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Journal of Physics A Mathematical and General

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

Dragan Slavkov Hajdukovic, S. Milošević

Published Nov 1, 1982• 16 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/15/11/034

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AIP conference proceedings•Jan 1, 2012•0 citations
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.
The phase diagram of the q=3q=3q=3 state Potts antiferromagnet in a field is studied by exact mappings and Monte Carlo methods, and it is shown that systems with macroscopic ground-state degeneracy may exhibit conventional-type critical ordering. Our Monte Carlo data for the square-lattice case indicate a finite-temperature ordering in a range of field values where the entropy of the ground state is extensive. In a particular limit this phase transition can be shown to be of the Ising type by mapping the Potts model onto a colored, nearest-neighbor exclusion lattice gas. We also discuss the phase diagram of the body-centered-cubic Ising antiferromagnet in a field in order to show that renormalization-group arguments alone are not sufficient to restrict the diversity of possible low-temperature phases occurring in systems with a macroscopically degenerate ground state.
The properties of the ground state of the simplest frustrated system, the dilute Ising chain in a magnetic field, are rigorously investigated over the entire range of concentrations of charged nonmagnetic impurities. Analytical methods are proposed for calculating the residual entropy of frustrated states, including states at phase boundaries, which are based on the Markov property of the system and involve solving a linear optimization problem for energy and a nonlinear optimization problem for entropy. These methods allow obvious generalizations for one-dimensional pseudospin models with anisotropic interactions. We calculate the composition, entropy, and magnetization for the ground state phases. We prove the absence of pseudotransitions in the dilute Ising chain, since the residual entropy of states at phase boundaries is always higher than the entropy of adjacent phases. The concentration dependencies of magnetization at the phase boundaries are obtained, and unlike linear dependencies for adjacent phases, they have nonlinear behavior. Field-induced transitions between ground states and entropy jumps associated with them are also considered, and in particular, it is shown that the field-induced transition from an antiferromagnetic state to a frustrated one is accompanied by charge ordering.
The properties of the ground state of one of the simplest models of frustrated magnetic systems, a dilute Ising chain in a magnetic field, are considered for all values of the concentration of charged non-magnetic impurities. An analytical method is proposed for calculating the residual entropy of frustrated states, including states at the boundaries between the phases of the ground state, which is based on the Markov property of the system under consideration and allows direct generalization to other one-dimensional spin models with Ising-type interactions. The properties of local distributions and concentration dependences of the composition, correlation functions, magnetization and entropy of the phases of the ground state of the model are investigated. It is shown that the field-induced transition from the antiferromagnetic ground state to the frustrated one is accompanied by charge ordering and the absence of pseudo-transitions in the dilute Ising chain is proved.
For a class of highly frustrated antiferromagnetic quantum spin lattices the ground state exhibits a huge degeneracy in high magnetic fields due to the existence of localized magnon states. For some of these spin lattices (in particular, the 1D dimer-plaquette, sawtooth and kagom\'elike chains as well as the 2D kagom\'e lattice) we calculate rigorously the ground-state entropy at the saturation field. We find that the ground-state entropy per site remains finite at saturation. This residual ground-state entropy produces a maximum in the field dependence of the isothermal entropy at low temperatures. By numerical calculation of the field dependence of the low-temperature entropy for the sawtooth chain we find that the enhancement of isothermal entropy is robust against small deviations in exchange constants. Moreover, the effect is most pronounced in the extreme quantum case of spin 12\frac{1}{2}21​.
The properties of the antiferromagnetic Ising model in the critical magnetic field are investigated by evaluating exactly its all possible ground states on L × L square lattice (L = 4∼16) with free boundary conditions for the first time. From the exact integer values for the number of ground states Ωc(L) on L × L square lattice, the exact ground-state entropy (in unit of the Boltzmann constant kB) per volume sc(L) = L−2lnΩc(L) and the exact ground-state magnetization mc(L;T = 0) = L−2〈Σiσi〉 are calculated in the critical magnetic field, where σi is the magnetic spin on a lattice site i. The values of the ground-state entropy sc(L) and the ground-state magnetization mc(L), in the limit L → ∞, are accurately estimated to be sc/kB = 0.40750±0.00108 and mc = 0.54696±0.00065. In addition, for the first time, we estimate the values of the scaling exponents ωs and ωm, controlling the finite-size scaling behaviors of the ground-state entropy and the ground-state magnetization as sc−sc(L)∼L−ωs and mc−mc(L)∼L−ωm respectively. Our results show clearly that ωs = ωm = 1.