4,360 research outputs found
Holographic Symmetries and Generalized Order Parameters for Topological Matter
We introduce a universally applicable method, based on the bond-algebraic
theory of dualities, to search for generalized order parameters in disparate
systems including non-Landau systems with topological order. A key notion that
we advance is that of {\em holographic symmetry}. It reflects situations
wherein global symmetries become, under a duality mapping, symmetries that act
solely on the system's boundary. Holographic symmetries are naturally related
to edge modes and localization. The utility of our approach is illustrated by
systematically deriving generalized order parameters for pure and
matter-coupled Abelian gauge theories, and for some models of topological
matter.Comment: v2, 10 pages, 3 figures. Accepted for publication in Physical Review
B Rapid Communication
Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter
Motivated by the prospect of attaining Majorana modes at the ends of
nanowires, we analyze interacting Majorana systems on general networks and
lattices in an arbitrary number of dimensions, and derive various universal
spin duals. Such general complex Majorana architectures (other than those of
simple square or other crystalline arrangements) might be of empirical
relevance. As these systems display low-dimensional symmetries, they are
candidates for realizing topological quantum order. We prove that (a) these
Majorana systems, (b) quantum Ising gauge theories, and (c) transverse-field
Ising models with annealed bimodal disorder are all dual to one another on
general graphs. As any Dirac fermion (including electronic) operator can be
expressed as a linear combination of two Majorana fermion operators, our
results further lead to dualities between interacting Dirac fermionic systems.
The spin duals allow us to predict the feasibility of various standard
transitions as well as spin-glass type behavior in {\it interacting} Majorana
fermion or electronic systems. Several new systems that can be simulated by
arrays of Majorana wires are further introduced and investigated: (1) the {\it
XXZ honeycomb compass} model (intermediate between the classical Ising model on
the honeycomb lattice and Kitaev's honeycomb model), (2) a checkerboard lattice
realization of the model of Xu and Moore for superconducting arrays,
and a (3) compass type two-flavor Hubbard model with both pairing and hopping
terms. By the use of dualities, we show that all of these systems lie in the 3D
Ising universality class. We discuss how the existence of topological orders
and bounds on autocorrelation times can be inferred by the use of symmetries
and also propose to engineer {\it quantum simulators} out of these Majorana
networks.Comment: v3,19 pages, 18 figures, submitted to Physical Review B. 11 new
figures, new section on simulating the Hubbard model with nanowire systems,
and two new appendice
Topological Order and Quantum Criticality
In this chapter we discuss aspects of the quantum critical behavior that
occurs at a quantum phase transition separating a topological phase from a
conventionally ordered one. We concentrate on a family of quantum lattice
models, namely certain deformations of the toric code model, that exhibit
continuous quantum phase transitions. One such deformation leads to a
Lorentz-invariant transition in the 3D Ising universality class. An alternative
deformation gives rise to a so-called conformal quantum critical point where
equal-time correlations become conformally invariant and can be related to
those of the 2D Ising model. We study the behavior of several physical
observables, such as non-local operators and entanglement entropies, that can
be used to characterize these quantum phase transitions. Finally, we briefly
consider the role of thermal fluctuations and related phase transitions, before
closing with a short overview of field theoretical descriptions of these
quantum critical points.Comment: 24 pages, 7 figures, chapter of the book "Understanding Quantum Phase
Transitions", edited by Lincoln D. Carr (CRC Press / Taylor and Francis,
2010); v2: updated reference
A quantum topological phase transition at the microscopic level
We study a quantum phase transition between a phase which is topologically
ordered and one which is not. We focus on a spin model, an extension of the
toric code, for which we obtain the exact ground state for all values of the
coupling constant that takes the system across the phase transition. We compute
the entanglement and the topological entropy of the system as a function of
this coupling constant, and show that the topological entropy remains constant
all the way up to the critical point, and jumps to zero beyond it. Despite the
jump in the topological entropy, the transition is second order as detected via
any local observable.Comment: (13 pages, 4 figures) v2: updated references and acknowledgments; v3:
final update (references) after publicatio
Dual Geometric Worm Algorithm for Two-Dimensional Discrete Classical Lattice Models
We present a dual geometrical worm algorithm for two-dimensional Ising
models. The existence of such dual algorithms was first pointed out by
Prokof'ev and Svistunov \cite{ProkofevClassical}. The algorithm is defined on
the dual lattice and is formulated in terms of bond-variables and can therefore
be generalized to other two-dimensional models that can be formulated in terms
of bond-variables. We also discuss two related algorithms formulated on the
direct lattice, applicable in any dimension. These latter algorithms turn out
to be less efficient but of considerable intrinsic interest. We show how such
algorithms quite generally can be "directed" by minimizing the probability for
the worms to erase themselves. Explicit proofs of detailed balance are given
for all the algorithms. In terms of computational efficiency the dual
geometrical worm algorithm is comparable to well known cluster algorithms such
as the Swendsen-Wang and Wolff algorithms, however, it is quite different in
structure and allows for a very simple and efficient implementation. The dual
algorithm also allows for a very elegant way of calculating the domain wall
free energy.Comment: 12 pages, 6 figures, Revtex
Consistency Conditions for an AdS/MERA Correspondence
The Multi-scale Entanglement Renormalization Ansatz (MERA) is a tensor
network that provides an efficient way of variationally estimating the ground
state of a critical quantum system. The network geometry resembles a
discretization of spatial slices of an AdS spacetime and "geodesics" in the
MERA reproduce the Ryu-Takayanagi formula for the entanglement entropy of a
boundary region in terms of bulk properties. It has therefore been suggested
that there could be an AdS/MERA correspondence, relating states in the Hilbert
space of the boundary quantum system to ones defined on the bulk lattice. Here
we investigate this proposal and derive necessary conditions for it to apply,
using geometric features and entropy inequalities that we expect to hold in the
bulk. We show that, perhaps unsurprisingly, the MERA lattice can only describe
physics on length scales larger than the AdS radius. Further, using the
covariant entropy bound in the bulk, we show that there are no conventional
MERA parameters that completely reproduce bulk physics even on super-AdS
scales. We suggest modifications or generalizations of this kind of tensor
network that may be able to provide a more robust correspondence.Comment: 38 pages, 9 figure
Topological phases and topological entropy of two-dimensional systems with finite correlation length
We elucidate the topological features of the entanglement entropy of a region
in two dimensional quantum systems in a topological phase with a finite
correlation length . Firstly, we suggest that simpler reduced quantities,
related to the von Neumann entropy, could be defined to compute the topological
entropy. We use our methods to compute the entanglement entropy for the ground
state wave function of a quantum eight-vertex model in its topological phase,
and show that a finite correlation length adds corrections of the same order as
the topological entropy which come from sharp features of the boundary of the
region under study. We also calculate the topological entropy for the ground
state of the quantum dimer model on a triangular lattice by using a mapping to
a loop model. The topological entropy of the state is determined by loop
configurations with a non-trivial winding number around the region under study.
Finally, we consider extensions of the Kitaev wave function, which incorporate
the effects of electric and magnetic charge fluctuations, and use it to
investigate the stability of the topological phase by calculating the
topological entropy.Comment: 17 pages, 4 figures, published versio
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