1,720,981 research outputs found
Integrable boundaries for the q-Hahn process
Full text linkTaking inspiration from the harmonic process with reservoirs introduced by
Giardin\`a, Kurchan and the author in arXiv:1904.01048, we propose integrable
boundary conditions for its trigonometric deformation which is known as the
q-Hahn process. Following the formalism established by Mangazeev and Lu in
arXiv:1903.00274 using the stochastic R-matrix, we argue that the proposed
boundary conditions can be derived from a transfer matrix constructed in the
framework of Sklyanin's extension of the quantum inverse scattering method and
consequently preserve the integrable structure of the model. The approach
avoids the explicit construction of the K-matrix.Comment: v1: One figure and 16 pages v2: added symmetric limit and improved
conclusio
The non-compact XXZ spin chain as stochastic particle process
Full text linkIn this note we relate the Hamiltonian of the integrable non-compact spin XXZ chain to the Markov generator of a stochastic particle process. The hopping rates of the continuous-time process are identified with the ones of a q-Hahn asymmetric zero range model. The main difference with the asymmetric simpleexclusion process (ASEP), which can be mapped to the ordinary XXZ spin chain, is that multiple particles can occupy one and the same site. For the non-compact spin XXZ chain the associated stochastic process reduces to the multiparticle asymmetric diffusion model introduced by Sasamoto-Wadati.<br
Eigenstates of triangularisable open XXX spin chains and closed-form solutions for the steady state of the open SSEP
Full text linkIn this article we study the relation between the eigenstates of openrational spin Heisenberg chains with different boundaryconditions. The focus lies on the relation between the spin chain with diagonalboundary conditions and the spin chain with triangular boundary conditions aswell as the class of spin chains that can be brought to such form by certainsimilarity transformations in the physical space. The boundary driven SymmetricSimple Exclusion Process (open SSEP) belongs to the latter. We derive atransformation that maps the eigenvectors of the diagonal spin chain to theeigenvectors of the triangular chain. This transformation yields an essentialsimplification for determining the states beyond half-filling. It allows tofirst determine the eigenstates of the diagonal chain through the Bethe ansatzon the fully excited reference state and subsequently map them to thetriangular chain for which only the vacuum serves as a reference state. Inparticular the transformed reference state, i.e. the fully excited eigenstateof the triangular chain, is presented at any length of the chain. It can bemapped to the steady state of the open SSEP. This results in a conciseclosed-form expression for the probabilities of particle distributions andcorrelation functions in the steady state. Further, the complete set ofeigenstates of the Markov generator is expressed in terms of the eigenstates ofthe diagonal open chain.<br
Algebraic Bethe ansatz for Q -operators: the Heisenberg spin chain
Full text linkWe diagonalize Q-operators for rational homogeneous sl(2)-invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang–Baxter equation we demonstrate that the Q-operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q-operators studied here are given by Baxterʼs Q-functions
The steady state of the boundary-driven multiparticle asymmetric diffusion model
Full text linkWe consider the multiparticle asymmetric diffusion model (MADM) introduced by Sasamoto and Wadati with integrability preserving reservoirs at the boundaries. In contrast to the open asymmetric simple exclusion process the number of particles allowed per site is unbounded in the MADM. Taking inspiration from the stationary measure in the symmetric case, i.e. the rational limit, we first obtain the length 1 solution and then show that the steady state can be expressed as an iterated product of Jackson q-integrals. In the proof of the stationarity condition, we observe a cancellation mechanism that closely resembles the one of the matrix product ansatz. To our knowledge, the occupation probabilities in the steady state of the boundary-driven MADM were not available before
Q-operators for the open Heisenberg spin chain
Full text linkAbstractWe construct Q-operators for the open spin-12 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang–Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots
Boundary Perimeter Bethe Ansatz
No full textInternational audienceWe study the partition function of the six-vertex model in the rational limit on arbitrary Baxter lattices with reflecting boundary. Every such lattice is interpreted as an invariant of the twisted Yangian. This identification allows us to relate the partition function of the vertex model to the Bethe wave function of an open spin chain. We obtain the partition function in terms of creation operators on a reference state from the algebraic Bethe ansatz and as a sum of permutations and reflections from the coordinate Bethe ansatz
A Family of Multiplicative Higgs Bundles on Rational Base
Full text linkInternational audienceIn this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at . The restriction of the family is that the matrix elements in the defining representation are linear functions of . We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs bundles on with prescribed singularities, (ii) moduli spaces of monopoles on with Dirac singularities, (iii) Coulomb branches of the moduli space of vacua of 4d supersymmetric quiver gauge theories compactified on a circle. While degree 1 symplectic leaves regular at (Coulomb branches of the superconformal quiver gauge theories) are isomorphic to co-adjoint orbits in and their Darboux parametrization and quantization is well known, the case irregular at infinity (asymptotically free quiver gauge theories) is novel. We also explicitly quantize the algebra of functions on these moduli spaces by presenting the corresponding solutions to the quantum Yang-Baxter equation valued in Heisenberg algebra (free field realization).</BR
Yangian-type symmetries of non-planar leading singularities
Full text linkWe take up the study of integrable structures behind non-planar contributions to scattering amplitudes in N=4N=4 super Yang-Mills theory. Focusing on leading singularities, we derive the action of the Yangian generators on color-ordered subsets of the external states. Each subset corresponds to a single boundary of the non-planar on-shell diagram. While Yangian invariance is broken, we find that higher-level Yangian generators still annihilate the non-planar on-shell diagram. For a given diagram, the number of these generators is governed by the degree of non-planarity. Furthermore, we present additional identities involving integrable transfer matrices. In particular, for diagrams on a cylinder we obtain a conservation rule similar to the Yangian invariance condition of planar on-shell diagrams. To exemplify our results, we consider a five-point MHV on-shell function on a cylinder
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