Abstract We investigate two one-dimensional tight-binding models with disorder that have extended states at zero energy. We use exact and partial diagonalisation of the Hamiltonian to obtain the eigenmodes and the associated participation ratios, and the transfer-matrix method to determine the localisation length. The first model has no on-site disorder, but random couplings. While the participation ratio remains finite at zero energy, the localisation length diverges logarithmically as the energy goes to zero. We provide an intuitive derivation of this logarithmic divergence based on the weak coupling of the two sublattices. The second model has a conserved quantity as the row sums of the Hamiltonian are zero. This model can be represented as a harmonic chain with random couplings, or as a diffusion model on a lattice with random links. We find, in agreement with existing analytical calculations, that the number of system-spanning eigenmodes increases proportionally to the square root of the system size, and we related this power law to other power laws that characterise the scaling behaviour of the eigenmodes, the participation ratio, the localisation length, and their dependence on energy and system size. When disorder is so strong that the smallest hopping terms can be arbitrarily close to zero, all these power laws change, and we show a crossover between the two scaling regimes. All these results are explained by intuitive arguments based on scaling.
We study Anderson localisation on high-dimensional graphs with spatial structure induced by long-ranged but distance-dependent hopping. To this end, we introduce a class of models that interpolate between the short-range Anderson model on a random regular graph and fully connected models with statistically uniform hopping, by embedding a random regular graph into a complete graph and allowing hopping amplitudes to decay exponentially with graph distance. The competition between the exponentially growing number of neighbours with graph distance and the exponentially decaying hopping amplitude positions our models effectively as power-law hopping generalisation of the Anderson model on random regular graphs. Using a combination of numerical exact diagonalisation and analytical renormalised perturbation theory, we establish the resulting localisation phase diagram emerging from the interplay of the lengthscale associated to the hopping range and the onsite disorder strength. We find that increasing the hopping range shifts the localisation transition to stronger disorder, and that beyond a critical range the localised phase ceases to exist even at arbitrarily strong disorder. Our results indicate a direct Anderson transition between delocalised and localised phases, with no evidence for an intervening multifractal phase, for both deterministic and random hopping models. A scaling analysis based on inverse participation ratios reveals behaviour consistent with a Kosterlitz-Thouless-like transition with two-parameter scaling, in line with Anderson transitions on high-dimensional graphs. We also observe distinct critical behaviour in average and typical correlation functions, reflecting the different scaling properties of generalised inverse participation ratios.
Abstract We focus on tight‐binding Hamiltonians on a regular one‐dimensional lattice with non‐random long‐range inter‐site coupling Jmn = J/|m−n|μ and random uncorrelated site energies. Within the model the localization‐delocalization transition occurs at one of the energy band edges provided 1 < μ < 3/2. Using the model we demonstrate that the ratio of the first two momenta of the participation number distribution for the critical states is a size invariant parameter at some value of the disorder magnitude Δc. We claim that the invariance manifests the transition. We find that Δc ≠ 0 at 1 < μ < 3/2, suggesting that the system undergoes the localization‐delocalization transition with respect to disorder magnitude. At μ ≥ 3/2, all states are localized. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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.
We perform numerical calculations on a simple cubic lattice for a standard diagonally disordered tight-binding Hamiltonian, whose random site energies are chosen from a Gaussian distribution with variance ∑2. From phenomenological renormalization group studies of the localization length, we determine that the critical disorder is σc≡∑c/J=6.00±0.17, which is in good agreement with previous results (J is the nearest neighbor transfer matrix element). From our calculations we can also determine the mobility edge trajectory, which appears to be analytic at the band center. Defining an order parameter exponent β, which determines how the fraction of extended states vanishes as the critical point is approached from below, this implies that β=1/2, in agreement with a previous study. From a finite-size scaling analysis we find that π2/ν=1.43±0.10, where π2 and ν are the inverse participation ratio and localization length critical exponents, respectively. This ratio of exponents can also be interpreted as the fractal dimension (also called the correlation dimension) D2 of the critical wave functions. Generalizations of the inverse participation ratio lead to a whole set of critical exponents πk, and corresponding generalized fractal dimensions Dk=πk/ν(k−1). From finite-size scaling results we find that D3=1.08±0.10 and D4=0.87±0.09. The inequality of the three dimensions D2, D3, and D4 shows that the critical wave functions have a multifractal structure. Using a generalized phenomenological renormalization technique on the participation ratios, we find that ν=0.99±0.04. This result is in agreement with experiments on compensated or amorphous doped semiconductors.
We study the one-dimensional nearest-neighbor tight-binding model of electrons with independently distributed random hopping and no on-site potential (i.e., off-diagonal disorder with particle-hole symmetry, leading to sublattice symmetry, for each realization). For nonsingular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states $\ensuremath{\rho}(E)\ensuremath{\sim}1/|E{ln}^{3}|E||$ and of the localization length $\ensuremath{\xi}(E)\ensuremath{\sim}|ln|E||$, near the band center
E=0. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain.) Simultaneously, the state at
E=0 shows a universal, subexponential decay at large distances $\ensuremath{\sim}exp[\ensuremath{-}\sqrt{r/{r}_{0}}]$. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small
t is of the form $\ensuremath{\sim}1/[t{ln}^{\ensuremath{\lambda}+1}(1/t)]$, characterized by a single, continuously tunable parameter $\ensuremath{\lambda}>0$. We find, using a combination of analytic and numerical methods, that while the universal result applies for $\ensuremath{\lambda}>2$, it no longer holds in the interval $0<\ensuremath{\lambda}<2$. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the nonuniversal parameter $\ensuremath{\lambda}$; simultaneously, the localization length shows a less divergent form at low energies and ceases to diverge below $\ensuremath{\lambda}=1$. For $\ensuremath{\lambda}<2$, the fall-off of the
E=0 state at large distances also deviates from the universal result and is of the form $\ensuremath{\sim}exp[\ensuremath{-}{(r/{r}_{0})}^{1/\ensuremath{\lambda}}]$, which decays faster than an exponential for $\ensuremath{\lambda}<1$.
The one-dimensional (1D) tight-binding model with random nearest-neighbor hopping is known to have a singularity of the density of states and of the localization length at the band center. We study numerically the effects of random long-range (power-law) hopping with an ensemble average magnitude $〈|{t}_{\mathrm{ij}}|〉\ensuremath{\propto}|i\ensuremath{-}j{|}^{\ensuremath{-}\ensuremath{\sigma}}$ in the 1D chain, while maintaining the particle-hole symmetry present in the nearest-neighbor model. We find, in agreement with results of real-space renormalization-group techniques applied to the random XY spin chain with power-law interactions, that there is a change of behavior when the power-law exponent $\ensuremath{\sigma}$ becomes smaller than 2.