arXiv subjects

Rouven Frassek

Publications and source records attributed to Rouven Frassek.

At least 19 recordsLinked to original sources

arXiv2026

The boundary-driven multispecies harmonic process

We introduce the multispecies version of the harmonic process on a one-dimensional chain in contact with boundary reservoirs. This process is a continuous-time Markov chain where each site can host an unbounded number of colored particles. The symmetric bulk dynamics is put in contact with reservoirs, which inject and remove particles driving the system out-of-equilibrium. The Markov generator of the process is identified with the integrable Hamiltonian of an open rational Heisenberg spin chain of higher rank. We derive the underlying R- and K-matrices in operator form and construct the double-row transfer matrix following Sklyanin. Similar to the monospecies case the R-matrix factorises into two R-operators, each factor corresponding to left and right moving particles. We further define three dual models: an absorbing particle model, a hidden parameter model and a heat conduction model.

math-ph↗

arXiv2026

Intertwining Markov Processes via Matrix Product Operators

Duality transformations reveal unexpected equivalences between seemingly distinct models. We introduce an out-of-equilibrium generalisation of matrix product operators to implement duality transformations in one-dimensional boundary-driven Markov processes on lattices. In contrast to local dualities associated with generalised symmetries, here the duality operator intertwines two Markov processes via generalised exchange relations and realises the out-of-equilibrium duality globally. We construct these operators exactly for the symmetric simple exclusion process with distinct out-of-equilibrium boundaries. In this case, out-of-equilibrium boundaries are dual to equilibrium boundaries satisfying Liggett's condition, implying that the Gibbs-Boltzmann measure captures out-of-equilibrium physics when leveraging the duality operator. We illustrate this principle through physical applications.

math-ph↗

arXiv2026

Steady state representations for the harmonic process

In this note we discuss how the matrix product solution for the steady state of the harmonic process is obtained from the solutions already known in the literature, i.e. the closed-form expression derived in arXiv:2107.01720 and the nested integral form obtained in arXiv:2307.02793 and arXiv:2307.14975. Our results clarify the relation between the three representations of the steady state and provide the matrix product solution that has not been available for this model before.

math-ph↗

arXiv2026

On the Quantum Spectral Curve for $\text{AdS}_3\times \text{S}^3\times \text{S}^3\times \text{S}^1$ strings and the $\mathfrak{d}(2,1;α)$ Q-system

In this paper, we put forward and discuss a proposal for a Quantum Spectral Curve (QSC) describing the planar spectrum of the holographic CFT dual to strings on AdS$_3\times$ S$^3\times$ S$^3\times$ S$^1$, a theory with global symmetry $\mathfrak{d}(2,1;α)^{\oplus 2}$. We focus mainly on the case when the radii of the two spheres are the same, i.e. $α= 1/2$, where the symmetry reduces to $\mathfrak{osp}(4|2)^{\oplus 2}$. In this case, our proposal is based on two copies of an $\mathfrak{osp}(4|2)$ Q-system, glued through the branch cuts of the Q-functions in a minimal way. We study in detail the ensuing analytic properties of the Q-functions in this proposal. Focusing on purely massive excitations, we consider the large worldsheet limit in which the QSC leads to a set of Asymptotic Bethe Ansatz (ABA) equations, yielding strong constraints on the (so-far unfixed) dressing factors of the worldsheet S-matrix. In a $\mathbb{Z}_2$-symmetric sector, our proposal is consistent with all previous results on the worldsheet S-matrix. However, in the non-symmetric case, we found a subtle incompatibility between the analytic constraints arising from the proposed QSC, the crossing equations present in the literature, and braiding unitarity. We discuss possible explanations for this mismatch: either our minimal QSC proposal does not hold beyond the symmetric sector, or the crossing unitarity equations receive a nontrivial correction that needs to be understood. Finally, we also propose a generalisation of the Q-system for the case of $α\neq 1/2$, corresponding to the superalgebra $\mathfrak{d}(2,1;α)$. This novel algebraic structure represents a significant step towards understanding the Quantum Spectral Curve of the entire theory.

hep-th↗

arXiv2025

Orthosymplectic Yangians

We study the RTT orthosymplectic super Yangians and present their Drinfeld realizations for any parity sequence, generalizing the results for non-super types BCD, a standard parity sequence, and super A-type.

math.RT↗

arXiv2024

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-Wadati], [Barraquand-Corwin] and [Povolotsky] in the context of KPZ universality class. We show that they may be mapped onto an integrable $\mathfrak{sl}(2)$ 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 $\mathcal{N}=4$ 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 $\mathfrak{sl}(2|1)$ superstring that has been derived directly from $\mathcal{N}=4$ SYM.

