Conductivities for hyperscaling violating geometries

Abstract

We show that many results about holographic conductivities in geometries with hyperscaling violating scaling can be reproduced from simple scaling laws in the dual field theory. We show that the electro-magnetic response of probe branes in these systems require at least one additional scaling parameter [Phi] beyond the usual dynamical exponent z and hyperscaling violating exponent [theta], as also pointed out in earlier work. We show that the scaling exponents can be chosen in such a way that the temperature dependence of DC conductivity and Hall angle in strange metals can be reproduced.

Published for SISSA by Springer

Received: May 20, 2014 Accepted: June 10, 2014

Published: June 24, 2014

JHEP06(2014)140

Conductivities for hyperscaling violating geometries

Andreas KarchDepartment of Physics, University of Washington, Seattle, WA 98195, U.S.A.

E-mail: [email protected]

Abstract: We show that many results about holographic conductivities in geometries with hyperscaling violating scaling can be reproduced from simple scaling laws in the dual eld theory. We show that the electro-magnetic response of probe branes in these systems require at least one additional scaling parameter beyond the usual dynamical exponent z and hyperscaling violating exponent , as also pointed out in earlier work. We show that the scaling exponents can be chosen in such a way that the temperature dependence of DC conductivity and Hall angle in strange metals can be reproduced.

Keywords: Holography and condensed matter physics (AdS/CMT), Gauge-gravity correspondence

ArXiv ePrint: 1405.2926

Open Access, c

The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP06(2014)140

Web End =10.1007/JHEP06(2014)140

Contents

1 Introduction 1

2 Experimental data points 32.1 Thermodynamics and dispersion relation 42.2 DC conductivities at zero temperature 42.3 Hall conductivity 52.4 DC conductivity at nite temperature 52.5 Probe brane thermodynamics 6

3 Holographic scaling relations 7

4 The Dd/Dq system 94.1 Brief review of the Dd/Dq system 104.2 Experimental facts 114.3 Deriving the anomalous scaling factors 12

5 Einstein-Maxwell-dilaton system 12

6 Discussion: connection to strange metals? 13

1 Introduction

Holography [13] has recently been used to construct putative novel phases of compressible matter. While many examples of theories with holographic dual are known, the simplest holographic duals correspond to scale invariant theories. The original examples of holography describe conformally invariant relativistic systems in terms of Einstein gravity on anti-de Sitter (AdS) space. These have been generalized to backgrounds incorporating a non-trivial dynamical critical exponent z [4] as well as hyperscaling violating (HV) exponent [57]. For = 0 the holographically calculated conductivity, and in particular its dependence on temperature, can be understood in the eld theory from a few very basic scaling laws [8]: the dynamical critical exponent is dened via the dispersion relation

kz. (1.1)

This forces us to assign scaling dimensions

[x] = 1, [t] = z (1.2) to space and time. In an equilibrium system this xes also the scaling of temperature

[T ] = z. (1.3)

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JHEP06(2014)140

These relations still are true in the presence of hyperscaling violation; the latter only manifests itself in an anomalous scaling of the energy and free energy density

[] = [f] = d + z (1.4) as well as the quantities that are derived from variations of f. In particular, one nds for the entropy density

[s] = [f] [T ] = (d ) s T

. (1.5)

Generalizing this scaling to the electro-magnetic sector one naively would postulate for vector potentials, chemical potential, electric and magnetic eld as well as charge density and current

[Ai] = 1, [A0] = [] = z, [E] = 1+z, [B] = 2, [n] = d, [j] = d+z1. (1.6) We will however see that these naive scalings are inconsistent with the holographic results for DC conductivities. In order to reproduce the properties of gauge elds in HV geometries we need to allow for anomalous scalings of the gauge elds1

[Ai] = 1 , [A0] = z . (1.7)

Unlike z and , not only depends on the bulk geometry but encodes properties of the bulk gauge eld action. For gauge elds described by a simple probe Dirac-Born-Infeld (DBI) action we will demonstrate that

= 2

d (1.8) where d is the number of spatial dimensions. We will also see the need to allow a di erent

HV exponent in the matter sector, m, as in the probe limit the probe and background energy densities can (and do) have a di erent scaling dimension. Di erent values for can be realized in theories coupled to a dilaton eld. We will in particular show that Dd/Dq intersections give a string theoretic realization of this latter scenario. Results similar to what we report here have appeared previously in [1016]. Especially in the more recent papers by Gouteraux and Gouteraux and Kiritsis the need for a separate scaling exponent governing the gauge elds (our above) had already been emphasized. We hope by succinctly summarizing the properties of these systems in terms of simple eld theory scaling laws we can add some clarity to the discussion.

