0912/0912.1061.md

Towards strange metallic holography

Sean A. Hartnoll^#,‡, Joseph Polchinski^‡, Eva Silverstein^‡,† and David Tong^‡,‡

^# Department of Physics, Harvard University,
Cambridge, MA 02138, USA

^‡ Kavli Institute for Theoretical Physics and Department of Physics,
University of California, Santa Barbara, CA 93106, USA

^† on leave from SLAC and Department of Physics, Stanford University,
Stanford, CA 94305, USA

^‡ Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge, CB3 0WA, UK

hartnoll@physics.harvard.edu, joep@kitp.ucsb.edu,
evas@stanford.edu, d.tong@damtp.cam.ac.uk

Abstract

We initiate a holographic model building approach to ‘strange metallic’ phenomenology. Our model couples a neutral Lifshitz-invariant quantum critical theory, dual to a bulk gravitational background, to a finite density of gapped probe charge carriers, dually described by D-branes. In the physical regime of temperature much lower than the charge density and gap, we exhibit anomalous scalings of the temperature and frequency dependent conductivity. Choosing the dynamical critical exponent $z$ appropriately we can match the non-Fermi liquid scalings, such as linear resistivity, observed in strange metal regimes. As part of our investigation we outline three distinct string theory realizations of Lifshitz geometries: from F theory, from polarised branes, and from a gravitating charged Fermi gas. We also identify general features of renormalisation group flow in Lifshitz theories, such as the appearance of relevant charge-charge interactions when $z \geq 2$ . We outline a program to extend this model building approach to other anomalous observables of interest such as the Hall conductivity.# Contents

1	Introduction	2
2	Dimensional analysis, $z$ and renormalization	4
3	Probe D-branes in IR scaling geometries	7
4	Renormalization and Lifshitz holography	14
5	Massless charge carriers	17
5.1	DC conductivity . . . . .	18
5.2	DC Hall conductivity . . . . .	21
5.3	AC conductivity . . . . .	22
6	Massive charge carriers	28
6.1	Drag calculation: conductivity in the dilute regime . . . . .	30
6.2	Finite densities . . . . .	33
6.3	DC conductivity . . . . .	37
6.4	AC conductivity . . . . .	38
6.4.1	Numerical results . . . . .	38
6.4.2	Maxwell fluctuations . . . . .	39
6.5	Model building . . . . .	43
7	Lifshitz from string theory	46
7.1	Lifshitz solutions from Landscape dual pairs . . . . .	46
7.2	Landscape of holographic Lifshitz superconductors . . . . .	49
7.3	Lifshitz from brane polarization . . . . .	51
7.3.1	Baryon-induced Lifshitz: top down considerations . . . . .	54
7.4	Backreaction in a Fermi surface model . . . . .	57
7.5	Charged black holes versus probe branes . . . . .	60
7.6	Fermi seasickness . . . . .	62

# 1 Introduction

Some of the most interesting challenges in condensed matter physics involve strongly interacting systems of fermions and other components. The difficulty is to understand ‘non-Fermi liquid’ (NFL) behavior, which is widely believed to require physics going beyond weakly interacting fermions. Of particular interest are the thermodynamic and transport properties of the ‘strange metal’ phases of heavy fermion compounds [1] and high temperature superconductors [2, 3]. A prime example of this is DC resistivity linear in temperature over several decades of temperature $T$ , with $T$ much less than the chemical potential $\mu$ of the system, e.g. [4]. Other aspects of strange metal phenomenology include possible nontrivial power-law tails in the AC conductivity ( $\sigma(\omega) \sim \omega^{-\nu}$ with $\nu \neq 1$ over a range of scales according to [5]) and anomalous behavior of the Hall conductivity, e.g. [6].

Even at the theoretical level, few (if any) calculations reproduce the observed behavior in a controlled quantum field theory. In this work, we present some basic results in this direction, exhibiting non-Fermi-liquid behaviors such as linear resistivity in a controllable – though unrealistic – class of field theories with a holographic dual description. Another, complementary, class of holographic systems with strange metallic behaviors appears in [7]. We will comment on some similarities and differences between the two classes below.¹

The holographic correspondence [8] provides powerful techniques for analyzing a class of strongly coupled quantum field theories. It is natural to explore these theories at finite charge density. At the very least this allows us to understand certain strongly correlated many-body systems much better at a theoretical level, and this investigation may ultimately lead to mechanisms for real world phenomena.² Therefore, although current holographic technology applies only to extreme limits of special quantum field theories, it is worthwhile to study the physics of strongly interacting fermions and to investigate mechanisms for strange metal behaviors in this context where reliable calculations can be made. What we learn this way may also back react on our understanding of holography

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¹In particular, we will address the backreaction of the bulk fermi sea in [7] on the black hole solution used in the analysis, and find a significant effect.

²For introductions to the holographic approach to finite density systems see [9, 10, 11, 12].and string theory.

The theories we will study involve a sector of (in general massive) charge carriers, in a state of nonzero charge density $J^t$ , interacting amongst themselves and with a larger set of neutral quantum critical degrees of freedom. The logical structure is illustrated in figure 1. The quantum critical sector has Lifshitz scale invariance with dynamical critical exponent

graph LR
    CC([Quantum critical  
z])
    CCs([Charge carriers  
Egap, Jt])
    CCs -- dissipation --> CC
    CCs -- self-interaction --> CCs
  

Figure 1: Our model will describe probe charge carriers interacting with a quantum critical Lifshitz bath. Parameters include the dynamical scaling exponent $z$ , the energy gap $E_{\text{gap}}$ and density $J^t$ of the carriers. Ultimately the charge-charge interactions are mediated by the Lifshitz sector.

$z$ . We begin with a brief summary of the scaling properties and renormalization-group (RG) structure of such quantum field theories in §2. In a dilute limit, $J^t \ll T^{2/z}$ , this structure leads to a simple formula for their resistivity in any dimension, $\rho \propto T^{2/z}/J^t$ , which is linear in temperature for $z = 2$ . This however is not the regime of physical interest – we will later recover the same formula dynamically in the opposite, physical, limit $J^t \gg T^{2/z}$ . The scaling symmetry also implies that for $z$ greater than or equal to the spatial dimensionality, a marginal or relevant interaction $\int dt d^d \vec{x} J^t J^t$ arises among charge carriers. We then turn to a holographic analysis of such systems. We use probe ‘flavor branes’ to model the sector of charge carriers, along the lines of the earlier work [13, 14, 15], but now applied to a bulk theory with Lifshitz scaling [16] (see also [17]). (In our final section, for (UV-)completeness we provide three methods for constructing Lifshitz fixed points from the top down, obtaining $z = 2$ in the simplest examples.)After analyzing the holographic manifestation of the renormalization-group structure, we compute the specific heat and the DC, Hall, and AC conductivities of our system and comment on their similarities and differences with respect to the corresponding results for strange metals.

This opens up some new directions, which we outline at various points in the present paper. For example, we can analyze the basic scales in holographic superconductors in this context, exploring the relationship between $T_c$ , the dynamical critical exponent $z$ which determines the strange metallic behaviors, and other parameters. New model-building possibilities suggest themselves as generalizations of our basic setup. In particular, having determined the results for the basic transport coefficients in our Lifshitz field theories coupled to charged flavors, we will find it useful to consider generalizations with running couplings arising from radially rolling scalars on the gravity side of the holographic duality. This suggests mechanisms for mixing and matching non-Fermi-liquid behaviors such as

$$\rho \sim T^{{\nu }_1}\text{and}\sigma (\omega )\sim {\omega }^{-{\nu }_2},(1.1)\backslash rho\; \backslash sim\; T^\{\backslash nu\_1\}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash sigma(\backslash omega)\; \backslash sim\; \backslash omega^\{-\backslash nu\_2\},\; \backslash quad\; (1.1)$$

for different nontrivial exponents $\nu_1$ and $\nu_2$ (though in our simplest setup, $\nu_1 = \nu_2$ ). Moreover, there are many possibilities for multiple flavor sectors subject to gauge and global symmetries which organize them into composites that might mock up various scenarios for fractionalization of the electron. We leave for future work the detailed construction of theories based on these mechanisms.

A key limitation of current holographic theories vis à vis the real world is that our theoretical control arises in the unrealistic limit of a large-rank gauge symmetry, for example $U(N_c)$ Yang-Mills theory at large $N_c$ . In the present case, we use an expansion in $N_f/N_c$ , where $N_f$ is the number of charged flavors, in order to control the calculations. One would ultimately hope for control of more realistic theories with mutually interacting sectors without such large disparities.

2 Dimensional analysis, $z$ and renormalization

We wish to study the thermodynamic and transport properties of charge carriers interacting with a strongly coupled and scale invariant quantum field theory. The quantumcritical theory will be neutral under the charge. We will work in a limit in which the neutral quantum critical theory has many more degrees of freedom than the charged ‘flavor’ sector. This can be measured for instance using the free energy. So long as we stay within a range of density and scales where this ‘probe flavor’ description is valid, then the charge-carrying flavors have a negligible effect on the state of the neutral sector. We will discuss regimes of validity below, as well as give a critical assessment of the phenomenological relevance of this limit.

Spatially isotropic scale invariance is characterized by the dynamical critical exponent $z$ [18]. The theory is invariant under space and time rescaling of the form

$$t\to {\lambda }^{z}t,\vec{x}\to \lambda \vec{x}\mathrm{.}(2.1)t\; \backslash rightarrow\; \backslash lambda^z\; t,\; \backslash quad\; \backslash vec\{x\}\; \backslash rightarrow\; \backslash lambda\; \backslash vec\{x\}.\; \backslash quad\; (2.1)$$

This scale invariance is often called a Lifshitz invariance. Invariance under this scaling forces physically meaningful observables to appear in specific ratios in order to be dimensionless. This is usefully implemented by assigning time and space the following dimensions of momentum

$$[t]=-z,[\vec{x}]=-1.(2.2)[t]\; =\; -z,\; \backslash quad\; [\backslash vec\{x\}]\; =\; -1.\; \backslash quad\; (2.2)$$

We can now work out the scaling dimension of various quantities of interest, which we collect here for future reference. Throughout we work with $\hbar = k_B = e = 1$ . The charge and current densities have

$$[J^t]=d,[\vec{J}]=d+z-1,(2.3)[J^t]\; =\; d,\; \backslash quad\; [\backslash vec\{J\}]\; =\; d\; +\; z\; -\; 1,\; \backslash quad\; (2.3)$$

where $d$ is the number of space dimensions. The former follows from the definition of $J^t$ as a density while the latter follows from charge conservation $\dot{J}^t + \nabla \cdot \vec{J} = 0$ . The dimensions of external scalar ( $\Phi$ ) and vector ( $\vec{A}$ ) potentials are fixed by the fact that these appear gauging derivatives. The dimensions of electric and magnetic fields then follow as

$$[\mathrm{\Phi }]=z,[\vec{A}]=1,[\vec{E}]=z+1,[\vec{B}]=2.(2.4)[\backslash Phi]\; =\; z,\; \backslash quad\; [\backslash vec\{A\}]\; =\; 1,\; \backslash quad\; [\backslash vec\{E\}]\; =\; z\; +\; 1,\; \backslash quad\; [\backslash vec\{B\}]\; =\; 2.\; \backslash quad\; (2.4)$$

The temperature and free energy both have dimensions of energy. This leads to the following dimensions for the specific heat and the magnetic susceptibility

$$[T]=z,[F]=z,[c_V]=d,[\chi ]=z-4.(2.5)[T]\; =\; z,\; \backslash quad\; [F]\; =\; z,\; \backslash quad\; [c\_V]\; =\; d,\; \backslash quad\; [\backslash chi]\; =\; z\; -\; 4.\; \backslash quad\; (2.5)$$Finally, the dimensions of conductivity follow from Ohm's law to be

$$[\sigma ]=d-2.(2.6)[\backslash sigma]\; =\; d\; -\; 2.\; \backslash quad\; (2.6)$$

In particular, the conductivity is dimensionless in $d = 2$ spatial dimensions.

