Vertex Model Research Articles

An Ising-type formulation of the six-vertex model

We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschütz-Jackson summation formula, originally discovered by J. F. Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models with only two-spin interactions.

Lattice local integrable regularization of the Sine-Gordon model

We study the local lattice integrable regularization of the Sine-Gordon model written down in terms of the lattice Bose-operators. We show that the local spin Hamiltonian obtained from the six-vertex model with alternating inhomogeneities in fact leads to the Sine-Gordon in the low-energy limit. We show that the Bethe Ansatz results for this model lead to the correct general relations for different critical exponents of the coupling constant.

Coloured stochastic vertex models and their spectral theory

Coloured stochastic vertex models and their spectral theory

On the Correspondence between Subshifts of Finite Type and Statistical Mechanics Models.

Several classical problems in symbolic dynamics concern the characterization of the simplex of measures of maximal entropy. For subshifts of finite type in higher dimensions, methods of statistical mechanics are ideal for dealing with these problems. R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics-a correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it and present some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way the possibilities of this correspondence.

Topologically-protected interior for three-dimensional confluent cellular collectives

Organoids are in vitro cellular collectives from which, for example, brain-like, or gut-like, or kidney-like structures emerge. To make quantitative predictions regarding the morphology and rheology of a cellular collective in its initial stages of development, we construct and study a three-dimensional vertex model. In such a model, the cells are represented as deformable polyhedrons with cells sharing faces such that there are no gaps between them, otherwise known as confluent. In a bulk model with periodic boundary conditions, we find a rigidity transition as a function of the target cell shape index ${s}_{0}$ with a critical value ${s}_{0}^{*}=5.39\ifmmode\pm\else\textpm\fi{}0.01$. For a confluent cellular collective with a finite boundary, and in the presence of lateral extensile and in-plane, radial extensile deformations, we find a significant boundary-bulk effect that is one-cell layer thick. More specifically, for lateral extensile deformations, the cells in the bulk are much less aligned with the direction of the lateral deformation than the cells at the boundary. For in-plane, radial deformations, the cells in the bulk exhibit much less reorientation perpendicular to the radial direction than the cells at the boundary. In other words, for both deformations, the bulk, interior cells are topologically protected from the deformations, at least over time scales much slower than the timescale for cellular rearrangements and up to reasonable amounts of strain. Our results provide an underlying mechanism for some observed cell shape patterning in organoids and in in vivo settings. Finally, we discuss the use of a cellular-based approach to designing organoids with new types of morphologies to study the intricate relationship between structure and function at the multicellular scale.

Classical vertex model dualities in a family of two-dimensional frustrated quantum antiferromagnets

We study a general class of easy-axis spin models on a lattice of corner sharing even-sided polygons with all-to-all interactions within a plaquette. The low energy description corresponds to a quantum dimer model on a dual lattice of even coordination number with a multi dimer constraint. At an appropriately constructed frustration-free Rokhsar-Kivelson (RK) point, the ground state wavefunction can be exactly mapped onto a classical vertex model on the dual lattice. When the dual lattice is bipartite, the vertex models are bonded and are self dual under Wegner's duality, with the self dual point corresponding to the RK point of the original multi-dimer model. We argue that the self dual point is a critical point based on known exact solutions to some of the vertex models. When the dual lattice is non-bipartite, the vertex model is arrowed, and we use numerical methods to argue that there is no phase transition as a function of the vertex weights. Motivated by these wavefunction dualities, we construct two other distinct families of frustration-free Hamiltonians whose ground states can be mapped onto these vertex models. Many of these RK Hamiltonians provably host $\mathbb{Z}_2$ topologically ordered phases.

Controlling the configuration space topology of mechanical structures.

Linkages are mechanical devices constructed from rigid bars and freely rotating joints studied both for their utility in engineering and as mathematical idealizations in a number of physical systems. Recently, there has been a resurgence of interest in designing linkages in the physics community due to the concurrent developments of mechanical metamaterials, topological mechanics, and the discovery of anomalous rigidity in fiber networks and vertex models. These developments raise a natural question: to what extent can the motion of a linkage or mechanical structure be designed? Here, we describe a method to design the topology of the configuration space of a linkage by first identifying the manifold of critical points, then perturbing around such critical configurations. Unlike other methods, our methods are tractable and provide a simple visual toolkit for mechanism design. We demonstrate our procedure by designing a mechanism to gate the propagation of a soliton in a Kane-Lubensky chain of interconnected rotors.

On the partition function of the Sp(4) integrable vertex model

In this paper we investigate certain fusion relations associated to an integrable vertex model on the square lattice which is invariant under Sp(4) symmetry. We establish a set of functional relations which include a transfer matrix inversion identity. The solution of these relations in the thermodynamic limit allows us to compute the partition function per site of the fundamental Sp(4) representation of the vertex model. As a byproduct we also obtain the partition function per site of a vertex model mixing the four and five dimensional representations of the Sp(4) symmetry.

