Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics Condensed Matter

Resonant localized states and quantum percolation on random chains with power-law-diluted long-range couplings

Samuel S. de Albuquerque, F. A. B. F. de Moura, M. L. Lyra

Published Apr 18, 2012• 4 citations
Publisher

Abstract

We investigate the nature of one-electron eigenstates in power-law-diluted chains for which the probability of occurrence of a bond between sites separated by a distance r decays as p(r) = p/r(1+σ). Using an exact diagonalization scheme and a phenomenological finite-size scaling analysis, we determine the quantum percolation transition phase diagram in the full parameter space (p,σ). We show that the density of states displays singularities at some resonance energies associated with degenerate eigenstates localized in a pair of sites with special symmetries. This model is shown to present an intermediate phase for which there is classical percolation but no quantum percolation. Quantum percolation only takes place for σ < 0.78, a value larger than the corresponding one for the Anderson transition in long-ranged coupled chains with random diagonal disorder. The fractality of critical wavefunctions is also characterized.

DOI
10.1088/0953-8984/24/20/205401

Related papers

Quantum percolation of one-electron states in power-law diluted chains

M. P. da Silva Jr., Samuel S. de Albuquerque, F. A. B. F. de Moura +1 more

We investigate the nature of one-electron eigenstates in power-law diluted chains for which the probability of occurrence of a bond between sites separated by a distance r decays as p(r) = 1/r1+sigma. Using an exact diagonalization scheme on finite chains, we compute the spreading of an initially localized wave-packet, the time dependent participation number as well as the return probability. Our results show the existence of a phase of extended states. By considering the scale invariance of the fluctuations of the participation number at the Anderson transition, we obtained that extended states emerges for sigma < 0.68. This limiting value is larger than the one reported in the literature for the emergence of extended states in one-dimensional Anderson models with power-law decaying couplings.

Brazilian Journal of Physics•Mar 1, 2008•3 citations

Fractality of largest clusters and the percolation transition in power-law diluted chains

Samuel S. de Albuquerque, F. A. B. F. de Moura, M. L. Lyra +1 more

Relying on the fractal character of the largest clusters at criticality, we employ a finite-size scaling analysis to obtain an accurate phase-diagram of the percolation transition in chains with bond concentration decaying as a power-law on the form 1/ r(1+sigma) . For the particular case of sigma=1, no percolation transition is observed to occur at a finite dilution, in contrast with the finite temperature Kosterlitz-Thouless transition exhibited in Ising and Potts chains with inverse square-law couplings. The fractal dimension of the critical percolation cluster is found to follow distinct dependencies on the decay exponent being numerically fitted by d(f) =0.35+4sigma/5 for 0<sigma<1/2 and d(f) = (1+sigma) /2 for 1/2<sigma<1 . We also compute average mass ratios of the two largest clusters at criticality.

Physical Review E•Jul 18, 2005•4 citations

Quantum percolation in power-law diluted chains

R. P. A. Lima, M. L. Lyra

We investigate the quantum percolation problem in a diluted chain with long-range hopping amplitudes. Each bond is activated with probability p(r)=p1/rα, where r is the distance between two sites and α characterizes the range of the interactions. The average participation ratio of all eigenstates is used as a measure of the wave-functions localization length. We found that, above a quantum percolation threshold p1(q), true extended states appear for α<1.5. In the regime of 1.5<α<2.0 there is no truly extended states even in the presence of a spanning cluster. Instead, a phase of critical wave-functions sets up.

•Jan 1, 2001•7 citations

One-dimensional long-range percolation: A numerical study

Giacomo Gori, M. Michelangeli, Nicolò Defenu +1 more

In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r^{d+σ}, where r is the distance length between distinct sites and d=1. We introduce and test an order-N Monte Carlo algorithm and we determine as a function of σ the critical value C_{c} at which percolation occurs. The critical exponents in the range 0<σ<1 are reported. Our analysis is in agreement, up to a numerical precision ≈10^{-3}, with the mean-field result for the anomalous dimension η=2-σ, showing that there is no correction to η due to correlation effects. The obtained values for C_{c} are compared with a known exact bound, while the critical exponent ν is compared with results from mean-field theory, from an expansion around the point σ=1 and from the ɛ-expansion used with the introduction of a suitably defined effective dimension d_{eff} relating the long-range model with a short-range one in dimension d_{eff}. We finally present a formulation of our algorithm for bond percolation on general graphs, with order N efficiency on a large class of graphs including short-range percolation and translationally invariant long-range models in any spatial dimension d with σ>0.

arXiv:1610.00200•Physical review. E•Jul 5, 2017•41 citations

The two-dimensional bond-diluted quantum Heisenberg model at the classical percolation threshold

Anders W. Sandvik

The two-dimensional antiferromagnetic S=1/2 Heisenberg model with random bond dilution is studied using quantum Monte Carlo simulation at the percolation threshold (50% of the bonds removed). Finite-size scaling of the staggered structure factor averaged over the largest connected clusters of sites on L*L lattices shows that long-range order exists within the percolating fractal clusters in the thermodynamic limit. This implies that the order-disorder transition driven by bond-dilution occurs exactly at the percolation threshold and that the exponents are classical. This result should apply also to the site-diluted system.

arXiv:cond-mat/9909230•Sep 15, 1999•0 citations

Finite-size scaling of power-law bond-disordered Anderson models

R. P. A. Lima, Heber R. da Cruz, J. C. Cressoni +1 more

We investigate numerically the nature of energy eigenstates in one-dimensional bond-disordered Anderson models with hopping amplitudes decreasing as ${H}_{\mathrm{ij}}\ensuremath{\propto}1/|i\ensuremath{-}j{|}^{\ensuremath{\alpha}}.$ The eigenstates become delocalized whenever the hopping amplitudes decay slower than $1/r.$ By performing an exact diagonalization scheme on finite chains, we compute the participation ratio of all energy eigenstates. Employing a finite-size scaling analysis, we report on the relevant scaling exponents characterizing this delocalization transition as well as the level-spacing distribution at the critical point $\ensuremath{\alpha}=1.$ The random hopping amplitudes are taken from both uniform and random sign distributions. We show that these models display similar critical behavior in the vicinity of $\ensuremath{\alpha}=1.$ However, the random sign model exhibits an asymptotic delocalization in the limit of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ and the universal scaling behavior in this regime is also reported.

