Random Hopping Disorder in Quantum Systems

• Random Hopping Disorder is characterized by spatially random hopping amplitudes with distance-dependent decay, leading to distinctive spectral, localization, and transport phenomena.
• Analytical and numerical methods such as exact diagonalization, strong-disorder renormalization, and Monte Carlo simulations reveal tunable critical scaling and phase transitions.
• The rich implications span quantum and classical systems, influencing entanglement entropy, transport laws, and the emergence of glassy phases in experimental realizations.

Random hopping disorder refers to models in which quantum or classical particles move through a system via hopping terms whose amplitudes are spatially random, often with nontrivial distance-dependent decay, and in which the disorder resides in the off-diagonal elements of the Hamiltonian or transition operator. Such disorder fundamentally alters spectral, transport, entanglement, and localization properties, distinguishing these models from their diagonal (on-site) disorder counterparts. This class encompasses free-fermion, bosonic, spin-chain, and random-walk formulations in 1D, 2D, and 3D, and supports a variety of universal and non-universal phenomena, critical scaling regimes, and phase transitions as the decay exponent and disorder statistics are tuned.

1. Model Architectures and Disorder Types

Random hopping disorder manifests in multiple architectures:

• 1D Lattice Models: Spinless fermion chains with Hamiltonians of the form $$H = \sum_{i<j} t_{ij}(c_i^\dagger c_j + c_j^\dagger c_i)$$, where $t_{ij}$ are random and may decay algebraically, $t_{ij} \sim w_{ij} |i-j|^{-\alpha}$, or be strictly nearest-neighbor, $t_{ij}=0$ unless $|i-j|=1$ (Juhász, 2021, Krishna et al., 2020, Krishna et al., 2021).
• 2D and 3D Networks: Generalizations to higher-dimensional graphs, including random long-range hopping on square or cubic lattices, with amplitude statistics $t_{ij} \sim \mathcal{N}(0,\varepsilon a(r))$; for example, $a(r)=1/[1+(r/b)^{2\alpha}]$ in banded random-matrix models (Ossipov et al., 2011, Deng et al., 2022). Empirical realizations include Rydberg gases with dipole-dipole couplings $J_{ij} \sim |r_i - r_j|^{-3}$ in 3D (Lippe et al., 2020).
• Classical Random Walks: Hopping-rate disorder arises in stochastic processes with random nearest-neighbor jump probabilities, generating spatially variable local biases and effective potentials (Holehouse et al., 2023).
• Bosonic Lattice Models: In the disordered Bose–Hubbard model, both site energy and hopping amplitudes may be disordered and correlated or uncorrelated, impacting quantum phase boundaries (Stasińska et al., 2014).

The disorder statistics—whether hopping amplitudes are drawn from smooth (e.g., uniform, Gaussian) or singular (fat-tailed, log-divergent) distributions—control universal versus non-universal behavior (Krishna et al., 2020, Krishna et al., 2021).

2. Spectral Properties and Universality

For one-dimensional models with sublattice symmetry and nearest-neighbor random hopping, the low-energy spectral singularities exhibit remarkable universality. The density of states near $E=0$ follows the celebrated Dyson form: $$\rho(E) \sim \frac{1}{|E \ln^3 |E||},\quad \xi(E)\sim |\ln|E||,$$ independent of the details of the hopping distribution provided its moments are finite (Krishna et al., 2020, Krishna et al., 2021).

However, when the hopping distribution becomes singular as $t\to0$ (e.g., $P(t)\sim 1/[t \ln^{\lambda+1}(1/t)]$, $0<\lambda<2$), universal scaling breaks down. Both the divergence in the density of states and the localization length exponent become tunable functions of $\lambda$, and the spectrum transitions to non-universal fixed points with enhanced singularity or even super-exponential localization for $\lambda<1$ (Krishna et al., 2020, Krishna et al., 2021).

In 2D long-range hopping models, a metal-insulator transition arises at a critical decay exponent $\alpha_c=2$, but the critical eigenfunctions lack multifractal self-similarity. Instead, the inverse participation ratio moments scale as powers of $\ln \ln L$ rather than $L$, signaling a distinct universality class (Ossipov et al., 2011). In random two-dimensional systems with hopping $V(r)\sim r^{-2}$ and time-reversal symmetry, there exist weak-disorder “superdiffusive” phases, diffusive regimes, and transitions characterized by modified, non-standard critical exponents and fractal eigenstate dimensions (Deng et al., 2022).

3. Localization, Entanglement, and Transport

Random hopping induces localization phenomena markedly different from Anderson (diagonal) disorder:

