Five-vertex Model Research Articles

Derivation of the matrix product ansatz for the Heisenberg chain from the algebraic Bethe ansatz

We derive a matrix product representation of the Bethe ansatz state for the XXX and XXZ spin- Heisenberg chains using the algebraic Bethe ansatz. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on , the tensor product of auxiliary spaces. By changing the basis in , we derive explicit finite-dimensional representations for the matrices. These matrices are the same as those appearing in the recently proposed matrix product ansatz by Alcaraz and Lazo (2006 J. Phys. A: Math. Gen. 39 11335) apart from normalization factors. We also discuss the close relation between the matrix product representation of the Bethe eigenstates and the six-vertex model with domain wall boundary conditions (Korepin 1982 Commun. Math. Phys. 86 391) and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.

Five-vertex model with fixed boundary conditions

The exactly solvable five-vertex model on a square lattice with fixed boundary conditions is considered. Application of the algebraic Bethe ansatz makes it possible to express the partition function and the boundary correlation functions of the nonhomogeneous model in the determinantal form. The relationship established between the homogeneous model and plane partitions helps to calculate its partition function.

The five-vertex model and boxed plane partitions

Boxed plane partitions are considered in terms of the five-vertex model on a finite lattice with fired boundary conditions. Assuming that all weights of the model have the same value, the one-point correlation function. describing the probability of having a given state on an arbitrary horizontal edge of the lattice is calculated. This is equivalent to the enumeration of boxed plane partitions that correspond to rhombus tilings of a hexagon with one fired rhombus of a particular type. The solution of the problem is given for the case of a boar of generic size. Bibliography: 24 titles.

Bethe Ansatz in the Bernoulli matching model of random sequence alignment

For the Bernoulli matching model of the sequence alignment problem we apply the Bethe Ansatz technique via an exact mapping to the five-vertex model on a square lattice. Considering the terracelike representation of the sequence alignment problem, we reproduce by the Bethe Ansatz the results for the averaged length of the longest common subsequence in the Bernoulli approximation. In addition, we compute the average number of nucleation centers of the terraces.

Exactly solvable interacting spin-ice vertex model

A special family of solvable five-vertex model is introduced on a square lattice. In addition to the usual nearest-neighbor interactions, the vertices defining the model also interact along one of the diagonals of the lattice. This family of models includes in a special limit the standard six-vertex model. The exact solution of these models is an application of the matrix product ansatz introduced recently and applied successfully in the solution of quantum chains. The phase diagram and the free energy of the models are calculated in the thermodynamic limit. The models exhibit massless phases, and our analytical and numerical analyses indicate that such phases are governed by a conformal field theory with central charge c=1 and continuously varying critical exponents.

Exactly solvable statistical model for two-way traffic

We generalize a recently introduced traffic model, where the statistical weights are associated with whole trajectories, to the case of two-way flow. An interaction between the two lanes is included which describes a slowing down when two cars meet. This leads to two coupled five-vertex models. It is shown that this problem can be solved by reducing it to two one-lane problems with modified parameters. In contrast to stochastic models, jamming appears only for very strong interaction between the lanes.

From the Bethe Ansatz to the Gessel-Viennot theorem

We state and prove several theorems that demonstrate how the coordinate Bethe Ansatz for the eigenvectors of suitable transfer matrices of a generalized inhomogeneous, five-vertex model on the square lattice, given certain conditions hold, is equivalent to the Gessel-Viennot determinant for the number of configurations ofN non-intersecting directed lattice paths, or vicious walkers, with various boundary conditions. Our theorems are sufficiently general to allow generalisation to any regular planar lattice.

Exact solution ofNdirected non-intersecting walks interacting with one or two boundaries

The partition function for the problem of an arbitrary number of directed non-intersecting walks interacting with one or two walls parallel to the direction of the walks is calculated exactly utilizing a theorem recently proved concerning the Bethe ansatz for the eigenvectors of the transfer matrix of the five-vertex model. This theorem shows that the completeness of the Bethe ansatz eigenvectors for the N-walk transfer matrix can be deduced from the completeness of the one-walk eigenvectors.

An Exactly Solvable Anisotropic Directed Percolation Model in Three Dimensions

We solve exactly a special case of the anisotropic directed bond percolation problem in three dimensions, in which the occupation probability is 1 along two spatial directions, by mapping it to a five-vertex model. We determine the asymptotic shape of the ininite cluster and hence the direction dependent critical probability. The exponents characterising the fluctuations of the boundary of the wetted cluster in d-dimensions are related to those of the (d-2)-dimensional KPZ equation.

A five-vertex model interpretation of one-dimensional traffic flow

Here we solve a discrete one-dimensional traffic flow problem by mapping the allowed sets of car trajectories onto a line representation of the five-vertex model configurations. The fundamental flow diagram, obtained previously in a grand canonical ensemble, is rederived. Fluctuations of the flow are described quantitatively and two critical exponents are defined. The zero-density limit is studied by considering an ensemble of single directed self-avoiding loops on a finite torus.

