Asymmetric Six-vertex Model Research Articles

Cylindric Hecke Characters and Gromov\u2013Witten Invariants via the Asymmetric Six-Vertex Model

We construct a family of infinite-dimensional positive sub-coalgebras within the Grothendieck ring of Hecke algebras, when viewed as a Hopf algebra with respect to the induction and restriction functor. These sub-coalgebras have as structure constants the 3-point genus zero Gromov–Witten invariants of Grassmannians and are spanned by what we call cylindric Hecke characters, a particular set of virtual characters for whose computation we give several explicit combinatorial formulae. One of these expressions is a generalisation of Ram’s formula for irreducible Hecke characters and uses cylindric broken rim hook tableaux. We show that the latter are in bijection with so-called ‘ice configurations’ on a cylindrical square lattice, which define the asymmetric six-vertex model in statistical mechanics. A key ingredient of our construction is an extension of the boson-fermion correspondence to Hecke algebras and employing the latter we find new expressions for Jing’s vertex operators of Hall–Littlewood functions in terms of the six-vertex transfer matrices on the infinite planar lattice.

Yang–Baxter algebras, convolution algebras, and Grassmannians

This paper surveys a new actively developing direction in contemporary mathematics which connects quantum integrable models with the Schubert calculus for quiver varieties: there is a purely geometric construction of solutions to the Yang–Baxter equation and their associated Yang–Baxter algebras which play a central role in quantum integrable systems and exactly solvable (integrable) lattice models in statistical physics. A simple but explicit example is given using the classical geometry of Grassmannians in order to explain some of the main ideas. The degenerate five-vertex limit of the asymmetric six-vertex model is considered, and its associated Yang–Baxter algebra is identified with a convolution algebra arising from the equivariant Schubert calculus of Grassmannians. It is also shown how our methods can be used to construct quotients of the universal enveloping algebra of the current algebra (so-called Schur-type algebras) acting on the tensor product of copies of its evaluation representation . Finally, our construction is connected with the cohomological Hall algebra for the -quiver. Bibliography: 125 titles.

A new exponent in the equilibrium shape of crystal surfaces**This paper is dedicated in fond remembrance of our PhD advisor Vladimir Rittenberg.

A new exponent characterizing the rounding of crystal facets is found by mapping a crystal surface onto the asymmetric six-vertex model (i.e. with external fields h and ) and using the Bethe ansatz to obtain appropriate expansions of the free energy close to criticality. Leading order exponents in , are determined along the whole phase boundary and in an arbitrary direction. A possible experimental verification of this result is discussed.

Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles

We discuss the correspondence between models solved by Bethe ansatz and classical integrable systems of Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric 6-vertex model parametrized by trigonometric (hyperbolic) functions.

Quantum integrability and generalised quantum Schubert calculus

We introduce and study a new mathematical structure in the generalised (quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six-vertex model. Our approach offers a new perspective on already established and well-studied special cases, for example equivariant K-theory, and in addition allows us to formulate a conjecture on the so-far unknown case of quantum equivariant K-theory.

Stochastic six-vertex model

We study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line. We show that the random height function of the model converges to an explicit deterministic limit shape as the mesh size tends to 0. We further prove that the one-point fluctuations around the limit shape are asymptotically governed by the GUE Tracy-Widom distribution. We also explain an equivalent formulation of our model as an interacting particle system, which can be viewed as a discrete time generalization of ASEP started from the step initial condition. Our results confirm an earlier prediction of Gwa and Spohn (1992) that this system belongs to the KPZ universality class.

General scalar products in the arbitrary six-vertex model

In this work we use the algebraic Bethe ansatz to derive the general scalar product inthe asymmetric six-vertex model for generic Boltzmann weights. We performedthis calculation using only the unitarity property, the Yang–Baxter algebra andthe Yang–Baxter equation. We have derived a recurrence relation for the scalarproduct. The solution of this relation was written in terms of the domain wallpartition functions. In turn, these partition functions were also obtained for genericBoltzmann weights, which provided us with an explicit expression for the general scalarproduct.

