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.
We investigate the ground-state property of the one-dimensional spin-1/2 antiferromagnetic X X Z chain in a symmetry-breaking field, by means of numerical diagonalization method for finite-size spin systems combined with the phenomenological renormalization group method. It is found that, with increasing field, there occurs the ground-state phase transition from an ordered phase to a disordered phase, whose critical property belongs to the universality class of the transverse Ising model. The ground-state phase diagram is determined in two parameter regions; (i) an Ising-like region and (ii) an X Y -like region. Our inspection of the field dependence of the energy gap and the static spin structure factors reveals that the ordered phase in both regions exhibits the characteristics of Néel order. Quantum nature of the ground state is also briefly discussed.
We study the random antiferromagnetic spin-1 chain by mapping to a random dimerized spin-1/2 chain with antiferromagnetic and ferromagnetic bonds and applying an asymptotically exact real space renormalization procedure. We find that the chain undergoes an impurity driven second order phase transition from the Haldane phase to the random singlet phase as the bond distribution broadens. In the Haldane phase and near the critical point, there is a Griffiths region in which the excitation gap is filled and the magnetic susceptibility diverges in a nonuniversal manner. The correlation length critical exponent is $\ensuremath{\nu}\ensuremath{\approx}2.3$.
The spin-1/2 quantum anisotropic XY spin chain in a transverse random magnetic field parallel to the z axis is numerically studied by means of the density-matrix renormalization group. The dependence of the spontaneous magnetization and the energy gap on both the strength of the random magnetic field and the factor of anisotropy is determined. The critical line for the order-disorder phase transition is obtained and the resulting phase diagram is drawn. Our results are compatible with the fact that models with different factors of anisotropy fall within the same universality class as the quantum Ising model in a transverse random field.
The spin-1/2 quantum anisotropic
XY spin chain in a transverse random magnetic field parallel to the z axis is numerically studied by means of the density-matrix renormalization group. The dependence of the spontaneous magnetization and the energy gap on both the strength of the random magnetic field and the factor of anisotropy is determined. The critical line for the order-disorder phase transition is obtained and the resulting phase diagram is drawn. Our results are compatible with the fact that models with different factors of anisotropy fall within the same universality class as the quantum Ising model in a transverse random field.
String correlations are investigated in an infinite-size XXZ spin-1 chain. By using the infinite matrix product state representation, we calculate a long-range string order. In the XY phase, the string correlations decay within a relatively very large lattice distance, which makes a finite-size study difficult to verify the non-existence of the string order. Thus, in the Haldane phase, the non-vanishing string correlations in the limit of a very large distance allow to characterize the phase boundaries to the XY phase and the Neel phase, which implies that the transverse long-range string order is the order parameter for the Haldane phase. In addition, the singular behaviors of the von Neumann entropy and the fidelity per lattice site are shown to capture clearly the phase transition points that are consistent with the results from the string order. The estimated critical points including a BKT transition from the XY phase to the Haldane phase agree well with the previous results:
Δc2=0 for the XY-Haldane phase transition and
Δc3=1.185 for the Haldane-N'eel phase transition from the density renormalization group. From a finite-entanglement scaling of the von Neumann entropy with respect to the truncation dimension, the central charges are found to be
c≃1.0 at
Δc2=0 and
c≃0.5 at
Δc3=1.185, respectively, which shows that the XY-Haldane phase transition at
Δc2=0 belongs to the Heisenberg universality class, while the Haldane-Neel phase transition at
Δc2=1.185 belongs to the two-dimensional classical Ising universality class. It is also shown that, the long-range order parameters and the von Neumann entropy, as well as the fidelity per site approach, can be applied to characterize quantum phase transitions as a universal phase transition indicator for one-dimensional lattice many-body systems.