This simple dimensional analysis leads to the following statement. Consider a system with an energy gap $E_{\text{gap}}$ to exciting charge carriers which is large compared to the temperature. If the conductivity in a Lifshitz system is linear in the density $J^t$ of charge carriers, then by the scaling given above we can conclude that the resistivity $\rho = 1/\sigma$ scales like

$$\rho \propto \frac{T^{2\mathrm{/}z}}{J^t}\mathrm{.}(2.7)\backslash rho\; \backslash propto\; \backslash frac\{T^\{2/z\}\}\{J^t\}.\; \backslash quad\; (2.7)$$

This result is independent of the spatial dimension $d$ . Here we are using the fact that increasing the energy gap should not lead to larger conductivity in order to exclude significant $E_{\text{gap}}$ -dependent contributions to (2.7). (Contributions to the conductivity which decrease with $E_{\text{gap}}$ are negligible in the limit of large $E_{\text{gap}}/T$ .)

When the chemical potential $\mu \ll T$ , linearity of the conductivity in the density is immediate: this regime corresponds to very low density, where the conductivity is linear in $J^t$ because it is simply the sum of the individual contributions of non-interacting charge carriers.

However, we will also find, using the approach of [13], that in an extreme limit of holographic systems with probe flavor branes the result (2.7) persists for $\mu \gg T$ , which is the regime of interest for strange metal phenomenology. Here the self interactions of the charge carriers are non-negligible. The linearity of the conductivity as a function of charge density in these more general cases may arise because in the probe limit $\mu J^t \ll F_{\text{QCT}}$ , where $F_{\text{QCT}}$ is the free energy of the quantum critical theory (QCT) into which the momentum of the charge carriers is dissipated. Roughly speaking, the interactions among the charge carriers may be a subdominant effect on the (DC) resistivity, even though these interactions are important enough to preclude a quasiparticle interpretation of the charge carriers. We will make this statement a little more precisely below, suggesting that it is related to the fact that without the neutral QCT ‘medium’ to carry away momentum, theDC conductivity would be infinite. The mobility $\sigma/J^t$ as a function of doping has been studied experimentally in e.g. [19], exhibiting weak dependence that may be consistent with (2.7).³ Note that in contrast to single-scale models such as that discussed in [20], where $J^t$ is taken to scale with temperature as $T^{d/z}$ , in our system $J^t$ is an independent scale.

In fact, independently of the holographic correspondence, we can see from the RG structure of our theory that interactions among charge carriers will necessarily be important in the case $z \geq d$ . The dimension of $J^t$ being $d$ , the operator $J^t J^t$ becomes marginal at $z = d$ , and relevant for $z > d$ . For $d = 2$ – the dimensionality of interest for many unconventional real materials such as high- $T_c$ superconductors – this transition happens at $z = 2$ , the value of $z$ for which the resistivity is linear. In general, for $z \geq d$ , this operator is important at low energies in our theory, leading to additional interactions among charge carriers. As a relevant operator for $z > d$ , its coefficient is naturally at the UV cutoff scale of the system.⁴

3 Probe D-branes in IR scaling geometries

We are primarily interested in the low temperature and low energy behaviour of the theory. Low temperatures and energies will be defined with respect to some energy scale: $T, E \ll \Lambda_{\text{UV}}$ . In particular, we will restrict attention to theories for which the neutral sector we defined above becomes quantum critical at these low energies. Here $\Lambda_{\text{UV}}$ should presumably be of order the lattice scale (i.e. electron volts), although this may be larger than the melting temperature of the solid, allowing scaling laws to persist up to the melting point, as in e.g. [4]. Quantum criticality means that there are no intrinsic scales in the low energy (IR) theory, the only scales will be external: temperature $T$ , electric and magnetic fields $E, B$ and the density of charge carriers $J^t$ . Later we will add an energy gap scale $E_{\text{gap}}$ for the charge carriers. In the systems we study, we will see that for $J^t \neq 0$ , our window of control in which the charge carriers do not back react on the geometry

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³We thank S. Kivelson for pointing this paper out to us.

⁴One could formally introduce counterterms to cancel this divergence, but this would constitute a fine tuning in our system. We discuss this fact in some detail in section 4 below.does not extend all the way into the infrared (as long as all parameters are finite), but still covers a wide range of scales in our probe approximation.

For concreteness, and with a view to ultimately connecting to interesting experimental systems, we focus on 2+1 dimensional field theories, with 3+1 dimensional bulk duals. The dual IR geometry therefore takes the following form at zero temperature [16]

$$d{s}_{\text{IR}}^2=L^2(-\frac{d{t}^2}{v^{2z}}+\frac{d{v}^2}{v^2}+\frac{d{x}^2+d{y}^2}{v^2})\mathrm{.}(3.1)ds\_\{\backslash text\{IR\}\}^2\; =\; L^2\; \backslash left(\; -\backslash frac\{dt^2\}\{v^\{2z\}\}\; +\; \backslash frac\{dv^2\}\{v^2\}\; +\; \backslash frac\{dx^2\; +\; dy^2\}\{v^2\}\; \backslash right).\; \backslash quad\; (3.1)$$

This metric realises the scaling symmetry (2.1) as an isometry, together with $v \rightarrow \lambda v$ . The radial coordinate therefore has dimensions of length and extends from the (singular) ‘horizon’ $v = \infty$ to the ‘boundary’ $v = 0$ . We will require the above metric to give the correct physics for a window of radial positions $v$ satisfying

$$v_{\text{br}}\gg v\gg ϵ\equiv \frac{1}{{\mathrm{\Lambda }}_{\text{UV}}^{1\mathrm{/}z}}\mathrm{.}(3.2)v\_\{\backslash text\{br\}\}\; \backslash gg\; v\; \backslash gg\; \backslash epsilon\; \backslash equiv\; \backslash frac\{1\}\{\backslash Lambda\_\{\backslash text\{UV\}\}^\{1/z\}\}.\; \backslash quad\; (3.2)$$

where $v_{\text{br}}$ is an infrared ‘backreaction’ radial scale at which our probe approximation breaks down; we will quantify this shortly. We will implement the UV cutoff approximately by taking the metric (3.1) to be valid up to $v = \epsilon$ and imposing boundary conditions there.

The full background will generally have nonzero matter fields supporting the metric (3.1), such as those described in [16]. Furthermore, when embedded into a consistent quantum gravity theory, such as string theory, there may be additional spatial dimensions to those shown. We will outline three classes of examples of string-theoretic constructions of infrared Lifshitz geometries later in the paper.

When placed at a finite temperature the metric can be written as

$$d{s}_{\text{IR}}^2=L^2(-\frac{f(v)d{t}^2}{v^{2z}}+\frac{d{v}^2}{f(v)v^2}+\frac{d{x}^2+d{y}^2}{v^2})\mathrm{.}(3.3)ds\_\{\backslash text\{IR\}\}^2\; =\; L^2\; \backslash left(\; -\backslash frac\{f(v)dt^2\}\{v^\{2z\}\}\; +\; \backslash frac\{dv^2\}\{f(v)v^2\}\; +\; \backslash frac\{dx^2\; +\; dy^2\}\{v^2\}\; \backslash right).\; \backslash quad\; (3.3)$$

The precise form of $f(v)$ will depend on the theory and various solutions of this form have been constructed [21, 22, 23, 24]. All we will require is the presence of a horizon, $f(v_+) = 0$ , which defines the temperature

$$T=\frac{\mathrm{\mid }{f}^{\mathrm{\prime }}(v_{+})\mathrm{\mid }}{4\pi v_{+}^{z-1}}\propto \frac{1}{v_{+}^z}\mathrm{.}(3.4)T\; =\; \backslash frac\{|f\text{'}(v\_+)|\}\{4\backslash pi\; v\_+^\{z-1\}\}\; \backslash propto\; \backslash frac\{1\}\{v\_+^z\}.\; \backslash quad\; (3.4)$$In the second relation there is an order one number which we do not know explicitly unless $f(v)$ is given. Our normalisation is such that at the boundary $f(0) = 1$ . In order to maintain control over our calculations, we may consider cases in which the infrared back reaction scale $v_{\text{br}}$ is cloaked by a black hole horizon: $v_{\text{br}} > v_+$ .

We now turn to the dynamics of a probe D-brane in the background (3.3). The background describes a quantum critical theory; we are interested in the physics of a small number of charge carriers interacting with this theory. By a ‘small number’ here we mean that the carriers do not backreact on the quantum critical system. As we will emphasize, this does not imply that the charge carriers are weakly interacting amongst themselves; in general they will have significant interactions mediated by the quantum critical sector. A probe D $q$ brane is described by the Dirac-Born-Infeld (DBI) action

$$S_q=-T_q\int d\tau d^q\sigma e^{-\varphi }\sqrt{\mathrm{\mid }\star g+2\pi {\alpha }^{\mathrm{\prime }}F\mathrm{\mid }}\mathrm{.}(3.5)S\_q\; =\; -T\_q\; \backslash int\; d\backslash tau\; d^q\; \backslash sigma\; e^\{-\backslash phi\}\; \backslash sqrt\{|\backslash star\; g\; +\; 2\backslash pi\backslash alpha\text{'}\; F|\}.\; \backslash quad\; (3.5)$$

The nonlinearity of this action in the field strength $F$ encodes the interactions between carriers. In general the D $q$ brane can also have Chern-Simons like couplings to bulk field strengths. We will ignore these for the moment. In (3.5) $\star g$ is the pullback of the metric (3.3), $F = dA$ is the field strength of a worldvolume $U(1)$ gauge field and $e^{-\phi}$ is the dilaton. In order for the background solution to respect the scale invariance, $\phi$ and $T_q$ must be constant in the IR region.

We look for an embedding given by

$$\tau =t,{\sigma }^1=x,{\sigma }^2=y,{\sigma }^3=v,\{{\sigma }^4\dots {\sigma }^q\}=\mathrm{\Sigma },(3.6)\backslash tau\; =\; t,\; \backslash quad\; \backslash sigma^1\; =\; x,\; \backslash quad\; \backslash sigma^2\; =\; y,\; \backslash quad\; \backslash sigma^3\; =\; v,\; \backslash quad\; \backslash \{\backslash sigma^4\; \backslash dots\; \backslash sigma^q\backslash \}\; =\; \backslash Sigma,\; \backslash quad\; (3.6)$$

together with the gauge potential

$$A=\mathrm{\Phi }(v)dt+Bxdy\mathrm{.}(3.7)A\; =\; \backslash Phi(v)dt\; +\; Bx\; dy.\; \backslash quad\; (3.7)$$

In (3.6), $\Sigma$ refers to a submanifold of an internal space. If the background spacetime is a direct product of (3.3) with an internal space $M$ , the simplest way to solve the equations of motion is for $\Sigma$ to be a stationary submanifold of $M$ , independent of ${t, x, y, v}$ . Many backgrounds of interest are not direct products and many probe brane embeddings of interest are not constant in the internal directions. Nonetheless, for the moment wewill take a ‘phenomenological’ approach and consider that the only effect of internal dimensions is to multiply the overall D $q$ brane action by the volume of $\Sigma$ . The effective brane in $3 + 1$ dimensions thus has tension

$${\tau }_{\text{eff.}}=T_q\text{Vol}(\mathrm{\Sigma })e^{-\varphi }\mathrm{.}(3.8)\backslash tau\_\{\backslash text\{eff.\}\}\; =\; T\_q\; \backslash text\{Vol\}(\backslash Sigma)\; e^\{-\backslash phi\}.\; \backslash quad\; (3.8)$$

The assumption in (3.6) that the D-brane does not bend into the transverse dimensions will shortly translate into the assumption that the charge carriers are gapless. While this may be relevant for materials with a Dirac-cone dispersion relation for electronic excitations (along the lines of graphene), or other situations in which there are emergent gapless charge carrying excitations, in general we will wish to consider massive charge carriers. We will consider the massless case first for simplicity, and generalise to the massive case in section 6.

