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Physical review. B, Condensed matter

Spectral and magnetic interplay in quantum spin chains: Stabilization of the critical phase due to long-range order

Indubala I. Satija

Published Feb 1, 1994• 17 citations
Publisher

Abstract

The anisotropic quantum XY spin chain in a modulating magnetic field incommensurate with the periodicity of the chain is studied. Using Jordan-Wigner transformation, the model is mapped to a tight-binding model in fermions. In the absence of symmetry-breaking terms, the fermion model is identical with the well-known Harper equation exhibiting extended and localized phases with the onset of transition being a critical point with fractal spectrum and wave functions. The effect of O(2) symmetry breaking is to fatten the critical point resulting in a critical phase with power-law localization existing in a finite window of size determined by the anisotropy. In this three-phase spectral diagram, the transition from critical to localized phase is accompanied by the magnetic transition to long-range order. Furthermore, the scaling properties of the fractal spectrum vary in the critical phase. This leads to interesting consequences on the low-temperature thermodynamical properties of the system.

DOI
10.1103/physrevb.49.3391

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The anisotropic quantum XY spin chain in a modulating magnetic field incommensurate with the periodicity of the chain is studied. Using Jordan-Wigner transformation, the model is mapped to a tight-binding model in fermions. In the absence of symmetry-breaking terms, the fermion model is identical with the well-known Harper equation exhibiting extended and localized phases with the onset of transition being a critical point with fractal spectrum and wave functions. The effect of O(2) symmetry breaking is to fatten the critical point resulting in a critical phase with power-law localization existing in a finite window of size determined by the anisotropy. In this three-phase spectral diagram, the transition from critical to localized phase is accompanied by the magnetic transition to long-range order. Furthermore, the scaling properties of the fractal spectrum vary in the critical phase. This leads to interesting consequences on the low-temperature thermodynamical properties of the system.
Anisotropic quantum XY spin chains in a quasiperiodic (QP) transverse field provide a very interesting and a rich class of one-dimensional models exhibiting extended (E), localized (L), and critical (C) phases in addition to a magnetic transition to long-range order. The interesting feature in this class of models is the fact that the intermediate C phase exists in a finite window of size determined by the magnitude of spin-space anisotropy. Furthermore, even a small amount of anisotropy in the spin space destroys the fractal phase boundary between the E and L phases of the isotropic model.
A renormalization scheme is developed to study an anisotropic quantum XY spin chain in a quasiperiodic transverse field. The critical phase of the quasi-particle excitations of the model with fractal wave functions exists in a finite parameter interval and is sandwiched between the extended and localized phases. The scaling properties of the critical phase fall into four distinct universality classes referred as spectral, subconformal, conformal and Harper. The spectral and conformal classes respectively describe the onsets of extended to critical and critical to localized transitions while the subconformal class describes the part of the phase diagram sandwiched between the conformal and spectral transitions. The Harper universality class describes the isotropic limit of the XY model. A decimation scheme is developed to compute the infinite sets of universal scaling ratios characterizing the wave functions in the four universality classes. The renormalization flow equations exhibit a limit cycle at the band center and at the band edges providing a new method for determining these energies with extremely high precision.
The XY-to-Ising crossover is studied in an anisotropic quantum XY spin chain where a modulating transverse magnetic field introduces a competing periodicity in the system. In the fermion representation, the model describes the perturbed Harper model, exhibiting a critical phase in addition to the extended and the localized phases. The XY-to-Ising crossover results in quadrupling of the butterfly spectrum, which lacks the full symmetry of the Harper butterfly spectrum. However, for infinitesimal field, the Ising butterfly spectrum asymptotically approaches the Harper butterfly spectrum. A duality transformation provides some insight into the various phases of the butterfly spectrum.
We study anisotropic quantum spin chains in an aperiodic magnetic field h n = λ cos(2πσ n v ); for 0 < v < 1. For v equal to unity and σ equal to the golden mean, the system is quasiperiodic. In the isotropic limit, the incommensurate behaviour is described by the Harper equation exhibiting the metal-insulator transition whereas in the anisotropic case a new fat critical phase is observed between the extended and localized phases. For v different from unity, the global extended phase of the quasiperiodic anisotropic model is replaced by the metal-insulator transition with mobility edges. The mobility edge corresponding to the isotropic model exhibits a novel intermittency. In addition, a new mobility edge with no analog in the isotropic model appears at energy corresponding to the spin space anisotropy g . Instead of the fat critical phase of the quasiperiodic model we now observe a phase which is localized everywhere except at the energy equal to g . There is no localization-delocalization transition in this case. Various high precision numerical tools suggest the existence of critical states at the onset of metal-insulator transition, along the line where the energy is equal to g as well as at the conformally invariant point. We conjecture that the existence of critical states at a conformal point is a universal phenomenon for all aperiodic potentials.
Abstract In this study, a spin-1/2 extended anisotropic XY chain has been introduced in which both time reversal and SU(2) symmetries are broken but Z 2 symmetry is preserved. Magnetic and topological phase diagrams in the parameter space have been drawn in the presence of transverse magnetic field. Entanglement measures like mutual information and quantum discord are also evaluated and it indicates that these transitions are second order in nature. Quantum phase transition is noted at zero magnetic field, as well as magnetic long range order is found to withstand magnetic field of any strength. Exact analytic results for spin-spin correlation functions have been obtained in terms of Jordan-Wigner fermionization. Existence of long range magnetic order has been investigated numerically by finding correlation functions as well as the Binder cumulant in the ground state. Dispersion relation, ground state energy, and energy gap are obtained analytically. In order to find the topologically nontrivial phase, sign of Pfaffian invariant and value of winding number have been evaluated. Both magnetic and topological phases are robust against the magnetic field and found to move coercively in the parameter space with the variation of its strength. Long range orders along two orthogonal directions and two different topological phases are found and their one-to-one correspondence has been found. Finally casting the spinless fermions onto Majorana fermions, properties of zero energy edge states are studied. Three different kinds of Majorana pairings are noted. In the trivial phase, next-nearest-neighbor Majorana pairing is found, whereas two different types of nearest-neighbor Majorana pairings are identified in the topological superconducting phase.
We study the phase diagram of a spin-1 antiferromagnetic chain with isotropic antiferromagnetic interactions decaying with a power law $\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ $(\ensuremath{\alpha}\ensuremath{\ge}1)$ accompanied by modulated single-ion anisotropy. Employing the techniques of the density-matrix renormalization group, the effects of long-range interactions and single-ion anisotropy on a variety of correlations are investigated. In order to check the consistency, the fidelity susceptibilities are evaluated across quantum phase transitions. The quantum critical points are faithfully detected and orders of phase transitions are determined. The correlation-length critical exponent is extracted from scaling functions of the fidelity susceptibility. The presence of long-range interactions leads to a quantitative change of the phase boundaries and reduces the order of the phase transition under certain conditions. A direct first-order transition between the periodic N\'eel phase and the large-DDD phase occurs for slowly decaying antiferromagnetic interactions.