In [8] it was demonstrated that the in the special case of = 0 the scaling laws imply that for z = 2 the DC resistivity grows linearly with temperature, raising the hopes that this

1A somewhat similar proposal has been recently made in [9]. This reference tried to introduce the anomalous scaling into [x] and [t] directly. While this can correctly reproduce some of the holographic conductivity results, it does not lead to a self-consistent assignment for several reasons. Maybe most importantly, the scalings of [9] are inconsistent with the basic dispersion relation ! kz. More concretely,

in order to reproduce the correct conductivities we determine in here one would need the version of the scalings in [9] which symmetrically distributes the anomalous scalings between [t] and [x]. In that case one obtains incorrect thermodynamics. Last but not least, we can derive our scaling from the properties of the holographic dual directly.

2

d z

JHEP06(2014)140

system may be related to the enigmatic strange metal phase of high-Tc superconductors. In the same work it was however also found that the very same scaling implies a temperature dependence of the Hall angle that is inconsistent with what is seen in strange metals. We will demonstrate that the more general scaling laws described in here allow for the observed strange metal scaling of both conductivities. Our scaling also makes rm predictions for the dimensions of all thermo-electric coe cients, so it would be curious to explore to what extend those can be matched against the properties of strange metals.

The organization of this note is as follows: in the next section we review and determine holographically properties of systems based on HV metrics using the probe brane approach to conductivities [17, 18]. In section 3 we will use the bulk form of the action to derive the correct scaling properties of the dual eld theory and show that they reproduce all the data-points collected in section 2. In section 4 we discuss the Dd/Dq system as an example of 6= 2/d; we once more can show perfect matching between bulk and boundary

calculations. In section 5 we briey comment on similar results that had been obtained previously for the Einstein-Maxwell-Dilaton system and show that they nicely t into our framework. In section 6 we conclude with a discussion of the putative connection to strange metals.

2 Experimental data points

In this section we collect a few facts about the thermodynamics and transport properties of bulk theories based on HV metrics, as well as charged matter described by a simple DBI probe brane.2 We can write a metric which allows for a non-trivial scale symmetry with dynamical critical exponent z and hyperscaling violating exponent in the form

gtt = r , gxx = r , grr = r . (2.1)

By reparametrizing r this metric has been cast in di erent forms in the literature, but the coordinate invariant statements we would like to incorporate are that the scaling transformation gives

Under this scaling r r and hence = 2

d 2, =

3

JHEP06(2014)140

x x, t zt, ds /dds. (2.2)

2d 2z. (2.3)

While these relations are su cient for our calculations, one convenient coordinate choice for this metric well sometimes use is

gtt = r2( /dz), grr = gxx = r2( /d1). (2.4) This is the form used, for example, in [7].

2There is some ambiguity how to include coupling to background scalars in the DBI action. In this section we simply take the background geometry as a given and put a purely geometric DBI action L g + F

on this background. In the earlier analysis of [16] the background dilaton already appeared in the DBI action, giving rise to di erent scaling relations. We will discuss the case of a non-trivial dilaton coupling to DBI in our section on the Dd/Dq system. For now the main purpose of our analysis is to nd the simplest holographic system that allows us to extract the pattern of scaling dimensions in the dual eld theory. The set-up is also di erent from our earlier study of HV probe systems [19] where the HV theory was set up by the probe elds themselves instead of the background geometry.

At nite temperature we need to modify the HV metric to include a horizon at a horizon radius rh. One family of such solutions has been found in [6, 15, 16], where we will mostly follow the last reference. They use a metric with = 2 (that is gxx r2)). In this coordinate system they nd rh T

(d ) dz

. (2.7)

These are indeed both properties of the background geometry (2.1) and its nite temperature extension [6].

