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Physical review. B, Condensed matter

Symmetry breaking and stabilization of critical phase

Indubala I. Satija

Published Aug 1, 1993• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.3511

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Physica Scripta•Dec 1, 1995•1 citation
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.
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.
The critical breakdown of a one-dimensional quantum magnet with long-range interactions is studied by investigating the high-field polarized phase of the anisotropic XY model in a transverse field for the ferromagnetic and antiferromagnetic cases. While for the limiting case of the isotropic long-range XY model we can extract the elementary one-quasiparticle dispersion analytically and calculate two-quasiparticle excitation energies quantitatively in a numerical fashion, for the long-range Ising limit as well as in the intermediate regime we use perturbative continuous unitary transformations on white graphs in combination with classical Monte Carlo simulations for the graph embedding to extract high-order series expansions in the thermodynamic limit. This enables us to determine the quantum-critical breakdown of the high-field polarized phase by analyzing the gap closing including associated critical exponents and multiplicative logarithmic corrections. In addition, for the ferromagnetic isotropic XY model we determined the critical exponents zzz and $\ensuremath{\nu}$ analytically by a bosonic quantum-field theory.
The zero-temperature phase diagram of a quasiperiodic anisotropic XY\mathrm{XY}XY quantum spin chain in a transverse magnetic field is studied. In the isotropic limit, the system exhibits an infinite number of quantum phase transitions with a devil's-staircase-like magnetization curve. Anisotropy decreases the number of transitions. A mechanism is proposed for the successive transitions based on the ordering of finite clusters of spins along the chain.
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 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.
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.