We explain a method for computing the bulk viscosity of strongly coupled thermal plasmas dual to supergravity backgrounds supported by one scalar field. Whereas earlier investigations required the computation of the leading dissipative term in the dispersion relation for sound waves, our method requires only the leading frequency dependence of an appropriate Green's function in the low-frequency limit. With a scalar potential chosen to mimic the equation of state of QCD, we observe a slight violation of the lower bound on the ratio of the bulk and shear viscosities conjectured in arXiv:0708.3459.
We study the ratio of the shear viscosity to the entropy density for various holographic superfluids. For the s-wave case, the ratio has the universal value 1/(4pi) as in various holographic models. For the p-wave case, there are two shear viscosity coefficients because of the anisotropic boundary spacetime, and one coefficient has the universal value. For the (p+ip)-wave case, the existing technique is not applicable since there is no tensor mode of metric perturbations which decouples from Yang-Mills perturbations. Our results indicate that the shear viscosity does not show a singular behavi…
Recently Eling and Oz [1] proposed a simple formula for the bulk viscosity of holographic plasma. They argued that the formula is valid in the high temperature (near-conformal) regime, but is expected to break down at low temperatures. We point out that the formula is in perfect agreement with the previous computations of the bulk viscosity of the cascading plasma [2,3], as well as with the previous computations of the bulk viscosity of N=2^* plasma [4,5]. In the latter case it correctly reproduces the critical behaviour of the bulk viscosity in the vicinity of the critical point with the vani…
A novel technique is used to compute the bulk viscosity of high temperature holographic gauge theory plasmas with softly broken conformal symmetry. Working in a black hole background which corresponds to a non-trivial solution to the Navier-Stokes equation, and using a Ward identity for the trace of the stress-energy tensor, it is possible to obtain an analytic expression for the bulk viscosity. This can be used to verify the high temperature limit of a conjectured bound on the bulk viscosity for these theories. The bound is saturated when the conformal symmetry-breaking operator becomes margi…
Motivated by gauge theory/string theory correspondence, a lower bound on the bulk viscosity of strongly coupled gauge theory plasma was proposed in arXiv:0708.3459. We consider strongly coupled N=4 supersymmetric Yang-Mills plasma compactified on a two-manifold of constant curvature beta. We show that the effective (1+1)-dimensional hydrodynamic description of the system is governed by the bulk viscosity violating the bound of arXiv:0708.3459, once beta<0.
Recently Eling and Oz (EO) proposed a formula for the holographic bulk viscosity, in arXiv:1103.1657, derived from the null horizon focusing equation. This formula seems different from that obtained earlier by Gubser, Pufu and Rocha (GPR) in arXiv:0806.0407 calculated from the IR limit of the two-point function of the trace of the stress tensor. The two were shown to agree only for some simple scaling cases. We point out that the two formulae agree in two non-trivial holographic theories describing RG flows. The first is the strongly coupled N=2* gauge theory plasma. The second is the semi-phe…
A holographic dual of a finite-temperature SU(N_c) gauge theory with a small number of flavours N_f << N_c typically contains D-branes in a black hole background. By considering the backreaction of the branes, we demonstrate that, to leading order in N_f/N_c, the viscosity to entropy ratio in these theories saturates the conjectured universal bound eta/s >= 1/4π. The contribution of the fundamental matter eta_fund is therefore enhanced at strong 't Hooft coupling lambda; for example, eta_fund ~ lambda N_c N_f T^3 in four dimensions. Other transport coefficients are analogously enhanced. These …
Based on the gauge/gravity correspondence, the hydrodynamic response coefficients, shear and Hall viscosities, have been studied. The holographic model of Einstein-Maxwell- AdS $(3+1)$-dimensional system additionally coupled with the another gauge field mimicking the {\it dark matter} sector, as well as, gravitational Chern-Simons term bounded with a dynamical scalar field, were taken into account. Condensation of the scalar field provides the parity violating term. Both shear and Hall viscosities have been calculated and their dependence on $α$ - the coupling constant between matter and {\it …
We propose a lower bound on bulk viscosity of strongly coupled gauge theory plasmas. Using explicit example of the N=2^* gauge theory plasma we show that the bulk viscosity remains finite at a critical point with a divergent specific heat. We present an estimate for the bulk viscosity of QGP plasma at RHIC.
