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JHEP07(2017)042

Published for SISSA by Springer

Received: May 25, 2017

Accepted: July 5, 2017

Published: July 10, 2017

Studies of a general ﬂat space/boson star transition

model in a box through a language similar to

holographic superconductors

Yan Peng

School of Mathematical Sciences, Qufu Normal University,

Qufu, Shandong 273165, China

E-mail:

yanpengphy@163.com

Abstract: We study a general ﬂat s pac e/boson star transition model in quasi-local en-

semble through approaches familiar from holographic superconductor theories. We manage

to ﬁnd a parameter ψ

, which is proved to be useful in disc l osi ng properties of phase tran-

sitions. In this work, we expl or e eﬀects of the scalar mass, scalar charge and St¨uckelber g

mechanism on the critical phase transition points and the order of transitions mainly from

behaviors of the parameter ψ

. We mention that properties of transitions in q uasi -l ocal

gravity are strikingly simil ar to those in hol ogr aph i c superconductor models. We also

obtain an analytical relation ψ

∝ (µ − µ

)

1/2

, which also holds for the condensed scalar

operator in the h ol ograph i c insulator/superconductor system in accordance with mean ﬁeld

theories.

Keywords: AdS-CFT Correspondence, Holography and condensed matter physics

(AdS/CMT)

ArXiv ePrint:

1705.08694

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP07(2017)042

JHEP07(2017)042

Contents

1 Introduction

2 Equations of motion and boundary conditions 2

3 Properties of phase transitions in a box 3

3.1 Scalar condensation with various q and m

3.2 Phase transitions with St¨uckelberg mechanism 6

4 Conclusions 7

1 Introduction

The schwarzschi l d black holes in ﬂat space with negative speciﬁc heat cannot be in equi-

librium with the thermal environment since there is inevitable Hawking radiati on. For this

case, York and other authors provided a way to bypass this problem by putting the black

hole in a box to constru ct a quasi-local thermodynam i c ensemble [

1, 2]. In contrast, the

AdS black holes are usually thermodynamical l y stable since in a certain sense the AdS

boundary plays a role of the box condition [

3]. W i t h the interest of the AdS/CFT corre-

spondence [

4–6], the AdS gravity has attracted a lot of attentions, such as there are many

literatures about holographic superconductors constructed in the AdS spacetime [

7]–[31].

In the Einstein-Maxwell theory, it was shown that phase structures of the system

in a box is similar to those of the AdS gr avity [

32–34]. Then, it is very interesting to

include a scalar ﬁeld to further compare the similarity between transitions in a box and

those of the holographi c superconductor system. Recently, P. Basu, C. Krishnan and

P.N.B. Subramanian init i at ed a thermodynamic study of such systems constructed by a

charged scalar ﬁeld coupled t o electromagnetic ﬁeld in asy mpt ot i c ﬂat space with box

boundary conditions [

35]. Besides the ﬂat space and normal RN black hole, it also admits

hairy solutions of boson star an d hairy RN black hole in this model. These two types

of hairy solutions are thermodynamically stable for certain region of parameters. Firstly,

it provides a way to evade the ﬂat space no-hai r theorems. The more interesting result

is that the overall phase structure of the system in a box is strikingly similar to that

of holographic sup e r con duc tor s in AdS gravity background. For example, there is ﬂat

space/black hole transitions corresponding to the classical Hawking-Page transition in AdS

gravity [

3]. And this model also admits black hole/hairy black hole transitions similar

to those in holographi c theori es . In addition, there are ﬂat space/boson star transiti ons

corresponding to holographic i ns ul at or/s uperconductor and global AdS/boson star systems

in AdS gravity [

36, 37]. As a further step along this line, it will be very interesting to

study phase transitions in a box with condensation diagrams from asymptotic behaviors

– 1 –

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