψa1uto be na1u, the equation of motion
fixes the down component Ψa1dto have normalization i times na1u. The same applies to the
up and down components of Ψa2. Switching to the Π± projectors (recall eq. (2.13)), the
statement is that once we fix the normalization of Ψa−dto be na−d, then Ψa+dmust have
normalization i times na−d. The same statement applies to Ψa−uand Ψa+u.
We thus need only fix 2N normalizations, for the up and down components of the Ψa−.
Let us arrange these normalizations into a row vector ? n
? n = (n1−u,n1−d,n2−u,n2−d,...,nN−u,nN−d). (2.34)
Following refs. [55], we use these horizon normalizations to construct a basis of solutions
as follows. We solve the equations of motion 2N times, each time with a different choice
of ? n. The first time we use ? n = (+1,+1,+1,...,+1,+1), the second time we use ? n =
(+1,−1,+1,...,+1,+1), the third time we use ? n = (+1,+1,−1,... ,+1,+1), and so on. We
label these choices ? n(i), with i = 1,...,2N. For each choice of normalizations, we obtain
solutions Ψ(i)
a±. We now have a basis of solutions, so we can write any particular solution as
a linear combination of these. To do so, we construct matrices that we will call˜P±
from the basis solutions, where each row corresponds to a field and each column corresponds
to a choice of normalization (the i index). For example, (suppressing the Ψa−’s dependence
aj(u,ω,k)
– 15 –
Page 17
on all variables)
?˜P−
aj(u,ω,k)
?
=
Ψ(1)
Ψ(1)
...
Ψ(1)
1−Ψ(2)
2−Ψ(2)
1−... Ψ(2N)
2−... Ψ(2N)
...
N−Ψ(2)
1−
2−
...
N−... Ψ(2N)
N−
,(2.35)
with˜P+
venience, we will factor out the leading asymptotic behavior of the solutions, defining new
matrices P±
aj(u,ω,k),
˜P±
aj(u,ω,k) ≡ u
We can now write any solution as a linear combination of the basis solutions:
aj(u,ω,k) defined similarly. The˜P±
aj(u,ω,k) are 2N × 2N matrices. For later con
d
2±mP±
aj(u,ω,k).(2.36)
Ψa+(u,ω,k) = u
d
2+mP+
aj(u,ω,k)?P+(ǫ,ω,k)−1?
jbcb+(ω,k),
Ψa−(u,ω,k) = u
d
2−mP−
aj(u,ω,k)?P−(ǫ,ω,k)−1?
jbcb−(ω,k), (2.37)
with a summation over the j index. Notice that we take the solutions Ψa±to be linear in the
“sources,” ca±. As emphasized in refs. [55], eq. (2.37) is simply saying that the sources ca±
will source various linear combinations of fields in the bulk, and that we can write those linear
combinations as linear combinations of our basis solutions. Notice that when we evaluate the
solutions at u = ǫ, we reproduce the leading asymptotic form, Ψa±∼ ca±u
Now we arrive at the main difference between bulk fermions and bulk bosons: ca+and
ca−are not independent. The equation of motion relates them [62, 63]. Indeed, we saw above
that only the ca−are sources, while the ca+give onepoint functions (roughly speaking). To
relate them, we follow refs. [10, 56]. We return to the equation of motion as written in eq.
(2.32). We focus only on the equation with Ψa+ on the lefthandside, and simply insert
solutions as written in eq. (2.37) (suppressing all ω and k dependence)
d
2±m.
∇ab+u
d
2+mP+
bj(u)
??P+(ǫ)−1?
jdcd+
?
= Mae+u
d
2−mP−
ej(u)
??P−(ǫ)−1?
jfcf−
?
,(2.38)
where the parentheses separate udependent factors from uindependent factors. We now
observe that the matrices P±
ajalso solve the equation of motion, by construction, since they
are built from solutions. We thus have
∇ab+u
d
2+mP+
bj(u) = Mac+u
d
2−mP−
cj(u). (2.39)
Here we have a free j index, so we actually have 2N such equations. (Recall that the index
j labels the choice of normalization vector ? n.) The above equation is just the statement that
one column of the P±
ajmatrices solves the equation of motion. We are free to act on the right
with the vector?P+(ǫ)−1?
∇ab+u
jdcd+, so that we obtain
d
2+mP+
bj(u)
??P+(ǫ)−1?
jdcd+
?
= Mac+u
d
2−mP−
cj(u)
??P+(ǫ)−1?
jdcd+
?
. (2.40)
– 16 –
Page 18
We now simply compare eqs. (2.38) and (2.40). The lefthand sides are identical, so we may
equate the righthand sides. Acting on the left with some inverse matrices, we obtain the
desired relation between the ca+and ca−,
ca+= P+(ǫ)aj
?P−(ǫ)−1?
?P−(ǫ)−1?
jbcb−. (2.41)
Invoking eq. (2.28), we now just need to perform two operations to extract the retarded
twopoint function GR
right with −iγt.
The effect of taking ǫ → 0 is easy to understand. From the definition of the˜P±
(2.35) and the definition of the P±
aj(u) in eq. (2.36), we can identify the ǫ → 0 limit of the
P±
aj(u) as
and similarly for limǫ→0P+
are simply matrices of the ca+and ca−.
Notice that the P−(ǫ)−1matrix will introduce a factor of detP−(ǫ) in the denominator of
the Green’s function. Generically, then, if detP−(ǫ) has a zero, the Green’s function will have
a pole, which means a quasinormal mode appears in the bulk spectrum, as in the bosonic
cases of ref. [55]. Given the identification in eq. (2.42), then, to identify quasinormal modes
we need only identify the zeroes of the matrix of c−’s.
Understanding how γtacts on P+(ǫ)aj
?P−(ǫ)−1?
(2.41) is written in a twocomponent form: here ca±are two component spinors. To restore
them to fourcomponent form, we take a direct product,
ab(ω,k) from P+(ǫ)aj
jb: we take ǫ → 0 and then act on the
aj(u) in eq.
lim
ǫ→0
?
P−
aj(ǫ)
?
=
c(1)
c(1)
...
c(1)
1−c(2)
2−c(2)
1−... c(2N)
2−... c(2N)
...
N−c(2)
1−
2−
...
N−... c(2N)
N−
, (2.42)
aj(ǫ). In short, the matrices P±
aj, when evaluated at the boundary,
jbis a little tricky. Luckily, the way we
have written P+(ǫ)aj
?P−(ǫ)−1?
jbmeans that −iγtacts trivially. To see that, notice that eq.
ca+=
?
ca+u
ca+d
?
→ ca+⊗
?
0
1
?
=
0
ca+u
0
ca+d
,ca−=
?
ca−u
ca−d
?
→ ca−⊗
?
1
0
?
=
ca−u
0
ca−d
0
. (2.43)
To restore the P±
shows that we should perform exactly the same direct products (suppressing the dependence
on all variables):
?
1
?
ca+⊗
1
ajmatrices to the same fourcomponent form, we recall recall eq. (2.35), which
P+
aj→ P+
aj⊗
0
?
,P−
aj→ P−
aj⊗
?
1
0
?
,(2.44)
which implies (P−)−1
ja→ (P−)−1
?
ja⊗
1 0
?
. Eq. (2.41) thus becomes
0
?
=
??
P+(ǫ)aj
?P−(ǫ)−1?
jb
?
⊗
?
1 0
0 0
?? ?
cb−⊗
?
1
0
??
.(2.45)
– 17 –
Page 19
We now simply observe that, in such a representation, −iγt= 1N⊗σ1. In the N×N subspace
we want, −iγtmerely acts as the identity.
In summary, the retarded Green’s function for coupled bulk fermions is
GR
ab(ω,k) = lim
ǫ→0
?
P+(ǫ)ajP−(ǫ)−1
jb
?
, (2.46)
with the matrices P±
ajdefined in eq. (2.36).
Finally, as an important check, let us use our prescription to reproduce the result for free
fermions, eq. (2.30). For illustration, we consider N = 2, so we have two bulk fermions, which
we will call Ψaand Ψb. We return to the equation of motion as written in eq. (2.32), and
assume the equations for Ψaand Ψbdecouple, so that ∇ab±and Mab±become diagonal in the
a and b indices. We can further decouple the equations of motion by using the projectors Π1,2.
Acting with these, we obtain equations similar to eq. (2.14). We thus find four decoupled
equations, for Ψa1, Ψa2, Ψb1and Ψb2.
We now solve the equations 2N = 4 times, each time with a different normalization
vector ? n for the Ψa− and Ψb− fields.In the first solution, all four fields have normal
izations ? n = (na−u,na−d,nb−u,nb−d) = (+1,+1,+1,+1). In the second solution, we use
? n = (+1,−1,+1,+1). The key observation is that the field Ψa−dwhose normalization we
change is Ψa−d= Ψa1u (recall eq. (2.13)), and hence couples only to Ψa1d= Ψa+d. The
change in normalization thus leaves the other three fields, Ψa−u, Ψb−u, and Ψb−dunchanged.
The solutions for these fields will thus be identical to what they were using the original +1
normalizations. The P−
ajmatrix thus takes the form (here we must write the up and down
components explicitly)
?˜P−
aj(u,ω,k)
?
=
Ψ(1)
Ψ(1)
Ψ(1)
Ψ(1)
1−uΨ(1)
1−dΨ(2)
2−uΨ(1)
2−dΨ(1)
1−uΨ(1)
1−dΨ(1)
2−uΨ(3)
2−dΨ(1)
1−uΨ(1)
1−dΨ(1)
2−uΨ(1)
2−dΨ(4)
1−u
1−d
2−u
2−d
, (2.47)
with˜P+
is that all the superscripts are the same, except on the diagonal. A straightforward exercise
(especially simple for 2 × 2 matrices) then shows that taking the inverse (P−)−1
contracting with P+
aj, and taking ǫ → 0, reproduces exactly the purely diagonal c+u/c−uand
c+d/c−dform of eq. (2.30).
In summary: by combining the methods of refs. [54, 55] and [10, 56], we have provided
a relatively simple prescription to compute the matrixvalued retarded twopoint function
from bulk solutions for coupled fermions. We simply solve the equations of motion (typically
numerically) 2N times, using a different normalization vector ? n each time, use those solutions
to construct the matrices P±
aj(u,ω,k) being identical except all − subscripts become +. The main feature here
jaand then
aj(ǫ), and then take limǫ→0P+(ǫ)aj
?P−(ǫ)−1?
jb.
– 18 –
Page 20
3. Probe Branes and Holographic pwave Superfluids
In this section we review how to obtain a holographic pwave phase transitions from simple
string theory constructions of intersecting Dqbranes and Dpbranes [32, 33, 34, 35] (see also
ref. [66]). We also present some new results for the particular D3/D5 system we subsequently
explore in later sections.
3.1 pwaves, Probe Branes, and Vector Meson Condensation
The minimal ingredients for a holographic pwave phase transition are gravity in a black hole
spacetime with holographic variable u (and some dual field theory), plus nonAbelian bulk
gauge fields. We will consider the simple example of SU(2) gauge fields Aa
labels the generators τaof SU(2), although other nonAbelian groups besides SU(2) work just
as well [18, 67]. As described in the introduction, the pwave superfluid transition appears in
the bulk as a charged black hole growing vector hair at low temperature.
As observed in refs. [32, 33, 34, 35], we can easily obtain holographic pwave superfluids
using wellknown intersections of Nccoincident Dqbranes with Nfcoincident Dpbranes in
type II string theory, which we will refer to as Dq/Dp systems. The idea is to take the
usual decoupling limit for the Dqbranes, which in particular means Nc→ ∞, to obtain type
II supergravity in the nearhorizon geometry of the Dqbranes. Starting with nonextremal
Dqbranes produces a black hole spacetime.