It is straightforward to solve the equations of motion for $\Phi$ to obtain

$$F_{vt}={\mathrm{\Phi }}^{\mathrm{\prime }}=\frac{1}{v^{1+z}}\frac{C}{\sqrt{v^{-4}+{\left(\frac{2\pi {\alpha }^{\mathrm{\prime }}}{L^2}\right)}^2(B^2+C^2)}},(3.9)F\_\{vt\}\; =\; \backslash Phi\text{'}\; =\; \backslash frac\{1\}\{v^\{1+z\}\}\; \backslash frac\{C\}\{\backslash sqrt\{v^\{-4\}\; +\; \backslash left(\backslash frac\{2\backslash pi\backslash alpha\text{'}\}\{L^2\}\backslash right)^2\; (B^2\; +\; C^2)\}\},\; \backslash quad\; (3.9)$$

where $C$ is a constant of integration. Near the boundary $v \rightarrow 0$ the potential is expanded as

$$\mathrm{\Phi }=\mu -\frac{1}{v^{z-2}}\frac{C}{z-2}+\dots ,(3.10)\backslash Phi\; =\; \backslash mu\; -\; \backslash frac\{1\}\{v^\{z-2\}\}\; \backslash frac\{C\}\{z-2\}\; +\; \backslash dots,\; \backslash quad\; (3.10)$$

for $z \neq 2$ and

$$\mathrm{\Phi }=\mu +C\log \frac{v}{\mathrm{\Lambda }}+\dots ,(3.11)\backslash Phi\; =\; \backslash mu\; +\; C\; \backslash log\; \backslash frac\{v\}\{\backslash Lambda\}\; +\; \backslash dots,\; \backslash quad\; (3.11)$$

when $z = 2$ . In this last case $\mu$ has a scheme dependence on a scale $\Lambda$ . We think of $\mu$ as the chemical potential, although for $z \geq 2$ it is not the largest mode near the boundary. We will discuss this phenomenon in detail in the following section. The coefficient $C$ is proportional to the charge density,

$$J^t={\tau }_{\text{eff}}(2\pi {\alpha }^{\mathrm{\prime }}{)}^{2}C,(3.12)J^t\; =\; \backslash tau\_\{\backslash text\{eff\}\}\; (2\backslash pi\backslash alpha\text{'})^2\; C,\; \backslash quad\; (3.12)$$

as one reads off from the boundary term that arises upon varying the action with respect to $\delta A_t^{(0)} = \delta\mu$ .Evaluating the action on this solution gives,

$$\frac{T{S}_q}{V_2}=-{\tau }_{\text{eff}}{L}^4{\int }_{ϵ}^{v_{+}}dv\frac{1}{v^{1+z}}\frac{v^{-4}+B^2}{\sqrt{v^{-4}+B^2+C^2}}\mathrm{.}(3.13)\backslash frac\{TS\_q\}\{V\_2\}\; =\; -\backslash tau\_\{\backslash text\{eff\}\}\; L^4\; \backslash int\_\{\backslash epsilon\}^\{v\_+\}\; dv\; \backslash frac\{1\}\{v^\{1+z\}\}\; \backslash frac\{v^\{-4\}\; +\; B^2\}\{\backslash sqrt\{v^\{-4\}\; +\; B^2\; +\; C^2\}\}.\; \backslash quad\; (3.13)$$

Here $V_2$ is the spatial volume. In this and the following few expressions, we will drop the factors of $\frac{2\pi\alpha'}{L^2}$ that appear multiplying $B$ and $C$ . Expanding the integrand for small $v$ , the contribution from the UV endpoint is

$$\frac{T{S}_q}{V_2}={\tau }_{\text{eff}}{L}^4(-\frac{1}{z+2}\frac{1}{{ϵ}^{z+2}}+\frac{C^2-B^2}{2(z-2)}\frac{1}{{ϵ}^{z-2}}+\frac{B^4-3C^4-2B^2{C}^2}{8(z-6)}\frac{1}{{ϵ}^{z-6}}+\dots )(3.14)\backslash frac\{TS\_q\}\{V\_2\}\; =\; \backslash tau\_\{\backslash text\{eff\}\}\; L^4\; \backslash left(\; -\backslash frac\{1\}\{z+2\}\; \backslash frac\{1\}\{\backslash epsilon^\{z+2\}\}\; +\; \backslash frac\{C^2\; -\; B^2\}\{2(z-2)\}\; \backslash frac\{1\}\{\backslash epsilon^\{z-2\}\}\; +\; \backslash frac\{B^4\; -\; 3C^4\; -\; 2B^2C^2\}\{8(z-6)\}\; \backslash frac\{1\}\{\backslash epsilon^\{z-6\}\}\; +\; \backslash dots\; \backslash right)\; \backslash quad\; (3.14)$$

for $z \neq 2$ . For $z = 2$ we have

$$\frac{T{S}_q}{V_2}={\tau }_{\text{eff}}{L}^4(-\frac{1}{4}\frac{1}{{ϵ}^4}+\frac{B^2-C^2}{2}\log \frac{ϵ}{\mathrm{\Lambda }}+\dots )\mathrm{.}(3.15)\backslash frac\{TS\_q\}\{V\_2\}\; =\; \backslash tau\_\{\backslash text\{eff\}\}\; L^4\; \backslash left(\; -\backslash frac\{1\}\{4\}\; \backslash frac\{1\}\{\backslash epsilon^4\}\; +\; \backslash frac\{B^2\; -\; C^2\}\{2\}\; \backslash log\; \backslash frac\{\backslash epsilon\}\{\backslash Lambda\}\; +\; \backslash dots\; \backslash right).\; \backslash quad\; (3.15)$$

For all positive $z$ the leading term is divergent as $\epsilon \rightarrow 0$ . This term is independent of the temperature and all other parameters, and reflects the fact that the energy density is dominated by UV physics. For the relativistic case, $z = 1$ , this is the only divergence, but for $z \geq 2$ the second term diverges as well. The coefficient of this divergence depends on the magnetic field and charge density. Again, naturalness requires that we include this as representing a UV sensitivity of the physics. In the next section we will analyze why a divergence appears at $z = 2$ , and why additional such effects appear as $z$ is increased further.

When we vary the action to obtain the specific heat and other observable quantities, depending on the application we may wish to hold fixed either the charge density $J^t$ or the chemical potential $\mu$ . In our setup we can implement a fixed $J^t$ by adding the familiar ‘Neumannizing’ term [25]. In the following section we will discuss boundary conditions in some detail and note that for $z > 2$ fixed charge is in fact the ‘natural’ boundary condition in a renormalisation group sense. The free energy is then

$$f\equiv \frac{F}{V_2}=\frac{T{S}_q}{V_2}+\mu J^t\mathrm{.}(3.16)f\; \backslash equiv\; \backslash frac\{F\}\{V\_2\}\; =\; \backslash frac\{TS\_q\}\{V\_2\}\; +\; \backslash mu\; J^t.\; \backslash quad\; (3.16)$$

By integrating (3.9) from the horizon, where $\Phi = 0$ , to near the boundary and comparing to (3.10), we obtain

$$\mu J^t=\frac{{\tau }_{\text{eff}}{L}^4{C}^2}{(z-2)v_{+}^{z-2}}{}_2{F}_1(\frac{1}{2},\frac{2-z}{4},\frac{6-z}{4},-(B^2+C^2)v_{+}^4)\mathrm{.}(3.17)\backslash mu\; J^t\; =\; \backslash frac\{\backslash tau\_\{\backslash text\{eff\}\}\; L^4\; C^2\}\{(z-2)v\_+^\{z-2\}\}\; \{\}\_2F\_1\; \backslash left(\; \backslash frac\{1\}\{2\},\; \backslash frac\{2-z\}\{4\},\; \backslash frac\{6-z\}\{4\},\; -(B^2\; +\; C^2)v\_+^4\; \backslash right).\; \backslash quad\; (3.17)$$For the case $z = 2$ one has instead

$$\mu J^t=\frac{{\tau }_{\text{eff}}{L}^4{C}^2}{2}\log \left(\frac{1+\sqrt{1+(B^2+C^2)v_{+}^4}}{C{v}_{+}^2}\right)\mathrm{.}(3.18)\backslash mu\; J^t\; =\; \backslash frac\{\backslash tau\_\{\backslash text\{eff\}\}\; L^4\; C^2\}\{2\}\; \backslash log\; \backslash left(\; \backslash frac\{1\; +\; \backslash sqrt\{1\; +\; (B^2\; +\; C^2)v\_+^4\}\}\{C\; v\_+^2\}\; \backslash right).\; \backslash quad\; (3.18)$$

where we have partially fixed the scheme dependence by requiring that this quantity remains finite as $v_+ \rightarrow \infty$ .

With fixed charge, the divergences appearing in the free energy (3.14) are temperature independent. The following difference of free energies is then finite

$$\mathrm{\Delta }f\equiv f(T)-f(0)(3.19)\backslash Delta\; f\; \backslash equiv\; f(T)\; -\; f(0)\; \backslash quad\; (3.19)$$

$$=-{\tau }_{\text{eff}}{L}^4{\int }_{\mathrm{\infty }}^{v_{+}}dv\frac{1}{v^{1+z}}\frac{v^{-4}+B^2}{\sqrt{v^{-4}+B^2+C^2}}+(\mu (T)-\mu (0))J^t(3.20)=\; -\backslash tau\_\{\backslash text\{eff\}\}\; L^4\; \backslash int\_\{\backslash infty\}^\{v\_+\}\; dv\; \backslash frac\{1\}\{v^\{1+z\}\}\; \backslash frac\{v^\{-4\}\; +\; B^2\}\{\backslash sqrt\{v^\{-4\}\; +\; B^2\; +\; C^2\}\}\; +\; (\backslash mu(T)\; -\; \backslash mu(0))J^t\; \backslash quad\; (3.20)$$

$$={\tau }_{\text{eff}}{L}^4(\frac{1}{z}\sqrt{B^2+C^2}\frac{1}{v_{+}^z}+\frac{1}{2(z+4)\sqrt{B^2+C^2}}\frac{1}{v_{+}^{4+z}}+\dots )\text{as }{v}_{+}\to \mathrm{\infty }=\; \backslash tau\_\{\backslash text\{eff\}\}\; L^4\; \backslash left(\; \backslash frac\{1\}\{z\}\; \backslash sqrt\{B^2\; +\; C^2\}\; \backslash frac\{1\}\{v\_+^z\}\; +\; \backslash frac\{1\}\{2(z+4)\backslash sqrt\{B^2\; +\; C^2\}\}\; \backslash frac\{1\}\{v\_+^\{4+z\}\}\; +\; \backslash dots\; \backslash right)\; \backslash quad\; \backslash text\{as\; \}\; v\_+\; \backslash rightarrow\; \backslash infty$$

$$\propto {\tau }_{\text{eff}}{L}^4(\sqrt{B^2+C^2}T+\frac{1}{\sqrt{B^2+C^2}}{T}^{1+4\mathrm{/}z}+\dots )\mathrm{.}(3.21)\backslash propto\; \backslash tau\_\{\backslash text\{eff\}\}\; L^4\; \backslash left(\; \backslash sqrt\{B^2\; +\; C^2\}\; T\; +\; \backslash frac\{1\}\{\backslash sqrt\{B^2\; +\; C^2\}\}\; T^\{1+4/z\}\; +\; \backslash dots\; \backslash right).\; \backslash quad\; (3.21)$$

In the last line we have not kept track of numerical coefficients, as we do not know the precise relation between $v_+$ and the temperature $T$ . The full integral in (3.20) may be performed in terms of hypergeometric functions. However it is clear, as emphasised in [15], that the low temperature free energy only depends on the zero temperature metric at $v = v_+$ .

