带电黑洞的范德华行为：QFT中磁滞效应的AdS/CFT解析

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本文主要探讨了在量子场论（Quantum Field Theory, QFT）与引力理论的背景下，特别是在AdS/CFT（Anti-de Sitter/Conformal Field Theory）对应关系的框架下，研究带电黑洞（charged black holes）的性质。焦点在于计算了在五维广义相对论中带有负宇宙常数的带电黑洞所对应的QFT中的剪切粘度（shear viscosity）的球形等效值。剪切粘度是衡量流体抵抗剪切变形的能力，其计算通过QFT的应力能量张量的延迟Green函数来实现。作者们发现了一个重要的物理现象，即粘度与熵密度比（η/s）之间存在温度依赖的磁滞（hysteresis）。磁滞通常在热力学系统中，如范德华气体或某些相变材料中观察到，其中系统在不同的温度路径上表现出不同的响应特性。在这个量子场论系统中，当电荷接近临界值时，小黑洞和大黑洞之间的平衡路径被亚稳态区域（即中间黑洞）形成的势垒所阻断，导致系统状态的变化不再遵循单一路径，而是呈现出路径依赖的η/s值。带电黑洞在这种情况下表现出类似于范德华气体的磁滞行为，这表明它们具有热力学系统的动态特性。这种现象被解释为，由于量子效应和引力相互作用，带电黑洞的热力学行为受到类似于经典范德瓦尔斯方程中分子间相互作用的影响，即在不同温度和电荷条件下，系统的相态变化呈现出非线性和记忆性。这篇研究揭示了带电黑洞在AdS/CFT对偶中的非平凡行为，不仅展示了粘度与熵密度比的动态关系，还提出了一个新的视角去理解黑洞作为热力学系统的行为。这一发现对于理解量子重力理论以及量子信息在极端环境下的物理过程具有重要意义。

Physics Letters B 768 (2017) 393–396

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Van der Waals-like behaviour of charged black holes and hysteresis

in the dual QFTs

Mariano Cadoni, Edgardo Franzin

∗

, Matteo Tuveri

Dipartimento di Fisica, Università di Cagliari & INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy

a r t i c l e i n f o a b s t r a c t

Article history:

Received

10 January 2017

Received

in revised form 22 February 2017

Accepted

27 February 2017

Available

online 4 March 2017

Editor:

M. Cveti

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,

η/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

η/s is generated.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

The investigation of black brane conﬁgurations with holo-

graphic

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 [1–5].

In this context, the hydrodynamic limit of holographic QFTs plays

a very important role, because it allows computing transport coef-

ﬁcients,

like the shear viscosity to entropy density ratio η/s, in the

strongly coupled regime of the QFT. This has led to the proposal

of a fundamental bound η/s ≥ 1/4π, 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 [8–16]. Typically, when

the KSS bound is violated, η/s exhibits a non-trivial dependence

on the temperature [17].

Until

now, these investigations have been restricted to pla-

nar

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

Corresponding author.

E-mail

addresses: mariano.cadoni@ca.infn.it (M. Cadoni),

edgardo.franzin@ca.infn.it (E. Franzin), matteo.tuveri@ca.infn.it (M. Tuveri).

translational symmetry in the dual QFT preventing the existence

of conserved charges. However, it is still possible to deﬁne a rela-

tivistic

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 descrip-

tion

of the system and associate this microscopic scale with the

inverse of the temperature T . Thus, the hydrodynamic limit cor-

responds

to consider excitations of the system with wavelength

λ  1/T . In this limit, the macroscopic behaviour of the QFT liv-

ing

in a curved background is described by a stress-energy tensor,

which can be written as [18,19]

(

 + P

)

+ Pg

+

, (1)

where  and P are the energy density and the thermodynami-

cal

pressure and u

is the ﬂuid velocity, usually considered in the

frame in which the ﬂuid is at rest. The tensor 

contains all the

dissipative contributions to the stress-energy tensor. At ﬁrst order

in the velocity expansion, it depends on the spacetime curvature κ,

the relaxation time τ



and the shear viscosity η.

The

previous considerations hold for a QFT in a generic curved

space. Working in the AdS/CFT framework, we can apply eq. (1) to

http://dx.doi.org/10.1016/j.physletb.2017.02.060

0370-2693/

SCOAP

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