Linear System Size Research Articles

Anomalous criticality coexists with giant cluster in the uniform forest model

We show by extensive simulations that the whole supercritical phase of the three-dimensional uniform forest model simultaneously exhibits an infinite tree and a rich variety of critical phenomena. Besides typical scalings like algebraically decaying correlation, power-law distribution of cluster sizes, and divergent correlation length, a number of anomalous behaviors emerge. The fractal dimensions for off-giant trees take different values when being measured by linear system size or gyration radius. The giant-tree size displays two-length scaling fluctuations, instead of following the central-limit theorem.

Eigen microstates in self-organized criticality.

We employ the eigen microstates approach to explore the self-organized criticality (SOC) in two celebrated sandpile models, namely the BTW model and the Manna model. In both models, phase transitions from the absorbing state to the critical state can be understood by the emergence of dominant eigen microstates with significantly increased weights. Spatial eigen microstates of avalanches can be uniformly characterized by a linear system size rescaling. The first temporal eigen microstates reveal scaling relations in both models. Furthermore, by finite-size scaling analysis of the first eigen microstates, we numerically estimate critical exponents, i.e., sqrt[σ_{0}w_{1}]/v[over ̃]_{1}∝L^{D} and v[over ̃]_{1}∝L^{D(1-τ_{s})/2}. Our findings could provide profound insights into eigen microstates of the universality and phase transition in nonequilibrium complex systems governed by self-organized criticality.

Structure of liquid-vapor interfaces: Perspectives from liquid state theory, large-scale simulations, and potential grazing-incidence x-ray diffraction.

Grazing-incidence x-ray diffraction (GIXRD) is a scattering technique that allows one to characterize the structure of fluid interfaces down to the molecular scale, including the measurement of surface tension and interface roughness. However, the corresponding standard data analysis at nonzero wave numbers has been criticized as to be inconclusive because the scattering intensity is polluted by the unavoidable scattering from the bulk. Here, we overcome this ambiguity by proposing a physically consistent model of the bulk contribution based on a minimal set of assumptions of experimental relevance. To this end, we derive an explicit integral expression for the background scattering, which can be determined numerically from the static structure factors of the coexisting bulk phases as independent input. Concerning the interpretation of GIXRD data inferred from computer simulations, we extend the model to account also for the finite sizes of the bulk phases, which are unavoidable in simulations. The corresponding leading-order correction beyond the dominant contribution to the scattered intensity is revealed by asymptotic analysis, which is characterized by the competition between the linear system size and the x-ray penetration depth in the case of simulations. Specifically, we have calculated the expected GIXRD intensity for scattering at the planar liquid-vapor interface of Lennard-Jones fluids with truncated pair interactions via extensive, high-precision computer simulations. The reported data cover interfacial and bulk properties of fluid states along the whole liquid-vapor coexistence line. A sensitivity analysis shows that our findings are robust with respect to the detailed definition of the mean interface position. We conclude that previous claims of an enhanced surface tension at mesoscopic scales are amenable to unambiguous tests via scattering experiments.

Diffusive persistence on disordered lattices and random networks.

To better understand the temporal characteristics and the lifetime of fluctuations in stochastic processes in networks, we investigated diffusive persistence in various graphs. Global diffusive persistence is defined as the fraction of nodes for which the diffusive field at a site (or node) has not changed sign up to time t (or, in general, that the node remained active or inactive in discrete models). Here we investigate disordered and random networks and show that the behavior of the persistence depends on the topology of the network. In two-dimensional (2D) disordered networks, we find that above the percolation threshold diffusive persistence scales similarly as in the original 2D regular lattice, according to a power law P(t,L)∼t^{-θ} with an exponent θ≃0.186, in the limit of large linear system size L. At the percolation threshold, however, the scaling exponent changes to θ≃0.141, as the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. Moreover, studying finite-size effects for 2D lattices at and above the percolation threshold, we find that at the percolation threshold, the long-time asymptotic value obeys a power law P(t,L)∼L^{-zθ} with z≃2.86 instead of the value of z=2 normally associated with finite-size effects on 2D regular lattices. In contrast, we observe that in random networks without a local regular structure, such as Erdős-Rényi networks, no simple power-law scaling behavior exists above the percolation threshold.

