Vertex Model Research Articles

Correlation functions of the six-vertex IRF model and its quantum spin chain

We consider the interaction-round-a-face version of the isotropic six-vertex model. The associated spin chain is made of two coupled Heisenberg spin chains with different boundary twists. The phase diagram of the model and the long distance correlations were studied in [Nucl. Phys. B, 995 (2023) 116333]. Here, we compute the short-distance correlation functions of the model in the ground state for finite system sizes via non-linear integral equations and in the thermodynamic limit. This was possible since the model satisfies the face version of the discrete quantum Knizhnik-Zamolodchikov (qKZ) equation. A suitable ansatz for the density matrix is proposed in the form of a direct sum of two Heisenberg density matrices, which allows us to obtain the discrete functional equation for the two-site function ω(λ1,λ2). Thanks to the known results on the factorization of correlation functions of the Heisenberg chain, we are able to compute the density matrix of the IRF model for up to four sites and its associated spin chain for up to three sites.

The theoretical study of pp¯→Λ¯Ση\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\bar{p}\\rightarrow \\bar{\\Lambda }\\Sigma \\eta $$\\end{document} reaction

We study the production of hyperon resonances in the pp¯→Λ¯Ση\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\bar{p}\\rightarrow \\bar{\\Lambda }\\Sigma \\eta $$\\end{document} reaction within an effective Lagrangian approach. The model includes the production of Σ(1750)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma (1750)$$\\end{document} and Λ(1670)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda (1670)$$\\end{document} in the intermediate state excited by the K and K∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^*$$\\end{document} meson exchanges between the initial proton and antiproton. Due to the large coupling of Σ(1750)Ση\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma (1750)\\Sigma \\eta $$\\end{document} vertex, Σ(1750)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma (1750)$$\\end{document} is found a significant contribution near the threshold in this reaction. We provide total and differential cross section predictions for the reaction and discuss the possible influence of Σ(1750)Ση\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma (1750)\\Sigma \\eta $$\\end{document} vertex coupling and model parameters, which will be useful in future experimental studies. This reaction can provide a platform for studying the features of Σ(1750)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma (1750)$$\\end{document} resonance, especially the coupling to Ση\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma \\eta $$\\end{document} channel.

Symmetric Functions from the Six-Vertex Model in Half-Space

Symmetric Functions from the Six-Vertex Model in Half-Space

Bayesian parameter inference for epithelial mechanics

Cell-based mechanical models, such as the Cell Vertex Model (CVM), have proven useful for studying the mechanical control of epithelial tissue dynamics. We recently developed a statistical method called image-based parameter inference for formulating CVM model functions and estimating their parameters from image data of epithelial tissues. In this study, we employed Bayesian statistics to improve the utility and flexibility of image-based parameter inference. Tests on synthetic data confirmed that both our non-hierarchical and hierarchical Bayesian models provide accurate estimates of model parameters. By applying this method to Drosophila wings, we demonstrated that the reliability of parameter estimation is closely linked to the mechanical anisotropies present in the tissue. Moreover, we revealed that the cortical elasticity term is dispensable for explaining force-shape correlations in vivo. We anticipate that the flexibility of the Bayesian statistical framework will facilitate the integration of various types of information, thereby contributing to the quantitative dissection of the mechanical control of tissue dynamics.

Finite-size spectrum of the staggered six-vertex model with antidiagonal boundary conditions

The finite-size spectrum of the critical staggered six-vertex model with antidiagonal boundary conditions is studied. Similar to the case of periodic boundary conditions, we identify three different phases. In two of those, the underlying conformal field theory can be identified to be related to the twisted U(1) Kac-Moody algebra. In contrast, the finite size scaling in the third regime, whose critical behaviour with the (quasi-)periodic BCs is related to the 2d black hole CFTs possessing a non-compact degree of freedom, is more subtle. Here with antidiagonal BCs imposed, the corrections to the scaling of the ground state grow logarithmically with the system size, while the energy gaps appear to close logarithmically. Moreover, we obtain an explicit formula for the Q-operator which is useful for numerical implementation.

