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arXiv:1404.1381 · Physical Review B

Strong-disorder renormalization-group study of the one-dimensional tight-binding model

H. Javan Mard, José A. Hoyos, E. Miranda, V. Dobrosavljević

Published Sep 23, 2014• 21 citations
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Abstract

We formulate a strong-disorder renormalization-group (SDRG) approach to study the $\ensuremath{\beta}$ function of the tight-binding model in one dimension with both diagonal and off-diagonal disorder for states at the band center. We show that the SDRG method, when used to compute transport properties, yields exact results since it is identical to the transfer matrix method. The $\ensuremath{\beta}$ function is shown to be universal when only off-diagonal disorder is present even though single-parameter scaling is known to be violated. A different single-parameter scaling theory is formulated for this particular (particle-hole symmetric) case. Upon breaking particle-hole symmetry (by adding diagonal disorder), the $\ensuremath{\beta}$ function is shown to crossover from the universal behavior of the particle-hole symmetric case to the conventional nonuniversal one in agreement with the two-parameter scaling theory. We finally draw an analogy with the random transverse-field Ising chain in the paramagnetic phase. The particle-hole symmetric case corresponds to the critical point of the quantum Ising model, while the generic case corresponds to the Griffiths paramagnetic phase.

DOI
10.1103/physrevb.90.125141

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We formulate a strong-disorder renormalization-group (SDRG) approach to study the \ensuremathβ\ensuremath{\beta}\ensuremathβ function of the tight-binding model in one dimension with both diagonal and off-diagonal disorder for states at the band center. We show that the SDRG method, when used to compute transport properties, yields exact results since it is identical to the transfer matrix method. The \ensuremathβ\ensuremath{\beta}\ensuremathβ function is shown to be universal when only off-diagonal disorder is present even though single-parameter scaling is known to be violated. A different single-parameter scaling theory is formulated for this particular (particle-hole symmetric) case. Upon breaking particle-hole symmetry (by adding diagonal disorder), the \ensuremathβ\ensuremath{\beta}\ensuremathβ function is shown to crossover from the universal behavior of the particle-hole symmetric case to the conventional nonuniversal one in agreement with the two-parameter scaling theory. We finally draw an analogy with the random transverse-field Ising chain in the paramagnetic phase. The particle-hole symmetric case corresponds to the critical point of the quantum Ising model, while the generic case corresponds to the Griffiths paramagnetic phase.
The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in d=1,2d=1,2d=1,2, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In d=1d=1d=1 the results are related to dimer-dimer correlations in the random antiferromagnetic XXXXXX chain and have been tested by numerical calculations using free-fermionic techniques.
The infinite-disorder fixed point of the random transverse-field Ising model is expected to control the critical behavior of a large class of random quantum and stochastic systems having an order parameter with discrete symmetry. Here we study the model on the square lattice with a very efficient numerical implementation of the strong disorder renormalization group method, which makes us possible to treat finite samples of linear size up to L=2048L=2048L=2048. We have calculated sample dependent pseudocritical points and studied their distribution, which is found to be characterized by the same shift and width exponent: $\ensuremath{\nu}=1.24(2)$. For different types of disorder the infinite-disorder fixed point is shown to be characterized by the same set of critical exponents, for which we have obtained improved estimates: x=0.982(15)x=0.982(15)x=0.982(15) and $\ensuremath{\psi}=0.48(2)$. We have also studied the scaling behavior of the magnetization in the vicinity of the critical point as well as dynamical scaling in the ordered and disordered Griffiths phases.
Motivated by long-range dispersal in ecological systems, we formulate and apply a general strong-disorder renormalization group (SDRG) framework to describe one-dimensional disordered contact processes with heavy-tailed (such as power law, stretched exponential, and log-normal) dispersal kernels, widely used in ecology. The focus is on the close-to-critical scaling of the order parameters, including the commonly used density, as well as the less known persistence, which is nonzero in the inactive phase. Our analytic and numerical results obtained by SDRG schemes at different levels of approximation reveal that the more slowly decaying dispersal kernels lead to faster-vanishing densities as the critical point is approached. The persistence, however, shows an opposite tendency: the broadening of the dispersal makes its decline sharper at the critical point, becoming discontinuous for the extreme case of power-law dispersal. The SDRG schemes presented here also describe the quantum phase transition of random transverse-field Ising chains with ferromagnetic long-range interactions, the density corresponding to the magnetization of that model.
We have developed a very efficient numerical algorithm of the strong disorder renormalization group method to study the critical behaviour of the random transverse field Ising model, which is a prototype of random quantum magnets. With this algorithm we can renormalize an N-site cluster within a time NlogN, independently of the topology of the graph, and we went up to N ∼ 4 × 10(6). We have studied regular lattices with dimension D ≤ 4 as well as Erdős-Rényi random graphs, which are infinite dimensional objects. In all cases the quantum critical behaviour is found to be controlled by an infinite disorder fixed point, in which disorder plays a dominant role over quantum fluctuations. As a consequence the renormalization procedure as well as the obtained critical properties are asymptotically exact for large systems. We have also studied Griffiths singularities in the paramagnetic and ferromagnetic phases and generalized the numerical algorithm for other random quantum systems.
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=0E=0E=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=0E=0E=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 ttt 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=0E=0E=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.