
ROUVEN FRASSEK
Ricercatore t.d. art. 24 c. 3 lett. B Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede exMatematica

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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 wellknown KipnisMarchioroPresutti (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 nonequilibrium 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 macroscale by a product measure. Integrability manifests itself through the description of the model via the open Heisenberg chain with noncompact 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 BGGType Resolutions
[Articolo su rivista]
Frassek, R; Karpov, I; Tsymbaliuk, A
abstract
We obtain BGGtype formulas for transfer matrices of irreducible finitedimensional 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 infinitedimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinitedimensional transfer matrices into the products of two Baxter Qoperators, arising from our previous study Frassek et al. (Adv. Math. 401:108283, 2022), Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545619, 2022) of the degenerate Lax matrices. Our approach is crucially based on the new BGGtype resolutions of the finitedimensional gmodules, 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 Borbits.
2022
 Algebraic Bethe ansatz for Qoperators of the open XXX Heisenberg chain with arbitrary spin
[Articolo su rivista]
Frassek, Rouven; M Szecsenyi, Istvan
abstract
In this note we construct Qoperators 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 Qoperators using the fundamental commutation relations. By acting on Bethe offshell states and explicitly evaluating the trace in the auxiliary space we compute the eigenvalues of the Qoperators in terms of Bethe roots and show that the unwanted terms vanish if the Bethe equations are satisfied.
2022
 Exact solution of an integrable nonequilibrium particle system
[Articolo su rivista]
Frassek, Rouven; Giardinà, Cristian
abstract
We consider the integrable family of symmetric boundarydriven interacting particle systems that arise from the noncompact XXX Heisenberg model in one dimension with open boundaries. In contrast to the wellknown 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 nonequilibrium steadystate 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. Longrange correlations are computed in the finitevolume system. The exact solution allows us to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A byproduct of the solution is the algebraic construction of a direct mapping between the nonequilibrium steady state and the equilibrium reversible measure.
2022
 Integrable boundaries for the qHahn 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 13571), we propose integrable boundary conditions for its trigonometric deformation, which is known as the qHahn process. Following the formalism established by Mangazeev and Lu in (2019 Nucl. Phys. B 945 114665) using the stochastic Rmatrix, 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 Kmatrix.
2022
 Lax matrices from antidominantly shifted Yangians and quantum affine algebras: Atype
[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
 QQsystem and Weyltype 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 Qfunctions (QQsystem) for the integrable spin chains with the symmetry of the Dr Lie algebra. We use this QQsystem to derive new Weyltype formulas expressing transfer matrices in all symmetric and antisymmetric (fundamental) representations through r + 1 basic Qfunctions. Our functional relations are consistent with the Qoperators 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 nonequilibrium states very naturally as the (usual) free energy of the mapped system.
2020
 Eigenstates of triangularisable open XXX spin chains and closedform 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 halffilling. 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 closedform 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
 Noncompact 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 onedimensional 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 nonequilibrium 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 imaginarytime. 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 (21) superstring that has been derived directly from =4 SYM.
2020
 Oscillator realisations associated to the Dtype Yangian: Towards the operatorial Qsystem of orthogonal spin chains
[Articolo su rivista]
Frassek, R.
abstract
We present a family of novel Lax operators corresponding to representations of the RTTrealisation of the Yangian associated with Dtype Lie algebras. These Lax operators are of oscillator type, i.e. one space of the operators is infinitedimensional 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 Qoperators for so(2r) spin chains by verifying certain QQrelations 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 noncompact 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 Qsystem for noncompact super spin chains
[Articolo su rivista]
Frassek, R; Marboe, C; Meidinger, D
abstract
2016
 An integrability primer for the gaugegravity correspondence: an introduction
[Articolo su rivista]
Bombardelli, D; Frassek, R; LevkovichMaslyuk, F; Loebbert, F; Negro, S; Szécsényi, I M; Sfondrini, A; Tongeren S, J van; Torrielli, A
abstract
2016
 Onshell diagrams, Graßmannians and integrability for form factors
[Articolo su rivista]
Frassek, R; Meidinger, D; Nandan, D; Wilhelm, M
abstract
2016
 Yangiantype symmetries of nonplanar 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
 Qoperators 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 Qoperators to local charges
[Articolo su rivista]
Frassek, R; Meneghelli, C
abstract
2011
 Baxter Qoperators and representations of Yangians
[Articolo su rivista]
Bazhanov, V; Frassek, R; Łukowski, T; Meneghelli, C; Staudacher, M
abstract
2011
 Oscillator construction of Qoperators
[Articolo su rivista]
Frassek, R; Łukowski, T; Meneghelli, C; Staudacher, M
abstract