arXiv subjects

Hisao Nakanishi

Publications and source records attributed to Hisao Nakanishi.

15 recordsLinked to original sources

arXiv2017

Two-dimensional quantum percolation on anisotropic lattices

In a previous work [Dillon and Nakanishi, Eur. Phys.J B {\bf 87}, 286 (2014)], we calculated the transmission coefficient of the two-dimensional quantum percolation model and found there to be three regimes, namely, exponentially localized, power-law localized, and delocalized. However, the existence of these phase transitions remains controversial, with many other works divided between those which claim that quantum percolation in 2D is always localized, and those which assert there is a transition to a less localized or delocalized state. It stood out that many works based on highly anisotropic two-dimensional strips fall in the first group, whereas our previous work and most others in the second group were based on an isotropic square geometry. To better understand the difference in our results and those based on strip geometry, we apply our direct calculation of the transmission coefficient to highly anisotropic strips of varying widths at three energies and a wide range of dilutions. We find that the localization length of the strips does not converge at low dilution as the strip width increases toward the isotropic limit, indicating the presence of a delocalized state for small disorder. We additionally calculate the inverse participation ratio of the lattices and find that it too signals a phase transition from delocalized to localized near the same dilutions.

cond-mat.stat-mech↗

arXiv2016

Two-dimensional quantum percolation with binary non-zero hopping integrals

In a previous work [Dillon and Nakanishi, Eur.Phys.J B 87, 286 (2014)], we numerically calculated the transmission coefficient of the two-dimensional quantum percolation problem and mapped out in detail the three regimes of localization, i.e., exponentially localized, power-law localized, and delocalized which had been proposed earlier [Islam and Nakanishi, Phys.Rev. E 77, 061109 (2008)]. We now consider a variation on quantum percolation in which the hopping integral ($w$) associated with bonds that connect to at least one diluted site is not zero, but rather a fraction of the hopping integral (V=1) between non-diluted sites. We study the latter model by calculating quantities such as the transmission coefficient and the inverse participation ratio and find the original quantum percolation results to be stable for $w>0$ over a wide range of energy. In particular, except in the immediate neighborhood of the band center (where increasing $w$ to just $0.02*V$appears to eliminate localization effects), increasing $w$ only shifts the boundaries between the 3 regimes but does not eliminate them until $w$ reaches 10%-40% of $V$.

cond-mat.stat-mech↗

arXiv2013

Localization phase diagram of two-dimensional quantum percolation

We examine quantum percolation on a square lattice with random dilution up to $q=38%$ and energy $0.001 \le E \le 1.6$ (measured in units of the hopping matrix element), using numerical calculations of the transmission coefficient at a much larger scale than previously. Our results confirm the previous finding that the two dimensional quantum percolation model exhibits localization-delocalization transitions, where the localized region splits into an exponentially localized region and a power-law localization region. We determine a fuller phase diagram confirming all three regions for energies as low as $E=0.1$, and the delocalized and exponentially localized regions for energies down to $E=0.001$. We also examine the scaling behavior of the residual transmission coefficient in the delocalized region, the power law exponent in the power-law localized region, and the localization length in the exponentially localized region. Our results suggest that the residual transmission at the delocalized to power-law localized phase boundary may be discontinuous, and that the localization length is likely not to diverge with a power-law at the exponentially localized to power-law localized phase boundary. However, further work is needed to definitively assess the characters of the two phase transitions as well as the nature of the intermediate power-law regime.

cond-mat.stat-mech↗

arXiv2007

Localization-delocalization transition in 2D quantum percolation model

We study the hopping transport of a quantum particle through randomly diluted percolation clusters in two dimensions realized both on the square and triangular lattices. We investigate the nature of localization of the particle by calculating the transmission coefficient as a function of energy (-2 < E < 2 in units of the hopping integral in the tight-binding Hamiltonian) and disorder, q (probability that a given site of the lattice is not available to the particle). Our study based on finite size scaling suggests the existence of delocalized states that depends on energy and amount of disorder present in the system. For energies away from the band center (E = 0), delocalized states appear only at low disorder (q < 15%). The transmission near the band center is generally very small for any amount of disorder and therefore makes it difficult to locate the transition to delocalized states if any, but our study does indicate a behavior that is weaker than power-law localization. Apart from this localization-delocalization transition, we also find the existence of two different kinds of localization regimes depending on energy and amount of disorder. For a given energy, states are exponentially localized for sufficiently high disorder. As the disorder decreases, states first show power-law localization before showing a delocalized behavior.

cond-mat.dis-nn↗

arXiv2007

Propagation of a quantum particle through two dimensional percolating systems

We study quantum percolation which is described by a tight-binding Hamiltonian containing only off-diagonal hopping terms that are generally in quenched binary disorder (zero or one). In such a system, transmission of a quantum particle is determined by the disorder and interference effects, leading to interesting sharp features in conductance as the energy, disorder, and boundary conditions are varied. To aid understanding of this phenomenon, we develop a visualization method whereby the progression of a wave packet entering the cluster through a lead on one side and exiting from another lead on the other side can be tracked dynamically. Using this method, we investigate the evolution of the wave packet through transmission and reflection under clusters of various disorder, sizes, boundary conditions, and energies. Our results indicate the existence of two different kinds of localized regimes, namely exponential and power law localization, depending on the amount of disorder. Our study further suggests that there may be a delocalized state in the 2D quantum percolation system at very low disorder. These results are based on a finite size scaling analysis of the systems of size up to 70 x 70 (containing 4900 sites) on the square lattice.