2.2 DC conductivities at zero temperature

In order to calculate conductivities we need to add a gauge eld to the bulk action. Calculations are particularly easy if we use a bulk gauge eld described by a probe brane, that is we use the DBI action for the gauge eld and ignore its backreaction on the spacetime geometry. In a spacetime with metric

ds2 = gttdt2 + grrdr2 + gxxd~x2

where d~x2 is a d spatial dimensional at space and all metric functions only depend on the radial direction the DBI action gives a non-linear DC conductivity [17]

=

qA2gds2xx + g2xxn2 (2.8)

A is a normalization constant that depends on the prefactor of the DBI and n is the appropriately normalized charge density. ds is the number of spatial dimensions occupied by the probe matter elds, which can be less than d: ds d. The metric functions are to

be evaluated at a critical position r which is determined by

gttgxx = E2. (2.9)

Already at zero temperature there are various limits in which the conductivity takes a simple scaling form as we only turn on one dimensionful quantity. The rst scenario is to work at zero density, n = 0. In this limit we only get a scale from the electric eld itself, so

= Ag(ds2)/2xx E . (2.10)

4

which in our preferred coordinates of (2.4), where grr gxx near the asymptotic boundary, implies

rh T 1/z. (2.5)

2.1 Thermodynamics and dispersion relation

The dening relations of a d spatial dimensional system with dynamical critical exponent z and hyperscaling violating exponent are that the zero temperature dispersion relation is given by

= kz (2.6)

and the entropy density scales as

s T

d z

JHEP06(2014)140

These non-linear conductivities, together with a microscopic model of how they could arise, where discussed in detail in [20].

Alternatively, we can study the system in the limit of large density where the second term dominates. As in [8] we can exploit the fact that in this limit the DBI conductivity is linear in density to nd a scaling form:

= ng1xx nE~ . (2.11) At rst sight this nite conductivity in a translationally invariant system at nite charge density should be impossible. Without a mechanism to dissipate momentum the charge carriers should be accelerated by the external eld without limit. The steady state described by (2.11) owes its existence to the probe limit. The gravitational bulk describes a neutral heat bath with a large number of degrees of freedom, N2 if the dual is a large N gauge theory. There are only order N charged degrees of freedom which dissipate their momentum to the heat bath. Over time of course this dissipation will heat up the background heat bath, but in the large N limit this backreaction can be ignored at least as long as one doesnt let the system evolve for times that are parametrically long in N.

The relevant information about the conductivity is encoded in the two scaling exponents and ~

. From (2.9) we get

r + = E2 (2.12)

which implies that, at r = r, we have

gxx = r = E

Reassuringly , our choice of radial variable, dropped out from the relation. The conductivity was smart enough not to pay attention to a choice of coordinate system. With this we can read of our two exponents

= (ds 2)

2.3 Hall conductivity

If we include nite magnetic elds, many more opportunities arise. The full non-linear conductivities can be mapped out similar to above [18]. One universal relation one nds at zero temperature in the small electric eld limit is

xy = n

B (2.15)

independent of d, ds, and z.

2.4 DC conductivity at nite temperature

To get some meaningful scaling relations at nite temperature it is best to look at the case of small electric elds, so that E does not set an additional scale the conductivities can depend on. For small E we have from (2.9) that gttgxx|r=r 0 and so

r = rh (2.16)

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JHEP06(2014)140

2 + = E

2( d)2 (z+1)d . (2.13)

( d)

2 (z + 1)d

, ~

= 2(d )

2 (z + 1)d

. (2.14)

since gtt(rh) = 0 and gxx is non-vanishing everywhere. In this case we can again look at the two contributions to the conductivity separately. At zero density we have T , at large density we have conductivities proportional to density nT ~ with

= (ds 2)

(d )

dz , ~

dz . (2.17)

Last but not least let as look at the nite temperature Hall conductivity. The general expression is [18]

xy = nBg2xx + B2 . (2.18)

The presence of non-zero B also modies the expression for r, but for small elds we can still set r = rh (temperature is the dominant scale). In this case we get

xy = nBg2xx nBT

4( d) dz

= 2( d)

JHEP06(2014)140

. (2.19)

2.5 Probe brane thermodynamics

One interesting aspect of the probe brane system is that we have a clear separation between bulk elds and probe elds. As such we can dene two separate free energy densities, the bulk and the probe free energy density, which do not have to have the same scaling behavior. This is most obvious in the case where ds < d. In this case the two contributions to the free energy density can be distinguished by living at di erent places in space. Even in the case without any HV, they have di erent dimension, d + z and ds + z respectively. In the presence of HV, where the scaling dimension of the bulk free energy density changes to d + z , the most naive expectation would be for the probe free energy to change to

d+z dsd . That is each spatial direction picks up the same share of the anomalous scaling.

