Large-scale Monte Carlo Simulations Research Articles

Anomalous phase separation in a correlated electron system: Machine-learning–enabled large-scale kinetic Monte Carlo simulations

SignificancePhase separation is crucial to the functionalities of many correlated electron materials with notable examples including colossal magnetoresistance in manganites and high-Tc superconductivity in cuprates. However, the nonequilibrium phase-separation dynamics in such systems are poorly understood theoretically, partly because the required multiscale modeling is computationally very demanding. With the aid of machine-learning methods, we have achieved large-scale dynamical simulations in a representative correlated electron system. We observe an unusual relaxation process that is beyond the framework of classical phase-ordering theories. We also uncover a correlation-induced freezing behavior, which could be a generic feature of phase separation in correlated electron systems.

Emergent Potts Order in a Coupled Hexatic-Nematic XY model

Addressing the nature of an unexpected smectic-A' phase in liquid crystal 54COOBC films, we perform large scale Monte Carlo simulations of a coupled hexatic-nematic XY model. The resulting finite-temperature phase diagram reveals a small region with composite Potts $\mathbb{Z}_3$ order above the vortex binding transition; this phase is characterized by relative hexatic-nematic ordering though both variables are disordered. The system develops algebraic hexatic and nematic order only at a lower temperature. This multi-step melting scenario agrees well with the experimental observations of a sharp specific heat anomaly that emerges above the onset of hexatic positional order. We therefore propose that the smectic-A' phase is characterized by composite Potts order and bound-states of fractional vortices.

Monte Carlo studies of the Blume–Capel model on nonregular two- and three-dimensional lattices: phase diagrams, tricriticality, and critical exponents

We perform Monte Carlo simulations, combining both the Wang–Landau and the Metropolis algorithms, to investigate the phase diagrams of the Blume–Capel model on different types of nonregular lattices (Lieb lattice (LL), decorated triangular lattice (DTL), and decorated simple cubic lattice (DSC)). The nonregular character of the lattices induces a double transition (reentrant behavior) in the region of the phase diagram at which the nature of the phase transition changes from first-order to second-order. A physical mechanism underlying this reentrance is proposed. The large-scale Monte Carlo simulations are performed with the finite-size scaling analysis to compute the critical exponents and the critical Binder cumulant for three different values of the anisotropy , showing thus no deviation from the standard Ising universality class in two and three dimensions. We report also the location of the tricritical point to considerable precision: (Δ t /J = 1.3457(1); k B T t /J = 0.309(2)), (Δ t /J = 1.5766(1); k B T t /J = 0.481(2)), and (Δ t /J = 1.5933(1); k B T t /J = 0.569(4)) for LL, DTL, and DSC, respectively.

QMC study of the chiral Heisenberg Gross-Neveu universality class

We investigate a quantum criticality of an antiferromagnetic phase transition in the Hubbard model on a square lattice with a d-wave pairing field by large-scale auxiliary-field quantum Monte Carlo simulations. Since the d-wave pairing filed induces Dirac cones in the non-interacting single-particle spectrum, the quantum criticality should correspond to the chiral Heisenberg universality class in terms of the Gross-Neveu theory, which is the same as those expected in the Hubbard model on the honeycomb lattice, despite the unit cells being different (e.g., they contain one and two sites, respectively). We show that both the two phase transitions, expected to occur on the square and on the honeycomb lattices, indeed have the same quantum criticality. We also argue that details of the models, i.e., the way of counting the total number N of fermion components and the anisotropy of the Dirac cones, do not change the critical exponents.

Equilibrium phases and domain growth kinetics of calamitic liquid crystals.