We revisit the membrane paradigm calculations of the holographic shear viscosity tensor of strongly coupled isotropic plasmas with Einstein gravity dual by emphasizing the fact which was overlooked in the previous literatures that the shear viscosity is a fourth-rank tensor. Using the membrane paradigm we show that depending on whether the holographic shear viscosity tensor to entropy density ratio is η^j_i^j_i/s or η^{jiji}/s or η_{jiji}/s, we can derive three distinct formulae for the holographic shear viscosity tensor to entropy density ratios given explicitly in terms of the background met…
We argue that the Kovtun--Son--Starinets (KSS) lower bound on the viscosity to entropy density ratio holds in fluid systems but is violated in solid materials with a non-zero shear elastic modulus. We construct explicit examples of this by applying the standard gauge/gravity duality methods to massive gravity and show that the KSS bound is clearly violated in black brane solutions whenever the massive gravity theories are of solid type. We argue that the physical reason for the bound violation relies on the viscoelastic nature of the mechanical response in these materials. We speculate on whet…
We construct a slowly varying space-time dependent holographic superfluid and compute its transport coefficients. Our solution is presented as a series expansion in inverse powers of the charge of the order parameter. We find that the shear viscosity associated with the motion of the condensate vanishes. The diffusion coefficient of the superfluid is continuous across the phase transition while its third bulk viscosity is found to diverge at the critical temperature. As was previously shown, the ratio of the shear viscosity of the normal component to the entropy density is 1/(4 pi). As a conse…
We study the spontaneous parity breaking and generating of Hall viscosity and angular momentum in holographic p+ip model, which can describe strongly-coupled chiral superfluid states in many quantum systems. The dual gravity theory, an SU(2) gauge field minimally coupled to Einstein gravity, is parity-invariant but allows a black hole solution with vector hair corresponding to a parity-broken superfluid state. We show that this state possesses a non-vanishing parity-odd transport coefficient -- Hall viscosity -- and an angular momentum density. We first develop an analytic method to solve this…
We present evidence for the universality of the shear viscosity of conformal gauge theory plasmas beyond infinite coupling. We comment of subtleties of computing the shear viscosity in effective models of gauge/gravity correspondence rather than in string theory.
We consider strongly coupled gauge theory plasma with conserved global charges that allow for a dual gravitational description. We study the shear viscosity of the gauge theory plasma in the presence of chemical potentials for these charges. Using gauge theory/string theory correspondence we prove that at large 't Hooft coupling the ratio of the shear viscosity to the entropy density is universal.
We develop analytic techniques to construct the leading dissipative terms in a derivative expansion of holographic fluids. Our basic ingredient is the Crnkovic-Witten symplectic current of classical gravity which we use to extract the dissipative transport coefficients of holographic fluids, assuming knowledge of the thermodynamics and the near horizon geometries of the bulk black hole geometries. We apply our techniques to non-conformal neutral fluids to reproduce previous results on the shear viscosity and generalise a known expression for the bulk viscosity.
In (2+1)-dimensional systems with broken parity, there exists yet another transport coefficient, appearing at the same order as the shear viscosity in the hydrodynamic derivative expansion. In condensed matter physics, it is referred to as "Hall viscosity". We consider a simple holographic realization of a (2+1)-dimensional isotropic fluid with broken spatial parity. Using techniques of fluid/gravity correspondence, we uncover that the holographic fluid possesses a nonzero Hall viscosity, whose value only depends on the near-horizon region of the background. We also write down a Kubo's formula…
We compute the bulk viscosity in holographic models dual to theories with Lifshitz scaling and/or hyperscaling violation, using a generalization of the bulk viscosity formula derived in arXiv:1103.1657 from the null focusing equation. We find that only a class of models with massive vector fields are truly Lifshitz scale invariant, and have a vanishing bulk viscosity. For other holographic models with scalars and/or massless vector fields we find a universal formula in terms of the dynamical exponent and the hyperscaling violation exponent.
