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Journal of Applied Physics

Random-exchange and random-field x y chain for s=1/2

T. Schneider, Antonio Politi

Published Apr 15, 1987• 1 citation
Publisher

Abstract

One of the points of interest in the study of random systems is the nature of the eigenstates, i.e., exponentially localized or extended, and the density of states. To identify the disorder-induced effects, we consider the one-dimensional s=1/2 xy models with random exchange, or random field. Using the transformation to spinless fermions, the problem is reduced to a nonlinear map, providing accurate numerical estimates for the integrated density of states and the exponential localization length. The results clearly reveal: (i) appearance of disorder-induced tails in the integrated density of states at the bottom and top of the spectrum, yielding to a characteristic field dependence of magnetization and susceptibility at zero temperature; and (ii) important differences between random-exchange and random-field models. In the random-exchange case, the middle of the band is found to be extended, while in the random-field case, corresponding to the standard Anderson model, all states are exponentially localized. This difference also affects the temperature dependence of zero-field specific heat and susceptibility as T→0.

DOI
10.1063/1.338569

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One of the points of interest in the study of random systems is the nature of the eigenstates, i.e., exponentially localized or extended, and the density of states. To identify the disorder-induced effects, we consider the one-dimensional s=1/2 xy models with random exchange, or random field. Using the transformation to spinless fermions, the problem is reduced to a nonlinear map, providing accurate numerical estimates for the integrated density of states and the exponential localization length. The results clearly reveal: (i) appearance of disorder-induced tails in the integrated density of states at the bottom and top of the spectrum, yielding to a characteristic field dependence of magnetization and susceptibility at zero temperature; and (ii) important differences between random-exchange and random-field models. In the random-exchange case, the middle of the band is found to be extended, while in the random-field case, corresponding to the standard Anderson model, all states are exponentially localized. This difference also affects the temperature dependence of zero-field specific heat and susceptibility as T→0.
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We present some exact results for the effect of disorder on the critical properties of an anisotropic XY\mathrm{XY}XY spin chain in a transverse field. The continuum limit of the corresponding fermion model is taken and in various cases results in a Dirac equation with a random mass. Exact analytic techniques can then be used to evaluate the density of states and the localization length. In the presence of disorder the ferromagnetic-paramagnetic or Ising transition of the model is in the same universality class as the random transverse field Ising model solved by Fisher using a real-space renormalization-group decimation technique (RSRGDT). If there is only randomness in the anisotropy of the magnetic exchange then the anisotropy transition (from a ferromagnet in the x direction to a ferromagnet in the y direction) is also in this universality class. However, if there is randomness in the isotropic part of the exchange or in the transverse field then in a nonzero transverse field the anisotropy transition is destroyed by the disorder. We show that in the Griffiths' phase near the Ising transition that the ground-state energy has an essential singularity. The results obtained for the dynamical critical exponent, typical correlation length, and for the temperature dependence of the specific heat near the Ising transition agree with the results of the RSRGDT and numerical work.
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We investigate equilibrium statistical properties of the isotropic quantum XY spin-1/2 model in an external magnetic field when the interaction and field parts are subjected to quenched or annealed disorder or both. The randomness present in the system are termed annealed or quenched depending on the relation between two different time scales---the time scale associated with the equilibration of the randomness and the time of observation. Within a mean-field framework, we study the effects of disorders on spontaneous magnetization, both by perturbative and numerical techniques. Our primary interest is to understand the differences between quenched and annealed cases, and also to investigate the interplay when both of them are present in a system. We find that the magnetization survives in the presence of a unidirectional random field, irrespective of its nature, i.e., whether it is quenched or annealed. However, the field breaks the circular symmetry of the magnetization, and the system magnetizes in specific directions, parallel or transverse to the applied magnetic field. Interestingly, while the transverse magnetization is affected by the annealed disordered field, the parallel one remains unfazed by the same. Moreover, the annealed disorder present in the interaction term does not affect the system's spontaneous magnetization and the corresponding critical temperature, irrespective of the presence or absence of quenched or annealed disorder in the field term. We carry out a comparative study of these and all other different combinations of the disorders in the interaction and field terms, and point out their generic features.
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