Search

We study the stirring process with N − 1 species on a generic graph G = (V , E) with reservoirs. The multispecies stirring process generalizes the symmetric exclusion process, which is recovered in the case N = 2. We prove the existence of a dual process defined on an extended graph G ~ = (V ~ , E ~) which includes additional sites in V ~ ∖ V where dual particles get absorbed in the long-time limit. We thus obtain a characterization of the non-equilibrium steady state of the boundary-driven system in terms of the absorption probabilities of dual particles. The process is integrable for the case of the one-dimensional chain with two reservoirs at the boundaries and with maximally one particle per site. We compute the absorption probabilities by relying on the underlying g l (N) symmetry and the matrix product ansatz. Thus one gets a closed-formula for (long-ranged) correlations and for the non-equilibrium stationary measure. Extensions beyond this integrable set-up are also discussed. [ABSTRACT FROM AUTHOR]

We construct Lax matrices of superoscillator type that are solutions of the RTT-relation for the rational orthosymplectic R-matrix, generalizing orthogonal and symplectic oscillator type Lax matrices previously constructed by the authors in Frassek (Nuclear Phys B, 2020), Frassek and Tsymbaliuk (Commun Math Phys 392 (2):545–619, 2022), Frassek et al. (Commun Math Phys 400 (1):1–82, 2023). We further establish factorisation formulas among the presented solutions. [ABSTRACT FROM AUTHOR]

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras g , whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian Y (g) . These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter Q-operators, arising from our previous study Frassek et al. (Adv. Math. 401:108283, 2022), Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional g -modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety G/P stratified by B - -orbits. [ABSTRACT FROM AUTHOR]

We consider a stochastic process of heat conduction where energy is redistributed along a chain between nearest neighbor sites via an improper beta distribution. Similar to the well-known Kipnis–Marchioro–Presutti (KMP) model, the finite chain is coupled at its ends with two reservoirs that break the conservation of energy when working at different temperatures. At variance with KMP, the model considered here is integrable, and one can write in a closed form the n-point correlation functions of the non-equilibrium steady state. As a consequence of the exact solution one, can directly prove that the system is in "local equilibrium," which is described at the macro-scale by a product measure. Integrability manifests itself through the description of the model via the open Heisenberg chain with non-compact spins. The algebraic formulation of the model allows us to interpret its duality relation with a purely absorbing particle system as a change of representation. [ABSTRACT FROM AUTHOR]

We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of N sites connected at its ends to two reservoirs can be solved exactly, i.e., the factorial moments of the non-equilibrium steady-state can be written in the closed form for each N. 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 and (ii) the solution of the dual dynamics exploiting a symmetry obtained from the quantum inverse scattering method. Long-range correlations are computed in the finite-volume system. The exact solution allows us 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 between the non-equilibrium steady state and the equilibrium reversible measure. [ABSTRACT FROM AUTHOR]

Generalizing Frassek et al. (Adv. Math. 401, 108283 (2022). https://doi.org/10.1016/j.aim.2022.108283), we construct a family of SO(2r), Sp(2r), S O (2 r + 1) rational Lax matrices T D (z) , polynomial in the spectral parameter z, parametrized by Λ + -valued divisors D on P 1 . To this end, we provide the RTT realization of the antidominantly shifted extended Drinfeld Yangians of g = so 2 r , sp 2 r , so 2 r + 1 , and of their coproduct homomorphisms. [ABSTRACT FROM AUTHOR]

We propose the full system of Baxter Q-functions (QQ-system) for the integrable spin chains with the symmetry of the Dr Lie algebra. We use this QQ-system to derive new Weyl-type formulas expressing transfer matrices in all symmetric and antisymmetric (fundamental) representations through r + 1 basic Q-functions. Our functional relations are consistent with the Q-operators proposed recently by one of the authors and verified explicitly on the level of operators at small finite length. [ABSTRACT FROM AUTHOR]

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 58:4181–4190, 1998), Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 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 sl (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 "dual model" 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 N = 4 super Yang–Mills 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 sl (2 | 1) superstring that has been derived directly from N = 4 SYM. [ABSTRACT FROM AUTHOR]

We present an approach to evaluate the full operatorial Q-system of all $$ \mathfrak{u}\left(p,q\Big|r+s\right) $$ -invariant spin chains with representations of Jordan-Schwinger type. In particular, this includes the super spin chain of planar $$ \mathcal{N}=4 $$ super Yang-Mills theory at one loop in the presence of a diagonal twist. Our method is based on the oscillator construction of Q-operators. The Q-operators are built as traces over Lax operators which are degenerate solutions of the Yang-Baxter equation. For non-compact representations these Lax operators may contain multiple infinite sums that conceal the form of the resulting functions. We determine these infinite sums and calculate the matrix elements of the lowest level Q-operators. Transforming the Lax operators corresponding to the Q-operators into a representation involving only finite sums allows us to take the supertrace and to obtain the explicit form of the Q-operators in terms of finite matrices for a given magnon sector. Imposing the functional relations, we then bootstrap the other Q-operators from those of the lowest level. We exemplify this approach for non-compact spin − s spin chains and apply it to $$ \mathcal{N}=4 $$ at the one-loop level using the BMN vacuum as an example. [ABSTRACT FROM AUTHOR]