Finite-size Scaling Research Articles

Charged critical behavior and nonperturbative continuum limit of three-dimensional lattice SU(Nc) gauge Higgs models

We consider three-dimensional (3D) lattice SU(Nc) gauge Higgs theories with multicomponent (Nf>1) degenerate scalar fields and U(Nf) global symmetry, focusing on systems with Nc=2, to identify critical behaviors that can be effectively described by the corresponding 3D SU(Nc) gauge Higgs field theory. The field-theoretical analysis of the RG flow allows one to identify a stable charged fixed point for large values of Nf, that would control transitions characterized by the global symmetry-breaking pattern U(Nf)→SU(2)⊗U(Nf−2). Continuous transitions with the same symmetry-breaking pattern are observed in the SU(2) lattice gauge model for Nf≥30. Here we present a detailed finite-size scaling analysis of the Monte Carlo data for several large values of Nf. The results are in substantial agreement with the field-theoretical predictions obtained in the large-Nf limit. This provides evidence that the SU(Nc) gauge Higgs field theories provide the correct effective description of the 3D large-Nf continuous transitions between the disordered and the Higgs phase, where the flavor symmetry breaks to SU(2)⊗U(Nf−2). Therefore, at least for large enough Nf, the 3D SU(Nc) gauge Higgs field theories with multicomponent scalar fields can be nonperturbatively defined by the continuum limit of lattice discretized models with the same local and global symmetries. Published by the American Physical Society 2024

Exact results on the dynamics of the stochastic Floquet-East model

Abstract We introduce a stochastic generalisation of the classical deterministic Floquet-East model, a discrete circuit with the same kinetic constraint as the East model of glasses. We prove exactly that, in the limit of long time and large size, this model has a large deviation phase transition between active and inactive dynamical phases. We also compute the finite time and size scaling of general space-time fluctuations, which for the case of inactive regions gives rise to dynamical hydrophobicity. We also discuss how, through the Trotter limit, these exact results also hold for the continuous-time East model, thus proving long-standing observations in kinetically constrained models. Our results here illustrate the applicability of exact tensor network methods for solving problems in many-body stochastic systems.

Tunable disorder on the S-state majority-voter model.

We investigate the nonequilibrium phase transition in the S-state majority-vote model for S=2,3, and 4. Each site, k, is characterized by a distinct noise threshold, qk, which indicates its resistance to adopting the majority state of its Nv nearest neighbors. Precisely, this noise threshold is governed by a hyperbolic distribution, P(k)∼1/k, bounded within the limits e-α/2<qk<1/2. Here, the parameter α plays a pivotal role as it determines the extent of disorder in the system through the spread of the threshold distribution. Through Monte Carlo simulations and finite-size scaling analyses on regular square lattices, we deduced that the model undergoes a continuous order-disorder phase transition at a specific α=αc. Interestingly, the critical threshold exhibits a power-law decay, αc∼Nv-δ, as the number Nv of neighboring sites increases. From the least square fits to the data sets results in δ=0.65±0.01 for S=2, δ=0.92±0.01 for S=3, and δ=0.93±0.01 for S=4. Furthermore, the critical exponents β/ν and γ/ν are consistent with those found in the S-state Potts model.

First Principles Numerical Demonstration of Emergent Decoherent Histories

Within the histories formalism the decoherence functional is a formal tool to investigate the emergence of classicality in isolated quantum systems, yet an explicit evaluation of it from first principles has not been reported. We provide such an evaluation for up to five-time histories based on exact numerical diagonalization of the Schrödinger equation. We find a robust emergence of decoherence for slow and coarse observables of a generic random matrix model and extract a finite-size scaling law by varying the Hilbert space dimension over 4 orders of magnitude. Specifically, we conjecture and observe an exponential suppression of coherent effects as a function of the particle number of the system. This suggests a solution to the preferred basis problem of the many-worlds interpretation (or the set selection problem of the histories formalism) within a minimal theoretical framework without relying on environmentally induced decoherence, quantum Darwinism, Markov approximations, low-entropy initial states, or ensemble averages. Published by the American Physical Society 2024

Partition Function Zeros of the Frustrated J1–J2 Ising Model on the Honeycomb Lattice