Full-wave Image Reconstruction in Transcranial Photoacoustic Computed Tomography using a Finite Element Method.

Transcranial photoacoustic computed tomography presents challenges in human brain imaging due to skull-induced acoustic aberration. Existing full-wave image reconstruction methods rely on a unified elastic wave equation for skull shear and longitudinal wave propagation, therefore demanding substantial computational resources. We propose an efficient discrete imaging model based on finite element discretization. The elastic wave equation for solids is solely applied to the hard-tissue skull region, while the soft-tissue or coupling-medium region that dominates the simulation domain is modeled with the simpler acoustic wave equation for liquids. The solid-liquid interfaces are explicitly modeled with elastic-acoustic coupling. Furthermore, finite element discretization allows coarser, irregular meshes to conform to object geometry. These factors significantly reduce the linear system size by 20 times to facilitate accurate whole-brain simulations with improved speed. We derive a matched forward-adjoint operator pair based on the model to enable integration with various optimization algorithms. We validate the reconstruction framework through numerical simulations and phantom experiments.

Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations

Scalable quantum algorithms for the simulation of quantum many-body systems in thermal equilibrium are important for predicting properties of quantum matter at finite temperatures. Here we describe and benchmark a quantum computing version of the minimally entangled typical thermal states (METTS) algorithm for which we adopt an adaptive variational approach to perform the required quantum imaginary time evolution. The algorithm, which we name AVQMETTS, dynamically generates compact and problem-specific quantum circuits, which are suitable for noisy intermediate-scale quantum (NISQ) hardware. We benchmark AVQMETTS on statevector simulators and perform thermal energy calculations of integrable and nonintegrable quantum spin models in one and two dimensions and demonstrate an approximately linear system-size scaling of the circuit complexity. We further map out the finite-temperature phase transition line of the two-dimensional transverse field Ising model. Finally, we study the impact of noise on AVQMETTS calculations using a phenomenological noise model.

Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models

We generalize the area-law violating models of Fredkin and Motzkin spin chains into two dimensions by building quantum six- and nineteen-vertex models with correlated interactions. The Hamiltonian is frustration free, and its projectors generate ergodic dynamics within the subspace of height configuration that are non negative. The ground state is a volume- and color-weighted superposition of classical bi-color vertex configurations with non-negative heights in the bulk and zero height on the boundary. The entanglement entropy between subsystems has a phase transition as the qq-deformation parameter is tuned, which is shown to be robust in the presence of an external field acting on the color degree of freedom. The ground state undergoes a quantum phase transition between area- and volume-law entanglement phases with a critical point where entanglement entropy scales as a function L\log LLlogL of the linear system size LL. Intermediate power law scalings between L\log LLlogL and L^2L2 can be achieved with an inhomogeneous deformation parameter that approaches 1 at different rates in the thermodynamic limit. For the q>1q>1 phase, we construct a variational wave function that establishes an upper bound on the spectral gap that scales as q^{-L^3/8}q−L3/8.

A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods.

Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.

Dephasing-enhanced Majorana zero modes in two-dimensional and three-dimensional higher-order topological superconductors

The 1D Kitaev model in the topological phase, with open boundary conditions, hosts strong Majorana zero modes. These are fermion parity-odd operators that almost commute with the Hamiltonian and manifest in long coherence times for edge degrees of freedom. We obtain higher-dimensional counterparts of such Majorana operators by explicitly computing their closed form expressions in models describing 2D and 3D higher-order superconductors. Due to the existence of such strong Majorana zero modes, the coherence time of the infinite temperature autocorrelation function of the corner Majorana operators in these models diverges with the linear system size. In the presence of a certain class of orbital-selective dissipative dynamics, the coherence times of half of the corner Majorana operators is enhanced, while the time correlations corresponding to the remaining corner Majoranas decay much faster as compared with the unitary case. We numerically demonstrate robustness of the coherence times to the presence of disorder.