If we keep Nf fixed as Nc→ ∞, so that Nf ≪ Nc, then we may neglect the effect of
the Dpbranes on the supergravity fields. The Dpbranes are then probes of the background
geometry, and their dynamics is described by the nonAbelian BornInfeld action (possibly
plus WessZumino terms) with gauge group U(Nf). If we introduce exactly Nf = 2 probe
branes, then we have U(2) gauge fields on the worldvolume of the Dpbranes. The SU(2)
subgroup gives us the SU(2) gauge fields we want. Given that the nonAbelian BornInfeld
action is not known to all orders in the field strength, we are typically limited to working at
the leading nontrivial order, which is the YangMills term. We have thus obtained SU(2)
gauge fields in a black hole spacetime.
Crucially, notice that these Dq/Dp systems give rise to probe gauge fields, rather than
gauge fields coming from the supergravity sector. The probe limit is sufficient to study many
properties of the pwave phase transition [16, 32, 33, 34, 35], however, the probe limit is known
to fail at low temperatures, because the solutions with nonzero A1
that increases as we cool the system, so that we can no longer neglect the backreaction (or
trust the YangMills approximation to the nonAbelian BornInfeld action) [32, 34]. To reach
zero temperature, we must solve the fully coupled equations of motion, which to date has
only been done in ad hoc models [17, 18, 23, 68].
The benefit of the Dq/Dp construction is that we can identify the dual theory, which
we will call the Dq/Dp theory. The NcDqbranes generically give rise to an SU(Nc) gauge
theory with fields only in the adjoint representation of SU(Nc). Open strings from the Nf
Dpbranes to the Dqbranes give rise to fields in the fundamental representation of SU(Nc),
M, where a = 1,2,3
x(u) have a field strength
– 19 –
Page 21
i.e. flavor fields. In analogy with (supersymmetric) QCD, we will call any flavor fermions or
scalars “quarks” or “squarks,” respectively. If the Dpbranes do not overlap with all q spatial
directions of the Dqbranes, then the flavor fields will be confined to propagate along some
defect of nonzero codimension.
In the field theory, the probe limit consists of neglecting quantum effects due to the flavors,
such as the effect on the running of the coupling, because such effects are parametrically
suppressed by Nf/Nc. In the language of perturbation theory, the probe limit consists of
discarding all diagrams involving quark or squark loops.
If we separate the Dbranes in an overall transverse direction, we may give the DqDp
strings a finite length and hence the flavor fields a finite mass, although in this paper we
consider only massless flavor fields (unless stated otherwise).
The U(Nf) gauge invariance on the Dpbranes’ worldvolume is dual to a U(Nf) flavor
symmetry, analogous to the vector symmetry of QCD. The overall U(1) we identify as baryon
(or really quark) number, and the SU(2) subgroup we identify as isospin.
We can thus easily see what the bulk transition looks like in the field theory. We have a
stronglycoupled, largeNcnonAbelian gauge theory coupled to Nf= 2 species of massless
flavor fields, which may be confined to a defect. We study thermal equilibrium states with
temperature T, and introduce an isospin chemical potential µ for U(1)3. For sufficiently large
µ, the system develops a nonzero ?Jx
flavor fields, valued in the adjoint of SU(Nf). For example, in twoflavor massless QCD, with
up and down quarks u and d, Jx
1?. The operator Jx
1is a gaugeinvariant bilinear in the
1∼ ¯ uγxd.
Such an operator is precisely what we would call a vector meson, and the phase transi
tion appears to be vector meson condensation. To be precise, the spectrum of the Dq/Dp
theory includes gaugeinvariant bound states of flavor fields. We will refer to such bosonic or
fermionic bound states as “mesons” or “mesinos,” respectively. For massive flavor fields, these
mesons/mesinos are typically the lightest flavor degrees of freedom in the theory [69, 70]. We
may thus imagine writing an effective theory for these degrees of freedom, analogous to the
chiral Lagrangian of QCD. An isospin chemical potential µ will act as a negative masssquared
for any mesons/mesinos charged under U(1)3. If we make µ sufficiently large, then we expect
BoseEinstein condensation of mesons. In QCD, we expect the lightest charged mesons, the
pions, to condense first producing a scalar condensate (and hence an swave superfluid), while
the heavier vector mesons may condense at higher µ [71, 72]. Which mesons condense first
in Dq/Dp systems depends on the details of the system. In Dq/Dp holographic models of
QCD, such as the SakaiSugimoto model [73], holographic calculations suggest that indeed
the pions condense first and the vector mesons second, as we increase µ [74, 75]. The general
lesson from these Dq/Dp systems is that the pwave superfluid phase transition appears to
be vector meson condensation, which is in line with our weakcouling intuition.
Moreover, thinking of the pwave states as a BoseEinstein condensate makes many poten
tially confusing features of the pwave state transparent. For example, the pwave transition
appears to involve the spontaneous generation of a persistent current ?Jx
1?, that is, at high
– 20 –
Page 22
density charges begin moving without experiencing dissipation.14While not impossible, such
a scenario naturally raises some questions. Why do charges start moving? How does that
lower the free energy? Vector meson condensation neatly accounts for all of the physics: we
merely see BoseEinstein condensation, i.e. bosons populating a zeromomentum state, the
main novelty being that the bosons are vectors, not scalars.
We are interested in condensed matter applications, and in particular quantum critical
theories, which are scaleinvariant, hence our Dqbranes will be D3branes, whose nearhorizon
geometry is AdS5× S5. The dual theory is then a CFT, namely (3+1)dimensional N = 4
SYM with large Ncand and large ’t Hooft coupling. Our Dpbranes will preserve half the
supersymmetry of the background, which means Dpbranes extended along AdSP× SQwith
P +Q = p+1, where supersymmetry requires P −Q = 2 [36]. Wellknown examples include
D7branes extended along AdS5× S3[76, 77] or AdS3× S5[78, 79, 80], D5branes extended
along AdS4×S2[41, 42], or D3branes along AdS3×S1[81].15For two coincident D7branes
with P = 5 and Q = 3, holographic calculations have shown that a pwave transition occurs
precisely when peaks in the Green’s function, namely those corresponding to vector mesons
charged under U(1)3in the vacuum state, cross into the upperhalf of the complex frequency
plane, indicating an instability toward BoseEinstein condensation.
With an eye toward condensed matter applications, and for technical reasons we explain
in section 4, we will work with (two coincident) D5branes with P = 4 and Q = 2. The
dual theory is thus N = 4 SYM coupled to Nf= 2 flavor fields that propagate only in 2+1
dimensions (a codimension one defect) and preserve (2+1)dimensional N = 4 supersymme
try (eight real supercharges). The field content and Lagrangian of the D3/D5 theory were
determined in refs. [41, 42], which we review in section 4.2. The D3/D5 system exhibits rich
thermodynamics, studied in detail (holographically) in [43, 44, 45, 46, 47, 48, 50, 51]. Here
we initiate the study of the D3/D5 theory with a finite isospin chemical potential and finite
temperature. As expected, we will find a pwave phase transition.
Our ultimate goal is to compute, holographically, fermionic retarded Green’s functions
in pwave superfluid states.We discuss fermionic excitations on the worldvolume of the
D5branes, and the dual mesinos, in section 4. We will be able to compare our numerical
results with previous studies, however, we cannot exploit the analytic results for the form of
fermionic Green’s functions derived in ref. [11]. The analysis of ref. [11] involved fermions
in an extremal ReissnerNordstr¨ om background, and in particular made great use of the
emergent nearhorizon AdS2factor. Without having access to the T = 0 finitedensity state,
14Crucially, however, no net momentum is flowing. In holographic calculations, in both the probe and fully
backreacted cases [17, 18, 23], the YangMills stressenergy tensor and the metric are diagonal, and indeed
the bulk spacetime is static, which indicates that the expectation value of the field theory stressenergy tensor
is strictly diagonal. The system thus has zero net momentum. If charges are moving, they must be doing so
in pairs that move in opposite directions. A static bulk spacetime also indicates that the energy density of the
field theory is not changing in time: the system is not heating up, consistent with the fact that the moving
charges experience no dissipation (no frictional forces).
15These are the cases in which the flavor fields propagate in at least one spatial dimension, and hence a
nonzero ?Jx
1? is possible.
– 21 –
Page 23
we do not know for sure whether the geometry exhibits an emergent AdS2 or something
similar. Nevertheless, the equations of motion for our fermions are formally similar to those
of refs. [9, 11], and hence we can recover similar finitetemperature results.
Despite the bad news that we cannot reach T = 0 in the probe limit, we do have good
news: we can study fermionic response near the pwave transition. The hightemperature
normal phase is rotationally symmetric, but the pwave phase is of course not. We will see
the breaking occur explicitly in fermionic spectral functions in section 5.
3.2 Probe Dpbranes in AdS5× S5
We now want to study a pwave superfluid transition for (2+1)dimensional flavor fields
described holographically by two coincident D5branes with P = 4 and Q = 2. Although our
(numerical) analysis will be for the D5brane, in the interest of generality, and to connect to
our discussion in section 4, we will write formulas for an arbitrary Dpbrane with P ≥ 3.
The background supergravity solution includes a metric and RamondRamond (RR) five
form. The fiveform will be important in section 4. The spacetime is (4+1)dimensional
AdSSchwarzschild times S5, with metric
ds2=
1
u2
?du2
f(u)− f(u)dt2+ d? x2
?
+ ds2
S5,(3.1)
with
f(u) = 1 −u4
u4
h
,uh=
1
πT.
(3.2)
In our units, where the AdS radius is one, we can convert between string theory and field
theory quantities using α′−2= 4πgsNc= 2g2
Y MNc= 2λ.
As we will be studying fermions in section 4, we will need the vielbeins and spin connection
associated with the metric above. We record these here for later use. The nonzero vielbeins
of (4+1)dimensional AdSSchwarzschild are (recall that upper index is local Lorentz, and
the lower index is general coordinate),
eu
u=
1
u√f,
et
t=
√f
u,
ei
j=1
uδi
j. (3.3)
The spin connection ω of (4+1)dimensional AdSSchwarzschild then has the nonzero com
ponents
?f(u)
where ω? xuindicates the three components ωxu, ωyu, and ωzu.
Next we introduce two coincident probe Dpbranes extended along AdSP× SQ. We will
only consider the trivial embedding of the Dpbranes, that is, we consider solutions in which
all the Dpbranes’ worldvolume scalars (including scalars in AdS5directions) are zero. The
dual flavor fields are then massless. The induced metric on the Dpbranes
ωtu=
u
−f′(u)
2
?
dt,ω? xu= −
?f(u)
u
d? x,(3.4)
ds2
Dp=
1
u2
?du2
f(u)− f(u)dt2+ d? x2
?
+ ds2
SQ, (3.5)
– 22 –
Page 24
where now d? x2represents the appropriatedimensional Euclidean metric.
We want nontrivial worldvolume SU(2) gauge fields. The action for the gauge fields, to
leading nontrivial order, is
SDp= −TDpNf
?
dp+1ξ?−gDp
is the tension of the Dpbrane, Nf= 2, the integral is over
?
1 + (2πα′)21
2Tr(FµνFµν)
?
,
where TDp= (2π)−pg−1
the worldvolume coordinates ξµ, gDpis the determinant of the induced metric, and the trace
is over gauge indices. We use SU(2) generators τa=1
s
(α′)−p+1
2
2σasuch that, with ǫ123= +1
[τa,τb] = iǫabcτc.(3.6)
The field strength Fµν= Fa
The equation of motion for the gauge field is simply the YangMills equation,
µντa.
∇µFµν
a
+ fabc(Aµ)bFµν
c
= 0. (3.7)
For probe Dpbranes wrapping AdSP ⊆ AdS5with P ≥ 3, we will consider solutions of eq.
(3.7) of the form
A = A1
x(u)τ1dx + A3
t(u)τ3dt,(3.8)
in which case the YangMills equation becomes
?A3
+
t
?′′
+4 − P
u
?A3
t
?′−
??A1
1
f(u)A3
t
?A1
1
x
?2= 0,
?A3
(3.9a)
?A1
x
?′′
?4 − P
u
+f′(u)
f(u)
x
?′+
f(u)2
t
?2A1
x= 0,(3.9b)
where primes denote ∂u. The equations of motion determine the asymptotic forms of the
solutions,16
A3
t(u) = µ − d3
tuP−3+ ...,A1
x(u) = d1
xuP−3+ ...,(3.10)
where ... represent terms that decay faster than uP−3as u → 0. Here the constant d3
related to ?Jt
?Jt
tis
3? as
3? = NfTDp
?2πα′?2d3
t= (2π)−p+32
p−3
4 NfNcλ
p−7
4 d3
t,(3.11)
where in the second equality we converted to field theory quantities.
