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arXiv:1003.1134v2 [hep-th] 25 May 2010
Preprint typeset in JHEP style - HYPER VERSION
MPP-2010-28
PUPT-2331
Fermionic Operator Mixing in Holographic p-wave
Superfluids
Martin Ammon,1∗Johanna Erdmenger,1†Matthias Kaminski,2‡and Andy O’Bannon1§
1Max-Planck-Institut f¨ ur Physik (Werner-Heisenberg-Institut)
F¨ ohringer Ring 6, 80805 M¨ unchen, Germany
2Department of Physics, Princeton University
Jadwin Hall, Princeton, NJ 08544, USA
Abstract: We use gauge-gravity duality to compute spectral functions of fermionic operators
in a strongly-coupled defect field theory in p-wave superfluid states. The field theory is (3+1)-
dimensional N = 4 supersymmetric SU(Nc) Yang-Mills 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 p-wave 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 D5-branes in AdS5× S5with non-trivial 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.
∗E-mail address: ammon@mppmu.mpg.de
†E-mail address: jke@mppmu.mpg.de
‡E-mail address: mkaminsk@princeton.edu
§E-mail 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 p-wave Superfluids
3.1p-waves, Probe Branes, and Vector Meson Condensation
3.2Probe Dp-branes 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 p-wave 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 Anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT) [1, 2, 3], and more
generally gauge-gravity duality, is a holographic duality between a weakly-coupled theory of
gravity in some spacetime and a strongly-coupled field theory living on the boundary of that
spacetime. Gauge-gravity duality thus provides a powerful new tool for studying strongly-
coupled, scale-invariant field theories in states with finite charge density, and hence may be
useful in condensed matter physics, for instance in understanding low-temperature 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].
Gauge-gravity 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 strongly-coupled theory with a global U(1) symmetry, in a zero-temperature 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 Reissner-Nordstr¨ 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 s-wave superfluid states, if a scalar
charged under the U(1) is present [13, 14, 15]. On the gravity side, the Reissner-Nordstr¨ om
black hole grows scalar hair at low temperature, that is, a solution with a non-trivial scalar
becomes thermodynamically preferred to Reissner-Nordstr¨ om. In the dual field theory, the
thermodynamically-preferred 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 mean-field exponents [14, 15].
Gauge-gravity duality can also describe p-wave 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 non-Abelian 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 non-trivial 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 p-wave, a major technical simplification is the so-
called probe limit, in which the charge of the bulk scalar, or the SU(2) Yang-Mills coupling, is
sent to infinity, so that the scalar or Yang-Mills stress-energy tensor on the right-hand 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 Yang-Mills equation of motion in a fixed Reissner-Nordstr¨ 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’ stress-energy
tensor grows and we can no longer trust the probe limit. Reaching zero temperature requires
solving the fully-coupled equations, as done in refs. [15, 17, 18].
The zero-temperature 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 zero-temperature s-wave su-
perfluid states [24, 25, 26, 27] exhibit the so-called ‘peak-dip-hump’ structure [24], expected
to be relevant in high-Tcsuperconductors [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, weak-coupling 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 D-branes (for which we know the
worldvolume theories). String (and M-) theory embeddings of holographic s-wave superfluids
appear in refs. [20, 30, 31].
A string theory embedding of holographic p-wave superfluids, in the probe limit, appears
in refs. [32, 33, 34, 35]. Here we begin with Nc Dq-branes and Nf Dp-branes. Taking
the usual decoupling limit for the Dq-branes, which in particular means Nc→ ∞, we obtain
supergravity in the near-horizon geometry of the Dq-branes. Non-extremal Dq-branes 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 Dp-branes is then described by the non-
1? the result of some pairing
2A good recent example is ref. [29].
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Abelian Born-Infeld action (plus Wess-Zumino terms) in the near-horizon Dq-brane geometry.
Truncating that action to leading order in the field strength, we obtain a Yang-Mills action
in a black hole geometry.
We then know precisely what the dual field theory is: the Dq-brane worldvolume theory,
with gauge group SU(Nc), in the large-Ncand 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 Dp-branes do not overlap with all q spatial Dq-brane
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 Dp-branes is dual to a global U(Nf) analogous to the isospin symmetry of QCD. In
such systems the p-wave 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 p-wave phase.
We choose our Dq-branes to be D3-branes. The dual field theory is then a CFT, N = 4
supersymmetric Yang-Mills (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 five-sphere, AdS5×S5, with Ncunits of Ramond-Ramond
(RR) five-form flux on the S5. At finite temperature AdS becomes AdS-Schwarzschild.
