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arXiv:2211.04737 · Physical review. E

Residual entropy of the dilute Ising chain in a magnetic field

Yu. D. Panov

Published Nov 4, 2022• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.106.054111

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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.
The features of the response of frustrated states to the external field are considered on the example of a diluted Ising chain. In the ferromagnetic case, partial ordering occurs, which leads to a decrease in entropy. In the antiferromagnetic case, the switching on of the field leads to the appearance of a long-range order in the system, although the state remains frustrated. A mapping of a one-dimensional spin model to a Markov chain is proposed, which makes it possible to study in detail the properties of correlation functions and local distributions of the states in the spin chain.
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.
This study provides a comprehensive analysis of the ground state and thermodynamic properties of a spin-pseudospin chain representing a model of a one-dimensional dilute magnet with two types of nonmagnetic charged impurities. For this purpose, a method utilizing the transfer-matrix properties is employed. Despite the wide variety of intriguing frustrated phase states, we show that the model showcases pseudotransitions solely between simple charge and magnetic quasiorders. These pseudotransitions are characterized by distinct features in the thermodynamic and magnetic quantities, resembling first- and second-order phase transitions. In addition to pseudotransitions for the "pure" system, similar to those observed in other one-dimensional spin models, this study also reveals the presence of "second-order" pseudotransitions for the dilute case. We show that the nature of these discovered pseudotransitions is associated with the phase separation in the chain into regions of (anti)ferromagnetic and charge-ordered phases. The ability to compare the results of an exact transfer-matrix calculation with a simple phenomenological description within the framework of Maxwell construction contributes to a deeper understanding of both the physical mechanisms underlying this phenomenon and the analytical methods used.
We analyze frustrated states of the one-dimensional dilute Ising chain with charged interacting impurities of two types with mapping of the system to some Markov chain. We perform classification and reveal two types of Markov chains: periodic with period 2 and aperiodic. Frustrated phases with various types of chains have different properties. In phases with periodic Markov chains, long-range order is observed in the sublattice while another sublattice remains disordered. This results in a conjunction of the non-zero residual entropy and the infinite correlation length. In frustrated phases with aperiodic chains, there is no long-range order, and the correlation length remains finite. It is shown that under the magnetic field the most significant change in the spin chain structure corresponds to the change of the Markov chain type.
Using the Kramers–Wannier transfer matrix method, we studied several decorated Ising chains. The exact expressions for thermodynamic characteristics, including the ground state characteristics, were obtained. We considered several modeling chains with different signs and absolute values of exchange constants for the nearest- and the next-nearest neighbors. For these models, we calculated the magnetization curves. The critical values of magnetic fields and corresponding magnetization plateau parameters were obtained. Analytic expressions for the ground state entropy were obtained for the chains with different interaction constants. The dependencies of the number of states with minimum energy (the degeneration of the ground state) as the function of the number of particles were found. It was shown that these dependencies are expressed in terms of well-known numerical sequences, namely Lucas numbers and Pell numbers, which, in the limit of a large number of particles, are proportional to the powers of the golden and silver sections. Therefore, the ground state entropy (per particle) of the systems under consideration can be described in terms of these sections and, therefore, is nonzero.