The anisotropic shape of calamitic liquid crystal (LC) particles results in distinct values of energy when the nematogens are placed side by side or end to end. This anisotropy in energy which is governed by a parameter κ^{'} has deep consequences on equilibrium and nonequilibrium properties. Using the Gay-Berne (GB) model, which exhibits the nematic (Nm) as well as the low-temperature smectic (Sm) order, we undertake large-scale Monte Carlo and molecular dynamics simulations to probe the effect of κ^{'} on the equilibrium phase diagram and the nonequilibrium domain growth following a quench in the temperature T or coarsening. There are two transitions in the GB model: (i) isotropic to Nm at T_{c}^{1} and (ii) Nm to Sm at T_{c}^{2}<T_{c}^{1}. κ^{'} decreases T_{c}^{1} significantly but has relatively little effect on T_{c}^{2}. Domain growth in the Nm phase exhibits the well-known Lifshitz-Allen-Cahn (LAC) law, L(t)∼t^{1/2} and the evolution is via annihilation of string defects. The system exhibits dynamical scaling that is also robust with respect to κ^{'}. We find that the Sm phase at the quench temperatures T (T>T_{c}^{1}→T<T_{c}^{2}) that we consider has SmB order with a hexatic arrangement of the LC molecules in the layers (SmB-H phase). Coarsening in this phase exhibits a striking two-timescale scenario: First, the LC molecules align and develop orientational order (or nematicity), followed by the emergence of the characteristic layering (or smecticity) along with the hexatic bond-orientational-order within the layers. Consequently, the growth follows the LAC law L(t)∼t^{1/2} at early times and then shows a sharp crossover to a slower growth regime at later times. Our observations strongly suggest that L(t)∼t^{1/4} in this regime. Interestingly, the correlation function shows dynamical scaling in both the regimes and the scaling function is universal. The dynamics is also robust with respect to changes in κ^{'}, but the smecticity is more pronounced at larger values. Further, the early-time dynamics is governed by string defects, while the late-time evolution is dictated by interfacial defects. We believe this scenario is generic to the Sm phase even with other kinds of local order within the Sm layers.

Metallic and Deconfined Quantum Criticality in Dirac Systems

Motivated by the physics of spin-orbital liquids, we study a model of interacting Dirac fermions on a bilayer honeycomb lattice at half filling, featuring an explicit global SO(3)×U(1) symmetry. Using large-scale auxiliary-field quantum MonteCarlo (QMC) simulations, we locate two zero-temperature phase transitions as function of increasing interaction strength. First, we observe a continuous transition from the weakly interacting semimetal to a different semimetallic phase in which the SO(3) symmetry is spontaneously broken and where two out of three Dirac cones acquire a mass gap. The associated quantum critical point can be understood in terms of a Gross-Neveu-SO(3) theory. Second, we subsequently observe a transition toward an insulating phase in which the SO(3) symmetry is restored and the U(1) symmetry is spontaneously broken. While strongly first order at the mean-field level, the QMC data are consistent with a direct and continuous transition. It is thus a candidate for a new type of deconfined quantum critical point that features gapless fermionic degrees of freedom.

An investigation of hypothesis testing procedures for circular and spherical mean vectors

Numerous testing procedures for directional data have been proposed over the years and a natural question that springs is which to use and when. The aim of this paper is to answer this question, via a large scale Monte Carlo simulation study that covers circular and spherical data for the two mean directions problem. The results evidently signify that tests assuming equal concentration parameters should be avoided as they tend to inflate the test size, while the heterogeneous test that does not make this assumption is to be preferred, but unfortunately only with large sample sizes. Permutation calibration does not improve the performance of any testing procedure, whereas bootstrap does. Specifically, bootstrap calibrated tests exhibited superior performance; they attain the type I error in the vast majority of the case scenarios examined and possess nearly indistinguishable empirical power levels. Finally, examples with real data illustrate the performance of the bootstrap calibrated tests.

On unified framework for nonlinear grey system models: An integro-differential equation perspective

Nonlinear grey system models, serving to time series forecasting, are extensively used in diverse areas of science and engineering. However, most research concerns improving classical models and developing novel models, relatively limited attention has been paid to the relationship among diverse models and the modelling mechanism. The current paper proposes a unified framework and reconstructs the unified model from an integro-differential equation perspective. First, we propose a methodological framework that subsumes various nonlinear grey system models as special cases, providing a cumulative sum series-orientated modelling paradigm. Then, by introducing an integral operator, the unified model is reduced to an equivalent integro-differential equation; on this basis, the structural parameters and initial value are estimated simultaneously via the integral matching approach. The modelling procedure comparison further indicates that the integral matching-based integro-differential equation provides a direct modelling paradigm. Next, large-scale Monte Carlo simulations are conducted to compare the finite sample performance, and the results show that the reduced model has higher accuracy and robustness to noise. Applications of forecasting the municipal sewage discharge and water consumption in the Yangtze River Delta of China further illustrate the effectiveness of the reconstructed nonlinear grey models.

