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Physical review. B, Solid state

Dynamical correlation functions of the transverse spin and energy density for the one-dimensional spin-½ Ising model with a transverse field

T. N. Tommet, David L. Huber

Published Jan 1, 1975• 28 citations
Publisher

Abstract

Dynamical correlation functions for the transverse (along the direction of the field) spin and energy densities are calculated for the one-dimensional spin-\textonehalf{} Ising model with a transverse field $H=\ensuremath{-}\ensuremath{\Gamma}{\ensuremath{\Sigma}}_{i}{S}_{i}^{x}\ensuremath{-}J{\ensuremath{\Sigma}}_{i}{S}_{i}^{z}{S}_{i+1}^{z}$. Explicit results are obtained for $〈{{S}^{x}(q, \ensuremath{\omega}){S}^{x}(\ensuremath{-}q)}〉$, $〈{E(q, \ensuremath{\omega}), E(\ensuremath{-}q)}〉$, and their Fourier transforms in the limits $T=\ensuremath{\infty}$ and $T=0$ and for the autocorrelation functions $〈{{S}_{i}^{x}, {S}_{i}^{x}(t)}〉$ and $〈{{E}_{i}, {E}_{i}(t)}〉$ in the same limits. The results are discussed with special attention given to the coupling of the transverse spin component and energy densities.

DOI
10.1103/physrevb.11.450

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Quantum Critical Dynamics of the Random Transverse Field Ising Spin Chain

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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.
The behavior at finite temperatures of the dynamical transverse autocorrelation of a spin is investigated for two special cases of the one-dimensional XY\mathrm{XY}XY model. We find for temperatures restricted by $\frac{J}{\mathrm{kT}}>{e}^{\ensuremath{-}2\mathrm{Jt}}$ that its large-time behavior is given by $〈{S}_{i}^{x}(t){S}_{i}^{x}〉\ensuremath{\sim}C(\mathrm{kT})\mathrm{exp}[\ensuremath{-}\frac{1}{2}{J}^{2}{t}^{2}+\frac{2{J}^{2}{t}^{2}}{\mathrm{ln}(\mathrm{Jt})}]$ for the transverse Ising model with field equal to exchange constant and by its square for the isotropic XY\mathrm{XY}XY model without field.
The dynamics of the three-dimensional anisotropic spin-\textonehalf{} Ising model in a transverse field is studied above Tc{T}_{c}Tc​ by a method in which the coupled-chain problem is reduced to that of a single chain in an effective field. The problem of a single chain is studied by means of Mori's continued-fraction expansion in a four-pole approximation. The moments are obtained by using the time-dependent correlation function calculated in the ccc-cyclic and in the improved ccc-cyclic approximations [J. Phys. A 13, 223 (1980)]. A comparison between these two approximations is done by calculating the critical temperature as function of the tunneling field. The one-dimensional correlation length is obtained, and it agrees well with the experimental results for the quasi-one-dimensional ferroelectric CsD2{\mathrm{D}}_{2}D2​PO4{\mathrm{O}}_{4}O4​.
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: