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International Journal of Modern Physics B

THERMAL PHASES IN FINITE QUANTUM SYSTEMS

David J. Dean

Published Nov 10, 2003• 0 citations
Publisher

Abstract

I will describe the behaviour of two different quantum-mechanical systems as a function of increasing temperature. While these systems are somewhat different, the questions addressed are very similar, namely, how does one describe transitions in phase of a finite many-body system; how does one recognise these transitions in practical calculations; and how may one obtain the order of the transition.

DOI
10.1142/s0217979203020223

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I will describe the behaviour of two different quantum-mechanical systems as a function of increasing temperature. While these systems are somewhat different, the questions addressed are very similar, namely, how does one describe transitions in phase of a finite many-body system; how does one recognise these transitions in practical calculations; and how may one obtain the order of the transition.
Many-body localized (MBL) systems lie outside the framework of statistical mechanics, as they fail to equilibrate under their own quantum dynamics. Even basic features of MBL systems, such as their stability to thermal inclusions and the nature of the dynamical transition to thermalizing behaviour, remain poorly understood. We study a simple central spin model to address these questions: a two-level system interacting with strength J with N ≫1 localized bits subject to random fields. On increasing J , the system transitions from an MBL to a delocalized phase on the vanishing scale J c ( N )∼1/ N , up to logarithmic corrections. In the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either ‘on’ (strongly entangled) or ‘off’ (weakly entangled) in eigenstates. The clusters of ‘on’ bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogeneous discontinuous transition out of the localized phase in single-site observables. We obtain these results by perturbative mapping to bond percolation on the hypercube at small J and by numerical exact diagonalization of the full many-body system. Our results support the arguments that the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions d that are high but finite. This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.
This article first gives a concise introduction to quantum phase transitions, emphasizing similarities with and differences to classical thermal transitions. After pointing out the computational challenges posed by quantum phase transitions, a number of successful computational approaches is discussed. The focus is on classical and quantum Monte Carlo methods, with the former being based on the quantum-to classical mapping while the latter directly attack the quantum problem. These methods are illustrated by several examples of quantum phase transitions in clean and disordered systems.
Abstract The uncertainty principle guarantees a non-zero value for the positional uncertainty, ⟨ Δ x 2 ⟩ > 0 , even without thermal fluctuations. This implies that quantum fluctuations inherently enhance positional uncertainty at zero temperature. A natural question then arises: what happens at finite temperatures, where the interplay between quantum and thermal fluctuations may give rise to complex and intriguing behaviors? To address this question, we systematically investigate the positional uncertainty, ⟨ Δ x 2 ⟩ , of a particle in equilibrium confined within a nonlinear potential of the form V ( x ) ∝ x n , where n = 2 , 4 , 6 , … represents an even exponent. Using path integral Monte Carlo simulations, we calculate ⟨ Δ x 2 ⟩ in equilibrium as a function of the thermal de Broglie wavelength Λ. Interestingly, for large values of n, ⟨ Δ x 2 ⟩ exhibits a non-monotonic dependence on Λ: it initially decreases with increasing Λ at small Λ but increases at larger Λ. To further understand this behavior, we employ a semiclassical approximation, which reveals that quantum fluctuations can reduce positional uncertainty for small Λ when the nonlinearity of the potential is sufficiently strong. Finally, we discuss the potential implications of this result for many-body phenomena driven by strong nonlinear interactions, such as glass transitions, where the transition densities exhibit a similar non-monotonic dependence on Λ.