Stripe order, impurities, and symmetry breaking in a diluted frustrated magnet

We investigate the behavior of the frustrated $J_1$-$J_2$ Ising model on a square lattice under the influence of random dilution and spatial anisotropies. Spinless impurities generate a random-field type disorder for the spin-density wave (stripe) order parameter. These random fields destroy the long-range stripe order in the case of spatially isotropic interactions. Combining symmetry arguments, percolation theory and large-scale Monte Carlo simulations, we demonstrate that arbitrarily weak spatial interaction anisotropies restore the stripe phase. More specifically, the transition temperature $T_c$ into the stripe phase depends on the interaction anisotropy $\Delta J$ via $T_c \sim 1/|\ln (\Delta J)|$ for small $\Delta J$. This logarithmic dependence implies that very weak anisotropies are sufficient to restore the transition temperature to values comparable to that of the undiluted system. We analyze the critical behavior of the emerging transition and find it to belong to the disordered two-dimensional Ising universality class, which features the clean Ising critical exponents and universal logarithmic corrections. We also discuss the generality of our results and their consequences for experiments.

A Framework for Analytical Power Flow Solution Using Gaussian Process Learning

This paper proposes a novel analytical solution framework for power flow (PF) solutions in active distribution networks under uncertainty. We use the Gaussian process (GP) regression to learn node voltage as a function of effective bus load or negative net-injection vector. The proposed approximation is valid over a subspace of load and provides an understanding of system behavior under uncertainty via GP interpretability. We interpret the relative variation extent of different node voltages using the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quality ratio</i> (QR) defined based on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hyper-parameters</i> of GP. Further, the application of the proposed framework in calculation of voltage limit violation probability and dominant voltage influencer ranking has also been presented. Through test simulations for 33-bus and 56-bus systems, the proposed method achieves low mean absolute error (MAE) of order E-05 (pu) in voltage magnitude and E-04 (rad) in angle. The discussions on salient features of the proposed method and comparative analysis with large-scale Monte-Carlo simulations, and state-of-art methods is also presented for the proposed applications.

Extensive networks would eliminate the demand for pricing formulas

In this study, we generate a large number of implied volatilities for the Stochastic Alpha Beta Rho (SABR) model using a graphics processing unit (GPU) based simulation and enable an extensive neural network to learn the volatilities. This model does not have any exact pricing formulas for vanilla options, and neural networks have an outstanding ability to approximate various functions. Surprisingly, the network reduces the simulation noise by itself, thereby achieving accuracy equal to large-scale Monte-Carlo simulation. Extremely high accuracy cannot be attained via existing approximate formulas. Moreover, the network is as efficient as the approaches based on the formulas. When evaluating accuracy and efficiency, extensive networks can eliminate the necessity of pricing formulas for the SABR model. Another significant contribution is that a novel method is proposed to examine the errors based on nonlinear regression. This approach is easily extendable to other pricing models for which it is hard to deduce analytic formulas.

Embedding Overhead Scaling of Optimization Problems in Quantum Annealing

In order to treat all-to-all connected quadratic binary optimization problems (QUBO) with hardware quantum annealers, an embedding of the original problem is required due to the sparsity of the hardware's topology. Embedding fully-connected graphs - typically found in industrial applications - incurs a quadratic space overhead and thus a significant overhead in the time to solution. Here we investigate this embedding penalty of established planar embedding schemes such as minor embedding on a square lattice, minor embedding on a Chimera graph, and the Lechner-Hauke-Zoller scheme using simulated quantum annealing on classical hardware. Large-scale quantum Monte Carlo simulation suggest a polynomial time-to-solution overhead. Our results demonstrate that standard analog quantum annealing hardware is at a disadvantage in comparison to classical digital annealers, as well as gate-model quantum annealers and could also serve as benchmark for improvements of the standard quantum annealing protocol.

