Contents
###### Abstract

We study generic types of holographic matter residing in Lifshitz invariant defect field theory as modeled by adding probe D-branes in the bulk black hole spacetime characterized by dynamical exponent and with hyperscaling violation exponent . Our main focus will be on the collective excitations of the dense matter in the presence of an external magnetic field. Constraining the defect field theory to 2+1 dimensions, we will also allow the gauge fields become dynamical and study the properties of a strongly coupled anyonic fluid. We will deduce the universal properties of holographic matter and show that the Einstein relation always holds.

HIP-2016-17/TH

Non-relativistic anyons from holography

Niko Jokela,niko.jokela@helsinki.fi Jarkko Järvelä,jarkko.jarvela@helsinki.fi and Alfonso V. Ramalloalfonso@fpaxp1.usc.es

Department of Physics and Helsinki Institute of Physics

P.O.Box 64

FIN-00014 University of Helsinki, Finland

Departamento de Física de Partículas

Universidade de Santiago de Compostela

and

Instituto Galego de Física de Altas Enerxías (IGFAE)

E-15782 Santiago de Compostela, Spain

## 1 Introduction

Gauge/gravity duality has achieved its stature in theorists’ arsenal to attack problems notoriously difficult to fight with perturbative tools. This applicability stems from the fact that the duality relates a theory at strong coupling to another theory at weak coupling, and vice versa. This is a particularly useful property when dealing with situations that one would normally describe using gauge field theory techniques, but when such systems are subject to conditions where strong interactions are expected and thus behave drastically differently. The prototypical example is the theory of strong interactions, QCD, at finite baryon chemical potentials. Here the implementation of the gauge/gravity duality, holography, has been successfully utilized both at high [1] and at low temperatures [2], natural environments for dense quark matter in heavy ion collisions and at the cores of neutron stars, respectively.

In its best understood scenario, the gauge/gravity correspondence relates string theory living in an Anti-de Sitter (AdS) spacetime (times a compact manifold) to a conformal field theory (CFT) in one less non-compact spatial dimension. Natural extensions consist of those bulk spacetimes which are still asymptotically AdS and act as dual geometries to relativistic matter. However, in many cases the configurations one deals with in the laboratories are not relativistic. The bulk geometries then ought to be warped products of Lifshitz spaces with compact manifolds. However, only a few examples of top-down constructions have been found to possess Lifshitz scaling. Moreover, it seems very subtle to nail down the precise holographic dictionary [3].

While there is no obvious obstruction to deriving generic metrics possessing Lifshitz scaling from string theory, the progress has been excruciatingly slow due to highly technical reasons. For this reason, most of the holographic studies related to Lifshitz geometries have been bottom-up, meaning that some broader form of the gauge/gravity correspondence is assumed while the string theory embedding of the background is lacking. In this paper, we will also follow this approach and start with a background metric possessing Lifshitz scaling with dynamical exponent . We will also allow hyperscaling violation, introduced via an additional parameter in the background metric. The matter in our model is introduced by adding flavor D-branes with appropriate bulk gauge fields turned on in the worldvolume of the brane, in particular, in such a manner that the matter has finite charge density. We note that this has been under systematic study also in the past [4, 5, 6], though essentially only at zero temperature. In this paper, we will also consider thermal effects on the collective excitations of the system, putting special focus on exploring how the system enters in the hydrodynamic regime. We will also discuss charge diffusion and establish the Einstein relation for all parameter values.

An important new development that we will report is the analysis of the matter in the background of an external magnetic field. The standard prescription of introducing an external magnetic field in holography is via introducing new non-vanishing components for the gauge field living on the brane, . In generic dimensions, one needs to keep fixed, corresponding to Dirichlet boundary conditions. However, when the bulk spacetime is four-dimensional, an alternative scheme for quantizing the gauge field opens up [7, 8]. In particular, one can implement combined Dirichlet/Neumann (or Robin) boundary conditions for the gauge field [9, 10], leading to dynamical gauge fields. In such a scenario, the magnetic field is not kept fixed, but one allows for it to adjust its own expectation value. This leads us to the study of matter which is not only charged electrically, but also carries magnetic charges. These are anyons, particles of fractional statistics, which are the subject of the latter part of our work.

