A Spinor calculation

# Emergent quantum criticality, Fermi surfaces, and AdS2

## Abstract

Gravity solutions dual to -dimensional field theories at finite charge density have a near-horizon region which is . The scale invariance of the region implies that at low energies the dual field theory exhibits emergent quantum critical behavior controlled by a -dimensional CFT. This interpretation sheds light on recently-discovered holographic descriptions of Fermi surfaces, allowing an analytic understanding of their low-energy excitations. For example, the scaling behavior near the Fermi surfaces is determined by conformal dimensions in the emergent IR CFT. In particular, when the operator is marginal in the IR CFT, the corresponding spectral function is precisely of the “Marginal Fermi Liquid” form, postulated to describe the optimally doped cuprates.

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March 7, 2018

## I Introduction

The AdS/CFT correspondence (1) has opened new avenues for studying strongly-coupled many-body phenomena by relating certain interacting quantum field theories to classical gravity (or string) systems. Compared to conventional methods of dealing with many-body problems, this approach has some remarkable features which make it particularly valuable:

1. Putting the boundary theory at finite temperature and finite density corresponds to putting a black hole in the bulk geometry. Questions about complicated many-body phenomena at strong coupling are now mapped to single- or few-body classical problems in a black hole background.

2. Highly dynamical, strong-coupling phenomena in the dual field theories can often be understood on the gravity side using simple geometric pictures.

3. At small curvature and low energies a gravity theory reduces to a universal sector: classical Einstein gravity plus matter fields. Through the duality, this limit typically translates into the strong-coupling and large- limit of the boundary theory, where characterizes the number of species. Thus by working with Einstein gravity (plus various matter fields) one can extract certain universal properties of a large number of strongly coupled quantum field theories.

In this paper following (2); (3) we continue the study of non-Fermi liquids using the AdS/CFT correspondence (see also (4)).

Consider a -dimensional conformal field theory (CFT) with a global symmetry that has an AdS gravity dual. Examples of such theories include the super-Yang-Mills (SYM) theory in , the M2 brane theory in , the multiple M5-brane theory in , and many others with less supersymmetry. With the help of the AdS/CFT correspondence, many important insights have been obtained into strongly coupled dynamics in these systems, both near the vacuum and at a finite temperature. In particular, as a relative of QCD, thermal SYM theory has been used as a valuable guide for understanding the strongly coupled Quark-Gluon Plasma of QCD.

It is also natural to ask what happens to the resulting many-body system when we put such a theory at a finite charge density (and zero temperature). Immediate questions include: What kind of quantum liquid is it? Does the system have a Fermi surface? If yes, is it a Landau Fermi liquid? A precise understanding of the ground states of these finite density systems at strong coupling should help expand the horizon of our knowledge of quantum liquids, and may find applications to real condensed matter systems.

On the gravity side, such a finite density system is described by an extremal charged black hole in -dimensional anti-de Sitter spacetime (AdS(5). The metric of the extremal black hole has two interesting features which give some clues regarding the nature of the system. The first is that the black hole has a finite horizon area at zero temperature, suggesting a large ground state degeneracy (or approximate degeneracy) in the large limit. The second is that the near horizon geometry is given by AdS, which appears to indicate that at low frequencies the boundary system should develop an enhanced symmetry group including scaling invariance. In particular, it is natural to expect that quantum gravity (or string theory) in this region may be described by a boundary CFT. It has been argued in (6) that the asymptotic symmetry group of the near horizon AdS region is generated by a single copy of Virasoro algebra with a nontrivial central charge, suggesting a possible description in terms of some chiral 2d CFT .

More clues to the system were found in (3) from studying spectral functions of a family of spinor operators (following earlier work of (2)):

1. The system possesses sharp quasi-particle-like fermionic excitations at low energies near some discrete shells in momentum space, which strongly suggests the presence of Fermi surfaces.2 In particular, the excitations exhibit scaling behavior as a Fermi surface is approached with scaling exponents different from that of a Landau Fermi-liquid3. The scaling behavior is consistent with the general behavior discussed by Senthil in (7) for a critical Fermi surface at the critical point of a continuous metal-insulator transition.

2. For a finite range of momenta, the spectral function becomes periodic in in the low frequency limit. Such log-periodic behavior gives rise to a discrete scaling symmetry which is typical of a complex scaling exponent.

