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Physica Scripta

Quantum spin chains in aperiodic magnetic fields

Juan Carlos Chaves, Indubala I. Satija, Jukka A. Ketoja

Published Dec 1, 1995• 1 citation

Abstract

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.

DOI: 10.1088/0031-8949/52/6/002

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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.

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.

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.

We study the one-dimensional quantum Ising model in a transverse magnetic field where the exchange couplings are ordered according to the Thue-Morse (TM) sequence. At zero temperature, this model is equivalent to a two-dimensional classical Ising model in a magnetic field with TM aperiodicity along one direction. We compute the order parameter (magnetization) of the chain and the scaling behavior of the energy spectrum when the system undergoes a phase transition. Analogous to the quasiperiodic (QP) quantum Ising chain, the onset of long-range order is signaled by a nonanaliticity in the exponent \ensuremath{\delta} which describes the scaling of the total bandwidth with the size of the chain. The critical spin-coupling can be computed analytically and it is found to be lower than the QP case. Furthermore, the energy bands are found to be narrower than the corresponding QP chain. The former and latter results are consistent with the fact that the present structure has a degree of ordering intermediate between QP and random.

The zero-temperature phase diagram of a quasiperiodic anisotropic

\mathrm{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.

Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems and disordered ones as well. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization-group transformation. These discrete sequences are therefore fixed points of a functional renormalization group. This observation allows for an asymptotically exact treatment of the critical points. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains, as well as a phenomenological model for the quasiperiodic many-body localization transition.