aa r X i v : . [ h e p - t h ] A ug Universal conductivity and central charges
Pavel Kovtun a,b and Adam Ritz a a Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada b Center for Theoretical Physics, Massachusetts Inst. of Technology, Cambridge, MA, 02139, USA (Dated: May 2008)We discuss a class of critical models in d > I. INTRODUCTION AND SUMMARY
It is not uncommon to find physical systems which are described by interacting conformal fieldtheories (CFTs). A simple example is the liquid-gas critical point whose static correlations are de-scribed by the Ising CFT in d = 3. Recently, CFTs which are formulated in space-time (rather thanjust space) have received attention, partly due to their appearance in quantum critical phenom-ena [1, 2]. Such CFTs are relativistic theories, even though their speed of “light” v is not necessarilyequal to 3 × m/s. As a result, charge transport in these systems at non-zero temperature obeyssimple scaling laws.At very short distances, the effects of temperature are irrelevant, and the natural physicalquestions involve the leading short-distance singularities of the correlation functions. On the otherhand, at long distances the effects of temperature become important, and the natural questions arerelated to thermodynamics and transport phenomena. In CFTs, however, short and long distancesare related by a scaling symmetry, and one may anticipate a universal relation between the long-distance transport coefficients and the parameters which describe the short-distance singularities.Unfortunately, this expectation seems to be quantitatively true only in 1+1 dimensions. Thesubject of this note is precisely the class of models where such universal relations between short-and long-distance transport parameters extend naturally to any dimension d > T µν and a U (1) conserved current J µ in a CFT are fixed to be h J µ ( x ) J ν (0) i = kx d − ω d − I µν , (1) h T µν ( x ) T αβ (0) i = cx d ω d − (cid:18) I µα I νβ + I µβ I να − d δ µν δ αβ (cid:19) , (2)where I µν ≡ δ µν − x µ x ν /x , and k and c are central charges, which are dimensionless constants.[We use units in which ~ = v =1 where v is the speed of “light” in the CFT.] The factors of ω d − ≡ π d/ / Γ( d/
2) are inserted for notational convenience. At non-zero temperature T , the equilibriumstate is characterized by pressure P , as well as by the charge susceptibility χ = h Q i / ( V T ),where Q is the conserved charge associated with the current J µ , and we take the thermodynamiclimit V →∞ . The susceptibility can be evaluated by introducing a small chemical potential µ , sothat χ ( T ) = ∂ρ/∂µ | µ =0 , where ρ ( T, µ ) = h Q i /V is the charge density. In a CFT, temperatureremains the only scale which dictates P ( T ) = c ′ T d , χ ( T ) = k ′ T d − , (3)where c ′ and k ′ are dimensionless constants. Physically, c and c ′ provide a measure of the totalnumber of degrees of freedom in the system, while k and k ′ measure the number of charged degreesof freedom. In two dimensions, c is uniquely related to c ′ [3, 4], while k is uniquely related to k ′ : c ′ = π c , k ′ = 12 π k , (4)which means that thermodynamics is uniquely fixed by the central charges. The reason is that intwo dimensional CFTs, the vacuum state is related to the thermal state by a symmetry transfor-mation. We review this argument in the next section. In d >
2, the conformal symmetry groupis not large enough to enforce a relation similar to Eq. (4), and therefore thermodynamics is notdetermined by the central charges. However, there does exist a large class of CFTs in d > c , resembling the two-dimensional case [5]. Thecrucial property of these models is that they admit a dual description in terms of classical gravityon a ( d +1) dimensional anti-de Sitter space (AdS). We will show that these CFTs also have theproperty that their susceptibility is determined by the central charge k , as for the two-dimensionalcase. Namely, we find the following relations: c ′ c = 14 π d/ (cid:18) πd (cid:19) d Γ( d/ Γ( d ) ( d − d ( d +1) , k ′ k = 12 π d/ (cid:18) πd (cid:19) d − Γ( d/ Γ( d ) . (5)Even though the ratios (5) are derived for integer d >
