Nearly Critical Holographic Superuids Aristomenis Donos and Polydoros Kailidis Centre for Particle Theory and Department of Mathematical Sciences

2025-04-24 0 0 756.8KB 39 页 10玖币

侵权投诉

Nearly Critical Holographic Superﬂuids

Aristomenis Donos and Polydoros Kailidis

Centre for Particle Theory and Department of Mathematical Sciences,

Durham University, Durham, DH1 3LE, U.K.

Abstract

We study the nearly critical behaviour of holographic superﬂuids at ﬁnite

temperature and chemical potential in their probe limit. This allows us to

examine the coupled dynamics of the full complex order parameter with

the charge density of the system. We derive an eﬀective theory for the

long wavelength limit of the gapless and pseudo-gapped modes by using

analytic techniques in the bulk. We match our construction with Model F

in the classiﬁcation of Hohenberg and Halperin and compute the complex

dissipative kinetic transport coeﬃcient in terms of thermodynamics and

black hole horizon data. We carry out an analysis of the corresponding

modes and argue that at ﬁnite density the dispersion relations are dis-

continuous between the normal and the broken phase. We compare and

contrast our results with earlier numerical work.

arXiv:2210.06513v3 [hep-th] 5 Dec 2022

1 Introduction

The holographic duality provides a laboratory to analyse the behaviour of large classes

of strongly coupled systems [1, 2]. In a certain large Nlimit, large classes of ﬁeld

theories become dual to classical theories of gravity. Using holography as a tool kit

is particularly helpful in dealing with real time physics when ﬁnite temperature and

chemical potential are involved [3]. More generally, holography is a powerful tool to

study ﬁeld theories deformed by relevant deformations.

The geometries dual to the ﬁeld theory thermal states are black holes of Hawking

temperature equal to the ﬁeld theory temperature. In the most well understood case

of conformal ﬁeld theories, the bulk geometries asymptote to Anti de-Sitter space

(AdS). The chemical potentials for the charges of global symmetries are ﬁxed by

the asymptotic behaviour of the gauge ﬁelds dual to the corresponding ﬁeld theory

Noether current operators. Likewise, the deformation parameters of other irrelevant

operators are ﬁxed by the boundary conditions of their bulk duals at the time-like

conformal boundary of AdS.

In this paper we will be particularly interested in the intersection two areas that

holography has already seen many applications. The ﬁrst one is the study of thermal

phase transitions and symmetry breaking. One of the most prominent examples is

the superﬂuid phase transition which was pioneered in [4, 5]. With applications in

condensed matter physics in mind, examples where spacetime symmetries are broken

were also realised holographically [6–9].

From the point of view of the bulk theory, continuous phase transitions are driven

by perturbative instabilities which can lead to spontaneous symmetry breaking in the

stable phase. In such a case, a new gapless mode appears in the theory, the dual of

the Goldstone mode. At the same time, the mode which drives the transition acquires

a small gap which closes to zero when approaching the critical point. This gapless

collective degree of freedom is precisely the Higgs/amplitude mode.

The second area of applied holography that we will be interested in this paper is

the eﬀective theory governing the dynamics of low energy modes close to the critical

point and incorporate the almost gapless Higgs mode. In the language of superﬂuids,

the usual description of hydrodynamics away from the critical point captures the

long wavelength behaviour of the phase of the order parameter [10–13]. Our aim is

to enlarge the eﬀective theory to include ﬂuctuations of its modulus.

Papers with similar questions have appeared in the past. However, they either

involved models which can be solved exactly close to the critical point [14, 15], or

numerical techniques [16–19]. We chose to employ analytic techniques as we want to

understand the universality of the underlying physics from a boundary theory point

of view. The main tool in our construction will be the techniques we have recently

developed in [20–23] to analyse dissipative eﬀects in holographic theories. These will

let us identify an equation of motion for the amplitude of the order parameter, a

constitutive relation for the conserved electric current of the theory and a Josephson

relation for a local chemical potential we will identify. In combination with the Ward

identity for the global U(1) of the theory, these will constitute a closed system of

equations.

Later, we compare the resulting equations with those resulting from the Model

F in the classiﬁcation of Hohenberg and Halperin [24] and ﬁnd exact agreement

after a certain identiﬁcation of the parameters in their model with our holographic

results. As part of the matching procedure, we produce a holographic formula for the

complex kinetic coeﬃcient Γ0in terms of black hole horizon data and thermodynamic

quantities of the state. It would be interesting to compare model F to holography

beyond linear response.1.

Using our eﬀective theory equations, we analyse the behaviour of quasinormal

modes in the broken phase. By incorporating the dynamics of the almost gapless

mode, we are able to commute the limits of zero gap and inﬁnite wavelength for the

ﬂuctuations. Naively, one might expect that when holding the wavelength ﬁxed, in

the limit of small gap the modes should match with the ones of the normal phase.

However, we show that this is true only at zero background chemical potential. More-

over, we analyse interesting pole collisions in the complex plane that happen in the

crossover region.

As we will see, the discontinuity mentioned above is related to the fact that the

modes of oscillation of the order parameter are diﬀerent between the normal and the

broken phase. In the normal phase we have two copies of the same mode coming

from its real and imaginary parts. However, even though the background of the

broken phase is continuously connected to that of the normal phase, the mode for

the ﬂuctuations of the order parameter involve its phase in a singular manner close

to the critical point. This was already evidenced from the analysis of [22], at inﬁnite

wavelengths. Interestingly, we ﬁnd that the mode for responsible charge diﬀusion is

continuous.

