Do hard spheres have natural boundaries?
I use recent advances in the study of the susceptibility of the Ising model to propose a new mechanism for the freezing transition which is observed in three dimensional hard spheres.
cond-mat.stat-mech↗
arXiv subjects
Barry M. McCoy
Publications and source records attributed to Barry M. McCoy.
Algebra versus analysis in statistical mechanics and quantum field theory
I contrast the profound differences in the ways in which algebra and analysis are used in physics. In particular I discuss the fascinating phenomenon that theoretical physicists devote almost all their efforts to algebraic problems even though all problems of experimental interest require some methods of analysis.
cond-mat↗
Bethe's equation is incomplete for the XXZ model at roots of unity
We demonstrate for the six vertex and XXZ model parameterized by $Δ=-(q+q^{-1})/2\neq \pm 1$ that when q^{2N}=1 for integer $N\geq 2$ the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl_2 loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl_2 loop algebra can be used to complete this specification.
cond-mat.stat-mech↗
The Baxter Revolution
I review the revolutionary impact Rodney Baxter has had on statistical mechanics beginning with his solution of the 8 vertex model in 1971 and the invention of corner transfer matrices in 1976 to the creation of the RSOS models in 1984 and his continuing current work on the chiral Potts model.
cond-mat.stat-mech↗
The sl_2 loop algebra symmetry of the six-vertex model at roots of unity
We demonstrate that the six vertex model (XXZ spin chain) with $Δ=(q+q^{-1})/2$ and $q^{2N}=1$ has an invariance under the loop algebra of $sl_2$ which produces a special set of degenerate eigenvalues. For $Δ=0$ we compute the multiplicity of the degeneracies using Jordan Wigner techniques
cond-mat.stat-mech↗
The 1999 Heineman Prize Address- Integrable models in statistical mechanics: The hidden field with unsolved problems
In the past 30 years there have been extensive discoveries in the theory of integrable statistical mechanical models including the discovery of non-linear differential equations for Ising model correlation functions, the theory of random impurities, level crossing transitions in the chiral Potts model and the use of Rogers-Ramanujan identities to generalize our concepts of Bose/Fermi statistics. Each of these advances has led to the further discovery of major unsolved problems of great mathematical and physical interest. I will here discuss the mathematical advances, the physical insights and extraordinary lack of visibility of this field of physics.
math-ph↗
Temperature dependent spatial oscillations in the correlations of the XXZ spin chain
We study the correlation $<σ^z_0σ^z_n>$ for the XXZ chain in the massless attractive (ferromagnetic) region at positive temperatures by means of a numerical study of the quantum transfer matrix. We find that there is a range of temperature where the behavior of the correlation for large separations is oscillatory with an incommensurate period which depends on temperature.
cond-mat.stat-mech↗
The perturbation $ϕ_{2,1}$ of the M(p,p+1) models of conformal field theory and related polynomial character identities
Using $q$-trinomial coefficients of Andrews and Baxter along with the technique of telescopic expansions, we propose and prove a complete set of polynomial identities of Rogers-Ramanujan type for M(p, p+1) models of conformal field theory perturbed by the operator $phi_{2,1}$. The bosonic form of our polynomials is closely related to corner transfer matrix sums which arise in the computation of the order parameter in the regime $1^+$ of $A_{p-1}$ dilute models. In the limit where the degree of the polynomials tends to infinity our identities provide new companion fermionic representations for all Virasoro characters of unitary minimal series.
math.QA↗
Quantum-classical crossover in the spin 1/2 XXZ chain
We compute, by means of exact diagonalization of systems of N=16 and 18 spins, the correlation function <σ^z_0σ^z_n> at nonzero temperature for the XXZ model with anisotropy Δ. In the gapless ferromagnetic region -1<Δ<0 for fixed separation the temperature can always be made sufficiently low so that the correlation is always negative for n \neq 0. However we find that for sufficiently large temperatures and fixed separation or for fixed temperature greater than some T0(\DElta) and sufficiently large separations the correlations are always positive. This sign changing effect has not been previously seen and we interpret it as a crossover from quantum to classical behavior.
cond-mat.stat-mech↗
Spin diffusion and the spin 1/2 XXZ chain at $T=\infty$ from exact diagonalization
We study the long time behavior of the zz and xx time dependent autocorrelation function of the spin 1/2 XXZ chain at $T=\infty$ by exact diagonalizations on a chain of 16 sites. We find that the numerical results for the zz correlation are very well fit by the formula $t^{-d}[A+Be^{-γ(t-t_0)}\cos Ω(t-t_0)].$ From this we estimate $d$ as a function of the anisotropy of the chain and study the crossover from ballistic to diffusive behavior.
