Properties of gapped systems in AdSBCFT Yan Liuab1 Hong-Da Lyuab2and Jun-Kun Zhaoca3 aCenter for Gravitational Physics Department of Space Science

2025-05-02 0 0 886.84KB 36 页 10玖币

侵权投诉

Properties of gapped systems in AdS/BCFT

Yan Liua,b1, Hong-Da Lyua,b2and Jun-Kun Zhaoc,a3

aCenter for Gravitational Physics, Department of Space Science

and International Research Institute of Multidisciplinary Science,

Beihang University, Beijing 100191, China

bPeng Huanwu Collaborative Center for Research and Education,

Beihang University, Beijing 100191, China

cCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing 100190, China

Abstract

We study the conductivities and entanglement structures of two diﬀerent holo-

graphic gapped systems at zero density in the presence of boundaries within AdS/BCFT.

The ﬁrst gapped system is described by the Einstein-scalar gravity and the second

one is the dual of AdS soliton geometry. We show that in both these two systems

the bulk and boundary conductivities along the spatial direction of the boundary of

BCFT are trivial. For the ﬁrst system, when we increase the size of the subsystem

the renormalized entanglement entropy is always non-negative and monotonically

decreasing with discontinuous, or continuous, or smooth behavior, depending on

the eﬀective tension of the brane. While for the AdS soliton with a boundary, the

renormalized entanglement entropy only exhibits a discontinuous drop when we

increase the size of the subsystem.

1Email: yanliu@buaa.edu.cn

2Email: hongdalyu@buaa.edu.cn

3Email: junkunzhao@itp.ac.cn

arXiv:2210.02802v2 [hep-th] 24 Mar 2023

Contents

1 Introduction 1

2 A gapped system in AdS4/BCFT33

2.1 Zero temperature ground state . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Conductivity ................................. 9

2.3 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 AdS Soliton in AdS/BCFT 23

3.1 Conductivity ................................. 24

3.2 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Conclusion and discussion 27

A Analysis of conductivity in terms of Schrodinger equation 29

A.1 Gapped geometry in Sec. 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 30

A.2 The AdS soliton geometry in Sec. 3.1 . . . . . . . . . . . . . . . . . . . . 30

B Derivation of (2.48) 31

1 Introduction

The AdS/CFT correspondence, also known as holographic duality, provides a novel way

to study strongly correlated quantum systems in terms of weakly coupled gravity. In

particular, it can describe strongly correlated gapped systems in terms of gravity duals.

One class of such models at zero density include the Girardello-Petrini-Porrati-Zaﬀaroni

gapped geometry [1], AdS soliton [2], AdS with cutoﬀs in IR [3] and so on. Another class

of models are for ﬁnite density systems with translational symmetry-breaking eﬀects; see,

e.g., [4].

Physical systems in the real world often have boundaries, and the boundary eﬀects

play important roles, ranging from D-branes in string theory to topological states in con-

densed matter physics. One well-known example of the topological states in a condensed

matter system is the topological insulator, which is gapped in the bulk while nontrivial

gapless charged excitations exist on the boundary [5]. Constructing a holographic model

of topological insulators is a diﬃcult question for the bottom-up holography.4Because of

the diﬃculty and as a preliminary step with the hope that we could obtain some impor-

tant hints toward constructing a model of topological insulators, we start from a simpler

while nontrivial question to analyze what happens to a holographic gapped system in the

presence of a boundary. In condensed matter physics, both the bulk of the topological

insulator and normal insulator have a hard gap in the band structure, while they have

diﬀerent boundary states. The holographic gapped system without a boundary shows

a gap. Though there has been no evidence showing that it is topologically nontrivial,

as the Hilbert space has fundamentally changed compared to the weakly coupled ﬁeld

theory, we still need to check the boundary states to see if it is topologically trivial or

nontrivial. Therefore, the question on the properties for such a system in the presence of

a boundary is natural and important.

We study this problem in the framework of AdS/BCFT. In AdS/CMT there are few

studies on the eﬀects of “soft” boundaries by considering matter ﬁelds with spatially

dependent proﬁles which separate two diﬀerent phases; see, e.g., [8,9]. Here, we use

AdS/BCFT to describe a “hard” boundary of the holographic system which might make

the model be more realistic. AdS/BCFT allows us to study the properties of ﬁeld the-

ories with boundaries from the holographic dual. In AdS/BCFT, the bulk geometry

terminates at the end-of-the-world (EOW) brane such that the boundary of the EOW

brane near AdS coincides with the boundary of BCFT [10–12]. AdS/BCFT has been

actively explored during the past decade. A far from complete list includes applications

to condensed matter physics [13,14], cosmology [15], black hole physics [16–18], quantum

information [19], and so on. However, so far, the studies of AdS/BCFT have been mainly

limited to critical gapless systems with boundaries. Studies of gapped systems in such a

framework might provide more insights on the properties of strongly interacting quantum

ﬁeld theories with boundaries.