We study the zeros of the partition function in the complex temperature plane (Fisher zeros) and in the complex external field plane (Lee–Yang zeros) of a frustrated Ising model with competing nearest-neighbor (J1&gt;0) and next-nearest-neighbor (J2&lt;0) interactions on the honeycomb lattice. We consider the finite-size scaling (FSS) of the leading Fisher and Lee–Yang zeros as determined from a cumulant method and compare it to a traditional scaling analysis based on the logarithmic derivative of the magnetization ∂ln⟨|M|⟩/∂β and the magnetic susceptibility χ. While for this model both FSS approaches are subject to strong corrections to scaling induced by the frustration, their behavior is rather different, in particular as the ratio R=J2/J1 is varied. As a consequence, an analysis of the scaling of partition function zeros turns out to be a useful complement to a more traditional FSS analysis. For the cumulant method, we also study the convergence as a function of cumulant order, providing suggestions for practical implementations. The scaling of the zeros convincingly shows that the system remains in the Ising universality class for R as low as −0.22, where results from traditional FSS using the same simulation data are less conclusive. Hence, the approach provides a valuable additional tool for mapping out the phase diagram of models afflicted by strong corrections to scaling.

Interplay of the complete-graph and Gaussian fixed-point asymptotics in finite-size scaling of percolation above the upper critical dimension

Interplay of the complete-graph and Gaussian fixed-point asymptotics in finite-size scaling of percolation above the upper critical dimension

Phase behavior of metastable water from large-scale simulations of a quantitatively accurate model near ambient conditions: The liquid-liquid critical point.

The molecular mechanisms of water's unique anomalies are still debated upon. Experimental challenges have led to simulations suggesting a liquid-liquid (LL) phase transition, culminating in the supercooled region's LL critical point (LLCP). Computational expense, small system sizes, and the reliability of water models often limit these simulations. We adopt the CVF model, which is reliable, transferable, scalable, and efficient across a wide range of temperatures and pressures around ambient conditions. By leveraging the timescale separation between fast hydrogen bonds and slow molecular coordinates, the model allows a thorough exploration of the metastable phase diagram of liquid water. Using advanced numerical techniques to bypass dynamical slowing down, we perform finite-size scaling on larger systems than those used in previous analyses. Our study extrapolates thermodynamic behavior in the infinite-system limit, demonstrating the existence of the LLCP in the 3D Ising universality class in the low-temperature, low-pressure side of the line of temperatures of maximum density, specifically at TC = 186 ± 4 K and PC = 174 ± 14 MPa, at the end of a liquid-liquid phase separation stretching up to ∼200 MPa. These predictions align with recent experimental data and sophisticated models, highlighting that hydrogen bond cooperativity governs the LLCP and the origin of water anomalies. We also observe substantial cooperative fluctuations in the hydrogen bond network at scales larger than 10nm, even at temperatures relevant to biopreservation. These findings have significant implications for nanotechnology and biophysics, providing new insights into water's behavior under varied conditions.

High-precision analysis of the critical point in heavy-quark QCD at Nt=6

Binder-cumulant analysis of the critical point in the heavy-quark region of QCD is performed by Monte-Carlo simulations with the hopping-parameter expansion at Nt=6. We extend our previous analysis at Nt=4 to finer lattices and perform high-precision analyses on large spatial volumes up to the aspect ratio LT=Ns/Nt=18. Higher order terms in the hopping-parameter expansion are incorporated effectively up to 14th order. The numerical results show that the violation of the finite-size scaling becomes more prominent on the finer lattice at a given aspect ratio. Published by the American Physical Society 2024

Superconductor to metal quantum phase transition with magnetic field in Josephson coupled lead islands on Graphene

Abstract Superconductor-to-metal transition with magnetic field and gate-voltage is studied in a Josephson junction array comprising of randomly distributed lead islands on exfoliated single-layer graphene. The superconductivity onset temperature at low magnetic-fields is fitted to the Werthamer-Helfand-Hohenberg theory to model the temperature dependence of the upper critical field. The magnetoresistance in the intermediate temperature and field regime is described using thermally activated flux flow dictated by field dependent activation barrier. The barrier also depends on the gate voltage which dictates the inter-island Josephson coupling and disorder. The magnetoresistance near the upper critical field at low temperatures shows signatures of a gate dependent continuous quantum phase transition between superconductor and metal. The finite size scaling analysis shows that this transition belongs to the $(2+1)$D-XY universality class without disorder.&amp;#xD;

Pseudogap Effects in the Strongly Correlated Regime of the Two-Dimensional Fermi Gas.