math-ph↗

arXiv2024

Exact solution of an integrable non-equilibrium particle system

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 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; 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 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.

math-ph↗

arXiv2024

Orthosymplectic superoscillator Lax matrices

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 arXiv:2001.06825, arXiv:2104.14518, and arXiv:2112.12065. We further establish factorisation formulas among the presented solutions.

math-ph↗

arXiv2024

The steady state of the boundary-driven multiparticle asymmetric diffusion model

We 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 (ASEP) 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.

math-ph↗

arXiv2023

Duality for the multispecies stirring process with open boundaries

We study the stirring process with $N-1$ species on a generic graph $G=(V,\mathcal{E})$ 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 $\widetilde{G}=(\widetilde{V},\widetilde{\mathcal{E})}$ which includes additional extra-sites $\widetilde{V}\setminus V$ 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 ${gl}(N)$ 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.

math-ph↗

arXiv2023

Large deviations and additivity principle for the open harmonic process

We consider the boundary driven harmonic model, i.e. the Markov process associated to the open integrable XXX chain with non-compact spins. Using the factorial moments we characterize the stationary measure as a mixture of product measures. For all spin values, we identify the law of the mixture in terms of the Dirichlet process. Next, by using the explicit knowledge of the non-equilibrium steady state we establish formulas predicted by Macroscopic Fluctuation Theory for several quantities of interest: the pressure (by Varadhan's lemma), the density large deviation function (by contraction principle), the additivity principle (by using the Markov property of the mixing law). To our knowledge, the results presented in this paper constitute the first rigorous derivation of these macroscopic properties for models of energy transport with unbounded state space, starting from the microscopic structure of the non-equilibrium steady state.

math.PR↗

arXiv2023

Integrable heat conduction model

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 a `local equilibrium' and 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 to interpret its duality relation with a purely absorbing particle system as a change of representation.

cond-mat.stat-mech↗

arXiv2023

Transfer matrices of rational spin chains via novel BGG-type resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. 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 (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-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.

math.RT↗

arXiv2022

Integrable boundaries for the q-Hahn process

Taking inspiration from the harmonic process with reservoirs introduced by Giardinà, 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.

math-ph↗

arXiv2022

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.

math-ph↗

arXiv2022

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 $SO(2r),Sp(2r),SO(2r+1)$ 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 $\mathfrak{so}_{2r}, \mathfrak{sp}_{2r}, \mathfrak{so}_{2r+1}$, 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.

math.RT↗

arXiv2022

Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type

We construct a family of $GL_n$ rational and trigonometric Lax matrices $T_D(z)$ parametrized by $Λ^+$-valued divisors $D$ on $\mathbb{P}^1$. To this end, we study the shifted Drinfeld Yangians $Y_μ(\mathfrak{gl}_n)$ and quantum affine algebras $U_{μ^+,μ^-}(L\mathfrak{gl}_n)$, which slightly generalize their $\mathfrak{sl}_n$-counterparts. Our key observation is that both algebras admit the RTT type realization when $μ$ (respectively, $μ^+$ and $μ^-$) are antidominant coweights. We prove that $T_D(z)$ are polynomial in $z$ (up to a rational factor) and obtain explicit simple formulas for those linear in $z$. This generalizes the recent construction by the first two authors of linear rational Lax matrices in both trigonometric and higher $z$-degree directions. Furthermore, we show that all $T_D(z)$ are normalized limits of those parametrized by $D$ supported away from $\{\infty\}$ (in the rational case) or $\{0,\infty\}$ (in the trigonometric case). The RTT approach provides conceptual and elementary proofs for the construction of the coproduct homomorphisms on shifted Yangians and quantum affine algebras of $\mathfrak{sl}_n$, previously established via rather tedious computations. Finally, we establish a close relation between a certain collection of explicit linear Lax matrices and the well-known parabolic Gelfand-Tsetlin formulas.

math.RT↗

arXiv2020

Duality in quantum transport models

We develop the `duality approach', that has been extensively studied for classical models of transport, for quantum systems in contact with a thermal `Lindbladian' bath. The method provides (a) a mapping of the original model to a simpler one, containing only a few particles and (b) shows that any dynamic process of this kind with generic baths may be mapped onto one with equilibrium baths. We exemplify this through the study of a particular model: the quantum symmetric exclusion process introduced in [D. Bernard, T. Jin, Phys. Rev. Lett. 123, 080601 (2019)]. As in the classical case, the whole construction becomes intelligible by considering the dynamical symmetries of the problem.

cond-mat.stat-mech↗