Van der Waals-like Behaviour of Charged Black Holes and Hysteresis in the Dual QFTs

TL;DR

This paper investigates the temperature-dependent hysteresis in the shear viscosity to entropy density ratio of dual QFTs, linking it to the Van der Waals-like thermodynamics of charged black holes in AdS space.

Contribution

It introduces a spherical analogue of shear viscosity in holography and explains hysteresis phenomena via black hole thermodynamics and phase transitions.

Findings

Hysteresis in /s observed in charged black holes.

Hysteretic behavior linked to Van der Waals-like phase transitions.

System evolution depends on path through meta-stable states.

Abstract

Using the rules of the AdS/CFT correspondence, we compute the spherical analogue of the shear viscosity, defined in terms of the retarded Green function for the stress-energy tensor for QFTs dual to five-dimensional charged black holes of general relativity with a negative cosmological constant. We show that the ratio between this quantity and the entropy density, $\tilde\eta/s$, exhibits a temperature-dependent hysteresis. We argue that this hysteretic behaviour can be explained by the Van der Waals-like character of charged black holes, considered as thermodynamical systems. Under the critical charge, hysteresis emerges owing to the presence of two stable states (small and large black holes) connected by a meta-stable region (intermediate black holes). A potential barrier prevents the equilibrium path between the two stable states; the system evolution must occur through the meta-stable region, and a path-dependence of $\tilde\eta/s$ is generated.

Equations

$$T^{ab}=\left(ϵ+P\right)u^a{u}^b+P{g}^{ab}+{\Pi }^{ab},$$

$$T^{ij}=-P{h}_{ij}-\eta {\dot{h}}_{ij}+\eta {\tau }_{\Pi }{\ddot{h}}_{ij}-\frac{\kappa }{2}\left[{\ddot{h}}_{ij}+L^2{\Delta }_L{h}_{ij}\right],$$

$$G_{T^{ij}{T}^{ij}}^R(\omega ,\ell )=-P-i\omega \eta -{\omega }^2\eta {\tau }_{\Pi }-\frac{\kappa }{2}\left({\omega }^2+L^2\gamma \right),$$

$$\tilde{\eta }=-\underset{\omega \to 0}{\lim }\frac{1}{\omega }\mathrm{Im}{G}_{T^{ij}{T}^{ij}}^R(\omega ,\ell \to {\ell }_0),$$

$$d{s}^2$$

$$f(r)$$

$$T(r_{+})=\frac{r_{+}}{\pi L^2}+\frac{1}{2\pi r_{+}}-\frac{2Q^2}{3r_{+}^5}.$$

$$\frac{1}{r^3}\frac{d}{dr}\left[r^{3}f(r)\frac{d\psi (r)}{dr}\right]+\left[\frac{{\omega }^2}{f(r)}-m^2(r)\right]\psi (r)=0,$$

$$\frac{\tilde{\eta }}{s}=\frac{1}{4\pi }{\psi }_0(r_{+}{)}^2.$$

Full text

††journal: Phys. Lett. B

The investigation of black brane configurations with holographic duals has become an important source of information both for the understanding of fundamental features of the gravitational interaction and for the description of strongly coupled QFTs [1, 2, 3, 4, 5]. In this context, the hydrodynamic limit of holographic QFTs plays a very important role, because it allows computing transport coefficients, like the shear viscosity to entropy density ratio $\eta/s$, in the strongly coupled regime of the QFT. This has led to the proposal of a fundamental bound $\eta/s\geqslant 1/4\pi$, known as the KSS bound [1], which found support both from string theory [2] and quark-gluon plasma experimental data [6]. By now, it is well-known that the KSS bound is violated by higher curvature terms in the Einstein-Hilbert action [7] or by breaking of translational or rotational symmetry of the black brane background [8, 9, 10, 11, 12, 13, 14, 15, 16]. Typically, when the KSS bound is violated, $\eta/s$ exhibits a non-trivial dependence on the temperature [17].

Until now, these investigations have been restricted to planar topologies in the bulk (black branes) and have not concerned spherical topologies (black holes). The main obstruction to this generalisation is the absence of the usual hydrodynamic limit for QFTs dual to spherical black holes. Indeed, differently from the black brane case, the spherical geometry of the horizon breaks the translational symmetry in the dual QFT preventing the existence of conserved charges. However, it is still possible to define a relativistic hydrodynamics in curved spacetimes without translational symmetry as an expansion in the derivatives of the hydrodynamic fields of the stress-energy tensor [18] and a related Kubo formula for the shear viscosity.