We consider supersymetric probe Dp-branes 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 Dp-brane worldvolume scalars are trivial.
Our bulk fermions will be the Dp-branes’ 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 gauge-covariant 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 D-brane action,
for D-branes in arbitrary backgrounds (including RR fields) is known to quadratic order
[37, 38, 39]. The form of the action is determined by supersymmetry and T-duality [39], as
we review in section 4. For our Dp-branes 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 five-form.
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 near-horizon gravity solution
1? possible.3We will study only trivial embeddings of such Dp-branes, that is,
3One exception is a D5-brane along AdS2× S4, which we study in section 4.1 (but not in p-wave 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 T-duality. 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 Dp-brane extended along
AdSP× SQ, with P ≥ 3, but one particular Dp-brane is attractive for a number of reasons,
namely the D5-brane extended along AdS4× S2(P = 4 and Q = 2). From the bulk point of
view, this D5-brane is the only Dp-brane 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 D5-branes, 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 D5-branes’ 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 D5-brane 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 Berezinskii-Kosterlitz-Thouless
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 D5-brane is again attractive, however, because we know that,
unlike many Dp-branes, with zero temperature and zero chemical potential, fully back-reacted
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 AdS-Schwarzschild.
For any Dp-brane, the three worldvolume fermions decouple in the normal (non-superfluid)
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 Dp-branes, 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 Euclidean-signature AdS and determine the countert-
erms needed to render the on-shell 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 on-shell action were simply discarded.
For the P = 4, Q = 2 D5-brane, 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 p-wave phase.
Furthermore, as we lower the temperature, the zero-frequency 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 vector-hairy 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 p-wave 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 p-wave state is supported by ferromagnetic fluctuations that
increase the propensity for electrons to form spin triplet Cooper pairs, with an odd (p-wave)
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 anti-ferromagnet. 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 p-wave 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 p-wave 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 p-wave superfluids and demonstrate a p-wave transition using the
P = 4, Q = 2 D5-brane. In section 4, we write the fermionic part of the Dp-brane action,
perform the reduction of the worldvolume Dirac equation to AdSP, and, for the P = 4,
Q = 2 D5-brane, 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
D5-brane. 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 two-point function from a solution for a bulk Dirac fermion,
following refs. [9, 11, 60].
In this section we will work mainly with Euclidean-signature AdS space, with the metric
written in Fefferman-Graham 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 on-shell 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 lower-case 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 d-dimensional field theory. The on-shell 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 on-shell 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 long-distance 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 on-shell 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 on-shell 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 on-shell 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 well-defined 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 well-defined [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 well-known 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 non-relativistic 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 half-integer (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
half-integer. 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 five-dimensional 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 Γ-matrices. 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 finite-temperature, real-time correla-
tion functions, in particular the retarded Green’s functions, in which case the bulk geometry
will be Lorentzian-signature AdS-Schwarzschild. 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
curved-space 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 Γ-matrices. We will use a basis in which all
the Γ-matrices 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 p-wave 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
two-component 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 p-wave superfluid phase, the most general momentum has nonzero
components in multiple directions. That means Ψ1 and Ψ2 will no longer decouple because other Γ-matrices,
such as γy, will appear in the equation of motion, and these do not commute with Π1,2. Nevertheless, when
studying the p-wave superfluid phase we use Ψ1,2 to make the comparison with the rotationally-symmetric
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 first-order equations give rise to the second-order 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 near-boundary 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 left-hand-side is the exponential of minus the action in eq. (2.2), evaluated on
a solution and properly renormalized (hence the subscript), and the right-hand-side 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 on-shell 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 first-order in derivatives, the only nonzero contribution
to the on-shell 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 on-shell action: we evaluate Sbdy
on a solution and take ǫ → 0 to obtain (for positive, non-half-integer m)
?
We can now easily compute the renormalized connected correlators of O and¯ O by taking
functional derivatives of Sren. For example, the renormalized one-point 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 on-shell 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 two-point 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 AdS-Schwarzschild 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 two-point function
in the finite-temperature, Lorentzian-signature case. Here the geometry is AdS-Schwarzschild,
with a horizon at some position uh. To obtain the retarded two-point 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 in-going wave. The asymptotic form for Ψ near the boundary
is the same as in eq. (2.19) (for positive, non-half-integer 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 anti-Hermitian,
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
AdS-Schwazrschild, 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 left-hand-side, while only the Ψa∓are on the right-hand-side.
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 un-diagonalized 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 –
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