Finite-size scaling (FSS) for a critical phase transition ( $t=0$ ) states that within a window of size $|t|\sim L^{-1/ν}$ , the scaling behavior of any observable $Q$ in a system of linear size $L$ asymptotically follows a scaling form as $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/ν})$ , where $ν$ is the correlation-length exponent, $Y_Q$ is an FSS exponent and ${\tilde Q}(x)$ is a function of the scaled distance-to-criticality $x \equiv tL^{1/ν}$ . We systematically study the asymptotic scaling behavior of ${\tilde Q}(|x|\to\infty)$ 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 $t \to 0$ and $|x|\to\infty$ . This crossover FSS theory predicts that when the criticality is approached at a slower speed as $|t|\sim L^{-λ}$ 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.Finite-size scaling (FSS) for a critical phase transition ($t=0$) states that within a window of size $|t|\sim L^{-1/ν}$, the scaling behavior of any observable $Q$ in a system of linear size $L$ asymptotically follows a scaling form as $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/ν})$, where $ν$ is the correlation-length exponent, $Y_Q$ is an FSS exponent and ${\tilde Q}(x)$ is a function of the scaled distance-to-criticality $x \equiv tL^{1/ν}$. We systematically study the asymptotic scaling behavior of ${\tilde Q}(|x|\to\infty)$ 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 $t \to 0$ and $|x|\to\infty$. This crossover FSS theory predicts that when the criticality is approached at a slower speed as $|t|\sim L^{-λ}$ 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.

arXiv:2412.06228Dec 9, 20241 citation

Dimensional crossover in the large-N limit

Denjoe O’Connor, Christopher Rhodes Stephens, Alan J. Bray

arXiv:cond-mat/9601146Journal of Statistical PhysicsApr 1, 19975 citations

Finite-Size Scaling Theory

V. Rittenberg

Springer eBooksJan 1, 19831 citation

Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime

C. Daniel Freeman, C. M. Herdman, Dylan J Gorman +1 more

We present an analysis of the relaxation dynamics of finite-size topological qubits in contact with a thermal bath. Using a continuous-time Monte Carlo method, we explicitly compute the low-temperature nonequilibrium dynamics of the toric code on finite lattices. In contrast to the size-independent bound predicted for the toric code in the thermodynamic limit, we identify a low-temperature regime on finite lattices below a size-dependent crossover temperature with nontrivial finite-size and temperature scaling of the relaxation time. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial two-dimensional classical random walks. The transition out of this low-temperature regime defines a dynamical finite-size crossover temperature that scales inversely with the log of the system size, in agreement with a crossover temperature defined from equilibrium properties. We find that both the finite-size and finite-temperature scaling are stronger in the low-temperature regime than above the crossover temperature. Since this finite-temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of memory lifetimes of possible physical realizations of topological qubits.We present an analysis of the relaxation dynamics of finite-size topological qubits in contact with a thermal bath. Using a continuous-time Monte Carlo method, we explicitly compute the low-temperature nonequilibrium dynamics of the toric code on finite lattices. In contrast to the size-independent bound predicted for the toric code in the thermodynamic limit, we identify a low-temperature regime on finite lattices below a size-dependent crossover temperature with nontrivial finite-size and temperature scaling of the relaxation time. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial two-dimensional classical random walks. The transition out of this low-temperature regime defines a dynamical finite-size crossover temperature that scales inversely with the log of the system size, in agreement with a crossover temperature defined from equilibrium properties. We find that both the finite-size and finite-temperature scaling are stronger in the low-temperature regime than above the crossover temperature. Since this finite-temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of memory lifetimes of possible physical realizations of topological qubits.

arXiv:1405.2315Physical Review BOct 7, 201417 citations

Finite-size scaling (FSS) for a critical phase transition (

t=0

) states that within a window of size

|t|\sim 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)=L^{Y_Q}\tilde{Q}(tL^{1/ν})

, where

ν

is the correlation-length exponent,

Y_Q

is an FSS exponent and

{\tilde Q}(x)

is a function of the scaled distance-to-criticality

x \equiv tL^{1/ν}

. We systematically study the asymptotic scaling behavior of

{\tilde Q}(|x|\to\infty)

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 \to 0

and

|x|\to\infty

. This crossover FSS theory predicts that when the criticality is approached at a slower speed as

|t|\sim 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.