Nested closed paths in two-dimensional percolation

For two-dimensional percolation on a domain with the topology of a disc, we introduce a nested-path (NP) operator and thus a continuous family of one-point functions , where ℓ is the number of independent (i.e., non-overlapping) nested closed paths surrounding the center, k is a path fugacity, and projects on configurations having a cluster connecting the center to the boundary. At criticality, we observe a power-law scaling , with L the linear system size, and we determine the exponent X NP as a function of k. On the basis of our numerical results, we conjecture an analytical formula, with k = 2 cos(πϕ), which reproduces the exact results for k = 0, 1 and agrees with the high-precision estimate of for other k values. In addition, we observe that W 2(L) = 1 for site percolation on the triangular lattice with any size L, and we prove this identity for all self-matching lattices.

Percolation of the two-dimensional XY model in the flow representation.

We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L=4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition K_{perc} from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the quasi-long-range order associated with spin properties. Near K_{perc}, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold K_{perc}=1.1053(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point K_{BKT}=1.1193(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed light on a variety of classical and quantum systems of BKT phase transitions.

The art of coarse Stokes: Richardson extrapolation improves the accuracy and efficiency of the method of regularized stokeslets.

The method of regularized stokeslets is widely used in microscale biological fluid dynamics due to its ease of implementation, natural treatment of complex moving geometries, and removal of singular functions to integrate. The standard implementation of the method is subject to high computational cost due to the coupling of the linear system size to the numerical resolution required to resolve the rapidly varying regularized stokeslet kernel. Here, we show how Richardson extrapolation with coarse values of the regularization parameter is ideally suited to reduce the quadrature error, hence dramatically reducing the storage and solution costs without loss of accuracy. Numerical experiments on the resistance and mobility problems in Stokes flow support the analysis, confirming several orders of magnitude improvement in accuracy and/or efficiency.

Critical polynomials in the nonplanar and continuum percolation models.

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently, the critical polynomial P_{B}(p,L) was introduced for planar-lattice percolation models, where p is the occupation probability and L is the linear system size. The solution of P_{B}=0 can reproduce all known exact thresholds and leads to unprecedented estimates for thresholds of unsolved planar-lattice models. In two dimensions, assuming the universality of P_{B}, we use it to study a nonplanar lattice model, i.e., the equivalent-neighbor lattice bond percolation, and the continuum percolation of identical penetrable disks, by Monte Carlo simulations and finite-size scaling analysis. It is found that, in comparison with other quantities, P_{B} suffers much less from finite-size corrections. As a result, we obtain a series of high-precision thresholds p_{c}(z) as a function of coordination number z for equivalent-neighbor percolation with z up to O(10^{5}) and clearly confirm the asymptotic behavior zp_{c}-1∼1/sqrt[z] for z→∞. For the continuum percolation model, we surprisingly observe that the finite-size correction in P_{B} is unobservable within uncertainty O(10^{-5}) as long as L≥3. The estimated threshold number density of disks is ρ_{c}=1.43632505(10), slightly below the most recent result ρ_{c}=1.43632545(8) of Mertens and Moore obtained by other means. Our work suggests that the critical polynomial method can be a powerful tool for studying nonplanar and continuum systems in statistical mechanics.

Size distributions of the largest hole in the largest percolation cluster and backbone