3, they can be analytically continued to anyreal positive d . In particular, in d = 2 they reproduce the universal relations (4).The CFTs which admit a dual gravitational description have many more universal propertiesbeyond the above relation between thermodynamics and the central charges. A surprising featureof these CFTs (and of their relevant deformations) is that momentum transport in these modelsis completely determined by thermodynamics. In particular, their viscosity is given by η = s/ π in any dimension [6, 7], where s = ( ∂P/∂T ) is the entropy density. This is surprising becausetransport coefficients are typically determined by the mean-free path even in CFTs [8], and are notfixed by thermodynamics. We will show that charge transport in these models is also completelydetermined by thermodynamics. Namely, we find that the dc electrical conductivity σ obeys asimilar relation, ηs = 14 π , σχ = 14 πT dd − . (6)Again, even though the ratio σ/χ was derived for integer d >
3, it can be analytically continuedto any real positive d >
2. We will see that the singularity at d = 2 is precisely what one expectsin 1+1 dimensional CFTs.The ratio of viscosity to entropy density was conjectured to be a universal lower bound, saturatedby models with a dual gravity description [6]. Motivated by the viscosity bound conjecture, wediscuss similar bounds on conductivity in relativistic CFTs which are saturated by models withgravity duals. II. NO HYDRODYNAMICS IN 1+1 DIMENSIONS
In this section, we review the argument [9] that 1+1 dimensional CFTs have no hydrodynamicregime, and derive the relation (4) between the susceptibility and the central charge k .In two dimensions, correlation functions at zero temperature and finite temperature can berelated to each other [10]. This is because the transformation which maps a plane to a cylinderis a conformal transformation, and therefore is a symmetry transformation in a CFT. The finitetemperature state is obtained by the exponential map z = e πi T w , where z = x + ix represents apoint on the plane, w = τ + iy represents a point on the cylinder, and τ is Euclidean time whichis periodic with period 1 /T .For a scalar operator of dimension ∆, a conformal transformation x → x ′ restricts the two-pointcorrelation function as follows h φ ( x ′ a ) φ ( x ′ b ) i = D ( x a ) − ∆ /d D ( x b ) − ∆ /d h φ ( x a ) φ ( x b ) i , (7)where D ( x ) = | det( ∂ x ′ /∂ x ) | is the Jacobian of the coordinate transformation. For the aboveexponential map in d =2, we have D ( x ) = 1 / (2 πT | x | ) , while the zero-temperature correlator issimply a power-law, h φ ( x a ) φ ( x b ) i = C φ / | x a − x b | . From the transformation relation (7) we findthe finite-temperature correlator of the scalar field, h φ ( τ, y ) φ (0) i = C φ (cid:20) ( πT ) sin[ πT ( τ + iy )] sin[ πT ( τ − iy )] (cid:21) ∆ . (8)This expression is periodic under τ → τ + 1 /T (as it should be), and reduces to the standardpower-law result in the limit T →
0. For models with a dual gravitational description, this form ofthe correlator was reproduced from small perturbations of the BTZ black hole in [11]. A similarargument can be applied to the density-density correlator on the plane in Eq. (1), which can bewritten as C ( x , x ) = − k π (cid:26) x + ix ) + 1( x − ix ) (cid:27) . (9)At finite temperature, we find C ττ ( τ, y ) = − k π ((cid:20) πT sin[ πT ( τ + iy )] (cid:21) + (cid:20) πT sin[ πT ( τ − iy )] (cid:21) ) . (10)Again, this expression is periodic under τ → τ + 1 /T , and reduces to Eq. (9) when T →