It is now simple from (3.21) to compute the specific heat. The specific heat divided by temperature is

$$\frac{c_V}{T}=-\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{T}^2}\mathrm{.}(3.22)\backslash frac\{c\_V\}\{T\}\; =\; -\backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; T^2\}.\; \backslash quad\; (3.22)$$

The linear term in (3.21) will drop out upon taking two derivatives, leaving the second term. Setting $B = 0$ , this gives the specific heat

$$\frac{c_V}{T}\propto -{\tau }_{\text{eff}}^2{L}^6{\alpha }^{\mathrm{\prime }}\frac{T^{4\mathrm{/}z-1}}{J^t}\mathrm{.}(3.23)\backslash frac\{c\_V\}\{T\}\; \backslash propto\; -\backslash tau\_\{\backslash text\{eff\}\}^2\; L^6\; \backslash alpha\text{'}\; \backslash frac\{T^\{4/z-1\}\}\{J^t\}.\; \backslash quad\; (3.23)$$

Here we restored the $\alpha'$ factors. This is the leading small- $T$ behavior at fixed $J^t$ . Within the probe approximation, to be made precise shortly, this scaling will always be subdominant to the thermodynamics of the Lifshitz sector.The magnetic susceptibility

$$\frac{\chi }{V_2}=-\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{B}^2},(3.24)\backslash frac\{\backslash chi\}\{V\_2\}\; =\; -\backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; B^2\},\; \backslash quad\; (3.24)$$

has a UV sensitivity through $f(0)$ if $z \geq 2$ . This gives a temperature independent term. At low temperatures from (3.19) and (3.21)

$$\frac{\chi }{V_2}\propto -{\tau }_{\text{eff.}}{L}^2{\alpha }^{\mathrm{\prime }2}(\frac{1}{L^2}{\mathrm{\Lambda }}_{UV}^{1-2\mathrm{/}z}+{\tau }_{\text{eff.}}{\alpha }^{\mathrm{\prime }}\frac{T}{J^t})\mathrm{.}(3.25)\backslash frac\{\backslash chi\}\{V\_2\}\; \backslash propto\; -\backslash tau\_\{\backslash text\{eff.\}\}\; L^2\; \backslash alpha\text{'}^2\; \backslash left(\; \backslash frac\{1\}\{L^2\}\; \backslash Lambda\_\{UV\}^\{1-2/z\}\; +\; \backslash tau\_\{\backslash text\{eff.\}\}\; \backslash alpha\text{'}\; \backslash frac\{T\}\{J^t\}\; \backslash right).\; \backslash quad\; (3.25)$$

where we have set $B$ to zero after differentiating. The dependence on the UV scale is logarithmic when $z = 2$ . In the absence of the UV divergence, the temperature independent term is proportional to $(J^t)^{z/2-1}$ . We have not been careful with the relative normalisation of the two terms in (3.25).

Let us consider the regime of control of our system (3.5). There are two issues to address. The first is the neglect of backreaction of the brane onto the metric. In our background, the effective action $S_q$ takes the form (with $2\pi\alpha' = 1$ )

$$S_q=-{\tau }_{\text{eff.}}\int dt{d}^{2}xdv\sqrt{-g}\sqrt{1+g^{tt}{g}^{vv}{F}_{vt}^2+g^{tt}{g}^{xx}{F}_{tx}^2+g^{vv}{g}^{xx}{F}_{vx}^2}\mathrm{.}(3.26)S\_q\; =\; -\backslash tau\_\{\backslash text\{eff.\}\}\; \backslash int\; dt\; d^2x\; dv\; \backslash sqrt\{-g\}\; \backslash sqrt\{1\; +\; g^\{tt\}g^\{vv\}F\_\{vt\}^2\; +\; g^\{tt\}g^\{xx\}F\_\{tx\}^2\; +\; g^\{vv\}g^\{xx\}F\_\{vx\}^2\}.\; \backslash quad\; (3.26)$$

To avoid back reaction of the probe on the metric, its stress-energy must be smaller than the stress energy generating the original background (3.1). The original energy density is of order $M_4^2|\Lambda| \sim M_4^2/L^2$ , where $M_4$ is the four-dimensional Planck mass and $\Lambda$ the four-dimensional cosmological constant. Varying (3.26) with respect to $g_{tt}$ , we find this condition to be

$$\gamma \equiv \frac{1+g^{vv}{g}^{xx}{F}_{vx}^2}{\sqrt{1+g^{tt}{g}^{vv}{F}_{vt}^2+g^{tt}{g}^{xx}{F}_{tx}^2+g^{vv}{g}^{xx}{F}_{vx}^2}}\ll \frac{M_4^2\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}{{\tau }_{\text{eff.}}}\mathrm{.}(3.27)\backslash gamma\; \backslash equiv\; \backslash frac\{1\; +\; g^\{vv\}g^\{xx\}F\_\{vx\}^2\}\{\backslash sqrt\{1\; +\; g^\{tt\}g^\{vv\}F\_\{vt\}^2\; +\; g^\{tt\}g^\{xx\}F\_\{tx\}^2\; +\; g^\{vv\}g^\{xx\}F\_\{vx\}^2\}\}\; \backslash ll\; \backslash frac\{M\_4^2|\backslash Lambda|\}\{\backslash tau\_\{\backslash text\{eff.\}\}\}.\; \backslash quad\; (3.27)$$

In our solutions $\gamma$ approaches one at the boundary and grows toward larger $v$ , the region corresponding to the infrared regime of the field theory. As long as the brane tension is sufficiently small, the right hand side allows for a window of scales in which $\gamma$ can grow larger than one (leading to nontrivial DBI dynamics, corresponding to interactions between the charge carriers) while satisfying the condition (3.27). This is the regime $v_{\text{br}} \gg v$ of equation (3.2).

In the simplest examples of brane probes, such as those discussed in [15], the probe limit requires a power law tune $N_c/N_f \gg 1$ where $N_c$ is the rank of the Yang-Mills gaugegroup of the field theory, and $N_f$ the number of charged matter fields (the number of probe branes). More generally in the landscape, however, low-tension branes can arise naturally – via an exponential hierarchy – in compactifications with strong warping (gravitational redshift) in the extra dimensions. This effectively makes the internal volume $\text{Vol}(\Sigma)$ small in the tension (3.8).

There is another way that the description can break down at large $\gamma$ . As $\gamma$ gets large the electric field on the brane is approaching its critical value, where the force on string endpoints exactly balances the string tension [26]. Beyond this point the system is unstable to creation of open strings. As the critical field is approached, the effective value of the open string tension falls as $1/\gamma^2$ [27, 28] and so the effective string length scale grows as $\gamma$ . When this exceeds the typical length scale of the geometry, supergravity will break down in an interesting way as string modes become important. The condition for supergravity to be valid is then

$${\alpha }^{\mathrm{\prime }}{\gamma }^2\ll 1\mathrm{/}\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }\mathrm{.}(3.28)\backslash alpha\text{'}\; \backslash gamma^2\; \backslash ll\; 1/|\backslash Lambda|\; .\; \backslash quad\; (3.28)$$

The conditions (3.27) and (3.28) are compatible with one another but independent. We can go to a regime where the first is satisfied but the second is not; this appears to correspond to a situation where the backreaction of the charges on the critical sector is small, but their interaction with each other due to finite density has become stronger than their 't Hooft coupling interactions (setting for instance $|\Lambda|\alpha' = \lambda^{-1/2}$ ). In general, as we approach this regime, an expansion in small perturbations about the DBI action (3.26) brings down inverse powers of the square root, i.e. powers of $\gamma$ , leading to strong non-Gaussian effects [29]. It would be interesting to understand the role of these effects in holographic condensed matter systems.

4 Renormalization and Lifshitz holography

We would like to understand in a more general way the divergences that we encountered with increasing $z$ in section 3.⁵ For simplicity we focus on the quadratic Maxwell action,

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⁵For a recent work on Lifshitz holography see [30].which governs the dominant behavior of the full action (3.26) near the boundary,

$$S\propto -\int dt{d}^{d}xdv{v}^{-d-z-1}(\frac{1}{2}{v}^{2+2z}{F}_{ti}^2-\frac{1}{2}{v}^4{F}_{iv}^2+\frac{1}{2}{v}^{2+2z}{F}_{tv}^2-\frac{1}{4}{v}^4{F}_{ij}^2)\mathrm{.}(4.1)S\; \backslash propto\; -\; \backslash int\; dt\; d^d\; x\; dv\; v^\{-d-z-1\}\; \backslash left(\; \backslash frac\{1\}\{2\}\; v^\{2+2z\}\; F\_\{ti\}^2\; -\; \backslash frac\{1\}\{2\}\; v^4\; F\_\{iv\}^2\; +\; \backslash frac\{1\}\{2\}\; v^\{2+2z\}\; F\_\{tv\}^2\; -\; \backslash frac\{1\}\{4\}\; v^4\; F\_\{ij\}^2\; \backslash right)\; .\; \backslash quad\; (4.1)$$

For the dominant boundary behavior we can ignore $i$ and $t$ derivatives in the field equations. With the pure Liftshitz background (3.1) this gives the equations of motion

$${\mathrm{\partial }}_v(v^{1+z-d}{\mathrm{\partial }}_v{A}_t)={\mathrm{\partial }}_v(v^{3-z-d}{\mathrm{\partial }}_v{A}_i)=0,(4.2)\backslash partial\_v(v^\{1+z-d\}\; \backslash partial\_v\; A\_t)\; =\; \backslash partial\_v(v^\{3-z-d\}\; \backslash partial\_v\; A\_i)\; =\; 0\; ,\; \backslash quad\; (4.2)$$

with solutions

$$A_t=\alpha +\beta v^{d-z},A_i={\alpha }^{\mathrm{\prime }}+{\beta }^{\mathrm{\prime }}{v}^{d+z-2}\mathrm{.}(4.3)A\_t\; =\; \backslash alpha\; +\; \backslash beta\; v^\{d-z\}\; ,\; \backslash quad\; A\_i\; =\; \backslash alpha\text{'}\; +\; \backslash beta\text{'}\; v^\{d+z-2\}\; .\; \backslash quad\; (4.3)$$

In this limit $A_v$ is pure gauge. The field strengths scale as

We now focus on $d = 2$ . In the relativistic case $z = 1$ , the $\alpha$ and $\alpha'$ solutions are larger at the boundary $v \rightarrow 0$ . We thus take the usual quantization, in which these are fixed while $\beta$ and $\beta'$ are dynamical. That is, we fix the potentials and field strengths tangent to the boundary. At $z = 1$ , all terms in the action are convergent at $v \rightarrow 0$ . As we increase $z$ , at $z \geq 2$ the $F_{ij}^2$ term has a divergence proportional to $\alpha'^2$ , and the $F_{tr}^2$ term has a divergence proportional to $\beta^2$ , and new boundary counterterms are needed to obtain a finite on shell action.

