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arXiv:0811.3127 · International Journal of Modern Physics B

FIDELITY APPROACH TO QUANTUM PHASE TRANSITIONS

SHI-JIAN GU

Published Sep 20, 2010• 659 citations
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Abstract

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.

DOI
10.1142/s0217979210056335

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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.
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.
Quantum phase transition is one of the most important issues in condensed matter physics,and quantum fidelity is an important concept emerging from quantum information theory.In this paper,we describe as simply as possible the role of fidelity and its leading term,i.e.the so-called fidelity susceptibility,in quantum phase transitions.As a purely geometric quantity,the key advantage of the fidelity approach is that no a priori knowledge of the order parameter is assumed.Specifically,we discuss definitions of the fidelity,the fidelity susceptibility,and the corresponding quantum adiabatic dimension,as well as their role in quantum phase transitions.We also use the Lipkin-Meshkov-Glick model and the Kitaev honeycomb model as examples to clarify these issues.
The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.
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.
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 NNN for the isotropic LMG model and N3{N}^{3}N3 for the Heisenberg model.