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

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

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

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 $S$ of the method used in [T. Matsui, Lett. Math. Phys. 37 (1996) 397] for the spin-1/2 ferromagnetic XXZ chain with $\Delta>1$. 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

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