NfTDp(2πα′)2d1
One solution of eqs. (3.9) has A1
Similarly, ?Jx
1? =
x. Notice in particular that both ?Jt
3? and ?Jx
1? are proportional to NfNc.
x(u) = 0 and
A3
t(u) = µ
?
1 −uP−3
uP−3
h
?
.(3.12)
16Notice that A1
theory: U(1)3 will be broken spontaneously.
x(u) has no leading constant the way At
3(u) does, so no source for Jx
1 is present in the field
– 23 –
Page 25
Such a solution corresponds in the field theory to the normal phase, in which the chemical po
tential µ explicitly breaks the SU(2) isospin symmetry down to the U(1)3, but no spontaneous
symmetry breaking occurs. These solutions exist for all values of µ.
Notice that these D3/Dp theories in the probe limit are scaleinvariant17, so the only
meaningful physical quantity is µ/T, so fixing T and increasing µ is equivalent to fixing
µ and reducing T. We will think in terms of the latter. Any transition must occur at a
temperature Tcset by the chemical potential, Tc∝ µ.
For sufficiently low T (or large µ), other solutions of eq. (3.7) exists in which A1
nonzero. For the D7brane with P = 5 and Q = 3, such solutions were found numerically
in refs. [32, 33, 34]. These solutions correspond in the field theory to superfluid states,
with nonzero ?J1
rotational symmetry18is also broken from SO(P − 2) down to SO(P − 3).
For sufficiently low T, we have two solutions, so we need to determine which is thermo
dynamically preferred. As shown in refs. [32, 33, 34] for the D7brane with P = 5 and Q = 3,
the superfluid phase is thermodynamically preferred relative to the normal phase for all T/µ
where the solutions with nonzero Ax
1(u) exist. The transition between the phases is second
order, with meanfield exponents. In particular, near the transition, the condensate has mean
field exponent 1/2: ?Jx
For two coincident D5 branes with P = 4 and Q = 2, the story is qualitatively the same.
For µ ≥ 3.81 × (πT) the state with nonzero ?J1
Tc=
rescaled temperature T/Tc. Near the transition, ?Jx
of 1/2 as in the D7brane case [32, 33, 34].
Notice that as T decreases, the condensate grows, which, as explained in refs. [14, 15],
suggests that we are leaving the probe limit. We will only present results in the pwave
phase for T/Tc ? 0.4 (as plotted in figure (1)), where we have some hope that the probe
approximation captures the essential physics faithfully.
x(u) is
x?, so U(1)3is spontaneously broken. For P ≥ 4, the field theory’s spatial
1? ∝ (1 − T/Tc)1/2.
x? has lower free energy. In other words,
x, which is proportional to ?Jx
1? appears to have a meanfield exponent
µ
3.81×π. In figure (1) we plot the constant d1
1?, versus the
4. The Worldvolume Fermions
We want to study fluctuations of fermionic operators of the D3/D5 theory with finite tem
perature and isospin chemical potential, in the two phases described above, the normal (non
superfluid) phase and the superfluid phase. On the field theory side, we will study mesino
operators valued in the adjoint representation of the SU(2) isospin symmetry. As we will see
below, we will thus have three mesinos, two with equal and opposite charges under the U(1)3,
and one that is neutral. We discuss the form of the mesino operators in section 4.2.
17In the probe limit we neglect the quantum effects that would cause the N = 4 SYM coupling to run, and
we are working with massless flavor fields. The theory is thus in a limit where no intrinsic scale appears.
18When P = 3 the flavor fields are confined to a (1+1)dimensional defect and hence have no spatial
rotational symmetry. Notice that in the P = 3,4 cases the large Nc limit is what permits spontaneous
symmetry breaking to occur, by suppressing the fluctuations that would destroy longrange order.
– 24 –
Page 26
0.0 0.20.40.60.81.0
0
5
10
15
20
23/2λ1/2
NfNcT2?J1
x?
T/Tc
Figure 1: The condensate ?J1
x?, times23/2λ1/2
NfNcT2, versus the rescaled temperature T/Tc.
To be specific, we will compute holographically the retarded twopoint function of mesinos
as a function of frequency and momentum. On the gravity side, that means studying fermionic
fluctuations of the Dpbranes, and in particular solving their linearized equation of motion
in the background we found in the last section, where the geometry is (4+1)dimensional
AdSSchwarzschild and the D5branes have nontrivial worldvolume gauge fields.
all supersymmetric Dpbranes, the worldvolume fermions are in a supermultiplet with the
worldvolume gauge field and scalars, and hence are in the adjoint of the worldvolume SU(2)
gauge group, which is dual to the statement that the mesinos are in the adjoint of the isospin
symmetry.19
As we will see, in the normal phase the three bulk fermions decouple, which is dual to the
statement that the retarded Green’s function is diagonal. In the superfluid phase, however,
where A1
x(u) is nonzero, the three fermions couple, indicating that the dual operators mix
under renormalization group flow in the field theory. The Green’s function is then a 6 × 6
matrix, where the 6 is the number of fermions times the two components of the fermions
(two, using the Π1,2projectors). We thus have a perfect testing ground for the method we
developed in section 2 for computing Green’s functions from coupled bulk fermions.
The fermionic part of general Dpbrane actions, to quadratic order in the fermionic fields
and in backgrounds with nontrivial RR forms, was determined in refs. [37, 38, 39]. The
general couplings were derived by starting with the action for a supermembrane in Mtheory,
written in a superspace formalism, expanding the action to second order in the Grassmann
variables, reducing to type IIA supergravity, and then performing a Tduality to type IIB.
As for
19Obviously, the mesinos carry no baryon number, which is dual to the statement that the worldvolume
fermions, like all of the worldvolume fields, do not couple via a gaugecovariant derivative to the diagonal U(1)
part of the worldvolume gauge field.
– 25 –
Page 27
The form of the quadratic fermionic action on the Dpbrane worldvolume is thus determined
completely by supersymmetry and Tduality.
Using the worldvolume fermion actions of refs. [37, 38, 39], the spectra of mesinos in the
D3/D7 theory (for the D7brane with P = 5 and Q = 3) and in the SakaiSugimoto model
were determined in refs. [40, 82]. We will very closely follow the D7brane analysis of ref.
[40], which in turn was the fermionic generalization of the analysis of ref. [69] for mesons. For
a Dpbrane extended along AdSP×SQ, we consider a worldvolume spinor that is a spherical
harmonic on the SQ. We reduce the worldvolume spinor on the SQ, obtaining an effective
Dirac action in AdSP. This procedure fixes the masses of the bulk fermionic excitations,
which allows us to identify the dimensions of the dual mesinos, and more generally to map
bulk fluctuations to mesino operators. As emphasized in ref. [40], the coupling to the RR
fiveform is crucial to obtain the correct bulk masses.20
One of our main points is: because we work with a particular string theory system, we do
not have the freedom to change the mass or the charge of our bulk fermion, in stark contrast
to the models of refs. [9, 10, 11]. Both the masses and charges are, ultimately, fixed by
supersymmetry and Tduality, as explained above.
4.1 Equation of Motion I: Reduction to AdS
We will now repeat the analysis of ref. [40], in which the fermionic action of a D7brane
extended along AdS5×S3was reduced to an effective Dirac action in AdS5, but now for more
general Dpbranes extended along AdSP× SQ, with emphasis on D5branes with P = 4 and
Q = 2.21
The quadratic action for fermionic fluctuations of the Dpbranes is (refs. [39, 40])
SDp= NfTDp
?
dp+1ξ?−gDp1
2Tr
?
ˆ ¯ΨP−Γˆ A
?
Dˆ A+1
8
i
2 ∗ 5!Fˆ NˆPˆQˆRˆSΓˆ NˆPˆQˆRˆSΓˆ A
?
ˆΨ
?
,
HereˆΨ is a tendimensional positivechirality MajoranaWeyl spinor of type IIB supergravity,
the Γˆ Aare the pullback of the tendimensional Γmatrices to the Dpbrane worldvolume,
Γˆ A= Γˆ
projector that ensures κsymmetry invariance of the action, DA is a (gauge and curved
space) covariant derivative, and Fˆ NˆPˆQˆRˆSis the fiveform of the background. Notice that
here Fˆ NˆPˆQˆRˆSis not the pullback of the fiveform to the Dpbrane worldvolume, rather, it
is the fiveform evaluated on the submanifold spanned by the Dpbrane. Indeed, no part of
the expression Fˆ Nˆ Pˆ Qˆ RˆSΓˆ NˆPˆQˆRˆSinvolves a pullback. HereˆA,ˆB,... denote all worldvolume
indices, while below A,B,... denote AdSSchwarzschild coordinates which are wrapped by
M∂ˆ Axˆ
M(we use a trivial embedding, so the pullback is trivial), P−is a κsymmetry
20The authors of ref. [83] appear to omit the coupling to the fiveform when they study fermionic fluctuations
of the probe D3brane extended along AdS3× S1.
21In section 3 we were interested in pwave states and hence required P ≥ 3. In this subsection we relax
that constraint. Our results will thus also apply for D5branes extended along AdS2× S4, which were used in
ref. [84] to construct a holographic model of fermions at lattice sites (that can pair to form dimers).
– 26 –
Page 28
the probe brane. Moreover, the indices of the coordinates on the sphere SQare labelled by
a,b,.... Notice that the fermionˆΨ is in the adjoint representation of SU(2).
The equation of motion for the fermion is (for now we suppress gauge indices)
?
Γˆ ADˆ A+1
8
i
2 ∗ 5!Γˆ AFˆ NˆPˆQˆRˆSΓˆ NˆPˆQˆRˆSΓˆ A
?
ˆΨ = 0
(4.1)
We will reduce the equation of motion for the fermion to a Dirac equation in AdSP, following
ref. [40] very closely. First we decompose every tendimensional spinor and Γmatrix into
parts associated with AdS5and S5. In a local Lorentz frame, the Γmatrices decompose as
ΓM= σ2⊗ 14⊗ γM,Γm= σ1⊗ γm⊗ 14,(4.2)
where 14is the 4 × 4 identity matrix, the index M runs over AdS5directions (which we will
generically call 01234), and the index m runs over S5directions (which we will call 56789).
The γmatrices are fivedimensional, obeying the usual relations
?γM,γN?= 2ηMN,
{γm,γn} = 2δmn. (4.3)
Given the above decompositions, we then have
Γ01234= iσ2⊗ 14⊗ 14,Γ56789= σ1⊗ 14⊗ 14
(4.4)
Γ11= Γ0123456789= σ3⊗ 14⊗ 14
(4.5)
The tendimensional spinorˆΨ has positive chirality, Γ11ˆΨ =ˆΨ, and decomposes as
ˆΨ =↑ ⊗χ ⊗ Ψ,(4.6)
where ↑=
on the tangent spaces of S5and AdS5, respectively. The spinor χ further decomposes as
χ = χ?⊗ χ⊥, where χ?is a spinor associated with the SQthat the Dpbrane wraps and χ⊥
is associated with the S5directions transverse to the SQ.
We parameterize the fiveform in terms of the volume forms of AdS5and S5, which we
denote as ΩAdS5and ΩS5,
?
1
0
?
, and χ and Ψ are fourcomponent spinors of SO(5) and SO(4,1), which act
FNPQRS= 4 (ΩAdS5)NPQRS,Fnpqrs= 4 (ΩS5)npqrs.
