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Physical review. B, Condensed matter

Effects of quenched disorder on spin-1/2 quantumXXZchains

Curtis A. Doty, Daniel S. Fisher

Published Feb 1, 1992• 123 citations
Publisher

Abstract

We analyze the zero-temperature phase diagram of the spin-1/2 quantum XXZ chain in the presence of weak disorder. The effects of various random perturbations are considered, including random fields and random exchange. For random perturbations that preserve the XY symmetry, we find a phase transition, as the anisotropy parameter is varied, from a ground state with quasi-long-range order (as in the pure system) to one in which typical correlation functions decay rapidly. The critical behavior at this transition is shown to be in the universality class of the Giamarchi-Schulz transition for one-dimensional bosons in a random potential. Random perturbations that break the XY symmetry are found always to destroy the quasi-long-range order of the ground state. Properties of the resulting random phases are also discussed.

DOI
10.1103/physrevb.45.2167

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We analyze the zero-temperature phase diagram of the spin-1/2 quantum XXZ chain in the presence of weak disorder. The effects of various random perturbations are considered, including random fields and random exchange. For random perturbations that preserve the XY symmetry, we find a phase transition, as the anisotropy parameter is varied, from a ground state with quasi-long-range order (as in the pure system) to one in which typical correlation functions decay rapidly. The critical behavior at this transition is shown to be in the universality class of the Giamarchi-Schulz transition for one-dimensional bosons in a random potential. Random perturbations that break the XY symmetry are found always to destroy the quasi-long-range order of the ground state. Properties of the resulting random phases are also discussed.
The one-dimensional XXZ model is studied in the presence of disorder in the Heisenberg exchange integral. Recent predictions obtained from renormalization-group calculations are investigated numerically using a Lanczos algorithm on chains of up to 18 sites. It is found that in the presence of strong X-Y-symmetric random-exchange couplings, a ``random singlet'' phase with quasi-long-range order in the spin-spin correlations persists. As the planar anisotropy is varied, the full zero-temperature phase diagram is obtained and compared with predictions of Doty and Fisher [Phys. Rev. B 45, 2167 (1992)]. In addition, we observed a novel reentrant transition of the ordered phases when exchange disorder is included.
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.
We analyze an anisotropic spin-1/2 two-leg ladder in the presence of various types of random perturbations. The generic phase diagram for the pure system, in a way similar to spin one chains, consists of four phases: an antiferromagnet, a Haldane gap or a singlet phase (depending on the sign of interchain coupling), and two XY\mathrm{XY}XY phases designated by XY1\mathrm{XY}1XY1 and XY2\mathrm{XY}2XY2. The effects of disorder depend crucially on whether it preserves XY\mathrm{XY}XY symmetry (random field along zzz and random exchange) or not (random anisotropy or random XY\mathrm{XY}XY fields). In all cases we computed the new phase diagram and the correlation length for the disordered system. The ladder exhibits a remarkable stability to disorder with XY\mathrm{XY}XY symmetry. Not only the singlet phases but also the massless XY1\mathrm{XY}1XY1 phase are totally unaffected by disorder in stark contrast with the single chain case. Even in the presence of an external magnetic field breaking the spin gap most of the phase diagram (including the XY\mathrm{XY}XY point) remains unaffected by the disorder, again in opposition with the single chain case. This stabilization towards disorder is similar, albeit stronger, to the one occurring for fermionic ladders. On the other hand, the XY2\mathrm{XY}2XY2 phase is very strongly suppressed. Disorder breaking XY\mathrm{XY}XY symmetry has much stronger effects and most of the phases at the exception of the singlet one are now destroyed. Interestingly, the disordered XY1\mathrm{XY}1XY1 phase has a much shorter correlation length than the disordered XY2\mathrm{XY}2XY2 phase contrarily with the case of perturbations preserving the rotation symmetry around the zzz axis. The ladder system is thus in fact much more anisotropic than its single chain counterpart with identical exchanges. Finally we examine the case of strong disorder or weak interchain exchange for XY\mathrm{XY}XY symmetric disorder. Close to the isotropic point, when the interchain exchange is increased the system undergoes a transition between two decoupled disordered spin-1/2 chains to a singlet ladder unaffected by disorder. For more XY\mathrm{XY}XY like systems, the transition occurs in two steps with a nondisordered XY1\mathrm{XY}1XY1 phase occurring between the decoupled chains and the singlet phase. Comparison with other disordered gapped systems such as spin one chains is discussed.
We study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
We investigate the effect of quenched bond disorder on the anisotropic antiferromagnetic spin-1/2 (XXZ) chain as a model for disorder-induced quantum phase transitions. We find nonuniversal behavior of the average correlation functions for weak disorder, followed by a quantum phase transition into a strongly disordered phase with only short-range xy correlations. We find no evidence for the universal strong-disorder fixed point predicted by the real-space renormalization group, suggesting a qualitatively different view of the relationship between quantum fluctuations and disorder.
The properties of spin-1/2 antiferromagnetic chains with various types of random exchange coupling are studied via an asymptotically exact decimation renormalization-group transformation, which is a generalization of that introduced by Dasgupta and Ma. Random-singlet phases occur in which each spin is paired with one other spin that may be very far away; more exotic phases also occur. The behavior of typical and mean correlation functions is analyzed and found to be very different, with very small sets of spins dominating the latter at long distances as well as the low-temperature thermodynamics. Some of the phase transitions that occur between antiferromagnetically ordered phases and random singlet or other antiferromagnetic phases are also analyzed. For example, if a small uniaxial anisotropy perturbation is added to a random Heisenberg antiferromagnetic chain, a transition occurs from a random-singlet phase to an Ising antiferromagnetic phase, as the anisotropy changes sign from easy plane to easy axis. The staggered magnetization vanishes at the transition with critical exponent \ensuremath{\beta}=8/(1+ \ensuremath{\surd}7 ). Possible implications for the properties of random quantum magnetic systems in higher dimensions are briefly discussed.