• Dyson sub-exponential localization: For well-behaved hopping statistics, the $E=0$ eigenstate decays as $|\psi(r)|\sim \exp(-\sqrt{r/r_0})$, reflecting weak localization (Krishna et al., 2020).
• Super-exponential localization: For strongly singular hopping distributions ($\lambda<1$), states decay as $\exp[-(r/r_0)^{1/\lambda}]$, faster than any exponential (Krishna et al., 2020).
• Power-law localization in interacting systems: Even with deterministic power-law decaying hopping and strong on-site disorder, multi-particle eigenfunctions are localized with uniform power-law decay $|\phi(x)|\le C \langle x\rangle^{-r/300}$ for sufficiently strong decay exponent $r$ and disorder amplitude $g$ (Jian et al., 25 Mar 2025).
• Random-Singlet State and Entanglement Scaling: In 1D chains with random hopping decaying as $|i-j|^{-\alpha}$, entanglement entropy $S_\ell$ grows logarithmically, $S_\ell\simeq (c_{\rm eff}/3)\ln \ell$, but the effective central charge $c_{\rm eff}(\alpha)$ varies with $\alpha$. The standard random-singlet consistency $c_{\rm eff}=3\ln 2\cdot b$ holds only for large $\alpha$ and breaks for $\alpha\leq 2$, with mutual crossover regimes (Juhász, 2021).
• Transport Laws: Random hopping forms the basis of variable range hopping (VRH) and nearest-neighbor hopping (NNH) transport in disordered materials. Random resistor networks simulated via Monte Carlo percolation quantitatively connect spatial/energetic disorder and localization length $\alpha$ to the evolution of the conductivity $\sigma(T)$, capturing transitions between NNH ($\sigma(T)\sim \exp[-E_a/k_BT]$) and VRH ($\sigma(T)\sim \exp[-(T_0/T)^{1/(d+1)}]$) as a function of temperature (Toral-Lopez et al., 3 Jan 2026).

4. Impact on Quantum Phase Diagrams and Topology

Random hopping disorder has nontrivial consequences on many-body phase diagrams and topological properties:

• Bose–Hubbard Model: The introduction of hopping disorder (in addition to diagonal disorder) frustrates coherence, shrinks Mott lobes, and eliminates direct Mott–superfluid transitions. An intermediate glassy phase (Bose glass) is mandatory, even in correlated or uncorrelated models. This expands glassy domains in parameter space and shifts critical points for superfluid onset (Stasińska et al., 2014).
• SSH and Topological Models: Extended SSH models with random off-diagonal disorder exhibit Anderson localization, but also link the Lyapunov exponent $\gamma(E)$ to topological fingerprints. In the topological phase, mid-gap states become less localized as disorder increases, in contrast to the trivial phase. Edge-mode contributions yield robust differences in transmission decay rates at $E=0$, with direct consequences for transport experiments (Pérez-González et al., 2018).
• Experimental Realization: 3D random hopping models have been realized in ultracold Rydberg gases, with tunable dipole-dipole couplings generating effective random hopping processes. Spectroscopic signatures reveal crossovers from delocalized to localized pair states, with direct experimental observation of transport and localization mechanisms (Lippe et al., 2020).

5. Landscape Theory, First-Passage, and Classical Random Walks

In both quantum and classical contexts, random hopping models support model-independent analytic approaches:

• Landscape Function and Ground State Bounds: The landscape function $u=H^{-1}1$ encodes tunneling barriers; the product $\lambda\|u\|_\infty$ for the ground state is universally bounded. In 1D random hopping chains with Bernoulli or uniform disorder, this product approaches $\pi^2/8$ in the thermodynamic limit, reflecting a generic connection to the Dirichlet Laplacian on the unit interval (Shou et al., 2022).
• First-Passage Statistics: Classical random walkers on disordered intervals with random hopping rates display highly non-self-averaging exit-time statistics. The generating-function approach provides closed-form moments and distributions. Bimodal first-passage time distributions can arise in environments with competing biases, violating homogeneous-medium intuition and requiring disorder-resolved predictions. Scaling laws connect with Sinai-type transport and barrier-dominated dynamics (Holehouse et al., 2023).

6. Experimental and Numerical Methods

Multiple advanced methodologies underpin the study of random hopping disorder:

• Exact Diagonalization: Chains of up to $L\sim 10^4$–$10^6$ sites (Juhász, 2021, Krishna et al., 2020).
• Strong-Disorder Renormalization Group (SDRG): Both full and minimal schemes to efficiently approximate ground states and entanglement scaling under strong disorder (Juhász, 2021).
• Monte Carlo Random Resistor Networks: For modeling hopping-conductance regimes and dimensional crossover in disordered materials (Toral-Lopez et al., 3 Jan 2026).
• Transfer-Matrix and Lyapunov Analysis: For extracting localization exponents and edge-mode fingerprints in topological chains (Pérez-González et al., 2018).
• Spectroscopic and Transport Measurements: Direct experimental access in Rydberg gases and graphene (Lippe et al., 2020, Pereira et al., 2011).

7. Physical Implications and Open Directions

Random hopping disorder generates a rich phenomenology with significant impact on condensed matter, cold atom, and statistical physics:

• Breakdown and tuning of universal singularities and fixed-point behavior.
• Emergence of new universality classes in spectral, localization, and entanglement scaling.
• Enhanced glassiness, criticality, and anomalous transport regimes, including superdiffusion and non-self-averaging statistics.
• Model-independent ground-state estimates via landscape theory and connections to tunneling barriers.
• Robust experimental platforms for probing many-body localization, mobility edges, and long-range transport in controlled settings.

A plausible implication is that further exploration of fat-tailed disorder distributions, dimensional crossover, and correlated hopping mechanisms will uncover novel transitions and critical phenomena beyond existing universality paradigms. Future research may synthesize landscape, quantum information, and transport metrics to constrain and classify localization across arbitrary random network structures.