Using the decomposition of the type IIB spinor in eq. (4.6) and of the γmatrices in eq. (4.2)
we obtain
Γa?Γ01234+ Γ56789?Γa(↑ ⊗χ ⊗ Ψ) = 2 ↓ ⊗χ ⊗ Ψ,
ΓA?Γ01234+ Γ56789?ΓA(↑ ⊗χ ⊗ Ψ) = −2 ↓ ⊗χ ⊗ Ψ,(4.7)
– 27 –
Page 29
In eq. (4.7) we have not summed over a or A. Using this result we can simplify the coupling
of the spinor to the fiveform,
1
8
i
2 ∗ 5!Γˆ AFˆ Nˆ Pˆ Qˆ RˆSΓˆ NˆPˆQˆRˆSΓˆ AˆΨ =i
where here we do sum overˆA. We can also extract the SQand the AdSPpart of the derivative
terms as
4Γˆ A((σ1+ iσ2) ⊗ 14⊗ 14)Γˆ AˆΨ = −i
2(P −Q)(↓ ⊗χ ⊗ Ψ)
(4.8)
Γˆ ADˆ AˆΨ = ΓADAˆΨ + ΓaDaˆΨ
=??σ2⊗ 14⊗ γADA
=?i?12⊗ 14⊗ γADA
≡?i?DAdSP+ ?DSQ?(↓ ⊗χ ⊗ Ψ),
where ?DAdSPand ?DSQ are the Dirac operators of AdSP and SQ, respectively. The Dirac
operator on a sphere SQhas spinor spherical harmonics χ±
?+ (σ1⊗ γaDa⊗ 14)?(↑ ⊗χ ⊗ Ψ)
?+ (12⊗ γaDa⊗ 14)?(↓ ⊗χ ⊗ Ψ)
(4.9)
ℓthat obey
?DSQχ±
ℓ= ∓
i
RQ
?
ℓ +Q
2
?
χ±
ℓ,(4.10)
where ℓ ≥ 0 and RQis the radius of the SQ. In our units, RQ= 1. For Q = 3, relevant for
the D7brane along AdS5×S3, the spinors χ+
that acts on S3, while the spinors χ−
for the D5brane along AdS4× S2, the spinors χ±
that acts on S2.
Inserting everything into eq. (4.1), we find
ℓare in the?ℓ+1
ℓare in the?ℓ +1
2,ℓ
2
?representation of the SO(4)
?of the SU(2) ≃ SO(3)
ℓare in the?ℓ
2,ℓ+1
2
?representation. For Q = 2, relevant
2
?
?DAdSP∓
?
ℓ +Q
2
?
−1
2(P − Q)
?
Ψ±
ℓ=
?
??DAdSP−?ℓ +1
2P??Ψ+
ℓ
??DAdSP+?ℓ −1
2P + Q??Ψ−
ℓ
?
= 0. (4.11)
The fermions22Ψ±thus have masses (in our units, where the radius of AdS is one)
m+
ℓ= ℓ +P
2,m−
ℓ= −
?
ℓ + Q −1
2P
?
.(4.12)
We collect the values of m±
Notice that since P and Q are integers, the m±
we review in the appendix, a bulk fermion with an integer or halfinteger mass m is dual to a
fermionic operator of dimension ∆ =P−1
2
in the table.
To get a rough idea of which operators correspond to which bulk fermion, we can do
some dimension counting. Let us denote a generic quark as ψ, a generic squark as q, a generic
ℓfor our Dpbranes of interest in the table below.
ℓwill always be integer or halfinteger. As
+ m. We include the values of ∆±
ℓ=P−1
2
+ m±
ℓ
22The ± superscript here refers to the sign of the eigenvalue of the Dirac operator in eq. (4.10), not to the
projectors Π± defined in section 2.
– 28 –
Page 30
DpPQ[ψ][q]m+
ℓ= ℓ + P/2
ℓ + 5/2
ℓ + 3/2
ℓ + 2
ℓ + 1
ℓ + 3/2
∆+
ℓ
m−
ℓ = ℓ + Q − P/2
ℓ + 1/2
ℓ + 7/2
ℓ
ℓ + 3
ℓ − 1/2
∆−
ℓ
D7
D7
D5
D5
D3
5
3
4
2
3
3
5
2
4
1
3/2
1/2
1
0
1/2
1

ℓ + 9/2
ℓ + 5/2
ℓ + 7/2
ℓ + 3/2
ℓ + 5/2
ℓ + 5/2
ℓ + 9/2
ℓ + 3/2
ℓ + 7/2
ℓ + 1/2
1/2

0
Table 1: Masses of fermionic excitations on the worldvolume of a Dpbrane extended along AdSP×SQ
inside AdS5× S5. We list Dpbranes that are known to preserve eight real supercharges (at zero
temperature and density), in which case P − Q = 2. Here ψ denotes a generic quark field and q
denotes a generic squark field. ∆±
ℓdenotes the dimension of the operator dual to the bulk fermion
with mass m±
2
+ m±
ℓ ≥ 1 only, whereas for ℓ = 0, m−
ℓ, with ∆±
ℓ=P−1
ℓ. For the D3brane, the values of m−
0 = 1/2 and ∆−
ℓand ∆−
ℓshown are for
0= 3/2.
adjoint Majorana fermion as λ, and a generic adjoint real scalar as X. The dimensions of the
fields are [ψ] =P−2
2, [λ] = 3/2, [X] = 1.
For the D7brane extended along AdS5× S3, the D5brane along AdS4× S2, and the
D3brane along AdS3× S1, all of which have P − Q = 2, the dual flavor fields comprise
a supermultiplet with both quarks ψ and squarks q. In these cases, we can build a gauge
invariant mesino in two ways [40]. One way is to construct an operator of the form¯ψλψ +
q†Xλq, with dimension ∆ = P − 1/2. We can additionally include some number ℓ of adjoint
scalars23as¯ψλXℓψ + q†XλXℓq, so that the dimension is ∆ = ℓ + P − 1/2. Inspecting the
table, these are precisely the ∆+
ℓ, so apparently these kinds of mesinos are dual to the bulk
fermions with masses m+
Hermitian conjugate), with dimension ∆ = ℓ+P −5/2. For Dpbranes with P −Q = 2, these
dimensions are precisely the ∆−
ℓ, so apparently mesinos of this type are dual to the fermions
with masses m−
ℓ.
For the D7brane extended along AdS3×S5and D5brane along AdS2×S4, which have
P − Q = −2, the dual flavor fields are quarks alone, with no squarks [78, 79, 80, 85]. The
mesinos with dimensions ∆+
∆−
ℓmust obviously have a different form. We leave a detailed study of these mesinos for the
future.
Looking at the table, we immediately notice that the D5branes are special: for these,
the masses of the worldvolume fermions are integers. The reason is that the D5branes wrap
evendimensional spheres, so the eigenvalue in eq. (4.10) is ±i times an integer.
In the next subsection we focus on the D5brane extended along AdS4× S2, explaining
in more detail the symmetries of the theory and the form of the mesinos. In the subsequent
sections, we focus on the single worldvolume fermion with mass m−
2, [q] =P−3
ℓ. The other way to build a mesino is to construct¯ψXℓq (plus the
ℓare of the same form,¯ψλXℓψ, but the mesinos with dimensions
0= 0, for a number of
23Notice these are not necessarily all the same scalar, i.e. Xℓcould represent ℓ distinct scalars. At the
moment we are just counting dimensions, ignoring this subtlety.
– 29 –
Page 31
reasons. First, of all the worldvolume fermions, these have the smallest mass, hence the
dual operator will have the lowest dimension, ∆−
mesino. Second, a numerical analysis is simpler when the fermion’s mass is zero. Third, with
a massless bulk fermion we can directly compare to the results of refs. [9, 11], where most
of the analysis focused on massless bulk fermions. Fourth, as we show in the appendix, a
massless fermion requires no counterterms.
0= 3/2, and hence be the most relevant
4.2 The Dual Operators
In this section we focus on the D5brane along AdS4× S2and study in detail the operators
in the D3/D5 theory dual to the fermionic fluctuations considered above. For the D7brane
along AdS5×S3, a similar analysis appears in ref. [40]. We begin with Nf= 1 and generalize
to Nf> 1 at the end.
The dual field theory is (3+1)dimensional N = 4 SYM coupled to defect flavor fields
preserving (2+1)dimensional N = 4 supersymmetry (eight real supercharges). The couplings
of the theory were determined in refs. [41, 42]. Coupling the defect fields to the ambient fields
requires decomposing the (3+1)dimensional N = 4 multiplet into two (2+1)dimensional
N = 4 multiplets, a vector multiplet and a hypermultiplet. The bosonic content of the (3+1)
dimensional N = 4 multiplet is the vector Aµand six scalars24X4,X5,...,X9. The bosonic
content of the (2+1)dimensional vector multiplet is the (2+1)dimensional vector field Ak
and the three scalars XV = (X7,X8,X9). The bosonic content of the (2+1)dimensional
hypermultiplet is the scalar A3and the three scalars XH = (X4,X5,X6). The flavor fields
form a (2+1)dimensional hypermultiplet with two fermions (quarks) ψ and two complex
scalars (squarks) q.
The classical Lagrangian preserves (2+1)dimensional SO(3,2) conformal symmetry but
breaks the SO(6) Rsymmetry down to a subgroup SU(2)H×SU(2)V, under which the scalars
in XH transform in the (1,0) representation and the scalars in XV transform in the (0,1).
We use an upper index to denote these representations: XA
λimtransform in the (1/2,1/2). Here i is the SU(2)V index and m is the SU(2)Hindex. The
quarks ψitransform in the (1/2,0) and the squarks qmtransform in the (0,1/2). In table 2
(borrowed from ref. [41]), we summarize the field content and quantum numbers, including
the conformal dimensions of the fields.
Let us now match fluctuations of the D5brane probe to dual field theory operators, build
ing on the matching of bosonic fields and operators in refs. [41, 70, 86]. These fluctuations
correspond to mesonic operators in the dual theory [69, 76], which can be arranged into a
(2+1)dimensional massive N = 4 supersymmetric multiplet.
First we consider the bosonic fluctuations of the D5brane, as studied in refs. [41, 70, 86].
The bosonic fluctuations consists of three real scalars, which in the notation of ref. [41] are
Vand XI
H. The adjoint fermions
24In the initial type IIB D3/D5 intersection, the D3branes are extended along 0123, and these scalars
represent fluctuations of the D3branes in the 456789 directions, hence our notation. The D5branes are
extended along 012456, so they break the SO(6) rotational symmetry in 456789 down to SO(3) × SO(3) ≃
SU(2) × SU(2), one rotating 456 and one rotating 789.
– 30 –
Page 32
Mode
Ak
XA
A3
XI
λim
qm
ψi
Spin
1
0
0
0
1
2
0
1
2
SU(2)H
0
0
0
1
1
2
1
2
0
SU(2)V
0
1
0
0
1
2
0
1
2
SU(Nc)
adj
adj
adj
adj
adj
N
N
∆
1
1
1
1
3
2
1
2
1
V
H
Table 2: The field content of the D3/D5 theory. (Adapted from ref. [41].) Here Ak, XA
and λimare the adjoint fields of (3+1)dimensional N = 4 SYM decomposed into (2+1)dimensional
N = 4 multiplets. Akand XA
XI
fields, which are in an N = 4 hypermultiplet.
V, A3, XI
H
Vare the bosons in a (2+1)dimensional vector multiplet while A3and
Hare the bosons in a (2+1)dimensional hypermultiplet. qmand ψiare the (2+1)dimensional flavor
Mode∆ SU(2)H
SU(2)V
OperatorOperator in lowest multiplet
i¯ qm← →
Dkqm+¯ψiρkψi
¯ψiσA
bk
l
l + 2 l,l ≥ 0
l ≥ 0
0
Jl
El
Cl
Dl
φl
l + 2l,1
ijψj+ 2¯ qmXAa
¯ qmσI
VTaqm
(b + z)(−)
(b + z)(+)
l+1
l + 1l + 1, l ≥ 0
l − 1,
0
mnqn
l−1
l + 3l ≥ 10—
Table 3: The bosonic fluctuations and their dual field theory operators for the D3/D5 system.
(Adapted from ref. [41].) Here σ are Pauli matrices, Taare the generators of SU(2)V, and ρkare the
(2+1)dimensional Γmatrices.