To understand these divergences from the point of view of the field theory, recall the momentum (inverse length) dimensions from section 2

$$[J^t]=d,[J^i]=d+z-1,[A_t]=z,[A_i]=1.(4.5)[J^t]\; =\; d\; ,\; \backslash quad\; [J^i]\; =\; d\; +\; z\; -\; 1\; ,\; \backslash quad\; [A\_t]\; =\; z\; ,\; \backslash quad\; [A\_i]\; =\; 1\; .\; \backslash quad\; (4.5)$$

The field theory contains an explicit $A_\mu J^\mu$ interaction, but will generate additional divergences for any gauge-invariant relevant interaction constructed from $A$ and $J$ . Here $A$ istreated as a nonfluctuating spurion field, while $J$ is a single trace operator, and the new counterterms will in general involve multiple traces. The field theory volume element has $[dt d^d x] = -d - z$ , so an interaction will be relevant if its momentum dimension is less than or equal to $d + z$ . The dimension of $F_{ij}^2$ is 4, so this becomes relevant at $d + z = 4$ . The dimension of $(J^t)^2$ is $2d$ , so this becomes relevant when $d = z$ . For $d = 2$ , the critical $z$ is 2 for both operators.

The divergence from $F_{ij}^2$ involves the fixed $\alpha'$ , so this is just an additive classical term. It reflects the fact that the dominant momentum-, temperature-, and frequency-independent magnetic susceptibility will come from the UV when $z > 2$ . We saw this in equation (3.25) above.

The divergence from $F_{tr}^2$ at $z > 2$ is more subtle. At the same time that $(J^t)^2$ becomes relevant, the $\alpha$ and $\beta$ solutions cross, and the latter dominates at the boundary $v \rightarrow 0$ . Thus we are in the situation discussed for relativistic scalars in Ref. [31]. For a generic UV theory we will flow to the more stable boundary condition in which $\alpha$ is dynamical and $\beta$ is fixed.⁶

It is tempting to ‘renormalize’ the low energy effective theory, adding boundary counterterms to cancel the divergences. In the range $2 < z < 4$ the following counterterms would do the job

$$S_{\text{bdy.}}=\frac{1}{g_{\text{eff.}}^2}{\int }_{ϵ}dt{d}^{2}x\sqrt{\mathrm{\mid }\star \gamma \mathrm{\mid }}[\frac{1}{z+2}+\frac{\zeta }{2}(F_{ij}{F}^{ij}-F_{tv}{F}^{tv})],(4.6)S\_\{\backslash text\{bdy.\}\}\; =\; \backslash frac\{1\}\{g\_\{\backslash text\{eff.\}\}^2\}\; \backslash int\_\{\backslash epsilon\}\; dt\; d^2\; x\; \backslash sqrt\{|\backslash star\backslash gamma|\}\; \backslash left[\; \backslash frac\{1\}\{z+2\}\; +\; \backslash frac\{\backslash zeta\}\{2\}\; (F\_\{ij\}\; F^\{ij\}\; -\; F\_\{tv\}\; F^\{tv\})\; \backslash right],\; \backslash quad\; (4.6)$$

In this expression $g_{\text{eff.}}^2 = 1/\tau_{\text{eff.}}(2\pi\alpha')^2$ , $\zeta = 1/(z-2)$ , and $\star\gamma$ is the induced metric on the $r = \epsilon$ surface. For the field-independent and $F_{ij}^2$ terms this just subtracts off the UV contribution and isolates that from the IR. For the $F_{tv} F^{tv}$ term, however, the boundary terms actually change the theory, from the IR stable $\beta = 0$ theory to the tuned $\alpha = 0$ theory. To see this, perform the variation of the bulk and boundary action with respect to $A_t$ , and insert the asymptotics (4.3) to obtain

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⁶Ref. [32] studied the 2+1 dimensional relativistic ( $z = 1$ ) gauge theory and showed that it had two IR stable realizations, the second corresponding to the gauging of the $U(1)$ symmetry.We see that precisely the value $\zeta = 1/(z - 2)$ , that cancels the divergence, also gives the tuned $\alpha$ -fixed theory. Throughout this paper we work with the untuned, $\beta$ -fixed, theory. This has no boundary counterterms. We argued in the previous section that this fixed charge ensemble is in fact the physically correct one for condensed matter applications of holography, for all values of $z$ .

The difference between the $\alpha$ - and $\beta$ -fixed theories is subtle: in the planar limit, only the correlators of $J^t$ are affected by the double trace deformation [33], so most observables are the same. Intuitively, a large $(J^t)^2$ interaction would inhibit local fluctuations of $J^t$ , explaining why $\beta$ must be fixed.

The expansions (4.3, 4.4) have higher order terms, e.g. at relative order $k^2 v^2$ , which will lead to further divergences as we increase $z$ . In the field theory, this is reflected by the operators $F_{ij,k}^2$ and $(J_{,k}^t)^2$ becoming relevant at $z = 4$ . As $z$ is further increased, higher spatial derivatives become relevant, so that in the $z \rightarrow \infty$ limit of $AdS_2 \times \mathbb{R}^2$ there is an infinite number of relevant operators. Note however that terms with additional time derivatives never become relevant. Higher powers of the fields can become relevant, e.g. $(F_{ij}F_{ij})^2$ , $F_{ij}F_{ij}(J^t)^2$ and $(J^t)^4$ at $z = 6$ .

5 Massless charge carriers

A key observable capturing strange metallic behavior is the conductivity. We will start by analyzing the DC conductivity of our system, following closely the work of Karch and O’Bannon [13], keeping the full nonlinear dependence of the current resulting from a given constant electric field. Using similar methods [34] we also compute the Hall conductivity. Then we will obtain the optical conductivity by computing the linearized response of the system to small oscillating electric field perturbations. We focus first in this section on the massless case, to illustrate the techniques. Then in the next section we will include a mass for the charge carriers, motivated by the large energy gap of carriers relative to temperature in real-world strange metals.## 5.1 DC conductivity

In order to compute the DC ( $\omega = 0$ ) conductivity, the strategy is to turn on an electric field $E \equiv F_{tx}$ on the D-brane probe and compute the resultant current $\langle J^x \rangle$ in the boundary theory. This then allows us to directly read off the field and temperature dependent conductivity $\sigma(E, T)$ from Ohm's law

$$J^x=\sigma (E,T)E\mathrm{.}(5.1)J^x\; =\; \backslash sigma(E,\; T)E.\; \backslash quad\; (5.1)$$

To this end, we revise the ansatz (3.7) for the worldvolume gauge field, now looking for solutions of the form

$$A=\mathrm{\Phi }(v)dt+(-Et+h(v))dx\mathrm{.}(5.2)A\; =\; \backslash Phi(v)dt\; +\; (-Et\; +\; h(v))dx.\; \backslash quad\; (5.2)$$

With this ansatz, the action (3.5) becomes

$$S=-{\tau }_{\text{eff}}\int dt{d}^{2}xdv\sqrt{g_{xx}}\sqrt{-g_{tt}{g}_{xx}{g}_{vv}-(2\pi {\alpha }^{\mathrm{\prime }}{)}^2(g_{vv}{E}^2+g_{xx}{\mathrm{\Phi }}^{\mathrm{\prime }2}+g_{tt}{h}^{\mathrm{\prime }2})}\mathrm{.}(5.3)S\; =\; -\backslash tau\_\{\backslash text\{eff\}\}\; \backslash int\; dt\; d^2x\; dv\; \backslash sqrt\{g\_\{xx\}\}\; \backslash sqrt\{-g\_\{tt\}g\_\{xx\}g\_\{vv\}\; -\; (2\backslash pi\backslash alpha\text{'})^2\; (g\_\{vv\}E^2\; +\; g\_\{xx\}\backslash Phi\text{'}^2\; +\; g\_\{tt\}h\text{'}^2)\}.\; \backslash quad\; (5.3)$$

The action depends only on the derivatives $\Phi'$ and $h'$ , resulting in two quantities which are independent of the radial direction $v$ ,

$$C=\frac{-g_{xx}^{3\mathrm{/}2}{\mathrm{\Phi }}^{\mathrm{\prime }}}{\sqrt{-g_{tt}{g}_{xx}{g}_{vv}-(2\pi {\alpha }^{\mathrm{\prime }}{)}^2(g_{vv}{E}^2+g_{xx}{\mathrm{\Phi }}^{\mathrm{\prime }2}+g_{tt}{h}^{\mathrm{\prime }2})}},C\; =\; \backslash frac\{-g\_\{xx\}^\{3/2\}\; \backslash Phi\text{'}\}\{\backslash sqrt\{-g\_\{tt\}g\_\{xx\}g\_\{vv\}\; -\; (2\backslash pi\backslash alpha\text{'})^2\; (g\_\{vv\}E^2\; +\; g\_\{xx\}\backslash Phi\text{'}^2\; +\; g\_\{tt\}h\text{'}^2)\}\},$$

and

$$H=\frac{-g_{tt}\sqrt{g_{xx}}{h}^{\mathrm{\prime }}}{\sqrt{g_{tt}{g}_{xx}{g}_{vv}-(2\pi {\alpha }^{\mathrm{\prime }}{)}^2(g_{vv}{E}^2+g_{xx}{\mathrm{\Phi }}^{\mathrm{\prime }2}+g_{tt}{h}^{\mathrm{\prime }2})}}\mathrm{.}(5.4)H\; =\; \backslash frac\{-g\_\{tt\}\backslash sqrt\{g\_\{xx\}\}\; h\text{'}\}\{\backslash sqrt\{g\_\{tt\}g\_\{xx\}g\_\{vv\}\; -\; (2\backslash pi\backslash alpha\text{'})^2\; (g\_\{vv\}E^2\; +\; g\_\{xx\}\backslash Phi\text{'}^2\; +\; g\_\{tt\}h\text{'}^2)\}\}.\; \backslash quad\; (5.4)$$

These are the same expressions found in [13] and, as pointed out by those authors, obey the relation $g_{tt}h'C = g_{xx}\Phi'H$ . The near-boundary profile of $\Phi$ is once again given by (3.10) for $z \neq 2$ and (3.11) for $z = 2$ . Meanwhile, the asymptotic behavior of $h(v)$ is

$$h(v)\to h_0+\frac{H}{z}{v}^z+\dots \mathrm{.}(5.5)h(v)\; \backslash rightarrow\; h\_0\; +\; \backslash frac\{H\}\{z\}\; v^z\; +\; \backslash dots.\; \backslash quad\; (5.5)$$

We set $h_0 = 0$ and identify the coefficient of the decaying term with the current

$$J^x={\tau }_{\text{eff}}(2\pi {\alpha }^{\mathrm{\prime }}{)}^{2}H\mathrm{.}(5.6)J^x\; =\; \backslash tau\_\{\backslash text\{eff\}\}(2\backslash pi\backslash alpha\text{'})^2\; H.\; \backslash quad\; (5.6)$$This equation is completely analogous to that defining the charge density in (3.12). The normalization factor of $\frac{1}{z}$ in (5.5) follows from computing $J^x$ as the derivative of the on-shell action with respect to $h_0$ . The on-shell bulk action (5.3) can be written as,

$$S={\tau }_{\text{eff}}\int dt{d}^{2}xdv{g}_{xx}^{3\mathrm{/}2}\sqrt{-g_{tt}{g}_{vv}}{\left[\frac{g_{tt}{g}_{xx}+(2\pi {\alpha }^{\mathrm{\prime }}{)}^2{E}^2}{(2\pi {\alpha }^{\mathrm{\prime }}{)}^2(g_{tt}{C}^2+g_{xx}{H}^2)+g_{xx}^2{g}_{tt}}\right]}^{1\mathrm{/}2}\mathrm{.}(5.7)S\; =\; \backslash tau\_\{\backslash text\{eff\}\}\; \backslash int\; dt\; d^2x\; dv\; \backslash \; g\_\{xx\}^\{3/2\}\; \backslash sqrt\{-g\_\{tt\}g\_\{vv\}\}\; \backslash left[\; \backslash frac\{g\_\{tt\}g\_\{xx\}\; +\; (2\backslash pi\backslash alpha\text{'})^2\; E^2\}\{(2\backslash pi\backslash alpha\text{'})^2\; (g\_\{tt\}C^2\; +\; g\_\{xx\}H^2)\; +\; g\_\{xx\}^2\; g\_\{tt\}\}\; \backslash right]^\{1/2\}.\; \backslash quad\; (5.7)$$

The divergences (UV sensitivities) of this expression are as discussed in the previous sections, and do not involve $E$ .