Note that the above scaling behavior is emergent, a consequence of collective dynamics of many particles, not related to the conformal invariance of the UV theory which is broken by finite density.

The results of (3) were obtained by solving numerically the Dirac equation for bulk spinor fields dual to boundary operators and it was not possible to identify the specific geometric feature of the black hole which is responsible for the emergence of the scaling behavior. Nevertheless, as speculated in (3), it is natural to suspect that the AdS region of the black hole may be responsible.

In this paper, we show that at low frequencies4, retarded Green functions5 of generic operators in the boundary theory exhibit quantum critical behavior. This critical behavior is determined by the AdS region of the black hole; assuming it exists, a CFT dual to this region of the geometry (which we will call the ‘IR CFT’) can be said to govern the critical behavior.

The spirit of the discussion of this paper will be similar to that of (3); we will not restrict to any specific theory. Since Einstein gravity coupled to matter fields captures universal features of a large class of field theories with a gravity dual, we will simply work with this universal sector, essentially scanning many possible CFTs.6

The role played by the IR CFT in determining the low-frequency form of the Green’s functions of the -dimensional theory requires some explanation. Each operator in the UV theory gives rise to a tower of operators in the IR CFT labeled by spatial momentum . The small expansion of the retarded Green function for contains an analytic part which is governed by the UV physics and a non-analytic part which is proportional to the retarded Green function of in the IR CFT. What kind of low-energy behavior occurs depends on the dimension of the operator in the IR CFT and the behavior of .7 For example, when is complex one finds the log-periodic behavior described earlier. When has a pole at some finite momentum  (with real), one then finds gapless excitations around indicative of a Fermi surface.

Our discussion is general and should be applicable to operators of any spin. In particular both types of scaling behavior mentioned earlier for spinors also applies to scalars. But due to Bose statistics of the operator in the boundary theory, this behavior is associated with instabilities of the ground state. In contrast, there is no instability for spinors even when the dimension is complex.

Our results give a nice understanding of the low-energy scaling behavior around the Fermi surface. The scaling exponents are controlled by the dimension of the corresponding operator in the IR CFT. When the operator is relevant (in the IR CFT), the quasi-particle is unstable. Its width is linearly proportional to its energy and the quasi-particle residue vanishes approaching the fermi surface. When the operator is irrelevant, the quasi-particle becomes stable, scaling toward the Fermi surface with a nonzero quasi-particle residue. When the operator is marginal the spectral function then has the form for a “marginal Fermi liquid” introduced in the phenomenological study of the normal state of high cuprates (8).

It is also worth emphasizing two important features of our system. The first is that in the IR, the theory has not only an emergent scaling symmetry but an conformal symmetry (maybe even Virasoro algebra). The other is the critical behavior (including around the Fermi surfaces) only appears in the frequency, not in the spatial momentum directions.

The plan of the paper is as follows. In §2, we introduce the charged AdS black hole and its AdS near-horizon region. In §3, we determine the low energy behavior of Green’s functions in the dual field theory, using scalars as illustration. The discussion for spinors is rather parallel and presented in Appendix A. In §IVVI, we apply this result to demonstrate three forms of emergent quantum critical behavior in the dual field theory: scaling behavior of the spectral density (§IV), periodic behavior in at small momentum (§V), and finally (§VI) the Fermi surfaces found in (3). We conclude in §VII with a discussion of various results and possible future generalizations. We have included various technical appendices. In particular in Appendix D we give retarded functions of charged scalars and spinors in the AdS/CFT correspondence.

## Ii Charged black holes in AdS and emergent infrared CFT

### ii.1 Black hole geometry

Consider a -dimensional conformal field theory (CFT) with a global symmetry that has a gravity dual. At finite charge density, the system can be described by a charged black hole in -dimensional anti-de Sitter spacetime (AdS(5) with the current in the CFT mapped to a gauge field in AdS.

