Session X22: Computational Methods for Statistical Mechanics: Advances and Applications - III

Live

The processes occurring at surfaces play a critical role in the manufacture and performance of advanced materials, e.g., electronic, magnetic, and optical devices, sensors, and catalysts. A prerequisite for analyzing and understanding the electronic properties and the function of surfaces is detailed knowledge of the atomic structure, i.e., the surface composition and geometry under realistic gas-phase conditions. The key quantity for studying the structure and function of surfaces/clusters in reactive atmospheres is the Gibbs free energy, as function of number of particles, pressure, and temperature. Here, I present a set of methods for the sampling of the configurational space of (nano)clusters and surfaces in reactive (e.g., O2, H2) atmosphere, in the canonical and grand-canonical ensembles, aiming at the unbiased determination of the phase diagrams as function of temperature and partial pressure of the reactive gas.
Applications to gold and metal-oxide nanoclusters, and silicon surfaces, described with first-principles potential-energy surfaces, will demonstrate the insight gained by the direct access to observables at finite temperature and pressure.   

Live

We provide numerical evidence that after a local quench in an isolated infinite quantum spin
chain, the quantum state locally relaxes to the ground state of the post-quenched Hamiltonian, i.e.
dissipates. This is a consequence of the unitary quantum dynamics. A mechanism similar to the
eigenstate thermalization hypothesis is shown to be responsible for the dissipation observed. We also
demonstrate that integrability obstructs dissipation. The numerical simulations are done directly in
the thermodynamic limit with a time-evolution algorithm based on matrix product states. The area
law of entanglement entropy is observed to hold after the local quench. As a result, the simulations
can be performed for long times with small bond dimensions. Various local quenches on the Ising
chain and the three-state Potts chain are studied.   

Live

Potts spin systems play a fundamental role in statistical mechanics and quantum field theory, and can be studied within the spin, the Fortuin-Kasteleyn (FK) bond or the q-flow (loop) representation. We introduce a Loop-Cluster (LC) joint model of bond-occupation variables interacting with q-flow variables, and formulate a LC algorithm that is found to be in the same dynamical universality as the celebrated Swendsen-Wang algorithm. This leads to a theoretical unification for all the representations, and numerically, one can apply the most efficient algorithm in one representation and measure physical quantities in others. Moreover, by using the LC scheme, we construct a hierarchy of geometric objects that contain as special cases the q-flow clusters and the backbone of FK clusters, the exact values of whose fractal dimensions in two dimensions remain as an open question. Our work not only provides a unified framework and an efficient algorithm for the Potts model, but also brings new insights into rich geometric structures of the FK clusters.   

Live

The analysis of complex physical systems hinges on ability to sift out the relevant degrees of freedom from among the many others. Though much hope is placed in machine learning, it also brings challenges, chief of which is interpretability. It is often unclear what relation, if any, the architecture and training-dependent learned "relevant" features bear to standard objects of physical theory.
Here we report on theoretical results which may help to systematically address this issue: we establish equivalence between the information-theoretic notion of relevancy defined in the Information Bottleneck (IB) formalism of compression theory, and the field-theoretic relevance of the Renormalization Group. We show analytically that for statistical physical systems described by a field theory the "relevant" degrees of freedom found using IB compression indeed correspond to primary operators with the lowest scaling dimensions. We confirm our field theoretic predictions numerically to high precision. We study dependence of the IB solutions on the physical symmetries of the data. Our findings provide a dictionary connecting two distinct theoretical toolboxes, and an avenue to constructively incorporate physical interpretability in applications of machine learning.   