φl, (b + z)(−)
the embedding in S5directions (transverse to the S2), (b + z)(−)
combinations of the fluctuations of the S2components of the worldvolume gauge field with the
fluctuation of the embedding in AdS5transverse to AdS4, and bk
to fluctuations of the worldvolume gauge field in the AdS4 directions. We summarize the
quantum numbers of these fluctuations in table 3 (borrowed from ref. [41]). Notice that our
definition of l differs from that in ref. [41]. In our notation, fluctuations with the same l have
the same mass. Later we will show that all operators with the same quantum number l fit
into a super multiplet. Note that (b + z)(+)
l
is not present in the lowest multiplet with l = 0.
We studied the fermionic fluctuations25Ψ±
fluctuations Ψ−
l
and (b + z)(+)
l
, as well as a vector bk
l. Here φlcorresponds to fluctuations of
and (b + z)(+)
ll
are linear
l(k = 0,1,2) corresponds
l
of the D5brane in the last section. The
lwith l ≥ 0 correspond to operators with dimensions ∆−
25In the last subsection we used a subscript ℓ, while here we use a subscript l. For Ψ−
ℓ = l. For Ψ+
l= l + 3/2 that are
ℓthe two are identical:
ℓwe take ℓ = l − 1.
– 31 –
Page 33
D5brane Mode
Ψ−
Ψ+
l−1
∆SU(2)H
l + 1/2,
l − 1/2,
SU(2)V
1/2
1/2
Operator
Fl
Gl
l
l + 3/2
l + 5/2
l ≥ 0
l ≥ 1
Table 4: Matching between fermionic fluctuations of the D5brane and field theory operators. The
explicit form of the fermionic operators Fland Glappear in the text below.
in the l + 1/2 representation of SU(2)H. The fluctuations Ψ+
operators with dimensions ∆+
l−1= l + 5/2 in the l − 1/2 representation of SU(2)H. Since
both fluctuations are fermionic they transform in the 1/2 representation 1/2 of SU(2)V. We
summarize the fermionic modes in table 4.
We first review the lowest multiplet, i.e. l = 0, which appears already, including the
fermionic operators, in ref. [41]. According to tables 3 and 4, the D5brane fluctuation
corresponding to the lowestdimension operator is (b + z)(−)
exists on the field theory side with the same quantum numbers as (b + z)(−)
match (b + z)(−)
1
with the operator CI
SU(2)H. C0transforms in the (1,0) representation of SU(2)H× SU(2)V. Moreover C0is the
lowest chiral primary in the multiplet since all other operators dual to D5brane fluctuations
have larger conformal dimensions. We can thus construct all operators in the same multplet
as C0by applying supersymmetry generators to C0. The supersymmetry generators form a
2 × 2 matrix of Majorana spinors ηim, which transforms like λim, i.e. in the representation
(1/2,1/2) of SU(2)H×SU(2)V. Applying the supersymmetry generators to C0we obtain the
fermionic operator Fim
SU(2)V quantum numbers (1/2,1/2). Fim
Applying another supersymmetry generator to Fim
which appear in table 3. Both J0and E0have conformal dimension ∆ = 2 and are singlets
under SU(2)H but can be distinguished by their SU(2)V quantum number: J0is a singlet
whereas E0is a triplet under SU(2)V.
Let us now discuss the general multiplet dual to the higherl fluctuations of the D5brane.
As in the l = 0 case, we construct the multiplet by applying supersymmetry generators to the
lowest chiral primary in the multiplet, Cl, which is dual to (b + z)(−)
the lowest chiral primary is CI0I1...Il
symmetric product of l copies of the field XI
H. Clhas conformal dimension ∆ = l + 1 and is
in the (l+1,0) representation of SU(2)H×SU(2)V. Applying a supersymmetry generator to
Clwe find the fermionic operator Flwith conformal dimension ∆ = l + 3/2, which is dual to
the D5brane fluctuation Ψ−
l. Flis in the (l +1/2,1/2) representation of SU(2)H×SU(2)V.
Explicitly, Flis of the form
=¯ψi?
Applying another supersymmetry generator to Flwe obatin Jlor El, which have the same
l−1with l ≥ 1 correspond to
l+1with l = 0. Only one operator
1
, so we can
0= q†mσI
mnqn, where σIare the Pauli matrices of
0
=¯ψiqm+ q†mψiwith conformal dimension ∆ = 3/2 and SU(2)H×
0
is dual to the fermionic D5brane fluctuation Ψ−
we obtain either J0or E0, the forms of
l=0.
0
l
H) stands for the traceless
. According to ref. [41],
l
= C(I0
0
?Xl
H
?I1...Il), where (Xl
FI1...Ilim
l
Xl
H
?I1...Ilqm+ q†m?
Xl
H
?I1...Ilψi.(4.13)
– 32 –
Page 34
conformal dimension ∆ = l + 2, but differ in the SU(2)H× SU(2)V representation.
transforms in the (l,0) representation whereas Elhas quantum numbers (l,1). To obtain the
precise form of Jlor Elwe insert the operator Xl
In contrast to the l = 0 multiplet, other operators also appear in the multiplet for
l ≥ 1, which we construct by applying three or four supersymmetry generators to Cl: a
fermionic operator Gland a bosonic operator Dl. Glhas conformal dimension ∆ = l + 5/2
and SU(2)H× SU(2)V quantum numbers (l − 1/2,1/2). These are precisely the quantum
numbers of the fermionic D5brane fluctuation Ψ+
Jl
Hinto the operator J0or E0, respectively.
l−1. Explicitly, Glhas the form
?I1...Il−1λimXH,IσI
GI1...Il−1im
l
=¯ψj?
Xl−1
H
?I1...Il−1λimψj+ q†n?
Xl−1
H
npqp. (4.14)
Finally Dlhas conformal dimension ∆ = l + 3 and SU(2)H× SU(2)V quantum numbers
(l − 1,0) and therefore can be identified with the D5brane fluctuation (b + z)(+)
We have constructed the supermultiplet for the cases l = 0 and l ≥ 1. For l = 0 they
multiplet consists of the bosonic operators C0,J0and E0and of the fermionic operator F0. The
multiplet includes are eight bosonic and eight fermionic degrees of freedom. The multiplet
containing Cl, l ≥ 1 has 16l + 1 fermionic and bosonic degrees of freedom:
• One real scalar Clin the (l + 1,0) representation with ∆ = l + 1
• One spinor Flin the (l + 1/2,1/2) representation with ∆ = l + 3/2,
• One massive vector Jlin the (l,0) representation with ∆ = l + 2,
• One real scalar Elin the (l,1) representation with ∆ = l + 2,
• One spinor Glin the (l − 1/2,1/2) representation with ∆ = l + 5/2,
• One real scalar Dlin the (l − 1,0) representation with ∆ = l + 3.
Moreover we mapped the operators in the supermultiplet to the fluctuations of the probe
brane summarized in tables 3 and 4.
Finally, we consider Nf> 1 coincident probe D5branes. The dual field theory then has
Nf massless flavors, with a global U(Nf) flavor symmetry. The overall U(1) we identify as
baryon (more accurately quark) number, while the SU(Nf) subgroup we identify as isospin.
The mesino operators Fland Glof course have zero baryon number charge and are valued in
the adjoint of SU(Nf). For example, in our case with Nf= 2 the mesinos acquire an SU(2)
isospin index, Fa
As explained at the end of the last subsection, for our numerical analysis we use the
D5brane fermions with zero mass, Ψ−
l−1.
land Ga
l.
0. The dual fermionic operator is F0∼¯ψq + q†ψ.
4.3 Equation of Motion II: Gauge Couplings
In this section we return to the equation of motion for the worldvolume fermions, eq. (4.11),
and specialize to our case of interest, namely two coincident Dpbranes in (4+1)dimensional
– 33 –
Page 35
AdSSchwarzschild with trivial worldvolume scalars but nontrivial worldvolume gauge fields
A3
x(u). More specifically, we will explicitly unpack the gauge and curvedspace co
variant Dirac operator ?DAdSPfor the AdSPsubmanifold of (4+1)dimensional AdSSchwarzschld
and see how, when A1
x(u) is nonzero, the three worldvolume fermions couple to one another.
In this subsection we assume P ≥ 3.
The linearized equation of motion for the worldvolume fermions in eq. (4.11) is26
t(u) and A1
??DAdSP− m±
l
?Ψ±
l= 0, (4.15)
where the masses m±
as well as Ψ instead of Ψ±
operator. The index A runs over the worldvolume directions inside AdS5. Notice that here
eM
space, which obey?γA,γB?= 2ηAB.
For the Dirac operator, we have (here a is a gauge index)
lappear in eq. (4.12). To simplify the notation we write m instead of m±
l. ?DAdSP= eM
l
AγADMis the gauge and curvedspace covariant Dirac
Aare the inverse vielbeins. The γAare the Γmatrices of (4+1)dimensional Minkowski
???DAdSP− m?Ψ?
a=
?
u
?
f γu∂u+
u
√fγt∂t+ uγi∂i+
?
−P − 1
2
?
f +1
4uf′
√f
?
γu
?
(4.16)
Ψa
+ eM
AiγA[AM,Ψ]a− mΨa,
where f′= ∂uf. When T = 0 and hence f(u) = 1, the operator in parentheses on the
righthand side in the first line is the Dirac operator of AdSP. At finite temperature, where
f(u) = 1 − u4/u4
Schwarzschild.27The coupling to the gauge field in the second line is of course fixed by gauge
invariance.
h, the operator is that of an AdSP submanifold of (4+1)dimensional AdS
We need an ansatz for Ψa. For the coordinate dependence, our ansatz will be similar to
the one in refs. [9, 10, 11],
Ψ = Ψaσa= u(P−1)/2f−1
4eikµxµψa(u)τa,(4.17)
where µ runs over field theory directions, the ψ(u)aare three spinor functions for which we
must solve, and we extract a factor of u(P−1)/2f−1
later (in the language of ref. [25] these factors will “remove the spin connection” from the
equation of motion).
4 to make the Dirac equation look nice
Using our ansatz for the fermion in eq. (4.17) and the ansatz for the gauge field in eq.
(3.8) (from which we can recover eq. (3.12) simply by setting A1
x= 0), we find three Dirac
26Here the ± does not refer to the projectors Π±of section 2, but rather to the ± sign labeling the eigenvalues
of the Dirac operator of SQin eq. (4.10).
27For a general (d+1)dimensional AdSSchwarzschild space, f(u) = 1 − ud/ud
h.
– 34 –
Page 36
equations,
0 =
??
??
??
fγu∂u−iω
fγu∂u−iω
fγu∂u−iω
√fγt+ ikiγi−1
√fγt+ ikiγi−1
√fγt+ ikiγi−1
um
?
?
?
ψ1+A3
t(u)
√f
γtψ2, (4.18)
0 =
um
ψ2−A3
t(u)
√f
γtψ1+ A1
x(u)γxψ3, (4.19)
0 =
um
ψ3− A1
x(u)γxψ2.(4.20)
In what follows we use the Lorentziansignature γAfrom section 2, in which all the γA
are Hermitian except for γt, which will be antiHermitian:28
γu=
?
−σ3
0
0
−σ3
?
,γt=
?
iσ1 0
0 iσ1
?
,γx=
?
−σ2 0
0σ2
?
,γy=
?
0 σ2
σ2 0
?
(4.21)
.
We will also use the projectors Π1,2, which in Lorentzian signature are defined as
Πα≡1
2
?1 − (−1)αγuγtγx?, (4.22)
with α = 1,2.
4.3.1 Normal Phase
First consider the normal phase, where A1
ψ2, and its equation of motion becomes that of a free neutral fermion, as expected. We can
then simplify the remaining two equations by taking linear combinations of them. Defining29
ψ±≡ ψ2± iψ1, we find three decoupled equations,
??
0 =
x(u) = 0. In that case, ψ3decouples from ψ1and
0 =
fγu∂u−iω
fγu∂u−iω
fγu∂u−iω
√fγt+ ikiγi−1
√fγt+ ikiγi−1
√fγt+ ikiγi−1
um
?
?
?
ψ++ iA3
t(u)
√f
γtψ+, (4.23)
??