Universal properties in the free energy of the asymmetricN-state vertex model

Electric-field-dependent free energy of the exactly solvable asymmetric N-state vertex model (i.e. in an arbitrary vertical and an arbitrary horizontal electric field v and h) in the low-temperature antiferroelectric phase is obtained in the form of the expansion in terms of small but nonzero polarization p. Exact mapping onto a microscopic surface model (solid-on-solid (SOS) model) is employed to study the vicinal-surface free energy below the roughening temperature TR which determines the equilibrium crystal shape (ECS) near the facet edge of a crystal. The obtained expansion of the free energy is of the well-established Gruber-Mullins-Pokrovsky-Talapov (GMPT) type: f(p,h) = f(0,h) + a(h)p + b(h)p3 + O(p4). It is found that the coefficients a(h) and b(h) are identical with those of the asymmetric six-vertex model. Based on this expansion, in cooperation with the Andreev construction of the ECS, universal properties are verified along the whole facet contour. First, directly from the GMPT-type expansion, the critical exponents governing the rounding of the crystal facet are obtained; the exponent is (3/2) in all direction except the tangential one in which case it is 3. Second, the universal relation between a(h) and b(h) is verified, leading to the universal Gaussian curvature jump at the facet edge and a universal relation between the critical amplitudes of the ECS profiles of the vicinal surface.

Symmetry Relations for the Six-Vertex Model

The exact solution of the asymmetric six-vertex model is cast in an algebraically simple form in which the extraction of physical quantities is transparent. This is used to derive a symmetry relation corresponding to the exchange of spatial x and y axes. As an application we study the field-induced phase transition of a two-dimensional analogue of spin ice, a frustrated Ising magnet.

Direction-Dependent Free Energy Singularity of the Antiferroelectric Asymmetric Six-Vertex Model

The transition from the ordered commensurate phase to the incommensurate gaussian phase of the antiferroelectric asymmetric six-vertex model is investigated by keeping the temperature constant below the roughening point and varying the external fields $(h,v)$. In the $(h,v)$ plane, the phase boundary is approached along straight lines $\delta v=k \delta h$, where $(\delta h,\delta v)$ measures the displacement from the phase boundary. It is found that the free energy singularity displays the exponent 3/2 typical of the Pokrovski-Talapov transition $\delta f \sim const (\delta h)^{3/2}$ for any direction other than the tangential one. In the latter case $\delta f$ shows a discontinuity in the third derivative.

Lax pair formulation for one-dimensional Hubbard open chain with chemical potential

The Lax pair for the one-dimensional Hubbard spin chain with open boundary conditions (BC) is explicitly constructed and two different kinds of boundary K-matrices compatible with the integrability of the model are obtained. Our construction provides an alternative and direct demonstration for the quantum integrability of the model and it is found that the model is related to a class of commuting transfer matrices of an equivalent coupled asymmetric six-vertex model with open BC. Our results show that the chemical potential term is indeed nontrivial for the underlying algebraic structure of the model.

The free energy singularity of the asymmetric six-vertex model and the excitations of the asymmetric XXZ chain

We consider the asymmetric six-vertex model, widely used to describe the equilibrium shape of crystals, and the relevant asymmetric XXZ chain. By means of the Bethe-ansatz solution we determine the free energy singularity, as function of the external field, at two special points on the phase boundary. We confirm the exponent 3 2 (checked experimentally, as the antiferroelectric ordered phase is reached from the incommensurate phase normally to this boundary, and we determine a new singularity along the tangential direction. Both singularities describe the rounding off of the crystal near a facet. At this point the hole excitations of the spin chain show dispersion relations ΔE ∼ (ΔP) 1 2 at small momenta, leading to a finite-size scaling ΔE ∼ N − 1 2 for the low-lying excited states, N being the chain size. We discuss the nature of the phase transition and the behavior of arrow-arrow correlation lengths in the ordered phase.

Finite-size scaling and the toroidal partition function of the critical asymmetric six-vertex model.

Finite-size corrections to the energy levels of the asymmetric six-vertex model transfer matrix are considered using the Bethe ansatz solution for the critical region. The non-universal complex anisotropy factor is related to the bulk susceptibilities. The universal Gaussian coupling constant $g$ is also related to the bulk susceptibilities as $g=2H^{-1/2}/\pi$, $H$ being the Hessian of the bulk free energy surface viewed as a function of the two fields. The modular covariant toroidal partition function is derived in the form of the modified Coulombic partition function which embodies the effect of incommensurability through two mismatch parameters. The effect of twisted boundary conditions is also considered.

