Classical Percolation Threshold Research Articles

Finite-size effect on quantum percolation in topological insulators

We study the finite-size effect on quantum percolation in two-dimensional topological insulators. We demonstrate that the percolation threshold in topological insulators strongly depends on the localization length of the edge states in small clusters due to the finite-size effect. Also, we explain why the percolation threshold in the corresponding classical model determines the lower bound of the quantum percolation threshold in topological insulators. In addition, we extend the percolation model to a more general scenario, where the system is composed of both topological and trivial clusters. We find that the quantum percolation threshold can be less than the classical percolation threshold due to quantum tunneling of the edge states.

Magnetic impurities in a charge-ordered background

We investigate how magnetic impurities may affect a system exhibiting a charge-density wave (CDW) in its ground state. We consider a disordered Hubbard-Holstein model with a homogeneous electron-phonon interaction but with a (randomly chosen) fraction of sites displaying a nonzero Coulomb repulsion $U$ and perform state-of-the-art finite-temperature quantum Monte Carlo simulations. For a single magnetic impurity, charge-charge correlations hamper the spin-spin ones around the repulsive site, thus requiring a strong enough value of $U$ to create nonnegligible antiferromagnetic (AFM) correlations. As the number of magnetic impurities increases, these AFM correlations become deleterious to CDW order and its features. First, the critical temperature is drastically reduced and seems to vanish $\ensuremath{\sim}40%$ of impurities (for fixed $U/\ensuremath{\lambda}=2$), which we correlate with the classical percolation threshold. We also notice that just a small amount of disorder suffices to create a bad insulating state, with the suppression of both Peierls and spin gaps, even within the charge-ordered phase. Finally, we have also found that pairing correlations are enhanced at large doping, driven by the competition between CDW and AFM tendencies.

Antiferromagnetism beyond the classical percolation threshold in the diluted half-filled one-band Hubbard model in three dimensions

We investigate the impact of site dilution by setting the on-site repulsion strength ($U$) to zero at a fraction of sites in the half-filled Hubbard model on a simple cubic lattice. We employ a semi-classical Monte-Carlo approach first to recover the zero dilution (undiluted $x=1$) properties, including $U$ dependence of insulator to metal crossover temperature scale $T^*$ and long-range staggered antiferromagnetic ordering temperature ($T_N$). For the non-perturbative regime of $U \sim$ bandwidth, we find a rapid suppression of $T^*$ with reducing $x$ from 1 to 0.7. However, $T_N$ remains unchanged in this dilution range, showing a weakening of the insulating state but not of the magnetic order. At $x \leq 0.7$, $T^*$ and $T_N$ coincide and are suppressed together with further increase in site-dilution. Finally, the system loses the magnetic order and the insulating state for $x=0.15$, significantly below the classical percolation threshold $x_p^{sc} (\sim 0.31$). We show that the induced moments on $U=0$ sites drive the magnetic order below the classical percolation limit by studying local moment systematics and finite-size analysis of magnetic order. At the end, we show that either increasing $U$ to large values or raising temperature beyond a $U$ dependent critical value, suppresses the induced local moments of the $U=0$ sites and recovers the classical percolation threshold.

Magnetic dilution of a honeycomb lattice XY magnet CoTiO3

We report our study of cobalt (II) titanate, CoTiO3, in which magnetic Co ions are replaced by non-magnetic ions. The antiferromagnetic ordering transition of CoTiO3 around 37 K is described with ferromagnetic honeycomb layers coupled antiferromagnetically along the crystallographic c-direction. The effect of magnetic dilution on the Néel temperature of this material is investigated through the doping of Zn2+ and Mg2+ in place of Co2+ for various dilution levels up to x + y = 0.46 in Co1−x−y Zn x Mg y TiO3. Single phase polycrystalline samples have been synthesized and their structural and magnetic properties have been examined. A linear relation between dilution and the Néel temperature is observed over a wide doping range. A linear extrapolation would suggest that the required dilution level to suppress magnetic order is around x + y ∼ 0.74, well beyond the classical percolation threshold. The implication of this observation for microscopic models for describing CoTiO3 is discussed.

