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 $\ensuremath{\lambda}<1$.
We introduce and implement a reformulation of the strong-disorder renormalization group method in real space, well suited to study bond-disordered antiferromagnetic power-law coupled quantum spin chains. We apply it to a chain of randomly placed spins coupled by power-law long-range interaction with power $\ensuremath{\alpha}$. First, keeping only interactions between adjacent spins, we derive the Master equation and confirm that it is solved by the infinite randomness fixed-point distribution. Then, we derive the master equation for power-law long-range interaction between all spins for any anisotropy $\ensuremath{\gamma}$ ranging from the XX limit, $\ensuremath{\gamma}=0$, to the isotropic Heisenberg limit, $\ensuremath{\gamma}=1,$ which corresponds to a tight-binding chain of disordered long-range interacting fermions with long-range hopping. We thereby show that the distribution function of couplings
J smaller than renormalization scale $\mathrm{\ensuremath{\Omega}}$ flows to the strong-disorder fixed-point distribution of finite width $2\ensuremath{\alpha}.$ We find only small corrections to that distribution, which depend on power exponent $\ensuremath{\alpha}$ and coupling anisotropy $\ensuremath{\gamma}.$ As a consequence, the low-temperature magnetic susceptibility diverges with an anomalous power law. The distribution of singlet lengths
l is found to decay as ${l}^{\ensuremath{-}2}$. The entanglement entropy of a subsystem of length
n increases in the ground state logarithmically for all $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$. After a global quantum quench, the entanglement entropy increases with time logarithmically as $S(t)\ensuremath{\sim}ln(t)/(2\ensuremath{\alpha})$.
The decay of (disorder-averaged) static spin correlation functions at T = 0 for the one-dimensional spin-1/2 XXZ antiferromagnet with uniform longitudinal coupling J ∆ and random transverse coupling Jλi is investigated by numerical calculations for ensembles of finite chains. At ∆ = 0 (XX model) the calculation is based on the Jordan-Wigner mapping to free lattice fermions for chains with up to N = 100 sites. At ∆ ̸ = 0 Lanczos diagonalizations are carried out for chains with up to N = 22 sites. The longitudinal correlation function 〈Sz 0Sz r 〉 is found to exhibit a power-law decay with an exponent that varies with ∆ and, for nonzero ∆, also with the width of the λi-distribution. The results for the transverse correlation function 〈S x 0 Sx r 〉 show a crossover from power-law decay to exponential decay as the exchange disorder is turned on. 1 Typeset using REVTEXThe combination of randomness and quantum fluctuations is well-known to be a fertile ground for interesting physical phenomena including Anderson localization. In onedimensional (1D) tight-binding systems the rule is that disorder always leads to localization. However, if the randomness is purely off-diagonal, the localization length diverges at the band center, 1 which is bound to affect the decay law of correlation functions. One particular off-diagonally disordered fermion model, the 1D half-filled tight-binding model with random hopping, is equivalent to the special XX case ( ∆ = 0) of the 1D s = 1/2 XXZ model with random transverse exchange coupling, described by the Hamiltonian H = J ∑
Local moment formation is ubiquitous in disordered semiconductors such as Si:P, where it is observed both in the metallic and the insulating regimes. Here, we focus on local moment behavior in disordered insulators, which arises from short-ranged, repulsive electron-electron interactions. Using density matrix renormalization group and strong-disorder renormalization group methods, we study paradigmatic models of interacting insulators: one-dimensional Hubbard chains with quenched randomness. In chains with either random fermion hoppings or random chemical potentials, both at and away from half-filling, we find exponential decay of disorder-averaged charge and fermion two-point correlations, but power-law decay of disorder-averaged spin correlations that are indicative of the random singlet phase. The numerical results can be understood qualitatively by appealing to the large-interaction limit of the Hubbard chain, in which a remarkably simple picture emerges.
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.
A real-space, zero-temperature renormalization-group method is used to study a Heisenberg antiferromagnetic spin chain with long-range interactions varying as ${(\mathrm{distance})}^{\ensuremath{-}p}$. Simple renormalization-group equations are obtained and studied analytically, and the results are checked against a more accurate numerical calculation. The calculations provide evidence for a phase transition at $p\ensuremath{\approx}1.85$ characterized by a change in the behavior of zero-temperature correlation functions. For $p\ensuremath{\gtrsim}1.85$ the large-distance physics is that of the nearest-neighbor antiferromagnet, while there is a line of fixed points for $p\ensuremath{\lesssim}1.85$. I conjecture that $p\ensuremath{\approx}1.85$ also marks an order-disorder transition at finite temperature such as occurs in the Ising model with power-law interactions.