At criticality in two dimensions, the size distribution of holes within a large percolation cluster or backbone was recently found to be governed by a power law with the exponent given by a hyperscaling relation. It was also found that, though the boundary of the largest hole is a fractal, the size of the largest hole itself is proportional to Ld, where L represents the linear system size and d=2 is the spatial dimension. In this work, we study the probability distribution of the size of the largest hole in the largest percolation cluster and that in the largest backbone, and compare them with the probability distribution of the size of the largest percolation cluster and that of the largest backbone themselves. We obtain these four size distributions in the critical region by Monte Carlo simulations for the two-dimensional bond percolation model on the square lattice with periodic boundary conditions. It is found that each of these distributions can be described by a different single-variable function P(C,L)dC=P̃(x)dx, with x≡C∕LdC, where C is the size of the structure whose (fractal) dimension is dC. The function P̃(x) is generally not unimodal, and interestingly, for the largest cluster hole, symmetric properties are found in its size distribution due to duality, e.g., P̃(x) is symmetric around the mean value x=1 at the percolation threshold pc. We characterize these size distributions by calculating the variance and different dimensionless cumulant ratios, which exhibit rich critical behaviors at the pseudocritical and critical points. For example, the extremum of the excess kurtosis and the zero point of the skewness locate at the same (pseudocritical) point, whose values are different for the four structures but approach pc in the limit L→∞. We also perform wrapping analyses for each size distribution, showing that different wrapping types contribute to the overall distribution in different forms.

Energetics and coarsening analysis of a simplified non-linear surface growth model

<p style='text-indent:20px;'>We study a simplified multidimensional version of the phenomenological surface growth continuum model derived in [<xref ref-type="bibr" rid="b6">6</xref>]. The considered model is a partial differential equation for the surface height profile <inline-formula><tex-math id="M1">\begin{document}$ u $\end{document}</tex-math></inline-formula> which possesses the following free energy functional:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ E(u) = \int_{\Omega} \left[ \frac{1}{2} \ln\left(1+\left|\nabla u \right|^2\right) - \left|\nabla u \right| \arctan\left(\left|\nabla u \right|\right) + \frac{1}{2} \left|\Delta u \right|^2 \right] {\rm d}x, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is the domain of a fixed support on which the growth is carried out. The term <inline-formula><tex-math id="M3">\begin{document}$ \left|\Delta u \right|^2 $\end{document}</tex-math></inline-formula> designates the standard surface diffusion in contrast to the second order term which phenomenologically describes the growth instability. The energy above is mainly carried out in regions of higher surface slope <inline-formula><tex-math id="M4">\begin{document}$ \left( \left|\nabla u \right| \right) $\end{document}</tex-math></inline-formula>. Hence minimizing such energy corresponds to reducing surface defects during the growth process from a given initial surface configuration. Our analysis is concerned with the energetic and coarsening behaviours of the equilibrium solution. The existence of global energy minimizers and a scaling argument are used to construct a sequence of equilibrium solutions with different wavelength. We apply our minimum energy estimates to derive bounds in terms of the linear system size <inline-formula><tex-math id="M5">\begin{document}$ \left| \Omega \right| $\end{document}</tex-math></inline-formula> for the characteristic interface width and average slope. We also derive a stable numerical scheme based on the convex-concave decomposition of the energy functional and study its properties while accommodating these results with 1d and 2d numerical simulations.</p>

Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices

We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with the code distance being the linear system size is decomposed by a local Clifford circuit of constant depth into a finite number of copies of the toric code stabilizer group (Abelian discrete gauge theory). This means that under local Clifford circuits, the number of toric code copies is the complete invariant of topological Pauli stabilizer codes. Previously, the same conclusion was obtained under the assumption of nonchirality for qubit codes or the Calderbank–Shor–Steane structure for prime qudit codes; we do not assume any of these.

Closure of the entanglement gap at quantum criticality: The case of the quantum spherical model

The study of entanglement spectra is a powerful tool to detect or elucidate universal behaviour in quantum many-body systems. We investigate the scaling of the entanglement (or Schmidt) gap $\delta\xi$, i.e., the lowest laying gap of the entanglement spectrum, at a two-dimensional quantum critical point. We focus on the paradigmatic quantum spherical model, which exhibits a second-order transition, and is mappable to free bosons with an additional external constraint. We analytically show that the Schmidt gap vanishes at the critical point, although only logarithmically. For a system on a torus and the half-system bipartition, the entanglement gap vanishes as $\pi^2/\ln(L)$, with $L$ the linear system size. The entanglement gap is nonzero in the paramagnetic phase and exhibits a faster decay in the ordered phase. The rescaled gap $\delta\xi\ln(L)$ exhibits a crossing for different system sizes at the transition, although logarithmic corrections prevent a precise verification of the finite-size scaling. Interestingly, the change of the entanglement gap across the phase diagram is reflected in the zero-mode eigenvector of the spin-spin correlator. At the transition quantum fluctuations give rise to a non-trivial structure of the eigenvector, whereas in the ordered phase it is flat. We also show that the vanishing of the entanglement gap at criticality can be qualitatively but not quantitatively captured by neglecting the structure of the zero-mode eigenvector.

