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arXiv:0809.3628 · Physical Review B

Fidelity, fidelity susceptibility, and von Neumann entropy to characterize the phase diagram of an extended Harper model

Longyan Gong, Peiqing Tong

Published Sep 17, 2008• 42 citations
arXivHTMLPDF

Abstract

In an extended Harper model, the fidelity for the two lowest band-edge states corresponding to different model parameters, the fidelity susceptibility and the von Neumann entropy of the lowest band-edge states, and the spectrum-averaged von Neumann entropy are studied numerically. The fidelity is near one when parameters are in the same phase or same phase boundary; otherwise it is close to zero. There are drastic changes in fidelity when one parameter is at phase boundary. For the fidelity susceptibility the finite scaling analysis is performed. The critical exponents $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\nu}$ depend on system sizes for the metal-metal phase transition, while this is not so for the metal-insulator phase transition. At both phase transitions $\ensuremath{\nu}/\ensuremath{\alpha}\ensuremath{\approx}2$. The von Neumann entropy is near one for the metallic phase, while it is small for the insulating phase. There are sharp changes in the von Neumann entropy at phase boundaries. According to the variations of the fidelity, fidelity susceptibility, and the von Neumann entropy with model parameters, the phase diagram, which includes two metallic phases and one insulating phase separated by three critical lines with one bicritical point, can be completely characterized. These numerical results indicate that the three quantities are suited for revealing all the critical phenomena in the model.

DOI
10.1103/physrevb.78.115114

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In an extended Harper model, the fidelity for the two lowest band-edge states corresponding to different model parameters, the fidelity susceptibility and the von Neumann entropy of the lowest band-edge states, and the spectrum-averaged von Neumann entropy are studied numerically. The fidelity is near one when parameters are in the same phase or same phase boundary; otherwise it is close to zero. There are drastic changes in fidelity when one parameter is at phase boundary. For the fidelity susceptibility the finite scaling analysis is performed. The critical exponents \ensuremathα\ensuremath{\alpha}\ensuremathα, \ensuremathβ\ensuremath{\beta}\ensuremathβ, and \ensuremathν\ensuremath{\nu}\ensuremathν depend on system sizes for the metal-metal phase transition, while this is not so for the metal-insulator phase transition. At both phase transitions \ensuremathν/\ensuremathα\ensuremath≈2\ensuremath{\nu}/\ensuremath{\alpha}\ensuremath{\approx}2\ensuremathν/\ensuremathα\ensuremath≈2. The von Neumann entropy is near one for the metallic phase, while it is small for the insulating phase. There are sharp changes in the von Neumann entropy at phase boundaries. According to the variations of the fidelity, fidelity susceptibility, and the von Neumann entropy with model parameters, the phase diagram, which includes two metallic phases and one insulating phase separated by three critical lines with one bicritical point, can be completely characterized. These numerical results indicate that the three quantities are suited for revealing all the critical phenomena in the model.
By using the measure of von Neumann entropy, we numerically investigate quantum entanglement of an electron moving in the one-dimensional Harper model and in the one-dimensional slowly varying potential model. The delocalized and localized eigenstates can be distinguished by von Neumann entropy of the individual eigenstates. There are drastic decreases in von Neumann entropy of the individual eigenstates at mobility edges. In the curve of the spectrum averaged von Neumann entropy as a function of potential parameter λ, a sharp transition exists at the metal–insulator transition point λ c = 2. It is found that the von Neumann entropy is a good quantity to reflect localization and metal–insulator transition.
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We study the quantum fidelity (ground-state overlap) near quantum phase transitions of the Ising universality class in one-dimensional (1D) systems of finite-size LLL. Prominent examples occur in magnetic systems (e.g., spin-Peierls, the anisotropic XYXYXY model) and in 1D topological insulators of any topologically nontrivial Altland-Zirnbauer-Kitaev universality class. The rescaled fidelity susceptibility is a function of the only dimensionless parameter LMLMLM, where 2M2M2M is the gap in the fermionic spectrum. We present analytic expressions for the fidelity susceptibility for periodic and open boundaries conditions with zero, one, or two edge states. The latter are shown to have a crucial impact and alter the susceptibility both quantitatively and qualitatively. We support our analytical solutions with numerical data.
Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two — a metal–insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.
Abstract We study the critical behaviors of the ground and first-excited states in the one-dimensional nonreciprocal Aubry-André-Harper (AAH) model using both the self-normal and biorthogonal fidelity susceptibilities. We demonstrate that fidelity susceptibility serves as a probe for the phase transition in the nonreciprocal AAH model. For ground states, characterized by real eigenenergies across the entire regime, both fidelity susceptibilities near the critical points scale as N 2 , akin to the Hermitian AAH model. However, for the first-excited states, the fidelity susceptibilities exhibit distinct scaling laws, contingent upon whether the lattice consists of even or odd sites. For even lattices, both the self-normal and biorthogonal fidelity susceptibilities near the critical points continue to scale as N 2 . In contrast, for odd lattices, the biorthogonal fidelity susceptibilities diverge, while the self-normal fidelity susceptibilities exhibit linear behavior, indicating a novel scaling law.
The one-dimensional extended quantum compass model in the presence of a transverse magnetic field is considered. Induced effects of the transverse magnetic field on the ground state of the system are studied from the viewpoint of fidelity. Using the numerical Lanczos method, the fidelity and susceptibility of fidelity are computed in finite chains. The critical exponent of the fidelity susceptibility is obtained in good agreement with the scaling behavior of the correlation length. In addition, the von Neumann entropy is calculated and its signature on the quantum phase transition is shown.