Vertex Model Research Articles

The frustration-free fully packed loop model

We consider a quantum fully packed loop model on the square lattice with a frustration-free projector Hamiltonian and ring-exchange interactions acting on plaquettes. A boundary Hamiltonian is added to favor domain-wall boundary conditions and link ground state properties to the combinatorics and six-vertex model literature. We discuss how the boundary term fractures the Hilbert space into Krylov subspaces, and we prove that the Hamiltonian is ergodic within each subspace, leading to a series of energy-equidistant exact eigenstates in the lower end of the spectrum. Among them we systematically classify both finitely entangled eigenstates and product eigenstates. Using a recursion relation for enumerating half-plane configurations, we compute numerically the exact entanglement entropy of the ground state, confirming area law scaling. Finally, the spectrum is shown to be gapless in the thermodynamic limit with a trial state constructed by adding a twist to the ground state superposition.

Emergence of Spatial Scales and Macroscopic Tissue Dynamics in Active Epithelial Monolayers

Migrating cells in tissues are often known to exhibit collective swirling movements. In this paper, we develop an active vertex model with polarity dynamics based on contact inhibition of locomotion (CIL). We show that under this dynamics, the cells form steady-state vortices in velocity, polarity, and cell stress with length scales that depend on polarity alignment rate (ζ), self-motility (v0), and cell-cell bond tension (λ). When the ratio λ/v0 becomes larger, the tissue reaches a near jamming state because of the inability of the cells to exchange their neighbors, and the length scale associated with tissue kinematics increases. A deeper examination of this jammed state provides insights into the mechanism of sustained swirl formation under CIL rule that is governed by the feedback between cell polarities and deformations. To gain additional understanding of how active forcing governed by CIL dynamics leads to large-scale tissue dynamics, we systematically coarse-grain cell stress, polarity, and motility and show that the tissue remains polar even on larger length scales. Overall, we explore the origin of swirling patterns during collective cell migration and obtain a connection between cell-level dynamics and large-scale cellular flow patterns observed in epithelial monolayers.

On the partition function of the Sp(2n) integrable vertex model

We study the partition function per site of the integrable vertex model on the square lattice. We establish a set of transfer matrix fusion relations for this model. The solution of these functional relations in the thermodynamic limit allows us to compute the partition function per site of the fundamental representation of the vertex model. In addition, we also obtain the partition function of vertex models mixing the fundamental with other representations.

Residual entropy of a two-dimensional Ising model with crossing and four-spin interactions

We study the residual entropy of a two-dimensional Ising model with crossing and four-spin interactions, both in the case of a zero magnetic field and in an imaginary magnetic field iπ/2kBT. The spin configurations of this Ising model can be mapped into the hydrogen configurations of square ice with the defined standard direction of the hydrogen bonds. Making use of the equivalence of this Ising system with the exactly solved eight-vertex model and taking the low temperature limit, we obtain the residual entropy. Two soluble cases in the zero field and one soluble case in the imaginary field are examined. In the case that the free-fermion condition holds in zero field, we find that the ground states in the low temperature limit include the configurations disobeying the ice rules. In another case in zero field where the four-spin interactions are −∞ and another case in imaginary field where the four-spin interactions are 0, the residual entropy exactly agrees with the result of square ice determined by Lieb in 1967. In the solutions to the latter two cases, we have shown alternative approaches to the residual entropy problem of square ice.

The spectrum properties of an integrable G2 invariant vertex model

This paper is concerned with the study of properties of the exact solution of the fundamental integrable G2 vertex model. The model R-matrix and respective spin chain are presented in terms of the basis generators of the G2 Lie algebra. This formulation permits us to relate the number of the Bethe roots of the respective Bethe equations with the eigenvalues of the U(1) conserved charges from the Cartan subalgebra of G2. The Bethe equations are solved by a peculiar string structure which combines complex three-strings with real roots allowing us to determine the bulk properties in the thermodynamic limit. We argue that G2 spin chain is gapless but the low-lying excitations have two different speeds of sound and the underlying continuum limit is therefore not strictly Lorentz invariant. We have investigated the finite-size corrections to the ground state energy and proposed that the critical properties of the system should be governed by the product of two c=1 conformal field theories. By combining numerical and analytical methods we have computed the bulk free-energy of the G2 vertex model. We found that there are three regimes in the spectral parameter in which the free-energy is limited and continuous. There exists however at least two sharp corner points in which the bulk-free energy is not differentiable.