As pointed out in [13], both the numerator and the denominator of $[\dots]^{1/2}$ change sign between the boundary $v = 0$ and the horizon $v = v_+$ (recall that $g_{tt} < 0$ ). The reality of the action means that this sign change must take place at the same point in the radial direction, $v_+ > v_* > 0$ , such that both

$$-g_{tt}{g}_{xx}{\mid }_{v=v_{\ast }}=(2\pi {\alpha }^{\mathrm{\prime }}{)}^2{E}^2,(5.8)-g\_\{tt\}g\_\{xx\}\; \backslash Big|\_\{v=v\_*\}\; =\; (2\backslash pi\backslash alpha\text{'})^2\; E^2,\; \backslash quad\; (5.8)$$

and

$$(2\pi {\alpha }^{\mathrm{\prime }}{)}^2(g_{tt}{C}^2+g_{xx}{H}^2)=-g_{xx}^2{g}_{tt}{\mid }_{v=v_{\ast }},(5.9)(2\backslash pi\backslash alpha\text{'})^2\; (g\_\{tt\}C^2\; +\; g\_\{xx\}H^2)\; =\; -g\_\{xx\}^2\; g\_\{tt\}\; \backslash Big|\_\{v=v\_*\},\; \backslash quad\; (5.9)$$

should be satisfied.⁷ Using the finite temperature metric (3.3), the first of these equations fixes $v_*$ in terms of the electric field,

$$f(v_{\ast })={\left(\frac{2\pi {\alpha }^{\mathrm{\prime }}}{L^2}\right)}^2{E}^2{v}_{\ast }^{2z+2}\mathrm{.}(5.10)f(v\_*)\; =\; \backslash left(\; \backslash frac\{2\backslash pi\backslash alpha\text{'}\}\{L^2\}\; \backslash right)^2\; E^2\; v\_*^\{2z+2\}.\; \backslash quad\; (5.10)$$

Meanwhile, the second equation can be rewritten to give the sought-after equation for the conductivity,

$$\sigma (E,T)=\sqrt{(2\pi {\alpha }^{\mathrm{\prime }}{)}^4{\tau }_{\text{eff}}^2+{\left(\frac{2\pi {\alpha }^{\mathrm{\prime }}}{L^2}\right)}^2{v}_{\ast }^4(J^t{)}^2}\mathrm{.}(5.11)\backslash sigma(E,\; T)\; =\; \backslash sqrt\{(2\backslash pi\backslash alpha\text{'})^4\; \backslash tau\_\{\backslash text\{eff\}\}^2\; +\; \backslash left(\; \backslash frac\{2\backslash pi\backslash alpha\text{'}\}\{L^2\}\; \backslash right)^2\; v\_*^4\; (J^t)^2\}.\; \backslash quad\; (5.11)$$

The right-hand-side of (5.11) is the root-mean-square of two terms: the first is a constant piece and arises from thermally produced pairs of charge carriers. It is expected to be

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⁷It might appear that a boundary condition is being imposed at the point $v_*$ , rather than the horizon, but in fact this is equivalent to the usual condition of ingoing b.c. on the horizon. One can study this by approximating the near-horizon geometry as Rindler, for which a finite ingoing wave satisfies the DBI field equation, and then taking the zero frequency limit. Outgoing b.c. give the opposite sign for $H$ .Boltzmann suppressed when the charge carriers have large mass, as we will see shortly. The surviving term exhibits the simple power-law (2.7) for the DC resistivity, namely

$$\rho \sim \frac{T^{2\mathrm{/}z}}{J^t}\mathrm{.}(5.12)\backslash rho\; \backslash sim\; \backslash frac\{T^\{2/z\}\}\{J^t\}.\; \backslash quad\; (5.12)$$

As discussed in §2, one situation in which this behavior is generic is a regime of dilute charge carriers which are coupled to a Lifshitz bath in such a way as to inherit its scaling symmetry (2.1). The diluteness of the charge carriers implies that the conductivity is approximately linear in $J^t$ and the rest follows from dimensional analysis. However, here we see that the linearity in $J^t$ arises in the massless case only in the concentrated (i.e. non-dilute) regime $J^t \gg T^{2/z}$ , while in the very massive case it arises for all $J^t/T^{2/z}$ in the DC conductivity. We will discuss this further below.

The first term in (5.11) is independent of both temperature $T$ and electric field $E$ . This is due to the fact we are in a $3 + 1$ dimensional bulk, rather than any Lifshitz scaling. This same constant term was seen in Section 5 of [13]. The second term on the right hand side contains the dependence on the temperature and on the electric field. Both of these arise through $v_*$ defined in (5.10). To compute the nonlinear, $E$ dependent corrections to the conductivity, we need to know the specific function $f(v)$ , which will depend on the matter content sourcing the Lifshitz background. On dimensional grounds, such corrections depend on the ratio $\frac{(2\pi\alpha')^2}{L^4} \frac{E^2}{T^{2+2/z}}$ . In the relativistic case ( $z = 1$ ), the nonlinearities of the conductivity in the electrical field could be elegantly understood by considering the drag force on a single string and using Lorentz invariance [13]. This does not appear to be the case at general $z$ .

A translationally invariant medium with a net charge density should have an infinite DC electrical conductivity. Specifically, the real part should have a delta function and the imaginary part should have a pole: $\sigma(\omega) \sim \delta(\omega) + i\omega^{-1}$ at small $\omega$ . This can be seen either directly from hydrodynamics or via the holographic correspondence (e.g. [35, 36]). The underlying reason is that the conserved momentum cannot relax and, combined with a net charge density, this gives a current that does not relax. Yet we have just obtained a finite DC conductivity. The reason for this [13] is that in the probe limit the momentum can be transferred to the quantum critical ‘bath’ without any backreaction on the charged probe system. Technically, the coefficient of the divergence goes to zero in the probe limit,so that the probe and $\omega \rightarrow 0$ limits do not commute. A physical circumstance in which this probe approximation could be legitimate is if the quantum critical excitations are more efficient at dissipating heat into the environment (via interaction with impurities etc.) than the charge carriers.

The fact that the DC conductivity would diverge in the absence of the Lifshitz bath suggests a heuristic understanding of why this conductivity is linear in the charge density, if the first, constant, term in (5.11) is taken to be Boltzmann suppressed. This is not an immediate result as interactions between the charges are important as evidenced, for instance, by the nonlinear dependence of the free energy (3.13) on the charge density. The fact that the Lifshitz medium is responsible for making the potentially infinite conductivity finite suggests that medium-carrier interactions are playing a dominant role in the DC limit. The diluteness of the carriers with respect to the medium then suggests that this interaction will be proportional to the density of carriers, leading to the linear dependence of the finite DC conductivity on $J^t$ .

5.2 DC Hall conductivity

The techniques of [13] can be extended to compute the conductivity tensor,

$$J^i={\sigma }^{ij}{E}_j\mathrm{.}(5.13)J^i\; =\; \backslash sigma^\{ij\}\; E\_j.\; \backslash quad\; (5.13)$$

The calculations are identical to those of [34] and we present only the results. The conductivity is once again expressed in terms of a function $v_*(T, E, B)$ , defined by the requirement that

$$-g_{tt}{g}_{xx}^2=(2\pi {\alpha }^{\mathrm{\prime }}{)}^2(g_{tt}{B}^2+g_{xx}{E}^2){\mid }_{v=v_{\ast }},(5.14)-g\_\{tt\}g\_\{xx\}^2\; =\; (2\backslash pi\backslash alpha\text{'})^2(g\_\{tt\}B^2\; +\; g\_\{xx\}E^2)\; \backslash Big|\_\{v=v\_*\},\; \backslash quad\; (5.14)$$

which generalizes (5.8). For $E$ and $B$ small, this gives $v_* \sim v_+ \sim 1/T^{1/z}$ . Corrections to this expression are functions of the dimensionless ratios $E/T^{1+1/z}$ and $B/T^{2/z}$ . The Hall conductivity has a simple expression in terms of $v_*$ ,

$${\sigma }^{xy}=-\frac{J^{t}B{v}_{\ast }^4}{{\left(\frac{L^2}{2\pi {\alpha }^{\mathrm{\prime }}}\right)}^2+B^2{v}_{\ast }^4}\mathrm{.}(5.15)\backslash sigma^\{xy\}\; =\; -\backslash frac\{J^t\; B\; v\_*^4\}\{\backslash left(\backslash frac\{L^2\}\{2\backslash pi\backslash alpha\text{'}\}\backslash right)^2\; +\; B^2\; v\_*^4\}.\; \backslash quad\; (5.15)$$Notice that the Hall conductivity is automatically linear in charge density. When both $B$ and $E$ are small, this becomes $\sigma^{xy} \sim T^{-4/z}$ . The expression for $\sigma^{xx}$ is

$${\sigma }^{xx}=\frac{1}{1+(\frac{2\pi {\alpha }^{\mathrm{\prime }}}{L^2}{)}^2{B}^2{v}_{\star }^4}\sqrt{(2\pi {\alpha }^{\mathrm{\prime }}{)}^4{\tau }_{\text{eff}}^2[1+(\frac{2\pi {\alpha }^{\mathrm{\prime }}}{L^2}{)}^2{B}^2{v}_{\star }^4]+(\frac{2\pi {\alpha }^{\mathrm{\prime }}}{L^2}{)}^2(J^t{)}^2{v}_{\star }^4}\mathrm{.}(5.16)\backslash sigma^\{xx\}\; =\; \backslash frac\{1\}\{1\; +\; (\backslash frac\{2\backslash pi\backslash alpha\text{'}\}\{L^2\})^2\; B^2\; v\_\backslash star^4\}\; \backslash sqrt\{(2\backslash pi\backslash alpha\text{'})^4\; \backslash tau\_\{\backslash text\{eff\}\}^2\; [1\; +\; (\backslash frac\{2\backslash pi\backslash alpha\text{'}\}\{L^2\})^2\; B^2\; v\_\backslash star^4]\; +\; (\backslash frac\{2\backslash pi\backslash alpha\text{'}\}\{L^2\})^2\; (J^t)^2\; v\_\backslash star^4\}.\; \backslash quad\; (5.16)$$

It's simple to see that this reduces to our previous expression when $B = 0$ . In particular, when the $J^t$ term dominates in the square root, and $B$ is small, then we reproduce the result $\sigma^{xx} \sim v_\star^2 \sim T^{-2/z}$ .