We study the AC optical and hall conductivities of Dp/Dq-branes intersections in the probe approximation and use sum-rules to study various associated transport coefficients. We determine that the presence of massive fundamental matter, as compared to massless fundamental matter described holographically by a theory with no dimensional defects, reduces the plasma frequency. We further show that this is not the case when the brane intersections include defects. We discuss in detail how to implement correctly the regularization of retarded Green's functions so that the dispersion relations are s…
We discuss string theory alpha' corrections in the dual description of the expanding boost invariant N=4 supersymmetric Yang-Mills plasma at strong coupling. We compute finite 't Hooft coupling corrections to the shear viscosity and find that it disagrees with the equilibrium correlation function computations. We comment on the possible source of the discrepancy.
We study holographic RG flow of the shear viscosity tensor of anisotropic, strongly coupled N=4 super-Yang-Mills plasma by using its type IIB supergravity dual in anisotropic bulk spacetime. We find that the shear viscosity tensor has three independent components in the anisotropic bulk spacetime away from the boundary, and one of the components has a non-trivial RG flow while the other two have a trivial one. For the component of the shear viscosity tensor with non-trivial RG flow, we derive its RG flow equation, and solve the equation analytically to second order in the anisotropy parameter …
Gauge theory/string theory holographic correspondence for ${\cal N}=4$ supersymmetric Yang-Mills theory is well under control in the planar limit, and for large (infinitely large) 't Hooft coupling, $λ\to \infty$. Certain aspects of the correspondence can be extended including ${\cal O}(λ^{-3/2})$ corrections. There are no reliable first principle computations of the ${\cal N}=4$ plasma non-equilibrium properties beyond the stated order. We show extreme sensitivity of the non-hydrodynamic spectra of holographic ${\cal N}=4$ SYM plasma to ${\cal O}(λ^{-3})$ corrections, challenging any conclusi…
We use gauge theory/string theory correspondence to study the bulk viscosity of strongly coupled, mass deformed SU(N_c) N=4 supersymmetric Yang-Mills plasma, also known as N=2^* gauge theory. For a wide range of masses we confirm the bulk viscosity bound proposed in arXiv:0708.3459. For a certain choice of masses, the theory undergoes a phase transition with divergent specific heat c_V ~ |1-T_c/T|^(-1/2). We show that, although bulk viscosity rapidly grows as T -> T_c, it remains finite in the vicinity of the critical point.
A heavy quark moving through a strongly coupled deconfined plasma has a holographic dual description as a string moving in a black brane geometry. We apply the holographic Wilsonian renormalization method to derive a holographic effective string action dual to the heavy quark. The effective action only depends on the geometry between the black brane horizon and a cutoff localized in the radial direction, corresponding to the IR of the dual theory. We derive RG flow equations for the coefficients in the effective action and show that the force acting on the heavy quark is independent of the pos…
In this paper we investigate the ratio of shear viscosity to entropy density, $η/s$, in hyperscaling violating geometry with lattice structure. We show that the scaling relation with hyperscaling violation gives a strong constraint to the mass of graviton and usually leads to a power law of temperature, $η/s\sim T^κ$. We find the exponent $κ$ can be greater than two such that the new bound for viscosity raised in arXiv:1601.02757 is violated. Our above observation is testified by constructing specific solutions with UV completion in various holographic models. Finally, we compare the boundedne…
We have modified the holographic model of Saremi and Son by using a charged black brane, instead of a neutral one, such that when the bulk pseudo scalar potential is made of quadratic and quartic terms, parity can still be broken spontaneously in the boundary theory. In our model, the 3+1 dimensional bulk has a pseudo scalar coupled to the gravitational Chern-Simons term in the anti de Sitter charged black brane back ground. Parity could be broken spontaneously in the bulk by the pseudo scalar hairy solution and give rise to non-zero Hall viscosity at the boundary theory.
Prior attempts to construct the gravity dual of boost-invariant flow of N=4 supersymmetric Yang-Mills gauge theory plasma suffered from apparent curvature singularities in the late time expansion. This Letter shows how these problems can be resolved by a different choice of expansion parameter. The calculations presented correctly reproduce the plasma energy-momentum tensor within the framework of second order viscous hydrodynamics.