??
um
ψ−− iA3
t(u)
√f
γtψ−, (4.24)
0 =
um
ψ3
(4.25)
which are precisely the equations of motion for fermions (in the fundamental representation
of the unbroken U(1)3 ⊂ SU(2)) with charges q = ∓1,0. As in section 4, we emphasize
that, because we consider a particular embedding of the Dirac equation into string theory,
the allowed values of the mass and charge of the fermions are constrained by supersymmetry
and Tduality.
28We only need γywhen P ≥ 4.
29Here the ± index refers to linear combinations of the worldvolume fermions that diagonalize the equations
of motion when A1
projectors Π± of section 2 or to the eigenvalues of the Dirac operator on SQof eq. (4.10).
x(u) = 0. Nowhere in this subsection or the next do we use a ± index to refer to the
– 35 –
Page 37
We will now follow appendix A of ref. [11] to simplify the equation of motion further.
First we rewrite the equation as30,
??
f γu∂u−1
um
?
ψ + iKµ(u)γµψ = 0, (4.26)
Kµ(u) = (−v(u),ki),v(u) =
1
√f(ω + qAt(u)),(4.27)
where the index i runs over spatial directions, q = ∓1 for ψ±and q = 0 for ψ3. Notice that
near the boundary, v(u) → ω + qµ, so the frequency ω is measured relative to (q times) the
chemical potential.
The system is rotationally invariant, so without loss of generality we can take only kxto
be nonzero. (Obviously, this will not be the case in the superfluid phase, where rotational
symmetry is broken.) The fermion’s equation of motion then depends only on γu, γtand
γx, so the projectors Παcommute with the operator acting on the ψ in eq. (4.26), hence
the equations for φα≡ Παψ decouple from one another. In terms of the φα, the equation of
motion becomes
?
∂u+
1
u√fmσ3
?
φα+
1
√f
(−iv(u)σ2− (−1)αkxσ1)φα= 0. (4.28)
We thus obtain six decoupled equations, four for the φ±αand two for the φ3α.
Eq. (4.28) is almost identical to eq. (A14) of ref. [11]. The biggest difference is the
function f(u), which for us is the f(u) of (4+1)dimensional AdSSchwarzschild and in ref.
[11] was the f(u) of (3+1)dimensional AdSSchwarzschild. Given that we will solve nearly
identical equations of motion, we will obtain qualitatively similar finitetemperature results.
As mentioned above, however, we cannot reach T = 0 within the probe approximation, so
we will not be able to reproduce the T = 0 results of refs. [9, 11], including in particular the
influence of an emergent AdS2.
4.3.2 Superfluid Phase
In the solution corresponding to the superfluid phase, where A1
write linear combinations of ψ1, ψ2 and ψ3 to diagonalize the system and produce three
decoupled equations. To make comparison to the normal phase easier, we will again work
with ψ±= ψ2± iψ1, so that eqs. (4.18)(4.20) become
??
0 =
um
??
30Starting now, we will use the notation ψ to refer to any of our three fermions, when we are making general
statements.
x(u) is nonzero, we cannot
0 =
fγu∂u−iω
fγu∂u−iω
fγu∂u−iω
√fγt+ ikiγi−1
√fγt+ ikiγi−1
√fγt+ ikiγi−1
um
?
?
?
ψ++ iA3
t(u)
√f
γtψ++ A1
x(u)γxψ3,(4.29)
??
ψ−− iA3
ψ3−1
t(u)
√f
γtψ−+ A1
x(u)γxψ3,(4.30)
0 =
um2A1
x(u)γx(ψ++ ψ−).(4.31)
– 36 –
Page 38
Clearly the three fermions ψ± and ψ3 couple to one another via a nonzero A1
we have a concrete example of the couplings described in section 2.2 (especially around eq.
(2.32)). We simplify the ψ+and ψ−equations again by writing them as
x(u). Here
0 =
??
??
??
f γu∂u−1
f γu∂u−1
f γu∂u−1
um
?
?
?
ψ++ iKµ(u)γµψ++ A1
x(u)γxψ3,(4.32)
0 =
um
ψ−+ iKµ(u)γµψ−+ A1
x(u)γxψ3,(4.33)
0 =
um
ψ3+ iKµ(u)γµψ3−1
2A1
x(u)γx(ψ++ ψ−), (4.34)
where Kµis defined the same way as in eq. (4.27). Recall that ψ+has charge q = −1, ψ−
has charge q = +1 and ψ3has charge q = 0.
Now we come to a big difference from the solution corresponding to the normal phase,
at least for Dpbranes wrapping AdSP with P ≥ 4. In the solution corresponding to the
superfluid phase, rotational symmetry is broken from SO(P − 2) to SO(P − 3). Using the
SO(P − 3) rotational symmetry, the most general momentum we can pick has nonzero kx
and nonzero ky. The equations for ψ+and ψ−then depend on γu, γt, γxand now also γy,
hence the Παprojectors no longer commute with the operators acting on the fermions in the
equations of motion, and the equations for the φα≡ Παψ will not decouple from each other.
Upon acting with the projectors Πα, the equations of motion become
0 =
?
∂u+
m
u√fσ3+
1
√f
(−iv(u)σ2− (−1)αkxσ1)
−ky
?
φ+α
√fσ1(−1)α+1ǫαβφ+β−A1
x
√f(−1)αiσ1φ3α, (4.35a)
0 =
?
∂u+
m
u√fσ3+
1
√f
(−iv(u)σ2− (−1)αkxσ1)
−ky
?
φ−α
√fσ1(−1)α+1ǫαβφ−β−A1
x
√f(−1)αiσ1φ3α,(4.35b)
0 =
?
∂u+
m
u√fσ3+
1
√f
(−iv(u)σ2− (−1)αkxσ1)
−ky
?
φ3α
√fσ1(−1)α+1ǫαβφ3β+1
2
A1
√f(−1)αiσ1(φ+α+ φ−α).
x
(4.35c)
Here ǫαβis antisymmetric with ǫ12= +1. Notice that when ky is nonzero, the φ1and φ2
couple.
Eq. (4.35) is the result for any Dpbrane extended along AdSP× SQ. What will change
from one Dpbrane to another are the allowed values of m and the solutions for A3
A1
and Q = 2), and to the massless worldvolume fermion.
t(u) and
x(u). In the next section we specialize to the D5brane extended along AdS4× S2(P = 4
– 37 –
Page 39
5. Emergence of the pwave Fermi surface
5.1 Properties of the Spectral Function
For the probe D5brane worldvolume fermions of the last section, we solved the linearized
equations of motion, eqs. (4.35), numerically, and used these solutions to extract the fermionic
spectral functions. In this section we present a selection of our numerical results.
We work with two D5branes extended along (when T = 0) AdS4×S2inside AdS5×S5.
As shown in section 4.1, we have many worldvolume fermions to choose from, with many
different masses. In our numerical analysis we work exclusively with the massless worldvolume
fermion. The dual operator is then the l = 0 case of the mesino operator Flwritten explicitly
in eq. (4.13). These mesinos are valued in the adjoint of the SU(2) isospin symmetry, so
we actually have three mesinos, F+
under U(1)3. These are dual to the three fermions ψ±and ψ0in subsection 4.3.2.
In more detail, our procedure is as follows. We first choose the values of T and µ that
we want, and solve for the background SU(2) gauge field functions A3
section 3.2. Plugging the gauge field solution into eqs. (4.35), we then solve for the bulk
fermions. Near the horizon the fermions have the form of an ingoing wave, eq. (2.33), with
the α in that equation being α =
couple to one another, hence we employ the technique of section 2 to compute the retarded
Green’s function, which then gives us the spectral function, as we explain below.
The normalization of our Green’s functions is fixed by the normalization of the fermionic
part of the D5brane action, eq. (4.1). The normalization includes various numerical factors,
and in particular depends on the normalization of the S5spinor χ defined in eq. (4.6).
We will omit the details, but we will mention that the normalization includes a factor of
NfTD5 ∝
normalization. In other words, we will divide the action by the normalization factor, so that
we obtain an effective AdS4Dirac action with a Lagrangian of the form i¯Ψ?DΨ.
As explained in section 2, with three bulk fermions, the field theory retarded Green’s
function will be a 6×6 matrix. In the normal phase where the three bulk fermions decouple,
the Green’s function will be diagonal in isospin indices and in the subspaces defined by the
Π1,2projectors defined in eq. (2.12) (see also eq. (2.13)). Explicitly, the retarded Green’s
function will have the form
0, F−
0, and F0
0, where the superscript denotes the charge
t(u) and A1
x(u), as in
ω
4πTfor our system. When A1
x(u) is nonzero, the fermions
√λNfNc. In what follows we will rescale our Green’s functions by the overall
?GR
AB(ω,kx,ky)?= diag?GR
−2,GR
−1,GR
+2,GR
+1,GR
02,GR
01
?,(5.1)
with A,B = 1,...,6, so that A = 1 corresponds to the components of the F−
the Π2subspace, A = 2 corresponds to F−
components of the F+
the bulk couplings, for generic momenta all the offdiagonal elements become nonzero.
The spectral function Rabis defined as the antiHermitian part of the retarded Green’s
function,
RAB(ω,kx,ky) ≡ i
0mesino in
0in the Π1subspace, A = 3 corresponds to the
0mesino in the Π2subspace, and so on. In the superfluid phase, due to
?
GR
AB(ω,kx,ky) − GR†
AB(ω,kx,ky)
?
.(5.2)
– 38 –
Page 40
We define the spectral measure R(ω,kx,ky) as the trace over RAB(ω,kx,ky) (a trace over
both flavor and spinor indices),
R(ω,kx,ky) ≡ trRAB(ω,kx,ky). (5.3)
Stability requires the eigenvalues of RAB, and hence both the diagonal elements of RAB
and the spectral measure (the sum of the eigenvalues), to be strictly nonnegative. Otherwise,
if we perturb the medium with one of the operators appearing in the spectral function, the
resulting excitation would experience negative energy dissipation into the medium, i.e. the
excitation would extract energy from the medium, signaling an instability. The eigenvalues
of the spectral function are a direct measure of the states of the theory that have an overlap
with the relevant operators. The offdiagonal elements of RAB, however, need not obey any
positivity requirement. Indeed, the offdiagonal elements are more similar to interference
effects than to a measure of a density of states. We have confirmed numerically that all of
our spectral functions obey the correct positivity requirements.
We have confirmed that our numerical result for the spectral function obeys the following
symmetries in both the normal and superfluid phases (in each case, any argument not shown
is invariant):
R11(−ω) = R44(ω),
R11(−kx) = R22(kx),
R11(−ω,−ky) = R44(ω,ky), R22(−ω,−ky) = R33(ω,ky), R55(−ω,−ky) = R66(ω,ky). (5.6)
R22(−ω) = R33(ω),
R33(−kx) = R44(kx),
R55(−ω) = R66(ω),
R55(−kx) = R66(kx),
(5.4)
(5.5)
An example for frequency and momentum symmetries in the offdiagonal elements is
RAB(−ky) = (−1)A+BRBA∗(ky),(5.7)
which is also true for all diagonal elements, stating their invariance under ky → −ky. We
have also confirmed that our numerical results obey
RAB= RBA∗,(5.8)
which follows directly from the definition of the spectral function in eq. (5.2).
Lastly, notice that because we study a fermionic operator of dimension 3/2, the retarded
Green’s function, and hence the spectral function and measure, are dimensionless.
5.2 Numerical Results
First we compute the spectral function for temperatures below Tcbut in the normal (non
superfluid) phase, which we know is thermodynamically disfavored. We do so for two reasons:
first, to compare later to the superfluid phase, and second, to reproduce some of the finite
temperature results of ref. [9], as a check of our methods. In practical terms, we use the
solution for the gauge field with A3
t(u) (from eq. (3.12) with P = 4) and zero A1
x(u). Figure 2
– 39 –
Page 41
shows two diagonal spectral function components, RAA(ω,kx,ky) with A = 1,5, as functions
of kx/πT, with ω = ky= 0, for T ≤ Tcin the normal phase.
