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arXiv:cond-mat/0506233 · Physical Review B

Quantum signatures of the classical topological nonconnectivity threshold

F. Borgonovi, G. L. Celardo, G. P. Berman

Published Dec 12, 2005• 4 citations
arXivHTMLPDF

Abstract

A quantum Heisenberg model with anisotropic coupling and all-to-all interaction has been analyzed using the Bose-Einstein statistics. In F. Borgonovi, G. L. Celardo, M. Maianti, and E. Pedersoli, J. Stat. Phys. 116, 1435 (2004), the existence of a disconnection in the classical phase space has been proved analytically. Such topological disconnection occurs when the energy exceeds a suitable threshold value, called the topological nonconnectivity threshold. We address here the problem of finding quantum signatures of such a threshold. An independent definition of the quantum nonconnectivity threshold, motivated by considerations strictly valid in the quantum regime, is given. We also discuss the dynamical relevance of the quantum border with respect to the quantum magnetic reversal. Contrary to the classical case the magnetization can flip even below the classical threshold through macroscopic quantum tunneling. We evaluate the time scale for magnetic reversal from statistical and spectral properties for a small number of particles and in the semiclassical limit.

DOI
10.1103/physrevb.72.224416

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A quantum Heisenberg model with anisotropic coupling and all-to-all interaction has been analyzed using the Bose-Einstein statistics. In F. Borgonovi, G. L. Celardo, M. Maianti, and E. Pedersoli, J. Stat. Phys. 116, 1435 (2004), the existence of a disconnection in the classical phase space has been proved analytically. Such topological disconnection occurs when the energy exceeds a suitable threshold value, called the topological nonconnectivity threshold. We address here the problem of finding quantum signatures of such a threshold. An independent definition of the quantum nonconnectivity threshold, motivated by considerations strictly valid in the quantum regime, is given. We also discuss the dynamical relevance of the quantum border with respect to the quantum magnetic reversal. Contrary to the classical case the magnetization can flip even below the classical threshold through macroscopic quantum tunneling. We evaluate the time scale for magnetic reversal from statistical and spectral properties for a small number of particles and in the semiclassical limit.
We demonstrate the existence of a topological disconnection threshold, recently found by Borgonovi [J. Stat. Phys. 116, 1435 (2004)], for generic 1-d anisotropic Heisenberg models interacting with an interparticle potential R(-alpha) when 0<alpha<1(here R is the distance among spins). We also show that alpha if is greater than the embedding dimension then the ratio between the disconnected energy region and the total energy region goes to zero when the number of spins becomes very large. On the other hand, numerical simulations in d=2,3 for the long-range case support the conclusion that such a ratio remains finite for large N values. The disconnection threshold can thus be thought of as a distinctive property of anisotropic long-range interacting systems.
Topological phase space disconnection has been recently found to be a general phenomenon in many-body spin system with anisotropic interaction. We show that the power law divergence of magnetic reversal time at the energy signaling such disconnection is generic for long-range interacting systems with an exponent proportional to the number of particles. We also study the modifications induced putting the system in contact with a thermal bath. Using the canonical formalism we analyze the magnetic reversal times at any temperature. Moreover, due to the divergence of reversal time at the energy disconnection threshold we can recover, using saddle point approximation, a simple exponential dependence on the inverse temperature showing the explicit relevance of the energy disconnection threshold for finite many-body interacting systems at finite temperature. This sets a general framework to understand the emergence of ferromagnetism in finite magnetic systems starting from microscopic models without phenomenological on-site barriers.
We study a spin-1 Bose-Einstein condensate small enough to be treated as a single magnetic domain: a system that we term a microcondensate. Because all particles occupy a single spatial mode, this quantum many-body system has a well-defined classical limit consisting of three degrees of freedom, corresponding to the three macroscopically occupied spin states. We study both the classical limit and its quantization, finding an integrable system in both cases. Depending on the sign of the ratio of the spin interaction energy and the quadratic Zeeman energy, the classical limit displays either a separatrix in phase space or Hamiltonian monodromy corresponding to nontrivial phase space topology. We discuss the quantum signatures of these classical phenomena using semiclassical quantization as well as an exact solution using the Bethe ansatz.
A proposal to study topological models beyond the standard topological classification and that exhibit breakdown of Lorentz invariance is presented. The focus of the investigation relies on their anisotropic quantum critical behavior. We study anisotropic effects on three-dimensional (3D) topological models, computing their anisotropic correlation length critical exponent [Formula: see text] obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. A generalized Weyl semimetal model with broken time-reversal symmetry is introduced and studied using a modified Dirac equation. An approach to characterize topological surface states in topological insulators when applied to Fermi arcs allows to capture the anisotropic critical exponent [Formula: see text]. We also consider the Hopf insulator model, for which the study of the topological surface states yields unusual values for [Formula: see text] and for the dynamic critical exponent z. From an analysis of the energy dispersions, we propose a scaling relation [Formula: see text] and [Formula: see text] for [Formula: see text] and z that only depends on the Hopf insulator Hamiltonian parameters p and q and the axis direction [Formula: see text]. An anisotropic quantum hyperscaling relation is also obtained.
Recently topological states of matter have witnessed a new physical phenomenon where both edge modes and gapless bulk coexist at topological quantum criticality. The presence and absence of edge modes on a critical line can lead to an unusual class of topological phase transition between the topological and nontopological critical phases. We explore the existence of this new class of topological phase transitions in a generic model representing the topological insulators and superconductors and we show that such transition occurs at a multicritical point, i.e., at the intersection of two critical lines. To characterize these transitions we reconstruct the theoretical frameworks, which include bound state solution of the Dirac equation, winding number, correlation factors, and scaling theory of the curvature function to work for the criticality. Critical exponents and scaling laws are discussed to distinguish between the multicritical points, which separate the critical phases. Entanglement entropy and its scaling in the real space provide further insights into the unique transition at criticality revealing the interplay between fixed point and critical point at the multicriticalities.