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder: II. Rotated lattice

This work continues the study started in Povolotsky (2021 J. Phys. A: Math. Theor. 54 22LT01), where the exact densities of loops in the O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions were obtained. These densities are also equal to the densities of critical percolation clusters on the 45∘ rotated square lattice rolled into a cylinder. Here, we extend those results to the square lattice with a tilt. This in particular allows us to obtain the densities of critical percolation clusters on the cylinder of the square lattice of standard orientation extensively studied before. We obtain exact densities of contractible and non-contractible loops or equivalently the densities of critical percolation clusters, which do not and do wrap around the cylinder, respectively. The solution uses the mapping of O(1) dense loop model to the six-vertex model in the Razumov–Stroganov point, while the effective tilt is introduced via the inhomogeneous transfer matrix proposed by Fujimoto. The further solution is based on the Bethe ansatz and Fridkin–Stroganov–Zagier’s solution of Baxter’s T–Q equation. The results are represented in terms of the solution of two explicit systems of linear algebraic equations, which can be performed either analytically for small circumferences of the cylinder or numerically for larger ones. We present exact rational values of the densities on the cylinders of small circumferences and several lattice orientations and use the results of high precision numerical calculations to study the finite-size corrections to the densities, in particular their dependence on the tilt of the lattice.

Refined Littlewood identity for spin Hall–Littlewood symmetric rational functions

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions F λ are multiparameter deformations of the classical Hall–Littlewood symmetric polynomials and can be viewed as partition functions in 𝔰𝔩(2) higher spin six vertex models.

Vertex models for Canonical Grothendieck polynomials and their duals

We study exactly solvable lattice models associated to canonical Grothendieck polynomials and their duals. We derive inversion relations and Cauchy identities.

Linear viscoelastic response of the vertex model with internal and external dissipation: Normal modes analysis

We use the normal mode formalism to study the shear rheology of the vertex model for epithelial tissue mechanics in the overdamped linear response regime. We consider systems with external (e.g., cell-substrate) and internal (e.g., cell-cell) dissipation mechanisms, and derive expressions for stresses on cells due to mechanical and dissipative forces. The semi-analytical method developed here is, however, general and can be directly applied to study the linear response of a broad class of soft matter systems with internal and external dissipation. It involves normal mode decomposition to calculate linear loss and storage moduli of the system. Specifically, displacements along each normal mode produce stresses due to elastic deformation and internal dissipation, which are in force balance with loads due to external dissipation. Each normal mode responds with a characteristic relaxation timescale, and its rheological behavior can be described as a combination of a standard linear solid element due to elastic stresses and a Jeffreys model element due to the internal dissipative stresses. The total response of the system is then fully determined by connecting in parallel all the viscoelastic elements corresponding to individual normal modes. This allows full characterization of the potentially complex linear rheological response of the system at all driving frequencies and identification of collective excitations. We show that internal and external dissipation mechanisms lead to qualitatively different rheological behaviors due to the presence or absence of Jeffreys elements, which is particularly pronounced at high driving frequencies. Our findings, therefore, underscore the importance of microscopic dissipation mechanisms in understanding the rheological behavior of soft materials and tissues, in particular.