For ds < d the net contribution of gets simply prorated. In order to conrm that this is born out for the probe, we need to calculate the free energy density of the probe.

The thermodynamics of simple DBI probes in HV geometries has been analyzed in [21]. At zero density the on-shell action of the brane is given by its area. This gives rise to a matter free energy (in the coordinates of (2.4))

fm [integraldisplay]rh dr rt1 T

t11

z

(ds + 2)

, t1 = ds + z + 1

d . (2.20)

To study the system at nite density, we need to include the At component of the gauge eld in the action. The DBI Lagrangian becomes

L rt1[radicalBig]1 r2+2z4

d

At . (2.21)

From this we can easily get the scaling of matter free energy and chemical potential at zero temperature:

fm [integraldisplay] dr L n

t11 ds ds

d

(At)2, n = L

, [integraldisplay] dr At n

z 2

d ds ds

d

. (2.22)

6

Note that even if ds = d we can meaningfully separate the two free energies by their scaling in N. In a large N gauge theory we can have a contribution to the free energy density scaling as N2 (the bulk) and one scaling as N (the probe). They are clearly separate entities and can scale di erently with temperature. At nite N the system only has a single free energy and unless all contributions to the free energy have the same scale dimension of course scaling will no longer be a good symmetry. So in a system with ds = d

with a matter free energy scaling di erently from the bulk free energy, scaling is only an approximate symmetry emerging in the large N limit.

3 Holographic scaling relations

We have seen that the holographic calculations from the previous sections give us simple scaling forms for, among other quantities, the conductivity whenever we are in the regime where it depends only on a single dimensionful quantity. Naively one would have expected these scaling relations to be given by assigning eld theory dimensions according to (1.6). Unfortunately these natural dimension assignments predict

= d 2

1 + z , ~

= 2

1 + z (3.1) which are much simpler than what the DBI nds and in gross contradiction to it.

The key to derive the correct scaling laws in the dual eld theory is to realize that for a HV metric scaling is in fact not an isometry in the bulk. This distinguishes HV metrics from their = 0 cousins. Under a scale transformation x x the metric scales by an

overall prefactor

ds

ds. (3.2)

With this the bulk action is actually not invariant, but scales with an overall prefactor as well. A transformation under which the action changes by an overall prefactor still gives rise to a symmetry of the classical equations of motion. Of course the on-shell action itself is no longer invariant, but picks up an extra scaling term which is responsible for the HV behavior of the free energy (1.4).

The fact that the metric is not invariant under scaling however puts strong restrictions on the matter we can add to the theory without spoiling the invariance of the classical equation of motions. Every new term we add to the action has to scale with the same overall prefactor. This requires that we have to assign separate scaling properties to the various matter elds, that is the eld theory operators and sources. For a gauge eld described by a DBI action this means in particular that FMNF MN has to have the same scaling as 1 so that gF 2 and g scale with the same overall prefactor.

With this insight, we can deduce the following scaling laws. In the gravitational sector, we use the natural (dening) scaling properties:

[x] = 1, [t] = z, [T ] = z, [f] = d + z . (3.3) This ensures that we get the correct behavior for the entropy density and dispersion:

[] = z, [k] = 1, [s] = [f] [T ] = (d ) s T

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d

d z

. (3.4)

In the gauge eld sector however we need to assign anomalous scaling properties to At (and hence the chemical potential) and Ai, that is they no longer have the same dimension as t1 and x1 respectively.

[Ai] = 1 , [A0] = z = [], [E] = 1 + z , [B] = 2 . (3.5)

For the DBI action we have = 2

d . (3.6)

That is each A cancels the transformation property of one inverse metric in F 2. Usually the scaling dimensions of Ai and x as well as A0 and t are linked with each other by gauge invariance, as they appear together in the covariant derivative. They can be divorced by a dimensionful coupling constant appearing in front of the gauge eld in the covariant derivative.

Thermodynamic quantities for the gauge eld sector introduce one additional free HV exponent

[n] = ds m. (3.7)

For the DBI probe brane we have

m = dsd . (3.8)

We will see holographic examples with di erent values of m in the next section. Note that in the case where ds < d, that is the eld theory currents are localized on a submanifold that is not lling all of space, n (and hence the matter free energy density with [fm] = [n] + [])

picks up only a fraction ds/d of the anomalous scaling implied by . In the bulk this can easily be seen from the fact the determinant of the induced metric simply picks up fewer metric factors than the determinant of the full spacetime metric.

