Loop Models and K-Theory

The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e., models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e., not free, not determinantal) six-vertex model, and provide evidence for the following two claims:(i) the critical fluctuations are given by a Gaussian Free Field (GFF), as in the free case, but (ii) contrarily to the free case, the GFF is inhomogeneous, meaning that its coupling constant K becomes position-dependent, K → K(x).The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of K and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green's function of the Laplacian ∆ = ∇ · ∇ inside the critical domain, but instead, of the Green's function of a generalized Laplacian ∆ = ∇ · 1 K ∇ parametrized by the function K. Surprisingly, we also find that there is a change of regime when ∆ ≤ −1/2, with K becoming singular at one point.

Section: J Stat Mech (2019) 013102mentioning

confidence: 99%

Inhomogeneous Gaussian free field inside the interacting arctic curve

Granet¹,

Budzynski²,

Dubail³

et al. 2019

“…Secondly, the summation term in (1.1) is that of the XXZ model at its 'combinatorial point' with anisotropy parameter ∆ = − 1 2 . The ground state of the closed XXZ chain at this point has been much studied [4] and it displays a rich combinatorial structure that is realised most famously in the Razumov-Stroganov theorem [5,6]. Recently, interesting combinatorial structure has also been observed in the ground state of the open Hamiltonian (1.1) [3].…”

Section: Introductionmentioning

confidence: 99%

“…Note that we swap notations q ↔ q † and Q ↔ Q † compared to the work of Fendley, Hagendorf and collaborators[2,12]-we like to think of Q † as creating a lattice site 4. We specialise the boundary parameters of[17] to be ξ + = ξ− = −η.…”

mentioning

confidence: 99%

Lattice supersymmetry in the open XXZ model: an algebraic Bethe Ansatz analysis

Weston¹,

Yang²

2017

We reconsider the open XXZ chain of Yang and Fendley. This model possesses a lattice supersymmetry which changes the length of the chain by one site. We perform an algebraic Bethe ansatz analysis of the model and derive the commutation relations of the lattice SUSY operators with the four elements of the open-chain monodromy matrix. Hence we give the action of the SUSY operator on off-shell and on-shell Bethe states. We show that this action generally takes one on-shell Bethe eigenstate to another. The exception is that a zero-energy vacuum state will be a SUSY singlet. The SUSY pairings of Bethe roots we obtain are analogous to those found previously for closed chains by Fendley and Hagendorf by analysing the Bethe equations.

“…The XXZ spin chain is a one-dimensional system of quantum spins 1 2 with nearestneighbour interactions. In this article, we consider this spin chain with open boundary conditions and diagonal boundary magnetic fields.…”

Section: Introductionmentioning

confidence: 99%

The open XXZ chain at Δ = −1/2 and the boundary quantum Knizhnik–Zamolodchikov equations

Hagendorf¹,

Liénardy²

2021

The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R-matrix and diagonal K-matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.