Statistical Mechanical Lattice Model Research Articles

On different approaches to integrable lattice models

Interaction-round the face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e. the models the Boltzmann weights of which satisfy the quantum Yang–Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras su(2)k and su(3)k are computed for fundamental and adjoint representations for some fixed levels k. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on su(3)k (for general k) and discuss how it can be used to find Boltzmann weights in terms of the quantum R^ matrix when the adjoint representation defines the admissibility conditions.

Conformal Field Theory at the Lattice Level: Discrete Complex Analysis and Virasoro Structure

Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333–380, 1984). Both exhibit exactly solvable structures in two dimensions. A long-standing question (Itoyama and Thacker in Phys Rev Lett 58:1395–1398, 1987) concerns whether there is a direct link between these structures, that is, whether the Virasoro algebra representations of CFT, the distinctive feature of CFT in two dimensions, can be found within lattice models of statistical mechanics. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the structures of discrete complex analysis in the lattice models with the Virasoro symmetry that is expected to describe their scaling limits. This allows for a tight connection of a number of objects from the lattice model world and the field theory one. In particular, our results link the CFT local fields with lattice local fields introduced in Gheissari et al. (Commun Math Phys 367(3):771–833, 2019) and the probabilistic formulation of the lattice model with the continuum correlation functions. Our construction is a decisive step towards establishing the conjectured correspondence between the correlation functions of the CFT fields and those of the lattice local fields. In particular, together with the upcoming (Chelkak et al. in preparation), our construction will complete the picture initiated in Hongler and Smirnov (Acta Math 211:191–225, 2013), Hongler (Conformal invariance of ising model correlations, 2012) and Chelkak et al. (Annals Math 181(3):1087–1138, 2015), where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven.

Multiple-SLE κ connectivity weights for rectangles, hexagons, and octagons

In previous work, two of the authors determined, completely and rigorously, a solution space for a homogeneous system of 2N + 3 linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm–Löwner evolution (SLE κ ). The system comprises 2N null-state equations and three conformal ward identities that govern CFT correlation functions of 2N one-leg boundary operators or SLE κ partition functions. M Bauer et al conjectured a formula, expressed in terms of ‘pure SLE κ partition functions,’ for the probability that the growing curves of a multiple-SLE κ process join in a particular connectivity. In a previous article, we rigorously define certain elements of , which we call ‘connectivity weights,’ argue that they are in fact pure SLE κ partition functions, and show how to find explicit formulas for them in terms of Coulomb gas contour integrals. Our formal definition of the connectivity weights immediately leads to a method for finding explicit expressions for them. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of κ ∈ (0, 8) corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights in for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called ‘rainbow connectivity weights’ in for all . We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix B, we investigate logarithmic singularities of some of these expressions, appearing for certain values of κ predicted by logarithmic CFT.

Achieving a realistic native protein ensemble by HDX-MS and computational modeling

Amide hydrogen-deuterium exchange (HDX) measurements pioneered in the 1950s and developed extensively by Englander in the 1980s (1), still give us more information than we understand about protein motions. Some backbone amides exchange within milliseconds, whereas others may take months. How amides exchange in a folded protein and why some exchange so quickly and others so slowly has puzzled biophysicists for several generations. What exactly are the underlying physicochemical phenomena that result in amide exchange and how do the rates of exchange relate to structural dynamics? Early investigators, including Ken Dill and Hue Sun Chan, worked in statistical mechanical lattice models of proteins to try to answer this question (2,3).

Interaction-round-a-face and consistency-around-a-face-centered-cube

There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations that satisfy multidimensional consistency, where the latter may be found in a quasi-classical expansion of the former. This paper extends this correspondence to interaction-round-a-face (IRF) models, resulting in a new formulation of the consistency-around-a-cube integrability condition applicable to five-point equations in the square lattice. Multidimensional consistency for these equations is formulated as consistency-around-a-face-centered-cube (CAFCC), which, namely, involves satisfying an overdetermined system of 14 five-point lattice equations for eight unknown variables on the face-centered cubic unit cell. From the quasi-classical limit of IRF models, which are constructed from the continuous spin solutions of the star–triangle relations associated with the Adler–Bobenko–Suris list, 15 sets of equations are obtained, which satisfy CAFCC.

