Boundary perimeter Bethe ansatz

Frassek, Rouven

doi:10.1088/1751-8121/aa7278

Cited by 5 publications

(5 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular we have obtained the more compact expression (4.16)-(4.17) for the partition function of the solid-on-solid model with domain walls and one reflecting end in terms of a crossingsymmetrized sum with 2 L terms featuring the elliptic domain-wall partition function of [37,38]. In the trigonometric case of the six-vertex model our result boils down to a relation between the reflecting-end partition function and the domain-wall partition function, see (4.18), which to the best of our knowledge is new, and can be matched with the outcome of the boundary perimeter Bethe ansatz by Frassek [7].…”

Section: Discussionsupporting

confidence: 63%

“…Since in the homogeneous limit x i → x we get a(x r i + x r j ) → 0 whenever r = r I with #(I ∩ {i, j}) = 1 it is not clear whether (4.18) might facilitate the computation of that limit. When we instead take the rational limit [w] → w, where one may set γ = 1, we recover the result of the boundary perimeter Bethe ansatz [7] for the lattice (4.9). 6 …”

Section: A New Expression For the Reflecting-end Partition Functionmentioning

confidence: 62%

“…, L} } ∼ = (Z 2 ) L for the group of all such 'reflections', cf. [7]. By the preceding observation we can rewrite…”

Section: A New Expression For the Reflecting-end Partition Functionmentioning

confidence: 84%

“…See the case m = L of (80) in[7], with zi := xi, vj := −yj, q := κ and finally τ = τI := r {1,...,L}\I , which is equivalent to always picking up the second term from (4.15b) to rewrite (4.15).…”

mentioning

confidence: 99%

See 3 more Smart Citations

The Functional Method for the Domain-Wall Partition Function

Lamers¹

2018

SIGMA

View full text Add to dashboard Cite

Abstract. We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author. We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domainwall boundary conditions. At the heart of the method lies a linear functional equation for the partition function. After deriving this equation we outline its analysis. The result is a closed expression in the form of a symmetrized sum -or, equivalently, multiple-integral formula -that can be rewritten to recover Izergin's determinant. Special attention is paid to the relation with other approaches. In particular we show that the Korepin-Izergin approach can be recovered within the functional method. We comment on the functional method's range of applicability, and review how it is adapted to the technically more involved example of the elliptic solid-on-solid model with domain walls and a reflecting end. We present a new formula for the partition function of the latter, which was expressed as a determinant by Tsuchiya-Filali-Kitanine. Our result takes the form of a 'crossing-symmetrized' sum with 2 L terms featuring the elliptic domain-wall partition function, which appears to be new also in the limiting case of the six-vertex model. Further taking the rational limit we recover the expression obtained by Frassek using the boundary perimeter Bethe ansatz.

show abstract

Section: Discussionsupporting

confidence: 63%

Section: A New Expression For the Reflecting-end Partition Functionmentioning

confidence: 62%

“…, L} } ∼ = (Z 2 ) L for the group of all such 'reflections', cf. [7]. By the preceding observation we can rewrite…”

Section: A New Expression For the Reflecting-end Partition Functionmentioning

confidence: 84%

mentioning

confidence: 99%

See 2 more Smart Citations

The Functional Method for the Domain-Wall Partition Function

Lamers¹

2018

SIGMA

View full text Add to dashboard Cite

show abstract

“…As for the domain wall boundary partition functions with reflecting end, the determinant formula was found by Tsuchiya [10] (see also Kuperberg [7,8] and Okada [9]), and the thermodynamic limit is investigated by Ribeiro-Korepin [50] following the idea of Korepin-Zinn-Justin [51]. As for the wavefunctions under reflecting boundary, there are studies on the reduced five-vertex model [52] and related q-boson model [25,26], boundary perimeter Bethe ansatz of the XXX chain [53], and solutions of the boundary qKZ equation [54]. We show in this paper that the Izergin-Korepin analysis can be applied to the wavefunctions of the six vertex model with reflecting end.…”

Section: Introductionmentioning

confidence: 99%

Izergin-Korepin analysis on the wavefunctions of the $U_q(sl_2)$ six-vertex model with reflecting end

Motegi

2017

Preprint

View full text Add to dashboard Cite

We extend the recently developed Izergin-Korepin analysis on the wavefunctions of the U q (sl 2 ) six-vertex model to the reflecting boundary conditions. Based on the Izergin-Korepin analysis, we determine the exact forms of the symmetric functions which represent the wavefunctions and its dual. Comparison of the symmetric functions with the coordinate Bethe ansatz wavefunctions for the open XXZ chain by Alcaraz-Barber-Batchelor-Baxter-Quispel is also made. As an application, we derive algebraic identities for the symmetric functions by combining the results with the determinant formula of the domain wall boundary partition function of the six-vertex model with reflecting end.

show abstract

Boundary overlaps from Functional Separation of Variables

Ekhammar,

Gromov,

Ryan

2024

J. High Energ. Phys.

View full text Add to dashboard Cite

In this paper we show how the Functional Separation of Variables (FSoV) method can be applied to the problem of computing overlaps with integrable boundary states in integrable systems. We demonstrate our general method on the example of a particular boundary state, a singlet of the symmetry group, in an $$ \mathfrak{su}(3) $$ su 3 rational spin chain in an alternating fundamental-anti-fundamental representation. The FSoV formalism allows us to compute in determinant form not only the overlaps of the boundary state with the eigenstates of the transfer matrix, but in fact with any factorisable state. This includes off-shell Bethe states, whose overlaps with the boundary state have been out of reach with other methods. Furthermore, we also found determinant representations for insertions of so-called Principal Operators (forming a complete algebra of all observables) between the boundary and the factorisable state as well as certain types of multiple insertions of Principal Operators. Concise formulas for the matrix elements of the boundary state in the SoV basis and $$ \mathfrak{su}(N) $$ su N generalisations are presented. Finally, we managed to construct a complete basis of integrable boundary states by repeated action of conserved charges on the singlet state. As a result, we are also able to compute the overlaps of all of these states with integral of motion eigenstates.

show abstract

Boundary perimeter Bethe ansatz

Cited by 5 publications

References 33 publications

The Functional Method for the Domain-Wall Partition Function

The Functional Method for the Domain-Wall Partition Function

Izergin-Korepin analysis on the wavefunctions of the $U_q(sl_2)$ six-vertex model with reflecting end

Boundary overlaps from Functional Separation of Variables

Contact Info

Product

Resources

About