Thermodynamics of black branes in asymptotically Lifshitz spacetimes

Gaetano Bertoldi, Benjamin A. Burrington, and Amanda W. Peet

Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7(Received 21 September 2009; published 4 December 2009)

Recently, a class of gravitational backgrounds in 3þ 1 dimensions have been proposed as holographic

duals to a Lifshitz theory describing critical phenomena in 2þ 1 dimensions with critical exponent z � 1.

We continue our earlier work [G. Bertoldi, B.A. Burrington, and A. Peet, preceding Article, Phys. Rev. D

80, 126003 (2009).], exploring the thermodynamic properties of the ‘‘black brane’’ solutions with horizon

topology R2. We find that the black branes satisfy the relation E ¼ 22þz Ts where E is the energy density, T

is the temperature, and s is the entropy density. This matches the expected behavior for a 2þ 1

dimensional theory with a scaling symmetry ðx1; x2Þ ! �ðx1; x2Þ, t ! �zt.

DOI: 10.1103/PhysRevD.80.126004 PACS numbers: 11.25.Tq

I. INTRODUCTION

Since the Maldacena conjecture [1], holography hasoffered an interesting new tool to explore strongly coupledfield theories (for a review, see [2]). In this framework,black hole backgrounds are dual to strongly coupledplasmas, and using these backgrounds, one can extracthydrodynamic and thermodynamic properties of theplasma [3–5].

Recently, much effort has gone into describing quantumcritical behavior in condensed matter systems using holo-graphic techniques (for a review, see [6]). Quantum criticalsystems exhibit a scaling symmetry

t ! �zt; xi ! �xi (1)

similar to the scaling invariance of pure AdS (z ¼ 1) in thePoincare patch. From a holographic standpoint, this sug-gests the form of the spacetime metric

ds2 ¼ L2

�r2zdt2 þ r2dxidxj�ij þ dr2

r2

�; (2)

where the above scaling is realized as an isometry of themetric along with r ! ��1r (for our purposes, i ¼ 1, 2).Other metrics exist with the above scaling symmetry, butalso with an added Galilean boost symmetry [7–9] whichwe will not consider here (black brane solutions in thesebackgrounds were discussed in [10]). There has also beensome success at embedding a related metric into stringtheory [11] with anisotropic (in space) scale invariance.This may serve as a template for embedding metrics of theform (2) into string theory. Here, however, wewill continueour study of the model in [12] (some analysis of general-izations of this model appear in [13–15]). The authors of[12] constructed a 4D action that admits the metric (2) as asolution,1 which is equivalent to the action [13]

S ¼ 1

16�G4

Zd4x

ffiffiffiffiffiffiffi�gp

��R� 2�� 1

4F ��F �� � c2

2A�A�

�(3)

(F ¼ dA) with terms in the action parameterized by

c ¼ffiffiffiffiffiffi2Z

p

L; � ¼ � 1

2

Z2 þ Zþ 4

L2: (4)

The solution discussed in [12] has the metric (2) and

A ¼ L2 rz

z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2zðz� 1Þ

L2

sdt (5)

with the identification z ¼ Z and L ¼ L or the identifica-

tion z ¼ 4=Z, L ¼ 2Z L and is defined for z � 1 (with z ¼ 1

giving AdS4).In our current work, we will analyze the thermodynam-

ics of black brane solutions which asymptote to (2).2 Theseblack brane solutions have been studied numerically in[17–19]. However, we will show that their energy densityE, entropy density s and temperature T are related by

E ¼ 2

2þ zTs (6)

using purely analytic methods.In the following section we will introduce the ingre-

dients needed to prove (6). Particularly important is theexistence of a conserved quantity, used to relate horizondata to boundary data. In the final section, we combinethese ingredients into the result (6). We also show that thisrelation is expected in the dual field theory as a result of thescaling symmetry ðx1; x2Þ ! �ðx1; x2Þ, t ! �zt.

1This background was earlier studied in [16], however, withoutan action principle that admits the above metric as a solution.

2We constructed numeric solutions for this system in [17](seen as the large r0 limit of this earlier work), however, ourcurrent work does not depend on any numeric analysis.

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II. SUMMARY OF EARLIER RESULTS

In our earlier work [17], the action (3) was reduced onthe Ansatz

ds2 ¼ �e2AðrÞdt2 þ e2BðrÞððdx1Þ2 þ ðdx2Þ2Þ þ e2CðrÞdr2

A ¼ eGðrÞdt (7)

to give a one-dimensional Lagrangian

L1D ¼ 4eð2BþA�CÞ@B@Aþ 2eð2BþA�CÞð@BÞ2þ 1

2eð�Aþ2B�Cþ2GÞð@GÞ2 � 2�eðAþ2BþCÞ

þ 12c

2eð�Aþ2BþCþ2GÞ: (8)

The equations of motion following from this action havesolutions given by (2) and (5). Further, there are the knownblack brane solutions that asymptote to AdS4,

ds2 ¼��3

�

���r2fðrÞdt2 þ r2ðdx21 þ dx22Þ þ

dr2

r2fðrÞ�;

fðrÞ ¼ 1� r30r3

A ¼ 0: (9)

For the remainder of the paper, we will be concerned withblack branes which asymptote to the Lifshitz background(2) and (5) for z � 1.