The action for a vector field coupled to AdS gravity can be written as

 S=12κ2∫dd+1x√−g[R+d(d−1)R2−R2g2FFMNFMN] (1)

where is an effective dimensionless gauge coupling8 and is the curvature radius of AdS. The equations of motion following from (1) are solved by the geometry of a charged black hole (9); (5)

 ds2≡gMNdxMdxN=r2R2(−fdt2+d→x2)+R2r2dr2f (2)

with

 f=1+Q2r2d−2−Mrd,At=μ(1−rd−20rd−2) . (3)

is the horizon radius determined by the largest positive root of the redshift factor

 f(r0)=0,→M=rd0+Q2rd−20 (4)

and

 μ≡gFQcdR2rd−20,cd≡√2(d−2)d−1 . (5)

The geometry (2)–(3) describes the boundary theory at a finite density with the charge, energy and entropy densities respectively given by

 ρ=2(d−2)cdQκ2Rd−1gF, (6) ϵ=d−12κ2MRd+1,s=2πκ2(r0R)d−1 . (7)

The temperature of system can be identified with the Hawking temperature of the black hole, which is

 T=dr04πR2(1−(d−2)Q2dr2d−20) (8)

and in (5) corresponds to the chemical potential. It can be readily checked from the above equations that the first law of thermodynamics is satisfied

 dϵ=Tds+μdρ . (9)

Note that has dimension of and it is convenient to parameterize it as

 Q≡√dd−2rd−1∗ . (10)

by introducing a length scale . In order for the metric (2) not to have a naked singularity one needs

 M≥2(d−1)d−2rd∗→r0≥r∗ . (11)

In terms of , the expressions for charge density , chemical potential , and temperature can be simplified as

 ρ=1κ2(r∗R)d−11ed, (12) μ=d(d−1)d−2r∗R2(r∗r0)d−2ed, (13) T=dr04πR2(1−r2d−2∗r2d−20) (14)

where we have introduced

 ed≡gF√2d(d−1) . (15)

Note that can be considered as fixed by the charge density of the boundary theory.

### ii.2 AdS2 and scaling limits

In this paper we will be mostly interested in the behavior of the system at zero temperature, in which limit the inequalities in (11) are saturated

 T=0→r0=r∗andM=2(d−1)d−2rd∗ . (16)

Note that the horizon area remains nonzero at zero temperature and thus this finite charge density system has a nonzero “ground state” entropy density9, which can be expressed in terms of charge density as

 s=(2πed)ρ . (17)

In the zero temperature limit (16) the redshift factor in (2) develops a double zero at the horizon

 f=d(d−1)(r−r∗)2r2∗+⋯ . (18)

As a result, very close to the horizon the metric becomes with the curvature radius of given by

 R2=1√d(d−1)R . (19)

More explicitly, considering the scaling limit

 r−r∗=λR22ζ,t=λ−1τ,λ→0 with ζ,τ finite (20)

we find that the metric (2) becomes :

 ds2=R22ζ2(−dτ2+dζ2)+r2∗R2d→x2 (21)

with

 Aτ=edζ . (22)

The scaling limit (20) can also be generalized to finite temperature by writing in addition to (20)

 r0−r∗=λR22ζ0withζ0finite (23)

after which the metric becomes a black hole in times :

 ds2=R22ζ2⎛⎜ ⎜ ⎜⎝−(1−ζ2ζ20)dτ2+dζ21−ζ2ζ20⎞⎟ ⎟ ⎟⎠+r2∗R2d→x2 (24)

with

 Aτ=edζ(1−ζζ0) (25)

and a temperature (with respect to )

 T=12πζ0 . (26)

Note that in the scaling limit (20), finite corresponds to the long time limit of the original time coordinate. Thus in the language of the boundary theory (21) and (24) should apply to the low frequency limit

 ωμ,Tμ→0,ω∼T (27)

where is the frequency conjugate to .

### ii.3 Emergent IR CFT

One expects that gravity in the near-horizon AdS region (21) of an extremal charged AdS black hole should be described by a CFT dual. Little is known about this AdS/CFT duality10. For example, it is not clear whether the dual theory is a conformal quantum mechanics or a chiral sector of a -dimensional CFT. It has been argued in (6) that the asymptotic symmetry group of the near horizon AdS region is generated by a single copy of Virasoro algebra with a nontrivial central charge, suggesting a possible description in terms of some chiral 2d CFT11. Some of the problems associated with , such as the fragmentation instability and the impossibility of adding finite-energy excitations (12) are ameliorated by the infinite volume of the factor in the geometry (21).

The scaling picture of the last subsection suggests that in the low frequency limit, the -dimensional boundary theory at finite charge density is described by this CFT, to which we will refer below as the IR CFT of the boundary theory. It is important to emphasize that the conformal symmetry of this IR CFT is not related to the microscopic conformal invariance of the higher dimensional theory (the UV theory) which is broken by finite charge density. It apparently emerges as a consequence of collective behavior of a large number of degrees of freedom.

