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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 947 (2019) 114728

www.elsevier.com/locate/nuclphysb

Holographic derivation of a class of short range

correlation functions

Hai Lin

, Haoxin Wang

a,b

Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, PR China

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China

Received 5

April 2019; received in revised form 15 July 2019; accepted 11 August 2019

Available

online 16 August 2019

Editor: Stephan

Stieberger

Abstract

construct a class of backgrounds with a warp factor and anti-de Sitter asymptotics, which are dual to

boundary systems that have a ground state with a short-range two-point correlation function. The solutions

of probe scalar ﬁelds on these backgrounds are obtained by means of conﬂuent hypergeometric functions.

The explicit analytical expressions of a class of short-range correlation functions on the boundary and the

correlation lengths ξ are derived from gravity computation. The two-point function calculated from gravity

side is explicitly shown to exponentially decay with respect to separation in the infrared. Such feature

inevitably appears in conﬁning gauge theories and certain strongly correlated condensed matter systems.

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

1. Introduction

The gauge/gravity correspondence [1–3] has given an extraordinary method to study a quan-

tum system by a higher dimensional gravity, which relates a theory with gravity to a quantum

system without gravity in a non-trivial way. This duality indicates the emergence of bulk space-

time geometry from the degrees of freedom li

ving in the boundary [4–7]. It further provides

us a way to compute interesting quantitative features of strongly-coupled quantum systems and

E-mail address: hailin@mail.tsinghua.edu.cn (Hai Lin).

https://doi.org/10.1016/j.nuclphysb.2019.114728

The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

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

2 H. Lin, H. Wang / Nuclear Physics B 947 (2019) 114728

non-perturbative effects of quantum ﬁeld theories, since it allows us to make predictions of ob-

servables pertaining to the boundary system, by working in the gravity side.

The g

auge/gravity correspondence enables us to compute correlation functions of a boundary

conformal ﬁeld theory by working in the gravity, and the details of this procedure were reviewed

in [8]. Most of the gravity computations in the literature are for long-range correlation functions

in the conformal ﬁeld theory. On the other hand, short-range correlation functions are also ve

interesting and important in both quantum ﬁeld theories and condensed matter systems. We focus

on a class of short-range correlation functions in a D-dimensional system and derive them from

gravity computation, using a new class of gravity backgrounds that we construct. The appearance

of a holographic direction for a boundary system is also closely related to the renormalization

group ﬂo

w of the boundary system, e.g. [9,10]. Our gravity ansatz is similar to the ansatz used

in the aforementioned holographic renormalization group.

We w

ant to construct a class of new backgrounds that enable us to compute short-range

exponential decay two-point functions around the ground state, with a correlation length. The

short-range correlation is distinguished from the long-range correlation. The short-range corre-

lation means that the correlation length is much smaller than the overall size of the boundary

system. If the ove

rall size of the boundary system is inﬁnite, a ﬁnite correlation length will lead

to a short-range correlation.

These short-range correlation functions are similar to those that occur in the v

acua of massive

quantum ﬁeld theories. On the other hand, vacua of massive quantum ﬁeld theories have gravity

dual descriptions, e.g. [11–16]. We compute short-range-correlated two-point correlation func-

tions from the gravity side with the backgrounds in our paper

, and our results are also potentially

related to conﬁning gauge theories.

The g

auge/gravity correspondence is very useful for studying condensed matter systems, since

it can give descriptions in strong coupling regimes, e.g. [17,18,4]. The feature that the boundary

systems in our case have a short-range correlation function near the ground state, is also sim-

ilar to that of many strongly correlated condensed matter systems. These aspects are also ve

interesting for investigations.

The or

ganization of this paper is as follows. In Section 2, we construct a class of backgrounds

with a warp factor and anti-de Sitter asymptotics, which we will show that the dual boundary sys-

tems have a short-range two-point correlation function around the ground state. In Section 3, we

solve basis solutions of a probe scalar ﬁeld on these curv

ed backgrounds, by means of conﬂuent

hypergeometric functions. Then in Section 4, we compute the bulk-to-bulk propagator and the

boundary-to-bulk propagator in these backgrounds. Afterward in Section 5, we derive from the

gravity side the short-range correlation function of the boundary system and the corresponding

correlation length. In Section 6, we analyze the implication of our gra

vity computation to the

short-range correlation and the correlation length. Finally, we discuss our results and draw some

conclusions in Section 7. In Appendix A, we include details of our derivation of the background

solutions in matter coupled gra

vity. In Appendix B, we describe relations between our ansatz

and that used in the holographic renormalization. In Appendices C and D, we include detailed

derivations for the basis solutions and the propagators, respectively.

