Spectral density of quantum Ising model in two fields for large but finite number of spins
N, is discussed in detail. When all coupling constants are of the same order, spectral densities in the bulk are well approximated by a Gaussian function which is typical behaviour for many-body models with short-range interactions. The main part of the paper is devoted to the investigation of a different characteristic case when spectral densities have peaks related with strong degeneracies of unperturbed states in certain limits of coupling constants. In the strict limit
N→∞, peaks overlap and disappear but for values of
N accessible in numerical calculations they often strongly influence spectral densities and other quantities as well. A simple method is developed which permits to find general approximation formulae for multi-peak structure of spectral density in good agreement with numerics.
Abstract The construction of a statistical model for eigenfunctions of the Ising model in transverse and longitudinal fields is discussed in detail for the chaotic case. When the number of spins is large, each wave function coefficient has the Gaussian distribution with zero mean and variance calculated from the first two moments of the Hamiltonian. The main part of the paper is devoted to the discussion of various corrections to the asymptotic result. One type of correction is related to higher order moments of the Hamiltonian, and can be taken into account by Gibbs-like formulae. Other corrections are due to symmetry contributions, which manifest as different numbers of non-zero real and complex coefficients. The statistical model with these corrections included agrees well with numerical calculations of wave function moments.
We formulate the spectral-density method in classical statistical mechanics in strict analogy with the known quantum version, and we apply it to the classical Heisenberg ferromagnetic model in an external field. A new formula for the magnetization in the classical formulation of spin-vector Green functions is derived for arbitrary spatial dimensionality. Furthermore, the static properties of the one- and three-dimensional cases are considered in detail. We obtain accurate results over a wide range of temperatures with the external field for the one-dimensional model and without field for the three-dimensional model. In the first case very good agreement with the exact numerical transfer-matrix data is also found in the region of higher temperature where the interacting-boson approach fails. In particular the zero-field susceptibility results are found to be consistent with the exact results obtained by Fisher.
In this article we review a less known unperturbative and powerful many-body method in the framework of classical statistical mechanics and then we show how it works by means of explicit calculations for a nontrivial classical model. The formalism of two-time Green functions in classical statistical mechanics is presented in a form parallel to the well known quantum counterpart, focusing on the spectral properties which involve the important concept of spectral density. Furthermore, the general ingredients of the classical spectral density method (CSDM) are presented with insights for systematic nonperturbative approximations to study conveniently the macroscopic properties of a wide variety of classical many-body systems also involving phase transitions. The method is implemented by means of key ideas for exploring the spectrum of elementary excitations and the damping effects within a unified formalism. Then, the effectiveness of the CSDM is tested with explicit calculations for the classical
d-dimensional spin-
S Heisenberg ferromagnetic model with long-range exchange interactions decaying as
r−p (
p>d) with distance
r between spins and in the presence of an external magnetic field. The analysis of the thermodynamic and critical properties, performed by means of the CSDM to the lowest order of approximation, shows clearly that nontrivial results can be obtained in a relatively simple manner already to this lower stage. The basic spectral density equations for the next higher order level are also presented and the damping of elementary spin excitations in the low temperature regime is studied. The results appear in reasonable agreement with available exact ones and Monte Carlo simulations and this supports the CSDM as a promising method of investigation in classical many-body theory.
We investigate several distinct spectral crossovers amongst various integrable (Poissonian) and quantum-chaotic (Wigner-Dyson) limits of a
1D disordered quantum spin (
S=1/2) model, by tuning the relative amplitudes of various Hamiltonian parameters to retain or break relevant unitary and antiunitary symmetries. The spin model consists of an isotropic and deterministic Heisenberg term, a random Ising term, an anisotropic and antisymmetric, but deterministic Dzyaloshinskii-Moriya (DM) term, and a Zeeman coupling to a random, inhomogeneous magnetic field. Since we are specially interested in crossovers involving a Gaussian symplectic ensemble (GSE) limit, we carry out all our calculations with an odd number of lattice sites (spins) that naturally results in eigenspectra with Kramers degeneracies (KD's). The various crossovers (viz., the reentrant Poissonian-to-GSE-to-Poissonian, Poissonian-to-GUE, GSE-to-GUE and the reentrant Poissonian-to-GOE-to-Poissonian crossovers) are investigated via detailed studies of both short-range (nearest-neighbour-spacings distribution, NNSD) and long-range [spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}(L)$ and number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}(L)$ ] spectral correlations, where
L is the spectral interval over which the long-range statistic is examined. The short-range studies show excellent agreement with RMT predictions. One of the highlights of this study is the systematic investigation of the consequences of retaining both eigenvalues corresponding to every Kramers doublet, in a crossover involving the GSE limit, and see how it evolves to a limit where the KD is naturally lifted. This is seen most clearly in the NNSD study of the GSE-to-GUE (Gaussian unitary ensemble) transition, achieved by gradually lifting the KD, using the random magnetic field. The NNSD plot in the GSE limit here exhibits a Dirac delta peak at zero splitting and a renormalized GSE hump at finite splitting, whose general analytical form and its asymptotic limit are derived. With an increasing symmetry breaking magnetic field the NNSD shows an interesting, dynamic two-peaked structure that finally converges to the standard GUE lineshape. We explain this trend in terms of a competition between the splittings amongst distinct Kramers doublets (related to unitary symmetries) and the Zeeman-like splittings induced by a breaking of the antiunitary time-reversal symmetry (TRS). This is investigated via the NNSD, the marginal spectral density (MSD) and the densities of states (DOS) for both spin models and RMT crossover matrix models. The first and the final short-range studies involve reentrant Poissonian-to-GOE(GSE)-to-Poissonian crossovers, where the final Poissonian is obtained by a many-body localization of states in the strongly disordered limit, whereas the initial Poissonian regime involve much more delocalized eigenstates. In the long-range spectral correlation studies, we shed light on the extent of agreement between our physical spin systems and RMT predictions. We find that the spin systems depart from the ideal RMT predictions for relatively finite $L\ensuremath{\sim}10\ensuremath{-}15$ at least, for the spectral rigidity and a much smaller $L\ensuremath{\sim}2\ensuremath{-}4$ for the number variance. It is further seen that the departure is usually sooner at the uncorrelated (Poissonian) upper end compared to the correlated (Wigner-Dyson) lower end. We carry out a detailed comparison between the local and the global crossover points, associated with the short-range and the long-range statistics respectively, and find that in most cases they seem to agree reasonably well, but for a few exceptions. Our studies also show that the long-range correlations may serve to distinguish between the two Poissonian limits (nonlocalized and localized) in the reentrant crossovers, which the short-range correlations fail to distinguish.