18 research outputs found
Symmetry protected topological phases of spin chains
Symmetry protected topological (SPT) phases are characterized by robust boundary features, which do not disappear unless passing through a phase transition. These boundary features can be quantified by a topological invariant which, in some cases, is related to a physical quantity, such as the spin conductivity for the quantum spin Hall insulators. In other cases, the boundary features give rise to new physics, such as the Majorana fermion. In all cases the boundary features can be analyzed with the help of an entanglement spectrum and their robustness make them promising candidates for storing quantum information. The topological invariant characterizing SPT phases is strictly only invariant under deformations which respect a certain
symmetry. For example, the boundary currents of the quantum spin Hall insulator are only robust against non-magnetic, i.e. time-reversal invariant, impurities. In this thesis we study the SPT phases of spin chains.
As a result of our work we find a topological invariant for SPT phases of spin chains which are protected by continuous symmetries. By means of a non-local order parameter we find a way to extract this invariant from the ground state wave function of the system. Using density-matrix-renormalization-group techniques we verify that this invariant is a tool to detect transitions between different topological phases. We find a non-local transformation that maps SPT phases to conventional phases characterized by a local order parameter. This transformation suggests an analogy between topological phases and conventional phases and thus give a deeper understanding of the role of topology in spin systems
symmetry breaking in Projected Entangled Pair State models
We consider Projected Entangled Pair State (PEPS) models with a global
symmetry, which are constructed from -symmetric
tensors and are thus -invariant wavefunctions, and study the
occurence of long-range order and symmetry breaking in these systems. First, we
show that long-range order in those models is accompanied by a degeneracy in
the so-called transfer operator of the system. We subsequently use this
degeneracy to determine the nature of the symmetry broken states, i.e., those
stable under arbitrary perturbations, and provide a succinct characterization
in terms of the fixed points of the transfer operator (i.e.\ the different
boundary conditions) in the individual symmetry sectors. We verify our findings
numerically through the study of a -symmetric model, and show that
the entanglement Hamiltonian derived from the symmetry broken states is
quasi-local (unlike the one derived from the symmetric state), reinforcing the
locality of the entanglement Hamiltonian for gapped phases.Comment: 11 page
Study of anyon condensation and topological phase transitions from a topological phase using Projected Entangled Pair States
We use Projected Entangled Pair States (PEPS) to study topological quantum
phase transitions. The local description of topological order in the PEPS
formalism allows us to set up order parameters which measure condensation and
deconfinement of anyons, and serve as a substitute for conventional order
parameters. We apply these order parameters, together with anyon-anyon
correlation functions and some further probes, to characterize topological
phases and phase transitions within a family of models based on a
symmetry, which contains quantum double, toric code, double
semion, and trivial phases. We find a diverse phase diagram which exhibits a
variety of different phase transitions of both first and second order which we
comprehensively characterize, including direct transitions between the toric
code and the double semion phase.Comment: 21+6 page
From symmetry-protected topological order to Landau order
Focusing on the particular case of the discrete symmetry group Z_N x Z_N, we
establish a mapping between symmetry protected topological phases and symmetry
broken phases for one-dimensional spin systems. It is realized in terms of a
non-local unitary transformation which preserves the locality of the
Hamiltonian. We derive the image of the mapping for various phases involved,
including those with a mixture of symmetry breaking and topological protection.
Our analysis also applies to topological phases in spin systems with arbitrary
continuous symmetries of unitary, orthogonal and symplectic type. This is
achieved by identifying suitable subgroups Z_N x Z_N in all these groups,
together with a bijection between the individual classes of projective
representations.Comment: 8 pages, 1 table. Version v2 corresponds to the published version. It
includes minor revisions and additional reference
Local Decoders for the 2D and 4D Toric Code
We analyze the performance of decoders for the 2D and 4D toric code which are
local by construction. The 2D decoder is a cellular automaton decoder
formulated by Harrington which explicitly has a finite speed of communication
and computation. For a model of independent and errors and faulty
syndrome measurements with identical probability we report a threshold of
for this Harrington decoder. We implement a decoder for the 4D toric
code which is based on a decoder by Hastings arXiv:1312.2546 . Incorporating a
method for handling faulty syndromes we estimate a threshold of for
the same noise model as in the 2D case. We compare the performance of this
decoder with a decoder based on a 4D version of Toom's cellular automaton rule
as well as the decoding method suggested by Dennis et al.
