Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

A general commutator equation in the algebra of spin operators

Dhirendra Kumar Jha

Published Nov 1, 1978• 1 citation
Publisher

Abstract

An algebraic method of spin operators was developed by Jha and Valatin to solve (a) the Hamiltonian of the isotropic and anisotropic xy model in a one-dimensional lattice of N spin 1/2 particles and (b) the partition function of the Ising model in the absence of magnetic field in two dimensions. The pair of fermion operators used to explain BCS theory in superconductivity were shown to be related to a set of spin operators of Jha and Valatin in a very simple way. Onsager’s Lie algebra for diagonalizing the partition function of the Ising model was found to be included within the said commutator algebra of the spin operators. The structure constants of the algebra are so simple as to allow the entire algebra to be casted in one general commutator equation. In the present paper, the author presents the proof of a general equation from which all sets of commutator relationships existing among the elements of the algebra directly follow.

DOI
10.1063/1.523599

Related papers

XY model and algebraic methods

D K Jha, Jean G. Valatin

A systematic study of properties of a set of operators leads to a new approach to solve the XY model. The algebraic solution refers only to a subalgebra of spin operators. Through the connection with the algebra of related fermion operators is apparent, these need not be referred to. As a by-product, Onsager's method for the Ising model results in a very simple form, and its relationship to the approach of Schultz, Mattis and Lieb (1961) becomes apparent.

Journal of Physics A Mathematical Nuclear and General•Nov 1, 1973•14 citations

Solvable Hamiltonians and Fermionization Transformations Obtained from Operators Satisfying Specific Commutation Relations

Kazuhiko Minami

It is shown that a solvable Hamiltonian can be obtained from a series of operators satisfying specific commutation relations. A transformation that diagonalize the Hamiltonian is obtained simultaneously. The two-dimensional Ising model with periodic interactions, the one-dimensional XY model with period 2, the transverse Ising chain, the one-dimensional Kitaev model and the cluster model, and other composite quantum spin chains are diagonalized following this procedure. The Jordan–Wigner transformation, the transformation from the Pauli spin operators to the Majorana fermion used by Shankar and Murthy, and the transformation introduced by Nambu, are special cases of this treatment.

Journal of the Physical Society of Japan•Jan 21, 2016•25 citations

Quasiclassical Methods in the Spin S, One Dimensional Ising Model

Robert L. Bowden, D. M. Kaplan, C. D. Graham +1 more

A Quasiclassical Spin Formalism is developed for spin algebras characteristic of Ising Models, that is, an algebra in which all elements commute. Spin operators and their products are mapped onto continous variables while traces are re‐expressed as integrals over these variables. Using the formalism, a certain Integral Equation is developed whose eigenfunctions and eigenvalues characterize the thermodynamical properties of the spin S, one dimensional Ising lattice. The kernel of the equation is the “classical” kernal in the limit as spin approaches infinity.

AIP conference proceedings•Jan 1, 1972•0 citations

Algebraic Derivation of the Partition Function of a Two-Dimensional Ising Model

Colin J. Thompson

In the algebraic formulation of the Ising model, the partition function is expressed as the trace of a power VM of the transfer operator V, or equivalently, as the sum of Mth powers of the eigenvalues of V. In the derivations of Kaufman, Onsager, and more recently of Schultz, Mattis, and Lieb (SML), the transfer operator is first reduced to a more amenable form for computation and then, in principle at least, diagonalized. For the infinite lattice, only the largest eigenvalue of V is needed, and this is all Onsager and SML compute. Kaufman finds all the eigenvalues and is thus able to write down the partition function for the finite lattice. In the present work we give an alternative derivation of the SML form for V, and show how the Kaufman result can be obtained from this form without actual diagonalization. Instead of diagonalizing V, the evaluation of the trace is done directly after assigning a simple representation to V.