MorphoSim: an efficient and scalable phase-field framework for accurately simulating multicellular morphologies

The phase field model can accurately simulate the evolution of microstructures with complex morphologies, and it has been widely used for cell modeling in the last two decades. However, compared to other cellular models such as the coarse-grained model and the vertex model, its high computational cost caused by three-dimensional spatial discretization hampered its application and scalability, especially for multicellular organisms. Recently, we built a phase field model coupled with in vivo imaging data to accurately reconstruct the embryonic morphogenesis of Caenorhabditis elegans from 1- to 8-cell stages. In this work, we propose an improved phase field model by using the stabilized numerical scheme and modified volume constriction. Then we present a scalable phase-field framework, MorphoSim, which is 100 times more efficient than the previous one and can simulate over 100 mechanically interacting cells. Finally, we demonstrate how MorphoSim can be successfully applied to reproduce the assembly, self-repairing, and dissociation of a synthetic artificial multicellular system - the synNotch system.

The six-vertex model on random planar maps revisited

We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In particular, in two special cases of combinatorial interest we rederive the formulae discovered by Bousquet-Mélou and the first author.

Structure and Rheology in Vertex Models under Cell-Shape-Dependent Active Stresses.

Biological cells can actively tune their intracellular architecture according to their overall shape. Here we explore the rheological implication of such coupling in a minimal model of a dense cellular material where each cell exerts an active mechanical stress along its axis of elongation. Increasing the active stress amplitude leads to several transitions. An initially hexagonal crystal motif is first destabilized into a solid with anisotropic cells whose shear modulus eventually vanishes at a first critical activity. Increasing activity beyond this first critical value, we find a re-entrant transition to a regime with finite hexatic order and finite shear modulus, in which cells arrange according to a rhombile pattern with periodically arranged rosette structures. The shear modulus vanishes again at a third threshold beyond which spontaneous tissue flows and topological defects of the nematic cell shape field arise. Flow and stress fields around the defects agree with active nematic theory, with either contractile or extensile signs, as also observed in several epithelial tissue experiments.

Shift-invariance of the colored TASEP and finishing times of the oriented swap process

We prove a new shift-invariance property of the colored TASEP. From the shift-invariance of the colored stochastic six-vertex model (proved in Borodin-Gorin-Wheeler or Galashin), one can get a shift-invariance property of the colored TASEP at one time, and our result generalizes this to multiple times. Our proof takes the single-time shift-invariance as an input, and uses analyticity of the probability functions and induction arguments. We apply our shift-invariance to prove a distributional identity between the finishing times of the oriented swap process and the point-to-line passage times in exponential last-passage percolation, which is conjectured by Bisi-Cunden-Gibbons-Romik and Bufetov-Gorin-Romik, and is also equivalent to a purely combinatorial identity related to the Edelman-Greene correspondence. With known results from last-passage percolation, we also get new asymptotic results on the colored TASEP and the finishing times of the oriented swap process.

Two-Point Convergence of the Stochastic Six-Vertex Model to the Airy Process

In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time T, it is known that the one-point height function fluctuations are of order $$T^{1/3}$$ and governed by the Tracy–Widom distribution. We prove that the two-point distribution of the height function, rescaled horizontally by $$T^{2/3}$$ and vertically by $$T^{1/3}$$ , converges to the two-point distribution of the Airy process. The starting point of this result is a recent connection discovered by Borodin–Bufetov–Wheeler between the stochastic six-vertex model and the ascending Hall–Littlewood process (a certain measure on plane partitions). Using the Macdonald difference operators, we obtain formulas for two-point observables for the ascending Hall–Littlewood process, which for the six-vertex model give access to the joint cumulative distribution function for its height function. A careful asymptotic analysis of these observables gives the two-point convergence result under certain restrictions on the parameters of the model.

Integrable boundary conditions for staggered vertex models

Yang–Baxter integrable vertex models with a generic -staggering can be expressed in terms of composite -matrices given in terms of the elementary R-matrices. Similarly, integrable open boundary conditions can be constructed through generalized reflection algebras based on these objects and their representations in terms of composite boundary matrices . We show that only two types of staggering yield a local Hamiltonian with integrable open boundary conditions in this approach. The staggering in the underlying model allows for a second hierarchy of commuting integrals of motion (in addition to the one including the Hamiltonian obtained from the usual transfer matrix), starting with the so-called quasi momentum operator. In this paper, we show that this quasi momentum operator can be obtained together with the Hamiltonian for both periodic and open models in a unified way from enlarged Yang–Baxter or reflection algebras in the composite picture. For the special case of the staggered six-vertex model, this allows constructing an integrable spectral flow between the two local cases.

