Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1904.02191 · Journal of Physics A Mathematical and Theoretical

The non-compact XXZ spin chain as stochastic particle process

Rouven Frassek

Published Jul 8, 2019• 9 citations
arXivHTMLPDF

Abstract

Abstract In this note we relate the Hamiltonian of the integrable non-compact spin s XXZ chain to the Markov generator of a stochastic particle process. The hopping rates of the continuous-time process are identified with the ones of a q-Hahn asymmetric zero range model. The main difference with the asymmetric simple exclusion process, which can be mapped to the ordinary XXZ spin chain, is that multiple particles can occupy one and the same site. For the non-compact spin XXZ chain the associated stochastic process reduces to the multiparticle asymmetric diffusion model introduced by Sasamoto–Wadati.

DOI
10.1088/1751-8121/ab2fb1

Related papers

The non-compact XXZ spin chain as stochastic particle process

Rouven Frassek

In this note we relate the Hamiltonian of the integrable non-compact spin $s$ XXZ chain to the Markov generator of a stochastic particle process. The hopping rates of the continuous-time process are identified with the ones of a q-Hahn asymmetric zero range model. The main difference with the asymmetric simple exclusion process (ASEP), which can be mapped to the ordinary XXZ spin chain, is that multiple particles can occupy one and the same site. For the non-compact spin $\\frac{1}{2}$ XXZ chain the associated stochastic process reduces to the multiparticle asymmetric diffusion model introduced by Sasamoto-Wadati.

•Jan 1, 2019•7 citations

Non-compact quantum spin chains as integrable stochastic particle processes

Rouven Frassek, Giardinà, C., Jorge Kurchan B

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.

•Jan 1, 2020•25 citations

A sufficient criterion for integrability of stochastic many-body dynamics and quantum spin chains

Vladislav Popkov, M. Ebrahim Fouladvand, Gunter Markus Schutz

We propose a dynamical matrix product ansatz describing the stochastic dynamics of two species of particles with excluded-volume interaction and the quantum mechanics of the associated quantum spin chains, respectively. The time-dependent algebra which is obtained from the action of the Markov generator of the exclusion process (or quantum Hamiltonian of the spin chain, respectively) is given in terms of a set of quadratic relations. By analysing the permutation consistency of the induced cubic relations, we obtain sufficient conditions on the hopping rates (i.e. the quantum mechanical interaction constants) which allow us to identify integrable models. From the dynamical algebra we construct the quadratic algebra of Zamolodchikov type, the associativity of which is a Yang–Baxter equation. We also obtain directly from the dynamical matrix product ansatz the Bethe ansatz equations for the spectra of these models.

arXiv:hep-th/0205169•Journal of Physics A Mathematical and General•Aug 7, 2002•25 citations

Integrable dissipative exclusion process: Correlation functions and physical properties

Nicolas Crampé, E. Ragoucy, V. Rittenberg +1 more

We study a one-parameter generalization of the symmetric simple exclusion process on a one-dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rates to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. We also show that the stationary state can be expressed in a matrix product form permitting to compute the multipoints correlation functions as well as the mean value of the lattice and the creation-annihilation currents. Finally, the variance of the lattice current is computed for a finite-size system. In the thermodynamic limit, it matches the value obtained from the associated macroscopic fluctuation theory.

arXiv:1603.06796•Physical review. E•Sep 1, 2016•18 citations

Asymmetric exclusion model with impurities

Matheus J. Lazo, Anderson A. Ferreira

An integrable asymmetric exclusion process with impurities is formulated. The model displays the full spectrum of the stochastic asymmetric XXZ chain plus new levels. We derive the Bethe equations and calculate the spectral gap for the totally asymmetric diffusion at half filling. While the standard asymmetric exclusion process without impurities belongs to the KPZ universality class with an exponent 3/2, our model has a scaling exponent 5/3.

