The authors demonstrate that all Ising antiferromagnets with arbitrary many-neighboured interactions and in the maximum critical field, have highly degenerate ground states accompanied with non-zero residual entropies. The residual entropies vanish when the range of interaction tends to infinity. The proof is realised by an explicit calculation in the case of a one-dimensional many-neighboured Ising antiferromagnet, and by establishing bounds for the residual entropy in the case of an Ising system situated on a lattice with arbitrary number of dimensions. They also show that the established bounds may serve as estimates of frequently unknown actual values of the residual entropy.

https://iopscience.iop.org/article/10.1088/0305-4470/15/11/034/pdf

Ground state degeneracy of the Ising antiferromagnets in the maximum critical field

We present an alternative solution of the Ising chain in a field under free and periodic boundary conditions, in the microcanonical, canonical, and grand canonical ensembles, from a unified combinatorial and topological perspective. In particular, the computation of the per-site entropy as a function of the energy un- veils a residual value for critical values of the magnetic field, a phenomenon for which we provide a topological interpretation and a connection with the Fibonacci se- quence. We also show that, in the thermodynamic limit, the per-site microcanonical entropy is equal to the loga- rithm of the per-site Euler characteristic. The canonical and grand canonical partition functions are identified as combinatorial generating functions of the micro- canonical problem, which allows us to evaluate them. A detailed analysis of the magnetic field-dependent thermodynamics, including positive and negative tem- peratures, reveals interesting features. Finally, we em- phasize that our combinatorial approach to the canon- ical ensemble allows exact computation of the ther- mally averaged value � χ� of the Euler characteristic associated with the spin configurations of the chain, which is discontinuous at the critical fields, and whose thermal behavior is expected to determine the phase transition of the model. Indeed, our results show that the conjecture � χ� (TC) = 0 ,w hereTC is the critical temperature, is valid for the Ising chain.

Combinatorial and Topological Analysis of the Ising Chain in a Field

Thermodynamic properties of the one-dimensional Ising model with magnetoelastic interaction

Residual entropy of the one-dimensional Ising antiferromagnet with general spin

An exact evaluation of the critical exponent γ associated with the average number of N-step walks of the piecewise directed random walks is presented in the case of the complete family of the Sierpinski gasket type of fractals.

S. Milošević

Papers

Piecewise directed random walk on the Sierpinski gasket family of fractals

Ising model and the third law of thermodynamics

Equal-time correlation functions of the dimerized Ising chain with next-nearest neighbour interactions

Phase diagrams of the ising spin-peierls transition model in a magnetic field