Domain Wall Boundary Conditions Research Articles

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.

Domain wall fluctuations of the six-vertex model at the ice point

We report on Monte-Carlo simulations of the six-vertex model with domain wall boundary conditions. In thermal equilibrium such boundary conditions force a fluctuating line separating the disordered region from the perfectly ordered ones. Specifically we study the ice point at which all vertex weights are equal. With high precision the one-point fluctuations of the line are confirmed to be of order and governed by the Tracy–Widom distribution. Furthermore, the non-universal scaling coefficients are computed for a wide range of interaction strengths.

Fluctuation of the phase boundary in the six-vertex model with domain wall boundary conditions: a Monte Carlo study

We consider the six-vertex model with domain wall boundary conditions in a square lattice of dimension N × N. Our main interest is the study of the fluctuations of the extremal lattice path about the arctic curves. We address the problem through Monte Carlo simulations. At , the fluctuations of the extremal path along any line parallel to the square diagonal were rigorously proven to follow the Tracy-Widom distribution. We provide strong numerical evidence that this is true also for other values of the anisotropy parameter Δ (). We argue that the typical width of the fluctuations of the extremal path about the arctic curves scales as and provide a numerical estimate for the parameters of the scaling random variable.

Construction of determinants for the six-vertex model with domain wall boundary conditions

We consider the problem of construction of determinant formulas for the partition function of the six-vertex model with domain wall boundary conditions that depend on two sets of spectral parameters. In the pioneering works of Korepin and Izergin a determinant formula was proposed and proved using a recursion relation. In later works, equivalent determinant formulas were given by Kostov for the rational case and by Foda and Wheeler for the trigonometric case. Here, we develop an approach in which the recursion relation is replaced by a system of algebraic equations with respect to one of the two sets of spectral parameters. We prove that this system has a unique solution. The result can be easily given as a determinant parametrized by an arbitrary basis of polynomials. In particular, the choice of the basis of Lagrange polynomials with respect to the remaining set of spectral parameters leads to the Izergin–Korepin representation, and the choice of the monomial basis leads to the Kostov and Foda–Wheeler representations.

Arctic curves of the 20V model on a triangle

We apply the Tangent Method of Colomo and Sportiello to predict the arctic curves of the Twenty Vertex model with specific domain wall boundary conditions on a triangle, in the Disordered phase, leading to a phase diagram with six types of frozen phases and one liquid one. The result relies on a relation to the Six Vertex model with domain wall boundary conditions and suitable weights, as a consequence of integrability. We also perform the exact refined enumeration of configurations.

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.

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.

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.

A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. II. The Polynomials p<sub>n</sub>

By specializing the parameters in the partition function of the 8VSOS model with domain wall boundary conditions and diagonal reflecting end, we find connections between the three-color model and certain polynomials $p_n(z)$, which are conjectured to be equal to certain polynomials of Bazhanov and Mangazeev, appearing in the eigenvectors of the Hamiltonian of the supersymmetric XYZ spin chain. This article is a continuation of a previous paper where we investigated the related polynomials $q_n(z)$, also conjectured to be equal to polynomials of Bazhanov and Mangazeev, appearing in the eigenvectors of the supersymmetric XYZ spin chain.

Exact results for the six-vertex model with domain wall boundary conditions and a partially reflecting end

The trigonometric six-vertex model with domain wall boundary conditions and one partially reflecting end on a lattice of size 2ntimes m, mle n, is considered. The partition function is computed using the Izergin–Korepin method, generalizing the result of Foda and Zarembo from the rational to the trigonometric case. Thereafter, we specify the parameters in Kuperberg’s way to get a formula for the number of states as a determinant of Wilson polynomials. We relate this to a new type of alternating sign matrices, similar to how the six-vertex model with domain wall boundary conditions is related to normal alternating sign matrices. In an appendix, we compute the partition function again, showing that it is also possible to find it with the method of Foda and Wheeler.

The two-point correlation function in the six-vertex model

We study numerically the two-point correlation functions of height functions in the six-vertex model with domain wall boundary conditions. The correlation functions and the height functions are computed by the Markov chain Monte-Carlo algorithm. Particular attention is paid to the free fermionic point (Δ = 0), for which the correlation functions are obtained analytically in the thermodynamic limit. A good agreement of the exact and numerical results for the free fermionic point allows us to extend calculations to the disordered (|Δ| < 1) phase and to monitor the logarithm-like behavior of correlation functions there. For the antiferroelectric (Δ < −1) phase, the exponential decrease of correlation functions is observed.

Non-Hermitian skin effect in a domain-wall system

The non-Hermitian skin effect is one of the most striking features in non-Hermitian physics. It reveals a novel phenomenon in a non-Hermitian system that the bulk wave function and energy spectrum are sensitively dependent on the boundary conditions. The concept of generalized Brillouin zones has been proposed to characterize bulk wave functions in such systems . Based on generalized Brillouin zones, non-Bloch topological invariants can reconstruct the non-Hermitian bulk-edge correspondence. Previous discussion of the non-Hermitian skin effect mainly focused on open boundary conditions, and the calculation of generalized Brillouin zones needs to be reconsidered under domain-wall boundary conditions. The paper introduces the related researches of the non-Hermitian skin effect in domain-wall systems, including the general form of the generalized Brillouin zone equation in a one-dimensional single-band model, non-Bloch topological invariants in non-Hermitian SSH (Su-Schieffer-Heeger) model, and the experimental realization of the non-Hermitian skin effect in one-dimensional quantum walk system.

