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arXiv:0907.3367 · Journal of Physics A Mathematical and Theoretical

Finite-temperature fidelity-metric approach to the Lipkin–Meshkov–Glick model

Daniel D. Scherer, Cord Axel Müller, Michael Kästner

Published Oct 22, 2009• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/42/46/465304

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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.
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.
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.
Abstract We study the global quantum discord (GQD) in the Lipkin–Meshkov–Glick (LMG) model at zero and finite temperatures, in which all spins are mutually interacted and introduced in an external magnetic field (denoted by h ). We confirm that the high coordinate number is one of the most distinguishing features of the LMG model, which directly results in the nontrivial behaviors of quantum correlations. We compare the GQD with other quantum correlations measures (such as concurrence, quantum discord, and global entanglement) and find the remarkable difference between them. For instance, we find that GQD spreads in the entire system and captures more information on quantum correlations when comparing with concurrence and quantum (pairwise) discord. We discover that GQD can characterize multipartite correlations in the both broken phase ( h < 1) and the symmetric phase ( h ⩾ 1), while global entanglement and its generalized fail. Moreover, we show that the ground-state GQD can identify second-order quantum phase transitions of the LMG model in the thermodynamic limit. By making the scaling behavior of the GQD in the LMG model analysis, we show that GQD (denoted by G ) scales as G ∼ k ⋅ N + c with k > 0 in the anisotropic cases for any fixed magnetic field. We further show that GQD behaves as G | s n ∼ k ⋅ 1 N + c with k < 0 in the isotropic cases for any Dicke state | s n ⟩. Herein k and c are the fitting parameters. We also find that the thermal stability of the GQD at low temperatures depends on the energy gap. We further reveal that the extraordinary behaviors of the thermal-state GQD in the isotropic LMG model are explained by the contribution theory of the energy levels.
Quantum phase transitions are currently studied in several fields of physics, as examples one finds: in nuclear physics the shape phase transitions within the Interacting Boson Model; in quantum optics two level atoms interacting with a one mode electromagnetic radiation in a cavity; and in condensed matter in the analysis of the behavior of spin systems. In this contribution we employ the fidelity and its susceptibility, two concepts widely used in quantum information theory, to determine the quantum phase transitions of the Lipkin-Meshkov-Glick (LMG) model together with its scaling properties. By means of these concepts, we propose a quantum method to determine the crossings and anti-crossings present in a model Hamiltonian as a function of the control parameters of the model. A review of the separatrix of the LMG model is done to compare the results obtained by means of quantum information concepts with those related with the singular behavior of the energy surface of the model, which is the expectation value of the Hamiltonian with respect to spin coherent states.
We study the classical analog of the quantum metric tensor and its scalar curvature for two well-known quantum physics models. First, we analyze the geometry of the parameter space for the Dicke model with the aid of the classical and quantum metrics and find that, in the thermodynamic limit, they have the same divergent behavior near the quantum phase transition, as opposed to their corresponding scalar curvatures which are not divergent there. On the contrary, under resonance conditions, both scalar curvatures exhibit a divergence at the critical point. Second, we present the classical and quantum metrics for the Lipkin-Meshkov-Glick model in the thermodynamic limit and find a perfect agreement between them. We also show that the scalar curvature is only defined on one of the system's phases and that it approaches a negative constant value. Finally, we carry out a numerical analysis for the system's finite sizes, which clearly shows the precursors of the quantum phase transition in the metric and its scalar curvature and allows their characterization as functions of the parameters and of the system's size.