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arXiv:quant-ph/0608059 · Physical Review B

Quantum phase transitions and quantum fidelity in free fermion graphs

Marco Cozzini, Paolo Giorda, Paolo Zanardi A

Published Jan 29, 2007• 150 citations
arXivHTMLPDF

Abstract

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 $XY$ 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.

DOI
10.1103/physrevb.75.014439

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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.
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.
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.
Abstract In this study, a spin-1/2 extended anisotropic XY chain has been introduced in which both time reversal and SU(2) symmetries are broken but Z 2 symmetry is preserved. Magnetic and topological phase diagrams in the parameter space have been drawn in the presence of transverse magnetic field. Entanglement measures like mutual information and quantum discord are also evaluated and it indicates that these transitions are second order in nature. Quantum phase transition is noted at zero magnetic field, as well as magnetic long range order is found to withstand magnetic field of any strength. Exact analytic results for spin-spin correlation functions have been obtained in terms of Jordan-Wigner fermionization. Existence of long range magnetic order has been investigated numerically by finding correlation functions as well as the Binder cumulant in the ground state. Dispersion relation, ground state energy, and energy gap are obtained analytically. In order to find the topologically nontrivial phase, sign of Pfaffian invariant and value of winding number have been evaluated. Both magnetic and topological phases are robust against the magnetic field and found to move coercively in the parameter space with the variation of its strength. Long range orders along two orthogonal directions and two different topological phases are found and their one-to-one correspondence has been found. Finally casting the spinless fermions onto Majorana fermions, properties of zero energy edge states are studied. Three different kinds of Majorana pairings are noted. In the trivial phase, next-nearest-neighbor Majorana pairing is found, whereas two different types of nearest-neighbor Majorana pairings are identified in the topological superconducting phase.
Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by complex many-body systems such as liquids or amorphous solids, classical spin chains with certain long-range interactions have been shown to demonstrate the same phenomenon. Such spin-chain systems are ideal models for exploring the effects of quantum mechanics on hyperuniformity. It is well-known that the transverse field Ising model shows a quantum phase transition (QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform an extensive analysis of these systems using density matrix renormalization group simulations to study the possibilities of phase transitions and the mechanism by which they lose hyperuniformity. Even for a spin chain of length 30, we see discontinuous changes in properties like the ``$\ensuremath{\tau}$ order metric'' of the ground state, the measure of hyperuniformity, and the second cumulant of the total magnetization along the xxx-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of a transverse magnetic field may lead remarkably to an increase in the order of the ground state as measured by the ``$\ensuremath{\tau}$ order metric,'' even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping onto a system of spinless fermions, showing that under the pairwise-interaction approximation and a mean-field treatment, there can be at most two quantum phase transitions. Based on these numerical and theoretical explorations, we conjecture that for spin chains with long-range pair interactions that have convergent cosine transforms, there can be a maximum of two quantum phase transitions at zero temperature.
Phase transitions in quantum systems, including symmetry breaking and topological types, always associate with gap closing and opening. We analyze the topological features of the quantum phase boundary of the XY model in a transverse magnetic field. Based on the results from graphs in the auxiliary space, we find that gapless ground states at boundary have different topological characters. On the other hand, in the framework of Majorana representation, the Majorana lattice is shown to be two coupled SSH chains. The analysis of the quantum fidelity for the Majorana eigen vector, which is shown to be identical to the square of that for ground states of the XY model, indicates the signature of the gapless phase transition (GPT). Furthermore analytical and numerical results for second-order derivative of groundstate energy density show that such a GPT obeys scaling behavior.
This chapter is devoted to a discussion of quantum phase transitions in regularly alternating spin-1/2 Ising chain in a transverse field. After recalling some generally-known topics of the classical (temperature-driven) phase transition theory and some basic concepts of the quantum phase transition theory I pass to the statistical mechanics calculations for a one-dimensional spin-1/2 Ising model in a transverse field, which is the simplest possible system exhibiting the continuous quantum phase transition. The essential tool for these calculations is the Jordan-Wigner fermionization. The latter technique being completed by the continued fraction approach permits to obtain analytically the thermodynamic quantities for a `slightly complicated' model in which the intersite exchange interactions and on-site fields vary regularly along a chain. Rigorous analytical results for the ground-state and thermodynamic quantities, as well as exact numerical data for the spin correlations computed for long chains (up to a few thousand sites) demonstrate how the regularly alternating bonds/fields effect the quantum phase transition. I discuss in detail the case of period 2, swiftly sketch the case of period 3 and finally summarize emphasizing the effects of periodically modulated Hamiltonian parameters on quantum phase transitions in the transverse Ising chain and in some related models.