Conservation laws now x the dimension of the current:

[n] = ds m [j] = ds m + z 1. (3.9)

For the DBI probe this evaluates to

[j] = ds + z 1

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dsd . (3.10)

We can easily deduce from this the scaling of the matter free energy

[fm] = [] + [n] = ds + z m. (3.11)

It is easy to check that for the DBI probe this scaling perfectly reproduces our holographic calculations (2.20) and (2.22).

Last but not least, let us match the results for the conductivity. The dimensions of the conductivity (and conductivity divided by particle density) are

[] = [j] [E] = ds + m 2, [bracketleftbigg]

n

[bracketrightbigg] = 2 (3.12)

8

For the DBI probe brane this evaluates to

[] = ds 2d (d ), [bracketleftbigg]

n

[bracketrightbigg] = 2

d

d . (3.13)

This scaling perfectly reproduces the zero temperature exponents from (2.14) as well as the nite temperature exponents from (2.17). Note that with our assignments we also get the zero T Hall conductivity right, since

[n] [B] =

d

d (ds 2) = []. (3.14)

It is also easy to see that this scaling perfectly reproduces the nite T Hall conductivity (2.19) as well. Note that for any choice of and m we have

[bracketleftbigg]

nB

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[bracketrightbigg] = 2 4 = 2[bracketleftbigg]

n

. (3.15)

This relation will become important for us later when we discuss potential applications to strange metals.

As we discussed before, for generic matter HV exponents and m the free energy of the matter sector has a di erent dimension from the free energy in the bulk sector. This is consistent only if ds < d, so we can physically separate the localized energy density from the bulk energy density, or when the matter is a probe eld, so we can physically separate the order N2 energy density of the bulk from the order N density of the matter elds. In both those case, f and fm are independent physical quantities which have their own dimension under scaling. If we are unable to separate the free energy into a bulk and a matter contribution, our system only has scaling as a symmetry if we can assign a unique dimension to the full free energy, that is when [f] = [fm], or in other words

[f] = [fm] = m + . (3.16)

In the last section of this paper we will discuss potential phenomenological applications of our scaling laws. In that case we will always insist on (3.16) being obeyed, leaving as the only new scaling parameter. But since all our experimental scaling facts have been obtained for DBI probe branes we for now can (and have to) use m and as independent parameters.

4 The Dd/Dq system

As we have seen in the previous section, in systems with HV violating metrics the scaling properties of the dual eld theory are not uniquely determined by the gravitational background, but also depend on the action we chose for the matter elds. The general rule is that we need to assign the matter elds anomalous transformation properties so that the matter action scales with a universal prefactor (giving rise to m). Of course it may not always be possible to nd any scaling rule for the matter elds in which this is true. In this case scaling is not a good symmetry of the system with the matter sources turned on

9

(or for correlation functions involving the matter elds). For DBI we needed to assign the vector potential A transformation properties that cancel that of a single inverse metric, giving rise to = 2/d. A simple example for a gauge eld action that has scaling as a symmetry but with a di erent is given by a Maxwell-Dilaton system with Lagrangian

L g 1

+ e F 2 + . . .

[parenrightbig]

. (4.1)

In a non-trivial dilaton background the scaling of A needs to compensate an inverse metric as well as the scaling e . With a dilaton, we can also get more interesting values for m. In this section we want to study a particular class of examples of this type where the scalings can easily be worked out along the lines above. The fact that we can reproduce the highly non-trivial bulk scalings in this case gives a great conrmation of our construction.

4.1 Brief review of the Dd/Dq system

A well studied string theory embedding of a system with an HV metric that also allows for probe branes non-trivially coupled with a dilaton prefactor is provided by the Dd/Dq system. That is, we study a Dq probe in the background geometry of a Dd brane. The holographically dual eld theory of these theories is well understood: maximally super-symmetric Yang-Mills theory in d + 1 dimensions coupled to fundamental representation matter, whose details vary with how the Dq is embedded [22, 23]. The full 9+1 dimensional Dd brane metric and dilaton background are given by

ds2 = H1/2(dt2 + dx2d) + H1/2 du2

+ u2d 28d

JHEP06(2014)140

[parenrightbig]

, e = H

3d

4 . (4.2)

In the near horizon limit we have (setting the curvature radius to 1)