In [17], a Noether charge was found which is associatedwith the shift

AðrÞBðrÞCðrÞGðrÞ

0BBB@

1CCCA !

AðrÞ þ �BðrÞ � �

2

CðrÞ þ 0GðrÞ þ �

0BBB@

1CCCA (10)

with � a constant. The above represents a diffeomorphismwhich preserves the volume element dtdx1dx2. This is whyit is inherited as a Noether symmetry in the reducedLagrangian. The associated conserved quantity is

ð2eðAþ2B�CÞ@A� 2eðAþ2B�CÞ@B� eð�Aþ2B�Cþ2GÞ@GÞ� D0: (11)

A. The perturbed solution near the horizon

We begin by first reviewing the results found in [17] forthe expansion near the horizon. We require that e2A goes tozero linearly, e2C has a simple pole, and eG goes to zerolinearly to make the flux dA go to a constant (in a localframe or not). Further, we take the gauge BðrÞ ¼ lnðLrÞ forthis section. We expand

AðrÞ ¼ lnðrzLða0ðr� r0Þ1=2 þ a0a1ðr� r0Þ3=2 þ � � �ÞÞ;BðrÞ ¼ lnðrLÞCðrÞ ¼ ln

�L

rðc0ðr� r0Þ�ð1=2Þ þ c1ðr� r0Þ1=2 þ � � �Þ

�;

GðrÞ ¼ ln

�L2rz

z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2zðz� 1Þ

L2

sða0g0ðr� r0Þ

þ a0g1ðr� r0Þ2 þ � � �Þ�: (12)

Note that by scaling time we can adjust the constant a0 byan overall multiplicative factor (note the use of a0 in theexpansion of GðrÞ as well, as eG multiplies dt for theoneformA). We will need to use this to fix the asymptoticvalue of AðrÞ to be exactly lnðrzLÞ with no multiplicativefactor inside the log.We plug this expansion into the equations of motion

arising from (8), and solve for the various coefficients. Wefind a constraint on the 0th order constants: as expected notall boundary conditions are allowed. We solve for c0 interms of the other g0 and r0, and find

c0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2zþ g20r0ðz� 1ÞÞ

qr3=20ffiffiffi

zp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðz2 þ zþ 4Þr20q : (13)

All further coefficients (e.g. g1) are determined from thetwo constants r0 and g0.We evaluate (11) at r ¼ r0 using the above expansion to

find

D0 ¼ rzþ30 L2a0c0

: (14)

This must be preserved along the flow in r. We will use thisto relate constants at the horizon to coefficients that appearin the expansion at r ¼ 1.

B. The perturbed solution near r ¼ 1We now turn to the question of the deformation space

around the solution given in (4) and (5). We take theexpansion of the functions

AðrÞ ¼ lnðrzLÞ þ �A1ðrÞ;BðrÞ ¼ lnðrLÞ þ �B1ðrÞCðrÞ ¼ ln

�L

r

�þ �C1ðrÞ;

GðrÞ ¼ ln

�L2rz

z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2zðz� 1Þ

L2

s �þ �G1ðrÞ:

(15)

Using straightforward perturbation theory, we may find thesolutions in the B1 ¼ 0 gauge

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A1ðrÞ ¼ C0ðz� 1Þðz� 2Þ

ðzþ 2Þ r�z�2

þ C2ðz2 þ 3zþ 2� ðzþ 1Þ�Þr�ðz=2Þ�1�ð�=2Þ (16)

B1ðrÞ ¼ 0 (17)

C1ðrÞ ¼ �C0ðz� 1Þr�z�2

þ C2ðz2 � 7zþ 6� ðz� 1Þ�Þr�ðz=2Þ�1�ð�=2Þ (18)

G1ðrÞ ¼ C02ðz2 þ 2Þzþ 2

r�z�2 þ C24zðzþ 1Þr�ðz=2Þ�1�ð�=2Þ

(19)

where we have defined the useful constant

� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9z2 � 20zþ 20

p: (20)

In the above, we have dropped certain terms in the expan-sion from [17]. We have dropped them so that we meet thecriterion of [20] to be sufficiently close to the Lifshitzbackground (2) and (5). We may evaluate the conservedquantity, and we find

D0 ¼ � 2ðz� 1Þðz� 2Þðzþ 2ÞL2

zC0: (21)

One may worry that nonlinearities may contribute to thevalue of this constant. However, one may examine thepowers of r available in [17], and quickly be convincedthat the higher nonlinear contributions will be zero once wemeet the criterion of [20] (i.e. dropping the ‘‘bad’’ modes).Hence, the above constant is the value ofD0 throughout theflow in r.