cond-mat↗
No spin diffusion in the spin 1/2 XXZ chain at $T=\infty$: Numerical asymptotics
We analyze the recent numerical computations made by Fabricius, L{\" o}w and Stolze to show that the long time behavior of the zz correlation function of the spin 1/2 XXZ chain at $T=\infty$ is very well fit by the formula $t^{-d}[A+Be^{-γ(t-t_0)}\cos Ω(t-t_0)]$ where $d$ is substantially greater than 1/2. This confirms the conclusion that there is no spin diffusion in this model.
cond-mat↗
Rogers-Schur-Ramanujan type identities for the $M(p,p')$ minimal models of conformal field theory
We present and prove Rogers-Schur-Ramanujan (Bose/Fermi) type identities for the Virasoro characters of the minimal model $M(p,p').$ The proof uses the continued fraction decomposition of $p'/p$ introduced by Takahashi and Suzuki for the study of the Bethe's Ansatz equations of the XXZ model and gives a general method to construct polynomial generalizations of the fermionic form of the characters which satisfy the same recursion relations as the bosonic polynomials of Forrester and Baxter. We use this method to get fermionic representations of the characters $χ_{r,s}^{(p,p')}$ for many classes of $r$ and $s.$
q-alg↗
Single Particle Excitations in the Lattice E_8 Ising Model
We present analytic expressions for the single particle excitation energies of the 8 quasi-particles in the lattice $E_8$ Ising model and demonstrate that all excitations have an extended Brillouin zone which, depending on the excitation, ranges from 0<P < 4πto 0< P< 12 π. These are compared with exact diagonalizations for systems through size 10 and with the E_8 fermionic representations of the characters of the critical system in order to study the counting statistics.
hep-th↗
N=2 Supersymmetry and Bailey Pairs
We demonstrate that the Bailey pair formulation of Rogers-Ramanujan identities unifies the calculations of the characters of $N=1$ and $N=2$ supersymmetric conformal field theories with the counterpart theory with no supersymmetry. We illustrate this construction for the $M(3,4)$ (Ising) model where the Bailey pairs have been given by Slater. We then present the general unitary case. We demonstrate that the model $M(p,p+1)$ is derived from $M(p-1,p)$ by a Bailey renormalization flow and conclude by obtaining the $N=1$ model $SM(p,p+2)$ and the unitary $N=2$ model with central charge $c=3(1-2/p).$
hep-th↗
Analyticity and Integrabiity in the Chiral Potts Model
We study the perturbation theory for the general non-integrable chiral Potts model depending on two chiral angles and a strength parameter and show how the analyticity of the ground state energy and correlation functions dramatically increases when the angles and the strength parameter satisfy the integrability condition. We further specialize to the superintegrable case and verify that a sum rule is obeyed.
hep-th↗
Generalizations of the Andrews-Bressoud Identities for the $N=1$ Superconformal Model $SM(2,4ν)$
We present generalized Rogers-Ramanujan identities which relate the fermi and bose forms of all the characters of the superconformal model $SM(2,4ν).$ In particular we show that to each bosonic form of the character there is an infinite family of distinct fermionic $q-$ series representations.
hep-th↗
Polynomial Identities, Indices, and Duality for the N=1 Superconformal Model SM(2,4ν)
We prove polynomial identities for the N=1 superconformal model SM(2,4ν) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q \rightarrow q^{-1} between SM(2,4ν) and M(2ν-1,4ν). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.
hep-th↗
Continued Fractions and Fermionic Representations for Characters of M(p,p') minimal models
We present fermionic sum representations of the characters $χ^{(p,p')}_{r,s}$ of the minimal $M(p,p')$ models for all relatively prime integers $p'>p$ for some allowed values of $r$ and $s$. Our starting point is binomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin ${1\over 2}$ chain of anisotropy $-Δ=-\cos(π{p\over p'})$. We use the Takahashi-Suzuki method to express the allowed values of $r$ (and $s$) in terms of the continued fraction decomposition of $\{{p'\over p}\}$ (and ${p\over p'}$) where $\{x\}$ stands for the fractional part of $x.$ These values are, in fact, the dimensions of the hermitian irreducible representations of $SU_{q_{-}}(2)$ (and $SU_{q_{+}}(2)$) with $q_{-}=\exp (i π\{{p'\over p}\})$ (and $q_{+}=\exp ( i π{p\over p'})).$ We also establish the duality relation $M(p,p')\leftrightarrow M(p'-p,p')$ and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented.
hep-th↗