There has been a tremendous amount of work devoted to the study of anyons, since their inception in the late seventies [11]. Yet, they are very mysterious and the field is still in its infancy. The main reason for the difficulties arise from the property that multi-anyon states cannot be expressed as a simple product of single particle states. The anyons are linked together via braiding, which might be suggestive of strong interactions. This is precisely where the holography applies and may help in rearranging thoughts in seeking answers to puzzles raised by anyonic fluids. The anyonic fluids have been studied in several holographic works [9, 15, 12, 14, 16, 17, 13]. The most recent work [10] was able to obtain the explicit equation of state for anyons (holographically modeled using a dyonic black brane), an achievement that has been extremely challenging to reach with perturbative methods. Clearly, one should try to implement the prescription given in [10] to other setups as well, in particular to those that are presented in this paper. This is, however, beyond the scope of current work.

The collective excitations of our system are dual to the quasinormal modes of the D-brane probes, which are obtained from the fluctuations of the Dirac-Born-Infeld action. At sufficiently high temperature, the system is in a hydrodynamical diffusive regime, characterized by a diffusion constant. We will find a closed expression for this constant and we will study its dependence on the scaling exponents and , as well as on the magnetic field . At low temperature, the dominant excitation is the so-called holographic zero sound [18, 19]. We will determine analytically the dispersion relation of the zero sound in the Lifshitz geometry for non-zero magnetic field. We will show that the zero sound mode is gapped when , generalizing similar previous results in other geometries [20, 21]. We will generalize this analysis to include alternative quantization conditions in the case of -dimensional field theories on the boundary. We will find that the effect of the new boundary conditions on the zero sound is similar to the one of a magnetic field. In particular, we will show that one can adjust them to make the zero sound gapless, as was found in [15, 16, 17] for relativistic backgrounds. We will also study the diffusion constant and the conductivities of the anyonic fluid.

The organization of the rest of this paper is the following. We will begin by introducing the background geometry in section 2. We embed a probe D-brane in a generic black hole metric possessing both Lifshitz scaling with dynamical exponent and hyperscaling violating exponent . We review the basic thermodynamic properties and then focus on deriving the fluctuation equations extracted from perturbing the flavor brane embedding function and the gauge fields. Section 3 solves the fluctuation equations in various limits of the background charge density, magnetic field, and temperature. We also compare the analytic results that we will obtain to those from numerics. In section 4, we switch gears by constraining to 2+1 dimensions to allow the gauge fields become dynamical via alternative quantization. We will discuss collective excitations of the resulting anyonic fluid. One important result that we can establish is the Einstein relation in the most generic case. This result we will take literally in section 5 and predict the conductivity of the matter at finite (and large) magnetic field strength, a regime where some of the other approximation schemes fall short. Section 6 contains a brief summary of our results together with remarks on a few open problems left in future works. The paper is complemented with several appendices which contain technical steps filling in the gaps in the calculations of the bulk text.

## 2 Set-up

We begin by introducing the bulk geometry which exhibits Lifshitz scaling and hyperscaling violation. We then embed flavor probe D-brane in this background, thus introducing massless quenched fundamental degrees of freedom localized on a lower-dimensional defect field theory with flavor symmetry . We analyze the thermodynamics of such holographic matter at non-zero baryonic charge density by introducing a chemical potential for the diagonal .

### 2.1 Background

Let us consider the following -dimensional metric,

 ds2p+2 = gtt(r)fp(r)dt2+gxx(dxi)2+grrfp(r)dr2 (2.1) = r−2θp[−fp(r)r2zdt2+r2(dxi)2+dr2fp(r)r2] , i=1,…,p ,

where the blackening factor reads

 fp=1−(rhr)pξ+z , (2.2)

and where the metric components we record separately for ease of reference

 gtt(r)=−r2ξ+2(z−1),gxx(r)=r2ξ,grr(r)=r2ξ−4 . (2.3)

In (2.2) we have defined a parameter , which is related to the hyperscaling violating parameter as follows:

 ξ=1−θp , (2.4)

and is the dynamical exponent. We note that the radial coordinate is defined as is standard, i.e., corresponds to the boundary, where the field theory lives and is the horizon radius of the black hole. By demanding the absence of conical singularity in the bulk, the horizon radius can be related to the field theory temperature according to

 rh=(4πTp+z−θ)1z . (2.5)

Here we are assuming that

 z≥1,θ≤0 . (2.6)

Realizations of seem pathological as they lead to violations of the null energy condition [22], whereas the latter requirement comes from thermodynamic stability (see below).

