218 research outputs found
Quantum Spin Systems
This article is a short introduction to the general topic of quantum spin
systems. After a brief sketch of the history of the subject, the standard
mathematical framework for formulating problems and results in quantum spin
systems is described. Then, three short sections are devoted to Spontaneaous
Symmetry Breaking, Phase transitions, and Dynamcis.Comment: Article for the Encyclopedia of Mathematical Physics (Elsevier
Interfaces and droplets in quantum lattice models
This paper is a short review of recent results on interface states in the
Falicov-Kimball model and the ferromagnetic XXZ Heisenberg model. More
specifically, we discuss the following topics: 1) The existence of interfaces
in quantum lattice models that can be considered as perturbations of classical
models. 2) The rigidity of the 111 interface in the three-dimensional
Falicov-Kimball model at sufficiently low temperatures. 3) The low-lying
excitations and the scaling of the gap in the 111 interface ground state in the
ferromagnetic XXZ Heisenberg model in three dimensions. 4) The existence of
droplet states in the XXZ chain and their properties.Comment: 7 pages, 1 figure (embedded eps). For the proceedings of the XIII
International Congress of Mathematical Physics, London, July 18-24, 200
Quantum Spin Systems after DLS1978
In their 1978 paper, Dyson, Lieb, and Simon (DLS) proved the existence of
Ne'el order at positive temperature for the spin-S Heisenberg antiferromagnet
on the d-dimensional hypercubic lattice when either S >= 1 and d >= 3 or S=1/2
and d is sufficiently large. This was the first proof of spontaneous breaking
of a continuous symmetry in a quantum model at finite temperature. Since then
the ideas of DLS have been extended and adapted to a variety of other problems.
In this paper I will present an overview of the most important developments in
the study of the Heisenberg model and related quantum lattice systems since
1978, including but not restricted to those directly related to the paper by
DLS.Comment: Dedicated to Barry Simon and to appear in a festschrift on the
occasion of his 60th birthday. v2: corrected typos and reference
Books Received
We investigate the persistence of spectral gaps of one-dimensional frustration free quantum lattice systems under weak perturbations and with open boundary conditions. Assuming that the interactions of the system satisfy a form of local topological quantum order, we prove explicit lower bounds on the ground state spectral gap and higher gaps for spin and fermion chains. By adapting previous methods using the spectral flow, we analyze the bulk and edge dependence of lower bounds on spectral gaps
Locality Estimates for Quantum Spin Systems
We review some recent results that express or rely on the locality properties
of the dynamics of quantum spin systems. In particular, we present a slightly
sharper version of the recently obtained Lieb-Robinson bound on the group
velocity for such systems on a large class of metric graphs. Using this bound
we provide expressions of the quasi-locality of the dynamics in various forms,
present a proof of the Exponential Clustering Theorem, and discuss a
multi-dimensional Lieb-Schultz-Mattis Theorem.Comment: Contribution for the proceedings of ICMP XV, Rio de Janeiro, 200
Interface states of quantum spin systems
We review recent results as well as ongoing work and open problems concerning
interface states in quantum spin systems at zero and finite temperature.Comment: 10 pages, LaTe
The complete set of ground states of the ferromagnetic XXZ chains
We show that the well-known translation invariant ground states and the
recently discovered kink and antikink ground states are the complete set of
pure infinite-volume ground states (in the sense of local stability) of the
spin-S ferromagnetic XXZ chains with Hamiltonian H=-sum_x [ S^1_x S^1_{x+1} +
S^2_x S^2_{x+1} + Delta S^3_x S^3_{x+1} ], for all Delta >1, and all
S=1/2,1,3/2,.... For the isotropic model (Delta =1) we show that all ground
states are translation invariant.
For the proof of these statements we propose a strategy for demonstrating
completeness of the list of the pure infinite-volume ground states of a quantum
many-body system, of which the present results for the XXX and XXZ chains can
be seen as an example. The result for Delta>1 can also be proved by an easy
extension to general of the method used in [T. Matsui, Lett. Math. Phys. 37
(1996) 397] for the spin-1/2 ferromagnetic XXZ chain with . However,
our proof is different and does not rely on the existence of a spectral gap. In
particular, it also works to prove absence of non-translationally invariant
ground states for the isotropic chains (Delta=1), which have a gapless
excitation spectrum.
Our results show that, while any small amount of the anisotropy is enough to
stabilize the domain walls against the quantum fluctuations, no boundary
condition exists that would stabilize a domain wall in the isotropic model
(Delta=1).Comment: 23 pages (LaTeX), typos corrected, references update
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