arXiv:1005.3780•Physical Review E•May 20, 2010•6 citations

Reaction-Diffusion Processes of Hard-Core Particles

Gunter Markus Schutz

We study a 12-parameter stochastic process involving particles with two-site interaction and hardcore repulsion on a d-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes and other processes, the stochastic variables are particle occupation numbers taking values n⃗x = 0, 1. We show that on a 10-parameter submanifold the k-point equaltime correlation functions 〈n⃗x1 · · · n⃗xk 〉 satisfy linear differential-difference equations involving no higher correlators. In particular, the average density 〈n⃗x 〉 satisfies an integrable diffusion-type equation. These properties are explained in terms of dual processes and various duality relations are derived. By defining the time evolution of the stochastic process in terms of a quantum Hamiltonian H, the model becomes equivalent to a lattice model in thermal equilibrium in d + 1 dimensions. We show that the spectrum of H is identical to the spectrum of the quantum Hamiltonian of a d-dimensional, anisotropic spin-1/2 Heisenberg model. In one dimension our results hint at some new algebraic structure behind the integrability of the system. Keywords: Reaction-diffusion processes, correlation functions, integrable models An interesting class of non-equilibrium problems with a rich dynamical behaviour are

•Jan 1, 1994•38 citations
Abstract In this note we relate the Hamiltonian of the integrable non-compact spin s XXZ chain to the Markov generator of a stochastic particle process. The hopping rates of the continuous-time process are identified with the ones of a q-Hahn asymmetric zero range model. The main difference with the asymmetric simple exclusion process, which can be mapped to the ordinary XXZ spin chain, is that multiple particles can occupy one and the same site. For the non-compact spin XXZ chain the associated stochastic process reduces to the multiparticle asymmetric diffusion model introduced by Sasamoto–Wadati.
In this note we relate the Hamiltonian of the integrable non-compact spin sss XXZ chain to the Markov generator of a stochastic particle process. The hopping rates of the continuous-time process are identified with the ones of a q-Hahn asymmetric zero range model. The main difference with the asymmetric simple exclusion process (ASEP), which can be mapped to the ordinary XXZ spin chain, is that multiple particles can occupy one and the same site. For the non-compact spin frac12\\frac{1}{2}frac12 XXZ chain the associated stochastic process reduces to the multiparticle asymmetric diffusion model introduced by Sasamoto-Wadati.
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 mathfraksl(2)\\mathfrak{sl}(2)mathfraksl(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 mathcalN=4\\mathcal{N}=4mathcalN=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 mathfraksl(2∣1)\\mathfrak{sl}(2|1)mathfraksl(2∣1) superstring that has been derived directly from mathcalN=4\\mathcal{N}=4mathcalN=4 SYM.
We propose a dynamical matrix product ansatz describing the stochastic dynamics of two species of particles with excluded-volume interaction and the quantum mechanics of the associated quantum spin chains, respectively. The time-dependent algebra which is obtained from the action of the Markov generator of the exclusion process (or quantum Hamiltonian of the spin chain, respectively) is given in terms of a set of quadratic relations. By analysing the permutation consistency of the induced cubic relations, we obtain sufficient conditions on the hopping rates (i.e. the quantum mechanical interaction constants) which allow us to identify integrable models. From the dynamical algebra we construct the quadratic algebra of Zamolodchikov type, the associativity of which is a Yang–Baxter equation. We also obtain directly from the dynamical matrix product ansatz the Bethe ansatz equations for the spectra of these models.
We study a one-parameter generalization of the symmetric simple exclusion process on a one-dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rates to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. We also show that the stationary state can be expressed in a matrix product form permitting to compute the multipoints correlation functions as well as the mean value of the lattice and the creation-annihilation currents. Finally, the variance of the lattice current is computed for a finite-size system. In the thermodynamic limit, it matches the value obtained from the associated macroscopic fluctuation theory.
We study a 12-parameter stochastic process involving particles with two-site interaction and hardcore repulsion on a d-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes and other processes, the stochastic variables are particle occupation numbers taking values n⃗x = 0, 1. We show that on a 10-parameter submanifold the k-point equaltime correlation functions 〈n⃗x1 · · · n⃗xk 〉 satisfy linear differential-difference equations involving no higher correlators. In particular, the average density 〈n⃗x 〉 satisfies an integrable diffusion-type equation. These properties are explained in terms of dual processes and various duality relations are derived. By defining the time evolution of the stochastic process in terms of a quantum Hamiltonian H, the model becomes equivalent to a lattice model in thermal equilibrium in d + 1 dimensions. We show that the spectrum of H is identical to the spectrum of the quantum Hamiltonian of a d-dimensional, anisotropic spin-1/2 Heisenberg model. In one dimension our results hint at some new algebraic structure behind the integrability of the system. Keywords: Reaction-diffusion processes, correlation functions, integrable models An interesting class of non-equilibrium problems with a rich dynamical behaviour are