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Topics in Disordered Systems

  • 1997
  • Book
Author
Charles M. Newman
Book Series
Lectures in Mathematics ETH Zürich
Publisher
Birkhäuser Basel
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About this book

Disordered systems are statistical mechanics models in random environments. This lecture notes volume concerns the equilibrium properties of a few carefully chosen examples of disordered Ising models. The approach is that of probability theory and mathematical physics, but the subject matter is of interest also to condensed matter physicists, material scientists, applied mathematicians and theoretical computer scientists. (The two main types of systems considered are disordered ferromagnets and spin glasses. The emphasis is on questions concerning the number of ground states (at zero temperature) or the number of pure Gibbs states (at nonzero temperature). A recurring theme is that these questions are connected to interesting issues concerning percolation and related models of geometric/combinatorial probability. One question treated at length concerns the low temperature behavior of short-range spin glasses: whether and in what sense Parisi's analysis of the meanfield (or "infinite-range") model is relevant. Closely related is the more general conceptual issue of how to approach the thermodynamic (i.e., infinite volume) limit in systems which may have many complex competing states. This issue has been addressed in recent joint work by the author and Dan Stein and the book provides a mathematically coherent presentation of their approach.)

Table of Contents

Frontmatter
Chapter 0. Introduction
Abstract
Let A be a finite subset of ℤ d , for example, Λ=ΛL= {−L, −L + 1, …, L} d . An Ising model on Λ is a family {S x : x ∈ Λ} of random variables (called spins) taking values +1 or −1, whose joint distribution P Λ,β depends on a parameter β ≥ 0 and has the form,
$$P \wedge,\beta \left( {\left\{ {\left. s \right\}} \right.} \right)\, = \,Z_{ \wedge,\beta }^{ - 1}\,\exp \left\{ {\left. { - \beta H \wedge \left( s \right)} \right\}} \right..\,(0.1)$$
Charles M. Newman
Chapter 1. Ground States of Disordered Ferromagnets
Abstract
When \(\mathop J\nolimits_e^\Lambda = \mathop J\nolimits_e\) (i.e., with no dependence on Λ), an infinite volume ground state for the couplings \(\left\{ {\mathop J\nolimits_e :e \in \mathop E\nolimits^d } \right\}\) may be defined as any \(s \in S = {\left\{ { - 1, + 1} \right\}^{{Z^d}}}\), such that there exists some sequence \({\bar s^{\left( L \right)}} \in {S_{\partial {\Lambda _L}}}\) (the restriction to cubes Λ L and to fixed b.c. ’s involves no loss of generality) and some ground state \({s^{\left( L \right)}} \in {S_{{\Lambda _L}}}{\rm{for}}\mathop H\nolimits_{{\Lambda _L}}^{{{\bar s}^{\left( L \right)}}}\) such that for each \(x \in {Z^d},{s_x} = \mathop s\nolimits_x^{\left( L \right)}\), for all large L.
Charles M. Newman
Chapter 2. Ground States of Highly Disordered Systems
Abstract
In this chapter, based on [NS94] and [NS96a] (see also [CiMB94]), we consider, for each cube Λ L , couplings \(\mathop J\nolimits_e^{\left( L \right)} = \mathop J\nolimits_e^{{\Lambda _L}}\) which, like in the EA spin glass model, are i.i.d. symmetric random variables, on a fixed probability space (Ω,ℱ, ν). However, unlike the EA model, these couplings will (strongly) depend on L. The dependence, explained precisely below, is in terms of a fixed set of random variables \(\left\{ {{K_{e,}}{\alpha _e}:e \in {E^d}} \right\}\) and a nonlinear scaling parameter λ( L ). The sign \(\mathop J\nolimits_e^{\left( L \right)}\) of will simply be α e for every L while the magnitude \(|\mathop J\nolimits_e^{\left( L \right)}|\) will be determined by K e and λ( L ). Infinite volume ground states, defined as the limits of finite volume ground states s L ) for \(H_{{\Lambda _L}}^{{{\bar s}^{\left( L \right)}}}\) with arbitrary boundary conditions \({\bar s^{\left( L \right)}}\), will again have an intrinsic characterization. But here the characterization, in terms of \(\left\{ {{K_e},{\alpha _e}:e \in {E^d}} \right\}\), will take some time to derive. Since flipping the b.c. \({\bar s^{\left( L \right)}}{\rm{to - }}{\bar s^{\left( L \right)}}\) clearly flips the ground state \({s^{\left( L \right)}}{\rm{ to - }}{s^{\left( L \right)}}\) it follows that infinite volume ground states always come in pairs. The main issue we will address is whether or not there is a.s. only a single pair
Charles M. Newman
Chapter 3. High Temperature States of Disordered Systems
Abstract
We return now to the more realistic models where \(\mathop J\nolimits_e^\Lambda = {J_e}\), with no dependence on Λ. The couplings J e , for \(e \in {E^d}\), are i.i.d. random variables on (Ω,ℱ,ν) with common distribution μ. Since we will not be considering ground states here, we do not require μ to be continuous. Indeed, except where otherwise noted, we make no particular restriction on μ. Ferromagnetic (where μ is supported on [0, ∞)) and spin glass (where μ is symmetric) models are of course important special cases. The topic of this chapter is a percolation-based approach to obtaining bounds for β c , the critical value of β (or equivalently bounds for the critical temperature), defined here as the supremum of β’s for which there is a unique infinite volume Gibbs distribution (regardless of b.c. ’s) for all inverse temperatures ≤ β.
Charles M. Newman
Chapter 4. Low Temperature States of Disordered Systems
Abstract
An Edwards-Anderson (EA) spin glass model [EA75] is a disordered Ising model on ℤ d whose nearest neighbor couplings \(J = \left( {{J_e}:e \in {E^d}} \right)\) are i.i.d. random variables (on some (Ω, ℱ, ν)) with a common symmetric distribution μ (i.e., J e and — J e are equidistributed). The most common examples are the Gaussian (where μ is a mean zero normal distribution) and the \(\pm \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over J}\) (where \(\mu = {1 \over 2}{\delta _{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over J} }} + {1 \over 2}{\delta _{ - \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over J} }}\)) models. We place no restrictions on μ beyond symmetry
Charles M. Newman
Backmatter
Metadata
Title
Topics in Disordered Systems
Author
Charles M. Newman
Copyright Year
1997
Publisher
Birkhäuser Basel
Electronic ISBN
978-3-0348-8912-4
Print ISBN
978-3-7643-5777-1
DOI
https://doi.org/10.1007/978-3-0348-8912-4