Oundary One-Point Function of the Rational Six-Vertex Model with Partial Domain Wall Boundary Conditions: Explicit Formulas and Scaling Properties

We consider the six-vertex model with the rational weights on an s by N square lattice with partial domain wall boundary conditions. We study the one-point function at the boundary where the free boundary conditions are imposed. For a finite lattice, it can be computed by the quantum inverse scattering method in terms of determinants. In the large N limit, the result boils down to an explicit terminating series in the parameter of the weights. Using the saddle-point method for an equivalent integral representation, we show that as s next tends to infinity, the one-point function demonstrates a step-wise behavior; at the vicinity of the step it scales as the error function. We also show that the asymptotic expansion of the one-point function can be computed from a second-order ordinary differential equation.

Six-vertex model on a finite lattice: Integral representations for nonlocal correlation functions

We consider the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions. To this aim, we formulate the model as a scalar product of off-shell Bethe states, and, by applying the quantum inverse scattering method, we derive three different integral representations for these states. By suitably combining such representations, and using certain antisymmetrization relation in two sets of variables, it is possible to derive integral representations for various correlation functions. In particular, focusing on the emptiness formation probability, besides reproducing the known result, obtained by other means elsewhere, we provide a new one. By construction, the two representations differ in the number of integrations and their equivalence is related to a hierarchy of highly nontrivial identities.

Arctic curves of the reflecting boundary six vertex and of the twenty vertex models

We apply the tangent method of Colomo and Sportiello to predict the arctic curves of the six vertex model with reflecting (U-turn) boundary and of the related twenty vertex model with suitable domain wall boundary conditions on a quadrangle, both in their disordered phase.

Refined Cauchy identity for spin Hall–Littlewood symmetric rational functions

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions Fλ arise in the context of sl(2) higher spin six vertex models, and are multiparameter deformations of the classical Hall–Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two Fλ's as a determinant. The determinant is of Izergin–Korepin type: it is the partition function of the six vertex model with suitably decorated domain wall boundary conditions. The proof of equality of two partition functions is based on the Yang–Baxter equation.We rewrite our Izergin–Korepin type determinant in a different form which includes one of the sets of variables in a completely symmetric way. This determinantal identity might be of independent interest, and also allows to directly link the spin Hall–Littlewood rational functions with (the Hall–Littlewood particular case of) the interpolation Macdonald polynomials. In a different direction, a Schur expansion of our Izergin–Korepin type determinant yields a deformation of Schur symmetric polynomials.In the spin-12 specialization, our refined Cauchy identity leads to a summation identity for eigenfunctions of the ASEP (Asymmetric Simple Exclusion Process), a celebrated stochastic interacting particle system in the Kardar–Parisi–Zhang universality class. This produces explicit integral formulas for certain multitime probabilities in ASEP.

The limit shape of the height function in the six-vertex model with domain-wall boundary conditions

The height function of the six-vertex model with the domain-wall boundary conditons in the free fermion point is computed by the Monte Carlo algorithm. The numerical results are in good agreement with the analytical expression for the limit shape height function. This paper is a “warm up” for the forthcoming one, where the two-point correlation function for the height function is calculated.

Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings

We consider the triangular lattice ice model (20-Vertex model) with four types of domain-wall type boundary conditions. In types 1 and 2, the configurations are shown to be equinumerous to the quarter-turn symmetric domino tilings of an Aztec-like holey square, with a central cross-shaped hole. The proof of this statement makes extensive use of integrability and of a connection to the 6-Vertex model. The type 3 configurations are conjectured to be in same number as domino tilings of a particular triangle. The four enumeration problems are reformulated in terms of four types of Alternating Phase Matrices with entries $0$ and sixth roots of unity, subject to suitable alternation conditions. Our result is a generalization of the ASM-DPP correspondence. Several refined versions of the above correspondences are also discussed.

Arctic Curves of the Twenty-Vertex Model with Domain Wall Boundaries

We use the tangent method to compute the arctic curve of the Twenty-Vertex (20V) model with particular domain wall boundary conditions for a wide set of integrable weights. To this end, we extend to the finite geometry of domain wall boundary conditions the standard connection between the bulk 20V and 6V models via the Kagome lattice ice model. This allows to express refined partition functions of the 20V model in terms of their 6V counterparts, leading to explicit parametric expressions for the various portions of its arctic curve. The latter displays a large variety of shapes depending on the weights and separates a central liquid phase from up to six different frozen phases. A number of numerical simulations are also presented, which highlight the arctic curve phenomenon and corroborate perfectly the analytic predictions of the tangent method. We finally compute the arctic curve of the Quarter-turn symmetric Holey Aztec Domino Tiling (QTHADT) model, a problem closely related to the 20V model and whose asymptotics may be analyzed via a similar tangent method approach. Again results for the QTHADT model are found to be in perfect agreement with our numerical simulations.

Integral Formulas and Antisymmetrization Relations for the Six-Vertex Model

We study the relationship between various integral formulas for nonlocal correlation functions of the six-vertex model with domain wall boundary conditions. Specifically, we show how the known representation for the emptiness formation probability can be derived from that for the so-called row configuration probability. A crucial ingredient in the proof is a relation expressing the result of antisymmetrization of some given function with respect to permutations in two sets of its variables in terms of the Izergin-Korepin partition function. This relation generalizes another one obtained by Tracy and Widom in the context of the asymmetric simple exclusion process.