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.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.

Physical Review LettersJul 3, 1995100 citations

Quantum dynamics as a stochastic process in phase space

P. L. Rubin

Theoretical and Mathematical PhysicsMar 1, 19972 citations

A dynamical universality class out of stochastic driving in interacting quantum many-body systems

Zi Cai

In this paper, we study the mean-field dynamics of a general class of quantum many-body systems with stochastically fluctuating interactions. Our findings reveal a universal algebraic decay of the order parameter $m(t)\sim t^{-\chi}$ with an exponent $\chi=\frac 13$ that is independent of most system details including the strength of the stochastic driving, the energy spectrum of the undriven systems, the initial states and even the driving protocols. The finite-size effect, as well as the relevance of our results with current experiments in high-finesse cavity QED systems are also discussed.In this paper, we study the mean-field dynamics of a general class of quantum many-body systems with stochastically fluctuating interactions. Our findings reveal a universal algebraic decay of the order parameter $m(t)\sim t^{-\chi}$ with an exponent $\chi=\frac 13$ that is independent of most system details including the strength of the stochastic driving, the energy spectrum of the undriven systems, the initial states and even the driving protocols. The finite-size effect, as well as the relevance of our results with current experiments in high-finesse cavity QED systems are also discussed.

Sep 8, 20210 citations

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.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/9601081Jan 19, 19968 citations

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.

m(t)\sim t^{-\chi}

with an exponent

\chi=\frac 13

that is independent of most system details including the strength of the stochastic driving, the energy spectrum of the undriven systems, the initial states and even the driving protocols. The finite-size effect, as well as the relevance of our results with current experiments in high-finesse cavity QED systems are also discussed.