We present a new family of percolation models. We show, using theory and computer simulations, that these represent a new universality class. Interestingly, systems in this class appear to violate the Harris criterion, making model systems within these class ideal systems for studying the influence of disorder on critical behavior. We argue that such percolative systems have already been realized in practice in strongly correlated electron systems that have been driven to the quantum critical point by means of chemical substitution.
Abstract We present computer simulations on a class of percolative systems that forms a new universality class. We determine the universal critical exponents for this new class from simulations on lattices consisting of up to one billion sites. These new percolative systems differ from standard systems in that once a cluster breaks off the lattice spanning cluster, its sites become protected and cannot be removed. We demonstrate that despite this restriction on the evolution of isolated clusters, the scaling relationships between the critical exponents remain valid. Protected percolation closely mimics the situation in heavily-doped quantum critical systems where isolated magnetic clusters are protected from Kondo screening. We show that protected percolation in three dimensions violates the Harris criterion, explaining why universal exponents for quantum phase transitions have been elusive.
We describe a new universality class - dubbed protected percolation - that we show to be relevant to quantum critical systems. Percolation theory describes phase transitions where long-range order is lost when parts of a system become disconnected from other parts; in the vicinity of the transition, critical behavior is observed, captured by universal power laws. Protected percolation has the added restriction that only sites from the system spanning connection can be removed. We developed a new technique to simulate protected percolation, and we used it to determine the critical exponents of this new universality class in 2, 3, and 4 dimensions. We relate the exponents analytically to those of standard percolation. The Harris criterion predicts whether a phase transition is stable against impurities. We prove that protected percolation violates this criterion in 3 dimensions and higher, implying that impurities result in the loss of universal behavior in systems governed by protected percolation. We investigated the change in critical exponents for various types of impurities, focusing on the case for three dimensions where protected percolation models quantum critical systems. We detail how our simulations can be used for direct comparison to experimental results on such quantum critical systems.
Finite-size scaling (FSS) for a critical phase transition (
t=0) states that within a window of size
∣t∣∼L−1/ν, the scaling behavior of any observable
Q in a system of linear size
L asymptotically follows a scaling form as
Q(t,L)=LYQQ~(tL1/ν), where
ν is the correlation-length exponent,
YQ is an FSS exponent and
Q~(x) is a function of the scaled distance-to-criticality
x≡tL1/ν. We systematically study the asymptotic scaling behavior of
Q~(∣x∣→∞) for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with
t→0 and
∣x∣→∞. This crossover FSS theory predicts that when the criticality is approached at a slower speed as
∣t∣∼L−λ with
λ<1/ν, the FSS becomes
λ-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality
t=0 can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.
Complex systems of classical physics in certain situations are characterized by intensive fluctuations of the quantities governing their dynamics. These include important phenomena such as (continuous) second-order phase transitions, fully developed turbulence, magnetic hydrodynamics, advective–diffusive processes, the kinetics of chemical reactions, percolation, and processes in financial markets. The theoretical goal of the study of such systems is to determine and predict the temporal and spatial dependence of statistical correlations of fluctuating variables. Modern methods of quantum field theory, originally developed in high-energy physics to describe the properties of elementary particles, allow for quantitative analysis of such correlations. Peculiarities of quantum field theory in solving two paradigmatic statistical problems related to percolation are reviewed, and new results on calculating representative universal parameters such as critical exponents that describe asymptotic behavior are presented.
A general method is given whereby
m-connectedness correlation functions can be studied in the percolation problem. The method for
m=1 involves calculating the kth power of the correlation function $ 〈\ensuremath{\sigma}(x)\ensuremath{\sigma}({x}^{\ensuremath{'}})〉 $ for a randomly dilute Ising model at nonzero temperature and subsequently averaging over configurations. The final step is to take the limit $k\ensuremath{\rightarrow}0$. This method is tested by reproducing the standard results for the percolation problem from an extension of the calculation of Stephen and Grest. For
m>1 an additional "color" index is introduced and a Hamiltonian is constructed in which different colors repel one another, thereby giving an exact prescription for
m-connectedness. Order parameters for
m-connectedness are identified. The
m=2 order parameter couples through a trilinear term to the
m=1 order parameter. The main result is that ${\ensuremath{\beta}}^{(m)}$ the exponent for
m-connectedness is given by ${\ensuremath{\beta}}^{(m)}=m\ensuremath{\beta}+\ensuremath{\nu}{\ensuremath{\psi}}^{(m)}$, where $\ensuremath{\beta}$ and $\ensuremath{\nu}$ are the usual exponents for percolation and ${\ensuremath{\psi}}^{(m)}$ is a new crossover exponent which, to lowest order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, is given by ${\ensuremath{\psi}}^{(m)}=\frac{m(m\ensuremath{-}1){\ensuremath{\epsilon}}^{2}}{49}$. This result implies that the fractal dimensionality of the biconnected part of the critically percolating cluster is given in terms of the percolation critical exponents as $\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}\ensuremath{-}{\ensuremath{\psi}}^{(2)}$. If "nodes" are defined as triconnected points, then ${\ensuremath{\beta}}^{(3)}$ is the critical exponent associated with their density in the infinite cluster. We also discuss evidence that the "node-link" model of Skal-Shklovskii-de Gennes breaks down for
d less than some critical value $\stackrel{^}{d}$. Numerically we adduce evidence that $\stackrel{^}{d}$ may be larger than 3.