0. Thecharge susceptibility follows from the thermal density-density correlator, χ = − T Z d d − y C ττ ( τ, y ) , (11)where the extra minus sign is due to the Euclidean signature. In d =2 dimensions, we use theexplicit expression (10), and find χ = k/ π , as stated earlier in Eq. (4).The imaginary-time result (10) can be Fourier transformed, C ττ ( ω n , q ) = Z /T d τ Z d y C ττ ( τ, y ) e − iω n τ − iqy , (12)where ω n = 2 πnT is the Matsubara frequency. When performing the Euclidean time integration,the domain can be extended to include the whole real axis, and one picks up contributions froman infinite sequence of poles in the complex τ plane. For the density-density correlator one finds asimple expression C ττ ( ω n , q ) = − k π q ω n + q . (13)Analytic continuation to real frequency ω produces the retarded correlator C ret tt ( ω, q ) which onlyhas light-cone singularities, but shows no hydrodynamic modes (as one would find in higher di-mensions). Formal application of the Kubo formula now gives σ ( ω ) = Im ωq C ret tt ( ω, q ) = k δ ( ω ) . The singularity in the dc limit ω → d =2. III. CHARGE SUSCEPTIBILITY
We will focus on quantum field theories which admit a dual description in terms of classical gravityin Anti-de Sitter (AdS) space within the AdS/CFT correspondence [12]. For such field theories,a large-volume thermal state in a d -dimensional CFT is described by a ( d +1)-dimensional blackhole in AdS. The black hole solution follows from the Einstein-Maxwell action, S = 116 πG Z d d +1 x √− g (cid:20) R + d ( d − L (cid:21) − g d +1 Z d d +1 x √− g F , (14)where L sets the value of the cosmological constant, and g d +1 is the ( d +1)-dimensional gaugecoupling constant, which has dimension of (length) d − . An equilibrium state at finite temperatureand density is described by the Reissner-Nordstrom black hole in AdS. The thermodynamics of theseblack holes has been studied extensively, see for example [13]. The metric in the thermodynamiclimit is given by ds = r L (cid:0) − V ( r ) dt + d x (cid:1) + L r dr V ( r ) , (15)where V ( r ) = 1 − m/r d + m q /r d − , and the boundary is at r →∞ . The parameter m determinesthe mass of the black hole, and m q determines its charge. The background gauge field is A t = µ − C/r d − , where the constant C is related to the charge density of the CFT. The chemicalpotential µ is fixed by the condition that A t vanishes at the horizon r = r , i.e. µ = Cr d − . (16)The charge density ρ is defined by the variation of the action with respect to the boundary valueof the bulk gauge field A ( b ) t = A t ( r →∞ ), ρ = δSδA ( b ) t = ( d − Cg d +1 L d − . (17)To find the susceptibility, we need the relation between ρ and µ to linear order in µ . This meansthat in (16) it suffices to express r in terms of temperature when µ →
0, and one finds T = r d/ (4 πL ). From the definition of the chemical potential (16) we find ρ ( T, µ ) = χ ( T ) µ , wherethe susceptibility is χ = ( d − L d − g d +1 (cid:18) πd (cid:19) d − T d − . (18)The value of the central charge k can be found from the results of Freedman et al. [17]: k = L d − g d +1 Γ( d )( d − π d/ Γ( d/ ω d − . (19)Comparing with the susceptibility in (18), we find our result for k ′ /k in Eq. (5). IV. ELECTRICAL CONDUCTIVITY