Live

Continuum descriptions of complex classical and quantum systems increasingly are alternatives to first-principles many-body analyses. The macroscopic momentum balance equation originates in the underlying exact conservation law, with a precisely defined operator for the associated momentum flux. Its average in a local equilibrium ensemble provides the basis for a hydrodynamic description. A local pressure can be defined in terms of the trace of the local equilibrium average momentum flux, or alternatively as a thermodynamic pressure obtained from the grand potential for the local equilibrium ensemble. Both have the same global pressure but can differ locally. For uniform temperature (equilibrium), the momentum flux trace agrees with the local thermodynamic pressure even for strong density inhomogeneity, providing a connection to density functional methods for hydrodynamic applications. For non-uniform temperatures (local equilibrium) and strong inhomogeneities, no simple relationship exists among the different local pressure definitions. However, agreement can be restored by adding a suitable divergence-free contribution to the local equilibrium average momentum flux*.
*J Dufty, J Wrighton, K Luo. AIChe J. doi:10.1002/aic.17037   

Live

Being one of the simplest models of magnetic systems with quenched disorder, the random-field Ising model shows surprisingly rich critical behavior. Only recently has it been possible with the help of large-scale numerical simulations to shed some light on a range of fundamental questions in three and higher dimensions, such as universality, critical scaling and dimensional reduction. The two-dimensional model
has received less attention, but is no less fascinating. We solve a long-standing puzzle by presenting compelling numerical evidence for the scaling behavior of the correlation length ξ. Results for two lattice geometries, square and triangular, consistently support the form ξ ~ exp[A/h²], where h denotes the random-field strength, in line with early theoretical work [1], but at variance with some more recent numerical and analytical results [2,3]. We also investigate the more widely used break-up length scale of the system, which we however find to be afflicted by much stronger scaling corrections and hence a rather less useful quantity.

[1] K. Binder, Z. Phys. B50, 343 (1983).
[2] G. P. Shrivastav, M. Kumar, V. Banerjee, and S. Puri,Phys. Rev. E90, 032140 (2014).
[3] L. Hayden, A. Raju, and J. P. Sethna, Phys. Rev. Res.1, 033060 (2019).   

Not Participating

We aim to develop and release a time-splitting numerical simulator of noisy driven-dissipative Gross-Pitaevskii equations (GPE), for use in general non-Hermitian problems. Our solver correctly addresses crucial method differences that other available GPE solvers neglect, when incorporating complex potential/coefficient and added noise term components for nonstandard GPEs, such as when considering the time-dependency of the solution density. This solver has applications to any area of research which would benefit from simulations of nonstandard GPEs, which include any of the following traits: complex Laplacian term coefficients, complex potentials, complex nonlinear term coefficients, coupled components, added noise terms. Examples include simulating dynamics involving (critical) exceptional point physics and modeling (noisy) interacting condensates, which are systems proposed as a basis for unconventional computing systems. Although previous research on hybrid polariton Bose-Einstein condensate computing systems has shown promising analytical results of an advantage over traditional computing systems, because of the approximations/limitations of the methods utilized in those approaches, those results should be verified through numerical analysis.   

Live

We present a self-contained theory for the exact calculation of particle number counting statistics of non-interacting indistinguishable particles in the canonical ensemble. This general framework introduces the concept of auxiliary partition functions, and represents a unification of previous distinct approaches with many known results appearing as direct consequences of the developed mathematical structure. In addition, we introduce a general decomposition of the correlations between occupation numbers in terms of the occupation numbers of individual energy levels, that is valid for both non-degenerate and degenerate spectra. To demonstrate the applicability of the theory in the presence of degeneracy, we compute energy level correlations up to fourth order in a bosonic ring in the presence of a magnetic field.   

Live

We study spectral properties and spectral-crossovers between different random matrix ensembles (Poissonian, GOE, GUE) in correlated spin chain systems, in the presence of random magnetic fields, and scalar spin-chirality term, as a function of competing Hamiltonian parameters. We have investigated these crossovers in the context of level-spacing distribution and level-spacing ratio distribution. Using interpolating functions between Poissonian-to-GOE and GOE-to-GUE, we have fitted the crossovers, to random matrix interpolating parameter ‘λ’. We find that once λ is properly scaled, the crossover behavior exhibits universality, in the sense that it becomes independent of lattice size in the large Hamiltonian-matrix dimension limit.   

