Kac type models of semi-finite spin systems

N. Angelescu, M. Bundaru, G. Costache +1 more

H. Y. Shik, Y.Q. Li, H. Q. Lin

Daniel M. Fröhlich, Benjamin Hinrichs

We prove a Feynman-Kac formula (FKF) for the self-energy renormalized spin boson Hamiltonian, describing a two-state quantum system linearly coupled to a bosonic quantum field. Similar to recent FKFs for the Fröhlich polaron and the non- and semi-relativistic Nelson models, it yields a probabilistic treatment of the spin as a jump process, but treats the field on the usual bosonic Fock space. As an application, we prove that the existence of ground states for infrared-regular models persists the removal of an ultraviolet cutoff.

Andreas Wipf

G. Costache

The free energy of the Kac-Baker model for spin systems with free surfaces is exactly calculated. The critical temperature is obtained and its shift index $\ensuremath{\lambda}$ evaluated. It is shown that for Ising-like interaction $\ensuremath{\lambda}=\frac{1}{\ensuremath{\nu}}$ ($\ensuremath{\nu}$ is the correlation-length critical index of the bulk), while for the spherical model with overall constraint $\ensuremath{\lambda}=1$; it is argued that the value $\ensuremath{\lambda}=1$ is determined by the imposed constraint. As a byproduct an exact analytical expression for the magnetization of an open Ising chain with random fields is obtained.

Paul A. Pearce, Elena Tartaglia, Romain Couvreur

We develop further the implementation and analysis of Kac boundary conditions in the general logarithmic minimal models with 1 ⩽ p < p ′ and p , p ′ coprime. Specifically, working in a strip geometry, we consider the ( r , s ) Kac boundary conditions. These boundary conditions are organized into infinitely extended Kac tables labeled by the Kac labels r , s = 1, 2, 3, …. They are conjugate to Virasoro Kac representations with conformal dimensions Δ r , s given by the usual Kac formula. On a finite strip of width N , built from a square lattice, the associated integrable boundary conditions are constructed by acting on the vacuum (1, 1) boundary with an s -type seam of width s − 1 columns and an r -type seam of width ρ − 1 columns. The r -type seam contains an arbitrary boundary field ξ . While the usual fusion construction of the r -type seam relies on the existence of Wenzl–Jones projectors restricting its application to r ⩽ ρ < p ′, this limitation was recently removed by Pearce et al who further conjectured that the conformal boundary conditions labeled by r are realized, in particular, for . In this paper, we confirm this conjecture by performing extensive numerics on the commuting double row transfer matrices and their associated quantum Hamiltonian chains. Letting [ x ] denote the fractional part, we fix the boundary field to the specialized values if and otherwise. For these boundary conditions, we obtain the Kac conformal weights Δ r , s by numerically extrapolating the finite-size corrections to the lowest eigenvalue of the quantum Hamiltonians out to sizes N ⩽ 32 − ρ − s . Additionally, by solving local inversion relations, we obtain general analytic expressions for the boundary free energies allowing for more accurate estimates of the conformal data.