Among the interesting, anomalous, results exhibited by strange metals is the ratio $\sigma^{xx}/\sigma^{xy}$ . The anomalous behavior $(\sigma^{xx})^{-1} \sim T$ is accompanied in the cuprates by the scaling $\sigma^{xx}/\sigma^{xy} \sim T^2$ (e.g. [6]). This is to be contrasted with Drude theory⁸ which implies $\sigma^{xx}/\sigma^{xy} \sim (\sigma^{xx})^{-1}$ . Within our probe calculation, this ratio is given by

$$\frac{{\sigma }^{xx}}{{\sigma }^{xy}}=-{\left(\frac{L^2}{2\pi {\alpha }^{\mathrm{\prime }}}\right)}^2\frac{1}{J^{t}B{v}_{\star }^4}\sqrt{(2\pi {\alpha }^{\mathrm{\prime }}{)}^4{\tau }_{\text{eff}}^2[1+(\frac{2\pi {\alpha }^{\mathrm{\prime }}}{L^2}{)}^2{B}^2{v}_{\star }^4]+(\frac{2\pi {\alpha }^{\mathrm{\prime }}}{L^2}{)}^2(J^t{)}^2{v}_{\star }^4}\mathrm{.}(5.17)\backslash frac\{\backslash sigma^\{xx\}\}\{\backslash sigma^\{xy\}\}\; =\; -\; \backslash left(\; \backslash frac\{L^2\}\{2\backslash pi\backslash alpha\text{'}\}\; \backslash right)^2\; \backslash frac\{1\}\{J^t\; B\; v\_\backslash star^4\}\; \backslash sqrt\{(2\backslash pi\backslash alpha\text{'})^4\; \backslash tau\_\{\backslash text\{eff\}\}^2\; [1\; +\; (\backslash frac\{2\backslash pi\backslash alpha\text{'}\}\{L^2\})^2\; B^2\; v\_\backslash star^4]\; +\; (\backslash frac\{2\backslash pi\backslash alpha\text{'}\}\{L^2\})^2\; (J^t)^2\; v\_\backslash star^4\}.\; \backslash quad\; (5.17)$$

The relevant experimental limit for the ratio is when the first term is subdominant in the square root, leading to $\sigma^{xx}/\sigma^{xy} \propto T^{2/z} \sim (\sigma^{xx})^{-1}$ . We see that this aspect of the probe computation does not reveal strange behavior, but rather mimics the Drude result. In a later model-building section we will consider generalizations of the setup which might evade this conclusion.

5.3 AC conductivity

Let us next calculate the frequency dependent conductivity. In this case, we will focus on the linear response rather than working out the full nonlinear dependence on the electric field as we did in the DC case. To do this, we will expand in small fluctuations about the background (3.9), working at zero magnetic field ( $B = 0$ ) and zero momentum for simplicity. As before we extract the conductivity from the ratio of non-normalizable and normalizable modes of $A_x$ near the boundary $v \rightarrow 0$ , having introduced a background electric field $E_x(t) \equiv \text{Re } E_x(\omega) e^{-i\omega t}$ :

$$A_x(\omega )=\frac{E_x(\omega )}{i\omega }+\frac{J_x(\omega )}{z{\tau }_{\text{eff}}(2\pi {\alpha }^{\mathrm{\prime }}{)}^2}{v}^z+\dots (5.18)A\_x(\backslash omega)\; =\; \backslash frac\{E\_x(\backslash omega)\}\{i\backslash omega\}\; +\; \backslash frac\{J\_x(\backslash omega)\}\{z\backslash tau\_\{\backslash text\{eff\}\}(2\backslash pi\backslash alpha\text{'})^2\}\; v^z\; +\; \backslash dots\; \backslash quad\; (5.18)$$

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⁸Recall that in Drude theory: $\sigma^{xx} = \frac{e^2 n}{m\tau} \frac{1}{\tau^{-2} + e^2 B^2 m^{-2}}$ while $\sigma^{xy} = \frac{e^3 n B}{m^2} \frac{1}{\tau^{-2} + e^2 B^2 m^{-2}}$ . Thus $\frac{\sigma^{xx}}{\sigma^{xy}} = \frac{m}{eB\tau} = \frac{Ben}{\sigma^{xx}(B=0)}$ .The coefficients in this expansion will be determined by solving the bulk equations of motion, with the ratio between the response $J_x$ and the source $E_x$ obtained from a boundary condition at the horizon ensuring that the former is determined causally (via the retarded Green's function) from the latter. The implementation of ingoing boundary conditions at the horizon [37] is by now standard, see e.g. [9] for a discussion.

The fluctuations of the probe gauge fields take the form

$$\delta A=(A_t(v)dt+A_x(v)dx+A_y(v)dy)e^{-i(\omega t-kx)}\mathrm{.}(5.19)\backslash delta\; A\; =\; (A\_t(v)dt\; +\; A\_x(v)dx\; +\; A\_y(v)dy)e^\{-i(\backslash omega\; t\; -\; kx)\}.\; \backslash quad\; (5.19)$$

The quadratic action for fluctuations about the background solution (3.9) is found to be

$$S^{(2)}=-\frac{{\tau }_{\text{eff.}}(2\pi {\alpha }^{\mathrm{\prime }}{)}^2}{2}\int dv{d}^{3}x{v}^{1-z}\gamma (f{F}_{vi}^2-\frac{v^{2z-2}}{f}{F}_{ti}^2-v^{2z-2}\gamma F_{tv}^2+\frac{1}{\gamma }{F}_{xy}^2),(5.20)S^\{(2)\}\; =\; -\backslash frac\{\backslash tau\_\{\backslash text\{eff.\}\}(2\backslash pi\backslash alpha\text{'})^2\}\{2\}\; \backslash int\; dv\; d^3x\; v^\{1-z\}\; \backslash gamma\; \backslash left(\; f\; F\_\{vi\}^2\; -\; \backslash frac\{v^\{2z-2\}\}\{f\}\; F\_\{ti\}^2\; -\; v^\{2z-2\}\; \backslash gamma\; F\_\{tv\}^2\; +\; \backslash frac\{1\}\{\backslash gamma\}\; F\_\{xy\}^2\; \backslash right),\; \backslash quad\; (5.20)$$

where $\gamma$ is defined as

$$\gamma =\sqrt{1+{\left(\frac{2\pi {\alpha }^{\mathrm{\prime }}}{L^2}\right)}^2{C}^2{v}^4},(5.21)\backslash gamma\; =\; \backslash sqrt\{1\; +\; \backslash left(\backslash frac\{2\backslash pi\backslash alpha\text{'}\}\{L^2\}\backslash right)^2\; C^2\; v^4\},\; \backslash quad\; (5.21)$$

as in (3.27) and $i$ runs over $x$ and $y$ .

For simplicity, we specialize to the case $k = 0$ , applicable when the applied field has wavelength much longer than the mean free path of charge carriers. In this case, the equations of motion for the transverse and longitudinal fields are the same; let us focus on the longitudinal ( $x$ -) component.

$$v^{1-z}f(v^{1-z}\gamma f{A}_x^{\mathrm{\prime }}{)}^{\mathrm{\prime }}=-\gamma {\omega }^2{A}_x\mathrm{.}(5.22)v^\{1-z\}\; f\; (v^\{1-z\}\; \backslash gamma\; f\; A\text{'}\_x)\text{'}\; =\; -\backslash gamma\; \backslash omega^2\; A\_x.\; \backslash quad\; (5.22)$$

It is useful to map this equation into a Schrödinger form

$$-\frac{d^2\mathrm{\Psi }}{d{s}^2}+U\mathrm{\Psi }={\omega }^2\mathrm{\Psi }\mathrm{.}(5.23)-\backslash frac\{d^2\backslash Psi\}\{ds^2\}\; +\; U\backslash Psi\; =\; \backslash omega^2\backslash Psi.\; \backslash quad\; (5.23)$$

This is achieved with the change of variables

$$A_x=\frac{1}{{\gamma }^{1\mathrm{/}2}}\mathrm{\Psi },(5.24)A\_x\; =\; \backslash frac\{1\}\{\backslash gamma^\{1/2\}\}\; \backslash Psi,\; \backslash quad\; (5.24)$$

with

$$\frac{d}{dv}=\frac{v^{z-1}}{f}\frac{d}{ds},(5.25)\backslash frac\{d\}\{dv\}\; =\; \backslash frac\{v^\{z-1\}\}\{f\}\; \backslash frac\{d\}\{ds\},\; \backslash quad\; (5.25)$$leading to the potential

$$U=\frac{(\gamma -1)f}{{\gamma }^2{v}^{2z}}((\gamma +1)(1-z+\frac{3}{{\gamma }^2})f+v{f}^{\mathrm{\prime }})\mathrm{.}(5.26)U\; =\; \backslash frac\{(\backslash gamma\; -\; 1)f\}\{\backslash gamma^2\; v^\{2z\}\}\; \backslash left(\; (\backslash gamma\; +\; 1)\; \backslash left(\; 1\; -\; z\; +\; \backslash frac\{3\}\{\backslash gamma^2\}\; \backslash right)\; f\; +\; v\; f\text{'}\; \backslash right).\; \backslash quad\; (5.26)$$

The AC conductivity is obtained by imposing ingoing boundary conditions at the horizon $s \rightarrow \infty$ , ensuring a causal relationship between $E_x$ and $J_x$ . We will shortly present numerical results for the conductivity, after commenting on some of the physics evident from the above formulae.

The potential $U$ (5.26) exhibits different behavior for different ranges of charge density $C$ and $z$ . At zero charge density, $C = 0$ , we have $\gamma = 1$ and the potential vanishes. It is then straightforward to solve analytically for $\sigma(\omega)$ , which is nonzero and constant at all temperatures

$$\sigma (\omega )={\tau }_{\text{eff}}(2\pi {\alpha }^{\mathrm{\prime }}{)}^2\equiv {\sigma }_0\mathrm{.}(5.27)\backslash sigma(\backslash omega)\; =\; \backslash tau\_\{\backslash text\{eff\}\}\; (2\backslash pi\backslash alpha\text{'})^2\; \backslash equiv\; \backslash sigma\_0.\; \backslash quad\; (5.27)$$

In the absence of any ambient charge density, the current can only arise from thermal fluctuations or particle production. The constancy of the result, technically following from the absence of scattering when $U = 0$ , reflects more than just the scaling symmetry of our system. General quantum critical theories in 2+1 dimensions have a conductivity $\sigma(\omega/T)$ that tends to different constant values as $\omega \rightarrow 0$ and $\omega \rightarrow \infty$ [38]. The frequency independence here is related to additional symmetries of the Maxwell and DBI actions [39].

The above $C = 0$ result is for massless charge carriers. However, in the real world systems we are ultimately interested in modeling, the effective energy gap $E_{\text{gap}}$ of the charge carriers is often greater than the temperature, and the current resulting from their thermal production should be Boltzmann suppressed. Their mass is also often greater than the frequency scale $\omega \sim 10^3 - 10^4 \text{ cm}^{-1} \sim 10^{-1} - 1 \text{ eV}$ of the electric field applied in the relevant measurements (see Fig 14 of [40] for an example in the cuprates). So their dynamical production by the oscillating electric field should also be suppressed. As a result, we do not expect the order one constant value we just obtained in our $C = 0$ massless calculation to survive in a more realistic treatment. We will study the massive case in the next section.Having made this cautionary remark, let us continue to analyze the physics of the solution. The potential $U$ of (5.26) approaches zero at the horizon. As we approach the boundary $v \rightarrow 0$ , $U$ becomes of order $C^2 v^{4-2z}$ . For $z < 2$ , the potential is everywhere bounded, and approaches zero at the boundary. For $z = 2$ , $U$ approaches a constant value of order $C^2$ at the boundary, and for $z > 2$ the potential blows up at the boundary. Again $z = 2$ is a marginal case separating two behaviors.