We now wish to embed probe D-branes in this background, so that the branes are extended in spatial dimensions of the Lifshitz spacetime (2.1). In the generic case then the flavor fields reside on a ()-dimensional defect. We will consider the following ansatz for the gauge field on the probes:

 F=A′tdr∧dt+Bdx1∧dx2 , (2.7)

where the prime denotes a derivative with respect to . The Dirac-Born-Infeld (DBI) action for massless probes thus reads

 S=−NfTDV∫dtdrdqx√−det(g+F)=−N∫dr√H√|gtt|grr−A′2t , (2.8)

where , is the tension, is the volume of the internal space which the D-branes may be wrapping, and the function is

 H=g2xx+gq−2xxB2=r2qξ+r2(q−2)ξB2 . (2.9)

Notice that we have not included a dilaton which is generically non-trivial in top-down string theory constructions dual to non-conformal field theories.

The equation of motion for , which follows from (2.8), can be integrated once to find

 A′t=d√grr|gtt|√H+d2 , (2.10)

where is an integration constant, proportional to the physical charge density of the field theory: . From now on, we will consider to be positive.

### 2.2 Thermodynamics

Let us now proceed with discussing some properties of the probe brane system. We are, in particular, interested in thermodynamic relations and how the parameters , , and affect them.

##### Zero temperature

Let us first consider the system at with vanishing magnetic field . From (2.10) we have that:

 A′t=dr2ξ+z−3√r2qξ+d2. (2.11)

Therefore, the (zero-temperature) chemical potential is:

 μ0=At(∞)=∫∞0drA′t=d∫∞0r2ξ+z−3√r2qξ+d2dr≡dI2ξ+z−3,2qξ(r=0) . (2.12)

The integrals of the kind (2.12) appear frequently in this paper, for sake of which we have defined two classes of integrals and collected their useful properties in Appendix A. The explicit form of the chemical potential follows

 μ0=γd2ξ+z−2ξq,γ=12ξqB(2ξ+z−22ξq,ξ(q−2)+2−z2ξq) . (2.13)

The on-shell action is:

 Son−shell=−N∫∞0dr√grr|gtt|√H+d2H=−N∫∞0r2ξq+2ξ+z−3√r2qξ+d2dr. (2.14)

This is a divergent integral. We regulate it by subtracting an on-shell action for probe branes at zero density:

 Sregon−shell=−N∫∞0drrξq+2ξ+z−3[rξq√r2qξ+d2−1]. (2.15)

To evaluate this integral we use the general result:

 ∫∞0rλ22[rλ12√rλ1+d2−1]dr=1λ1B(−λ2+22λ1,12+λ2+22λ1)dλ2+2λ1, (2.16)

which is valid for . We get:

 Sregon−shell=−N2qξB(−qξ+2ξ+z−22qξ,2qξ+2ξ+z−22qξ)d1+2ξ+z−2qξ. (2.17)

The (zero temperature) grand potential reads,

 Ω0=−2ξ+z−2qξ+2ξ+z−2Nγd1+2ξ+z−2qξ=−2ξ+z−2qξ+2ξ+z−2Nγ−ξq2ξ+z−2μ1+ξq2ξ+z−20 . (2.18)

It is now straightforward to obtain the density as:

 ρ=−∂Ω0∂μ0=Nd, (2.19)

i.e., is proportional to , as promised. The energy density can be obtained by Legendre transformation :

 ϵ=qξqξ+2ξ+z−2Nγd1+2ξ+z−2qξ. (2.20)

For the pressure we find:

 P=−Ω0=2ξ+z−2qξ+2ξ+z−2Nγd1+2ξ+z−2qξ=2ξ+z−2qξϵ. (2.21)

Therefore, the speed of first sound is:

 u2s=∂P∂ϵ=2ξ+z−2qξ. (2.22)

This result agrees with the one found in [6].