Journal of Mathematical Physics•Sep 1, 1965•6 citations

Recurrence relations in one-dimensional Ising models

C. M. S. da Conceição, Raphael N. P. Maia

The exact finite-size partition function for the nonhomogeneous one-dimensional (1D) Ising model is found through an approach using algebra operators. Specifically, in this paper we show that the partition function can be computed through a trace from a linear second-order recurrence relation with nonconstant coefficients in matrix form. A relation between the finite-size partition function and the generalized Lucas polynomials is found for the simple homogeneous model, thus establishing a recursive formula for the partition function. This is an important property and it might indicate the possible existence of recurrence relations in higher-dimensional Ising models. Moreover, assuming quenched disorder for the interactions within the model, the quenched averaged magnetic susceptibility displays a nontrivial behavior due to changes in the ferromagnetic concentration probability.

Physical review. E•Sep 13, 2017•5 citations

Two-Dimensional Ising Model as a Soluble Problem of Many Fermions

T. D. Schultz, Daniel Charles Mattis, Élliott H. Lieb

The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.

Reviews of Modern Physics•Jul 1, 1964•935 citations
An algebraic method of spin operators was developed by Jha and Valatin to solve (a) the Hamiltonian of the isotropic and anisotropic xy model in a one-dimensional lattice of N spin 1/2 particles and (b) the partition function of the Ising model in the absence of magnetic field in two dimensions. The pair of fermion operators used to explain BCS theory in superconductivity were shown to be related to a set of spin operators of Jha and Valatin in a very simple way. Onsager’s Lie algebra for diagonalizing the partition function of the Ising model was found to be included within the said commutator algebra of the spin operators. The structure constants of the algebra are so simple as to allow the entire algebra to be casted in one general commutator equation. In the present paper, the author presents the proof of a general equation from which all sets of commutator relationships existing among the elements of the algebra directly follow.
It is shown that a solvable Hamiltonian can be obtained from a series of operators satisfying specific commutation relations. A transformation that diagonalize the Hamiltonian is obtained simultaneously. The two-dimensional Ising model with periodic interactions, the one-dimensional XY model with period 2, the transverse Ising chain, the one-dimensional Kitaev model and the cluster model, and other composite quantum spin chains are diagonalized following this procedure. The Jordan–Wigner transformation, the transformation from the Pauli spin operators to the Majorana fermion used by Shankar and Murthy, and the transformation introduced by Nambu, are special cases of this treatment.
A Quasiclassical Spin Formalism is developed for spin algebras characteristic of Ising Models, that is, an algebra in which all elements commute. Spin operators and their products are mapped onto continous variables while traces are re‐expressed as integrals over these variables. Using the formalism, a certain Integral Equation is developed whose eigenfunctions and eigenvalues characterize the thermodynamical properties of the spin S, one dimensional Ising lattice. The kernel of the equation is the “classical” kernal in the limit as spin approaches infinity.
In the algebraic formulation of the Ising model, the partition function is expressed as the trace of a power VM of the transfer operator V, or equivalently, as the sum of Mth powers of the eigenvalues of V. In the derivations of Kaufman, Onsager, and more recently of Schultz, Mattis, and Lieb (SML), the transfer operator is first reduced to a more amenable form for computation and then, in principle at least, diagonalized. For the infinite lattice, only the largest eigenvalue of V is needed, and this is all Onsager and SML compute. Kaufman finds all the eigenvalues and is thus able to write down the partition function for the finite lattice. In the present work we give an alternative derivation of the SML form for V, and show how the Kaufman result can be obtained from this form without actual diagonalization. Instead of diagonalizing V, the evaluation of the trace is done directly after assigning a simple representation to V.
The exact finite-size partition function for the nonhomogeneous one-dimensional (1D) Ising model is found through an approach using algebra operators. Specifically, in this paper we show that the partition function can be computed through a trace from a linear second-order recurrence relation with nonconstant coefficients in matrix form. A relation between the finite-size partition function and the generalized Lucas polynomials is found for the simple homogeneous model, thus establishing a recursive formula for the partition function. This is an important property and it might indicate the possible existence of recurrence relations in higher-dimensional Ising models. Moreover, assuming quenched disorder for the interactions within the model, the quenched averaged magnetic susceptibility displays a nontrivial behavior due to changes in the ferromagnetic concentration probability.
The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.