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Open and Periodic Boundary Conditions in Statistical Mechanics: A Case Study of the Antiferromagnetic Ising Chain

Katarína Karľová, Jozef Strečka

Published Jun 24, 2026• 0 citations
Publisher

Abstract

The transfer-matrix method is employed to investigate a spin-1/2 Ising chain under open and periodic boundary conditions. It is demonstrated that finite-size Ising chains with antiferromagnetic coupling may exhibit significantly distinct magnetic behavior under open and periodic boundary conditions. While the open Ising chains display intriguing magnetic features regardless of the system size, mainly due to a specific contribution of boundary spins, the magnetic behavior of closed Ising chains depends basically on the number of spins. The closed Ising chains with an odd number of spins are subject to a geometric spin frustration leading to an additional plateau in the magnetization curve, which is naturally absent in the closed Ising chains with an even number of spins. Despite different microscopic origins, the magnetization curves of open and closed Ising chains with an odd number of spins exhibit an identical intermediate plateau, with only small quantitative differences appearing at moderate temperatures, which means that a geometric spin frustration of odd-membered rings is somewhat similar to the effect of open boundaries. The magnetization curves of the open Ising chains with an even number of spins differ drastically from those of the closed Ising chains due to the presence of an additional intermediate magnetization plateau. Furthermore, the initial susceptibility, inverse initial susceptibility, and susceptibility-temperature product are examined in detail as functions of temperature. These magnetic response functions demonstrate that the Curie constant and Weiss temperature represent fundamental characteristics of the magnetic system that are independent of the choice of boundary conditions.

DOI
10.3390/e28070727

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The transfer-matrix method is employed to investigate a spin-1/2 Ising chain under open and periodic boundary conditions. It is demonstrated that finite-size Ising chains with antiferromagnetic coupling may exhibit significantly distinct magnetic behavior under open and periodic boundary conditions. While the open Ising chains display intriguing magnetic features regardless of the system size, mainly due to a specific contribution of boundary spins, the magnetic behavior of closed Ising chains depends basically on the number of spins. The closed Ising chains with an odd number of spins are subject to a geometric spin frustration leading to an additional plateau in the magnetization curve, which is naturally absent in the closed Ising chains with an even number of spins. Despite different microscopic origins, the magnetization curves of open and closed Ising chains with an odd number of spins exhibit an identical intermediate plateau, with only small quantitative differences appearing at moderate temperatures, which means that a geometric spin frustration of odd-membered rings is somewhat similar to the effect of open boundaries. The magnetization curves of the open Ising chains with an even number of spins differ drastically from those of the closed Ising chains due to the presence of an additional intermediate magnetization plateau. Furthermore, the initial susceptibility, inverse initial susceptibility, and susceptibility-temperature product are examined in detail as functions of temperature. These magnetic response functions demonstrate that the Curie constant and Weiss temperature represent fundamental characteristics of the magnetic system that are independent of the choice of boundary conditions.
Full magnetization curves of S=1 antiferromagnetic Heisenberg chains with periodic and open boundaries are investigated by a quantum Monte Carlo method. The boundary condition has a significant effect on the magnetization process especially in the field range where the Zeeman energy is less than the Haldane gap. The magnetization of the periodic chain in the range is strongly suppressed at low temperatures because the ground state is nonmagnetic. On the other hand, the magnetic ground of the open chain bring about finite magnetization. The difference between both the magnetization curves is also investigated in detail by subtracting the magnetization of the periodic chain with L spins from one of the open chains with L+1 spins. It agrees well with the double of the Brillouin function for S=1/2, rather than the Brillouun function for S=1.
The one-dimensional Ising model with general spin S and either open or periodic boundary conditions has been formulated as a difference equation of order 2S+1 with non-zero magnetic field B. This difference equation has the characteristic equation of the transfer matrix, V, as its indicial equation. When B=0, for the open chain, the order of the difference equation is reduced to [S+1]. It is shown that the correlation functions for the open chain satisfy similar difference equations. This result explicitly displays the lack of translational invariance of the open chain. For several cases it is shown that the partition function can be represented as the difference of two Chebyshev polynomials of the second kind.
We investigate the statistical mechanics of the periodic one-dimensional Ising chain when the number of positive spins is constrained to be either an even or an odd number. We calculate the partition function using a generalization of the transfer matrix method. On this basis, we derive the exact magnetization, susceptibility, internal energy, heat capacity and correlation function. We show that in general the constraints substantially slow down convergence to the thermodynamic limit. By taking the thermodynamic limit together with the limit of zero temperature and zero magnetic field, the constraints lead to new scaling functions and different probability distributions for the magnetization. We demonstrate how these results solve a stochastic version of the one-dimensional voter model.
Exact solution of the magnetic susceptibility for an S = 1/2 defective finite Ising chain is obtained by employing the transfer matrix method. The thermal dependence of susceptibility multiplied by temperature for the chain coupled to different magnetic impurities is investigated at all T ≥ 0. The results show that in the zero temperature limit the product of these two quantities equals the square of the net spin, demonstrating that Curie’s constants are truly invariant at low temperatures. The intrinsic properties of net spin are also reflected through their different responses to the parity of chain length at low temperature. In addition, the impacts of various host–impurity couplings as well as single-ion anisotropies on the susceptibility are discussed as well.
We connected the two ends of a finite spin-1/2 antiferromagnetic Ising chain with a magnetic impurity at one end to form a closed ring, and studied its magnetic susceptibility exactly by using the transfer matrix method. We calculated the magnetic susceptibility in the whole temperature range and gave the phase diagram at ground state of the system about the anisotropy of the impurity and strength of the connection exchange interaction for spin-1 and -3/2 impurities. We also gave the ground state entropy of the system and derived the asymptotic expression of the magnetic susceptibility multiplied by temperature at the zero-temperature limit and the high-temperature limit. It was found that a degenerate phase may exist in some parameter region at zero temperature for the odd spin number system, and the ground state entropy is ln(2) in the nondegenerate phase and is dependent on the spin number in the degenerate phase. The magnetic susceptibility of the system at low temperature exhibits ferromagnetic behavior, and the Curie constant is related to the spin configuration at ground state. When the ground state is nondegenerate, the Curie constant is equal to the square of the net spin, regardless of the parity of the number of the spin. When the spin number is odd and the ground state is degenerate, the Curie constant may be related to the total spin number. At the high-temperature limit, the magnetic susceptibility multiplied by temperature is related to the spin quantum number of the impurity and the spin number in the ring.
A central tenant in the classification of phases is that boundary conditions cannot affect the bulk properties of a system. In this work, we show striking, yet puzzling, evidence of a clear violation of this assumption. We use the prototypical example of an XYZ chain with no external field in a ring geometry with an odd number of sites and both ferromagnetic and antiferromagnetic interactions. In such a setting, even at finite sizes, we are able to calculate directly the spontaneous magnetizations that are traditionally used as order parameters to characterize the system's phases. When ferromagnetic interactions dominate, we recover magnetizations that in the thermodynamic limit lose any knowledge about the boundary conditions and are in complete agreement with standard expectations. On the contrary, when the system is governed by antiferromagnetic interactions, the magnetizations decay algebraically to zero with the system size and are not staggered, despite the AFM coupling. We term this behavior ferromagnetic mesoscopic magnetization. Hence, in the antiferromagnetic regime, our results show an unexpected dependence of a local, one-- spin expectation values on the boundary conditions, which is in contrast with predictions from the general theory.