##### Non-zero temperature

To extract more useful information, we commit to heat up the system. Let us begin by analyzing the chemical potential at :

 μ=d∫∞rhr2ξ+z−3√r2qξ+d2dr=μ0−r2ξ+z−2h2ξ+z−2F(12,2ξ+z−22qξ;1+2ξ+z−22qξ;−r2qξhd2) , (2.23)

where is the chemical potential at zero temperature (2.12). The grand potential at is:

 Ω=N∫∞rhdrrξq+2ξ+z−3[rξq√r2qξ+d2−1]. (2.24)

We evaluate this integral using the formula:

 ∫∞rhrλ22[rλ12√rλ1+d2−1]dr=1λ1B(−λ2+22λ1,12+λ2+22λ1)dλ2+2λ1+22+λ2rλ2+22h −2λ1+λ2+2r1+λ1+λ22hF(12,2+λ1+λ22λ1;2+3λ1+λ22λ1;−rλ1hd2). (2.25)

We find

 ΔΩ=Ω0−N2qξ+2ξ+z−2r2qξ+2ξ+z−2hdF(12,1+2ξ+z−22qξ;2+2ξ+z−22qξ;−r2qξhd2), (2.26)

where is the grand potential at zero temperature (2.18) and is the density-dependent part of , defined as:

 ΔΩ=Ω−rqξ+2ξ+z−2hqξ+2ξ+z−2. (2.27)

Notice that the natural variable of is , as depends on it through (2.23). At low temperature we can explicitly invert (2.23). Indeed, let us consider the low temperature case in which is small. The chemical potential can then be expanded as:

 μ=μ0−r2ξ+z−2h2ξ+z−2+1212qξ+2ξ+z−2r2qξ+2ξ+z−2hd2+… . (2.28)

The expansion of is:

 ΔΩ=−2ξ+z−2qξ+2ξ+z−2Nγd1+2ξ+z−2qξ−N2qξ+2ξ+z−2r2qξ+2ξ+z−2hd+…. (2.29)

Plugging in the expression for from (2.28), we can write at leading order in temperature:

 ΔΩ=−2ξ+z−2qξ+2ξ+z−2Nγ−qξ2ξ+z−2[μ+r2ξ+z−2h2ξ+z−2]qξ2ξ+z−2+… . (2.30)

Let us then compute the entropy

 s=−∂Ω∂T∣∣μ=−∂Ω∂rh∣∣μ∂rh∂T. (2.31)

After some calculation we get (at low temperature):

 s≈Nq(p−θ)z(p(q+z)−(2+q)θ)dμ[4πp+z−θ]1−2θpzT−2θpz. (2.32)

Notice that the behavior coincides with the one found in [5] but the coefficient is different. The specific heat at low temperature thus scales as:

 cv=T∂s∂T∣∣d∼T−2θpz . (2.33)

The stability of the system (i.e., ) then requires .

### 2.3 Fluctuations

Having understood the thermodynamics of the underlying holographic fluid and the scaling upon varying and , we now wish to lay out a framework to exploring the response of the fluid under small perturbations. The relevant physics we are after are due to vector (gauge) fluctuations; the scalar deformations turn out to decouple as we focus on massless flavor degrees of freedom. We will thus consider fluctuations of the form:

 A=A(0)+a(r,xμ) , (2.34)

where and . The total gauge field strength is:

 F=F(0)+f , (2.35)

where is the two-form written in (2.7) and . We note that we consider the fluctuations to depend only on as we can always choose the momentum vector to align along one of the spatial directions.

The powerful method to fluctuating the DBI action is the approach introduced in [15]. In fact, we can just quote the corresponding results in [15] by first stating the relevant elements of the open string metric:

 Gtt = −1fpgrrgrr|gtt|−A′2t=−H+d2fp|gtt|H Grr = fp|gtt|grr|gtt|−A′2t=H+d2grrHfp Gx1x1 = Gx2x2=gxxg2xx+B2 , (2.36)

while those of the antisymmetric matrix are:

 Jtr = −Jrt=−A′tgrr|gtt|−A′2t=−d√|gtt|grr√H+d2H Jx1x2 = −Jx2x1=−Bg2xx+B2. (2.37)

The Lagrangian for the fluctuations is:

