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Canadian Journal of Physics

The Ground State of the One-Dimensional Ising Chain

Byron K. Jennings, R. K. Bhaduri

Published Mar 1, 1974• 0 citations

Abstract

We consider a linear spin 1/2 Ising chain with pair interactions extending up to the nth neighbor. The following general theorem is proved: 'The energy of any arbitrary spin sequence may be written as a linear sum of cluster energies, the upper limit of the number of Ising spins in a cluster being 2n'. The term cluster energy here is used to mean the energy of a group of spins in a certain configuration, evaluated as if the same configuration repeated itself throughout the chain. The structure of the ground state of the system is investigated using the above theorem. It is shown that the ground state is a repetition of a certain cluster, and that the upper limit of the cluster size is 2n, except for some specific combinations of the interaction strengths when the ground state may admit mixing of different clusters. The possible ground state configurations are worked out explicitly for n = 2, 3, and 4 in the absence of the magnetic field using the above theorem. Previous attempts were confined only up to n = 3.

DOI: 10.1139/p74-060

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We consider a linear spin 1/2 Ising chain with pair interactions extending up to the nth neighbor. The following general theorem is proved: 'The energy of any arbitrary spin sequence may be written as a linear sum of cluster energies, the upper limit of the number of Ising spins in a cluster being 2n'. The term cluster energy here is used to mean the energy of a group of spins in a certain configuration, evaluated as if the same configuration repeated itself throughout the chain. The structure of the ground state of the system is investigated using the above theorem. It is shown that the ground state is a repetition of a certain cluster, and that the upper limit of the cluster size is 2n, except for some specific combinations of the interaction strengths when the ground state may admit mixing of different clusters. The possible ground state configurations are worked out explicitly for n = 2, 3, and 4 in the absence of the magnetic field using the above theorem. Previous attempts were confined only up to n = 3.

The concept of the irreducible block is introduced for the one-dimensional Ising system with an arbitrary interaction of finite range. It is proved that the ground state energy of the system occurs for a regular chain of one of the irreducible blocks or for coexistence of the regular chains of the irreducible blocks. It is shown that the ground state energy of the linear Ising magnet with pair interactions up to third neighbours is effected by the seven spin orderings recently given for the case of spin 1 / 2 by Katsura and Narita (1973), and that the ground state energy of the Ising magnet of spin larger than 1 / 2 is effected by the same set of orderings as for the corresponding Ising magnet of spin 1 / 2 .

We consider interactions of Ising spins three at a time, five at a time, etc., as distinguished from the conventional interactions (two at a time). To illustrate, we calculate the exact thermodynamics of three-spin-cluster interactions in one dimension with and without a magnetic field. We find regular structures except at a certain value of the coupling constant ratio ($\frac{{J}_{1}}{{J}_{3}}=\ensuremath{-}3$) where the ground state becomes chaotic. The $T\ensuremath{\rightarrow}0$ limit of the magnetization agrees with the exact ground-state calculation.

We study the properties of a linear chain of spins governed by the Hamiltonian H=J1 ∑ j=1NSj·Sj+1+J2 ∑ j=1NSj·Sj+2and derive equations for an upper bound of the free energy by means of a temperature dependent Hartree-Fock approximation. These equations can be solved at zero temperature, yielding an approximate wavefunction for the ground state, of which the energy is an upper bound for the exact ground state energy. The upper bounds obtained are improvements of the results of Majumdar and Ghosh. There are several different cases to be considered, depending on the relative values of J1 and J2. In none of these is there a gap between the ground state and the first excited state in the thermodynamic limit. Finally, correlation functions are shortly discussed in connection with spiral structures in classical spin systems.

The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.

The approach to determination of the order parameter in the finite 1D Ising model was proposed. For such systems a root mean square magnetization and an average size of the maximal ground state cluster which is formed by neighboring spins with an energy {-2J} does not disappear at T!=0. The temperature dependence of heat capacity and its derivative with respect to temperature from the average size of the maximum quasi ground state cluster, formed by the neighboring spins with an energy {-2J,0} in the 1D Ising chain were considered. The percolation approach in Curie-Weiss does not work due the internal structure of the model.