C. Gauge invariance

In the previous sections, we have gauge fixed by takingB1ðrÞ ¼ 0. Here, we write down the linearized gauge trans-formations that will allow us to switch to other gauges inperturbation theory (used near r ¼ 1). The transformation

A1ðrÞ ! A1ðrÞ þ z

r�ðrÞ; B1ðrÞ ! B1ðrÞ þ 1

r�ðrÞ

C1ðrÞ ! C1ðrÞ þ r@r

��ðrÞr

�; G1ðrÞ ! G1ðrÞ þ z

r�ðrÞ(22)

corresponds to infinitesimal coordinate transformationsr ! rþ ��ðrÞ. One can see that such a shift leaves theequations invariant to leading order in � when expandingabout the solution (2) and (5).

III. THE BLACK BRANE THERMODYNAMICS

In the above, we have calculated the conserved quantityD0 in two regions: near the horizon r ¼ r0 and in theasymptotic region r ¼ 1. We may use this to solve for

C0 in terms of the horizon data

C 0 ¼ � 1

2

rzþ30 a0z

c0ðz� 1Þðz� 2Þðzþ 2Þ : (23)

We will see that C0 is proportional to the energy density Eof the background.Indeed, the authors of [20] identified the energy density

of the background in terms of the coefficient of the r�z�2

term in the expansion at infinity for any z. This mode wasalso identified in [17] as the mass mode using backgroundsubtraction. However, for 1 � z � 2 there were additionaldivergences that were not canceled. These were curedusing the local counter terms of [20].One may not, however, directly use the results of [20]

because of a different choice of gauge: above we useB1ðrÞ ¼ 0 and the authors of [20] use C1ðrÞ ¼ 0. Wemay easily switch to this gauge by taking

�ðrÞ ¼ �rZ C1ðrÞ

rdr (24)

and transforming the other fields appropriately (near r ¼1). In the above integral, we make sure to take the constantof integration so that at large r the correction fields vanish.We find that in the C1ðrÞ ¼ 0 gauge

A1ðrÞ ¼ �2ðz� 1ÞC0rð�z�2Þ

ð2þ zÞ þ � � � (25)

where � � � are the other terms we are not concerned about.This allows us to compare our results directly to thecalculation of [20]. We identify 2A1ðrÞ ¼ fðrÞ where fðrÞappears in equation (5.27) [20]. Therefore, we identify

c1;RS ¼ �ðz� 1ÞC0 (26)

where c1;RS appears in [20] as the coefficient of r�z�2.

Therefore, equation (5.31) of [20] becomes

E ¼ � 4ðz� 2Þðz� 1Þz

C0 ¼ 2rzþ30 a0

ð2þ zÞc0 : (27)

where E is the energy density. However, we should notethat this is a unitless energy. Restoring units we find

E ¼ 2rzþ30 a0

ð2þ zÞc01

16�G4L: (28)

From the metric, it is easy to read off the area of thehorizon, and therefore the entropy density

s ¼ 4�r2016�G4

: (29)

Further, one can easily read off the temperature from theexpansion at the horizon [17]

T ¼ rzþ10 a04�c0L

: (30)

From this, we may read an interesting thermodynamic

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relationship

E ¼ 2

2þ zTs; (31)

which is the main result of our work.We compare this expression with the known z ¼ 1 black

brane solution in AdS4. For this we have E ¼ 2r30

16�G4L, T ¼

3r04�L and s ¼ 4�r2

0

16�G4. These satisfy the relations dE ¼ Tds,

E ¼ 23Ts. The second relation agrees with our expression

above for z ¼ 1.In general, one may use 2 relations of the form

dE ¼ Tds (32)

E ¼ KTs (33)

(K a constant, which for our purposes is a function of z) tofind the functional forms Eðr0Þ, EðTÞ and sðr0Þ, sðTÞ. First,one may use (33) to eliminate ds in the relation (32),directly relating dE and dT. One may integrate this to

find EðTÞ. One may then use this in (33) to find sðTÞ ¼4�r2

0

16�G4. Such a procedure will furnish Tðr0Þ and so we can

find Eðr0Þ and sðr0Þ. Doing so, we find

sðr0Þ ¼ 4�r2016�G4

; Eðr0Þ ¼ �

�K4�r2016�G4�

�1=K

;