Finally, we carry out a few numerical checks in order to verify some of our ana-

lytic results. In particular, we focus on reproducing the dispersion relations for the

quasinormal modes that our theory predicts. The model that we chose to apply our

analysis to has been studied before in [16] and we chose to use exactly the same set of

1See e.g. [25] for some recent numerical work in a direction along those lines.

parameters that was used there. Both our analytical and numerical results indicate

that the original suggestion of [16] regarding the ”diﬀusion” constant of the pseudo-

diﬀusive mode is not accurate for small wavevectors. Interestingly, it only holds true

for wavevectors of norm much larger than the gap and below any other UV scale of

the theory.

Our paper is organised in six main sections. In section 2 we present our holo-

graphic setup along with the necessary thermodynamics of the bulk geometries. In

section 3 we employ our holographic techniques to extract all the necessary ingre-

dients for our eﬀective theory. In section 4 we state our theory in two equivalent

ways and we write the constitutive relations of the current in terms of our hydrody-

namic variables. In a separate subsection, we carry out the comparison with Model

F of [24]. In section 5 we examine the behaviour of the quasinormal modes of our

system in various limits and point out at its discontinuities. Section 6 is devoted to

our numerical checks. We conclude with some discussion and conclusions in section

2 Setup

Our bulk theory will have to contain a complex scalar ψwhich is dual to the operator

Oψwhose VEV will play the role of the order parameter in our system. The global

U(1) under which the boundary operator Oψtransforms, corresponds to a local sym-

metry in the bulk gauged by the one-form Aµ. Moreover, we will include a relevant

operator Oφwhich will introduce an additional deformation parameter φs. As we will

see, the phase transition we wish to study will be driven by either tuning the chemical

potential µor the deformation parameter φ(s). Our results will be valid both at ﬁnite

and at zero charge density.

For our purposes, it is suﬃcient to consider the bulk action,

Sbulk =Zd4x√−g−τ

4Fµν Fµν −1

2Dµψ Dµψ∗−1

2∇µφ∇µφ−V(2.1)

where τand Vare in general functions of φ,|ψ|2. The covariant derivative acts on

the complex scalar according to Dψ =∇ψ+iqeA ψ and the ﬁeld strength of the

gauge ﬁeld is simply F=dA. For small values of our scalar ﬁelds, we will assume

that the functions τand Vbehave according to,

V≈1

2m2

ψ|ψ|2+1

2m2

φφ2+··· ,

τ≈1 + cψ|ψ2|+cφφ+··· .(2.2)

For the bulk geometry dual to the thermal state, we will consider a general metric

which preserves the Euclidean subgroup and time translations. Without any loss of

generality, this is captured by the general metric,

ds2=−U(r)dt2+dr2

U(r)+e2g(r)(dx2+dy2).(2.3)

One can arrive to this background in a variety of ways and the details will not be

important to our analysis. As we will see, what matters is the general properties of

the background geometry (2.3).

The conformal boundary is at r→ ∞ and we can use the coordinate invariance

of the background theory to ﬁx the horizon r= 0. In the asymptotic region, the

functions that appear in our metric can be taken to approach,

U(r) = (r+R)2+··· , g(r) = ln (r+R) + ··· .(2.4)

We will set the Hawking temperature of the horizon to be T, ﬁxing the near horizon

Taylor expansion,

U(r) = 4πT r +··· , g(r) = g(0) +··· .(2.5)

Since we will be primarily interested in the broken phase of our probe theory,

the complex scalar ψwill be non-trivial in the bulk geometry. In this case, the ﬁeld

redeﬁnitions ψ=ρ eiqeθand Bµ=∂µθ+Aµare legitimate. This, brings our bulk

action (2.1) to the form,

S=Zd4x√−g−τ

4F2−1

2∇µρ∇µρ−1

2∇µφ∇µφ−1

2q2

eρ2B2−V,(2.6)

where the ﬁeld strength now reads F=dB. The resulting equations of motions from

a variation of the action (2.6) are given by,

∇µ∇µρ−∂ρV−q2ρB2−1

4∂ρτF 2= 0 ,

∇µ∇µφ−∂φV−1

4∂φτF 2= 0 ,

∇µ(τF µν )−q2

eρ2Bν= 0 .(2.7)

In our construction we will consider background solutions of these equations that

correspond to deforming the theory by a chemical potential µand scalar deformation

parameter φs. To achieve this, we will consider backgrounds with

ρ=ρ(r), φ =φ(r), B =Bt(r)dt . (2.8)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

NearlyCriticalHolographicSuperuidsAristomenisDonosandPolydorosKailidisCentreforParticleTheoryandDepartmentofMathematicalSciences,DurhamUniversity,Durham,DH13LE,U.K.AbstractWestudythenearlycriticalbehaviourofholographicsuperuidsatnitetemperatureandchemicalpotentialintheirprobelimit.Thisallowsustoexa...

展开>> 收起<<

Nearly Critical Holographic Superuids Aristomenis Donos and Polydoros Kailidis Centre for Particle Theory and Department of Mathematical Sciences.pdf

共39页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Nearly Critical Holographic Superuids Aristomenis Donos and Polydoros Kailidis Centre for Particle Theory and Department of Mathematical Sciences

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: