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Optical Memory and Neural Networks

Influence of Long-Range Interaction on Eigenvalues of Connection Matrix in One-Dimensional Ising Model

Boris Kryzhanovsky, Leonid B. Litinskii

Published Oct 1, 2020• 0 citations
Publisher

Abstract

We analyze a finite one-dimensional Ising system with periodic boundary conditions taking into account an arbitrary long-range interaction. We examine a discrete spectrum of eigenvalues of the spin connection matrix and a spectrum density of a continuous distribution obtained in the limit $$L \to \infty $$ (L is the linear size of the system). We apply our results to particular cases of long-range interactions decreasing with distance.

DOI
10.3103/s1060992x20040049

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We analyze a finite one-dimensional Ising system with periodic boundary conditions taking into account an arbitrary long-range interaction. We examine a discrete spectrum of eigenvalues of the spin connection matrix and a spectrum density of a continuous distribution obtained in the limit L→∞L \to \infty L→∞ (L is the linear size of the system). We apply our results to particular cases of long-range interactions decreasing with distance.
Abstract The connection matrix of the Ising model on a ddd-dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.
The author considers an Ising chain where the random interaction matrix has the symmetrised form J=PV+VP. The matrix elements of V and P are V ij =exp(- gamma mod i-j mod ) and P ij =P i delta j,i-1 +P i+1 delta j,i+1 , where i, j indicate sites on the chain. The quantities P i are independent random variables with the same probability distribution. Their mean and variance are fixed to be (P i )=J 0 gamma /4; (P i 2 )-(P i ) 2 =J 2 gamma /4, where gamma -1 is the range of the interaction. The author then obtains Sigma j (J ij )=2J 0 , Sigma j (J ij 2 )=J 2 in the limit gamma to 0, as in a model Hamiltonian that describes a spin-glass transition. The author evaluates exactly in this limit the density of eigenvalues of the random matrix PV. An integral equation for the distribution probability of new local random variables is derived by using the continued fraction method and the asymptotic solution in the limit gamma to 0 is obtained analytically. The largest eigenvalue of PV is found to be J 0 independent of J. However the largest eigenvalue of J=PV+VP is found to be mu M =J 0 +(J 0 2 +NJ 2 ) 1 2/, when gamma =0 and N is the total number of sites. An independent evaluation of the ground state energy shows that E 0 <-J square root NN, hence E 0 is not extensive. These results show that the scaling of the variance of the random interactions as J 2 N -1 gives highly unphysical results for the present model.
We present a possible way to study long-range interacting particles in finite-infinite systems with periodic boundary conditions. A symmetrical lattice and its contributions overall space are being used. In this context, we study the Ising model with ferromagnetic interaction that decays as a 1/r a law. We verify by Monte Carlo heat-bath simulations in the D (dimension) = 1 case that the thermodynamics quantities scale in a way proposed by Tsallis, and we confirm that mean field theory is exact in the last model for all 0 < a < D suggested by Cannas and Tamarit.
The equilibrium properties of the two-dimensional Ising model with nearest-neighbour interactions uniformly distributed between -J and J have been determined using a numerical transfer matrix method. The calculations are performed on a long strip of width L(3<or=L<or=11) with periodic boundary conditions and finite-size scaling arguments are used to extrapolate to L= infinity . The spin-glass correlation length diverges as T - nu at low temperature with nu =2.96+or-0.22. The correlation length scaling function differs from that for the random Ising model with interactions of magnitude J and random sign showing that the two models with different distribution functions belong to different universality classes. The heat capacity is linear at low temperature and can be interpreted using a continuous distribution of two-level states which is finite at zero energy and increases approximately linearly with energy. At zero temperature the entropy is zero and the internal energy extrapolated to L= infinity is -(0.7943+or-0.0010)J.