Quantum Monte Carlo study of topological phases on a spin analogue of Benalcazar–Bernevig–Hughes model

We study the higher-order topological spin phases based on a spin analogue of Benalcazar–Bernevig–Hughes model in two dimensions using large-scale quantum Monte Carlo simulations. A continuous Néel-valence bond solid quantum phase transition is revealed by tuning the ratio between dimerized spin couplings, namely, the weak and strong exchange couplings. Through the finite-size scaling analysis, we identify the phase critical points, and consequently, map out the full phase diagrams in related parameter spaces. Particularly, we find that the valence bond solid phase can be a higher-order topological spin phase, which has a gap for spin excitations in the bulk while demonstrates characteristic gapless spin modes at corners of open lattices. We further discuss the connection between the higher-order topological spin phases and the electronic correlated higher-order phases, and find both of them possess gapless spin corner modes that are protected by higher-order topology. Our result exemplifies higher-order physics in the correlated spin systems and will contribute to further understandings of the many-body higher-order topological phenomena.

Possible Superconductivity with a Bogoliubov Fermi Surface in a Lightly Doped Kagome U(1) Spin Liquid.

Whether the doped t-J model on the Kagome lattice supports exotic superconductivity has not been decisively answered. In this Letter, we propose a new class of variational states for this model and perform a large-scale variational MonteCarlo simulation on it. The proposed variational states are parameterized by the SU(2)-gauge rotation angles, as the SU(2)-gauge structure hidden in the Gutzwiller-projected mean-field Ansatz for the undoped model is broken upon doping. These variational doped states smoothly connect to the previously studied U(1) π-flux or 0-flux states, and energy minimization among them yields a chiral noncentrosymmetric nematic superconducting state with 2×2-enlarged unit cell. Moreover, this pair density wave state possesses a finite Fermi surface for the Bogoliubov quasiparticles. We further study experimentally relevant properties of this intriguing pairing state.

Hierarchical single-ion anisotropies in spin-1 Heisenberg antiferromagnets on the honeycomb lattice

We examine the thermal properties of the spin-1 Heisenberg antiferromagnet on the honeycomb lattice in the presence of an easy-plane single-ion anisotropy as well as the effects of an additional weak in-plane easy-axis anisotropy. In particular, using large-scale quantum Monte Carlo simulations, we analyze the scaling of the correlation length near the thermal phase transition into the ordered phase. This allows us to quantify the temperature regime above the critical point in which -- in spite of the additional in-plane easy-axis anisotropy -- characteristic easy-plane physics, such as near a Berezinskii-Kosterlitz-Thouless transition, can still be accessed. Our theoretical analysis is motivated by recent neutron scattering studies of the spin-1 compound BaNi${}_2$V${}_2$O${}_8$ in particular, and it addresses basic quantum spin models for generic spin-1 systems with weak anisotropies, which we probe over the full range of experimentally relevant correlation length scales.

Domain growth and aging in the random field XY model: A Monte Carlo study.

We use large-scale Monte Carlo simulations to obtain comprehensive results for domain growth and aging in the random field XY model in dimensions d=2,3. After a deep quench from the paramagnetic phase, the system orders locally via annihilation of topological defects, i.e., vortices and antivortices. The evolution morphology of the system is characterized by the correlation function and the structure factor of the magnetization field. We find that these quantities obey dynamical scaling, and their scaling function is independent of the disorder strength Δ. However, the scaling form of the autocorrelation function is found to be dependent on Δ, i.e., superuniversality is violated. The large-t behavior of the autocorrelation function is explored by studying aging and autocorrelation exponents. We also investigate the characteristic growth law L(t,Δ) in d=2,3, which shows an asymptotic logarithmic behavior: L(t,Δ)∼Δ^{-φ}(lnt)^{1/ψ}, with exponents φ,ψ>0.