2. A class of geometries for short range correlation

We desire a short-range correlation function evaluated around the ground state with a correla-

tion length ξ for a D-dimensional spacetime. This D-dimensional spacetime can be constructed

as an asymptotic boundary of a gravity system in higher dimensions. Let us consider that this

H. Lin, H. Wang / Nuclear Physics B 947 (2019) 114728 3

gravity system is on the spacetime that we denote as M. The gravity system contains a holo-

graphic direction which we denote by z here. Using a warp factor a

(z), the metric ansatz of M

can have the form

(z)(η

μν

+dz

), (2.1)

where μ = 0, ···, D − 1, and z>0is the holographic radial direction and x is denoted as the

spacetime position vector on the D-dimensional boundary ∂M. This metric ansatz (2.1)is used

extensively in the holographic analysis of the renormalization group ﬂow of boundary systems,

for a review see e.g. [8].

We w

ant to construct a class of new backgrounds of the form (2.1), that are dual to a quantum

theory on the boundary with exponential decay two-point function around the ground state, with

leading behavior

O(x)O(x



)∼e

−|x−x



|/ξ

(2.2)

when the separation |x − x



| is much larger than the correlation length ξ . The short-range cor-

relation is distinguished from the long-range correlation. Consider that the boundary system has

an overall size of the system l

sys

. The short-range correlation means that ξ  l

sys

. Hence if the

overall size of the boundary system is inﬁnite, a ﬁnite correlation length ξ will lead to a short-

range correlation. In a many-body condensed matter system, l

sys

is of order the ﬁnite size of

the material. In this paper, we focus on a class of short-range correlations and derive them from

gravity computation.

Since the ener

gy scale of the boundary system is related to the inverse of the radial direction z,

there must exists a special radius scale z = z

in the gravity dual characterizing the energy scale

−1

in the infrared of the boundary system. On general grounds, we expect that ξ is a function

ξ(z

) of z

, and ξ may also depend on other parameters.

Here we still wo

rk on the asymptotically AdS background, where the asymptotic boundary is

at z =0. We consider that a

(z) can be expanded in powers of z/L, namely,

(z) =

η(z

+γ(z

) +O(

), (2.3)

where η(z

) and γ(z

) are functions of z

, abbreviated as η and γ in this paper. In the later

sections, we will show that above warp factor is a nice choice we desire.

The a

bove metric with the warp factors (2.3) can be solved in matter coupled gravity. The

details of our derivation are in Appendix A. As an example, they can be obtained in scaler coupled

gravity with the action

S =

2κ



D+1

√

−g



R −

∂

ϕ∂

ϕ −V(ϕ)



, (2.4)

where κ is the gravitational coupling constant, and M =0, ···, D. The proﬁle of the scalar ﬁeld

ϕ deforms the AdS background.

For

(z) =

ηL

+γ +O(

), (2.5)

to the ﬁrst three orders in z, the scalar and its potential are

4 H. Lin, H. Wang / Nuclear Physics B 947 (2019) 114728

V =−

D(D −1)

+(D −1)(2D −1)

ηz

+(D −1)(12Dγ − 12γ −13Dη

+11η

)

, (2.6)

ϕ =ϕ

∓



(D −1)z

−2ηL



24η +(12γ −7η

)



. (2.7)

Here we require η<0. The meaning of ϕ

is that it is the value of ϕ at z = 0. Note that,

Eq. (2.6)–(2.7 )gives a parametric form of V(ϕ), where V is a function of ϕ, written in a para-

metric representation.

We also ﬁnd an e

xact solution, with the warp factor

a(z) = L



−

z +2z



, (2.8)

with z ≥0 and 2z

> 0, and the corresponding scalar and its potential are

V(ϕ)=−

(D −1)



(2D −1)e

ϕ−ϕ

√

D−1

+(2D −1)e

−ϕ

√

D−1

+2(2D +1)



, (2.9)

ϕ =ϕ

±4

√

D −1log





+1 +





. (2.10)

The solution (2.8)–(2.10)is an exact solution. If expanded, it is a special case of the solution

(2.5)–(2.7)for η(z

) =−Lz

−1

, γ(z

) =

−2

In the above analysis, the case for maximally symmetric AdS geometry is V =−

D(D−1)

and

ϕ =ϕ

, corresponding to η = 0 and γ =0.

Our metrics may be relevant for holographic normalization schemes, e.g. [9,10] and the re-

lations between our ansatz and that used in the holographic renormalization are described in

Appendix B.

3. The basis solutions

In this section, we consider a probe scalar ﬁeld φ on these curved backgrounds (2.3), (2.8)

whose boundary value is regarded as the source coupling to O(x) which has a short-range corre-

lation function as (2.2)in the large separation |x − x



|. We do not consider the back-reaction of

the probe scalar ﬁeld φ to the backgrounds. The action of φ in the curved background (2.3) reads

S =−



D+1



|g|[g

∂

φ∂

φ + m

], (3.1)

where y parametrizes the coordinates

(

z, x

)

in D + 1 dimensions. The equation of motion

derived from the action (3.1)is

√

|g|

∂

(



|g|g

∂

φ) − m

φ = 0 . (3.2)

More explicitly, after substituting the metric (2.1), it becomes

−2



−∂

−(D −1)

(

ln a

)



∂

− ∂

∂



φ(z,x

) =0. (3.3)

One may perform the Fourier transform of φ in the x

coordinates

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