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Journal of Physics A Mathematical and General

Langevin dynamics, stochastic quantization and the supersymmetric systems

Alexander Punnoose, Rudolf A. Römer

Published Apr 21, 1996• 1 citation
Publisher

Abstract

We establish a novel correspondence between the spacetime correlators of one-dimensional (1D) N-particle classical stochastic models described by a Langevin equation with that of the ground-state dynamics of a class of integrable 1D interacting many-body quantum models of the supersymmetric elliptic type. We show that these seemingly different concepts of stochastic systems, supersymmetry and quantum integrability can be viewed in a unified framework. Starting with an Lax matrix, we show that row (column) sums driven by a Gaussian noise term may be interpreted as a set of forward (backward) Langevin equations. Then, following functional path integral methods of stochastic quantization, we straightforwardly find an associated supersymmetric 1D quantum Hamiltonian. If, further, the classical stochastic system consists of two-body interactions only and we also want the quantum interactions to be of two-body type, we find that the only class of interactions permissible for the quantum models corresponds to the elliptic models. The algebraic structure that emerges very naturally reproduces the proof of integrability and allows the identification of the ground-state wavefunction of these quantum models.

DOI
10.1088/0305-4470/29/8/013

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We establish a novel correspondence between the spacetime correlators of one-dimensional (1D) N-particle classical stochastic models described by a Langevin equation with that of the ground-state dynamics of a class of integrable 1D interacting many-body quantum models of the supersymmetric elliptic type. We show that these seemingly different concepts of stochastic systems, supersymmetry and quantum integrability can be viewed in a unified framework. Starting with an Lax matrix, we show that row (column) sums driven by a Gaussian noise term may be interpreted as a set of forward (backward) Langevin equations. Then, following functional path integral methods of stochastic quantization, we straightforwardly find an associated supersymmetric 1D quantum Hamiltonian. If, further, the classical stochastic system consists of two-body interactions only and we also want the quantum interactions to be of two-body type, we find that the only class of interactions permissible for the quantum models corresponds to the elliptic models. The algebraic structure that emerges very naturally reproduces the proof of integrability and allows the identification of the ground-state wavefunction of these quantum models.
We use the stochastic quantization method to construct a supersymmetric version of the quantum spherical model. This is based on the equivalence between the Brownian motion described by a Langevin equation and the supersymmetric quantum mechanics, which is connected with the existence of the Nicolai map. We investigate the critical behavior of the supersymmetric model at finite as well as at zero temperatures, showing that it exhibits both classical and quantum phase transitions, and determine the critical dimensions.
We present an algebraic structure that provides an interesting and novel link between supersymmetry and quantum integrability. This structure underlies two classes of models that are exactly solvable in one dimension and belong to the 1/r2{\mathit{r}}^{2}r2 family of interactions. The algebra consists of the commutation between a ``super-Hamiltonian'' and two other operators, in an enlarged Hilbert space. These reduce to quantal Ordered Lax equations when projected onto the original subspace, and to a statement about the ``Harmonic Lattice Potential'' structure of the Lax operator, leading to a highly automatic proof of the integrability of these models and to an interesting hierarchy of new models.
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We give a new description of quantum Brownian motion in terms of stochastic pure states. The corresponding path integral propagator allows us to establish a direct connection to the classical Langevin equation, in the Schrodinger picture. We show that in the quantum domain, one is naturally led to consider two stochastic processes driving the Brownian dynamics, one of them representing thermal fluctuations, as in the classical case. The second process reflects growing entanglement between the Brownian particle and its environment, and is therefore a truly quantum noise process. Technically, our result rests on the representation of the full propagator of the Brownian particle and its environment in a coherent state basis. For open system dynamics that may be described by a master equation, such stochastic schemes have already been proven to be efficient Monte Carlo methods. Here, we give a new stochastic scheme that covers low temperatures and strong friction, the latter limit being the case Einstein originally investigated.
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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.