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International Journal of Modern Physics B

EXACT RESULTS FOR MANY-BODY PROBLEMS USING FEW-BODY METHODS

John Cardy

Published Jul 30, 2006• 0 citations
Publisher

Abstract

Recently there has been developed a new approach to the study of critical quantum systems in 1+1 dimensions which reduces them to problems in one-dimensional Brownian motion. This goes under the name of stochastic, or Schramm, Loewner Evolution (SLE). I review some of the recent progress in this area, from the point of view of many-body theory. Connections to random matrices also emerge.

DOI
10.1142/s0217979206035072

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Recently there has been developed a new approach to the study of critical quantum systems in 1+1 dimensions which reduces them to problems in one-dimensional Brownian motion. This goes under the name of stochastic, or Schramm, Loewner Evolution (SLE). I review some of the recent progress in this area, from the point of view of many-body theory. Connections to random matrices also emerge.
In the last years, substantial progress has been made in many areas of mathematical physics. The goal of this workshop was to bring together researchers working on analytic and probabilistic aspects of many-body quantum systems and quantum statistical mechanics, to discuss recent developments, exchange ideas and propose new challenges and research directions. Among the questions addressed during the workshop were the derivation of effective equations, the analysis of physically interesting nonlinear partial differential equations emerging from microscopic theories, the study of open quantum systems in and out of equilibrium, and the investigation of the ground state properties and of the dynamics of quantum spin systems.
The concept of Schramm-Loewner evolution provides a unified description of domain boundaries of many lattice spin systems in two dimensions, possibly even including systems with quenched disorder. Here, we study domain walls in the random-field Ising model. Although, in two dimensions, this system does not show an ordering transition to a ferromagnetic state, in the presence of a uniform external field spin domains percolate beyond a critical field strength. Using exact ground-state calculations for very large systems, we examine ground-state domain walls near this percolation transition finding strong evidence that they are conformally invariant and satisfy the domain Markov property, implying compatibility with Schramm-Loewner evolution (SLE κ ) with parameter κ = 6. These results might pave the way for new field-theoretic treatments of systems with quenched disorder.