Coherence effects in disordered geometries with a field-theory dual
Accepted: March
Coherence e ects in disordered geometries with a eld-theory dual
Keble Road 0 1 3 6
Oxford OX 0 1 3 6
Shanghai Jiao Tong University 0 1 3 6
Tomas Andrade 0 1 3 5 6
Antonio M. Garc a-Garc a 0 1 3 4 6
Bruno Loureiro 0 1 2 3 6
0 Shanghai 200240 , China
1 Mart i Franques 1 , E-08028 Barcelona , Spain
2 TCM Group, Cavendish Laboratory, University of Cambridge
3 Universitat de Barcelona
4 Shanghai Center for Complex Physics, Department of Physics and Astronomy
5 Rudolf Peierls Centre for Theoretical Physics, University of Oxford
6 JJ Thomson Avenue , Cambridge, CB3 0HE , U.K
We investigate the holographic dual of a probe scalar in an asymptotically Anti-de-Sitter (AdS) disordered background which is an exact solution of Einstein's equations in three bulk dimensions. Unlike other approaches to model disorder in holography, we are able to explore quantum wave-like interference e ects between an oscillating or random source and the geometry. In the weak-disorder limit, we compute analytically and numerically the one-point correlation function of the dual eld theory for di erent choices of sources and backgrounds. The most interesting feature is the suppression of the one-point function in the presence of an oscillating source and weak random background. We have also computed analytically and numerically the two-point function in the weak disorder limit. We have found that, in general, the perturbative contribution induces an additional power-law decay whose exponent depends on the distribution of disorder. For certain choices of the gravity background, this contribution becomes dominant for large separations which indicates breaking of perturbation theory and the possible existence of a phase transition induced by disorder.
AdS-CFT Correspondence; Duality in Gauge Field Theories; Holography and
Geometry Scalar eld and equations of motion
Perturbative analytical calculation of one-point and two-point scalar
cor
1 Introduction
Setup
2.1
2.2
relation functions
3.1
3.2
Zeroth order
Second order
3.2.1
3.2.2
3.2.3
3.2.4
2
3
4
5
6
Constant geometry
Oscillating geometries
Disordered geometries
Comments on other masses and correlated disorder
Numerical analysis
4.1
4.2
One-point correlation function
Two-point correlation function
Comparison with previous results in the literature
Conclusions and outlook
A Notes on random elds
A.1 Implementation
A.2 Cuto s
A.3 Discrete
B
Holographic renormalisation C Boundary-to-bulk propagator and boundary two-point function 1 4
description of disorder, which is ubiquitous in realistic systems and directly responsible for
a broad variety of phenomena ranging from momentum relaxation to quantum interference
leading to di erent forms of localization [1{5]. The introduction of disorder in gravity
backgrounds with a negative cosmological constant relevant for holography requires the
solution of spatially inhomogeneous Einstein's equations, in general a di cult task.
Di erent approximation schemes have been proposed to make the problem technically
tractable while keeping some of the expected phenomenology related to the introduction
of disorder. For instance, momentum relaxation, a rather general consequence of any form
of disorder, can be achieved by adding a massless scalar [6{9] that depends linearly on the
boundary coordinates. Since the scalar eld only couples to gravity through its derivatives,
translation invariance is broken by the background but the equations of motion are still
independent of the spatial coordinates which facilitates substantially the calculation, in
many cases analytical, of transport properties.
As was expected, the electrical conductivity is always nite and depends directly on
the strength of the translational symmetry breaking characterized by the slope, with
respect to the spatial coordinates, of the scalar eld in the boundary. For weak momentum
relaxation, the electrical conductivity reproduces the expected phenomenology of Drude's
model, which includes a peak at low frequencies, remnant of the broken translational
symmetry, followed by a decay for higher frequencies. For stronger relaxation the Drude peak
is suppressed, leading to an incoherent 'bad-metal' behaviour [10{13]. However even in
the limit of in nite relaxation no insulating behaviour is observed. Recently, models that
consider the coupling of the scalar eld to the gravity and the Maxwell term managed
to reproduce a vanishing conductivity in the limit of in nite relaxation [14{16]. Yet, in
this limit the e ective charge in these models vanish, so strictly speaking they cannot be
considered insulators. Other e ective models of momentum relaxation in holography
include the memory matrix formalism [17{20], helical and Q-lattices [21{23] and massive
gravity [24{27]. Similar results [28{34] have been obtained even for Maxwell elds with
a spatially oscillating chemical potentials in the boundary leading to inhomogeneous
Einstein's equations. We note that e ects suc (...truncated)