Solvability of the system of integral equations of lattice models of statistical mechanics

Solvability of the system of integral equations of lattice models of statistical mechanics

Large deviations in renewal models of statistical mechanics

In Zamparo (2019 (arXiv:1903.03527)) the author has recently established sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a separable Banach space. The renewal model has been there identified with constrained and non-constrained pinning models of polymers, which amount to Gibbs changes of measure of a classical renewal process. In this paper we show that the constrained pinning model is the common mathematical structure to the Poland–Scheraga model of DNA denaturation and to some relevant one-dimensional lattice models of statistical mechanics, such as the Fisher–Felderhof model of fluids, the Wako–Saitô–Muñoz–Eaton model of protein folding, and the Tokar–Dreyssé model of strained epitaxy. Then, in the framework of the constrained pinning model, we develop an analytical characterization of the large deviation principles for cumulative rewards corresponding to multivariate deterministic rewards that are uniquely determined by, and at most of the order of magnitude of, the time elapsed between consecutive renewals. In particular, we outline the explicit calculation of the rate functions and successively we identify the conditions that prevent them from being analytic and that underlie affine stretches in their graphs. Finally, we apply the general theory to the number of renewals. From the point of view of equilibrium statistical physics and statistical mechanics, cumulative rewards of the above type are the extensive observables that enter the thermodynamic description of the system. The number of renewals, which turns out to be the commonly adopted order parameter for the Poland–Scheraga model and for also the renewal models of statistical mechanics, is one of these observables.

Continuum limits of sparse coupling patterns

We exhibit simple lattice systems, motivated by recently proposed cold atom experiments, whose continuum limits interpolate between real and $p$-adic smoothness as a spectral exponent is varied. A real spatial dimension emerges in the continuum limit if the spectral exponent is negative, while a $p$-adic extra dimension emerges if the spectral exponent is positive. We demonstrate Holder continuity conditions, both in momentum space and in position space, which quantify how smooth or ragged the two-point Green's function is as a function of the spectral exponent. The underlying discrete dynamics of our model is defined in terms of a Gaussian partition function as a classical statistical mechanical lattice model. The couplings between lattice sites are sparse in the sense that as the number of sites becomes large, a vanishing fraction of them couple to one another. This sparseness property is useful for possible experimental realizations of related systems.

Integral Lattice Models from Gauge Theory

These notes provide an introduction to recent work by Kevin Costello in which integrable lattice models of classical statistical mechanics in two dimensions are understood in terms of quantum gauge theory in four dimensions. This construction will be compared to the more familiar relationship between quantum knot invariants in three dimensions and Chern-Simons gauge theory. (Based on a Whittaker Colloquium at the University of Edinburgh and a lecture at Strings 2016 in Beijing.)

SLE Boundary Visits

We study the probabilities with which chordal Schramm–Loewner evolutions (SLE) visit small neighborhoods of boundary points. We find formulas for general chordal SLE boundary visiting probability amplitudes, also known as SLE boundary zig-zags or order refined SLE multi-point Green’s functions on the boundary. Remarkably, an exact answer can be found to this important SLE question for an arbitrarily large number of marked points. The main technique employed is a spin chain–Coulomb gas correspondence between tensor product representations of a quantum group and functions given by Dotsenko–Fateev type integrals. We show how to express these integral formulas in terms of regularized real integrals, and we discuss their numerical evaluation. The results are universal in the sense that apart from an overall multiplicative constant the same formula gives the amplitude for many different formulations of the SLE boundary visit problem. The formula also applies to renormalized boundary visit probabilities for interfaces in critical lattice models of statistical mechanics: we compare the results with numerical simulations of percolation, loop-erased random walk, and Fortuin–Kasteleyn random cluster models at Q = 2 and Q = 3, and find good agreement.