We study the conductivity and shear viscosity tensors of a strongly coupled N=4 super-Yang-Mills plasma which is kept anisotropic by a theta parameter that depends linearly on one of the spatial dimensions. Its holographic dual is given by an anisotropic axion-dilaton-gravity background and has recently been proposed by Mateos and Trancanelli as a model for the pre-equilibrium stage of quark-gluon plasma in heavy-ion collisions. By applying the membrane paradigm which we also check by numerical evaluation of Kubo formula and lowest lying quasinormal modes, we find that the shear viscosity pure…
We study the thermodynamics and the jet quenching parameter of a black hole solution dual to a SQCD-like plasma which includes the backreaction of fundamental flavors. The free energy is calculated in several ways, including some recently proposed holographic renormalization prescriptions. The validity of the latter is confirmed by the consistency with the other methods. The resulting thermodynamic properties are similar to the Little String Theory ones: the temperature is fixed at the Hagedorn value and the free energy is vanishing. Finally, an accurate analysis of the relevant string configu…
We construct a gravity dual of a boost-invariant flow of an N=4 SU(N) supersymmetric Yang-Mills gauge theory plasma with chemical potential. We present both a first-order corrected late-time solution in Eddington-Finkelstein coordinates and a zeroth-order solution in parametric form in Fefferman-Graham coordinates. The resulting background takes the form of a time-dependent AdS Reissner-Nordstroem-type black hole whose horizons move into the bulk of the AdS space. The solution correctly reproduces the energy and charge density as well as the viscosity of the plasma previously computed in the l…
In this work, we perform a holographic analysis to study non local observables associated to a uniformly \textit{boosted} strongly coupled large $N$ thermal plasma in $d$-dimensions. In order to accomplish the holographic analysis, the appropriate dual bulk theory turns out to be $d+1$ dimensional \textit{boosted} AdS-Schwarzschild blackhole background. In particular, we compute entanglement entropy of the boosted plasma at high temperature living inside a strip geometry with entangling width $l$ in the boundary at a particular instant of time. We also study the two-point correlators in the bo…
We study time-dependent perturbations to a family of five-dimensional black hole spacetimes constructed as a holographic model of the QCD phase diagram. We use the results to calculate two transport coefficients, the bulk viscosity and conductivity, as well as the associated baryon diffusion constant, throughout the phase diagram. Near the critical point in the T-mu plane, the transport coefficients remain finite, although their derivatives diverge, and the diffusion goes to zero. This provides further evidence that large-N_c gauge theories suppress convective transport. We also find a diverge…
We use alternative quantisation of the $D3/D5$ system to explore properties of a strongly coupled charged plasma and strongly coupled anyonic fluids. The $S$-transform of the $D3/D5$ system is used as a model for charged matter interacting with a U(1) gauge field in the large coupling regime, and we compute the dispersion relationship of the propagating electromagnetic modes as the density and temperature are changed. A more general $SL(2,\mathbb{Z})$ transformation gives a strongly interacting anyonic fluid, and we study its transport properties as we change the statistics of the anyons and t…
We use the holographic approach to compare the Hall viscosity $η_H$ and the angular momentum density ${\cal J}$ in gapless systems in $2+1$ dimensions at finite temperature. We start with a conformal fixed point and turn on a perturbation which breaks the parity and time reversal symmetries via gauge and gravitational Chern-Simons couplings in the bulk. While the ratio of $η_H$ and ${\cal J}$ shows some universal properties when the perturbation is slightly relevant, we find that the two quantities behave differently in general. In particular, $η_H$ depends only on infrared physics, while ${\c…
Thin-shell AdS-Vaidya spacetimes can be considered as holographic models of the thermalization process in strongly-coupled conformal field theories following a rapid injection of energy from an external source. While the expected thermalization time is the inverse temperature, Bhattacharyya and Minwalla have pointed out that bulk causality implies that expectation values of local field-theory observables actually take on their thermal values immediately following the injection. In this paper we study two-point functions, for which the causality argument does not apply. Specifically, we study t…
We study phase diagram of the dense holographic gauge matter in the Sakai-Sugimoto model in the presence of the magnetic field above the deconfinement temperature. Even above the deconfinement, quarks could form colour bound states through the remaining strong interaction if the density is large. We demonstrate that in the presence of the magnetic field for a sufficiently large baryon density, the multiquark-pion gradient (MQ-$\mathcal{5}ϕ$) phase is more thermodynamically preferred than the chiral-symmetric quark-gluon plasma. The phase diagrams between the holographic multiquark and the chir…