The two-species Fermi gas with attractive short-range interactions in two spatial dimensions provides a paradigmatic system for the understanding of strongly correlated Fermi superfluids in two dimensions. It is known to exhibit a BEC to BCS crossover as a function of ln(k_{F}a), where a is the scattering length, and to undergo a Berezinskii-Kosterlitz-Thouless superfluid transition below a critical temperature T_{c}. However, the extent of a pseudogap regime in the strongly correlated regime of ln(k_{F}a)∼1, in which pairing correlations persist above T_{c}, remains largely unexplored with controlled theoretical methods. Here, we use finite-temperature auxiliary-field quantum MonteCarlo methods on discrete lattices in the canonical ensemble formalism to calculate thermodynamical observables in the strongly correlated regime. We extrapolate to continuous time and the continuum limit to eliminate systematic errors and present results for particle numbers ranging from N=42 to N=162. We estimate T_{c} by a finite-size scaling analysis, and observe clear pseudogap signatures above T_{c} and below a temperature T^{*} in both the spin susceptibility and free-energy gap. We also present results for the contact, a fundamental thermodynamic property of quantum many-body systems with short-range interactions.

Topological data analysis of monopole current networks in $U(1)$ lattice gauge theory

In 44-dimensional pure compact U(1)U(1) lattice gauge theory, we analyse topological aspects of the dynamics of monopoles across the deconfinement phase transition. We do this using tools from Topological Data Analysis (TDA). We demonstrate that observables constructed from the zeroth and first homology groups of monopole current networks may be used to quantitatively and robustly locate the critical inverse coupling \beta_{c}βc through finite-size scaling. Our method provides a mathematically robust framework for the characterisation of topological invariants related to monopole currents, putting on firmer ground earlier investigations. Moreover, our approach can be generalised to the study of Abelian monopoles in non-Abelian gauge theories.

Test of universality at first order phase transitions: The Lebwohl-Lasher model.

Finite size scaling for a first order phase transition, where a continuous symmetry is broken, is tested using an approximation of Gaussian probability distributions with a phenomenological "degeneracy" factor. Predictions are compared to the data from Monte Carlo simulations of the Lebwohl-Lasher model on L × L × L simple cubic lattices. The data show that the intersection of the fourth-order cumulant of the order parameter for different lattice sizes can be expressed in terms of the relative degeneracy q = 4π of the ordered and disordered phases. This result further supports the concept of universality at first order transitions developed recently.

Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.

Phase transitions in the node, edge, bootstrap, and diffusion percolation models on the Sierpiński carpet

We investigate four types of percolation models—node, edge, bootstrap, and diffusion percolation—in three fractal graphs constructed on the Sierpiński carpet, employing the Monte Carlo method based on the Newman–Ziff algorithm. For each case, we calculate the percolation threshold and critical exponents (ν, γ, and β) through the crossing of percolation probabilities and the finite-size scaling analysis, incorporating correction-to-scaling effects. Our results reveal that critical exponents of the percolation phase transition in the three fractal graphs exhibit universality across all four percolation models. Furthermore, we demonstrate that the hyperscaling relation dν=γ+2β is also valid in the percolation phase transition on the Sierpiński carpet if the spatial dimension d is replaced by the Hausdorff dimension.

Collective oscillations in a three-dimensional spin model with non-reciprocal interactions

Abstract We study the onset of collective oscillations at low temperature in a three-dimensional spin model with non-reciprocal short-range interactions. Performing numerical simulations of the model, the presence of a continuous phase transition to global oscillations is confirmed by a finite-size scaling analysis, yielding values of the exponents β and ν compatible with both the three-dimensional XY and Ising equilibrium universality classes. By systematically varying the interaction range, we show that collective oscillations in this spin model actually result from two successive phase transitions: a mean-field phase transition over finite-size neighborhoods, which leads to the emergence of local noisy oscillators, and a synchronization transition of local noisy oscillators, which generates coherent macroscopic oscillations. Using a Fokker–Planck equation under a local mean-field approximation, we derive from the spin dynamics coupled Langevin equations for the complex amplitudes describing noisy oscillations on a mesoscopic scale. The phase diagram of these coupled equations is qualitatively obtained from a fully-connected (mean-field) approximation. This analytical approach allows us to clearly disentangle the onset of local and global oscillations, and to identify the two main control parameters, expressed as combinations of the microscopic parameters of the spin dynamics, that control the phase diagram of the model.

