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Estudo da densidade espectral de uma cadeia de Ising com campo transverso

Conrado Luiz da Silveira Folly

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

In this work, we study the frequency spectrum of a spins chain in the presence of a transverse magnetic field B using the Ising model for the interaction between spins in one dimension, considering only the magnetic interactions, and assuming that the system is all the time at high temperatures. We show that for a chain of 11 spins in the presence of an external transverse magnetic field B, the dynamics of the system closely approximate the asymptotic limit of a system of infinite particles. The methodology consists of the diagonalization of the Hamiltonian of the system, by taking into account the interaction J between neighboring spins in a chain, together with the interaction of each spin with an external field transverse field B. We exactly determine the eigenvectors and eigenvalues of the system. Then we calculate the time-dependent autocorrelation function. Finally, we perform a Fourier transform of the autocorrelation function to find the frequency dependent spectral density. We obtained exact analytic results for the case of two spins, and numerically exact results for systems with up to 11 spins. Our results for the spectral densities covering three significative values of the energy parameters J and B are already very close to the thermodynamic limit. For B > J the dynamics is of collective-mode type where the spins tend to align with the external field.

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Physical review. E•Apr 3, 2020•19 citations
In this work, we study the frequency spectrum of a spins chain in the presence of a transverse magnetic field B using the Ising model for the interaction between spins in one dimension, considering only the magnetic interactions, and assuming that the system is all the time at high temperatures. We show that for a chain of 11 spins in the presence of an external transverse magnetic field B, the dynamics of the system closely approximate the asymptotic limit of a system of infinite particles. The methodology consists of the diagonalization of the Hamiltonian of the system, by taking into account the interaction J between neighboring spins in a chain, together with the interaction of each spin with an external field transverse field B. We exactly determine the eigenvectors and eigenvalues of the system. Then we calculate the time-dependent autocorrelation function. Finally, we perform a Fourier transform of the autocorrelation function to find the frequency dependent spectral density. We obtained exact analytic results for the case of two spins, and numerically exact results for systems with up to 11 spins. Our results for the spectral densities covering three significative values of the energy parameters J and B are already very close to the thermodynamic limit. For B > J the dynamics is of collective-mode type where the spins tend to align with the external field.
In this work we present a study of the dynamics of an Ising chain ( in the presence of a transverse magnetic field. The model involves only short-range interactions and is described by the interaction of nearest neighbors between two bodies in the chain. We calculate the time-dependent autocorrelation function using the exact diagonalization method. The method consists in considering finite spin chains with increasing sizes to determine the time-dependent autocorrelation function of endless chain. The results for the biggest system studied indicate that the asymptotic behavior in thermodynamical systems for large time has been reached. This means that if we consider larger chains, the results will be the same within the numerical accuracy presented here. The autocorrelation function is oscillatory for large values of the transverse field and is monotonically decreasing for small field values. The system undergoes a crossover from a collective mode excitation scheme to a dynamics of central mode type as the magnetic field is reduced.
We study the effects of next-nearest-neighbor (NNN) interactions on the dynamics of the one-dimensional spin-1/2 transverse Ising model in the high-temperature limit. We use exact diagonalization to obtain the time-dependent transverse correlation function and the corresponding spectral density for a tagged spin. Our results for chains of 13 spins with periodic boundary conditions produce results which are valid in the infinite-size limit. In general we find that the NNN coupling produces slower dynamics accompanied by an enhancement of the central mode behavior. Even in the case of a strong transverse field, if the NNN coupling is sufficiently large, then there is a crossover from collective mode to central mode behavior. We also obtain several recurrants for the continued fraction representation of the relaxation function.
We study the time dependent behavior of the four-spin interactions Ising chain in the presence of a disordered transverse magnetic field. In order to analyze the effects of field randomness in the dynamical behavior of the system we calculate the time-dependent correlation function and the longitudinal relaxation function of the infinite chain. We consider a bi-modal distribution for the field energy strength, where one of its values equals the exchange coupling energy while the other value can be lower or higher than the coupling energy. We find that an increasing concentration of weaker fields produces an enhancement of the central mode behavior. However, when the concentration of stronger fields increases, the system undergoes a crossover from central mode to collective-type dynamics.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian:
We present new results on the time-dependent correlation functions Ξn(t) =4〈Sξ0(t)Sξn〉, ξ=x,y at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising model (TI) at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−J∑l[(1+γ)SxlSxl+1 +(1−γ)SylSyl+1 +hSzl]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions (ACF’s) Ξ0(t) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y have singularities at the infinite sequence of frequencies ω=mω0, m=0, 1, 2, 3, ... where ω0=J for the XY model and ω0=2J for the TI model. In both models the singularities in Φxx0 (ω) for m=0, 1 are divergent, whereas the nonanalyticities at higher frequencies become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis.
Multiple-spin coupling is a key tool for investigating magnetic quantum systems. In particular, models with four-spin interactions became of great interest since they have been applied to describe material properties such as superconductivity. In this framework, the dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction in a transverse field is investigated. By means of the recurrence relations method, the first four recurrants are determined exactly for the general model, while five exact recurrants are calculated for two particular models. In all cases, higher-order recurrants are computed through extrapolation to obtain time-dependent autocorrelation functions and spectral densities in the long-time regime and infinite-temperature limit. It is found that the relaxation functions decay slower than the expected behavior of the one-dimensional transverse Ising model. The spectral lines, characterized by a Lorentzian behavior in the central body and a Gaussian shape in the tails, display exchange narrowing as the coupling intensities increase.