Figure 2 (a) shows the component R11, which corresponds to the components of the
mesino F−
point is that figure 2 (a) is qualitatively similar to figure 4 of ref. [9]. As we lower the
temperature, the peaks in the figure move to larger momenta and additional peaks appear
near zero momentum. In fact, in figure 2 (a), R11 along negative momenta (dashed blue
curve) already displays a small bump near kx/πT = 0, which grows into a peak as we cool
the system. Similar effects were observed in ref. [9], and were interpreted as the emergence
of multiple Fermi surfaces at different momenta. Additionally, the spectral functions for our
other charged fermions in the lowtemperature normal phase are similar to those in refs.
[9, 11] (so we will not present them).
Figure 2 (b) shows the component R55, which corresponds to the neutral operator F0
in the Π2subspace, at temperatures T = Tc, 0.75Tc, and 0.61Tc. R55is featureless here, but
will not be so in the pwave phase. Notice that R55also does not change as the temperature
decreases, or equivalently as the chemical potential increases, since when q = 0 the chemical
potential does not enter the relevant bulk fermion’s equation of motion: see eq. (4.27).
0in the Π2 subspace, at temperature T = 0.61Tc. For the moment, our main
0,
012345
0
2
4
6
8
R11
kx/πT
(a)
012345
0.0
0.5
1.0
1.5
2.0
R55
kx/πT
(b)
Figure 2: Two spectral function components RAA(ω,kx,ky), for A = 1,5, plotted versus the rescaled
momentum kx/πT, with ω = ky= 0, at temperatures T ≤ Tcbut in the normal (nonsuperfluid) phase,
i.e. in the thermodynamically disfavored phase. (a) R11, corresponding to the fermionic operator
with charge q = −1, at T = 0.61Tc. The two curves are for positive momentum kx(solid red curve)
and negative momentum −kx (dashed blue curve), the latter case being equal to R22(ω,kx,ky) at
positive momentum due to the symmetries of the spectral function. Multiple peaks are visible, just
as in ref. [9]. (b) R55, corresponding to a fermionic operator with charge q = 0, at temperatures
T = Tc(solid curve), T = 0.75Tc, (grey dotted curve), T = 0.61Tc(dashed curve). The curves remain
coincident since changing the chemical potential does not affect the uncharged operator.
Next we plot the essentially the same thing as in figure 2, but now in the thermodynam
ically favored superfluid phase. More precisely, figure 3 shows two diagonal spectral function
– 40 –
Page 42
components, RAA(ω,kx,ky) with A = 1,5, as a function of the rescaled momentum kx/πT,
with ω = ky= 0, for T ≤ Tcin the superfluid phase, i.e. now with nonzero A1
3 (a) shows the same component of the spectral function as in figure 2 (a), R11, again with
T = 0.61Tc. Figure 3 (b) shows the same component of the spectral function as in figure 2
(b), R55, at the same temperatures T = Tc, 0.75Tc, and 0.61Tc.
The operator mixing is obvious in figure 3: the spectral function for a neutral fermion,
R55, develops a bump that grows into a small peak located at the same momentum as the
peak in R11, kx/πT = 3.87. In bulk terms, the coupling between φ3and φ±in eq. (4.35)
is allowing the peak in the charged fermions’ spectral functions to “leak” into the spectral
functions of the neutral fermions. That coupling is proportional to A1
should grow as the temperature decreases and A1
figure 3 (b). The method we developed in section 2 for computing retarded Green’s functions
for coupled bulk fermions seems to work very well.
x(u). Figure
x(u), hence the peak
x(u) grows, which is indeed what we see in
012345
0
2
4
6
8
R11
kx/πT
(a)
012345
0.0
0.5
1.0
1.5
R55
kx/πT
(b)
Figure 3: Two spectral function components RAA(ω,kx,ky) plotted versus the rescaled momentum
kx/πT with ω = ky= 0 in the superfluid phase. (a) Exactly the same spectral function as in figure
2 (a), R11, at the same temperature T = 0.61Tc, but now in the superfluid phase. The two curves
correspond to positive momentum kx(solid red curve) or negative momentum −kx(dashed blue curve).
(b) Exactly the same spectral function component as in 2 (b), R55, at the same temperatures Tc(solid
curve), T = 0.75Tc(grey dotted curve), T = 0.61Tc(dashed curve). Here we see operator mixing: a
feature develops in the neutral fermion’s spectral function in the pwave phase. A bump grows into a
peak at the same momentum kx/πT ≈ 3.87 as the peak in R11.
As another comparison between the normal and broken phases when T ≤ Tc, we focus on
the pole in the retarded Green’s function GR
11(ω,kx,ky), for the mesino with charge q = −1
in the Π2subspace, that is closest to the origin of the complex frequency plane, ω = 0, and
follow the movement of the pole in the frequency plane as we change the momentum.31
31Poles in the retarded Green’s function are holographically equivalent to the bulk fermion’s quasinormal
modes [87, 88, 89]. As explained in section 2, we can detect these quasinormal modes from the vanishing of
detP−(ǫ), where the matrix P−is defined in eqs. (2.35) and (2.36).
– 41 –
Page 43
?8
?6
?4
?20
?0.0070
?0.0065
?0.0060
?0.0055
?0.0050
?0.0045
?0.0040
Re
ω
πT
Im
ω
πT
(a)
?8
?6
?4
ω
πT
?20
?7.5?10?6
?7.4?10?6
?7.3?10?6
?7.2?10?6
Re
Im
ω
πT
(b)
?8
?6
?4
?20
?0.018
?0.016
?0.014
?0.012
?0.010
?0.008
?0.006
Re
ω
πT
Im
ω
πT
(c)
?3.5
?3.0
?2.5
?2.0
?1.5
?1.0
?0.50.0
?0.10
?0.08
?0.06
?0.04
?0.02
Re
ω
πT
Im
ω
πT
(d)
Figure 4:
of charge q = −1, that is closest to ω = 0, as a function of momentum, for T < Tc, in both the
normal (thermodynamically disfavored) phase and superfluid (thermodynamically favored) phase. (a.)
The movement of the pole in the normal phase at T = 0.91Tc, for kx/πT ∈ [2.58,12.58]. As the
momentum increases, the pole moves from the lower right, near Reω/πT = 0, toward the upper
left. (The same applies for the following three figures.) The pole asymptotically approaches the real
frequency axis, Imω/πT = 0, as kxincreases. (b) The movement of the pole in the normal phase at
T = 0.48Tc, for kx/πT ∈ [6.15,16.15]. Here we see that the distance to the real axis does not decrease
monotonically, but rather a local minimum develops at (Reω/πT,Imω/πT) = (−1.79,−7.21× 10−6)
when kx/πT = 8.15. (c.) The movement of the pole in the superfluid phase at T = 0.91Tc, for
kx/πT ∈ [2.49,12.29] and ky= 0. The movement is qualitatively similar to (a). (d) The movement
of the pole in the superfluid phase at T = 0.48Tc, for kx/πT ∈ [4.83,8.98] and ky= 0. The movement
is again qualitatively similar to (a), in particular, the distance to the real frequency axis does not
develop a local minimum, in contrast to the normal phase result in (b). We see qualitatively similar
behavior when we set kx= 0 and increase ky.
The movement of the pole in the retarded Green’s function GR
11(ω,kx,ky), for a mesino
Figure 4 (a) shows the movement of the pole in the normal phase when T = 0.91Tc
for values of kx/πT ∈ [2.579,12.580], which in the figure corresponds to starting at the point
nearest Re[ω/πT] = 0 and moving toward the upper left. The pole asymptotically approaches
the real frequency axis Imω/πT = 0, as kxincreases.
– 42 –
Page 44
At a temperature T∗≈ 0.6Tc, however, the distance to the real axis develops a local
minimum. Figure 4 (b) shows the movement of the same pole as figure 4 (a) at a temperature
T = 0.48Tc < T∗, still in the normal phase. Here we see that the distance to the real
frequency axis has a local minimum at (Reω/πT,Imω/πT) = (−1.79,−7.21 × 10−6) when
kx/πT = 8.15. Such behavior persists to lower temperatures, and indeed, the distance to the
real frequency axis decreases. The lowest temperature we studied was T = 0.19Tc, where the
local minimum occurred at (Reω/πT,Imω/πT) = (5.32,2.64 × 10−18) when kx/πT = 23.22.
We seem to be seeing the emergence of a Fermi surface, which, as in ref. [9], would occur
at T = 0 when the pole would reach the origin of the complex frequency plane at some finite
momentum kF, the Fermi momentum. Let us consider low temperature, and define k∗as
the value of momentum where the closest approach to the real frequency axis occurs. When
T = 0, k∗would be the Fermi momentum, k∗= kF. At our lowest temperature, T = 0.19Tc,
k∗
(Reω∗,Imω∗) = (5.32,2.64×10−18)πT. Letting k∗play the role of kF, we see behavior similar
to the results of ref. [9]: for small but nonzero temperature, as we change the momentum the
frequency of the pole behaves as
x/πT ≡ 23.22, and the closest approach to the real frequency axis occurs at an ω∗given by
ω − ω∗∼ (k − k∗)z,(5.9)
where our numerical results indicate that the exponent z = 1.00 ± 0.01, and the spectral
function behaves as
R11∼ (k − k∗)−α,
where our numerical results indicate that the exponent α = 2.0 ± 0.1. In fact, these results
are rather robust: we find the same z and α for many values of T < T∗, and for several other
poles. These results suggest that the lowtemperature normal phase may not be a Landau
Fermi liquid, which would have z = α = 1.
Figures 4 (c) and (d) show the movement of the same pole, at the same temperatures,
but in the superfluid phase.32Figure 4 (c) shows the movement of the pole at T = 0.91Tcfor
kx/πT ∈ [2.49,12.29] and ky= 0. Figure 4 (d) shows the movement of the pole at T = 0.48Tc
for kx/πT ∈ [4.83,8.98] and ky= 0. Unlike the normal phase result, here the distance to the
real frequency axis does not develop a local minimum. In other words, here we do not see a
Fermi surface emerge in the same fashion as in the normal phase.
To see the emergence of the pwave superfluid Fermi surface, we study the spectral
measure (largely following ref. [18]), which as mentioned above, provides a direct measure of
the density of states that have overlap with our fermionic operators.
Our main results concern the evolution of the spectral measure R(ω,kx,ky) as we cool
the system through the superfluid phase transition. In the spectral measure we will see the
breaking of rotational symmetry as we take T below Tc, and we will see the emergence of the
Fermi surface as we approach T = 0, although in the probe limit we will not reach T = 0.
(5.10)
32Our independent caluclations in the normal and superfluid phases yield the same position for the pole at
T = Tc to within 0.1%.
– 43 –
Page 45
Our results agree qualitatively with the T = 0 results of ref. [18], in which the Fermi surface
in the pwave phase consists of isolated points.
Figure 5 provides a road map for the evolution of the spectral measure as we lower the
temperature. Here we set ω = 0, so we are studying fluctuations with zero energy above the
chemical potential. Figure 5 (a) indicates the locations of peaks in the spectral measure, in
the (kx/πT,ky/πT) plane, with solid lines and the locations of small bumps as the dashed
grey line. At T = Tcwe see rotational symmetry: the black solid line indicates peaks for
any values of (kx/πT,ky/πT) on the black circle. At T = 0.7Tcthe rotational symmetry is
mildly broken: the green line is not a perfect circle. At T = 0.43Tc, sharp peaks only appear
at isolated points on the axes, denoted by the red and blue dots (and also at the blue dot at
the origin), while the dashed line indicates a small bump, rather than a sharp peak. Here we
are clearly seeing the emergence of the Fermi surface at isolated points.
To illustrate what the peaks and bumps look like, we choose a representative slice of the
(kx/πT,ky/πT) plane, namely the line given by the polar angle φ = π/8, which is drawn in
figure 5 (a), and plot the (ω = 0) spectral measure versus the magnitude of the momentum
k =
curve) and T = 0.7Tc(green curve), while for this generic (offaxis) value of φ the only feature
of the spectral measure at T = 0.43Tc (dashed grey curve) is a small bump. On the axes
(φ = 0,π/2), the picture is similar, except the bump becomes a sharp peak, corresponding to
the red or blue dots in figure 5 (a).