In section III we will elucidate the role of this IR CFT by examining the low frequency limit of two-point functions of the full theory. Our discussion will not depend on the specific nature of the IR CFT, but only on its existence. In Appendix D we give correlation functions for a charged scalar and spinor in the IR CFT as calculated from the standard AdS/CFT procedure in AdS (19). They will play an important role in our discussion of section III.

## Iii Low frequency limit of retarded functions

In this section we elucidate the role of the IR CFT by examining the low frequency limit of correlation functions in the full theory. We will consider two-point retarded functions for simplicity leaving the generalization to multiple-point functions for future work. We will mostly focus on zero temperature.

Our discussion below should apply to generic fields in AdS including scalars, spinors and tensors. We will use a charged scalar for illustration. The results for spinors will be mentioned at the end with calculation details given in Appendix A. Vector fields and stress tensor will be considered elsewhere.

Consider a scalar field in AdS of charge and mass , which is dual to an operator in the boundary CFT of charge and dimension

 Δ=d2+√m2R2+d24 . (28)

In the black hole geometry (2), the quadratic action for can be written as

 S=−∫dd+1x√−g[(DMϕ)∗DMϕ+m2ϕ∗ϕ] (29)

with

 DMϕ=(∂M−iqAM)ϕ . (30)

Note that the action (29) depends on only through

 μq≡μq (31)

which is the effective chemical potential for a field of charge . Writing12

 ϕ(r,xμ)=∫ddk(2π)dϕ(r,kμ)eikμxμ,kμ=(−ω,→k) (32)

the equation of motion for is given by (below )

 −1√−g∂r(√−ggrr∂rϕ)+(gii(k2−u2)+m2)ϕ=0 (33)

where various metric components are given in (2) and

 u(r)≡√gii−gtt(ω+μq(1−rd−20rd−2)) . (34)

In (3) we have chosen the gauge so that the scalar potential is zero at the horizon. As a result for and . This implies that should correspond to the difference of the boundary theory frequency from . Thus the low frequency limit really means very close to the effective chemical potential .

The retarded Green function for in the boundary theory can be obtained by finding a solution which satisfies the in-falling boundary condition at the horizon, expanding it near the boundary as

 ϕ(r,kμ)r→∞≈A(kμ)rΔ−d+B(kμ)r−Δ, (35)

and then (10)

 GR(kμ)=KB(kμ)A(kμ) (36)

where is a positive constant which depends on the overall normalization of the action, and is independent of .

### iii.1 Low frequency limit

At , expanding (36) in small is not straightforward, as the limit of equation (33) is singular. This is because has a double pole at the horizon. As a result, the -dependent terms in equation (33) always dominates sufficiently close to the horizon and thus cannot be treated as small perturbations no matter how small is. To deal with this we divide the axis into two regions

 Inner:r−r∗=ωR22ζforϵ<ζ<∞ (37) Outer:ωR22ϵ

and consider the limit

 ω→0,ζ=finite,ϵ→0,ωR22ϵ→0 . (39)

Using as the variable for the inner region and as that for the outer region, small perturbations in each region can now be treated straightforwardly, with

 inner:ϕI(ζ)=ϕ(0)I(ζ)+ωϕ(1)I(ζ)+⋯ (40) outer:ϕO(r)=ϕ(0)O(r)+ωϕ(1)O(r)+⋯ . (41)

We obtain the full solution by matching and in the overlapping region, which is with . Note that since the definition of involves , the matching will reshuffle the perturbation series in two regions.

While the above scaling limit is defined for small , all our later manipulations and final results can be analytically continued to generic complex values of .

#### Inner region: scalar fields in AdS2

The scaling limit (37), (39) is in fact identical to that introduced in (20) (with replacing ) in which the metric reduces to that of AdS with a constant electric field. It can then be readily checked that in the inner region at leading order, equation (33) (i.e. the equation for ) reduces to equation (216) in Appendix D for a charged scalar field in AdS with an effective AdS mass

 m2k=k2R2r2∗+m2,k2=|→k2| . (42)

A single scalar field in AdS gives rise a tower of fields in AdS labeled by the spatial momentum . From the discussion of Appendix D, the conformal dimension for the operator in the IR CFT dual to is given by

 δk=12+νk,νk≡√m2kR22−q2e2d+14 . (43)

Note that momentum conservation in implies that operators corresponding to different momenta do not mix, i.e.