cond-mat.dis-nn↗

arXiv1999

Scaling for vibrational modes of fractals tethered at the boundaries

Using a recently introduced mapping between a scalar elastic network tethered at its boundaries and a diffusion problem with permanent traps, we study various vibrational properties of progressively tethered disordered fractals. Different scaling forms are proposed for different types of boundary tethering and numerical investigation of the scaling is performed by the approximate diagonalization of the corresponding large, sparse transition probability matrices. Rather different localization behaviors are found for the leading modes depending on the types of tethering.

cond-mat.stat-mech↗

arXiv1999

Geometry of fully coordinated, two-dimensional percolation

We study the geometry of the critical clusters in fully coordinated percolation on the square lattice. By Monte Carlo simulations (static exponents) and normal mode analysis (dynamic exponents), we find that this problem is in the same universality class with ordinary percolation statically but not so dynamically. We show that there are large differences in the number and distribution of the interior sites between the two problems which may account for the different dynamic nature.

cond-mat.stat-mech↗

arXiv1995

Diffusion with critically correlated traps and the slow relaxation of the longest wavelength mode

We study diffusion on a substrate with permanent traps distributed with critical positional correlation, modeled by their placement on the perimeters of a critical percolation cluster. We perform a numerical analysis of the vibrational density of states and the largest eigenvalue of the equivalent scalar elasticity problem using the method of Arnoldi and Saad. We show that the critical trap correlation increases the exponent appearing in the stretched exponential behavior of the low frequency density of states by approximately a factor of two as compared to the case of no correlations. A finite size scaling hypothesis of the largest eigenvalue is proposed and its relation to the density of states is given. The numerical analysis of this scaling postulate leads to the estimation of the stretch exponent in good agreement with the density of states result.

cond-mat↗

arXiv1994

Eigenspectrum and Localization for Diffusion with Traps

We investigate the {\em survival-return} probability distribution and the eigenspectrum for the transition probability matrix, for diffusion in the presence of perfectly absorbing traps distributed with critical disorder in two and three dimensions. The density of states is found to have a Lifshitz tail in the low frequency limit, consistent with a recent investigation of the long time behavior of the {\em survival} probability. The localization properties of the eigenstates are found to be very different from diffusion with no traps.

cond-mat↗

arXiv1993

Statistics of self-avoiding walks on randomly diluted lattice

A comprehensive numerical study of self-avoiding walks (SAW's) on randomly diluted lattices in two and three dimensions is carried out. The critical exponents $ν$ and $χ$ are calculated for various different occupation probabilities, disorder configuration ensembles, and walk weighting schemes. These results are analyzed and compared with those previously available. Various subtleties in the calculation and definition of these exponents are discussed. Precise numerical values are given for these exponents in most cases, and many new properties are recognized for them.

cond-mat↗

arXiv1993

Markov chain analysis of random walks on disordered medium

We study the dynamical exponents $d_{w}$ and $d_{s}$ for a particle diffusing in a disordered medium (modeled by a percolation cluster), from the regime of extreme disorder (i.e., when the percolation cluster is a fractal at $p=p_{c}$) to the Lorentz gas regime when the cluster has weak disorder at $p>p_{c}$ and the leading behavior is standard diffusion. A new technique of relating the velocity autocorrelation function and the return to the starting point probability to the asymptotic spectral properties of the hopping transition probability matrix of the diffusing particle is used, and the latter is numerically analyzed using the Arnoldi-Saad algorithm. We also present evidence for a new scaling relation for the second largest eigenvalue in terms of the size of the cluster, $|\lnλ_{max}|\sim S^{-d_w/d_f}$, which provides a very efficient and accurate method of extracting the spectral dimension $d_s$ where $d_s=2d_f/d_w$.

cond-mat↗

arXiv1993

A New Finite-Size Scaling Approach to Random Walks

We present a new finite-size scaling method for the random walks (RW) superseeding a previously widely used renormalization group approach, which is shown here to be inconsistent. The method is valid in any dimension and is based on the exact solution for the two-point correlation function and on finite size scaling. As an application, the phase diagram is derived for random walks with a surface-bulk interaction where the system has either a surface or a defect. Possible extensions to disordered systems are also discussed.

cond-mat↗

arXiv1993

Effect of Loops on the Vibrational Spectrum of Percolation Network

We study the effects of adding loops to a critical percolation cluster on the diffusional, and equivalently, (scalar) elastic properties of the fractal network. From the numerical calculations of the eigenspectrum of the transition probability matrix, we find that the spectral dimension $d_s$ and the walk dimension $d_w$ change suddenly as soon as the floppy ends of a critical percolation cluster are connected together to form relatively large loops, and that the additional inclusion of successively smaller loops only change these exponents little if at all. This suggests that there is a new universality class associated with the loop-enhanced percolation problem.

cond-mat↗

arXiv1992

Fractal Properties of the Distribution of Earthquake Hypocenters

We investigate a recent suggestion that the spatial distribution of earthquake hypocenters makes a fractal set with a structure and fractal dimensionality close to those of the backbone of critical percolation clusters, by analyzing four different sets of data for the hypocenter distributions and calculating the dynamical properties of the geometrical distribution such as the spectral dimension $d_s$. We find that the value of $d_s$ is consistent with that of the backbone, thus supporting further the identification of the hypocenter distribution as having the structure of the percolation backbone.

cond-mat↗

arXiv1992

Diffusion and spectral dimension on Eden tree

We calculate the eigenspectrum of random walks on the Eden tree in two and three dimensions. From this, we calculate the spectral dimension $d_s$ and the walk dimension $d_w$ and test the scaling relation $d_s = 2d_f/d_w$ ($=2d/d_w$ for an Eden tree). Finite-size induced crossovers are observed, whereby the system crosses over from a short-time regime where this relation is violated (particularly in two dimensions) to a long-time regime where the behavior appears to be complicated and dependent on dimension even qualitatively.

cond-mat↗