Solving the one-dimensional Ising chain via mathematical induction: An intuitive approach to the transfer matrix

Calculations in Ising model can be cumbersome and non-intuitive. Here we provide a formulation that addresses these issues for 1-D scenario. We represent the microstates of spin interactions as a diagonal matrix. This is done using two operations: Kronecker sum and Kronecker product. The calculations thus become simple matter of manipulating diagonal matrices. We address the following problems in this work: spins in the magnetic field, open-chain 1-D Ising model, closed-chain...

In this paper, we first rework B. Kaufman's 1949 paper, "Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis", by using representation theory. Our approach leads to a simpler and more direct way of deriving the spectrum of the transfer matrix for the finite periodic Ising model. We then determine formulas for the spin correlation functions that depend on the matrix elements of the induced rotation associated with the spin operator in a basis of eigenvector...

We study the classical 1D Heisenberg spin glasses. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from first principles of classical mechanics lead to NP hard problem, that however in the limit of the statistical equilibrium can be calculated by P algorithm. For the partition function of the ensemble a new representation is offered in the form of on...

Recently, we developed and implemented the bond propagation algorithm for calculating the partition function and correlation functions of random bond Ising models in two dimensions. The algorithm is the fastest available for calculating these quantities near the percolation threshold. In this paper, we show how to extend the bond propagation algorithm to directly calculate thermodynamic functions by applying the algorithm to derivatives of the partition function, and we deriv...

In this note we consider several kind of partition functions of one-dimensional models with nearest - neighbor interactions $I_n, n\in \mathbf{Z}$ and spin values $\pm 1$. We derive systems of recursive equations for each kind of such functions. These systems depend on parameters $I_n, n\in \mathbf{Z}$. Under some conditions on the parameters we describe solutions of the systems of recursive equations.

We propose a generic framework to describe classical Ising-like models defined on arbitrary graphs. The energy spectrum is shown to be the Hadamard transform of a suitably defined sparse "coding" vector associated with the graph. We expect that the existence of a fast Hadamard transform algorithm (used for instance in image ccompression processes), together with the sparseness of the coding vector, may provide ways to fasten the spectrum computation.. Applying this formalism ...

We present detailed analytical calculations for an 1D Ising ring of arbitrary number of spin-1/2 particles, in order to reveal entanglement properties of the stationary states. We show that the ground state and specific eigenstates of the Ising Hamiltonian posses remarkable entanglement properties that can reveal new insight into quantum correlations present in the Ising model. This correlations might be exploited in quantum information processing. We propose an intuitive pic...

We represent a general procedure for calculating the partition function of an Ising model on a one dimensional Fibonacci lattice in presence of magnetic field.This partition function can be written as a sum of partition functions of usual open Ising chains in presence of magnetic field with coefficients having Fibonacci symmetries.An exact expression for the partition function of the usual open Ising model is found.We observe that "Mirror Symmetry" is a characteristic propert...

We present an exact solution of a one-dimensional Ising chain with both nearest neighbor and random long-range interactions. Not surprisingly, the solution confirms the mean field character of the transition. This solution also predicts the finite-size scaling that we observe in numerical simulations.

The one-dimensional Ising model is easily generalized to a \textit{genuinely nonequilibrium} system by coupling alternating spins to two thermal baths at different temperatures. Here, we investigate the full time dependence of this system. In particular, we obtain the evolution of the magnetisation, starting with arbitrary initial conditions. For slightly less general initial conditions, we compute the time dependence of all correlation functions, and so, the probability dist...