Abstract Critical states are studied by a real space RG in the problem with strong diagonal disorder and long range power law hopping. The RG ow of the distribution of coupling parameters is characterized by a family of non‐trivial fix points. We consider the RG flow of the distribution of participation ratios of eigenstates. Scaling of participation ratios is sensitive to the nature of the RG fix point. For some fix points, scaling of participation ratios is characterized by a distribution of exponents, rather than by a single exponent. The RG method can be generalized to treat certain fermionic Hamiltonians with disorder and long range hopping. We derive the RG for a model of interacting two‐level systems. Besides couplings, in this problem the RG includes the density of states. The density of states is renormalized so that it develops a singularity near zero energy.
Abstract Critical states are studied by a real space RG in the problem with strong diagonal disorder and long range power law hopping. The RG ow of the distribution of coupling parameters is characterized by a family of non‐trivial fix points. We consider the RG flow of the distribution of participation ratios of eigenstates. Scaling of participation ratios is sensitive to the nature of the RG fix point. For some fix points, scaling of participation ratios is characterized by a distribution of exponents, rather than by a single exponent. The RG method can be generalized to treat certain fermionic Hamiltonians with disorder and long range hopping. We derive the RG for a model of interacting two‐level systems. Besides couplings, in this problem the RG includes the density of states. The density of states is renormalized so that it develops a singularity near zero energy.
Critical states are studied by a real space RG in the problem with strong diagonal disorder and long range power law hopping. The RG ow of the distribution of coupling parameters is characterized by a family of non-trivial fix points. We consider the RG flow of the distribution of participation ratios of eigenstates. Scaling of participation ratios is sensitive to the nature of the RG fix point. For some fix points, scaling of participation ratios is characterized by a distribution of exponents, rather than by a single exponent. The RG method can be generalized to treat certain fermionic Hamiltonians with disorder and long range hopping. We derive the RG for a model of interacting two-level systems. Besides couplings, in this problem the RG includes the density of states. The density of states is renormalized so that it develops a singularity near zero energy.
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.
The renormalization-group (RG) method is applied to the problem of formation of a localized state of a particle moving in a given potential. It is shown that RG transformation on the particle's Green's function can be performed exactly. The fixed-point equations yield information on the critical binding strength, while the transformation equations near the fixed points give scaling laws and critical exponents, e.g., for the dependence of the localization radius on the energy and the potential strength. The general theory is illustrated by considering the simplest case of the Slater-Koster problem in detail. We find that it is possible to take all the irrelevant variables into account and thus obtain the exact result for the critical binding point. Critical exponents and scaling laws are, however, universal in character. The method is then extended to the case of motion in random potentials, and some applications are pointed out. In particular it is shown that the RG properties of the averaged absolute value squared (〈∣G∣2〉) of the Green's function do not correspond to a simple second-order phase transition. The analysis illustrates certain features of RG methods which have not heretofore been encountered in other problems and which are likely to be found in treatments of disordered materials. In particular, the runaway behavior observed in the RG transformation of 〈G〉 and 〈∣G∣2〉 as well as in many random-spin systems, might be associated with the formation of bound states in the band tail for any amount of randomness.
We investigate a one-dimensional tight-binding model in which onsite potentials [Formula: see text] exhibit power-law spatial correlations (with exponent α) and the hopping amplitudes decay as [Formula: see text]. This two-parameter family interpolates continuously between short-range Anderson-like disorder, correlated disorder with conventional hopping, and long-range hopping models with nontrivial delocalization tendencies. Using large-scale exact diagonalization, we construct a comprehensive phase map in the [Formula: see text] plane by combining spectral statistics, density-of-states analysis, and energy-resolved localization indicators such as the participation ratio, single-particle entanglement entropy, level-spacing ratio r, and the ratio of the geometric to arithmetic density of states. From these observables we define phase-indicator functions that compactly quantify localization behavior across the spectrum. Our analysis reveals robust mobility edges and multiple regimes of spectral coexistence between localized, extended, resonant, and critical states. Finite-size scaling, implemented via an explicit smoothness-based cost function, enables extraction of critical exponents and delineation of transition lines across the [Formula: see text] parameter space. To validate and complement these physics-based diagnostics, we employ a supervised autoencoder that learns high-level representations of eigenstate structure directly from raw features and reliably reproduces the phase classification defined by the indicator functions. Together, these approaches provide a coherent and internally consistent picture of the spectral transitions driven by correlated disorder and long-range hopping, establishing a unified framework for characterizing mobility edges in long-range one-dimensional systems.
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.
Many of the most interesting electronic behaviours currently being studied are associated with strong correlations. In addition, many of these materials are disordered either intrinsically or due to doping. Solving interacting systems exactly is extremely computationally expensive, and approximate techniques developed for strongly correlated systems are not easily adapted to include disorder. As a non-interacting disordered model, it makes sense to consider the Anderson model as a first step in developing an approximate method of solution to the interacting and disordered Anderson-Hubbard model. Our renormalization group (RG) approach is modeled on that proposed by Johri and Bhatt [23]. We found an error in their work which we have corrected in our procedure. After testing the execution of the RG, we benchmarked the density of states and inverse participation ratio results against exact diagonalization. Our approach is significantly faster than exact diagonalization and is most accurate in the limit of strong disorder. Author Keywords: disorder, localization, real-space renormalization, strong correlations