Shannon Starr

Extended variational principle for the Sherrington-Kirkpatrick spin-glass model

The recent proof by F. Guerra that the Parisi ansatz provides a lower bound on the free energy of the SK spin-glass model could have been taken as offering some support to the validity of the purported solution. In this work we present a broader variational principle, in which the lower bound, as well as the actual value, are obtained through an optimization procedure for which ultrametic/hierarchal structures form only a subset of the variational class. The validity of Parisis ansatz for the SK model is still in question. The new variational principle may be of help in critical review of the issue.

Thermodynamic limit for the Mallows model on Sn

The Mallows model on Sn is a probability distribution on permutations, qd(π,e)/Pn(q), where d(π,e) is the distance between π and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs (i,j) where 1≤i πj. Analyzing the normalization Pn(q), Diaconis and Ram calculated the mean and variance of d(π,e) in the Mallows model, which suggests that the appropriate n→∞ limit has qn scaling as 1−β/n. We calculate the distribution of the empirical measure in this limit, u(x,y)dxdy=limn→∞(1/n)∑i=1nδ(i,πi). Treating it as a mean-field problem, analogous to the Curie–Weiss model, the self-consistent mean-field equations are (∂2/∂x∂y)ln u(x,y)=2βu(x,y), which is an integrable partial differential equation, known as the hyperbolic Liouville equation. The explicit solution also gives a new proof of formulas for the blocking measures in the weakly asymmetric exclusion process and the ground state of the Uq(sl2)-symmetric XXZ ferromagnet.

On the existence of the dynamics for anharmonic quantum oscillator systems

We construct a W*-dynamical system describing the dynamics of a class of anharmonic quantum oscillator lattice systems in the thermodynamic limit. Our approach is based on recently proved Lieb–Robinson bounds for such systems on finite lattices [19].

Ferromagnetic ordering of energy levels

We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the Lieb–Mattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest energies in each subspace of fixed total spin are strictly ordered according to the total spin, with the lowest, i.e., the ground state, belonging to the maximal total spin subspace. Our main result is a proof of this conjecture for the spin-1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof has two main ingredients. The first is an extension of a result of Koma and Nachtergaele which shows that monotonicity as a function of the total spin follows from the monotonicity of the ground state energy in each total spin subspace as a function of the length of the chain. For the second part of the proof we use the Temperley–Lieb algebra to calculate, in a suitable basis, the matrix elements of the Hamiltonian restricted to each subspace of the highest weight vectors with a given total spin. We then show that the positivity properties of these matrix elements imply the necessary monotonicity in the volume. Our method also shows that the first excited state of the XXX ferromagnet on any finite tree has one less than maximal total spin.

Quantum Spin Systems at Positive Temperature

We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be used to prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature β and the magnitude of the quantum spins

Droplet Excitations for the Spin-1/2 XXZ Chain with Kink Boundary Conditions

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Antiferromagnetic Potts model on the Erdős-Rényi random graph

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