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arXiv:0806.4633 · Physical Review E

Quantum fidelity and thermal phase transitions

H. T. Quan, Fernando Cucchietti

Published Mar 4, 2009• 68 citations

Abstract

We study the quantum fidelity approach to characterize thermal phase transitions. Specifically, we focus on the mixed-state fidelity induced by a perturbation in temperature. We consider the behavior of fidelity in two types of second-order thermal phase transitions (based on the type of nonanaliticity of free energy), and we find that usual fidelity criteria for identifying critical points is more applicable to the case of lambda transitions (divergent second derivatives of free energy). Our study also reveals that for fixed perturbations, the sensitivity of fidelity at high temperatures (where thermal fluctuations wash out information about the transition) is reduced. From the connection to thermodynamical quantities we propose slight variations to the usual fidelity approach that allow us to overcome these limitations. In all cases we find that fidelity remains a good precriterion for testing thermal phase transitions, and we use it to analyze the nonzero temperature phase diagram of the Lipkin-Meshkov-Glick model.

DOI: 10.1103/physreve.79.031101

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We study the quantum fidelity approach to characterize thermal phase transitions. Specifically, we focus on the mixed-state fidelity induced by a perturbation in temperature. We consider the behavior of fidelity in two types of second-order thermal phase transitions (based on the type of nonanaliticity of free energy), and we find that usual fidelity criteria for identifying critical points is more applicable to the case of lambda transitions (divergent second derivatives of free energy). Our study also reveals that for fixed perturbations, the sensitivity of fidelity at high temperatures (where thermal fluctuations wash out information about the transition) is reduced. From the connection to thermodynamical quantities we propose slight variations to the usual fidelity approach that allow us to overcome these limitations. In all cases we find that fidelity remains a good precriterion for testing thermal phase transitions, and we use it to analyze the nonzero temperature phase diagram of the Lipkin-Meshkov-Glick model.

In quantum phase transitions induced by continuous level crossing, global-state fidelity drops to zero at every level crossing point, making finite-size scaling not applicable to study the criticality. In this paper, we calculate the partial-state fidelity in two models, i.e., the Lipkin-Meshkov-Glick (LMG) model and one-dimensional Heisenberg model. We show that it can spot the second-order phase transition point in the thermodynamic limit, where continuous level crossing occurs. The finite-size scaling shows the partial-state fidelity susceptibility diverges as

N

for the isotropic LMG model and

{N}^{3}

for the Heisenberg model.

We review the quantum fidelity approach to quantum phase transitions in a pedagogical manner. We try to relate all established but scattered results on the leading term of the fidelity into a systematic theoretical framework, which might provide an alternative paradigm for understanding quantum critical phenomena. The definition of the fidelity and the scaling behavior of its leading term, as well as their explicit applications to the one-dimensional transverse-field Ising model and the Lipkin–Meshkov–Glick model, are introduced at the graduate-student level. Besides, we survey also other types of fidelity approach, such as the fidelity per site, reduced fidelity, thermal-state fidelity, operator fidelity, etc; as well as relevant works on the fidelity approach to quantum phase transitions occurring in various many-body systems.

The fidelity metric has recently been proposed as a useful and elegant approach to identify and characterize both quantum and classical phase transitions. We study this metric on the manifold of thermal states for the Lipkin–Meshkov–Glick (LMG) model. For the isotropic LMG model, we find that the metric reduces to a Fisher–Rao metric, reflecting an underlying classical probability distribution. Furthermore, this metric can be expressed in terms of derivatives of the free energy, indicating a relation to Ruppeiner geometry. This allows us to obtain exact expressions for the (suitably rescaled) metric in the thermodynamic limit. The phase transition of the isotropic LMG model is signalled by a degeneracy of this (improper) metric in the paramagnetic phase. Due to the integrability of the isotropic LMG model, ground-state level crossings occur, leading to an ill-defined fidelity metric at zero temperature.

We study the critical properties of the Lipkin-Meshkov-Glick model in terms of the fidelity susceptibility. By using the Holstein-Primakoff transformation, we obtain explicitly the critical exponent of the fidelity susceptibility around the second-order quantum phase transition point. Our results provide a rare analytic case for the fidelity susceptibility in describing the universality class in quantum critical behavior. The different critical exponents in two phases are nontrivial results, indicating that the fidelity susceptibility is not always extensive.

We present a phase-space study of first-, second- and third-order quantum phase transitions in the Lipkin–Meshkov–Glick model by means of the Husimi function. By analyzing the distribution of zeros of the ground state Husimi function we have characterized each phase and each type of quantum phase transition in this model. We show that Renyi–Wehrl entropies of the ground state Husimi function give a good description of quantum phase transitions. The study has been done using a numerical treatment and a variational approximation in terms of coherent states. Additionally, we have analyzed quantum phase transitions using the fidelity and fidelity susceptibility concepts.

We derive a general formula for the reduced fidelity susceptibility when the reduced density matrix is 2x2 block diagonal. By using this result and a continuous unitary transformation, we study finite-size scaling of the reduced fidelity susceptibility in the Lipkin-Meshkov-Glick model. It is found that it can well characterize quantum phase transitions, that is, we can extract information about quantum phase transitions from only the fidelity of a subsystem, which is of practical use in experiments.