5,077 research outputs found
Disordered Topological Insulators via -Algebras
The theory of almost commuting matrices can be used to quantify topological
obstructions to the existence of localized Wannier functions with time-reversal
symmetry in systems with time-reversal symmetry and strong spin-orbit coupling.
We present a numerical procedure that calculates a Z_2 invariant using these
techniques, and apply it to a model of HgTe. This numerical procedure allows us
to access sizes significantly larger than procedures based on studying twisted
boundary conditions. Our numerical results indicate the existence of a metallic
phase in the presence of scattering between up and down spin components, while
there is a sharp transition when the system decouples into two copies of the
quantum Hall effect. In addition to the Z_2 invariant calculation in the case
when up and down components are coupled, we also present a simple method of
evaluating the integer invariant in the quantum Hall case where they are
decoupled.Comment: Added detail regarding the mapping of almost commuting unitary
matrices to almost commuting Hermitian matrices that form an approximate
representation of the sphere. 6 pages, 6 figure
Inference from Matrix Products: A Heuristic Spin Glass Algorithm
We present an algorithm for finding ground states of two dimensional spin
glass systems based on ideas from matrix product states in quantum information
theory. The algorithm works directly at zero temperature and defines an
approximate "boundary Hamiltonian" whose accuracy depends on a parameter .
We test the algorithm against exact methods on random field and random bond
Ising models, and we find that accurate results require a which scales
roughly polynomially with the system size. The algorithm also performs well
when tested on small systems with arbitrary interactions, where no fast, exact
algorithms exist. The time required is significantly less than Monte Carlo
schemes.Comment: 4 pages, 1 figure, minor typos fixe
Tip Splittings and Phase Transitions in the Dielectric Breakdown Model: Mapping to the DLA Model
We show that the fractal growth described by the dielectric breakdown model
exhibits a phase transition in the multifractal spectrum of the growth measure.
The transition takes place because the tip-splitting of branches forms a fixed
angle. This angle is eta dependent but it can be rescaled onto an
``effectively'' universal angle of the DLA branching process. We derive an
analytic rescaling relation which is in agreement with numerical simulations.
The dimension of the clusters decreases linearly with the angle and the growth
becomes non-fractal at an angle close to 74 degrees (which corresponds to eta=
4.0 +- 0.3).Comment: 4 pages, REVTex, 3 figure
Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model
A fast method is presented for simulating the dielectric-breakdown model
using iterated conformal mappings. Numerical results for the dimension and for
corrections to scaling are in good agreement with the recent RG prediction of
an upper critical , at which a transition occurs between branching
fractal clusters and one-dimensional nonfractal clusters.Comment: 5 pages, 7 figures; corrections to scaling include
Solving Gapped Hamiltonians Locally
We show that any short-range Hamiltonian with a gap between the ground and
excited states can be written as a sum of local operators, such that the ground
state is an approximate eigenvector of each operator separately. We then show
that the ground state of any such Hamiltonian is close to a generalized matrix
product state. The range of the given operators needed to obtain a good
approximation to the ground state is proportional to the square of the
logarithm of the system size times a characteristic "factorization length".
Applications to many-body quantum simulation are discussed. We also consider
density matrices of systems at non-zero temperature.Comment: 13 pages, 2 figures; minor changes to references, additional
discussion of numerics; additional explanation of nonzero temperature matrix
product for
A short proof of stability of topological order under local perturbations
Recently, the stability of certain topological phases of matter under weak
perturbations was proven. Here, we present a short, alternate proof of the same
result. We consider models of topological quantum order for which the
unperturbed Hamiltonian can be written as a sum of local pairwise
commuting projectors on a -dimensional lattice. We consider a perturbed
Hamiltonian involving a generic perturbation that can be written
as a sum of short-range bounded-norm interactions. We prove that if the
strength of is below a constant threshold value then has well-defined
spectral bands originating from the low-lying eigenvalues of . These bands
are separated from the rest of the spectrum and from each other by a constant
gap. The width of the band originating from the smallest eigenvalue of
decays faster than any power of the lattice size.Comment: 15 page
Quasi-Adiabatic Continuation in Gapped Spin and Fermion Systems: Goldstone's Theorem and Flux Periodicity
We apply the technique of quasi-adiabatic continuation to study systems with
continuous symmetries. We first derive a general form of Goldstone's theorem
applicable to gapped nonrelativistic systems with continuous symmetries. We
then show that for a fermionic system with a spin gap, it is possible to insert
-flux into a cylinder with only exponentially small change in the energy
of the system, a scenario which covers several physically interesting cases
such as an s-wave superconductor or a resonating valence bond state.Comment: 19 pages, 2 figures, final version in press at JSTA
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