Colored vertex models and Iwahori Whittaker functions

We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group G over a nonarchimedean local field F. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case G=GLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=\ extrm{GL}_r$$\\end{document} we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here ‘solvable’ means that the models have a family of Yang–Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang–Baxter equations come from a Drinfeld twist of the quantum group Uq(gl^(r|1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_q(\\widehat{\\mathfrak {gl}}(r|1))$$\\end{document}, which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials.

A two-dimensional vertex model for curvy cell-cell interfaces at the subcellular scale.

Cross-sections of cell shapes in a tissue monolayer typically resemble a tiling of convex polygons. Yet, examples exist where the polygons are not convex with curved cell-cell interfaces, as seen in the adaxial epidermis. To date, two-dimensional vertex models predicting the structure and mechanics of cell monolayers have been mostly limited to convex polygons. To overcome this limitation, we introduce a framework to study curvy cell-cell interfaces at the subcellular scale within vertex models by using a parametrized curve between vertices that is expanded in a Fourier series and whose coefficients represent additional degrees of freedom. This extension to non-convex polygons allows for cells with the same shape index, or dimensionless perimeter, to be, for example, either elongated or globular with lobes. In the presence of applied, anisotropic stresses, we find that local, subcellular curvature or buckling can be energetically more favourable than larger scale deformations involving groups of cells. Inspired by recent experiments, we also find that local, subcellular curvature at cell-cell interfaces emerges in a group of cells in response to the swelling of additional cells surrounding the group. Our framework, therefore, can account for a wider array of multicellular responses to constraints in the tissue environment.

Mechanochemical patterning and wave propagation in multicellular tubes

Multicellular tubes are fundamental tissues for transporting and distributing liquids and gases in living organisms. Although the molecular, cellular and mechanical aspects in tube formation have been addressed experimentally, how these factors are coupled to control tube patterning and dynamics at the tissue level remains incompletely understood. Here, we propose a three-dimensional (3D) vertex model that incorporates a mechanochemical feedback loop correlating cell deformation and actomyosin signaling pathway to probe the morphodynamics of multicellular tubes. We show that diverse patterns, including ring, helix, double helix, and labyrinth, are generated in tubes through pitchfork bifurcation, where spatial fluctuations of both biochemical signaling and 3D cell deformation are remarkably involved. The mechanochemical feedback loop enables cell oscillations via Hopf bifurcation, which induces the mechanical and chemical patterns to propagate successively as either traveling or pulse waves while their spatial configurations are maintained, strikingly distinct from the classical Turing instability. Our simulations, together with stability analysis of a minimal model, uncover the essential role of mechanochemical principles in sculpting biological tubes.

Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin - [formula omitted] chain

It is known that for the Heisenberg XXZ spin - 12 chain in the critical regime, the scaling limit of the vacuum Bethe roots yields an infinite set of numbers that coincide with the energy spectrum of the quantum mechanical 3D anharmonic oscillator. The discovery of this curious relation, among others, gave rise to the approach referred to as the ODE/IQFT (or ODE/IM) correspondence. Here we consider a multiparametric generalization of the Heisenberg spin chain, which is associated with the inhomogeneous six-vertex model. When quasi-periodic boundary conditions are imposed the lattice system may be explored within the Bethe Ansatz technique. We argue that for the critical spin chain, with a properly formulated scaling limit, the scaled Bethe roots for the ground state are described by second order differential equations, which are multi-parametric generalizations of the Schrödinger equation for the anharmonic oscillator.

Transition of the Simple Random Walk on the Ice Model Graph

The 6-vertex model holds significance in various mathematical and physical domains. The configurations of the 6-vertex model correspond to the paths in multigraphs. This article focuses on calculating the transition probability for the simple random walk on these multigraphs. An intriguing aspect of the findings is the utilization of continued fractions in the computation of the transition probability.

