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

Physical Review Letters

Application of Operator Algebras to Stochastic Dynamics and the Heisenberg Chain

Robin B. Stinchcombe, Gunter Markus Schutz

Published Jul 3, 1995• 100 citations
Publisher

Abstract

Algebraic manipulations are used to reduce a new description of stochastic dynamics and quantum spin chains involving two nonlocal operators $C,D$. For the symmetric hopping of hard-core particles, and its associated Heisenberg chain, the operator algebra may be written in the reduced form $2\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{D}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}[[C,D]{,C}^{\ensuremath{-}1}]$, ${2D}^{2}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}CDC}^{\ensuremath{-}1}{D+DC}^{\ensuremath{-}1}\mathrm{DC}$. These equations are shown to describe diffusive dynamics and phase change on interchange, respectively, and to lead to Bethe ansatz equations for the spectrum of the isotropic Heisenberg chain with symmetry-breaking boundary fields. This yields new exact results for the dynamics of boundary-driven systems.

DOI
10.1103/physrevlett.75.140

Related papers

Stochastic Reaction-Diffusion Processes, Operator Algebras and Integrable Quantum Spin Chains

Gunter Markus Schutz

We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Using the Bethe ansatz and similarity transformations this yields new exact results. In a complementary approach we generalize previous work and present a new description of these and other processes and the related quantum chains in terms of an operator algebra with quadratic relations. The full solution of the master equation of the process is thus turned into the problem of finding representations of this algebra. We find a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. We obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal boundary fields.

arXiv:cond-mat/9601081•Jan 19, 1996•8 citations

Operator Algebra for Stochastic Dynamics and the Heisenberg Chain

Robin B. Stinchcombe, Gunter Markus Schutz

The operator algebra C ( S − ) = CD − DC = ( S + ) C , D + D = DS − SD is shown to represent the stochastic dynamics of symmetric hopping of hard-core particles in one dimension and to describe the Heisenberg quantum chain. The particle or spin state is specified by strings of the operators C and D , and S is related to a current. Recursive reductions and matrix representations are used to obtain stationary and time-dependent properties, including the evolving profile for a system driven by a density gradient between open boundaries. Generalizations to other models are outlined.

Europhysics Letters (EPL)•Mar 20, 1995•67 citations

Dynamic Matrix Ansatz for Integrable Reaction-Diffusion Processes

Gunter Markus Schutz

We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Generalizing earlier work [1, 2] we present an alternative description of these processes in terms of a time-dependent operator algebra with quadratic relations. These relations generate the Bethe ansatz equations for the spectrum and turn the calculation of time-dependent expectation values into the problem of either finding representations of this algebra or of solving functional equations for the initial values of the operators. We use both strategies for the study of two specific models: (i) We construct a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. In this way we obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal, symmetry-breaking boundary fields. (ii) We consider the non-equilibrium spin relaxation of Ising spins with zero-temperature Glauber dynamics and an additional coupling to an infinite-temperature heat bath

•Jan 1, 1998•14 citations

Stochastic non-equilibrium systems

Robin B. Stinchcombe

Recent advances in the theory of non-equilibrium systems are reviewed. The emphasis is on collective behaviour, particularly non-equilibrium stochastic dynamics and steady state phase transitions. The models discussed are largely lattice-based 'exclusion' models (with hard core interactions) which can include a wide variety of collective stochastic processes and phenomena. The techniques employed range from intuitive reasoning and approximate mean field approaches to scaling and renormalization group methods, and to exact descriptions in terms of operator algebras, where many new results are included. Mappings to quantum systems, particularly lattice spin models, are also exploited, which provide new viewpoints and allow the use of standard techniques including Bethe ansatz and free fermion solutions.

Advances In Physics•Jul 1, 2001•113 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

The Bethe ansatz

Fedor Levkovich-Maslyuk

Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.

arXiv:1606.02950•Journal of Physics A Mathematical and Theoretical•Jul 20, 2016•70 citations
Algebraic manipulations are used to reduce a new description of stochastic dynamics and quantum spin chains involving two nonlocal operators C,DC,DC,D. For the symmetric hopping of hard-core particles, and its associated Heisenberg chain, the operator algebra may be written in the reduced form 2D\ifmmode˙\else˙\fi=[[C,D],C\ensuremath−1]2\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{D}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}[[C,D]{,C}^{\ensuremath{-}1}]2D\ifmmode˙\else˙\fi​=[[C,D],C\ensuremath−1], 2D2=CDC\ensuremath−1D+DC\ensuremath−1DC{2D}^{2}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}CDC}^{\ensuremath{-}1}{D+DC}^{\ensuremath{-}1}\mathrm{DC}2D2=CDC\ensuremath−1D+DC\ensuremath−1DC. These equations are shown to describe diffusive dynamics and phase change on interchange, respectively, and to lead to Bethe ansatz equations for the spectrum of the isotropic Heisenberg chain with symmetry-breaking boundary fields. This yields new exact results for the dynamics of boundary-driven systems.
We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Using the Bethe ansatz and similarity transformations this yields new exact results. In a complementary approach we generalize previous work and present a new description of these and other processes and the related quantum chains in terms of an operator algebra with quadratic relations. The full solution of the master equation of the process is thus turned into the problem of finding representations of this algebra. We find a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. We obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal boundary fields.
The operator algebra C ( S − ) = CD − DC = ( S + ) C , D + D = DS − SD is shown to represent the stochastic dynamics of symmetric hopping of hard-core particles in one dimension and to describe the Heisenberg quantum chain. The particle or spin state is specified by strings of the operators C and D , and S is related to a current. Recursive reductions and matrix representations are used to obtain stationary and time-dependent properties, including the evolving profile for a system driven by a density gradient between open boundaries. Generalizations to other models are outlined.
We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Generalizing earlier work [1, 2] we present an alternative description of these processes in terms of a time-dependent operator algebra with quadratic relations. These relations generate the Bethe ansatz equations for the spectrum and turn the calculation of time-dependent expectation values into the problem of either finding representations of this algebra or of solving functional equations for the initial values of the operators. We use both strategies for the study of two specific models: (i) We construct a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. In this way we obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal, symmetry-breaking boundary fields. (ii) We consider the non-equilibrium spin relaxation of Ising spins with zero-temperature Glauber dynamics and an additional coupling to an infinite-temperature heat bath
Recent advances in the theory of non-equilibrium systems are reviewed. The emphasis is on collective behaviour, particularly non-equilibrium stochastic dynamics and steady state phase transitions. The models discussed are largely lattice-based 'exclusion' models (with hard core interactions) which can include a wide variety of collective stochastic processes and phenomena. The techniques employed range from intuitive reasoning and approximate mean field approaches to scaling and renormalization group methods, and to exact descriptions in terms of operator algebras, where many new results are included. Mappings to quantum systems, particularly lattice spin models, are also exploited, which provide new viewpoints and allow the use of standard techniques including Bethe ansatz and free fermion solutions.
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.
Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.