Dynamical symmetry approach and topological field theory for path integrals of quantum spin systems

Abstract

We develop a dynamical symmetry approach to path integrals for general interacting quantum spin systems. The time-ordered exponential obtained after the Hubbard-Stratonovich transformation can be disentangled into the product of a finite number of the usual exponentials. This procedure leads to a set of stochastic differential equations on the group manifold, which can be further formulated in terms of the supersymmetric effective action. This action has the form of the Witten topological field theory in the continuum limit. As a consequence, we show how it can be used to obtain the exact results for a specific quantum many-body system which can be otherwise solved only by the Bethe ansatz. To our knowledge this represents the first example of a many-body system treated exactly using the path integral formulation. Moreover, our method can deal with time-dependent parameters, which we demonstrate explicitly.

14 pages, 4 figures

References in corpus (9)

• The large deviation approach to statistical mechanics
• Strongly-correlated multi-particle transport in one-dimension through a quantum impurity: an outline of exact and complete solutions
• Dynamic modulation of photonic crystal nanocavities using gigahertz acoustic phonons
• Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach
• A field-theoretic approach to nonequilibrium work identities
• An Exact Solution for Spin and Charge Correlations in Quantum Dots: The Effect of Level Fluctuations and Zeeman Splitting
• Topological field theory of dynamical systems. II
• Liouville coherent states
• Topological field theory of dynamical systems

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