Fractionalized conductivity and emergent self-duality near topological phase transitions

The experimental discovery of the fractional Hall conductivity in two-dimensional electron gases revealed new types of quantum particles, called anyons, which are beyond bosons and fermions as they possess fractionalized exchange statistics. These anyons are usually studied deep inside an insulating topological phase. It is natural to ask whether such fractionalization can be detected more broadly, say near a phase transition from a conventional to a topological phase. To answer this question, we study a strongly correlated quantum phase transition between a topological state, called a {{mathbb{Z}}}_{2} quantum spin liquid, and a conventional superfluid using large-scale quantum Monte Carlo simulations. Our results show that the universal conductivity at the quantum critical point becomes a simple fraction of its value at the conventional insulator-to-superfluid transition. Moreover, a dynamically self-dual optical conductivity emerges at low temperatures above the transition point, indicating the presence of the elusive vison particles. Our study opens the door for the experimental detection of anyons in a broader regime, and has ramifications in the study of quantum materials, programmable quantum simulators, and ultra-cold atomic gases. In the latter case, we discuss the feasibility of measurements in optical lattices using current techniques.

Scaling of the disorder operator at (2+1)d U(1) quantum criticality

We study disorder operator, defined as a symmetry transformation applied to a finite region, across a continuous quantum phase transition in $(2+1)d$. We show analytically that, at a conformally invariant critical point with U(1) symmetry, the disorder operator with a small U(1) rotation angle defined on a rectangle region exhibits power-law scaling with the perimeter of the rectangle. The exponent is proportional to the current central charge of the critical theory. Such a universal scaling behavior is due to the sharp corners of the region and we further obtain a general formula for the exponent when the corner is nearly smooth. To probe the full parameter regime, we carry out systematic computation of the U(1) disorder parameter in the square lattice Bose-Hubbard model across the superfluid-insulator transition with large-scale quantum Monte Carlo simulations, and confirm the presence of the universal corner correction. The exponent of the corner term determined from numerical simulations agrees well with the analytical predictions.

Nonlocal Effects of Low-Energy Excitations in Quantum-Spin-Liquid Candidate Cu3Zn(OH)6FBrSupported by the National Key Research and Development Program of China (Grant Nos. 2017YFA0302900, 2016YFA0300500, and 2020YFA0406003), the National Natural Science Foundation of China (Grant Nos. 11874401, 11674406, 11961160699, 11774399, and 11804383), the Strategic Priority Research Program(B) of the Chinese Academy of Sciences (Grant Nos. XDB33000000, XDB28000000, XDB25000000, and XDB07020000), the K.

We systematically study the low-temperature specific heats for the two-dimensional kagome antiferromagnet, Cu3Zn(OH)6FBr. The specific heat exhibits a T 1.7 dependence at low temperatures and a shoulder-like feature above it. We construct a microscopic lattice model of Z 2 quantum spin liquid and perform large-scale quantum Monte Carlo simulations to show that the above behaviors come from the contributions from gapped anyons and magnetic impurities. Surprisingly, we find the entropy associated with the shoulder decreases quickly with grain size d, although the system is paramagnetic to the lowest temperature. While this can be simply explained by a core-shell picture in that the contribution from the interior state disappears near the surface, the 5.9-nm shell width precludes any trivial explanations. Such a large length scale signifies the coherence length of the nonlocality of the quantum entangled excitations in quantum spin liquid candidate, similar to Pippard’s coherence length in superconductors. Our approach therefore offers a new experimental probe of the intangible quantum state of matter with topological order.

Phase diagram of a frustrated Heisenberg model: From disorder to order and back again

We study the effects of bond and site disorder in the classical $J_{1}$-$J_{2}$ Heisenberg model on a square lattice in the order-by-disorder frustrated regime $2J_{2}>\left|J_{1}\right|$. Combining symmetry arguments, numerical energy minimization and large scale Monte Carlo simulations, we establish that the finite temperature Ising-like transition of the clean system is destroyed in the presence of any finite concentration of impurities. We explain this finding via a random-field mechanism which generically emerges in systems where disorder locally breaks the same real-space symmetry spontaneously globally broken by the associated order parameter. We also determine that the phase replacing the clean one is a paramagnet polarized in the nematic glass order with non-trivial magnetic response. This is because disorder also induces non-collinear spin-vortex-crystal order and produces a conjugated transverse dipolar random field. As a result of these many competing effects, the associated magnetic susceptibilities are non-monotonic functions of the temperature. As a further application of our methods, we show the generation of random axes in other frustrated magnets with broken SU(2) symmetry. We also discuss the generality of our findings and their relevance to experiments.