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arXiv:0809.3856 · Chinese Physics Letters

Density-Functional Fidelity Approach to Quantum Phase Transitions

Shi-Jian Gu

Published Feb 1, 2009• 18 citations
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Abstract

We propose a new approach to quantum phase transitions in terms of the density-functional fidelity, which measures the similarity between density distributions of two ground states in parameter space. The key feature of the approach is such that the density-functional fidelity can be measured easily in experiments. Both the validity and versatility of the approach are checked by the Lipkin–Meshkov–Glick model and the one-dimensional Hubbard model.

DOI
10.1088/0256-307x/26/2/026401

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We propose a new approach to quantum phase transitions in terms of the density-functional fidelity, which measures the similarity between density distributions of two ground states in parameter space. The key feature of the approach is such that the density-functional fidelity can be measured easily in experiments. Both the validity and versatility of the approach are checked by the Lipkin–Meshkov–Glick model and the one-dimensional Hubbard model.
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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 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 show that for quantum phase transitions with a single bosonic zero mode at the critical point, like the Dicke model and the Lipkin-Meshkov-Glick model, metric quantities such as fidelity, that is, the overlap between two ground states corresponding to two values ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ of the controlling parameter $\ensuremath{\lambda}$, depend only on the ratio $\ensuremath{\eta}=({\ensuremath{\lambda}}_{1}\ensuremath{-}{\ensuremath{\lambda}}_{c})/({\ensuremath{\lambda}}_{2}\ensuremath{-}{\ensuremath{\lambda}}_{c})$, where $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$ at the critical point. This scaling property is valid also for time-dependent quantities such as the Loschmidt echo, provided time is measured in units of the inverse frequency of the critical mode.
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.
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.