Explosive percolation in finite dimensions

Explosive percolation (EP) has received significant research attention due to its rich and anomalous phenomena near criticality. In our recent study [], we demonstrated that the correct critical behaviors of EP in infinite dimensions (complete graph) can be accurately extracted using the event-based method, with finite-size scaling behaviors still described by the standard finite-size scaling theory. We perform an extensive simulation of EPs on hypercubic lattices ranging from dimensions d=2 to 6, and find that the critical behaviors consistently obey the standard finite-size scaling theory. Consequently, we obtain a high-precision determination of the percolation thresholds and critical exponents, revealing that EPs governed by the product and sum rules belong to different universality classes. Remarkably, despite the mean of the dynamic pseudocritical point TL deviating from the infinite-lattice criticality by a distance determined by the d-dependent correlation-length exponent, TL follows a normal (Gaussian) distribution across all dimensions, with a standard deviation proportional to 1/V, where V denotes the system volume. A theoretical argument associated with the central-limit theorem is further proposed to understand the probability distribution of TL. These findings offer a comprehensive understanding of critical behaviors in EPs across various dimensions, revealing a different dimension-dependence compared to standard bond percolation. Published by the American Physical Society 2024

1/fα noise in the Robin Hood model

Abstract We consider the Robin Hood dynamics, a one-dimensional extremal self-organized critical model that describes the low-temperature creep phenomenon. One of the key quantities is the time evolution of the state variable (force noise). To understand the temporal correlations, we compute the power spectra of the local force fluctuations and apply finite-size scaling to get scaling functions and critical exponents. We find a signature of the 1 / f α noise for the local force with a nontrivial value of the spectral exponent 0 &lt; α &lt; 2 . We also examine temporal fluctuations in the position of the extremal site and a local activity signal. We present results for different local interaction rules of the model.

Quasiperiodic disorder induced critical phases in a periodically driven dimerized p-wave Kitaev chain

The intricate relationship between topology and disorder in non-equilibrium quantum systems presents a captivating avenue for exploring localization phenomenon. Here, we look for a suitable platform that enables an in-depth investigation of the topic. To this end, we delve into the nuanced analysis of the topological and localization characteristics exhibited by a one-dimensional dimerized Kitaev chain under periodic driving and perform detailed analyses of the Floquet Majorana modes. Such a non-equilibrium scenario is made further interesting by including a spatially varying quasiperiodic potential with a temporally modulated amplitude. Apriori, the motivation is to explore an interplay between dimerization and a quasiperiodic disorder in a topological setting which is also known to demonstrate unique (re-entrant) localization properties. While the topological properties of the driven system confirm the presence of zero and π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} Majorana modes, the phase diagram obtained by constructing a pair of topological invariants (Z×Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z} \ imes \\mathbb {Z} $$\\end{document}), also referred to as the real space winding numbers, at different driving frequencies reveal intriguing features that are distinct from the static scenario. In particular, at either low or intermediate frequency regimes, the phase diagram concerning the zero mode involves two distinct phase transitions, one from a topologically trivial to a non-trivial phase, and another from a topological phase to an Anderson localized phase. On the other hand, the study of the Majorana π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} mode unveils the emergence of a unique topological phase, characterized by complete localization of both the bulk and the edge modes, which may be called as the Floquet topological Anderson phase. Moreover, different frequency regimes showcase distinct localization features which can be examined via the localization toolbox, namely, the inverse and the normalized participation ratios. Specifically, the low and high-frequency regimes demonstrate the existence of completely extended and localized phases, respectively. While at intermediate frequencies, we observe the critical (multifractal) phase of the model which is further investigated via a finite-size scaling analysis of the fractal dimension. Finally, to add depth into our study, we have performed a mean level spacing analyses and computed the Hausdorff dimension which yields specific characteristics inherent to the critical phase, offering profound insights into its underlying properties.

Finite-size spectrum of the staggered six-vertex model with antidiagonal boundary conditions

The finite-size spectrum of the critical staggered six-vertex model with antidiagonal boundary conditions is studied. Similar to the case of periodic boundary conditions, we identify three different phases. In two of those, the underlying conformal field theory can be identified to be related to the twisted U(1) Kac-Moody algebra. In contrast, the finite size scaling in the third regime, whose critical behaviour with the (quasi-)periodic BCs is related to the 2d black hole CFTs possessing a non-compact degree of freedom, is more subtle. Here with antidiagonal BCs imposed, the corrections to the scaling of the ground state grow logarithmically with the system size, while the energy gaps appear to close logarithmically. Moreover, we obtain an explicit formula for the Q-operator which is useful for numerical implementation.

Finite-size scaling of the Kuramoto model at criticality.

The asymptotic scaling behavior of the Kuramoto model with finite populations has been notably elusive, despite comprehensive investigations employing both analytical and numerical methods. In this paper, we explore the Kuramoto model with "deterministic" sampling of natural frequencies, employing extensive numerical simulationsand reporting the asymptotic values of the finite-size scaling exponents, which deviate significantly from the previously reported values in the literature. Additionally, we observe that these exponents are sensitive to the specifics of the sampling method. We discuss the origins of this variability through the self-consistent theory of entrained oscillators.