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ROUVEN FRASSEK

Ricercatore t.d. art. 24 c. 3 lett. B
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

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Pubblicazioni

2023 - Integrable heat conduction model [Articolo su rivista]
Franceschini, Chiara; Frassek, Rouven; Giardina, Cristian
abstract

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.

2023 - Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions [Articolo su rivista]
Frassek, R; Karpov, I; Tsymbaliuk, A
abstract

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 Yan-gian 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 correspond-ing 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.

2022 - Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin [Articolo su rivista]
Frassek, Rouven; M Szecsenyi, Istvan
abstract

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.

2022 - Exact solution of an integrable non-equilibrium particle system [Articolo su rivista]
Frassek, Rouven; Giardinà, Cristian
abstract

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.

2022 - Integrable boundaries for the q-Hahn process [Articolo su rivista]
Frassek, R.
abstract

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

2022 - Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type [Articolo su rivista]
Frassek, Rouven; Pestun, Vasily; Tsymbaliuk, Alexander
abstract

2022 - Rational Lax Matrices from Antidominantly Shifted Extended Yangians: BCD Types [Articolo su rivista]
Frassek, Rouven; Tsymbaliuk, Alexander
abstract

2021 - Duality in quantum transport models [Articolo su rivista]
Frassek, R.; Giardina', C.; Kurchan, J.
abstract

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 [1]. As in the classical case, the whole construction becomes intelligible by considering the dynamical symmetries of the problem.

2021 - QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains [Articolo su rivista]
Ferrando, G.; Frassek, R.; Kazakov, V.
abstract

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.

2020 - Duality and hidden equilibrium in transport models [Articolo su rivista]
Frassek, R.; Giardina', C.; Kurchan, J.
abstract

A large family of diffusive models of transport that have been considered in the past years admit a transformation into the same model in contact with an equilibrium bath. This mapping holds at the full dynamical level, and is independent of dimension or topology. It provides a good opportunity to discuss questions of time reversal in out of equilibrium contexts. In particular, thanks to the mapping one may define the free energy in the non-equilibrium states very naturally as the (usual) free energy of the mapped system.

2020 - Eigenstates of triangularisable open XXX spin chains and closed-form solutions for the steady state of the open SSEP [Articolo su rivista]
Frassek, R
abstract

In this article we study the relation between the eigenstates of open rational Heisenberg spin chains with different boundary conditions. The focus lies on the relation between the spin chain with diagonal boundary conditions and the spin chain with triangular boundary conditions as well as the class of spin chains that can be brought to such form by certain similarity transformations in the physical space. The boundary driven Symmetric Simple Exclusion Process (open SSEP) belongs to the latter. We derive a transformation that maps the eigenvectors of the diagonal spin chain to the eigenvectors of the triangular chain. This transformation yields an essential simplification for determining the states beyond half-filling. It allows to first determine the eigenstates of the diagonal chain through the Bethe ansatz on the fully excited reference state and subsequently map them to the triangular chain for which only the vacuum serves as a reference state. In particular the transformed reference state, i.e. the fully excited eigenstate of the triangular chain, is presented at any length of the chain. It can be mapped to the steady state of the open SSEP. This results in a closed-form expression for the probabilities of particle distributions and correlation functions in the steady state. Further, the complete set of eigenstates of the Markov generator is expressed in terms of the eigenstates of the diagonal open chain.

2020 - Non-compact quantum spin chains as integrable stochastic particle processes [Articolo su rivista]
Frassek, R.; Giardinà, C; Kurchan, J
abstract

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 (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 =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 (2|1) superstring that has been derived directly from =4 SYM.

2020 - Oscillator realisations associated to the D-type Yangian: Towards the operatorial Q-system of orthogonal spin chains [Articolo su rivista]
Frassek, R.
abstract

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.

2019 - A Family of GLr Multiplicative Higgs Bundles on Rational Base [Articolo su rivista]
Frassek, R; Pestun, V
abstract

2019 - The non-compact XXZ spin chain as stochastic particle process [Articolo su rivista]
Frassek, R
abstract

2017 - Boundary perimeter Bethe ansatz [Articolo su rivista]
Frassek, R
abstract

2017 - Evaluation of the operatorial Q-system for non-compact super spin chains [Articolo su rivista]
Frassek, R; Marboe, C; Meidinger, D
abstract

2016 - An integrability primer for the gauge-gravity correspondence: an introduction [Articolo su rivista]
Bombardelli, D; Frassek, R; Levkovich-Maslyuk, F; Loebbert, F; Negro, S; Szécsényi, I M; Sfondrini, A; Tongeren S, J van; Torrielli, A
abstract

2016 - On-shell diagrams, Graßmannians and integrability for form factors [Articolo su rivista]
Frassek, R; Meidinger, D; Nandan, D; Wilhelm, M
abstract

2016 - Yangian-type symmetries of non-planar leading singularities [Articolo su rivista]
Frassek, R; Meidinger, D
abstract

2015 - Algebraic Bethe ansatz for Q -operators: the Heisenberg spin chain [Articolo su rivista]
Frassek, R
abstract

2015 - Q-operators for the open Heisenberg spin chain [Articolo su rivista]
Frassek, R; Szécsényi, I
abstract

2014 - Bethe ansatz for Yangian invariants: Towards super Yang–Mills scattering amplitudes [Articolo su rivista]
Frassek, R; Kanning, N; Ko, Y; Staudacher, M
abstract

2013 - Baxter operators and Hamiltonians for “nearly all” integrable closed spin chains [Articolo su rivista]
Frassek, R; Lukowski, T; Meneghelli, C; Staudacher, M
abstract

2013 - From Baxter Q-operators to local charges [Articolo su rivista]
Frassek, R; Meneghelli, C
abstract

2011 - Baxter Q-operators and representations of Yangians [Articolo su rivista]
Bazhanov, V; Frassek, R; Łukowski, T; Meneghelli, C; Staudacher, M
abstract

2011 - Oscillator construction of Q-operators [Articolo su rivista]
Frassek, R; Łukowski, T; Meneghelli, C; Staudacher, M
abstract