 L∼√grr|gtt|√H+d2H(GacGbd−JacJbd+12JcdJab)fcdfab ,   a,b,c,d∈{t,x,y,r} . (2.38)

The corresponding equation of motion for then follows:

 ∂c[√grr|gtt|√H+d2H(GcaGdb−JcaJdb+12JcdJab)fab]=0. (2.39)

From the equation of motion for (with ) we get the transversality condition:

 ∂ta′t−u2(r)∂xa′x=0, (2.40)

where is the function:

 u2(r)=−GxxGtt=gxx|gtt|fpg2xx+B2HH+d2=fpr2qξ+2z−2r2qξ+r2(q−2)ξB2+d2 . (2.41)

The next step is to Fourier transform the fields:

 aν(r,t,x)=∫dωdk(2π)2aν(r,ω,k)e−iωt+ikx . (2.42)

We now define the electric field as the gauge-invariant combination:

 E=kat+ωax. (2.43)

Using the transversality condition (2.40) in the momentum space, we obtain and in terms of as follows:

 a′t=−ku2ω2−k2u2E′,a′x=ωω2−k2u2E′. (2.44)

Using these relations, we obtain the equation of motion for the electric field , as follows:

 E′′+∂rlog[√|gtt|√grrgxxfpg2xx+B2√H+d2ω2−k2u2]E′+grr|gtt|f2p(ω2−k2u2)E =iBd√grr√|gtt|g2xx+B2gxxfpω2−k2u2√H+d2∂r(1g2xx+B2)ay . (2.45)

Let us write more explicitly this expression by plugging in the value of the function and the following relation:

 ω2−k2u2=(ω2−r2z−2fpk2)r2ξq+ω2B2r2ξ(q−2)+ω2d2r2ξq+r2ξ(q−2)B2+d2. (2.46)

The equation for the fluctuation of the electric field becomes:

 E′′+∂rlog[r2ξ+z+1r4ξ+B2(r2ξq+r2ξ(q−2)B2+d2)32fp(ω2−r2z−2fpk2)r2ξq+ω2B2r2ξ(q−2)+ω2d2]E′ +1r2z+2f2p(ω2−r2z−2fpk2)r2ξq+ω2B2r2ξ(q−2)+ω2d2r2ξq+r2ξ(q−2)B2+d2E =−4iξBdr2ξ−z−2(r4ξ+B2)fp(ω2−r2z−2fpk2)r2ξq+ω2B2r2ξ(q−2)+ω2d2(r2ξq+r2ξ(q−2)B2+d2)32ay. (2.47)

Moreover, the equation for can be written as:

 a′′y+∂rlog[gxx√|gtt|√grrfp√H+d2g2xx+B2]a′y+grrf2p|gtt|(ω2−k2u2)ay =−iBd√grrgxx√|gtt|1fpg2xx+B2√H+d2∂r(1g2xx+B2)E. (2.48)

Using the expressions for and the metric elements, this equation becomes:

 a′′y+∂rlog[r2ξ+z+1r4ξ+B2(r2ξq+r2ξ(q−2)B2+d2)12fp]a′y +1r2z+2f2p(ω2−r2z−2fpk2)r2ξq+ω2B2r2ξ(q−2)+ω2d2r2ξq+r2ξ(q−2)B2+d2ay =4iξBdr2ξ−z−2(r4ξ+B2)fpE(r2ξq+r2ξ(q−2)B2+d2)12. (2.49)

The rest of the paper analyzes the solutions to (2.3) and (2.3) in different regimes, at high temperature in section 3.1 and at low temperature in section 3.2. A particularly interesting setting can be obtained in the special case of as one can make use of mixed boundary conditions for the gauge fluctuations. This leads us to the study of anyons and is the topic of section 4.