The regime $\frac{(2\pi\alpha')^2}{L^4} C \ll 1$ is the dilute limit, in the sense that the charge density is small compared to the temperature scale. Specifically

$$\frac{1}{{\tau }_{\text{eff.}}2\pi {\alpha }^{\mathrm{\prime }}{L}^2}\frac{J^t}{T^{2\mathrm{/}z}}\ll 1.(5.28)\backslash frac\{1\}\{\backslash tau\_\{\backslash text\{eff.\}\}\; 2\backslash pi\backslash alpha\text{'}\; L^2\}\; \backslash frac\{J^t\}\{T^\{2/z\}\}\; \backslash ll\; 1.\; \backslash quad\; (5.28)$$

In this limit it is immediate that the potential (5.26) becomes proportional to $C^2$ . The conductivity $\sigma(\omega)$ is directly related to the reflection amplitude for scattering off the potential [41]. The correction to the constant result (5.27) will therefore be proportional to $C^2$ . This simplifies the general scaling form of the conductivity when the charge density is small,

$$\sigma (\frac{\omega }{(J^t{)}^{z\mathrm{/}2}},\frac{J^t}{T^{2\mathrm{/}z}})\sim {\tau }_{\text{eff.}}(2\pi {\alpha }^{\mathrm{\prime }}{)}^2+\frac{1}{{\tau }_{\text{eff.}}^2(2\pi {\alpha }^{\mathrm{\prime }}{)}^2{L}^4}\frac{(J^t{)}^2}{T^{4\mathrm{/}z}}\mathcal{F}\left(\frac{\omega }{T}\right)+\dots \mathrm{.}(5.29)\backslash sigma\backslash left(\backslash frac\{\backslash omega\}\{(J^t)^\{z/2\}\},\; \backslash frac\{J^t\}\{T^\{2/z\}\}\backslash right)\; \backslash sim\; \backslash tau\_\{\backslash text\{eff.\}\}\; (2\backslash pi\backslash alpha\text{'})^2\; +\; \backslash frac\{1\}\{\backslash tau\_\{\backslash text\{eff.\}\}^2\; (2\backslash pi\backslash alpha\text{'})^2\; L^4\}\; \backslash frac\{(J^t)^2\}\{T^\{4/z\}\}\; \backslash mathcal\{F\}\backslash left(\backslash frac\{\backslash omega\}\{T\}\backslash right)\; +\; \backslash dots.\; \backslash quad\; (5.29)$$

In this limit, DBI nonlinearities due to the charge density have been made small, the interactions between charge carriers are negligible and one might have expected the conductivity to be proportional to the density. However, in this case of massless charge carriers, the leading correction to the constant conductivity is found to be quadratic in the density. The observable we are computing here would be perhaps best characterized as a mobility rather than a conductivity. The result (5.29) is consistent with the previous DC result (5.11). In (5.11) the linear dependence on the charge density arises in the opposite limit to (5.28), in which interactions between the charges are important. It is worth emphasizing again therefore that the calculation of the conductivity which we are using does capture nonlinearities in the charge density, and the linearity emerging in the DC limit at large density is nontrivial.

We now move away from the dilute limit and explore numerically the dependence of the dissipative conductivity $\text{Re } \sigma(\omega)$ for different values of $z$ . To proceed, we need to makea choice for $f(v)$ . This will depend on the particular matter fields sourcing the Lifshitz background. For generic examples, one can expect the asymptotic behavior $f \sim 1 - v^{2+z}$ near the boundary $v \rightarrow 0$ . This is because the normalisable mode of $g_{tt}$ is dual to the energy density $T^{tt}$ , which has scaling dimension $[T^{tt}] = z + 2$ . Indeed such asymptotic behavior was found in the simple models of [42, 43], which used a massive vector field coupled to gravity. Thus, for illustration we will take

$$f=1-{\left(\frac{v}{v_{+}}\right)}^{2+z}\mathrm{.}(5.30)f\; =\; 1\; -\; \backslash left(\backslash frac\{v\}\{v\_+\}\backslash right)^\{2+z\}.\; \backslash quad\; (5.30)$$

The temperature of these backgrounds is, from (3.4), $T = (2+z)/4\pi v_+^z$ . We will comment below on the sensitivity of the results to this choice of $f$ . The resulting conductivities are shown in figure 2 below.

Figure 2: The real part of the conductivity as a function of frequency for $z = 1$ (left) and $z = 2$ (right). The four curves in each graph correspond to $\frac{1}{\tau_{\text{eff}} 2\pi \alpha' L^2} \frac{J^t}{T^{2/z}}$ equal to ${0, 10, 20, 30}$ (left) and ${0, 2, 5, 10}$ (right), with $J^t = 0$ giving the expected constant lines.

For all values of $z$ , the conductivity exhibits a peak reminiscent of Drude theory at $\omega = 0$ , approaches a nonzero constant value $\sigma_0$ at $\omega \rightarrow \infty$ , and exhibits a dip in the middle. Using the form for $f$ in (5.30), it is easy to evaluate (5.11) in the large charge density limit to obtain

$$\frac{\sigma (\omega =0)}{{\tau }_{\text{eff}}(2\pi {\alpha }^{\mathrm{\prime }}{)}^2}=\frac{1}{{\tau }_{\text{eff}}2\pi {\alpha }^{\mathrm{\prime }}{L}^2}\frac{J^t}{T^{2\mathrm{/}z}}{\left(\frac{2+z}{4\pi }\right)}^{2\mathrm{/}z}\mathrm{.}(5.31)\backslash frac\{\backslash sigma(\backslash omega\; =\; 0)\}\{\backslash tau\_\{\backslash text\{eff\}\}\; (2\backslash pi\; \backslash alpha\text{'})^2\}\; =\; \backslash frac\{1\}\{\backslash tau\_\{\backslash text\{eff\}\}\; 2\backslash pi\; \backslash alpha\text{'}\; L^2\}\; \backslash frac\{J^t\}\{T^\{2/z\}\}\; \backslash left(\backslash frac\{2+z\}\{4\backslash pi\}\backslash right)^\{2/z\}.\; \backslash quad\; (5.31)$$The final $z$ dependent term leads to the peak being bigger for a given $J^t/T^{2/z}$ at larger $z$ , as seen in the figure. All the conductivities exhibit a dip at intermediate frequencies. This can be understood from a sum rule: one can straightforwardly show using the Kramers-Krönig relations that $\int_0^\infty d\omega \text{Re } \sigma(\omega)$ is independent of the dimensionless ratio $J^t/T^{2/z}$ . In using the Kramers-Krönig relations it is important that the conductivity tends to its asymptotic value $\sigma_0$ sufficiently quickly in $\omega$ . For $1 < z < 2$ , $\text{Re } \sigma(\omega)$ approaches $\sigma_0$ from above; that is, it has a second peak (albeit much smaller than the Drude-like peak). For $z > 2$ , $\text{Re } \sigma(\omega)$ approaches $\sigma_0$ from below.

Plotting the Schrödinger potential one can see that there is dip close to the horizon where the potential becomes negative. The dip is not sufficiently big to allow negative energy bound states (which would lead to an instability of the spacetime), but it does allow for a low energy resonance. This is the ‘Drude peak’. This observation gives some indication of how sensitive our numerical results are to the form of $f$ chosen in (5.30). Experimentation shows that the overall shape of the conductivity plots is fairly robust if we do not modify $f$ drastically. That is, if we do not modify the minimal form of the potential in which there is a dip at the horizon that is then connected smoothly onto the asymptotic boundary behavior discussed above. However, if we introduce oscillations into $f$ in such a way that additional dips are introduced into the potential, then we can get additional peaks in the conductivity. Typically one finds a second or more peaks at low frequencies that are smaller than the peak at $\omega = 0$ . Curiously, such additional peaks also arise if one takes (5.30) with $z < 1$ .

In real-world strange metals, $\text{Re } \sigma(\omega)$ has a Drude-like peak at $\omega = 0$ , and approaches zero at large $\omega$ more slowly than in Drude theory (see e.g. Figure 14 of [40]). In our case, as discussed above, the massiveness of charge carriers as compared to the temperature and $\omega$ should lead to suppression of the $C = 0$ conductivity $\sigma_0$ . Before turning to the massive case, we make one final observation.

Although we have set the momentum $k$ to zero in our computations, in it straightforward in principle to work with a finite momentum. One interesting observation is that the combination of momentum and energy appearing in the Schrödinger equation is

$$v^{2-2z}f{k}^2-{\gamma }^2{\omega }^2\mathrm{.}(5.32)v^\{2-2z\}\; f\; k^2\; -\; \backslash gamma^2\; \backslash omega^2.\; \backslash quad\; (5.32)$$The feature of interest in this combination is that the coefficient of $k^2$ goes to zero at the horizon, while that of $\omega^2$ does not. This leads to the existence of low energy modes with arbitrary momentum, manifested for instance in a nonzero spectral density at zero temperature. This has something of the flavor of a Fermi surface; with a weakly coupled Fermi surface there are zero energy modes with finite momentum connecting different points on the Fermi surface. These exist for $k < 2k_F$ and lead to a sharp feature at $k = 2k_F$ . While interesting properties of the finite $k$ perturbation spectrum were found in [14, 15, 44], no such sharp feature was observed. This suggests that D-brane probe theories in the relativistic regime $\gamma \gg 1$ do not describe weakly coupled fermions. It is worth scrutinizing these systems more closely, taking into account effects that appear as $\gamma$ grows, cf (3.2) and (3.27). In addition to the back reaction on the metric, there are important effects in the open string sector that arise as the electric field $F_{tr}$ approaches the critical value, $\gamma \rightarrow \infty$ [28, 27]. Fermi surfaces have been found directly in other holographic systems in [7] (see [45] for an early approach to this problem).

6 Massive charge carriers

We will now study the effect of including a nonzero mass $m$ for the charge carriers described by the flavor brane, as in [13]. As discussed above, this is the case of interest in modeling some features of real-world strange metals, whose energy gap is large compared to the temperature:

$$E_{\text{gap}}\gg T\mathrm{.}(6.1)E\_\{\backslash text\{gap\}\}\; \backslash gg\; T.\; \backslash quad\; (6.1)$$

For instance $E_{\text{gap}}$ might be at the lattice eV scale which is larger than the melting temperature of the relevant materials.

As discussed in previous works on flavor branes [46] and on finite-density holography [47], a finite mass scale involves a configuration in which the volume of the internal cycle the flavor brane wraps varies with radial position, shrinking toward the horizon. In the case of massive flavors at $T = \mu = 0$ [46], the volume shrinks smoothly to zero at a finite radial position $v = v_0$ ; the brane forms a cigar-like shape with its tip at $v_0$ . Charge carrierscorrespond to strings stretching from the tip of the cigar down to the Poincaré horizon $v = \infty$ . At finite temperature, a black hole horizon arises at a finite radial position $v_+$ . For large enough mass (i.e. small enough $v_0$ ), the flavor brane still shrinks to a point outside the black hole horizon. Charge carriers in this $0 < T \ll E_{\text{gap}}, \mu = 0$ theory correspond to strings stretching from the flavor brane at $v_0$ to the horizon at $v_+$ . To obtain finite density and temperature, in a dilute limit one can simply introduce a small density of such strings and ignore back reaction on the brane configuration. We will refer to this as the string regime. For larger densities, the back reaction of the charge density on the brane is important; the upshot of this will be that the brane forms a ‘spike’ or ‘tube’ stretching to the horizon from $v_0$ in place of the bundle of strings that pertained in the dilute limit [47]. While the spike is string-like in some senses, it has a finite (order one in general) extent into the transverse internal space. An artist’s rendering of these two possibilities is shown in figure 3.

Figure 3: Schematic depiction of the massive flavor brane in the string regime (left) and spike regime (right).

In the following subsections, we will generalize these considerations to our Lifshitz backgrounds and use the resulting brane and string configurations to compute the AC conductivity for massive charge carriers. First we will consider the dilute limit in which the strings do not back react on the shape of the flavor brane, and extend standard calculations of drag forces [48, 49] to the Lifshitz case. This will lead to analytic results