H = ud7. (4.3)

The Dd brane allows a near-extremal generalization with a horizon radius uh that scales with temperature as

uh T

25d . (4.4)

The entropy density of this near-extremal brane scales as

s T

9d5d . (4.5)

To see that this is indeed a HV scaling background, one wants to reduce on the internal 8 d sphere and go to the d + 2 dimensional Einstein frame [7]. Since the prefactor of the

d + 2 dimensional Einstein Hilbert term in the string frame is e2 H

8d

4 u8d = H

d+2

4 u8d

the rescaling to Einstein frame is given by

g g H

d+2

2d u

2(8d) d

= g u

(d3)(d6)

2d . (4.6)

The resulting Einstein frame metric

ds2d+2 = u(162d)H1/d dt2 + dx2d + Hdu2[parenrightbig]

(4.7)

10

can easily be seen to be of HV scaling form [7] with

= d

9 d

5 d

=

(d 3)2

5 d

, z = 1 (4.8)

This scaling is in perfect agreement with the thermodynamics (4.5) of the Dd system. To reach our preferred coordinate system with gxx = grr we need to change to a new radial coordinate that absorbs the prefactor H in front of u:

r =

Hdu u

[integraldisplay] 2 . (4.9)

In this coordinate system the horizon radius simply scales as rh 1/T .For a Dq probe brane wrapping q ds 1 internal dimension, the ds + 2 dimensional

volume and Maxwell terms on the brane worldvolume we get after reduction on the internal space and going to the Einstein frame are given by

L = ra1g [parenleftBig]

1 + r F F + . . .

d5

JHEP06(2014)140

[parenrightBig]

(4.10)

with

(d 3)(d 6)

d . (4.11) sets the anomalous scaling of the gauge eld . a1 gives the overall scaling of the free energy and so will be responsible for m. But it is easier to calculate fm directly from the 10d metric as well do momentarily, so we will not worry about the value of a1.

4.2 Experimental facts

The thermodynamics associated with the background geometry of the Dd brane has already been accounted for by the background of (4.8). What we are concerned with here are the properties of the probe Dq brane extending over ds spatial dimensions of the d dimensional eld theory. Its free energy is given by the on-shell action, which is given by a DBI action multiplied with e . From this one nds that, at zero density, the free energy scales as [23, 24]

f T

25d ( +1) (4.12)

whereas at zero temperature, nite density it scales as

f n1+

1

=

2d 5

(4.13)

with

= (d 7)(q 2ds 4 + d)

4 + q ds 1. (4.14)

The DC conductivities for this system have been worked out in [24]. One once more nds simple scaling laws at zero density as well as in the density dominated limit. They are given by

T

25d ( + d72 ) (4.15)

and

nT

d75d (4.16)

respectively.

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4.3 Deriving the anomalous scaling factors

For the gauge eld in the action (4.10) to transform properly under scaling when compared to the area term, we need to assign the gauge elds the anomalous scaling exponent

= 2

d +

( + 1) (4.18)

This implies (according to (3.11) and using from (4.17) above) a matter HV exponent

m = (d 3)(d + q 8)

2(d 5)

2 =

d 3

d 5

. (4.17)

On the other hand, the scaling of the free energy with temperature (4.12) tells us that

[fm] = 2 5 d

JHEP06(2014)140

. (4.19)

It is straightforward to check that with these two assignments for the new HV exponents introduced in section 3, our dimension assignments accurately reproduce the holographic results from eqs. (4.13) through (4.16).

5 Einstein-Maxwell-dilaton system

A large class of HV solutions have been obtained in [1016] based on the Einstein-Maxwell-Dilaton (EMD) system. In the EMD system the matter elds in the bulk are fully back-reacted and so it is nice to compare and contrast these solutions to our analysis in the previous sections, which was mostly based on probe branes. Since in EMD there is no defect present in the eld theory these theories work with ds = d. For the system to have a scaling symmetry, all terms in the action need to scale the same in we need to enforce (3.16). Eliminating m in terms of this implies

[] = d + 2 2. (5.1) Comparing in particular with [10, 11] we see that this scaling agrees with the behavior found in there as long as we identify our scaling exponent with their via

= + d

2 . (5.2)

It is straightforward to verify that with this assignment indeed all terms in the EMD scale the same. The non-trivial scaling of the gauge eld implied by is forced upon us by the coupling to the dilaton. The gauge coupling in the bulk itself is dimensionful, but these dimensions can be completely accounted for by powers of the AdS curvature radius L and do not correspond to a non-trivial . In order for the gauge coupling to scale non-trivially in the sense that 6= 0, we need it to be set by a scalar eld that itself has a non-trivial

radial prole. It is indeed easy to see that if we set the dilaton potential and gauge coupling to a constant, that is chose = = 0 in [10, 11], we get = 0, = d and hence = 0: scaling is actually an isometry in that case.