The methods of the AdS/CFT correspondence also allow us to compute the electrical conductivityof CFTs with a dual gravity description. Since these models typically do not have dynamical U (1)gauge fields, we first need to say what we mean by the conductivity. We imagine gauging a global U (1) symmetry of the theory with a small coupling e , and work to leading order in e . The electricalconductivity is then defined with respect to this U (1) gauge field. To leading order in e , the effectsof the gauge field can be ignored, and the electromagnetic response can be determined from theoriginal theory [18]. This essentially amounts to sending J µ → eJ µ , and a factor of e will appearin both the conductivity and the susceptibility. The conductivity is determined from the real-timecurrent-current correlation function in thermal equilibrium, σ ( ω ) δ ij = e Im 1 ω C ret ij ( ω, q =0) . (20)Here C ret ij ( ω, q ) is the retarded correlation function of the global U (1) symmetry currents. The dcconductivity is σ ( ω =0).To evaluate C ret ij ( ω, q ), we use the standard AdS/CFT recipe of [11, 14], and consider Maxwellfields propagating on the ( d +1) dimensional background, ds = L z (cid:18) − f ( z ) dt + d x + dz f ( z ) (cid:19) , (21)where f ( z ) = 1 − ( z/z ) d , and the temperature of the CFT is T = d/ (4 πz ). This metric is obtainedfrom (15) at m q =0, changing the radial coordinate to z = L /r . The bulk action for the Maxwellfield is given by (14). Translation invariance allows us to take the bulk gauge field proportional The value of the susceptibility can also be deduced from the hydrodynamic current-current correlators. Namely,one has for the retarded charge density-charge density correlation function: C ret tt ( ω, q ) = χDq / ( iω − Dq ) , where D is the charge diffusion constant. Comparing this with the hydrodynamic correlators found in [14] for d =4, andin [15] for d =3 ,
6, one finds the susceptibility in d = 3 , , to e − iωt + i q · x , and q =0 is sufficient to find the conductivity using the Kubo formula (20). Thecomponent A i satisfies the equation u d − (cid:20) f ( u ) u d − A ′ i ( u ) (cid:21) ′ + w f ( u ) A i ( u ) = 0 , (22)where u = z/z , and w = ωz . The computation of the retarded correlation function requires thechoice of an outgoing boundary condition at the horizon, i.e. A i ( u ) = (1 − u ) − iw/d a ( u ), where a ( u )is regular at u =1. To find the dc conductivity, we solve the equation for a ( u ) as a power expansionin frequency, a ( u ) = a + iwa h ( u ) + O ( w ). For arbitrary dimension d , the solution for h ( u ) canbe expressed in terms of Gauss’ hypergeometric function, and the integration constants are fixedby requiring that h ( u ) vanishes at the horizon. The current-current retarded correlation functionis evaluated from the on-shell boundary action, S = ( L/z ) d − g d +1 Z d ω π d d − q (2 π ) d − A ′ i ( ω, u ) A i ( − ω, u ) z u d − , (23)with the implicit limit u →
0. The near-boundary expansion for h ( u ) has the form h ( u ) = h (0) +ln(1 − u ) /d + u d − / ( d −
2) + O ( u d − ) which allows us to read off C ret ij ( ω, q =0) to leading order in ω .The Kubo formula (20) then gives the conductivity, σ = e g d +1 (cid:18) Lz (cid:19) d − . (24)On the other hand, the susceptibility (18) can be written as χ = ( e /g d +1 ) ( L/z ) d − ( d − /z andwe arrive at the simple result (6) for the conductivity to susceptibility ratio. For systems in whichcharge transport proceeds by diffusion, conductivity is related to the diffusion constant D by theEinstein relation σ = χD . Therefore, our result can be interpreted as a remarkably simple diffusionconstant in d spacetime dimensions, D = 14 πT dd − . (25)One readily verifies that it agrees with the known results in d =4 [14], and d =3 , Theelectrical conductivity takes a particularly simple form in 2+1 dimensional CFTs. In this casethe equation (22) can easily be solved for all ω , and one finds a frequency-independent opticalconductivity [9], σ ( ω ) = e g . It is a peculiar feature of these models that because of strong quantum fluctuations the opticalconductivity in 2+1 dimensions is frequency-independent, and shows no crossover regime at ~ ω ∼ k B T . It would be very exciting to find two-dimensional materials which have this property. We were informed by A. Starinets that he has independently obtained Eq. (25) [16].