The hydrodynamic limit of a QFT living on a curved spacetime can be defined in the same way as for a QFT in the plane. We just consider the system at large relaxation times (small frequencies) and large scales compared to the microscopic scale of the system. When the latter is unknown, we can still give a thermal description of the system and associate this microscopic scale with the inverse of the temperature $T$. Thus, the hydrodynamic limit corresponds to consider excitations of the system with wavelength $\lambda\gg 1/T$. In this limit, the macroscopic behaviour of the QFT living in a curved background is described by a stress-energy tensor, which can be written as [18, 19]

[TABLE]

where $\epsilon$ and $P$ are the energy density and the thermodynamical pressure and $u^{a}$ is the fluid velocity, usually considered in the frame in which the fluid is at rest. The tensor $\Pi^{ab}$ contains all the dissipative contributions to the stress-energy tensor. At first order in the velocity expansion, it depends on the transport coefficient $\kappa$, the relaxation time $\tau_{\Pi}$ and the shear viscosity $\eta$.

The previous considerations hold for a QFT in a generic curved space. Working in the AdS/CFT framework, we can apply eq. 1 to a four dimensional CFT dual to a five dimensional AdS bulk spherical black hole [20, 21, 22, 23]. To derive a Kubo formula for CFTs living on the boundary of $\text{AdS}_{5}$, whose spatial section is the three-sphere, we consider small perturbations around the boundary background metric, i.e. $g_{ab}=\bar{g}_{ab}+h_{ab}$. Without loss of generality, we use transverse and traceless perturbations, which in turn, depend on time and the angular directions. Under these assumptions and in linear approximation, eq. 1 becomes

[TABLE]

where $\Delta_{L}$ is the Lichnerowicz operator and $L$ is the $\text{AdS}_{5}$ length. We choose a harmonic time dependence for the perturbation and we expand it in hyperspherical harmonics. We now extract the retarded Green function for the spatial components of the stress-energy tensor $T^{ij}$ in the tensor channel and, from eq. 2, we read

[TABLE]

where $\gamma\equiv\ell(\ell+2)-2$ is the eigenvalue of the Lichnerowicz operator and $\ell=1,2,3,\ldots$ is the first number associated with the hyperspherical harmonic expansion. Eq. 3 allows us to derive a Kubo formula for the analogue of the shear viscosity $\tilde{\eta}$ for a relativistic QFT on a spatial spherical background as

[TABLE]

where $\ell_{0}$ is the smallest eigenvalue of the Lichnerowicz operator and $\omega$ is the frequency of the perturbation. Notice that the only difference of eq. 4 with the planar case is the evaluation of the retarded Green function in $\ell\to\ell_{0}$ instead of wavenumber $k\to 0$.

It is important to stress that, with respect to the planar case, we have an additional contribution to the stress-energy tensor (2). This is rather expected in view of the breaking of translational invariance. However, this $\kappa$-term does not contribute to the shear viscosity.

Let us now compute the spherical analogue viscosity to entropy density ratio $\tilde{\eta}/s$ for the QFT dual to a five-dimensional spherically symmetric charged black hole of general relativity with a negative cosmological constant. Differently from black branes, black holes have a rich thermodynamical phase structure, characterised by different stable or metastable phases (small and large black holes, thermal AdS). As a consequence, one naturally expects the correlators (3) and even more the ratio $\tilde{\eta}/s$ to keep track of this rich phase structure. We find a hysteretic behaviour of $\tilde{\eta}/s$ as a function of the temperature and we explain it in terms of the Van der Waals-like behaviour of this class of black holes when considered as thermodynamical systems [24, 25]. Detailed calculations and generalisation to Gauss-Bonnet neutral and charged black holes are presented in Ref. [26].

The line element of the five-dimensional anti de Sitter-Reissner-Nordström (AdS-RN) black hole is

[TABLE]

where $d\Omega_{3}^{2}$ is the line element of the 3-sphere, $L$ is the AdS length, $M$ is the black hole mass and $Q$ its charge.

The Van der Waals-like liquid/gas phase transition for AdS-RN black holes can be understood by discussing the black hole free energy or by considering the relation between the temperature and the radius of the black hole

[TABLE]

As the charge of the black hole decreases to the critical charge $Q_{c}=L^{2}/6\sqrt{5\pi}$, the black hole undergoes a second-order phase transition. Below $Q_{c}$ the system is characterised by the presence of two stable states (small and large black holes) connected through a meta-stable region of intermediate black holes — see fig. 1. The phase transition small/large black holes is a first-order one [24, 25].

