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Journal of Physics Conference Series

Quantum information approach to the description of quantum phase transitions

Octavio Castaños, Ramón López–Peña, Eduardo Nahmad-Achar, Jorge G. Hirsch

Published Dec 18, 2012• 13 citations
Publisher

Abstract

The fidelity, linear entropy and entanglement entropy, concepts of information theory, are used to determine the localization of quantum phase transitions of the Lipkin–Meshkov–Glick (LMG) and the Dicke models for a finite number of particles. Another concept, the so-called fidelity susceptibility, gives us information of the scaling behavior of the quantum phase transitions of the models. In particular universal curves are presented for the specific susceptibility. It is shown that the control parameters of both models have a similar dependence in the number of particles when the systems approach to the thermodynamic limit.

DOI
10.1088/1742-6596/403/1/012003

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The fidelity, linear entropy and entanglement entropy, concepts of information theory, are used to determine the localization of quantum phase transitions of the Lipkin–Meshkov–Glick (LMG) and the Dicke models for a finite number of particles. Another concept, the so-called fidelity susceptibility, gives us information of the scaling behavior of the quantum phase transitions of the models. In particular universal curves are presented for the specific susceptibility. It is shown that the control parameters of both models have a similar dependence in the number of particles when the systems approach to the thermodynamic limit.
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.
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.
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.
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 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.
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.