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arXiv:2011.00760 · European Journal of Physics

1D Ising model using the Kronecker sum and Kronecker product

Sourabh Magare, Abhinash Kumar Roy, Varun Srivastava

Published Feb 17, 2022• 6 citations
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Abstract

Abstract Calculations in the Ising model can be cumbersome and non-intuitive. Here we provide a formulation that addresses these issues for 1D scenarios. We represent the microstates of spin interactions as a diagonal matrix. This is done using two operations: the Kronecker sum and Kronecker product. The calculations thus become a simple matter of manipulating diagonal matrices. We address the following problems in this work: spins in the magnetic field, open-chain 1D Ising model, closed-chain 1D Ising model and the 1D Ising model in an external magnetic field. We believe that this representation will help provide students and experts with a simple yet powerful technique to carry out calculations in this model.

DOI
10.1088/1361-6404/ac5637

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Abstract Calculations in the Ising model can be cumbersome and non-intuitive. Here we provide a formulation that addresses these issues for 1D scenarios. We represent the microstates of spin interactions as a diagonal matrix. This is done using two operations: the Kronecker sum and Kronecker product. The calculations thus become a simple matter of manipulating diagonal matrices. We address the following problems in this work: spins in the magnetic field, open-chain 1D Ising model, closed-chain 1D Ising model and the 1D Ising model in an external magnetic field. We believe that this representation will help provide students and experts with a simple yet powerful technique to carry out calculations in this model.
In this paper, we provide a method to represent the microstates of spin interactions as a diagonal matrix. This is done using two operations: Kronecker sum and Kronecker product. These operations are not usually introduced in a physics textbook. We introduce these operations in this paper and develop a method to represent following problems: spins in Magnetic field, open chain 1-d Ising model, closed chain 1-d Ising model, Ising model in external Magnetic field. We think that this representation will help students to bridge the gap between Basic statistical mechanics and advance techniques to solve spin interaction problems
The properties of the Kronecker product are revisited in terms of Hubbard operators. The simplest representation of a Hubbard operator X i,j n is a square matrix of size n with an entry equal to 1 and zero elsewhere. This framework simplifies the calculation of the Kronecker product of arbitrary matrices no matter the size or the number of the involved factors. Some applications are presented, these include the algebra of permutation matrices, the Hadamard matrix, the XXX Heisenberg model and the interaction of an atom with radiation fields.
Abstract In a typical introductory course on quantum mechanics the student has to face two quite different operator products. One of them leads to, for example, commutators and the uncertainty principle, and the other appears when one goes from a one-particle system to a multi-particle one. The latter gives rise to the direct or tensor product of operators and vector spaces, and to the Kronecker product of the matrix representations of the operators. We illustrate the practical utility of the Kronecker product by means of two spin systems commonly discussed in introductory courses on quantum mechanics and also by means of a simple model for the interpretation of the NMR signal produced by two interacting protons.
We present here various techniques to work with clean and disordered quantum Ising chains, for the benefit of students and non-experts. Starting from the Jordan-Wigner transformation, which maps spin-1/2 systems into fermionic ones, we review some of the basic approaches to deal with the superconducting correlations that naturally emerge in this context. In particular, we analyze the form of the ground state and excitations of the model, relating them to the symmetry-breaking physics, and illustrate aspects connected to calculating dynamical quantities, thermal averages, correlation functions, and entanglement entropy. A few problems provide simple applications of the techniques.