Determinant evaluations inspired by Di Francesco's determinant for twenty-vertex configurations

In his work on the twenty vertex model, Di Francesco (2021) found a determinant formula for the number of configurations in a specific such model, and he conjectured a closed form product formula for the evaluation of this determinant. We prove this conjecture here. Moreover, we actually generalize this determinant evaluation to a one-parameter family of determinant evaluations, and we present many more determinant evaluations of similar type — some proved, some left open as conjectures.

On the equivalence between n-state spin and vertex models on the square lattice

In this paper we investigate a correspondence among spin and vertex models with the same number of local states on the square lattice with toroidal boundary conditions. We argue that the partition functions of an arbitrary n-state spin model and of a certain specific n-state vertex model coincide for finite lattice sizes. The equivalent vertex model has n3 non-null Boltzmann weights and their relationship with the edge weights of the spin model is explicitly presented. In particular, the Ising model in a magnetic field is mapped to an eight-vertex model whose weights configurations combine both even and odd number of incoming and outcoming arrows at a vertex. We have studied the Yang-Baxter algebra for such mixed eight-vertex model when the weights are invariant under arrows reversing. We find that while the Lax operator lie on the same elliptic curve of the even eight-vertex model the respective R-matrix can not be presented in terms of the difference of two rapidities. We also argue that the spin-vertex equivalence may be used to imbed an integrable spin model in the realm of the quantum inverse scattering framework. As an example, we show how to determine the R-matrix of the 27-vertex model equivalent to a three-state spin model devised by Fateev and Zamolodchikov.

Vimentin promotes collective cell migration through collagen networks via increased matrix remodeling and spheroid fluidity.

The intermediate filament (IF) protein vimentin is associated with many diseases with phenotypes of enhanced cellular migration and aggressive invasion through the extracellular matrix (ECM) of tissues, but vimentin's role in in-vivo cell migration is still largely unclear. Vimentin is important for proper cellular adhesion and force generation, which are critical to cell migration; yet the vimentin cytoskeleton also hinders the ability of cells to squeeze through small pores in ECM, resisting migration. To identify the role of vimentin in collective cell migration, we generate spheroids of wide-type and vimentin-null mouse embryonic fibroblasts (mEFs) and embed them in a 3D collagen matrix. We find that loss of vimentin significantly impairs the ability of the spheroid to collectively expand through collagen networks and remodel the collagen network. Traction force analysis reveals that vimentin null spheroids exert less contractile force than their wild-type counterparts. In addition, spheroids made of mEFs with only vimentin unit length filaments (ULFs) exhibit similar behavior as vimentin-null spheroids, suggesting filamentous vimentin is required to promote 3D collective cell migration. We find the vimentin-mediated collective cell expansion is dependent on matrix metalloproteinase (MMP) degradation of the collagen matrix. Further, 3D vertex model simulation of spheroid and embedded ECM indicates that wild-type spheroids behave more fluid-like, enabling more active pulling and reconstructing the surrounding collagen network. Altogether, these results signify that VIF plays a critical role in enhancing migratory persistence in 3D matrix environments through MMP transportation and tissue fluidity.

Scaling limit of the staggered six-vertex model with $U_q\big(\mathfrak{sl}(2)\big)$ invariant boundary conditions

We study the scaling limit of a statistical system, which is a special case of the integrable inhomogeneous six-vertex model. It possesses U_q\big(\mathfrak{sl}(2)\big)Uq(𝔰𝔩(2)) invariance due to the choice of open boundary conditions imposed. An interesting feature of the lattice theory is that the spectrum of scaling dimensions contains a continuous component. By applying the ODE/IQFT correspondence and the method of the Baxter QQ operator the corresponding density of states is obtained. In addition, the partition function appearing in the scaling limit of the lattice model is computed, which may be of interest for the study of nonrational CFTs in the presence of boundaries. As a side result of the research, a simple formula for the matrix elements of the QQ operator for the general, integrable, inhomogeneous six-vertex model was discovered, that has not yet appeared in the literature. It is valid for a certain one parameter family of diagonal open boundary conditions in the sector with the zz-projection of the total spin operator being equal to zero.

Triangular solutions to the reflection equation for Uq(sln^)

We study solutions of the reflection equation related to the quantum affine algebra Uq(sln^) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of Uq(sln^) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained.

