# Fermionic operator mixing in holographic p-wave superfluids

**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 \mathcal{N} = 4 supersymmetric SU(N

c

) Yang-Mills theory, in the ’t Hooft limit and with large coupling, coupled to two massless flavors of (2+1)-dimensional N = 4 \mathcal{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 AdS

5 × S

5 with 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.

KeywordsGauge-gravity correspondence-AdS-CFT Correspondence-Brane Dynamics in Gauge Theories

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**ABSTRACT:**Using a string inspired model we construct gravity duals generalizing px and px+ipy superconductors. Introducing a Chern-Simons coupling in the gravity side we demonstrate the ability to control which phase dominates at low temperatures and focus on the chiral px+ipy phase. We study the fermionic spectral function and establish that the behavior is rather different from the standard p-wave two-nodes model.Physical review D: Particles and fields 05/2012; 85(10). - SourceAvailable from: de.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We calculate the two-point Green's functions of operators dual to fermions of maximal gauged supergravity in four and five dimensions, in finite temperature backgrounds with finite charge density. The numerical method used in these calculations is based on differential equations for bilinears of the supergravity fermions rather than the equations of motion for the fermions themselves. The backgrounds we study have vanishing entropy density in appropriate extremal limits. Holographic Fermi surfaces are observed when the scalar field participating in the dual field theory operator has an expectation value, which makes sense from the point of view that the quasi-particles near the Fermi surfaces observed carry non-singlet gauge quantum numbers in the dual field theory.11/2014; - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We analyze fermionic response in the geometry holographically dual to zero-temperature N=4 Super-Yang-Mills theory with two equal nonvanishing chemical potentials, which is characterized by a singular horizon and zero ground state entropy. We show that fermionic fluctuations are completely stable within a gap in energy around a Fermi surface singularity, beyond which non-Fermi liquid behavior returns. This gap disappears abruptly once the final charge is turned on, and is associated to a discontinuity in the corresponding chemical potential. We also show that the singular near-horizon geometry lifts to a smooth AdS_3 x R^3, and interpret the gap as a region where the quasiparticle momentum is spacelike in six dimensions due to the momentum component in the Kaluza-Klein direction, corresponding to the final charge.12/2013;

Page 1

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

Page 2

Contents

1.Introduction and Summary1

2.Holographic Fermionic Operator Mixing

2.1Review: Free Fermions

2.2Coupled 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

4.1 Equation of Motion I: Reduction to AdS

4.2 The Dual Operators

4.3 Equation of Motion II: Gauge Couplings

4.3.1Normal Phase

4.3.2Superfluid 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

– 1 –

Page 3

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].

– 2 –

Page 4

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].

– 3 –

Page 5

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).

– 4 –

Page 6

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

– 5 –

Page 7

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.

– 6 –

Page 8

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.

– 7 –

Page 9

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.

– 8 –

Page 10

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.

– 9 –

Page 11

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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