Physical Review B•Apr 28, 2004•22 citations
We investigate the nature of one-electron eigenstates in power-law-diluted chains for which the probability of occurrence of a bond between sites separated by a distance r decays as p(r) = p/r(1+σ). Using an exact diagonalization scheme and a phenomenological finite-size scaling analysis, we determine the quantum percolation transition phase diagram in the full parameter space (p,σ). We show that the density of states displays singularities at some resonance energies associated with degenerate eigenstates localized in a pair of sites with special symmetries. This model is shown to present an intermediate phase for which there is classical percolation but no quantum percolation. Quantum percolation only takes place for σ < 0.78, a value larger than the corresponding one for the Anderson transition in long-ranged coupled chains with random diagonal disorder. The fractality of critical wavefunctions is also characterized.
We investigate the nature of one-electron eigenstates in power-law diluted chains for which the probability of occurrence of a bond between sites separated by a distance r decays as p(r) = 1/r1+sigma. Using an exact diagonalization scheme on finite chains, we compute the spreading of an initially localized wave-packet, the time dependent participation number as well as the return probability. Our results show the existence of a phase of extended states. By considering the scale invariance of the fluctuations of the participation number at the Anderson transition, we obtained that extended states emerges for sigma < 0.68. This limiting value is larger than the one reported in the literature for the emergence of extended states in one-dimensional Anderson models with power-law decaying couplings.
Relying on the fractal character of the largest clusters at criticality, we employ a finite-size scaling analysis to obtain an accurate phase-diagram of the percolation transition in chains with bond concentration decaying as a power-law on the form 1/ r(1+sigma) . For the particular case of sigma=1, no percolation transition is observed to occur at a finite dilution, in contrast with the finite temperature Kosterlitz-Thouless transition exhibited in Ising and Potts chains with inverse square-law couplings. The fractal dimension of the critical percolation cluster is found to follow distinct dependencies on the decay exponent being numerically fitted by d(f) =0.35+4sigma/5 for 0<sigma<1/2 and d(f) = (1+sigma) /2 for 1/2<sigma<1 . We also compute average mass ratios of the two largest clusters at criticality.
We investigate the quantum percolation problem in a diluted chain with long-range hopping amplitudes. Each bond is activated with probability p(r)=p1/rα, where r is the distance between two sites and α characterizes the range of the interactions. The average participation ratio of all eigenstates is used as a measure of the wave-functions localization length. We found that, above a quantum percolation threshold p1(q), true extended states appear for α<1.5. In the regime of 1.5<α<2.0 there is no truly extended states even in the presence of a spanning cluster. Instead, a phase of critical wave-functions sets up.
In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r^{d+σ}, where r is the distance length between distinct sites and d=1. We introduce and test an order-N Monte Carlo algorithm and we determine as a function of σ the critical value C_{c} at which percolation occurs. The critical exponents in the range 0<σ<1 are reported. Our analysis is in agreement, up to a numerical precision ≈10^{-3}, with the mean-field result for the anomalous dimension η=2-σ, showing that there is no correction to η due to correlation effects. The obtained values for C_{c} are compared with a known exact bound, while the critical exponent ν is compared with results from mean-field theory, from an expansion around the point σ=1 and from the ɛ-expansion used with the introduction of a suitably defined effective dimension d_{eff} relating the long-range model with a short-range one in dimension d_{eff}. We finally present a formulation of our algorithm for bond percolation on general graphs, with order N efficiency on a large class of graphs including short-range percolation and translationally invariant long-range models in any spatial dimension d with σ>0.
The two-dimensional antiferromagnetic S=1/2 Heisenberg model with random bond dilution is studied using quantum Monte Carlo simulation at the percolation threshold (50% of the bonds removed). Finite-size scaling of the staggered structure factor averaged over the largest connected clusters of sites on L*L lattices shows that long-range order exists within the percolating fractal clusters in the thermodynamic limit. This implies that the order-disorder transition driven by bond-dilution occurs exactly at the percolation threshold and that the exponents are classical. This result should apply also to the site-diluted system.
We investigate numerically the nature of energy eigenstates in one-dimensional bond-disordered Anderson models with hopping amplitudes decreasing as ${H}_{\mathrm{ij}}\ensuremath{\propto}1/|i\ensuremath{-}j{|}^{\ensuremath{\alpha}}.$ The eigenstates become delocalized whenever the hopping amplitudes decay slower than 1/r.1/r.1/r. By performing an exact diagonalization scheme on finite chains, we compute the participation ratio of all energy eigenstates. Employing a finite-size scaling analysis, we report on the relevant scaling exponents characterizing this delocalization transition as well as the level-spacing distribution at the critical point $\ensuremath{\alpha}=1.$ The random hopping amplitudes are taken from both uniform and random sign distributions. We show that these models display similar critical behavior in the vicinity of $\ensuremath{\alpha}=1.$ However, the random sign model exhibits an asymptotic delocalization in the limit of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ and the universal scaling behavior in this regime is also reported.