We compute the strong coupling limit of the shear viscosity for the N=4 super-Yang-Mill theory with a chemical potential. We use the five-dimensional Reissner-Nordstrom-anti-deSitter black hole, so the chemical potential is the one for the R-charges U(1)_R^3. We compute the quasinormal frequencies of the gravitational and electromagnetic vector perturbations in the background numerically. This enables one to explicitly locate the diffusion pole for the shear viscosity. The ratio of the shear viscosity eta to the entropy density s is eta/s=1/(4pi) within numerical errors, which is the same resu…
We study a bottom-up holographic model of large-Nc Yang-Mills theory, in which conformal invariance is broken through the introduction of a dilaton potential on the gravity side. We use the model to calculate the spectral densities of the shear and bulk channels at finite temperature. In the shear channel, we compare our results to those derived in strongly coupled N=4 Super Yang-Mills theory as well as in weakly coupled ordinary Yang-Mills. In the bulk channel, where the conformal result is trivial, we make comparisons with both perturbative and lattice QCD. In both channels, we pay particula…
It was shown in arXiv:0810.4094 [hep-th] that in the presence of the magnetic field the sound waves in (2+1) dimensional plasma disappear and are replaces by a diffusive mode. Similarly, the shear and charge diffusion fluctuations form a subdiffusive mode. However, since the limit of small magnetic field does not commute with the hydrodynamic limit it is not obvious whether or not these modes are stable under higher order corrections. Using AdS/CFT correspondence we show that in the case of M2-brane plasma these modes do exist as we find the corresponding supergravity solutions. This allowed u…
We study odd viscosity in a holographic model of a Weyl semimetal. The model is characterised by a quantum phase transition from a topological semimetal to a trivial semimetal state. Since the model is axisymmetric in three spatial dimensions there are two independent odd viscosities. Both odd viscosity coefficients are non-vanishing in the quantum critical region and non-zero only due to the mixed axial gravitational anomaly. It is therefore a novel example in which the mixed axial gravitational anomaly gives rise to a transport coefficient at first order in derivatives at finite temperature.…
We compute the shear viscosity of the non-commutative N=4 super Yang-Mills quantum field theory at strong coupling using the dual supergravity background. Special interest derives from the fact that the background presents an intrinsic anisotropy in space through the distinction of commutative and non-commutative directions. Despite this anisotropy the analysis exhibits the ubiquitous result η/s = 1/4πfor two different shear channels. In order to derive this result, we show that the boundary energy momentum tensor must couple to the open string metric. As a byproduct we compute the renormalise…
The finite temperature physics of the pure glue sector in the improved holographic QCD model of arXiv:0707.1324 and arXiv:0707.1349 is addressed. The thermodynamics of 5D dilaton gravity duals to confining gauge theories is analyzed. We show that they exhibit a first order Hawking-Page type phase transition. In the explicit background of arXiv:0707.1349, we find T_c = 235 MeV. The temperature dependence of various thermodynamic quantities such as the pressure, entropy and speed of sound is calculated. The results show a good agreement with the corresponding lattice data.
We study the shear viscosity in an effective hydrodynamic theory and holographic model where the translational symmetry is broken by massless scalar fields. We identify the shear viscosity, $η$, from the coefficient of the shear tensor in the modified constitutive relation, constructed from thermodynamic quantities, fluid velocity and the scalar fields, which break the translational symmetry explicitly. Our construction of constitutive relation is inspired by those derived from the fluid/gravity correspondence in the weakly disordered limit $m/T \ll 1$. We show that the shear viscosity from th…
We review the progress in the holographic calculation of shear viscosity for strongly coupled field theories. We focus on the calculation of shear viscosity from the effective coupling of transverse gravitons and present some explicit examples.
We study the Schwinger effect with a moving D3-brane in a $\mathcal{N}$=4 SYM plasma with the aid of AdS/CFT correspondence. We discuss the test particle pair moving transverse and parallel to the plasma wind respectively. It is found that for both cases the presence of velocity tends to increase the Schwinger effect. In addition, the velocity has a stronger influence on the Schwinger effect when the pair moves transverse to the plasma wind rather than parallel.