Dynamical resilience to disorder: The dilute Hubbard model on the Lieb lattice

In itinerant systems, electron-electron interactions may lead to the formation of local magnetic moments and their effective exchange coupling, which in turn gives rise to long-range magnetic order. Therefore, when moment formation is weakened, such as in the single-band Hubbard model on a square lattice with the on-site repulsion being randomly switched off on a fraction $x$ of sites, magnetic order is suppressed beyond some critical $x_c$, which was found to lie below the classical percolation threshold, $x_c^\text{(perc,sq)}$. Here we study dilute magnetism in flat band systems, namely in the Hubbard model on a `Lieb' lattice. Interestingly, we show that magnetic order persists to $x$ almost twice as large as the classical percolation threshold for the lattice, thus emphasizing the central role of electron itinerancy to the magnetic response. The analysis of the orbital-resolved order parameters reveals that the contribution of the four-fold coordinated `d' sites to magnetism is dramatically affected by dilution, while the localized `p' states of the flat band provide the dominant contribution to long-range correlations. We also examine the transport properties, which suggest the existence of an insulator-to-metal transition in the same range of the critical magnetic dilution.

Analytical percolation theory for topological color codes under qubit loss

Quantum information theory has shown strong connections with classical statistical physics. For example, quantum error correcting codes like the surface and the color code present a tolerance to qubit loss that is related to the classical percolation threshold of the lattices where the codes are defined. Here we explore such connection to study analytically the tolerance of the color code when the protocol introduced in [Phys. Rev. Lett. $\textbf{121}$, 060501 (2018)] to correct qubit losses is applied. This protocol is based on the removal of the lost qubit from the code, a neighboring qubit, and the lattice edges where these two qubits reside. We first obtain analytically the average fraction of edges $ r(p) $ that the protocol erases from the lattice to correct a fraction $ p $ of qubit losses. Then, the threshold $ p_c $ below which the logical information is protected corresponds to the value of $ p $ at which $ r(p) $ equals the bond-percolation threshold of the lattice. Moreover, we prove that the logical information is protected if and only if the set of lost qubits does not include the entire support of any logical operator. The results presented here open a route to an analytical understanding of the effects of qubit losses in topological quantum error codes.

Topological phase transitions in glassy quantum matter

Amorphous systems have rapidly gained promise as novel platforms for topological matter. In this work we establish a scaling theory of amorphous topological phase transitions driven by the density of lattice points in two dimensions. By carrying out a finite-size scaling analysis of topological invariants averaged over discrete and continuum random geometries, we discover unique critical properties of Chern and $\mathbb{Z}_2$ glass transitions. Even for short-range hopping models the Chern glass phase may persist down to the fundamental lower bound given by the classical percolation threshold. While the topological indices accurately satisfy the postulated one-parameter scaling, they do not generally flow to the closest integer value in the thermodynamic limit. Furthermore, the value of the critical exponent describing the diverging localization length varies continuously along the phase boundary and is not fixed by the symmetry class of the Hamiltonian. We conclude that the critical behaviour of amorphous topological systems exhibit characteristic features not observed in disordered systems, motivating a wealth of new research directions.

Elastic Backbone Defines a New Transition in the Percolation Model.

The elastic backbone is the set of all shortest paths. We found a new phase transition at p_{eb} above the classical percolation threshold at which the elastic backbone becomes dense. At this transition in 2D, its fractal dimension is 1.750±0.003, and one obtains a novel set of critical exponents β_{eb}=0.50±0.02, γ_{eb}=1.97±0.05, and ν_{eb}=2.00±0.02, fulfilling consistent critical scaling laws. Interestingly, however, the hyperscaling relation is violated. Using Binder's cumulant, we determine, with high precision, the critical probabilities p_{eb} for the triangular and tilted square lattice for site and bond percolation. This transition describes a sudden rigidification as a function of density when stretching a damaged tissue.