 ⟨O†→k(t)O→k′(0)⟩∝δ(→k−→k′)t−2δk . (44)

To compute the retarded function (36) for the full theory, we impose the boundary condition that should be in-falling at the horizon. Near the boundary of the inner region (AdS region), i.e. , can then be expanded as (see (218))13

 ϕ(0)I(ω,→k;ζ) = (R22r−r∗)12−νk(1+O(ζ)) (45) + Gk(ω)(R22r−r∗)12+νk(1+O(ζ)) . (46)

The coefficient of the second term in (153) is precisely the retarded Green function for operator in the IR CFT. From (223) it can be written as

 Gk(ω)=2νke−iπνkΓ(−2νk)Γ(12+νk−iqed)Γ(2νk)Γ(12−νk−iqed)(2ω)2νk (47)

with given by (43). Equation (46) will be matched to the outer solution next.

#### Outer region and matching

The leading order equation in the outer region is obtained by setting in (33). Examining the resulting equation near , one finds that it is identical to the inner region equation for in the limit . It is thus convenient to choose the two linearly-independent solutions in the outer region using the two linearly independent terms in (46), i.e.  are specified by the boundary condition

 η(0)±(r)≈(r−r∗R22)−12±νk+⋯,r−r∗→0 . (48)

The matching to the inner region solution (46) then becomes trivial and the leading outer region solution can be written as

 ϕ(0)O=η(0)+(r)+Gk(ω)η(0)−(r) (49)

with given by (47).

One can easily generalize (49) to higher orders in . The two linearly independent solutions to the full outer region equation can be expanded as

 η±=η(0)±+ωη(1)±+ω2η(2)±+⋯ (50)

where higher order terms can be obtained using the standard perturbation theory and are uniquely specified by requiring that when expanded near , they do not contain any terms proportional to the zeroth order solutions. Each of the higher order terms satisfies an inhomogenous linear equation. The requirement amounts to choosing a specific special solution of the homogeneous equation. Note that it is important that the equations are linear. Given that higher order terms in (50) are uniquely determined by , to match the full solution to the inner region it is enough to match the leading order term which we have already done. We thus conclude that perturbatively

 ϕO=η++Gk(ω)η− . (51)

#### Small ω expansion of Gr

We first look at the retarded function at . At the inner region does not exist and the outer region equation reduces to that satisfied by . In (49) we have chosen the normalization so that at , . For real , this follows from the fact that gives the regular solution at . When is pure imaginary (i.e. when is sufficiently large) we will define the branch of the square root by taking so that with positive. Then is the in-falling solution at the horizon as is required by the prescription for calculating retarded functions. Expanding near as

 η(0)±(r,k)=a(0)±(k)rΔ−d(1+⋯)+b(0)±(k)r−Δ(1+⋯) (52)

then from (36) we find that

 GR(ω=0,k)=Kb(0)+a(0)+ . (53)

Now consider a small nonzero . Expanding various functions in (50) () near as

 η(n)±(r,k)=a(n)±(k)rΔ−d(1+⋯)+b(n)±(k)r−Δ(1+⋯), (54)

from (51) and (36) we find that for small

 GR(ω,k)=Kb(0)++ωb(1)++O(ω2)+Gk(ω)(b(0)−+ωb(1)−+O(ω2))a(0)++ωa(1)++O(ω2)+Gk(ω)(a(0)−+ωa(1)−+O(ω2)) . (55)

Equation (55) is our central technical result. In next few sections we explore its implications for the low energy behavior of the finite density boundary system. While its expression is somewhat formal, depending on various unknown functions which can only be obtained by solving the full outer region equations order by order (numerically), we will see that a great deal about the low energy behavior of the system can be extracted from it without knowing those functions explicitly.