The purpose of this paper is to study the properties of gapped systems in the presence

of boundaries in the framework of AdS/BCFT. We will focus on the vacuum states of

the ﬁrst class of models as mentioned in the ﬁrst paragraph at zero temperature and

zero density. Here, we consider two diﬀerent holographic models of gapped systems.

The ﬁrst one is the gapped geometry in Einstein-scalar theory. We choose the Neumann

boundary condition for the ﬁelds on the EOW brane. Taking a proper scalar potential

term localized on the brane, we can get a consistent background for the gapped geometry

with an EOW brane. Then we will study the transport properties and entanglement

entropies of the BCFT. The second gapped system is described by the AdS soliton [2].

The AdS soliton can be obtained by analytic continuation of the AdS Schwarzschild

4Previous attempts to study the holographic model of topological insulator from the top-down ap-

proach in the probe approximation (via probe branes) include, e.g., [6,7].

black hole. At ﬁnite temperature, there is a ﬁrst-order phase transition between the

AdS Schwarzschild black hole and the AdS soliton, which describes the conﬁnement-

deconﬁnement phase transition. There is a compact spatial dimension in the AdS soliton

which sets the scale of the transition. We consider the presence of a boundary for the

dual ﬁeld theory of the AdS soliton along one noncompact spatial direction and study

its transport properties and entanglement entropies. We will make comparisons on the

proﬁles and the properties between these two diﬀerent gapped systems in the presence of

boundaries.

Our paper is organized as follows. In section 2, we ﬁrst construct a gapped system in

the presence of a boundary in Einstein-scalar theory using AdS/BCFT, and then study

its conductivity along the spatial direction of the boundary as well as its entanglement

entropy. In section 3, we study the properties of a gapped system which is described by

the AdS soliton in AdS/BCFT. We summarize our results in section 4and discuss the

possible open questions. Some calculation details are collected in the appendices.

2 A gapped system in AdS4/BCFT3

In this section, we study the holographic gapped system with boundaries in the Einstein-

scalar gravity and consider its properties in the framework of AdS/BCFT [10,11]. We

focus on the case of three-dimensional ﬁeld theories with two-dimensional boundaries and

it is straightforward to generalized to other dimensions.

The conﬁguration under consideration is shown in Fig. 1. The three-dimensional

boundary ﬁeld theory is deﬁned on the manifold Mwith boundary Palong the ydirection.

The gravity dual lives in the bulk Nwith the EOW brane Qwhich anchors to the BCFT

boundary P. Note that uis the holographic direction and the boundary Mlives at u= 0.

We consider the Einstein-scalar gravitational theory

Sbulk =SN+SQ,(2.1)

with

SN=ZN

d4x√−g1

2κ2R+ 6 −1

2(∂φ)2−V(φ)−Z(φ)

4e2F2,

SQ=ZQ

d3x√−γ1

κ2(K−T+v(φ),

(2.2)

where the bulk gauge ﬁeld Aais dual to the electric current on the boundary and it has

ﬁeld strength Fab =∂aAb−∂bAa.κand eare the gravitational constant and the bulk

gauge coupling constant, respectively. Note that the scalar ﬁeld φis real. The induced

P M

{t, y}

Figure 1: The conﬁguration under consideration. The ﬁeld theory lives in the manifold M

with boundary P. The dual gravity lives in the bulk Nwith boundary Q.

metric on the EOW brane Qis denoted as γµν , where Kand Tare the extrinsic curvature

and the tension of the EOW brane Q, respectively. Note that on Qwe also consider a

potential term v(φ), and it contributes to the eﬀective tension of the brane.

We set 2κ2=e2= 1. The equations of motion in the bulk Nare

Rab −1

2gabR+ 6−1

2Tab = 0 ,

∇bZ(φ)Fba= 0 ,

∇a∇aφ−∂φZ(φ)

4F2−∂φV(φ) = 0 ,

(2.3)

where

Tab =Z(φ)FacFc

b−1

4gabF2+∇aφ∇bφ−gab1

2(∂φ)2+V(φ).

The equations of motion on the EOW brane Qcan be obtained from the variations.

The variation for metric ﬁelds, scalar ﬁeld, and vector ﬁeld yields

δSQ=ZQ

d3x√−γ1

2κ2Kµν −(K−T)γµν −1

2v(φ)γµν δγµν

+ZQ

d3x√−γ[−na∇aφ+v0(φ)] δφ

+ZQ

d3x√−γ na−Z(φ)

e2FabδAb.

(2.4)

Note that nais the outward unit vector of Q. Here, γµν should be understood as the

metric from the Gaussian normal coordinate on the EOW brane. Following the stan-

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PropertiesofgappedsystemsinAdS/BCFTYanLiua;b1,Hong-DaLyua;b2andJun-KunZhaoc;a3aCenterforGravitationalPhysics,DepartmentofSpaceScienceandInternationalResearchInstituteofMultidisciplinaryScience,BeihangUniversity,Beijing100191,ChinabPengHuanwuCollaborativeCenterforResearchandEducation,BeihangUniversit...

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Properties of gapped systems in AdSBCFT Yan Liuab1 Hong-Da Lyuab2and Jun-Kun Zhaoca3 aCenter for Gravitational Physics Department of Space Science

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