V. A CONDUCTIVITY BOUND?
We have shown that in all CFTs with a classical gravity dual, the ratio of electrical conductivity tothe static charge susceptibility is given by a very simple form (6). Given that in these models thethermodynamics is fixed by the central charges, and transport coefficients are fixed by thermody-namics, it follows that transport coefficients are uniquely fixed by the central charges, η ∼ c T d − ,and σ ∼ k T d − . Therefore, the ratio of viscosity to conductivity is proportional to the ratio of thecentral charges, η e σT = 8 π ( d − d − d ( d +1) ck . (26)If c and k indeed provide a suitable measure of the number of degrees of freedom in the system,it is not unreasonable to assume that the right-hand side of (26) is bounded from below becausethe number of charged degrees of freedom must be smaller than the total number of degrees offreedom. As a result, one could imagine that the conductivity and viscosity obey a bound of thekind σT λ d η e , with some order one constant λ d . However, one should keep in mind that thedefinition of σ (or η ) involves an arbitrary choice of normalization for the corresponding current.[E.g., if the electromagnetic U (1) is chosen as a subgroup of a larger global symmetry group G ,this translates to an arbitrary choice of normalization for the generators of G .] Therefore, anyuniversal bound on conductivity will more naturally involve a quantity which is independent of thenormalization, such as σ/χ .The universal relation for σ/χ in (6) looks similar to the universal relation for η/s : both ratiosbecome large at weak coupling due to a large mean-free path, and saturate at strong coupling inCFTs with a dual gravity description. Alternatively, one may wonder if there could be a lowerbound for conductivity, similar to the conjectured lower bound [6] for viscosity, σχ > ~ v πT dd − , (27)where we have now restored ~ and the speed of “light” v . There is an important differencebetween the two ratios in Eq. (6): while η/s = ~ / π only contains ~ (suppressing the Boltzmannconstant), the corresponding ratio for the conductivity also contains the speed of “light”. The non-relativistic limit corresponds to v →∞ , and therefore the bound (27) cannot hold in non-relativisticsystems. Within the AdS/CFT correspondence, this expectation is confirmed by several explicitcomputations [21, 22] where σ/χ falls below the bound (27) once conformal symmetry is broken.In practice, the speed of “light” v is different in different materials, and is not bounded from below. There are known counter-examples to the decrease of c and k along renormalization group trajectories in super-symmetric theories [19], which suggests that c and k do not always unambiguously measure the number of degreesof freedom in the theory. However, in these examples k corresponds to the R -current (which is related to theenergy-momentum tensor by supersymmetry), and therefore c and k are proportional to each other. Alternatively, the ratio of viscosity to conductivity can be expressed as the ratio of the d +1 dimensional gaugecoupling constant to the d +1 dimensional Newton’s constant, e η/ ( σT ) = ( π/d ) ( L g d +1 ) /G N . From the dualgravitational point of view, such an inequality between c and k is related to a version of the “weak gravityconjecture” of Ref. [20] in AdS space. We thank John McGreevy for pointing out Ref. [20] to us. Once the speed of light is fixed, it is not unreasonable to guess that Eq. (27) does represent a lowerbound on conductivity in relativistic CFTs such as the Wilson-Fisher fixed point in the O ( N )model. VI. DISCUSSION
We have argued that in a large class of CFTs in d >
2, there are universal relations between thethermodynamic and transport properties, and the central charges which dictate the short distancebehaviour of current-current correlators. One way of defining this class of theories is that theypossess dual descriptions within AdS at the level of classical gravity and Maxwell electrodynamics.For example, this universality determines the shear viscosity η and “electrical” conductivity σ in terms of the corresponding central charges and naturally leads to a conjectured bound onconductivity in physical systems, given in Eq. (27), in analogy with the well-studied viscositybound conjecture.It is natural to ask about the regime of validity, or alternatively the constraints on the CFTswhich may enter such universality classes. Indeed, the analysis we have performed using theAdS/CFT duality required the validity of a classical gravity approximation, and thus some kindof a