Tðr0Þ ¼�K4�r2016�G4�

�ð1�KÞ=K(34)

where � is a constant of integration independent of r0. �has units of length 3K�2

1�K so that the units of s and E are

canonical. We can rewrite � in the following way. r0 is ametric parameter, and the temperature should only dependon metric parameters, not 16�G4. Further, the energydensity should depend on 16�G4, with one power of thisfactor in the denominator. Therefore, we can say that � /

116�G4

. To make up the rest of the units, there is only one

dimensionful parameter that one can use: L. Therefore, we

take that � ¼ LK=ð1�KÞ16�G4

nðzÞ. This gives

sðr0Þ ¼ 4�r2016�G4

;

sðTÞ ¼ LKðzÞ=ð1�KðzÞÞ

16�G4KðzÞ nðzÞTKðzÞ=ð1�KðzÞÞ;

Eðr0Þ ¼ 1

16�G4LnðzÞ

�KðzÞ4�r20

nðzÞ�1=KðzÞ

;

EðTÞ ¼ LKðzÞ=ð1�KðzÞÞ

16�G4

nðzÞT1=ð1�KðzÞÞ;

Tðr0Þ ¼ 1

L

�KðzÞ4�r20

nðzÞ�ð1�KðzÞÞ=KðzÞ

(35)

where we now explicitly write that K is a function of z

(KðzÞ ¼ 22þz ). To the right of each of these expressions, we

have given the relation in terms of only the thermodynamicvariable T, rather than referring to the geometric variabler0.One may use the above relations to show that the pres-

sure P is related to the energy density by 2P ¼ zE, whichmatches the holographic result of [20]. Actually, startingfrom the equation of state 2P ¼ zE one can derive therelation (31).The value of nðzÞ, which is a unitless number, may be

determined numerically by plotting logðLTÞ vs logðr0Þ, butwe did not attempt to do this here.The slope of the graph of logðTðr0ÞÞ vs logðr0Þ is

logðTðr0ÞÞ ¼ 2ð1� KðzÞÞKðzÞ logðr0Þ þ constant

¼ z logðr0Þ þ constant; (36)

which can be quantitatively checked using our earlier data[17].We may easily compare the thermodynamic relationship

(31) with the expected behavior from a system with ascaling symmetry ðx1; x2Þ ! �ðx1; x2Þ, t ! �zt. This scal-ing symmetry (along with SOð2Þ symmetry rotating the xi)implies that the dispersion relation is

!2 ¼ �2ðk21 þ k22Þz � �2k2z (37)

where � is some parameter to restore canonical units. Wewill work with a finite system (a box with sides of length‘), and assume that the occupation number of a given mode(with energy En) in the box is Q0ðe�EnÞe�En . This way,the energy of the system is written as

E ¼ � @

@Q; Q ¼ X

n

Qðe�EnÞ: (38)

Inside the box, ki ¼ 2�‘ ni, and we have d ¼ 2 spatial

dimensions, so there are two ni. We approximate the sumby an integral, and realize that the density of integerðn1; n2Þ lattice points is uniform in the ðn1; n2Þ plane to find

Q ¼ 2�Z

n2dn

nQðe��ðð2�=‘ÞnÞzÞ; n21 þ n22 � n2:

(39)

Redefining the integration variable, we find

‘2

2�z��ð2=zÞ�ð2=zÞ Z 1

0d��ð2=zÞ�1Qðe��Þ

� z

2�V�ð2=zÞ; (40)

where the constant � is independent of , and we haveassumed that the function Q is well behaved at infinity(this is used to construct�). This yields all thermodynamicquantities

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E ¼ �VTð2þzÞ=z;

S ¼ZdV¼0

dT1

T

�dE

dT

�V¼ �VT2=z 2þ z

2

(41)

and so indeed relation (31) holds, as well as all subsequentformulas. Using the above argument, we can generalize theresult to arbitrary spatial dimension d where the relation(31) becomes

E ¼ Tsd

dþ z: (42)

which generalizes the function KðzÞ ! Kðz; dÞ ¼ ddþz (we

also promote nðzÞ ! nðz; dÞ). For d ¼ 3 and z ¼ 1 (giving

K ¼ 34 ) one can check that (35) and (42) are correct by

comparing to the known results for black D3 branes [3] forthe functions sðTÞ and EðTÞ up to the normalization nðz; dÞ.The functions of sðr0Þ, Eðr0Þ, Tðr0Þ need to be modified bypromoting r20 ! rd0 , so that the ‘‘area’’ is measured appro-

priately. With this modification, the expressions for sðr0Þ,Eðr0Þ, Tðr0Þ also agree; see for example [21], where similarcoordinates are used.

ACKNOWLEDGMENTS

This work has been supported by NSERC of Canada.

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