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arXiv:0911.5531 · Physical Review B

Localization-delocalization transitions in a two-dimensional quantum percolation model: von Neumann entropy studies

Longyan Gong, Peiqing Tong

Published Nov 20, 2009• 19 citations
arXivHTMLPDF

Abstract

In two-dimensional quantum site-percolation square lattice models, the von Neumann entropy is extensively studied numerically. At a certain eigenenergy, the localization-delocalization transition is reflected by the derivative of von Neumann entropy which is maximal at the quantum percolation threshold ${p}_{q}$. The phase diagram of localization-delocalization transitions is deduced in the extrapolation to infinite system sizes. The nonmonotonic eigenenergies dependence of ${p}_{q}$ and the lowest value ${p}_{q}\ensuremath{\simeq}0.665$ are found. At localized-delocalized transition points, the finite scaling analysis for the von Neumann entropy is performed and it is found the critical exponents $\ensuremath{\nu}$ not to be universal. These studies provide an evidence that the existence of a quantum percolation threshold ${p}_{q}<1$ in the two-dimensional quantum percolation problem.

DOI
10.1103/physrevb.80.174205

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In two-dimensional quantum site-percolation square lattice models, the von Neumann entropy is extensively studied numerically. At a certain eigenenergy, the localization-delocalization transition is reflected by the derivative of von Neumann entropy which is maximal at the quantum percolation threshold pq{p}_{q}pq​. The phase diagram of localization-delocalization transitions is deduced in the extrapolation to infinite system sizes. The nonmonotonic eigenenergies dependence of pq{p}_{q}pq​ and the lowest value pq\ensuremath≃0.665{p}_{q}\ensuremath{\simeq}0.665pq​\ensuremath≃0.665 are found. At localized-delocalized transition points, the finite scaling analysis for the von Neumann entropy is performed and it is found the critical exponents \ensuremathν\ensuremath{\nu}\ensuremathν not to be universal. These studies provide an evidence that the existence of a quantum percolation threshold pq<1{p}_{q}<1pq​<1 in the two-dimensional quantum percolation problem.
We theoretically investigate the quantum percolation problem on Lieb lattices in two and three dimensions. We study the statistics of the energy levels through random matrix theory and determine the level spacing distributions, which, with the aid of finite-size scaling theory, allows us to obtain accurate estimates for site- and bond-percolation thresholds and critical exponents. Our numerical investigation supports a localized-delocalized transition at finite threshold, which decreases as the average coordination number increases. The precise determination of the localization length exponent enables us to claim that quantum site- and bond-percolation problems on Lieb lattices belong to the same universality class, with ν decreasing with lattice dimensionality d, similarly to the classical percolation problem. In addition, we verify that, in three dimensions, quantum percolation on Lieb lattices belongs to the same universality class as the Anderson impurity model.
We present a numerical study of the two-dimensional quantum percolation model, revealing that a critical region with multifractal eigenstates mediates the transition from localized to delocalized states. By analyzing the mean level ratio and participation entropy, we identify two distinct transitions: a Berezinskii-Kosterlitz-Thouless (BKT) transition at the classical percolation threshold, separating the localized and critical phases, and a power-law-type transition at a larger concentration, marking the onset of full delocalization. The critical phase is characterized by multifractal eigenstates, as evidenced by the generalized fractal dimension and multifractal spectrum. Altogether, our results establish that in the marginal two-dimensional case, the Anderson impurity model and the quantum percolation model belong to different universality classes.
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.
We numerically study spectral and wave-function properties, as well as time evolution, in finite-size commensurate approximations of the 3D Aubry-André model. We explicitly show, through scaling analysis, the existence of two critical points. One of the critical points is a localization-delocalization transition in real-space, similar to the critical point in the 3D Anderson model. The other is a localization-delocalization transition in a dual momentum space, which does not have any analog in the 3D Anderson model. We estimate their values from spectral measures, such as the gap ratio and the spectral form factor, as well as from wave-function measures, such as the inverse participation ratio and the participation entropy. Additionally, we demonstrate that both critical points exhibit hallmarks of critical points between quantum ergodicity and localization. First, the eigenstates and the eigenspectrum at the critical point fulfill the mutual universal relation between the eigenspectrum compressibility χχχ and the eigenstate fractal dimension d(1)d^{(1)}d(1) [S. Jiricek et al., Proc. Nat. Acad. Sci. 123, e2518027123 (2026)]. Second, the dynamics of the wave packet at the critical point is governed by the critical dynamical exponent zzz, characterized by the spectral fractal dimension and the dimensionality of the lattice [M. Hopjan and L. Vidmar, Phys. Rev. Lett. 135, 060401 (2025)]. Thus, similarly to the 3D Anderson model, the critical dynamical exponent is sub-diffusive with the value of z=3z=3z=3.
Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).
The von Neumann entanglement entropy is a useful measure to characterize a quantum phase transition. We investigate the nonanalyticity of this entropy at disorder-dominated quantum phase transitions in noninteracting electronic systems. At these critical points, the von Neumann entropy is determined by the single particle wave function intensity, which exhibits complex scale invariant fluctuations. We find that the concept of multifractality is naturally suited for studying von Neumann entropy of the critical wave functions. Our numerical simulations of the three dimensional Anderson localization transition and the integer quantum Hall plateau transition show that the entanglement at these transitions is well described using multifractal analysis.