Following the rules of the AdS/CFT correspondence, to calculate $\tilde{\eta}$ for the QFT dual to the five-dimensional AdS-RN black hole we consider transverse and traceless perturbations about the background metric (5), $g_{ab}=g^{(0)}_{ab}+h_{ab}$ with $h_{ab}=0$ unless $(a,b)=(i,j)$ and $h_{ij}(r,t,x)=r^{2}\,\phi(r,t)\,\bar{h}_{ij}(x)$, being $\bar{h}_{ij}$ the eigentensor of the Laplacian operator built on the $3$-sphere. Such perturbations are gauge-invariant and by linearising the Einstein field equations, the angular part decouples [27, 28, 29]. By assuming a harmonic time dependence for the perturbation, $\phi(r,t)=\psi(r)\,e^{-i\omega t}$, one finds the linear second-order differential equation for $\psi(r)$

[TABLE]

where $f(r)$ is given by eq. 6, the mass term for the perturbation is $m^{2}(r)={[4-\ell(\ell+2)]}/{r^{2}}$ and $\ell$ are the eigenvalues of the Laplace operator on the 3-sphere. The presence of a non-vanishing mass term in eq. 8 is a consequence of the breaking of translational symmetry due to the spherical geometry of the horizon. In the black brane context, this term is responsible for the violation of the KSS bound [12] and generates a dependence of the shear viscosity to entropy ratio on the temperature. Eq. 8 admits as solutions a non-normalisable and a normalisable mode that behave asymptotically as $\psi_{0}\approx 1,\,\psi_{1}\approx 1/r^{4}$.

The retarded Green function in eq. 4 can be found using the method proposed in Refs. [30, 12] which gives a very simple and elegant way for computing correlators in a QFT dual to a gravitational bulk theory. The spherical analogue viscosity to density entropy ratio is determined by the non-normalisable mode $\psi_{0}(r)$ evaluated at the horizon

[TABLE]

To determine $\psi_{0}(r_{+})$, we numerically integrate eq. 8 with $\omega=0$ (supplied by regularity boundary conditions at the horizon) outwards from the horizon to infinity and then we use a shooting method requiring that $\psi_{0}(\infty)=1$. Finally, we compute $\tilde{\eta}/s$ as a function of $T$ using eqs. 7 and 9. In fig. 2 we plot our results for $\tilde{\eta}/s$ for selected values of the charge $Q$ and we observe that, for $Q<Q_{c}$, it exhibits a temperature-dependent hysteresis, after that the second-order Van der Waals-like phase transition occurs.

In the hydrodynamic, holographic context, a hysteretic behaviour in the shear viscosity has been already observed for AdS black branes with broken rotational symmetry and with a p-wave holographic superfluid dual [8]. Moreover, it is known that real fluids may exhibit hysteresis in the $\eta$-$T$ plane, this is, for instance, the case of nanofluids [31].

A quite general thermodynamical explanation of hysteresis associated with phase transitions has been given and is related to meta-stabilities [32, 33]. Whenever we have at least two stable states — say $A$ and $B$, respectively on the left of the maximum and on the right of the minimum of fig. 1 — connected by a meta-stable region, a potential barrier prevents the evolution of the system from occurring as an equilibrium path between the two stable states. Evolution must take place through the meta-stable region and the path $A\to B$ goes from the maximum directly to the state $B$, whereas the path $B\to A$ goes directly from the minimum to the state $A$ (see the inset in fig. 1). This is exactly what happens to $\tilde{\eta}/s$ in the case under consideration. The state $A$ corresponds to small black holes and the state $B$ to large black holes, while the meta-stable region (generated by the first-order phase transition) to intermediate black holes. With this general mechanism, hysteresis i.e. path-dependence of $\tilde{\eta}/s$ is generated.

We conclude this letter with some comments about the relation between our spherical analogue viscosity $\tilde{\eta}$ and the usual hydrodynamic viscosity for QFTs in the plane. By definition, the shear viscosity is a transport coefficient that measures the momentum diffusivity due to a strain in a fluid. Its definition is strictly related to the translational symmetries of the system which lead to the conservation of momentum and to an associated conserved current from which one can derive the Fick’s law of diffusion [34]. This is no longer true for systems that break translational invariance where the hydrodynamic interpretation in terms of conserved quantities falls. However, as shown in Ref. [18], hydrodynamics can be defined as an expansion in the derivatives of hydrodynamic fields (like the fluid velocity). This allows one to define the shear viscosity through a Kubo formula also for QFTs on a spatially curved background, where the stress-energy tensor is only covariantly conserved. There is an additional conceptual difficulty in defining the hydrodynamic limit of a QFT on the sphere due to the compactness of the space. In fact, in a compact space, the usual hydrodynamic limit as an effective theory describing the long-wavelength modes of the QFT has not a straightforward interpretation. Our proposal is that for QFTs living on a sphere dual to bulk black holes, the hydrodynamical, long wavelength modes can be described by the $\ell\to\ell_{0}$ modes that probe large angles on the sphere. This is in analogy with the $k\to 0$ modes for a QFT dual to bulk black branes which probe large scales on the plane.

We thank Luciano Colombo, Riccardo Dettori and Piero Olla for valuable discussions and comments.