Couple stresses and discrete potentials in the vertex model of cellular monolayers.

The vertex model is widely used to simulate the mechanical properties of confluent epithelia and other multicellular tissues. This inherently discrete framework allows a Cauchy stress to be attributed to each cell, and its symmetric component has been widely reported, at least for planar monolayers. Here, we consider the stress attributed to the neighbourhood of each tricellular junction, evaluating in particular its leading-order antisymmetric component and the associated couple stresses, which characterise the degree to which individual cells experience (and resist) in-plane bending deformations. We develop discrete potential theory for localised monolayers having disordered internal structure and use this to derive the analogues of Airy and Mindlin stress functions. These scalar potentials typically have broad-banded spectra, highlighting the contributions of small-scale defects and boundary layers to global stress patterns. An affine approximation attributes couple stresses to pressure differences between cells sharing a trijunction, but simulations indicate an additional role for non-affine deformations.

Generalised Onsager Algebra in Quantum Lattice Models

The Onsager algebra is one of the cornerstones of exactly solvable models in statistical mechanics. Starting from the generalised Clifford algebra, we demonstrate its relations to the graph Temperley--Lieb algebra, and a generalisation of the Onsager algebra. We present a series of quantum lattice models as representations of the generalised Clifford algebra, possessing the structure of a special type of the generalised Onsager algebra [1]. The integrability of those models is presented, analogous to the free fermionic eight-vertex model. We also mention further extensions of the models and physical properties related to the generalised Onsager algebras, hinting at a general framework that includes families of quantum lattice models possessing the structure of the generalised Onsager algebras.

A structural stiffness matrix-based computational mechanics method of epithelial monolayers

Epithelial monolayers are the simplest tissues in multicellular organisms, yet they play a vital role in many physiological processes, including development, growth, and wound healing. During these processes, the epithelia are usually subjected to mechanical cues, and involve multiple scales acting at molecular, cellular, and tissue levels. Computational techniques provide a useful and convenient means to investigate biological and mechanical behaviors of cell monolayers. Here, we propose a structural stiffness matrix-based computational method that can accurately and rapidly evaluate the mechanical properties of cell monolayers. By harmoniously calculating spatial movements of cellular positions and changes in cellular shapes over time, the present method holds higher computational efficiency than orthodox vertex model, especially the time ratio of these two methods exceeds an order of magnitude for large-scale cell monolayers. Furthermore, the structural features of cells, neglected in other methods, can be naturally integrated in the present approach. Then, this method is employed to examine the effects of cell division direction and molecular and cellular level changes on mechanical behaviors of cell monolayers. Our predictions are in good agreement with a broad range of experiments. The proposed method with high accuracy and efficacy holds great promise for application in simulating large-scale biological tissues.

Logarithmic Variance for the Height Function of Square-Ice

In this article, we prove that the height function associated with the square-ice model (i.e. the six-vertex model with $$a=b=c=1$$ on the square lattice), or, equivalently, of the uniform random homomorphisms from $$\mathbb {Z}^2$$ to $$\mathbb {Z}$$ , has logarithmic variance. This establishes a strong form of roughness of this height function.

On the Six-Vertex Model’s Free Energy

In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime Delta <1. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when a=b=1 and cge 1, and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.

Pulsations and flows in tissues as two collective dynamics with simple cellular rules

SummaryCollective motions of epithelial cells are essential for morphogenesis. Tissues elongate, contract, flow, and oscillate, thus sculpting embryos. These tissue level dynamics are known, but the physical mechanisms at the cellular level are unclear. Here, we demonstrate that a single epithelial monolayer of MDCK cells can exhibit two types of local tissue kinematics, pulsations and long range coherent flows, characterized by using quantitative live imaging. We report that these motions can be controlled with internal and external cues such as specific inhibitors and substrate friction modulation. We demonstrate the associated mechanisms with a unified vertex model. When cell velocity alignment and random diffusion of cell polarization are comparable, a pulsatile flow emerges whereas tissue undergoes long-range flows when velocity alignment dominates which is consistent with cytoskeletal dynamics measurements. We propose that environmental friction, acto-myosin distributions, and cell polarization kinetics are important in regulating dynamics of tissue morphogenesis.

Virtual spherical-shaped multicellular platform for simulating the morphogenetic processes of spider-like body axis formation.

Remodeling of multicellular architecture is a critical developmental process for shaping the axis of a bilaterally symmetric animal body and involves coordinated cell–cell interactions and cell rearrangement. In arthropods, the early embryonic process that leads to the segmented body axis varies at the cellular and molecular levels depending on the species. Developmental studies using insect and spider model species have provided specific examples of these diversified mechanisms that regulate axis formation and segmentation in arthropod embryos. However, there are few theoretical models for how diversity in the early embryonic process occurred during evolution, in part because of a limited computational infrastructure. We developed a virtual spherical-shaped multicellular platform to reproduce body axis-forming processes. Each virtual cell behaves according to the cell vertex model, with the computational program organized in a hierarchical order from cells and tissues to whole embryos. Using an initial set of two different mechanical states for cell differentiation and global directional signals that are linked to the planar polarity of each cell, the virtual cell assembly exhibited morphogenetic processes similar to those observed in spider embryos. We found that the development of an elongating body axis is achieved through implementation of an interactive cell polarity parameter associated with edge tension at the cell–cell adhesion interface, with no local control of the cell division rate and direction. We also showed that modifying the settings can cause variation in morphogenetic processes. This platform also can embed a gene network that generates waves of gene expression in a virtual dynamic multicellular field. This study provides a computational platform for testing the development and evolution of animal body patterns.