arXiv:quant-ph/0110143 .Comment: 22 pages, 21 figures; fixed typos, updated Figures 6,7,8,
Entanglement phases as holographic duals of anyon condensates
Anyon condensation forms a mechanism which allows to relate different
topological phases. We study anyon condensation in the framework of Projected
Entangled Pair States (PEPS) where topological order is characterized through
local symmetries of the entanglement. We show that anyon condensation is in
one-to-one correspondence to the behavior of the virtual entanglement state at
the boundary (i.e., the entanglement spectrum) under those symmetries, which
encompasses both symmetry breaking and symmetry protected (SPT) order, and we
use this to characterize all anyon condensations for abelian double models
through the structure of their entanglement spectrum. We illustrate our
findings with the Z4 double model, which can give rise to both Toric Code and
Doubled Semion order through condensation, distinguished by the SPT structure
of their entanglement. Using the ability of our framework to directly measure
order parameters for condensation and deconfinement, we numerically study the
phase diagram of the model, including direct phase transitions between the
Doubled Semion and the Toric Code phase which are not described by anyon
condensation.Comment: 20+7 page
On topological phases of spin chains
Symmetry protected topological phases of one-dimensional spin systems have
been classified using group cohomology. In this paper, we revisit this problem
for general spin chains which are invariant under a continuous on-site symmetry
group G. We evaluate the relevant cohomology groups and find that the
topological phases are in one-to-one correspondence with the elements of the
fundamental group of G if G is compact, simple and connected and if no
additional symmetries are imposed. For spin chains with symmetry
PSU(N)=SU(N)/Z_N our analysis implies the existence of N distinct topological
phases. For symmetry groups of orthogonal, symplectic or exceptional type we
find up to four different phases. Our work suggests a natural generalization of
Haldane's conjecture beyond SU(2).Comment: 18 pages, 7 figures, 2 tables. Version v2 corresponds to the
published version. It includes minor revisions, additional references and an
application to cold atom system
A discriminating string order parameter for topological phases of gapped SU(N) spin chains
One-dimensional gapped spin chains with symmetry PSU(N) = SU(N)/Z_N are known
to possess N different topological phases. In this paper, we introduce a
non-local string order parameter which characterizes each of these N phases
unambiguously. Numerics confirm that our order parameter allows to extract a
quantized topological invariant from a given non-degenerate gapped ground state
wave function. Discontinuous jumps in the discrete topological order that arise
when varying physical couplings in the Hamiltonian may be used to detect
quantum phase transitions between different topological phases.Comment: 15 pages, 4 figure
Performance and structure of single-mode bosonic codes
The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit
in an oscillator has recently been followed by cat- and binomial-code
proposals. Numerically optimized codes have also been proposed, and we
introduce new codes of this type here. These codes have yet to be compared
using the same error model; we provide such a comparison by determining the
entanglement fidelity of all codes with respect to the bosonic pure-loss
channel (i.e., photon loss) after the optimal recovery operation. We then
compare achievable communication rates of the combined encoding-error-recovery
channel by calculating the channel's hashing bound for each code. Cat and
binomial codes perform similarly, with binomial codes outperforming cat codes
at small loss rates. Despite not being designed to protect against the
pure-loss channel, GKP codes significantly outperform all other codes for most
values of the loss rate. We show that the performance of GKP and some binomial
codes increases monotonically with increasing average photon number of the
codes. In order to corroborate our numerical evidence of the cat/binomial/GKP
order of performance occurring at small loss rates, we analytically evaluate
the quantum error-correction conditions of those codes. For GKP codes, we find
an essential singularity in the entanglement fidelity in the limit of vanishing
loss rate. In addition to comparing the codes, we draw parallels between
binomial codes and discrete-variable systems. First, we characterize one- and
two-mode binomial as well as multi-qubit permutation-invariant codes in terms
of spin-coherent states. Such a characterization allows us to introduce check
operators and error-correction procedures for binomial codes. Second, we
introduce a generalization of spin-coherent states, extending our
characterization to qudit binomial codes and yielding a new multi-qudit code.Comment: 34 pages, 11 figures, 4 tables. v3: published version. See related
talk at https://absuploads.aps.org/presentation.cfm?pid=1351