Live

Self-avoiding walks on lattices are used to model the statistics of polymer chains. Here we consider growing self-avoiding walks (GSAWs) on a lattice, which grow by taking their Nth step into a randomly chosen unoccupied site adjacent to the N-1th step. It is known from simulations that on a square lattice, a GSAW will become trapped after a mean of 71 steps, but this is a purely empirical fact. We have extended the square lattice GSAW to include nearest-neighbor attractive interactions, similar to those used to model polymers in poor solvents, which lead to a non-monotonic trend in the mean trapping length. To gain additional insight into the statistics of GSAW trapping, we consider simplified cases of geometrically restricted lattices that are two to three sites wide. Using recursion relations and generating functions, we are able to derive exact expressions for parameters such as the mean trapping length, the asymptotic behavior of the trapping probability distribution, and the effect of nearest-neighbor attraction, finding, for example, that walks on a restricted square lattice are trapped after a mean of exactly 17 steps. Our findings provide mathematical insight into a phenomenon that has been known only empirically since the 1980s.   

Live

Advanced quantum Monte Carlo algorithms such as the loop algorithm [1] are based on the cluster update, and their relaxation has been considered too fast to apply the nonequilibrium relaxation (NER) method based on the power-law critical relaxation [2]. The present authors revealed that the critical cluster NER in classical spin systems is described by the stretched-exponential relaxation [3-6]. They generalized this procedure to the loop algorithm, and analyzed quantum phase transitions with the cluster NER [7]. Recently, the present authors proposed that the off-critical behaviors in classical spin systems can also be analyzed with the NER framework [8], and called this scheme as the "temperature scaling". In the present talk, we generalize this scheme to quantum phase transitions. As an example, we consider the Néel-dimer quantum phase transition in the two-dimensional S=1/2 antiferromagnetic Heisenberg model with columnar dimerization.

[1] H. G. Evertz et al., PRL 70, 875 (1993). [2] Y. Ozeki and N. Ito, J. Phys. A 40, R149 (2007). [3] Y. Nonomura, J. Phys. Soc. Jpn. 83, 113001 (2014). [4] YN and Y. Tomita, PRE 92, 062121 (2015). [5] YN and YT, PRE 93, 012101 (2016). [6] YT and YN, PRE 98, 052110 (2018). [7] YN and YT, PRE 101, 032105 (2020). [8] YN and YT, arXiv:2007.14356.   

Live

A simple method to calculate density, two-point correlation function, and other statistical quantities of interest for a class of random matrix ensembles is presented. We reproduce some known results for well-studied random matrix ensembles with varying potentials and two-body interaction. New results for spectral properties of the Muttalib-Borodin ensemble and recently introduced gamma-ensemble were also obtained numerically for which no analytical method is available yet.   

Live

The eigenvalue density for Generalized Muttalib-Borodin ensembles (also called γ-ensembles) can be computed by solving equivalent equilibrium problem for Muttalib-Borodin (MB) ensembles with γ-dependent effective potentials. Ideally, an exponent γ-induced hard-edge to soft-edge transition in eigenvalue density could describe a disorder induced metal to insulator transition in mesoscopic conductors. Such a transition in density has previously been produced only by especially designed non-monotonic confining potentials of MB ensembles. We find that the effective potentials computed for a range of parameter of hard-edge γ–ensembles with standard monotonic potentials show increasing non-monotonic behavior near the origin as γ is decreased (or disorder is increased) systematically. While this non-monotonicity is not sufficient to produce a hard-edge to soft-edge transition in density in this toy model, it suggests that with appropriate combination of the additional interaction and the confining potential, such a transition can indeed occur. As a byproduct of the above calculations we also obtain the eigenvalue densities of Laguerre β-ensembles for any β>1.