Interacting dimers on the honeycomb lattice: an exact solution of the five-vertex model

The problem of close-packed dimers on the honeycomb lattice was solved by Kasteleyn in 1963. Here we extend the solution to include interactions between neighboring dimers in two spatial lattice directions. The solution is obtained by using the method of Bethe ansatz and by converting the dimer problem into a five-vertex problem. The complete phase diagram is obtained and it is found that a new frozen phase, in which the attracting dimers prevail, arises when the interaction is attractive. For repulsive dimer interactions a new first-order line separating two frozen phases occurs. The transitions are continuous and the critical behavior in the disorder regime is found to be the same as in the case of noninteracting dimers characterized by a specific heat exponent α = 1 2 .

Mapping of crystal surfaces onto the five-vertex model.

Within the solid-on-solid framework we map certain surfaces of a simple cubic crystal onto a five-vertex model. The mapping scheme used differs from the more common body-centered solid-on-solid mapping, which uses a six-vertex model and is only suitable for interfaces whose lattice splits naturally into two sublattices. In the five-vertex model there is no touching of step lines and the arrow-reversal symmetry is broken by construction. After reviewing the exact solution of the five-vertex model and studying its phase diagram in the presence of an external field, we establish an isomorphism between some phases of the vertex model with the (100), (110), and ([bar 1]01) faces of the simple cubic crystal. We determine exactly the mean square height difference. If no [ital field] is applied the flat surfaces undergo a transition at a critical temperature, where the inclination of the surfaces begins to change, and roughening sets in. If, however, the inclination is kept constant by applying a suitable field, the low-index flat (or high-index vicinal) surfaces remain smooth (or rough) from zero up to infinite temperature, without undergoing any roughening transition.

Interacting domain walls and the five-vertex model.

We investigate the thermodynamic and critical properties of an interacting domain wall model which is derived from the triangular lattice antiferromagnetic Ising model with the anisotropic nearest and next nearest neighbor interactions. The model is equivalent to the general five--vertex model. Diagonalizing the transfer matrix exactly by the Bethe Ansatz method, we obtain the phase diagram displaying the commensurate and incommensurate (IC) phases separated by the Pokrovsky--Talapov transitions. The phase diagram exhibits commensurate phases where the domain wall density $q$ is locked at the values of $0$, $1/2$ and $1$. The IC phase is a critical state described by the Gaussian fixed point. The effective Gaussian coupling constant is obtained analytically and numerically for the IC phase using the finite size scaling predictions of the conformal field theory. It takes the value $1/2$ in the non-interacting limit and also at the boundaries of $q=0$ or $1$ phase and the value $2$ at the boundary of $q=1/2$ phase, while it varies smoothly throughout the IC region. (TO APPEAR IN PHY. REV. E)

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.

Absence of anomalous dimension in vertex models: Semidilute solution of directed polymers

We develop a continuum path-integral approach for the ferroelectric five-vertex model in arbitrary d dimensions by mapping it to a directed polymer problem. A renormalization-group approach with an \ensuremath{\epsilon}=3-d expansion, 3 being the upper critical dimension, is used to study the polymer solution. The free-energy change due to the interaction of the chains has been computed to O(\ensuremath{\epsilon}), and the exact expression for the second virial coefficient has been obtained. The fixed point of the problem is found to be exactly 2\ensuremath{\pi}\ensuremath{\epsilon}. By use of finite-size-scaling theory and thermodynamics, the exponents for the vertex model are obtained from those of the polymeric system as the specific-heat exponent \ensuremath{\alpha}=(3-d)/2, and the incommensuration exponent \ensuremath{\beta}\ifmmode\bar\else\textasciimacron\fi{}=(d-1)/2. The model is anisotropic with two length-scale exponents ${\ensuremath{\nu}}_{\mathrm{?}}$=1 in one direction and ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$ =1/2 in the remaining d-1 directions. It is shown that there are no anomalous dimensions so that the exponents we obtain are exact.

Vertex models and directed walks: length scale exponents from finite size scaling

By using the mapping between ferroelectric five-vertex (or the modified KDP) model and directed walks, the authors prove, via finite size scaling, that the length scale exponents nu /sub /// and nu perpendicular to are related by nu /sub ///=2 nu perpendicular to in all dimensions.

Nonintersecting string model and graphical approach: Equivalence with a Potts model

Using a graphical method we establish the exact equivalence of the partition function of aq-state nonintersecting string (NIS) model on an arbitrary planar, even-valenced, lattice with that of a q2-state Potts model on a related lattice. The NIS model considered in this paper is one in which the vertex weights are expressible as sums of those of basic vertex types, and the resulting Potts model generally has multispin interactions. For the square and Kagome lattices this leads to the equivalence of a staggered NIS model with Potts models with anisotropic pair interactions, indicating that these NIS models have a first-order transition forq > 2. For the triangular lattice the NIS model turns out to be the five-vertex model of Wu and Lin and it relates to a Potts model with two- and three-site interactions. The most general model we discuss is an oriented NIS model which contains the six-vertex model and the NIS models of Stroganov and Schultz as special cases.