On Descriptions of All Translation Invariant p−adic Gibbs Measures for the Potts Model on The Cayley Tree of Order Three

Unlike the real number field, a set of p−adic Gibbs measures of p−adic lattice models of statistical mechanics has a complex structure in a sense that it is strongly tied up with a Diophantine problem over p−adic fields. Recently, all translation-invariant p−adic Gibbs measures of the p−adic Potts model on the Cayley tree of order two were described by means of roots of a certain quadratic equation over some domain of the p−adic field. In this paper, we consider the same problem on the Cayley tree of order three. In this case, we show that all translation-invariant p−adic Gibbs measures of the p−adic Potts model can be described in terms of roots of some cubic equation over $\mathbb {Z}_{p}\setminus \mathbb {Z}_{p}^{*}$ . In own its turn, we also provide a solvability criterion of a general cubic equation over $\mathbb {Z}_{p}\setminus \mathbb {Z}_{p}^{*}$ for p > 3.

Exact matrix treatment of statistical mechanical lattice model of adsorption induced gate opening in metal-organic frameworks

Here we present a statistical mechanical lattice model which is exactly solvable using a matrix method and allows treatment of adsorption induced gate opening structural transformations of metal-organic frameworks which are nanoporous materials with exceptional adsorption properties. Modelling of these structural changes presents a serious theoretical challenge when the solid and gas species are treated in an even handed way. This exactly solvable model complements other simulation based approaches. The methodology presented here highlights the competition between the potential for adsorption and the energy required for structural transition as a driving force for the features in the adsorption isotherms.

Some inequalities for theoretical spatial ecology

An important open problem in fully-structured spatial population dynamics, particularly those of competing plant communities, is a rigorous justification of the key assumption required for the pair-approximations of lattice models in statistical mechanics introduced to theoretical ecology by H. Matsuda, K. Sato and Y. Iwasa, among others. A similar assumption is made in the derivation of the spatially continuous moment equations introduced by B.M. Bolker and S. Pacala (1997, Theor. Popul. Biol., 52: 179-197). Towards this aim, upper bounds of the k-th central moment in the contact process of a single spatial dimension are precisely derived. The proof of this result explains, from an analytical perspective, why moment closure methodologies of spatial ecology can be so effective. doi:10.1017/S1446181113000266

Microscopic interpretation of sign reversal in the electrocaloric effect in a ferroelectric PbMg1/3Nb2/3O3-30PbTiO3 single crystal

With increasing temperature, PbMg1/3Nb2/3O3-30PbTiO3 (PMN-30PT) crystals change from pseudo-rhombohedral to tetragonal to cubic phases. In addition to the usual positive electrocaloric effect (ECE), a negative ECE, whose origin is uncertain, is observed. Here, these two types of the ECE contributions in PbMg1/3Nb2/3O3-30PbTiO3 crystals are modelled theoretically using a one dimensional statistical mechanical lattice model, which is solved by an exact matrix method. The quasi one-dimensional model reproduces the trends in the experimental behaviour and attributes the electrocaloric sign reversal to free energy changes induced by the electric field.

Exactly solvable lattice models with crossing symmetry

We show how to compute the exact partition function for lattice statistical–mechanical models whose Boltzmann weights obey a special ‘crossing’ symmetry. The crossing symmetry equates partition functions on different trivalent graphs, allowing a transformation to a graph where the partition function is easily computed. The simplest example is counting the number of nets without ends on the honeycomb lattice, including a weight per branching. Other examples include an Ising model on the Kagomé lattice with three-spin interactions, dimers on any graph of corner-sharing triangles, and non-crossing loops on the honeycomb lattice, where multiple loops on each edge are allowed. We give several methods for obtaining models with this crossing symmetry, one utilizing discrete groups and another anyon fusion rules. We also present results indicating that for models which deviate slightly from having crossing symmetry, a real-space decimation (renormalization-group-like) procedure restores the crossing symmetry.

Cylindric Versions of Specialised Macdonald Functions and a Deformed Verlinde Algebra

We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew diagram of two partitions, viewed as a subset of the two-dimensional integer lattice, by the period vector (n,-k). Imposing a periodicity condition one defines cylindric skew tableaux as a map from the periodically continued skew diagram into the integers. The resulting cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra, which is a particular quotient of the spherical Hecke algebra. We realise this Frobenius algebra as a commutative subalgebra in the endomorphisms over a Kirillov-Reshetikhin module of the quantum affine sl(n) algebra. Acting with special elements of this subalgebra, which are noncommutative analogues of Macdonald polynomials, on a highest weight vector, one obtains Lusztig's canonical basis. In the limit q=0, one recovers the sl(n) Verlinde algebra, i.e. the structure constants of the Frobenius algebra become the WZNW fusion coefficients which are known to be dimensions of moduli spaces of generalized theta-functions and multiplicities of tilting modules of quantum groups at roots of unity. Further motivation comes from exactly solvable lattice models in statistical mechanics: the cylindric Macdonald functions arise as partition functions of so-called vertex models obtained from solutions to the quantum Yang-Baxter equation. We show this by stating explicit bijections between cylindric tableaux and lattice configurations of non-intersecting paths. Using the algebraic Bethe ansatz the idempotents of the Frobenius algebra are computed.

