22 research outputs found
Ground states in relatively bounded quantum perturbations of classical lattice systems
We consider ground states in relatively bounded quantum perturbations of
classical lattice models. We prove general results about such perturbations
(existence of the spectral gap, exponential decay of truncated correlations,
analyticity of the ground state), and also prove that in particular the AKLT
model belongs to this class if viewed at large enough scale. This immediately
implies a general perturbation theory about this model.Comment: 25 page
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
Polynomial-time algorithm for simulation of weakly interacting quantum spin systems
We describe an algorithm that computes the ground state energy and
correlation functions for 2-local Hamiltonians in which interactions between
qubits are weak compared to single-qubit terms. The running time of the
algorithm is polynomial in the number of qubits and the required precision.
Specifically, we consider Hamiltonians of the form , where
H_0 describes non-interacting qubits, V is a perturbation that involves
arbitrary two-qubit interactions on a graph of bounded degree, and
is a small parameter. The algorithm works if is below a certain
threshold value that depends only upon the spectral gap of H_0, the maximal
degree of the graph, and the maximal norm of the two-qubit interactions. The
main technical ingredient of the algorithm is a generalized Kirkwood-Thomas
ansatz for the ground state. The parameters of the ansatz are computed using
perturbative expansions in powers of . Our algorithm is closely
related to the coupled cluster method used in quantum chemistry.Comment: 27 page
Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
Gapped ground states of quantum spin systems have been referred to in the
physics literature as being `in the same phase' if there exists a family of
Hamiltonians H(s), with finite range interactions depending continuously on , such that for each , H(s) has a non-vanishing gap above its
ground state and with the two initial states being the ground states of H(0)
and H(1), respectively. In this work, we give precise conditions under which
any two gapped ground states of a given quantum spin system that 'belong to the
same phase' are automorphically equivalent and show that this equivalence can
be implemented as a flow generated by an -dependent interaction which decays
faster than any power law (in fact, almost exponentially). The flow is
constructed using Hastings' 'quasi-adiabatic evolution' technique, of which we
give a proof extended to infinite-dimensional Hilbert spaces. In addition, we
derive a general result about the locality properties of the effect of
perturbations of the dynamics for quantum systems with a quasi-local structure
and prove that the flow, which we call the {\em spectral flow}, connecting the
gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a
result, we obtain that, in the thermodynamic limit, the spectral flow converges
to a co-cycle of automorphisms of the algebra of quasi-local observables of the
infinite spin system. This proves that the ground state phase structure is
preserved along the curve of models .Comment: Updated acknowledgments and new email address of S