Killing the Hofstadter butterfly, one bond at a time

Electronic bands in a square lattice when subjected to a perpendicular magnetic field form the Hofstadter butterfly pattern. We study the evolution of this pattern as a function of bond percolation disorder (removal or dilution of lattice bonds). With increasing concentration of the bonds removed, the butterfly pattern gets smoothly decimated. However, in this process of decimation, bands develop interesting characteristics and features. For example, in the high disorder limit, some butterfly-like pattern still persists even as most of the states are localized. We also analyze, in the low disorder limit, the effect of percolation on wavefunctions (using inverse participation ratios) and on band gaps in the spectrum. We explain and provide the reasons behind many of the key features in our results by analyzing small clusters and finite size rings. Furthermore, we study the effect of bond dilution on transverse conductivity($\sigma_{xy}$). We show that starting from the clean limit, increasing disorder reduces $\sigma_{xy}$ to zero, even though the strength of percolation is smaller than the classical percolation threshold. This shows that the system undergoes a direct transition from a integer quantum Hall state to a localized Anderson insulator beyond a critical value of bond dilution. We further find that the energy bands close to the band edge are more stable to disorder than at the band center. To arrive at these results we use the coupling matrix approach to calculate Chern numbers for disordered systems. We point out the relevance of these results to signatures in magneto-oscillations.

Mott glass phase in a diluted bilayer Heisenberg quantum antiferromagnet

We use quantum Monte Carlo simulations to study a dimer-diluted S = 1/2 Heisenberg model on a bilayer square lattice with intralayer interaction J1 and interlayer interaction J2. Below the classical percolation threshold pc, the system has three phases reachable by tuning the interaction ratio g = J2/J1: a Néel ordered phase, a gapless quantum glass phase, and a gapped quantum paramagnetic phase. We present the ground-state phase diagram in the plane of dilution p and interaction ratio g. The quantum glass phase is certified to be of the gapless Mott glass type, having a uniform susceptibility vanishing at zero temperature T and following a stretched exponential form at T > 0; χu exp(-b/Tα) with α < 1. At the phase transition point from Neel ordered to Mott glass, we find that the critical exponents are different from those of the clean system described by the standard O(3) universality class in 2+1 dimensions.

Role of local and global geometry in entanglement percolation

We prove that enhanced entanglement percolation via lattice transformation is possible even if the new lattice is more poorly connected in that (i) the coordination number (a local property) decreases, or (ii) the classical percolation threshold (a global property) increases. In searching for protocols to transport entanglement across a network, it seems reasonable to try transformations that increase connectivity. In fact, all examples that we are aware of violate both conditions (i) and (ii). One might therefore conjecture that all good transformations must violate them. Here we introduce a method, partial entanglement swapping, and use it to construct a counterexample that satisfies conditions (i) and (ii). The example lowers the threshold, relative to all known protocols, of the amount of initial entanglement required for deterministic long-range entanglement. This result shows that a transformation may not be rejected on the basis of satisfying conditions (i) or (ii). Both the result and the method constitute steps toward answering basic questions, such as whether there is a minimum amount of local entanglement required to achieve long-range entanglement.

Three-dimensional localization of ultracold atoms in an optical disordered potential

We report a study of three-dimensional (3D) localization of ultracold atoms suspended against gravity, and released in a 3D optical disordered potential with short correlation lengths in all directions. We observe density profiles composed of a steady localized part and a diffusive part. Our observations are compatible with the self-consistent theory of Anderson localization, taking into account the specific features of the experiment, and in particular the broad energy distribution of the atoms placed in the disordered potential. The localization we observe cannot be interpreted as trapping of particles with energy below the classical percolation threshold.

Giant Hall effect in nonmagneticMo/SnO2granular films

We have measured the resistivities and Hall coefficients, ${R}_{H}$, of a series of nonmagnetic ${\text{Mo}}_{x}{({\text{SnO}}_{2})}_{1\ensuremath{-}x}$ nanogranular films with the Mo volume fraction $x$ ranging from $\ensuremath{\sim}0.29$ to 1. We found that the magnitude of ${R}_{H}(2\text{ }\text{K})$ largely increased by a factor of $\ensuremath{\sim}800$ as $x$ was reduced from $\ensuremath{\sim}0.8$ to $\ensuremath{\simeq}0.36$. Then, it slightly decreased with a further decrease in $x$ down to $\ensuremath{\simeq}0.32$, which was determined to be the classical percolation threshold, ${x}_{c}$, from the resistivity dependence on $x$. This nearly 3 orders of magnitude enhancement in ${R}_{H}$ at a metal volume fraction $x&gt;{x}_{c}$ is explained in terms of the recent theoretical concept of the local quantum-interference effect induced giant Hall effect in granular systems.