12

6 Discussion: connection to strange metals?

In this work we have demonstrated that the electromagnetic properties of critical points with non-trivial HV are characterized by two novel exponents, and m introduced in section 3, together with the familiar exponents and z. While we gave explicit holographic examples of theories realizing various values of these scaling parameters (and additional examples in a similar spirit can e.g. be found in [25, 26]), phenomenologically the most reasonable approach seems to be to just treat and m as free parameters characterizing any putative critical point and to then investigate whether such scaling can give rise to phenomenological acceptable predictions.

One system that one may try to apply our results to are strange metals. One of the enigmatic properties of these particular non-Fermi liquids is their resistivitys linear growth with temperature. In [8] it was pointed out that this behavior could be reproduced from a scale invariant theory with non-trivial dynamical critical exponent z. According to (3.12), the dimension of /n is 2. So if we assume our strange metal operates in the regime

where the conductivity is linear in density, this tells us that already for = 0 we get 1 T 2/z and so for z = 2 one gets a linear grows of the resistivity [8]. Allowing for

the more general scaling studied in this paper, we see that linear resistivity of the density dominated conductivity arises whenever3 z = 2 . It was however also pointed out in [8]

that the very same scaling fails to reproduce the correct behavior of the Hall conductivity. In strange metals, the linear temperature rise in 1xx comes along with a T 3 scaling of 1xy.

Assuming that the Hall conductivity is both linear in density and magnetic eld (as it is in holographic probe brane constructions) one can see from (3.15) that the scaling with temperature of the Hall conductivity is always twice that of the density dominated xx.

So if the latter goes as 1/T , the former scales as 1/T 2, not 1/T 3. The introduction of the new free parameter did not help resolve this tension. The only way to reconcile the density dominated xx and xy with experiment is to chose di erent values of for Ai and

A0. One way to accomplish this is to start with a genuinely non-relativistic gravitational theory in the bulk as recently advertised in [2830]

Of course the other option is to operate the system in a regime where xx is not

density dominated, but temperature dominated. The Hall conductivity still is linear in both density and magnetic eld, so we will still have

xy BnT

ds+ m2 z

32z. (6.3)

3The fact that hyperscaling violation can give rise to linear resistivity for z = 2 2 d is also already

implicit in [12, 21, 27].

13

JHEP06(2014)140

. (6.1)

Since the dimension of xx itself according to (3.12) is ds + m 2 we have xx T

2 4 z

. (6.2)

In this case, our new scaling exponents allow us to get the correct scalings for both conductivities. Getting xy to scale as T 3 requires

= 2

Using this in (6.2) we see that xx T

long as

ds mz 32 . This is inversely proportional to T as

z2. (6.4) So the properties of the strange metal can easily be accommodated. If we in addition want to avoid fm and f to have di erent scaling dimension, we can still impose (3.16) by xing itself to be

= m + = ds + 2 2z. (6.5)

As a concrete realization, we can e.g. chose a d = ds = 2 dimensional system with z = 2 and chose = 1, m = 1, = 0. Whether such a scaling can be obtained either from a

holographic model or some microscopic Lagrangian will be left as an open question.

It should be noted that once our HV coe cients are xed, there is no more freedom in setting the scaling laws for all thermoelectric coe cients. Dimensional analysis now can be applied to all thermoelectric phenomena, such as the Nernst and Seebeck e ect. The scaling dimensions also constrain the frequency and wavenumber dependence of current correlation functions. This gives in principle many more data points one can look at in order to determine whether our generalized HV critical points can give a correct physical description of strange metals or any other quantum critical system.

Acknowledgments

Id like to thank Sean Hartnoll for very useful correspondence and encouragement. Thanks also to Blaise Gouteraux for helping to clarify the connection between the approach followed in here and his original work, as discussed in section 5. This work was supported, in part, by the US Department of Energy under grant number DE-FG02-96ER40956.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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