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Physical review. B, Condensed matter

Dynamic critical behavior of random spin chains in a field

Iveta R. Pimentel, Robin B. Stinchcombe

Published Sep 1, 1989• 9 citations
Publisher

Abstract

The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-${H}_{c}$), where ${H}_{c}$ is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=${k}^{z}$f(\ensuremath{\Delta}/${k}^{\mathrm{cphi}}$), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors: ${k}^{z}$ for \ensuremath{\Delta}\ensuremath{\ll}${k}^{\mathrm{cphi}}$ and ${h}^{\ensuremath{\delta}}$ for \ensuremath{\Delta}\ensuremath{\gg}${k}^{\mathrm{cphi}}$, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.

DOI
10.1103/physrevb.40.4947

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The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-Hc{H}_{c}Hc​), where Hc{H}_{c}Hc​ is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=kz{k}^{z}kzf(\ensuremath{\Delta}/kcphi{k}^{\mathrm{cphi}}kcphi), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors: kz{k}^{z}kz for \ensuremath{\Delta}\ensuremath{\ll}kcphi{k}^{\mathrm{cphi}}kcphi and h\ensuremathδ{h}^{\ensuremath{\delta}}h\ensuremathδ for \ensuremath{\Delta}\ensuremath{\gg}kcphi{k}^{\mathrm{cphi}}kcphi, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.
The transverse dynamics of a one-dimensional vector spin glass at zero temperature is treated by a new transfer-matrix scaling technique, and also by evaluating static critical properties related to dynamics by a crossover argument. A nontrivial dynamic exponent is found which implies that the system at zero temperature has dynamic critical behavior w\ensuremath{\propto}k3/2{k}^{3/2}k3/2, in contradiction to the hydrodynamic behavior normally assumed and to the results of previous work.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
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 transverse dynamics of a vector spin glass chain at zero temperature is studied by a numerical scaling method which provides a new physical insight into the nature of the spin excitations. In the range of low energies considered we find that the states are all localized with the coupling between spins V decaying asymptotically for large lengths separating the sites l , like V ∼ exp[- l /ξ], where ξ measures the localization length. A dispersion relation is found between the energies and the associated inverse localization lengths of the form ω ∝ (1/ξ) z with the dynamic exponent z = 1.501 ± 0.044. This implies that the system at zero temperature has nontrivial dynamic critical behaviour, in agreement with our recent analytical results; it moreover gives for the first time a clear interpretation for the characteristic length of the spin fluctuations as the localization length.
Low-energy excitations of a classical XY\mathrm{XY}XY ferromagnetic chain in a random sign transverse field are investigated using the coherent-potential approximation (CPA). Near the critical field strength, the dynamics of the system is similar to that of the high-field spin-glass chain system investigated earlier. The CPA reproduced the anomalous dispersion law and nontrivial dynamic and crossover exponents which were obtained by the renormalization-group technique. The CPA also predicts an explicit form of the scaling function. For symmetric and asymmetric distributions of the field, different dynamics and exponents are obtained. A brief analysis is given about the thermodynamics of the system.