Finite elasticity of the vertex model and its role in rigidity of curved cellular tissues.

Using a mean field approach and simulations, we study the non-linear mechanical response of the vertex model (VM) of biological tissue to compression and dilation. The VM is known to exhibit a transition between solid and fluid-like, or floppy, states driven by geometric incompatibility. Target perimeter and area set a target shape which may not be geometrically achievable, thereby engendering frustration. Previously, an asymmetry in the linear elastic response was identified at the rigidity transition between compression and dilation. Here we show that the asymmetry extends away from the transition point for finite strains. Under finite compression, an initially solid VM can completely relax perimeter tension, resulting in a drop discontinuity in the mechanical response. Conversely, an initially floppy VM under dilation can rigidify and have a higher response. These observations imply that re-scaling of cell area shifts the transition between rigid and floppy states. Based on this insight, we calculate the re-scaling of cell area engendered by intrinsic curvature and write a prediction for the rigidity transition in the presence of curvature. The shift of the rigidity transition in the presence of curvature for the VM provides a new metric for predicting tissue rigidity from image data of curved tissues in a manner analogous to the flat case.

Strong law of large numbers for the stochastic six vertex model

Strong law of large numbers for the stochastic six vertex model

The role of non-affine deformations in the elastic behavior of the cellular vertex model.

The vertex model of epithelia describes the apical surface of a tissue as a tiling of polygonal cells, with a mechanical energy governed by deviations in cell shape from preferred, or target, area, A0, and perimeter, P0. The model exhibits a rigidity transition driven by geometric incompatibility as tuned by the target shape index, . For with p*(6) the perimeter of a regular hexagon of unit area, a cell can simultaneously attain both the preferred area and preferred perimeter. As a result, the tissue is in a mechanically soft compatible state, with zero shear and Young's moduli. For p0 < p*(6), it is geometrically impossible for any cell to realize the preferred area and perimeter simultaneously, and the tissue is in an incompatible rigid solid state. Using a mean-field approach, we present a complete analytical calculation of the linear elastic moduli of an ordered vertex model. We analyze a relaxation step that includes non-affine deformations, leading to a softer response than previously reported. The origin of the vanishing shear and Young's moduli in the compatible state is the presence of zero-energy deformations of cell shape. The bulk modulus exhibits a jump discontinuity at the transition and can be lower in the rigid state than in the fluid-like state. The Poisson's ratio can become negative which lowers the bulk and Young's moduli. Our work provides a unified treatment of linear elasticity for the vertex model and demonstrates that this linear response is protocol-dependent.

Continuum description of confluent tissues with spatial heterogeneous activity.

A continuum description is built to characterize the stationary and transient deformations of confluent tissues subject to heterogeneous activities. By defining a coarse-grained texture matrix field to represent the shape and size of cells, we derive the coarse-grained stress tensor for the vertex model. Activity in the tissue takes the form of inhomogeneous apical contractions, which can be modeled as reductions of the vertex model reference areas or perimeters representing activity in the medial and perimeter regions of the cells, respectively. For medial activity, the extra stress is just an isotropic pressure, while for perimeter activity, it also has a deviatoric component, which is aligned with the texture matrix. The predictions of the continuum description are compared with the average spatiotemporal deformations obtained in simulations of the vertex model subject to localized apical contractions, showing an excellent agreement, even if the active patch is as small as one cell. The fluctuations around the average are more prominent when the activity is in the medial region due to the lack of negative active shape feedback, which, coupled with the confluent property, increases cellular shape and size variations.

Irreversible Markov dynamics and hydrodynamics for KPZ states in the stochastic six vertex model

Irreversible Markov dynamics and hydrodynamics for KPZ states in the stochastic six vertex model