A statically condensed discontinuous Galerkin spectral element method on Gauss-Lobatto nodes for the compressible Navier-Stokes equations

We present a static-condensation method for time-implicit discretizations of the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points (GL-DGSEM). We show that, when solving the compressible Navier-Stokes equations, it is possible to reorganize the linear system that results from the implicit time-integration of the GL-DGSEM as a Schur complement problem, which can be efficiently solved using static condensation. The use of static condensation reduces the linear system size and improves the condition number of the system matrix, which translates into shorter computational times when using direct and iterative solvers.The statically condensed GL-DGSEM presented here can be applied to linear and nonlinear advection-diffusion partial differential equations in conservation form with the standard and the split-form variants of the DGSEM. To test it we solve the compressible Navier-Stokes equations with direct and Krylov subspace solvers, and we show for a selected problem that using the statically condensed GL-DGSEM leads to speed-ups of up to 200 when compared to the time-explicit GL-DGSEM, and speed-ups of up to three when compared with the time-implicit GL-DGSEM that solves the global system.The GL-DGSEM has gained increasing popularity in recent years because it satisfies the summation-by-parts property, which enables the construction of provably entropy stable schemes, and because it is computationally very efficient. In this paper, we show that the GL-DGSEM has an additional advantage: It can be statically condensed.

Magnetic Properties of Inverse Spinel: (Fe3+)A(Fe3+Fe2+)BO42− Magnetite

We used a Monte Carlo with Metropolis algorithm in the framework of a three-dimensional Ising model with interactions of closest magnetic neighbours for energy minimization, and we calculated the magnetization, the magnetic susceptibility and the magnetic hysteresis cycle of bulk inverse spinel $$ {\left({\mathrm{Fe}}^{3+}\right)}_A{\left({\mathrm{Fe}}^{3+}{\mathrm{Fe}}^{2+}\right)}_B{\mathrm{O}}_4^{2-} $$ magnetite, in which eight Fe3+ ions are located in tetrahedral sites (A-sites), whereas eight Fe2+ and eight Fe3+ ions are distributed in octahedral sites (B-sites) per unit cell. The critical temperatures have been obtained for several linear system sizes. The effect of temperatures on the magnetic hysteresis cycles has been investigated.

Many-body reactive force field development for carbon condensation in C/O systems under extreme conditions.

We describe the development of a reactive force field for C/O systems under extreme temperatures and pressures, based on the many-body Chebyshev Interaction Model for Efficient Simulation (ChIMES). The resulting model, which targets carbon condensation under thermodynamic conditions of 6500 K and 2.5 g cm-3, affords a balance between model accuracy, complexity, and training set generation expense. We show that the model recovers much of the accuracy of density functional theory for the prediction of structure, dynamics, and chemistry when applied to dissociative condensed phase systems at 1:1 and 1:2 C:O ratios, as well as molten carbon. Our C/O modeling approach exhibits a 104 increase in efficiency for the same system size (i.e., 128 atoms) and a linear system size scalability over standard quantum molecular dynamics methods, allowing the simulation of significantly larger systems than previously possible. We find that the model captures the condensed-phase reaction-coupled formation of carbon clusters implied by recent experiments, and that this process is susceptible to strong finite size effects. Overall, we find the present ChIMES model to be well suited for studying chemical processes and cluster formation at pressures and temperatures typical of shock waves. We expect that the present C/O modeling paradigm can serve as a template for the development of a broader high pressure-high temperature force-field for condensed phase chemistry in organic materials.