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Theoretical and Mathematical Physics

Free energy, entropy, and magnetization of a one-dimensional Ising model of a diluted magnet

S. V. Semkin, Виктор Павлович Смагин

Published Nov 1, 2023• 2 citations
Publisher

Abstract

We consider a one-dimensional Ising model (chain) with the the nearest-neighbor interaction and with a random nonmagnetic dilution. We find the exact free energy of such a chain as a function of the impurity concentration, temperature, and the external magnetic field. In the case of antiferromagnetic interaction in the chain, we find the specific magnetization, the mean value of the product of neighboring spins, and the entropy as functions of these parameters. We study the residual system entropy.

DOI
10.1134/s0040577923110132

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We consider a one-dimensional Ising model (chain) with the the nearest-neighbor interaction and with a random nonmagnetic dilution. We find the exact free energy of such a chain as a function of the impurity concentration, temperature, and the external magnetic field. In the case of antiferromagnetic interaction in the chain, we find the specific magnetization, the mean value of the product of neighboring spins, and the entropy as functions of these parameters. We study the residual system entropy.
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.
Abstract We reconsider the problem of a one‐dimensional Ising model with an arbitrary nearest‐neighbor random exchange integral, temperature, and random magnetic field in each site. A convenient formalism is developed that reduces the partition function to a recurrence equation, which is convenient both for numerical as well as for analytical approaches. We have calculated asymptotic expressions for an ensemble averaged free energy and the averaged magnetization in the case of strong and weak couplings in external constant magnetic field. With a random magnetic field at each site in addition to nearest‐neighbor random exchange integrals we also evaluated the free energy. We show that the zeros of the partition function for the Ising model in the complex external magnetic field plane formally coincide with the singularities of the real part of electron's transmission amplitude through the chain of δ‐function potentials.
Consider a closed, linear chain of NNN localized spins (each s=12s=\frac{1}{2}s=21​) with nearest-neighbor Ising interaction. Such a chain, exposed to a uniform external magnetic field, will be called the pure host. Replace a single spin in the pure host by an impurity spin which is also subjected to nearest-neighbor Ising interaction and to the external field. The magnitude of the impurity spin, the magnitude of its magnetic moment, and the magnitude and sign of its interaction with the host are allowed to differ from the corresponding values characterizing the host. For the bulk system ($N\ensuremath{\rightarrow}\ensuremath{\infty}$, for constant linear density of spins), the thermodynamic properties, such as the impurity magnetization, the (position-dependent) magnetization of the impurity-host system and the impurity-host spin correlation functions, are obtained exactly in terms of conventional, tabulated functions. Numerical results are presented for impurity spins of magnitude S=32,52S=\frac{3}{2},\frac{5}{2}S=23​,25​.