### 2.4 Scalings

Let us now see how one can eliminate in the equations of motion by rescaling. First of all, we define the new rescaled radial coordinate as:

 r=rh^r. (2.50)

This rescaling eliminates from the blackening factor . Moreover, it is easy to see that the different factors in (2.3) and (2.3) transform homogeneously if , , , and are rescaled as:

 ω=rzh^ω,k=rh^k,d=rξqh^d,B=r2ξh^B . (2.51)

Notice that and are rescaled differently for , in agreement with the Lifshitz nature of the metric. For this same reason the different components of the gauge field must transform differently. As the fluctuation equations are linear, we can simply assume that the electric field does not transform. As , it is clear that and should be rescaled as:

 at=r−1h^at,ax=r−zh^ax. (2.52)

Due to symmetry of the indices, should transform as . Thus:

 ay=r−zh^ay. (2.53)

It is now straightforward to verify that the two equations of motion scale homogeneously and that working with the hatted variables is equivalent to taking . Of course, instead of the temperature, one could have chosen to scale out or , too.

## 3 Collective excitations

In this section, we will analyze the collective excitations of the magnetized brane probes in the Lifshitz background. These collective excitations are dual to the quasinormal modes of the fluctuation equations of section 2.3. We consider first the system at non-zero temperature and we will look for hydrodynamic diffusive modes. By employing analytical techniques, we obtain the expression of the diffusion constant, which we compare with the result obtained from the numerical integration of the fluctuation equations. We then consider the system at zero temperature and find the dispersion relation of the zero sound mode in the collisionless regime. Again, we obtain analytic results which we then compare with the numerical values. The transition between the collisionless and hydrodynamic regime is studied numerically.

### 3.1 Diffusion constant

Let us start by analyzing the equation (2.3) for the fluctuation of the electric field near the horizon . The blackening factor behaves near as:

 fp=z+pξrh(r−rh)+…. (3.1)

The coefficients of and in (2.3) can be expanded near as:

 ∂rlog[r2ξ+z+1r4ξ+B2(r2ξq+r2ξ(q−2)B2+d2)32fp(ω2−r2z−2fpk2)r2ξq+ω2B2r2ξ(q−2)+ω2d2]=1r−rh+c1+… 1r2z+2f2p(ω2−r2z−2fpk2)r2ξq+ω2B2r2ξ(q−2)+ω2d2r2ξq+r2ξ(q−2)B2+d2=A(r−rh)2+c2r−rh+… , (3.2)

where , , and are the following constant coefficients:

 A = ω2r−2zh(ξp+z)2 c1 = r2z−3h(ξp+z)(B2r−4ξh+d2r−2ξqh+1)k2ω2 +ξrh[(q−2)B2r−4ξh+qB2r−4ξh+d2r−2ξqh+1−4B2r−4ξh+1]+z+1+(4−p)ξ2rh c2 = −r−3h(ξp+z)(B2r−4ξh+d2r−2ξqh+1)k2−r−2z−1h(−ξp+z+1)(ξp+z)2ω2. (3.3)

We want to solve (2.3) in the hydrodynamic diffusive regime in which and are small and . In this regime, we can neglect the right-hand side of (2.3) and the equations for and decouple. Near the horizon we solve (2.3) in a Frobenius series of the type:

 E=Enh(r−rh)α[1+β(r−rh)+…] , (3.4)

where is a constant. One can easily show that the exponent in (3.4) is given by:

 α=−iω(z+pξ)rzh, (3.5)

whereas the coefficient , at leading order in , is given by:

 β≈−αc1=ik2ωrz−3h1+B2r−4ξh+d2r−2ξqh . (3.6)

Notice from (3.5) that and, therefore, we can neglect the prefactor in (3.4). Thus we write:

 E≈Enh[1+β(r−rh)]. (3.7)

We now analyze (2.3) by taking the limit at low frequencies first. In this limit, we can neglect the terms without derivatives and (2.3) becomes:

 E′′+∂rlog⎛⎜⎝rξ(q−2)−z+3(1+B2r−4ξ+d2r−2ξq)3/21+B2r−4ξ⎞⎟⎠E′=0 , (3.8)

which can be readily integrated to give

 E=E(0)+cE∞∫r(1+B2ρ−4ξ)ρξ(2−q)+z−3(1+B2ρ−4ξ+d2ρ−2ξq)3/2dρ, (3.9)

where and are constants. Notice that . The integral in (3.9) does not have a closed analytic form in general but we can easily study its properties in the UV and IR limits. Near the horizon, it has the form

 E=E(0)+cEI−cE(1+B2r−4ξh)rξ(2−q)+z−3h(1+B2r−4ξh+d2r−2ξq