Monte Carlo Simulation and Exact Statistical Mechanical Lattice Models as a Development Tool for Zeolite Multi-Component Adsorption Isotherm Derivation

Developing a suitable separation and purification procedure for multi-component gas mixtures is a ubiquitous problem. Very little is known experimentally about multi-component adsorption isotherms, yet these are crucially important for designing separation processes. Multi-component adsorption isotherms over a wide range of composition, pressure and temperature are virtually unknown. Adsorption onto zeolites is widely used in adsorptive separation processes due to the unique characteristics of its structural pores which may be engineered to permit adsorption to take place on the basis of molecular shape and size. In this article, we discuss the methodological development of rapidly obtaining multicomponent adsorption isotherms using the interplay between the essentially exact (in the statistical mechanical sense) lattice model and configurational bias Monte Carlo simulations. The latter technique has the known capability of being able to reproduce experimental isotherms reliably, but can be very time co...

Molecular simulations of H2 adsorption in metal-porphyrin frameworks: A potential new material evaluation

Path integral grand canonical Monte Carlo (PI-GCMC) simulations using standard force fields are carried out to calculate the adsorption of H2 in five metal-porphyrin frameworks (MPFs), a new class of metal organic framework (MOF)-type materials. These simulations are performed at 77 K and room temperature (300 K). The adsorption isotherms of H2 in IRMOF-1 and IRMOF-10 are also calculated as a comparison. All calculations indicate that all MPFs adsorbed a higher weight fraction of H2 than both IRMOF-1 and IRMOF-10, with one exception (MPF-2). The gravimetric hydrogen capacities are still well short of practical goals. The MPFs provide additional adsorption sites due to the porphyrin. A statistical mechanical lattice model predicts the adsorption well at room temperature. The prediction by this model showed that a weight fraction of hydrogen of 6 wt. % adsorbed in pores of the size found in IRMOF-1 at ambient temperature and modest pressures required a binding energy of about 17 kJ/mole, which is consistent with other findings.

Statistical mechanical lattice model of the dual-peak electrocaloric effect in ferroelectric relaxors and the role of pressure

Despite considerable effort, the microscopic origin of the electrocaloric (EC) effect in ferroelectric relaxors is still intensely discussed. Ferroelectric relaxors typically display a dual-peak EC effect, whose origin is uncertain. Here we present an exact statistical mechanical matrix treatment of a lattice model of polar nanoregions forming in a neutral background and use this approach to study the characteristics of the EC effect in ferroelectric relaxors under varying electric field and pressure. The dual peaks seen in the EC properties of ferroelectric relaxors are due to the formation and ordering of polar nanoregions. The model predicts significant enhancement of the EC temperature rise with pressure which may have some contribution to the giant EC effect.

Statistical–mechanical lattice models for protein–DNA binding in chromatin

Statistical–mechanical lattice models for protein–DNA binding are well established as amethod to describe complex ligand binding equilibria measured in vitro with purified DNAand protein components. Recently, a new field of applications has opened up for thisapproach since it has become possible to experimentally quantify genome-wide proteinoccupancies in relation to the DNA sequence. In particular, the organization of theeukaryotic genome by histone proteins into a nucleoprotein complex termed chromatinhas been recognized as a key parameter that controls the access of transcriptionfactors to the DNA sequence. New approaches have to be developed to derivestatistical–mechanical lattice descriptions of chromatin-associated protein–DNAinteractions. Here, we present the theoretical framework for lattice models ofhistone–DNA interactions in chromatin and investigate the (competitive) DNAbinding of other chromosomal proteins and transcription factors. The results have anumber of applications for quantitative models for the regulation of gene expression.