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arXiv:quant-ph/0606130 · Journal of Statistical Mechanics Theory and Experiment

Ground state fidelity and quantum phase transitions in free Fermi systems

Paolo Zanardi A, Marco Cozzini, Paolo Giorda

Published Feb 16, 2007• 76 citations
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Abstract

We compute the fidelity of the ground states of general quadratic fermionic Hamiltonians and analyse its connections with quantum phase transitions. Each of these systems is characterized by an L × L real matrix whose polar decomposition, into a non-negative Λ Φ and a unitary T , contains all the relevant ground state (GS) information. The boundaries between different regions in the GS phase diagram are given by the points of, possibly asymptotic, singularity of Λ Φ . This latter in turn implies a critical drop of the fidelity function. We present general results as well as their exemplification by a model of fermions on a totally connected graph.

DOI
10.1088/1742-5468/2007/02/l02002

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Ground state fidelity and quantum phase transitions in free Fermi systems

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arXiv:2002.02491•Physica Scripta•Dec 23, 2020•13 citations
We compute the fidelity of the ground states of general quadratic fermionic Hamiltonians and analyse its connections with quantum phase transitions. Each of these systems is characterized by an L × L real matrix whose polar decomposition, into a non-negative Λ Φ and a unitary T , contains all the relevant ground state (GS) information. The boundaries between different regions in the GS phase diagram are given by the points of, possibly asymptotic, singularity of Λ Φ . This latter in turn implies a critical drop of the fidelity function. We present general results as well as their exemplification by a model of fermions on a totally connected graph.
We compute the fidelity between the ground states of general quadratic fermionic hamiltonians and analyze its connections with quantum phase transitions. Each of these systems is characterized by a LtimesLL\\times LLtimesL real matrix whose polar decomposition, into a non-negative Lambda\\LambdaLambda and a unitary TTT, contains all the relevant ground state (GS) information. The boundaries between different regions in the GS phase diagram are given by the points of, possibly asymptotic, singularity of Lambda\\LambdaLambda. This latter in turn implies a critical drop of the fidelity function. We present general results as well as their exemplification by a model of fermions on a totally connected graph.
In this paper we analyze the ground-state phase diagram of a class of fermionic Hamiltonians by looking at the fidelity of ground states corresponding to slightly different Hamiltonian parameters. The Hamiltonians under investigation can be considered as the variable range generalization of the fermionic Hamiltonian obtained by the Jordan-Wigner transformation of the XYXYXY spin chain in a transverse magnetic field. Under periodic boundary conditions, the matrices of the problem become circulant and the models are exactly solvable. Their free-ends counterparts are instead analyzed numerically. In particular, we focus on the long-range model corresponding to a fully connected directed graph, providing asymptotic results in the thermodynamic limit, as well as the finite-size scaling analysis of the second-order quantum phase transitions of the system. A strict relation between fidelity and single particle spectrum is demonstrated, and a peculiar gapful transition due to the long-range nature of the coupling is found. A comparison between fidelity and another transition marker borrowed from quantum information, i.e., single site entanglement, is also considered.
The notion of fidelity in quantum information science has been recently applied to analyze quantum phase transitions from the viewpoint of the ground-state (GS) overlap for various many-body systems. In this work, we unveil the intrinsic relation between the GS fidelity and the derivatives of GS energy and find that they play an equivalent role in identifying the quantum phase transition. The general connection between the two approaches enables us to understand the different singularity and scaling behaviors of fidelity exhibited in various systems on general grounds. Our general conclusions are illustrated via several quantum spin models which exhibit different kinds of quantum phase transitions.
Matrix product states, a key ingredient of numerical algorithms widely employed in the simulation of quantum spin chains, provide an intriguing tool for quantum phase transition engineering. At critical values of the control parameters on which their constituent matrices depend, singularities in the expectation values of certain observables can appear, in spite of the analyticity of the ground state energy. For this class of generalized quantum phase transitions, we test the validity of the recently introduced fidelity approach, where the overlap modulus of ground states corresponding to slightly different parameters is considered. We discuss several examples, successfully identifying all the present transitions. We also study the finite size scaling of fidelity derivatives, pointing out its relevance in extracting critical exponents.
We study ground-state fidelity defined as the overlap between two ground states of the same quantum system obtained for slightly different values of the parameters of its Hamiltonian. We focus on the thermodynamic regime of the XYXYXY model and the neighborhood of its critical points. We describe extensively fidelity when it is dominated by the universal contribution reflecting the quantum criticality of the phase transition. We show that proximity to the multicritical point leads to anomalous scaling of fidelity. We also discuss fidelity in a regime characterized by pronounced oscillations resulting from the change in either the system size or the parameters of the Hamiltonian. Moreover, we show when fidelity is dominated by non-universal contributions, study fidelity in the extended Ising model, and illustrate how our results provide additional insight into dynamics of quantum phase transitions. Special attention is given to studies of fidelity from the momentum space perspective. All our main results are obtained analytically. They are in excellent agreement with numerics.
A general procedure is established to calculate the quantum phase diagrams for finite matter-field Hamiltonian models. The minimum energy surface associated to the different symmetries of the model is calculated as a function of the matter-field coupling strengths. By means of the ground state wave functions, one looks for minimal fidelity or maximal Bures distance surfaces in terms of the parameters, and from them the critical regions of those surfaces characterize the finite quantum phase transitions. Following this procedure for N a = 1 and N a = 4 particles, the quantum phase diagrams are calculated for the generalised Tavis-Cummings and Dicke models of 3-level systems interacting dipolarly with 2 modes of electromagnetic field. For N a = 1, the reduced density matrix of the matter allows us to determine the phase regions in a 2-simplex (associated to a general three dimensional density matrix), on the different 3-level atomic configurations, together with a measurement of the quantum correlations between the matter and field sectors. As the occupation probabilities can be measured experimentally, the existence of a quantum phase diagram for a finite system can be established.