Critical exponents for the long-range Ising chain using a transfer matrix approach

It is often assumed that for treating numerical (or experimental) data on continuous transitions the formal analysis derived from the Renormalization Group Theory can only be applied over a narrow temperature range, the "critical region"; outside this region correction terms proliferate rendering attempts to apply the formalism hopeless. This pessimistic conclusion follows largely from a choice of scaling variables and scaling expressions which is traditional but which is ver...

In this work we present a thorough analysis of the phase transitions that occur in a ferromagnetic 2D Ising model, with only nearest-neighbors interactions, in the framework of the Tsallis nonextensive statistics. We performed Monte Carlo simulations on square lattices with linear sizes L ranging from 32 up to 512. The statistical weight of the Metropolis algorithm was changed according to the nonextensive statistics. Discontinuities in the m(T) curve are observed for $q\leq ...

We study the ferromagnetic Ising model with long-range interactions in two dimensions. We first present results of a Monte Carlo study which shows that the long-range interactions dominate over the short-range ones in the intermediate regime of interaction range. Based on a renormalization group analysis, we propose a way of computing the influence of the long-range interactions as a dimensional change.

This comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power law interactions, $r^{-\alpha}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has been earlier argued [1, 2] that this model obeys the dimensional constraint suggesting the delocalization of all finite temperature states in thermodynamic limit for $\alpha \leq 2d$ in a $d$-dim...

The critical behavior of Ising model on a one-dimensional network, which has long-range connections at distances $l>1$ with the probability $\Theta(l)\sim l^{-m}$, is studied by using Monte Carlo simulations. Through studying the Ising model on networks with different $m$ values, this paper discusses the impact of the global correlation, which decays with the increase of $m$, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the ord...

We numerically study the one-dimensional long-range Transverse Field Ising Model (TFIM) in the antiferromagnetic (AFM) regime at zero temperature using Generalized Hartree-Fock (GHF) theory. The spin-spin interaction extends to all spins in the lattice and decays as $1/r^\alpha$, where $r$ denotes the distance between two spins and $\alpha$ is a tunable exponent. We map the spin operators to Majorana operators and approximate the ground state of the Hamiltonian with a Fermion...

We propose the finite-size scaling of correlation function in a finite system near its critical point. At a distance ${\bf r}$ in the finite system with size $L$, the correlation function can be written as the product of $|{\bf r}|^{-(d-2+\eta)}$ and its finite-size scaling function of variables ${\bf r}/L$ and $tL^{1/\nu}$, where $t=(T-T_c)/T_c$. The directional dependence of correlation function is nonnegligible only when $|{\bf r}|$ becomes compariable with $L$. This finit...

The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension $\log_{4} 12 \approx 1.792$, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed throu...

We present a status report on the ongoing analysis of the 3D Ising model with nearest-neighbor interactions using the Monte Carlo Renormalization Group (MCRG) and finite size scaling (FSS) methods on $64^3$, $128^3$, and $256^3$ simple cubic lattices. Our MCRG estimates are $K_{nn}^c=0.221655(1)(1)$ and $\nu=0.625(1)$. The FSS results for $K^c$ are consistent with those from MCRG but the value of $\nu$ is not. Our best estimate $\eta = 0.025(6)$ covers the spread in the MCRG ...

An efficient Monte Carlo algorithm for the simulation of spin models with long-range interactions is discussed. Its central feature is that the number of operations required to flip a spin is independent of the number of interactions between this spin and the other spins in the system. In addition, critical slowing down is strongly suppressed. In order to illustrate the range of applicability of the algorithm, two specific examples are presented. First, some aspects of the Ko...