?
k2
x+ k2
ydivided by πT. In figure 5 (b) a distinct peak is visible for both T = Tc(black
To illustrate the evolution of the spectral measure in more detail, we present three
dimensional plots of R(ω,kx,ky), for ω = 0, over the (kx/πT,ky/πT) plane, for temperatures
T = Tc(figure 6 (a)), T = 0.91Tc (figure 6 (b)), T = 0.69Tc(figure 6 (c)) and T = 0.4Tc
(figure 6 (d)). In figure 6 (a) we see the peaks corresponding to the black circle in figure 5
(a). Clearly here the spectral measure is rotationally symmetric. When we cool the system
to T = 0.69Tc(figure 6 (c)), we clearly see the emergence of five distinct peaks, two on the
kx/πT axis, two on the ky/πT axis, and one at the origin. The circle of peaks corresponds
to the green circle in figure 5 (a). When we further cool the system to T = 0.4Tc, the five
peaks are still present, although the resolution of our threedimensional plot is insufficient to
resolve the two on the ky/πT axis away from the origin.
Although these peaks have a much smaller footprint in the (kx/πT,ky/πT) plane than
the peaks on the kx/πT axis, they are much taller. The spectral measure R is of order 5×105
at the peaks on the ky/πT axis, but only order 102at the peaks on the kx/πT axis, and order
5 × 104at the peak at the origin. Apparently a large number of states are piling up at two
precise locations on the ky/πT axis.
For a bulk theory with SU(2) gauge fields and fermions in the fundamental representation,
a combination of analytic and numerical results reveal that the ω = 0 spectral measure at
T = 0 consists of two isolated points on the kxaxis, located symmetrically about the origin
[18]. For the same bulk theory but for fermions in the adjoint representation, as we consider
here, the analytic arguments of ref. [18] indicate that the ω = 0 spectral measure at T = 0
– 44 –
Page 46
Φ?Π?8
Φ?Π?8
Tc Tc
0.7 Tc
0.7 Tc
0.4 Tc
0.4 Tc
?4
?2024
?4
?2
0
2
4
kx
πT
ky
πT
(a)
024
k/πT
(b)
68
0
2
4
6
8
10
12
14
R
Figure 5:
(kx/πT,ky/πT) plane at ω = 0, as we lower the temperature. (a.) The position of peaks in the
spectral measure R are indicated by the curves as we lower the temperature from Tc(black curve)
through 0.7Tc (green curve) to 0.43Tc (dashed grey curve). The T = 0.43Tc case exhibits a small
bump rather than a sharp peak, except for points on the kx and ky axes. We have indicated the
bump with the grey dashed curve and the peaks with red and blue dots, including the blue dot at the
origin. (b.) The spectral measure R plotted for a representative slice of the (kx/πT,ky/πT) plane
(still with ω = 0), namely along the line given by the polar angle φ = π/8 drawn in (a.). We plot
R versus the magnitude of the momentum k =
T = 0.7Tc(green curve), and T = 0.43Tc(dashed grey curve).
The evolution of the spectral measure R(ω,kx,ky) = trRAB(ω,kx,ky) in the
?
k2
x+ k2
ydivided by πT, at T = Tc(black curve),
should consist of three isolated points, one at the origin and two on the kx axis, located
symmetrically about the origin.
Therefore, what we see appears to be consistent with the results of ref. [18] for the
structure of the spectral measure in the superfluid phase. The main obstacle to a direct
comparison is the probe limit, which restricts us to finite temperatures: we do not know
which peaks in our spectral measure persist to T = 0. Nevertheless, given that we see the
three peaks on the kx/πT that we generically expect, and that the two peaks on the ky/πT
axis have a shrinking footprint as we cool the system, we have good reason to believe that
the results of ref. [18] may apply to our system. To answer the question fully requires
computing the backreaction of the D5branes in the bulk. Whether that produces a domain
wall geometry of the kind used in ref. [18] is not guaranteed.
We begin to explore the ω dependence of the spectral measure R(ω,kx,ky) in figure 7.
Figure 7 (a) simply reproduces the T = 0.4Tcpart of figure 5 (a), where ω = 0. Figure 7 (b)
shows the spectral function along three lines in the (kx/πT,ky/πT) plane: along the positive
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Page 47
(a) T = Tc, ω = 0 (b) T = 0.91Tc, ω = 0
(c) T = 0.69Tc, ω = 0 (d) T = 0.43Tc, ω = 0
Figure 6: Threedimensional plots of the spectral measure R(ω,kx,ky) in the superfluid phase over
the (kx/πT,ky/πT) plane at zero frequency, ω = 0, and for distinct temperatures T ≤ Tc. (a) The
T = Tccase, which is clearly rotationally invariant. The peaks correspond to the black circle in figure 5
(a). (b) The T = 0.91Tccase, where the spectral measure does not yet display any dramatic breaking
of rotational symmetry. Notice the peak a the origin of the (kx/πT,ky/πT) plane. (c) The T = 0.69Tc
case, where the breaking of rotational symmetry is obvious. We see that the “cylinder” of (a) breaks
into five distinct peaks on the kx/πT and ky/πT axes. The peaks correspond to the green circle in
figure 5 (a). (d) The T = 0.4Tccase, which still has five peaks, labeled by the red and blue dots in
figure 5 (a). The peaks along the ky/πT axis are too narrow to appear in the threedimensional plot
with the resolution we use.
kx/πT axis (red curve), along the positive ky/πT axis (blue curve), and along the line given
by the polar angle φ = π/8 in figure 5 (a). Here we see explicitly the difference in widths of
the peaks on the kx/πT and ky/πT axes (the red and blue peaks).
To explore the ω dependence, we choose a few representative points in the (kx/πT,ky/πT)
plane and, for each point, plot the spectral measure versus ω. For these points, we consider
not only T = 0.4Tc, as in figure 7 (a), but also the slightly higher temperature T = 0.55Tc,
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Page 48
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πT
(a)
024
k/πT
(b)
68
0
1
2
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6
R
02468 1012
0
20
40
60
80
100
R
ω/πT
(1)
02468 10
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25
R
ω/πT
(2)
02468101214
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R
ω/πT
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Figure 7: (a) Peaks in the ω = 0 spectral measure in the (kx
is the same as figure 5 (a). (b) The ω = 0 spectral measure along three lines in (a): along the
(red curve) and the
πTaxis (blue curve), so the peaks at nonzero momenta correspond to (3) and (4)
in (a), respectively, and along the polar angle φ = π/8 (dashed grey curve), drawn in figure 5 (a), so
the bump corresponds to where φ = π/8 intersects the dashed grey line in (a). In each case the peak
at zero momentum corresponds to (0) in (a). (1) R as a function of ω/πT for the point labeled (1) in
(a). The dotted and solid lines are for T = 0.55Tcand T = 0.4Tc, respectively. (2) and (3) show the
same thing for the corresponding points (2) and (3) in (a).
πT,
ky
πT) plane at T = 0.4Tc. The labeling
kx
πTaxis
ky
– 47 –
Page 49
in order to study the behavior as we cool the system.
Our points are similar to those in figure 8 of ref. [18], where the same quantities were
plotted (for the slightly different system of ref. [18]): the spectral measure versus ω for fixed
kxand ky. We will thus compare our results to those of figure 8 of ref. [18] along the way.
In figure 7 (1) we plot the ω dependence of R(ω,kx,ky) for the point (1) labeled in figure
7 (a). The dotted line is for T = 0.55Tcand the solid line is for T = 0.40Tc. Here we see that
as we cool the system, a small gap (a depletion of states) opens near ω = 0, while a sharp
peak emerges near ω/πT ≈ 1.7. Such behavior at least appears to be approaching that of
figure 8 (1) in ref. [18], where a genuine gap (zero states) appeared at ω = 0.
In figure 7 (2) we plot the ω dependence of R(ω,kx,ky) for the point (2) labeled in figure
7 (a). The dotted line is for T = 0.55Tcand the solid line is for T = 0.40Tc. Here we see
that a sharp peak near ω/πT ≈ 1.5 when T = 0.55Tc shrinks and begins moving toward
ω = 0 as we lower the temperature to T = 0.4Tc. Moreover, the small peak near ω/πT ≈ 3.5
when T = 0.55Tcgrows much sharper at T = 0.4Tc. As in ref. [18], here we seem to see the
emergence of the wellknown “peakdiphump” shape, with the peak being at ω/πT ≈ 3.5,
the dip being at ω/πT ≈ 5.2, and the hump being at ω/πT ≈ 6.5.
In figure 8 (2) of ref. [18], a gap was present in the spectral measure for small frequencies,
except for a single genuine deltafunction peak at finite frequency, and at larger frequency
a continuum of states appears (the “hump”). As argued in ref. [18], at finite temperature
the deltafunction peak will acquire a finite width and merge with the hump, producing the
peakdiphump. Our spectral measure appears to be approaching the form of the spectral
measure in figure 8 (2) of ref. [18] (with the usual caveat that we cannot actually reach
T = 0).
In figure 7 (3) we plot the ω dependence of R(ω,kx,ky) for the point (3) labeled in figure
7 (a), which is sitting right on top of the peak on the positive kx/πT axis. The dotted line is
for T = 0.55Tcand the solid line is for T = 0.40Tc. At the higher temperature (the dotted
line), the primary feature is the peak near ω/πT ≈ 1.2, which moves toward ω = 0 and also
shrinks (the peak is lower) as we lower the temperature, becoming the peak in the solid line.
Assuming such a trend continues, our results would be consistent with figure 8 (3) of ref. [18],
where, sitting right on top of the peak on the kxaxis, the spectral measure went to a finite
constant at ω = 0.
Figure 8 shows R(ω,kx,ky) versus ω/πT for the points (0) and (4) in figure 7 (a), sitting
right on top of the peaks at the origin and on the positive ky/πT axis, respectively. Here we
use only T = 0.4Tc. We clearly see a gap developing at low frequency in both cases. Such
behavior is similar to the gap that develops in the spectral function of vector fluctuations
[32, 34], which is immediately related (via a Kubo formula) to a gap in the conductivity.
Whether these two gaps are related is unclear, but deserves further study.
Finally, to explore further the ω dependence of R(ω,kx,ky), we do not restrict to points in
the (kx/πT,ky/πT) plane, but rather restrict to a single nonzero value of frequency, ω/πT =
0.25, and plot the spectral me asure over the entire (kx/πT,ky/πT) plane. The result appears
in figure 9, where (a) is for T = Tc, (b) is for T = 0.91Tc, (c) is for T = 0.54Tcand (d) is
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Page 50
?4??4?
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25
R
ω/πT
(a)
Figure 8:
corresponding to points (0) (dotted blue curve) and (4) (solid blue curve) in figure 7 (a). These points
sit right on top of the peaks at the origin and on the positive ky/πT axis, respectively. Here T = 0.4Tc.
The spectral measure R(ω,kx,ky) as a function of ω/πT for (kx/πT,ky/πT) values
for T = 0.4Tc. We see a number of differences from the ω = 0 case of figure 6. At the
transition, T = Tc, the spectral measure is again rotationally symmetric, but now with two
concentric “cylinders.” By the time we cool the system to T = 0.54Tc, we see a number of
peaks clumped near the momentum axes. If we compare the spectral measure at T = 0.4Tc
at ω = 0 and ω/πT = 0.25, figures 6 (d) and 9 (d), respectively, then we see that the ω = 0
peaks on the ky/πT axis each split into a number of peaks at ω/πT = 0.25 which then move
apart along the kx/πT axis.
To summarize, in the superfluid phase we observe isolated peaks in the spectral measure,
whose locations appear to be consistent with the pwave nature of the condensate, as well as
with the fact that our fermions are in the adjoint representation of SU(2). Such a structure
suggests nodes in the energy gap on the normalphase Fermi surface. We plan to investigate
this further in the near future, for instance by identifying the appropriate Dirac cones and by
studying the spectral measure for fixed values of ω, as well as for further values of fixed kx,
ky.
6. Conclusions
The main results of this paper are:
– 49 –
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