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CQUeST-2013-0593

TAUP-2963/13

Aging Logarithmic Galilean Field Theories

Seungjoon Hyun1, Jaehoon Jeong2 and Bom Soo Kim3

1Department of Physics, College of Science, Yonsei University, Seoul 120-749, Korea 2Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea

3Raymond and Beverly Sackler School of Physics and Astronomy,

Tel Aviv University, 69978, Tel Aviv, Israel

sjhyun@yonsei.ac.kr, jhjeong@sogang.ac.kr, bskim@post.tau.ac.il

Abstract

We analytically compute correlation and response functions of scalar operators

for the systems with Galilean and corresponding aging symmetries for general spatial

dimensions d and dynamical exponent z, along with their logarithmic and logarithmic

squared extensions, using the gauge/gravity duality. These non-conformal extensions

of the aging geometry are marked by two dimensionful parameters, eigenvalue M of an internal coordinate and aging parameter α.

We further perform systematic investigations on two-time response functions for

general d and z, and identify the growth exponent as a function of the scaling dimensions

∆ of the dual field theory operators and aging parameter α in our theory. The initial

growth exponent is only controlled by ∆, while its late time behavior by α as well

as ∆. These behaviors are separated by a time scale order of the waiting time. We

attempt to make contact our results with some field theoretical growth models, such

as Kim-Kosterlitz model at higher number of spatial dimensions d.

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Contents

1 Introduction 2

2 Logarithmic Galilean Field Theories 3

2.1 ALCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Response functions with log extension . . . . . . . . . . . . . . . . . 6

2.1.3 Response functions with log2 extensions . . . . . . . . . . . . . . . . 8

2.2 Schrödinger backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Correlation and response functions with log & log2 extensions . . . . 11

2.2.2 Conformal Schrödinger backgrounds with z = 2 . . . . . . . . . . . . 12

2.2.3 Schrödinger backgrounds with z = 3/2 . . . . . . . . . . . . . . . . . 13

2.3 Two-time response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Aging Logarithmic Galilean Field Theories 15

3.1 Constructing aging geometry for general z . . . . . . . . . . . . . . . . . . . 16

3.2 Aging in ALCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Aging backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Aging Response functions with log & log2 extensions . . . . . . . . . . . . . 20

4 Connection to KPZ 21

4.1 Response functions for z and d : ALCF . . . . . . . . . . . . . . . . . . . . . 22

4.2 Critical exponents for general z and d : ALCF . . . . . . . . . . . . . . . . . 24

4.2.1 Negative ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Response functions with log extensions : ALCF . . . . . . . . . . . . . . . . 26

4.3.1 With log extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.2 With log2 extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.4 Critical exponents for z = 3 2

and d = 1 : Aging backgrounds . . . . . . . . . 29

5 Conclusion 30

1

1 Introduction

Non-equilibrium growth and aging phenomena are of great interest due to their wide ap-

plications across various scientific fields of study, including many body statistical systems,

condensed matter systems, biological systems and so on [1]-[4]. They are complex physical

systems, and details of microscopic dynamics are widely unknown. Thus it is best to describe

these systems with a small number of variables, their underlying symmetries and correspond-

ing universality classes, which have been focus of nonequilibrium critical phenomena.

One particular interesting class is described by the Kardar-Parisi-Zhang (KPZ) equation

[5][3]. Recently, this class is realized in a clean experimental setup [6][7], and their exponents

for one spatial dimension d = 1 is confirmed: the roughness a = 1 2 , the growth b = 1

3 and the

dynamical z = 3 2

exponents. Along with the experimental developments, there have been also

theoretical developments in the context of aging. The KPZ class reveals also simple aging

in the two-time response functions [8][9]. In these works, it is shown that the autoresponse

function of the class is well described by the logarithmic (log) and logarithmic squared (log2)

extensions of the scaling function with local scale invariance for d = 1.

In a recent paper [10] based on [11][12], we have considered log extensions of the two-time

correlation and response functions of the scalar operators with the conformal Schrödinger

and aging symmetries for the spatial dimension d = 2 and the dynamical exponent z = 2, in

the context of gauge/gravity duality [13][14]. The power-law and log parts are determined

by the scaling dimensions of the dual field theory operators, the eigenvalue of the internal

coordinate and the aging parameter, which are explained below. Interestingly, our two-time

response functions show several qualitatively different behaviors: growth, aging (power-law

decaying) or both behaviors for the entire range of our scaling time, depending on the

parameters in our theory [10].

We further have made connections to the phenomenological field theory model [8] in

detail. The two-time response functions and their log corrections of our holographic model

[10] are completely fixed by a few parameters and are valid for z = 2, d = 2, while the field

theory model [8] has log2 extensions and is valid for z = 3 2 , d = 1. Closing the gap between

these two models from the holographic side is one of the main motivation of this work.

In this paper, we generalize our analysis [10] in two different directions: (1) by applying to

general dynamical exponent z (not conformal) as well as to any number of spatial dimensions

d and (2) by including log2 corrections in two-time response functions. In §2, we first analytically compute the correlation and response functions for general z and d along with

their log and log2 extensions. Then we add the aging generalizations of our non-conformal

results in §3. We try to make contact with KPZ class in §4, and conclude in §5.

2

2 Logarithmic Galilean Field Theories

Logarithmic conformal field theory(LCFT) is a conformal field theory(CFT) which contains

correlation functions with logarithmic divergences.1 They typically appear when two primary

operators with the same conformal dimensions are indecomposable and form a Jordan cell.

The natural candidates for the bulk fields, in the holographic dual descriptions of LCFT, of

the pair of two primary operators forming Jordan cell are given by a pair of fields with the

same spin and a special coupling. After integrating out one of them, it becomes the model

with higher derivative terms.

The AdS dual construction of the LCFT was first considered in [17][18][19] using a higher

derivative scalar field on AdS background. Recently, higher derivative gravity models on AdS

geometry with dual LCFT have got much attention, starting in three dimensional gravity

models [20]-[24]. In four and higher dimensional AdS geometry, the gravity models with

curvature-squared terms typically contain massless and ghostlike massive spin two fields.2

When the couplings of the curvature-squared terms are tuned, the massless and massive

modes become degenerate and turn into the massless and logarithmic modes [28]. This,

so called, critical gravity has the boundary dual LCFT which contains stress-energy tensor

operator and its logarithmic counterpart.

More recently, studies on generalizations of LCFT in the context of AdS/CFT corre-

spondence [13][14], in particular the correlation functions of a pair of scalar operators, have

been made in two different directions. One is on the non-relativistic LCFT. In [29] the dual

LCFT to the scalar field in the Lifshitz background has been investigated. The study on the

dual LCFT to the scalar field in the Schrödinger and Aging background was made in [12].

The other is on the LCFT with log2 divergences. Correlation functions with log2 corrections

have been investigated in several works. In the context of the gravity modes of tricritical

point, they are interpreted as rank-3 logarithmic conformal field theories (LCFT) with log2

boundary conditions [30]. Explicit action for the rank-3 LCFT is considered in [31]. See also

a recent review on these developments in [32].

In this section we would like to accomplish two different things motivated by these de-

velopments along with those explained in the introduction. First, we compute correlation

and response functions for AdS in light-cone (ALCF) and Schro