Dilute Bose gas with correlated disorder: a path integral Monte Carlo study

We investigate the thermodynamic properties of a dilute Bose gas in a correlated random potential using exact path integral Monte Carlo methods. The study is carried out in continuous space and disorder is produced in the simulations by a three-dimensional (3D) speckle pattern with tunable intensity and correlation length. We calculate the shift of the superfluid transition temperature due to disorder and we highlight the role of quantum localization by comparing the critical chemical potential with the classical percolation threshold. The equation of state of the gas is determined in the regime of strong disorder, where superfluidity is suppressed and the normal phase exists down to very low temperatures. We find a T2 dependence of the energy in agreement with the expected behavior in the Bose glass phase. We also discuss the major role played by the disorder correlation length and we make contact with a Hartree–Fock mean-field approach that holds if the correlation length is very large. The density profiles are analyzed as a function of temperature and interaction strength. Effects of localization and the depletion of the order parameter are emphasized in the comparison between local condensate and total density. At very low temperature, we find that the energy and the particle distribution of the gas are very well described by the T=0 Gross–Pitaevskii theory, even in the regime of very strong disorder.

Proton Conduction Through Interface Phase of CPAA/KDP Composites

AbstractIn this paper we present the results of the studies on proton conduction in composites polyantimonic acid (PAA)-potassium dihydrophosphate (KDP) obtained by precipitation technique as well as mechanical mixing. The highest conductivity (~10-2 Sm/sm at 300 K) is achieved in the precipitated samples with volume fraction of KDP close to f = 13 %. The results of X-ray diffraction showed the absence of reflections from KDP for the samples with f &lt; 20% and slight change in the intensities of odd and even reflections from PAA, which is explained by the ion exchange between the phases through the interface. These indicate that conduction is occurred through the interface regions of the composite. The value of interface conductivity is estimated from the dependence of the composite conductivity on the volume fraction of KDP based according to the model by Uvarov et al. This gives σs ≈ 5 Sm/sm, however the equation derived in this model doesn't represent the experimental data properly. Probably the maximum of the dependence simply corresponds to the percolation through the interface's continuous cluster – it is close to the classical percolation threshold.

Tunnel percolation and current path switching in a granular metal

By means of electron beam induced deposition we have prepared a series of granular metals consisting of tungsten nanocrystallites embedded in an insulating matrix. The metal content varied between 19 at% and 34 at%. The samples have been analyzed by voltage-current and temperature-dependent conductivity measurements. Within the insulating matrix the nm-sized metallic nanocrystallites are irregularly distributed so that the classical percolation threshold can be assumed to be at approximately 50 at% metal volume fraction. Consequently, within the classical limit no current transport at low temperatures would be expected. Nevertheless, for samples with metal volume fraction above 24 at% the temperature-dependent conductivity extrapolates to a finite value for T = 0. We argue that clustering of nanocrystallites to larger aggregates of enhanced tunneling propability between the cluster members may be the reason for the observed behaviour. The transport can then be considered as a tunnel percolation process between clusters. As a consequence the current is inhomogeneously distributed in the samples. From analyzing different voltage regions of the voltage-current characteristics, separated by a sample-dependent threshold voltage VC, showing either reversible (at lower voltages) or irreversible (at higher voltages) behaviour, we conclude that only one current path is active at any given moment and that this current path need not be that of the highest possible conductance. In particular, in some instances it proved possible to increase the conductance by a factor of 106 after having passed VC.