### iii.2 Generalization to fermions

Our discussion above only hinges on the fact that in the low frequency limit the inner region wave equation becomes that in AdS. It applies also to spinors and other tensor fields even though the equations involved are more complicated. In Appendix A we discuss equations and matching for a spinor in detail. After diagonalizing the spinor equations one finds that eigenvalues of the retarded spinor Green function (which is now a matrix) are also given exactly by equation (55) with now given by equation (155), which we copy here for convenience

 Gk(ω) = e−iπνkΓ(−2νk)Γ(1+νk−iqed)Γ(2νk)Γ(1−νk−iqed) (56) × (m+ikRr∗)R2−iqed−νk(m+ikRr∗)R2−iqed+νk(2ω)2νk (57)

with

 νk=√m2kR22−q2e2d,m2k=k2R2r2∗+m2 . (58)

Note the above scaling exponent can also be expressed as (using (13))

 νk=gFq√2d(d−1) ⎷2m2R2g2Fq2+d(d−1)(d−2)2k2μ2q−1 . (59)

The conformal dimension of the operator in the IR CFT is again given by

 δk=12+νk . (60)

### iii.3 Analytic properties of Gk

The analytic properties of will play an important role in our discussion of the next few sections. We collect some of them here for future reference. Readers should feel free to skip this subsection for now and refer back to it later.

We first introduce some notations, writing

 Gk(ω)≡c(k)ω2νk,c(k)≡|c(k)|eiγk (61)

where denotes the prefactor in (56) for spinor and that in (47) for scalars.

For real , the ratios in (241) and (240) of Appendix D become a pure phase and we find that14

 γk=⎧⎪⎨⎪⎩arg(Γ(−2νk)(e−2πiνk−e−2πqed))spinorarg(Γ(−2νk)(e−2πiνk+e−2πqed))scalar (62)

It can be readily checked by drawing and on the complex plane that the following are true:

• For both scalars and spinors, (and thus ) always lies in the upper-half complex plane.

• For scalars always lies in the lower-half complex plane, while for spinors always lies in the upper-half complex plane.

• For ,

 spinor:π−γk>2πνkscalar:π−γk<2πνk . (63)

For pure imaginary (), the ratios in (241) and (240) of Appendix D become real and give

 |ck|2=e−2πλk−e−2πqede2πλk−e−2πqed

and

 |ck|2=e−2πλk+e−2πqede2πλk+e−2πqed>e−4πλkscalar . (65)

It is also manifest from the above expressions that for both scalars and spinors.

Also note that for generic , and accordingly in (55) have a logarithmic branch point at . We will define the physical sheet to be , i.e. we place the branch cut along the negative imaginary axis. This choice is not arbitrary. As discussed in Appendix D, when going to finite temperature, the branch cut resolves into a line of poles along the negative imaginary axis.

### iii.4 Renormalization group interpretation of the matching

The matching procedure described above has a natural interpretation in terms of the renormalization group flow of the boundary theory. The outer region can be interpreted as corresponding to UV physics while the inner AdS region describes the IR fixed point. The matching between in the inner and outer regions can be interpreted as matching of the IR and UV physics at an intermediate scale. More explicitly, coefficients from solving the equations in the outer region thus encode the UV physics, but is controlled by the IR CFT.

In this context can be considered as a control parameter away from the IR fixed point. Equation (55) then shows a competition between analytic power corrections (in ) away from the fixed point and contribution from operator . In particular when (i.e. ), becomes irrelevant in the IR CFT and its contribution becomes subleading compared to analytic corrections. Nevertheless the leading non-analytic contribution is still given by and as we will see below in various circumstances does control the leading behavior of the spectral function and other important physical quantities like the width of a quasi-particle.

It is interesting to note the similarity of our matching discussion to those used in various black hole/brane emission and absorption calculations (see e.g. (13)) which were important precursors to the discovery of AdS/CFT. The important difference here is that in this asymptotically-AdS case we can interpret the whole process (including the outer region) in terms of the dual field theory.

## Iv Emergent quantum critical behavior I: Scaling of spectral functions

In this and the next two sections we explore the implications of equation (55) for the low energy behavior of the finite density boundary system. In this section we look at the behavior of (55) at a generic momentum for which is real and is nonzero. Imaginary will be discussed in section V and what happens when will be discussed in section VI.