Curvature Jump at the Facet Edge of a Crystal for Arbitrary Surface Orientation

The equilibrium crystal shape (ECS) for the body-centered solid-on-solid (BCSOS) model in the low-temperature phase is studied. Based on the equivalence between the BCSOS model and the asymmetric six-vertex model, the surface free energy f (ρ) (ρ: step density) is calculated for general mean surface tilt angle. In the expansion, f (ρ)= f (0)+α ρ+βρ 3 + O (ρ 4 ), the universal relation between a and b is verified. Accordingly, the Gaussian curvature of the ECS is shown to assume a universal finite jump at any position of the facet edge of the crystal.

Corner transfer matrix of an asymmetric six-vertex model

We consider a free Fermion six-vertex model with asymmetrically chosen vertex weights from the point of view of its corner transfer matrix. The generator of the corner transfer matrix then corresponds to a quantum XY spin chain Hamiltonian in an inhomogeneous magnetic field. The diagonalisation of the Hamiltonian employs orthogonal polynomials. The eigenenergy spectrum is related to the zeros of the asymptotic expansion of these polynomials. Lastly we identify a critical region in the phase diagram of the model which is spanned by the magnetic field and the parameter describing the strength of the inhomogeneity.

Equilibrium shape of bcc crystals: Thermal evolution of the facets.

The equivalence between the body-centered solid-on-solid model and the asymmetric six-vertex model is used to study the surface free energy and the equilibrium shape of bcc crystals in the solid-on-solid approximation. The thermal evolution of the (001)- and (011)-type facets is investigated analytically and illustrated for different values of the anisotropy of the next-nearest-neighbor couplings. It is shown that the (001) facet undergoes a roughening transition of the Kosterlitz-Thouless universality class and that the asymptotic regime below the critical point T=${\mathit{T}}_{\mathit{R}}$ is large. The occurrence of a universal jump in surface stiffness at the point where this facet vanishes at ${\mathit{T}}_{\mathit{R}}$ is reconfirmed.

The conical point in the ferroelectric six-vertex model

We examine the last unexplored regime of the asymmetric six-vertex model: the low-temperature phase of the so-called ferroelectric model. The original publication of the exact solution, by Sutherland, Yang, and Yang, and various derivations and reviews published afterwards, do not contain many details about this regime. We study the exact solution for this model, by numerical and analytical methods. In particular, we examine the behavior of the model in the vicinity of an unusual coexistence point that we call the ``conical'' point. This point corresponds to additional singularities in the free energy that were not discussed in the original solution. We show analytically that in this point many polarizations coexist, and that unusual scaling properties hold in its vicinity.

Phase diagram of the five-vertex model.

Within the ice-type models, the solution of the five-vertex model is obtained with the use of the Bethe ansatz. Since the allowed number of vertex types is odd, the arrow-reversal symmetry of the system is broken by construction. Due to this, the exact solution obtained and the phase diagram are very different from those of the symmetric six-vertex model. A connection to the asymmetric six-vertex model (of which the five-vertex model is an extreme case) is made. The different regions of the phase diagram are described and the transitions between them are analyzed. Several aspects of the phase diagram are unusual, i.e., the ordered phases (both ferroelectric and antiferroelectric) are frozen-in phases and the disordered phase is replaced by a ferrielectric phase. In the free-fermion case, the known results of the modified KDP model are recovered.

The asymmetric six-vertex model

The exact solution of the asymmetric six-vertex model, published nearly without derivation by Sutherlandet al. in 1967, is rederived in detail. The transfer matrix method and the Bethe Ansatz solution for the free energy (which can be calculated from an integral equation) are discussed. For some special cases (zero or maximal polarization) the integral equation can be solved exactly. In addition, an asymptotic analysis, valid for small but nonzero polarization, is carried out. The analytical properties of the results and their relevance for the BCSOS model are discussed.

Finite-size scaling amplitudes in a random tiling model

We study finite-size corrections to the free energy in a two-dimensional random tiling model. This model is equivalent to an asymmetric six-vertex model and exhibits commensurate-incommensurate phase transitions of the Pokrovsky-Talapov type. We calculate finite-size scaling amplitudes of the free energy inside the incommensurate phase, analytically and numerically, by employing a transfer matrix method. Both periodic and free boundary conditions are considered. These amplitudes are consistent with predictions of the theory of conformal invariance with the conformal charge c = 1.