METAL–INSULATOR TRANSITION IN 3D QUANTUM PERCOLATION

We present the metal–insulator transition study of a quantum site percolation model on simple cubic lattice. Transfer matrix method is used to calculate transport properties — Landauer conductance — for the binary distribution of energies. We calculate the mobility edge in disorder (ratio of insulating sites) — energy plane in detail and we find the extremal critical disorder somewhat closer to the classical percolation threshold than formerly reported. We calculate the critical exponent ν along the mobility edge and find it constant and equal to the one of 3D Anderson model, confirming common universality class. Possible exception is the center of the conduction band, where either the single-parameter scaling is not valid anymore, or finite size effects are immense. One of the reasons for such statement is the difference between results from arithmetic and geometric averaging of conductance at special energies. Only the geometric mean yields zero critical disorder in band center, which was theoretically predicted.

Site dilution of quantum spins in the honeycomb lattice

We consider the effect of random site dilution on a honeycomb lattice of quantum spins described by the antiferromagnetic Heisenberg spin-$S$ model. Using linear spin-wave theory, we compute the zero-temperature magnetization and density of states as a function of dilution up to the classical percolation threshold. Several subtle issues regarding the treatment of quasidivergent zero-energy modes, which appear in the real-space formulation of the spin-wave problem, are clarified. For $S&gt;1∕2$, the spin-wave theory is well defined in the sense that results at all dilution concentrations are consistent with the underlying assumptions of the theory. For $S=1∕2$, however, the approximation breaks down. In this case, we have studied the effect of dilution on the staggered magnetization using the stochastic series expansion Monte Carlo method. Two main results are to be stressed from the Monte Carlo calculation: (i) an improved estimate for the staggered magnetization of the undiluted system ${m}_{\mathrm{av}}(L\ensuremath{\rightarrow}\ensuremath{\infty})=0.2677(6)$ and (ii) a finite value of the staggered magnetization of the percolating cluster at the classical percolation threshold, showing that there is no quantum critical transition driven by dilution in the Heisenberg model. We have used the computed staggered magnetization and density of states to analyze neutron scattering experiments and N\'eel temperature measurements of two quasi-two-dimensional diluted honeycomb systems: (i) ${\mathrm{Mn}}_{p}{\mathrm{Zn}}_{1\ensuremath{-}p}\mathrm{P}{\mathrm{S}}_{3}$ (a diluted $S=5∕2$ system) and (ii) $\mathrm{Ba}{({\mathrm{Ni}}_{p}{\mathrm{Mg}}_{1\ensuremath{-}p})}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$ (a diluted $S=1$ system). We have found that our calculations are in good agreement with the experimental data.

Quantum Localization in Bilayer Heisenberg Antiferromagnets with Site Dilution

The field-induced antiferromagnetic ordering in systems of weakly coupled S = 1/2 dimers at zero temperature can be described as a Bose-Einstein condensation of triplet quasiparticles (singlet quasiholes) in the ground state. For the case of a Heisenberg bilayer, it is here shown how the above picture is altered in the presence of site dilution of the magnetic lattice. Geometric randomness leads to quantum localization of the quasiparticles or quasiholes and to an extended Bose-glass phase in a realistic disordered model. This localization phenomenon drives the system towards a quantum-disordered phase well before the classical geometric percolation threshold is reached.

Excitations and quantum fluctuations in site-diluted two-dimensional antiferromagnets

We study the effect of site dilution and quantum fluctuations in an antiferromagnetic spin system on a square lattice within the linear spin-wave approximation. By performing numerical diagonalization in real space and finite-size scaling, we characterize the nature of the low-energy spin excitations for different dilution fractions up to the classical percolation threshold. We find nontrivial signatures of fractonlike excitations at high frequencies. Our simulations also confirm the existence of an upper bound for the amount of quantum fluctuations in the ground state of the system, leading to the persistence of long-range order up to the percolation threshold. This result is in agreement with recent neutron-scattering experimental data and quantum Monte Carlo numerical calculations. We also show that the absence of a quantum critical point below the classical percolation threshold holds for a large class of systems whose Hamiltonians can be mapped onto a system of coupled noninteracting massless bosons.