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arXiv:1203.4599 · Physical Review B

Level statistics for quantum k -core percolation

L. Cao, Jennifer M. Schwarz

Published Aug 24, 2012• 1 citation
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Abstract

Quantum $k$-core percolation is the study of quantum transport on $k$-core percolation clusters where each occupied bond must have at least $k$ occupied neighboring bonds. As the bond occupation probability $p$ is increased from zero to unity, the system undergoes a transition from an insulating phase to a metallic phase. When the length scale for the disorder ${l}_{d}$ is much greater than the coherence length ${l}_{c}$, earlier analytical calculations of quantum conduction on the Bethe lattice demonstrated that for $k=3$ the metal-insulator transition (MIT) is discontinuous, suggesting a new type of disorder-driven quantum MITs. Here, we numerically compute the level spacing distribution as a function of bond occupation probability $p$ and system size on a Bethe-like lattice. The level spacing analysis suggests that for $k=0$, ${p}_{q}$, the quantum percolation critical probability, is greater than ${p}_{c}$, the geometrical percolation critical probability, and the transition is continuous. In contrast, for $k=3$, ${p}_{q}={p}_{c}$, and the transition is discontinuous such that these numerical findings are consistent with our previous work to reiterate a new random first-order phase transition and therefore a new universality class of disorder-driven quantum MITs.

DOI
10.1103/physrevb.86.064206

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Quantum kkk-core percolation is the study of quantum transport on kkk-core percolation clusters where each occupied bond must have at least kkk occupied neighboring bonds. As the bond occupation probability ppp is increased from zero to unity, the system undergoes a transition from an insulating phase to a metallic phase. When the length scale for the disorder ld{l}_{d}ld​ is much greater than the coherence length lc{l}_{c}lc​, earlier analytical calculations of quantum conduction on the Bethe lattice demonstrated that for k=3k=3k=3 the metal-insulator transition (MIT) is discontinuous, suggesting a new type of disorder-driven quantum MITs. Here, we numerically compute the level spacing distribution as a function of bond occupation probability ppp and system size on a Bethe-like lattice. The level spacing analysis suggests that for k=0k=0k=0, pq{p}_{q}pq​, the quantum percolation critical probability, is greater than pc{p}_{c}pc​, the geometrical percolation critical probability, and the transition is continuous. In contrast, for k=3k=3k=3, pq=pc{p}_{q}={p}_{c}pq​=pc​, and the transition is discontinuous such that these numerical findings are consistent with our previous work to reiterate a new random first-order phase transition and therefore a new universality class of disorder-driven quantum MITs.
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We consider the cluster-size statistics of two types of site-bond-correlated percolation. In model A (B) [see Phys. Rev. B 40, 10 986 (1989)], the activation of a bond between two first-neighbor occupied sites is also dependent on the occupancy of both (either) of their own nearest neighbors, along the line joining them. The critical concentration for models A and B, obtained from simulations on square lattices of sizes up to $2000\ifmmode\times\else\texttimes\fi{}2000$, is found to be $0.7405\ifmmode\pm\else\textpm\fi{}0.0005$ and $0.6145\ifmmode\pm\else\textpm\fi{}0.0005$, respectively. The resulting critical exponents indicate that both models are in the same universality class as ordinary independent (site or bond) percolation.
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We investigate a class of dynamic percolation models on a lattice, in which connections are added and removed by a set of local stochastic rules. In particular, the removal of any connection that can result in one connected cluster becoming two (or more) separate clusters is strictly forbidden. These models include and are inspired by a model for local quantum error correction for a toric code studied by Chirame, et al., PRX Quantum 6, 030363 (2025), who identified a phase transition with first-order characteristics, such as bistability and a discontinuous order parameter. We show numerically that this transition is a hybrid percolation transition (HPT), having characteristics of a continuous percolation transition such as diverging critical clusters while at the same transition having discontinuities like a first-order transition. We then modify this model to be fully isotropic, showing that the HPT remains. In both of these models, the steady state on one side of the HPT is an absorbing state; by turning on additional isotropic local stochastic moves, we remove the absorbing state and strongly suppress or remove the first-order transition, leaving an apparently continuous percolation transition. Curiously, the numerically-observed percolation critical exponents do not change significantly between the models with hybrid transitions and the model with an apparently continuous transition. These exponents are significantly different from known percolation transitions, indicating that this may be a new universality class.
We investigate quantum percolation in a honeycomb lattice with site dilution and random spin-orbit coupling. Using exact diagonalization combined with finite-size scaling analysis, we study the metal-insulator transition, extracting the quantum percolation threshold pqp_qpq​, and the correlation-length exponent, ννν. In the absence of spin-orbit coupling, we find that pqp_qpq​ remains finite and demonstrate that the quantum threshold is significantly higher than the classical site-percolation threshold pcp_cpc​ of the honeycomb lattice. When spin-orbit coupling is present, the spectral statistics exhibit a crossover from the Gaussian orthogonal ensemble to the Gaussian symplectic ensemble, reflecting the change in symmetry class. Simultaneously, the quantum percolation threshold shifts systematically to lower occupation probabilities, indicating that the spin-orbit coupling favors delocalization. For sufficiently strong spin-orbit coupling, pqp_qpq​ tends to saturate, while the critical exponent approaches the expected one of the two-dimensional symplectic universality class.