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physica status solidi (b)

Short‐Time Behaviour of the Transverse Auto‐Correlation Function of the 1D XY‐Model

G. A. Farias, L.L. Gonçalves

Published Jan 1, 1987• 3 citations
Publisher

Abstract

Abstract The dynamics of the spin 1/2 one‐dimensional XY‐model in a transverse field is considered. The short‐time behaviour of the auto‐correlation function 〈S (t) S (0)〉 for the closed chain is obtained numerically by considering the exact results known for the open chain which are given in terms of Pfaffians. The bulk auto‐correlation is identified with the correlation 〈S (t) S (0)〉 for the open chain, and this identification is shown to be valid up to a characteristic time Jt ≅ 20. It is also shown, as expected, that the boundary effects are more pronounced for fields smaller than the exchange constant (h < |J|) and in the low temperature limit. The results are obtained for different temperatures and field strengths, and the numerical convergence is discussed in each case.

DOI
10.1002/pssb.2221390131

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Abstract The dynamics of the spin 1/2 one‐dimensional XY‐model in a transverse field is considered. The short‐time behaviour of the auto‐correlation function 〈S (t) S (0)〉 for the closed chain is obtained numerically by considering the exact results known for the open chain which are given in terms of Pfaffians. The bulk auto‐correlation is identified with the correlation 〈S (t) S (0)〉 for the open chain, and this identification is shown to be valid up to a characteristic time Jt ≅ 20. It is also shown, as expected, that the boundary effects are more pronounced for fields smaller than the exchange constant (h < |J|) and in the low temperature limit. The results are obtained for different temperatures and field strengths, and the numerical convergence is discussed in each case.
The authors study the time-dependent correlations of the free-ends isotropic XY model in a transverse field. The boundary effects on the longitudinal correlation, namely (S j z (t)S l z (0)), and bulk dynamic properties are considered at the temperature limits T= infinity and T=0- (for h>or= mod J mod ) where the results are known in closed form. The transverse correlation namely (S j x (t)S l x (0)) is calculated exactly at T=0 (for h>or= mod J mod ), and the boundary effects discussed. At T= infinity the known result for the bulk correlation is rederived by using a new combination of fermion operators.
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:
The paper presents a numerical approach for the calculation of thermodynamics and spin correlations for spin-½ XY chains. To illustrate the approach we performed exact finite-chain calculations of all time-dependent two-spin correlation functions for the spin-½ Ising chain in the transverse field. The computed correlation functions permit to evaluate the frequency-dependent susceptibilities and structure factors for this model. The obtained results are discussed in detail. They may have an application for the interpretation of dynamic experiments on quasi-one-dimensional crystals that can be treated within the frames of the spin-½ transverse Ising model.
Results are presented for the static and time-dependent behavior of the classical one-dimensional XY-model at finite temperatures. Equilibrium properties were determined using an importance sampling Monte Carlo method. A spin-dynamics computer simulation method was then used to evaluate spin-spin and energy-energy time-displaced correlation functions out to time t=10J−1 using initial states chosen from the equilibrium distribution. The transverse spin-spin auto correlation function decreases smoothly as a function of time. The longitudinal spin-spin correlation functions, however, show temperature dependent oscillatory behavior. At moderately high temperature the energy-energy correlations show diffusive t−1/2 behavior at long times with only a slow temperature dependence of the diffusion constant.
Dynamic xx and xy spin pair correlation functions for the isotropic spin-½ XY chain are calculated numerically for long open chains in the presence of a transverse ( z ) magnetic field at finite temperature. From these data the first numerical results for the dynamic structure factors S xx (κ,ω) and S xy (κ,ω) are obtained and compared with the known results for S zz (κ,ω). While S zz (κ,ω) is restricted to certain regions in the (κ,ω)-plane S xx (κ,ω) and S xy (κ,ω) are not. Nevertheless, the numerical results show that the latter structure factors are quite small outside the regions defined by S zz (κ,ω) and moreover, that they may in many circumstances, especially at low temperature, be approximately described by a small number of broadened excitation branches.