Instabilities and Geometry of Growing Tissues.

We present a covariant continuum formulation of a generalized two-dimensional vertexlike model of epithelial tissues which describes tissues with different underlying geometries, and allows for an analytical macroscopic description. Using a geometrical approach and out-of-equilibrium statistical mechanics, we calculate both mechanical and dynamical instabilities of a tissue, and their dependences on various variables, including activity, and cell-shape heterogeneity (disorder). We show how both plastic cellular rearrangements and the tissue elastic response depend on the existence of mechanical residual stresses at the cellular level. Even freely growing tissues may exhibit a growth instability depending on the intrinsic proliferation rate. Our main result is an explicit calculation of the cell pressure in a homeostatic state of a confined growing tissue. We show that the homeostatic pressure can be negative and depends on the existence of mechanical residual stresses. This geometric model allows us to sort out elastic and plastic effects in a growing, flowing, tissue.

Arctic curves of the 6V model with partial DWBC and double Aztec rectangles

Previous numerical studies have shown that in the disordered and anti-ferroelectric phases the six-vertex (6V) model with partial domain wall boundary conditions (DWBC) exhibits an arctic curve whose exact shape is unknown. The model is defined on a s × n square lattice (s ⩽ n). In this paper, we derive the analytic expression of the arctic curve, for a = b = 1 and (Δ = 0), while keeping the ratio s/n ∈ [0, 1] as a free parameter. The computation relies on the tangent method. We also consider domino tilings of double Aztec rectangles and show via the tangent method that, for particular parameters, the arctic curve is identical to that of the 6V model with partial DWBC. Our results are confirmed by extensive numerical simulations.

Anomalous elasticity of a cellular tissue vertex model.

Vertex models, such as those used to describe cellular tissue, have an energy controlled by deviations of each cell area and perimeter from target values. The constrained nonlinear relation between area and perimeter leads to new mechanical response. Here we provide a mean-field treatment of a highly simplified model: a uniform network of regular polygons with no topological rearrangements. Since all polygons deform in the same way, we only need to analyze the ground states and the response to deformations of a single polygon (cell). The model exhibits the known transition between a fluid/compatible state, where the cell can accommodate both target area and perimeter, and a rigid/incompatible state. We calculate and measure the mechanical resistance to various deformation protocols and discover that at the onset of rigidity, where a single zero-energy ground state exists, linear elasticity fails to describe the mechanical response to even infinitesimal deformations. In particular, we identify a breakdown of reciprocity expressed via different moduli for compressive and tensile loads, implying nonanalyticity of the energy functional. We give a pictorial representation in configuration space that reveals that the complex elastic response of the vertex model arises from the presence of two distinct sets of reference states (associated with target area and target perimeter). Our results on the critically compatible tissue provide a new route for the design of mechanical metamaterials that violate or extend classical elasticity.

Six-Vertex Model as a Grassmann Integral, One-Point Function, and the Arctic Ellipse

We formulate the six-vertex model with domain wall boundary conditions in terms of an integral over Grassmann variables. Relying on this formulation, we propose a method of calculation of correlation functions of the model for the case of the weights satisfying the free-fermion condition. We consider here in details the one-point correlation function describing the probability of a given state on an arbitrary edge of the lattice. We show that in the thermodynamic limit performed in such a way that the lattice is scaled to the square of unit side length, this function exhibits the “arctic ellipse” phenomenon, in agreement with the previous studies on random domino tilings of Aztec diamonds: it approaches its limiting values outside of an ellipse inscribed into this square, and takes continuously intermediate values inside the ellipse. We derive also the scaling properties of the one-point function in the vicinity of an arbitrary point of the arctic ellipse and in the vicinities of the points where the ellipse touches the boundary.

Stochastic symplectic ice

In this paper, we construct solvable ice models (six-vertex models) with stochastic weights and U-turn right boundary, which we term “stochastic symplectic ice”. The models consist of alternating rows of two types of vertices. The probabilistic interpretation of the models leads to novel interacting particle systems where particles alternately jump to the right and then to the left. Two colored versions of the models and related stochastic dynamics are also introduced. Using the Yang–Baxter equations, we establish functional equations and recursive relations for the partition functions of these models. In particular, the recursive relations satisfied by the partition function of one of the colored models are closely related to Demazure–Lusztig operators of type C.