121 research outputs found
Integrable boundaries for the q-Hahn process
Taking inspiration from the harmonic process with reservoirs introduced by Frassek, GiardinĂ and Kurchan in (2020 J. Stat. Phys. 180 135-71), 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 (2019 Nucl. Phys. B 945 114665) using the stochastic R-matrix, we argue that the proposed boundary conditions can be derived from a transfer matrix constructed in the framework of Sklyaninâs 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
Baxter Operators and Hamiltonians for "nearly all" Integrable Closed gl(n) Spin Chains
We continue our systematic construction of Baxter Q-operators for spin
chains, which is based on certain degenerate solutions of the Yang-Baxter
equation. Here we generalize our approach from the fundamental representation
of gl(n) to generic finite-dimensional representations in quantum space. The
results equally apply to non-compact representations of highest or lowest
weight type. We furthermore fill an apparent gap in the literature, and provide
the nearest-neighbor Hamiltonians of the spin chains in question for all cases
where the gl(n) representations are described by rectangular Young diagrams, as
well as for their infinite-dimensional generalizations. They take the form of
digamma functions depending on operator-valued shifted weights.Comment: 26 pages, 1 figur
Oscillator realisations associated to the D-type Yangian: Towards the operatorial Q-system of orthogonal spin chains
We present a family of novel Lax operators corresponding to representations of the RTT-realisation of the Yangian associated with D-type Lie algebras. These Lax operators are of oscillator type, i.e. one space of the operators is infinite-dimensional while the other is in the first fundamental representation of so(2r). We use the isomorphism between the first fundamental representation of D3 and the 6 of A3, for which the degenerate oscillator type Lax matrices are known, to derive the Lax operators for r=3. The results are used to generalise the Lax matrices to arbitrary rank for representations corresponding to the extremal nodes of the simply laced Dynkin diagram of Dr. The multiplicity of independent solutions at each extremal node is given by the dimension of the fundamental representation. We further derive certain factorisation formulas among these solutions and build transfer matrices with oscillators in the auxiliary space from the introduced degenerate Lax matrices. Finally, we provide some evidence that the constructed transfer matrices are Baxter Q-operators for so(2r) spin chains by verifying certain QQ-relations for D4 at low lengths
Rational Lax matrices from antidominantly shifted extended Yangians: BCD types
Generalizing our recent joint paper with Vasily Pestun (arXiv:2001.04929), we
construct a family of rational Lax matrices,
polynomial in the spectral parameter, parametrized by the divisors on the
projective line with coefficients being dominant integral coweights of
associated Lie algebras. To this end, we provide the RTT realization of the
antidominantly shifted extended Drinfeld Yangians of , and of their coproduct
homomorphisms. This establishes some of the recent conjectures in the physics
literature by Costello-Gaiotto-Yagi (arXiv:2103.01835) in the classical types.Comment: v3: 67 pages, minor corrections, details added. v2: 65 pages, minor
corrections, some details added, Remark 4.37 added. v1: 64 pages, comments
are welcome
From Baxter Q-operators to local charges
We discuss how the shift operator and the Hamiltonian enter the hierarchy of Baxter Q-operators in the example of gl(n) homogeneous spin-chains. Building on the construction that was recently carried out by the authors and their collaborators, we find that a reduced set of Q-operators can be used to obtain local charges. The mechanism relies on projection properties of the corresponding -operators on a highest/lowest weight state of the quantum space. It is intimately related to the ordering of the oscillators in the auxiliary space. Furthermore, we introduce a diagrammatic language that makes these properties manifest and the results transparent. Our approach circumvents the paradigm of constructing the transfer matrix with equal representations in quantum and auxiliary space and underlines the strength of the Q-operator construction. © 2013 IOP Publishing Ltd and SISSA Medialab srl
Exact solution of an integrable non-equilibrium particle system
We consider the boundary-driven interacting particle systems introduced in
[FGK20a] related to the open non-compact Heisenberg model in one dimension. We
show that a finite chain of sites connected at its ends to two reservoirs
can be solved exactly, i.e. the non-equilibrium steady state has a closed-form
expression for each . The solution relies on probabilistic arguments and
techniques inspired by integrable systems. It is obtained in two steps: i) the
introduction of a dual absorbing process reducing the problem to a finite
number of particles; ii) the solution of the dual dynamics exploiting a
symmetry obtained from the Quantum Inverse Scattering Method. The exact
solution allows to prove by a direct computation that, in the thermodynamic
limit, the system approaches local equilibrium. A by-product of the solution is
the algebraic construction of a direct mapping (a conjugation) between the
generator of the non-equilibrium process and the generator of the associated
reversible equilibrium process.Comment: 46 pages, 2 figure
Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin
In this note we construct Q-operators for the spin s open Heisenberg XXX
chain with diagonal boundaries in the framework of the quantum inverse
scattering method. Following the algebraic Bethe ansatz we diagonalise the
introduced Q-operators using the fundamental commutation relations. By acting
on Bethe off-shell states and explicitly evaluating the trace in the auxiliary
space we compute the eigenvalues of the Q-operators in terms of Bethe roots and
show that the unwanted terms vanish if the Bethe equations are satisfied.Comment: 17 page
Orthosymplectic superoscillator Lax matrices
We construct Lax matrices of superoscillator type that are solutions of the
RTT-relation for the rational orthosymplectic -matrix, generalizing
orthogonal and symplectic oscillator type Lax matrices previously constructed
by the authors in arXiv:2001.06825, arXiv:2104.14518 and arXiv:2112.12065. We
further establish factorisation formulas among the presented solutions.Comment: 25 page
Non-compact quantum spin chains as integrable stochastic particle processes
In this paper we discuss a family of models of particle and energy diffusion on a one-dimensional lattice, related to those studied previously in Sasamoto and Wadati (Phys Rev E 58:4181\u20134190, 1998), Barraquand and Corwin (Probab Theory Relat Fields 167(3\u20134):1057\u20131116, 2017) and Povolotsky (J Phys A 46(46):465205, 2013) in the context of KPZ universality class. We show that they may be mapped onto an integrable (2) Heisenberg spin chain whose Hamiltonian density in the bulk has been already studied in the AdS/CFT and the integrable system literature. Using the quantum inverse scattering method, we study various new aspects, in particular we identify boundary terms, modeling reservoirs in non-equilibrium statistical mechanics models, for which the spin chain (and thus also the stochastic process) continues to be integrable. We also show how the construction of a \u201cdual model\u201d of probability theory is possible and useful. The fluctuating hydrodynamics of our stochastic model corresponds to the semiclassical evolution of a string that derives from correlation functions of local gauge invariant operators of \ue23a=4 super Yang\u2013Mills theory (SYM), in imaginary-time. As any stochastic system, it has a supersymmetric completion that encodes for the thermal equilibrium theorems: we show that in this case it is equivalent to the (2|1) superstring that has been derived directly from \ue23a=4 SYM
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