Local Mechanical Modulation-Driven Evagination in Invaginated Epithelia.

Local cells can actively create reverse bending (evagination) in invaginated epithelia, which plays a crucial role in the formation of elaborate organisms. However, the precise physical mechanism driving the evagination remains elusive. Here, we present a three-dimensional vertex model, incorporating the intrinsic cell polarity, to explore the complex morphogenesis induced by local mechanical modulations. We find that invaginated tissues can spontaneously generate local reverse bending due to the shift of the apicobasal polarity. Their exact shapes can be analytically determined by the local apicobasal differential tension and the internal stress. Our continuum theory exhibits three regions in a phase diagram controlled by these two parameters, showing curvature transitions from ordered to disordered states. Additionally, we delve into epithelial curvature transition induced by the nucleus repositioning, revealing its active contribution to the apicobasal force generation. The uncovered mechanical principles could potentially guide more studies on epithelial folding in diverse systems.

Morphological instability at topological defects in a three-dimensional vertex model for spherical epithelia

Epithelial monolayers are a central building block of complex organisms. Topological defects have emerged as important elements for single cell behavior in flat epithelia. Here we theoretically study such defects in a three-dimensional vertex model for spherical epithelia like cysts or intestinal organoids. We find that they lead to the same generic morphological instability to an icosahedral shape as it is known from spherical elastic shells like virus capsids, polymerized vesicles, or buckyballs. We derive analytical expressions for the effective stretching and bending moduli as a function of the parameters of the vertex model, in excellent agreement with computer simulations. These equations accurately predict both the buckling of a flat epithelial monolayer under uniaxial compression and the faceting transition around the topological defects in spherical epithelia. We further show that localized apico-basal tension asymmetries allow them to reduce the transition threshold to small system sizes. Published by the American Physical Society 2024

Evaluation of integrals for the emptiness formation probability in the square-ice model

We study the emptiness formation probability (EFP) in the six-vertex model with domain wall boundary conditions. We present a conjecture according to which at the ice point, i.e., when all the Boltzmann weights are equal, the known multiple integral representation (MIR) for the EFP can be given as a finite-size matrix determinant of Fredholm type. Our conjecture is based on the explicit evaluation of the MIR for particular values of geometric parameters and on two kinds of identities for the boundary correlation function. The obtained representation can be further written as the Fredholm determinant of some linear integral operator. We show that as the geometric parameters of the EFP are tuned to the vicinity of the arctic curve arising in the scaling limit, the conjectured determinant turns into the GUE Tracy–Widom distribution.

A distant descendant of the six-vertex model

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.

Automated machine learning model for fundus image classification by health-care professionals with no coding experience

To assess the feasibility of code-free deep learning (CFDL) platforms in the prediction of binary outcomes from fundus images in ophthalmology, evaluating two distinct online-based platforms (Google Vertex and Amazon Rekognition), and two distinct datasets. Two publicly available datasets, Messidor-2 and BRSET, were utilized for model development. The Messidor-2 consists of fundus photographs from diabetic patients and the BRSET is a multi-label dataset. The CFDL platforms were used to create deep learning models, with no preprocessing of the images, by a single ophthalmologist without coding expertise. The performance metrics employed to evaluate the models were F1 score, area under curve (AUC), precision and recall. The performance metrics for referable diabetic retinopathy and macular edema were above 0.9 for both tasks and CFDL. The Google Vertex models demonstrated superior performance compared to the Amazon models, with the BRSET dataset achieving the highest accuracy (AUC of 0.994). Multi-classification tasks using only BRSET achieved similar overall performance between platforms, achieving AUC of 0.994 for laterality, 0.942 for age grouping, 0.779 for genetic sex identification, 0.857 for optic, and 0.837 for normality with Google Vertex. The study demonstrates the feasibility of using automated machine learning platforms for predicting binary outcomes from fundus images in ophthalmology. It highlights the high accuracy achieved by the models in some tasks and the potential of CFDL as an entry-friendly platform for ophthalmologists to familiarize themselves with machine learning concepts.