We consider the effect of a periodic perturbation with frequency $ω$ on the holographic N=4 plasma represented by the planar AdS black hole. The response of the system is given by exponentially decaying waves. The corresponding complex wave numbers can be found by solving wave equations in the AdS black hole background with infalling boundary conditions on the horizon in an analogous way as in the calculation of quasinormal modes. The complex momentum eigenvalues have an interpretation as poles of the retarded Green's functions, where the inverse of the imaginary part gives an absorption lengt…
We use the holographic principle to study quark jets with trailing strings in an expanding plasma that asymptotes Bjorken hydrodynamics. We make use of the fact that the trailing string is the locus of the light delay in bulk to obtain the explicit form for quark jets in the expanding plasma. From the trailing string solution we calculate the drag coefficient of a heavy quark in the strongly coupled expanding plasma. The energy scaling of the maximum penetration length of an ultrarelativistic light quark jet using light rays in bulk is estimated.
The phase diagram of quark gluon plasma (QGP) formed at a very early stage just after the heavy ion collision is obtained by using a holographic dual model for the heavy ion collision. In this dual model colliding ions are described by the charged shock gravitational waves. Points on the phase diagram correspond to the QGP or hadronic matter with given temperatures and chemical potentials. The phase of QGP in dual terms is related to the case when the collision of shock waves leads to formation of trapped surface. Hadronic matter and other confined states correspond to the absence of trapped s…
We review construction of the improved holographic models for QCD-like confining gauge theories and their applications to the physics of the Quark-gluon plasma. We also review recent progress in this area of research. The lecture notes start from the vacuum structure of these theories, then develop thermodynamic and hydrodynamic observables and end with more advanced topics such as the holographic QCD in the presence of external magnetic fields. This is a summary of the lectures presented at the 56th Cracow School of Theoretical Physics in Spring 2016 at Zakopane, Poland.
A computation of the quotient of shear viscosity to entropy density, or KSS number $η/s$ is performed, in the non-relativistic and classical regime, first in Chiral Perturbation Theory, and then in the $SO(g+1)/SO(g)$ Non-Linear Sigma Model in the large $g$ limit. Both are field theories stemming from a renormalizable Sigma Model but in spite of that, we explicitly calculate how one undercomes the KSS bound by increasing the number of degenerate pions sufficiently. However we argue that particle production could still keep the validity of the KSS bound in the weak sense. We also show how a lar…
We consider a holographic model of chiral superfluidity whose bulk is Einstein Yang-Mills and compute viscosity and conductivity responses away from the probe limit. We calculate Hall viscosity and analyze its relationship to the superfluid density and the shift. We find that the relationship between these quantities derived from effective field theory at zero temperature persists for all temperatures: for $p\pm ip$ their ratio is equal to $\mp1/2$. At low temperatures the system develops a Lifshitz throat, indicating an anisotropic scaling symmetry in the infrared dynamics.
We calculate the bulk viscosity, drag force and jet quenching parameter in Improved Holographic QCD. We find that the bulk viscosity rises near the phase transition but does not exceed the shear viscosity. The drag force shows the effects of asymptotic freedom both as a function of velocity and temperature. It indicates diffusion times of heavy quarks in rough agreement with data. The jet quenching parameter values computed via the light-like Wilson loop are in the lower range suggested by data.
Based on AdS/CFT correspondence, we build a deep neural network to learn black hole metrics from the complex frequency-dependent shear viscosity. The network architecture provides a discretized representation of the holographic renormalization group flow of the shear viscosity and can be applied to a large class of strongly coupled field theories. Given the existence of the horizon and guided by the smoothness of spacetime, we show that Schwarzschild and Reissner-Nordström metrics can be learned accurately. Moreover, we illustrate that the generalization ability of the deep neural network can …
Using the framework of higher-form global symmetries, we examine the regime of validity of force-free electrodynamics by evaluating the lifetime of the electric field operator, which is non-conserved due to screening effects. We focus on a holographic model which has the same global symmetry as that of low energy plasma and obtain the lifetime of (non-conserved) electric flux in a strong magnetic field regime. The lifetime is inversely correlated to the magnetic field strength and thus suppressed in the strong field regime.
In this study, the gauge/string duality is used to investigate the dynamics of a moving heavy quark in a strongly coupled, non-conformal plasma. The drag force in this non-conformal model is smaller than that of $\mathcal{N} = 4$ supersymmetric Yang-Mills (SYM) plasma and decreases as the level of non-conformality is increased. At intermediate temperatures, the world-sheet temperature, which is derived numerically by calculating the world-sheet horizon, deviates from its conformal value, but at high temperatures, it tends to its conformal value.