large- N limit. On the gravity side of the duality, universality follows from the uniqueness ofthe lowest dimension operator which determines the dynamics of the metric and/or the gauge fielddual to the current in question, i.e. the uniqueness of the Einstein-Hilbert and Maxwell actionsrespectively. From this point of view, once we move to finite N , it appears that a large numberof higher derivative corrections will also be required, thus limiting the possibility for universalbehaviour. Nonetheless, it would be interesting if additional symmetries on the bulk side couldconstrain the possible classes that might arise. Some hint in this direction is provided by the blackhole solutions in string theory beyond the leading classical Einstein term [23].With these issues in mind, it is clearly useful to have a concrete example with which to contrastthe general holographic results. We will consider the 3-dimensional O ( N ) model at large N withfields φ α , α = 1 , .., N subject to a constraint φ α φ α = 1. In this system, the ratio of c ′ /c wascomputed in the large- N limit by Sachdev [24], with the result, (cid:18) c ′ c (cid:19) O ( N ) = 8 ζ (3)15 π ≈ . , (28)which differs by only a few percent from the holographic answer, c ′ /c = π / ≈ . k ′ /k , where a natural variant of the vector currentstudied above lies in the adjoint, J αβµ = ( φ α ∂ µ φ β − φ β ∂ µ φ α ), and we can write h J αβµ ( x ) J γδν (0) i = kx (cid:18) δ µν − x µ x ν x (cid:19) δ αγ δ βδ − δ αδ δ βγ (4 π ) . (29)A straightforward calculation of the central charge at large N leads to k = 2 [25], while the charge Alternatively, one can think of the central charge k as the dynamical conductivity in the regime ω ≫ T . For highfrequencies, we can use the zero-temperature correlation functions (1), and obtain σ ( ω →∞ ) = e k/ susceptibility in this case was computed by Chubukov et al. [26]. Using these results, we find atlarge N , (cid:18) k ′ k (cid:19) O ( N ) = √ π ln √ ! ≈ . , (30)which differs by about 24% from the result k ′ /k = π/ ≈ . O ( N ) model is large, σ/χ is O ( N ) [2],reflecting the fact that the model becomes weakly coupled at large N . Therefore, the comparisonwith the O ( N ) model in d = 3 provides an example of a situation where two systems have verysimilar static thermodynamic properties, but vastly different transport properties.As another aspect of the constraints defining these universality classes, it is possible to considereven more restrictive models. Namely, for theories with four supercharges, e.g. N =1 supersymme-try in d = 4, there is a global U(1) R -symmetry whose current lies in the same supermultiplet as theenergy momentum tensor. If we use the R -current to determine k , it follows that c and k are not in-dependent, and specifically that the ratio c/k depends only on the dimension d . Therefore, for thesesystems the viscosity and the R -current conductivity are related by a simple dimension-dependentconstant as given in Eq. (26). As an interesting corollary, in such models the thermodynamicproperties, the viscosity, and the conductivity are fully determined by a single number, the centralcharge c .In this paper we focused on a universality class of models in d > N limit.As examples, we have in mind pairs of “parent” and “daughter” theories where the daughter isobtained by projection onto a sector invariant under a global discrete symmetry, such as thosestudied in [27, 28]. As a simple example, consider a parent U ( kN ) gauge theory with matter fieldsin the adjoint representation, and project out by a global Z k symmetry to form a daughter theorywith U ( N ) k gauge group, and matter fields in the bi-fundamental representation. It was shownin [28, 29] that the parent and daughter theories are completely equivalent in the Z k -invariantsector at large N , provided that the Z k symmetry is not spontaneously broken. In this and similarexamples, since the energy-momentum tensor does not carry any global charges it follows that thecorrelation functions of T µν in the parent and daughter theories are proportional to each other.For example, viscosity is proportional to the two-point correlation function of T µν , and thus alldaughters have the same η/s ratio. For CFTs, all daughters would also have the same c ′ /c ratio.One can say that the universality class consists of all daughter theories, which (even though theymay have different global symmetries and contain matter fields in different representations) sharethe same universal ratios, similar to Eqs. (5) and (6). 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