When is real, the boundary condition (48) is real. Since the differential equation satisfied by is also real, one concludes that both and are real, which implies that

 ImGR(ω=0,k)=0,forrealνk . (66)

Similarly we can conclude that all coefficients in (54) are also real. Thus the only complex quantity in (55) is the Green function of the IR CFT, . When is nonzero we can expand the denominator of (55) and the spectral function for can be written at small as

 ImGR(ω,k)=GR(k,ω=0)d0ImGk(ω)+⋯∝ω2νk . (67)

with

 d0=b(0)−b(0)+−a(0)−a(0)+ . (68)

We thus see that the spectral function of the full theory has a nontrivial scaling behavior at low frequency with the scaling exponent given by the conformal dimension of operator in the IR CFT. Note that the -dependent prefactor in (67) depends on and thus the metric of the outer region. This is consistent with the RG picture we described at the end of last section; the scaling exponent of the spectral function is universal, while the amplitude does depend on UV physics and is non-universal. By “universal” here, we mean the following. We can imaging modifying the metric in the outer region without affecting the near horizon AdS region. Then will change, but the exponent will remain the same. The real part of is dominated by a term linear in when and is non-universal, but the leading nonanalytic term is again controlled by .

## V Emergent quantum critical behavior II: Log-periodicity

In this section we examine the implication of (55) when the becomes pure imaginary. We recover the log-oscillatory behavior for spinors first found numerically in (3).

### v.1 Log-periodic behavior: complex conformal dimensions

When the charge of the field is sufficiently large (or too small)

 scalar:m2R22+14

there exists a range of momenta

 k2

for which is pure imaginary

 νk=−iλk,λk=⎧⎪⎨⎪⎩√q2e2d−m2kR22−14scalar√q2e2d−m2kR22spinor . (71)

We have chosen the branch of the square root of by taking . The effective dimension of the operator in the IR CFT is thus complex. Following (3) we will call this region of momentum space (70) the oscillatory region.15 For spinors we always have and the existence of the oscillatory region requires . For scalars, equation (69) can be satisfied for for in the range

 −d24

where the lower limit comes from the BF bound in AdS and the upper limit is the BF bound for the near horizon AdS region.

For a charged field an imaginary reflects the fact that in the constant electric field (22) of the AdS region, particles with sufficiently large charge can be pair produced. It can be checked that equations (69) indeed coincide with the threshold for pair production in AdS (14).

With an imaginary , the boundary condition (48) for is now complex. As a result is complex and

 ImGR(ω=0,k)≠0 . (73)

Thus there are gapless excitations (since ) for a range of momenta . This should be contrasted with discussion around (66).

The leading small behavior (55) is now given by

 GR(ω,k)≈b(0)++b(0)−c(k)ω−2iλka(0)++a(0)−c(k)ω−2iλk+O(ω) (74)

where was introduced in (61). Note that here

 b(0)−=(b(0)+)∗,a(0)−=(a(0)+)∗ (75)

since at the horizon and that the differential equation the satisfy is real. Equation (74) is periodic in with a period given by

 τk=πλk . (76)

In other words (74) is invariant under a discrete scale transformation

 ω→enτkω,n∈Z,ω→0 . (77)

We again stress that while the retarded function (and the spectral function) depends on UV physics (i.e. solutions of the outer region), the leading nonanalytic behavior in and in particular the period (76) only depends on the (complex) dimension of the operator in the IR CFT.

Here we find that the existence of log-periodic behavior at small frequency is strongly correlated with (73), i.e. existence of gapless excitations. It would be desirable to have a better understanding of this phenomenon from the boundary theory side.

### v.2 (In)stabilities and statistics

It is natural to wonder whether the complex exponent (71) implies some instability. We will show now that it does for scalars but not for spinors. The scalar instability arises because the scalar becomes tachyonic in the AdS region due to the electric field or reduced curvature radius. At zero momentum this is precisely the superconducting instability discussed before in (15); (16); (17); (18).16 That the log-oscillatory behavior does not imply an instability for spinors was observed before in (3) by numerically showing there are no singularities in the upper half -plane. Below we will give a unified treatment of both scalars and spinors, showing that the difference between them can be solely attributed to statistics even though we have been studying classical equations.

The spectral function following from (74) can be written as

 ImGR(ω,k)ImGR(ω=0,k)ω>0=1−|c(k)|2|1+|c(k)|eiX|2ω<0=1−|c(k)|2e4πλk|1+|c(k)|e2πλkeiX|2 (78)

where we have introduced

 X≡γk−2α−2λklog|ω|,a(0)+=|a(0)+|eiα (79)

and was defined in (61). In the boundary theory retarded Green functions for bosons are defined by commutators, while those for fermions by anti-commutators, which implies that for

 scalars:ImGR(−ω,k)<0,% ImGR(ω,k)>0 (80) spinors: