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CRC Press eBooks

Grassmann Variables

I. J. R. Aitchison, Anthony J.G. Hey

Published Apr 19, 2024• 0 citations
Publisher

Abstract

In the path integral representation of quantum amplitudes ( chapter 16 ) the fields are regarded as classical functions. Matrix elements of time-ordered products of bosonic operators could be satisfactorily represented (see the discussion following ( 16.79 )). But something new is needed to represent, for example, the time-ordered product of two fermionic operators: there must be a sign difference between the two orderings, since the fermionic operators anticommute . Thus it seems that to represent amplitudes involving fermionic operators by path integrals we must think in terms of ‘classical’ anticommuting variables.

DOI
10.1201/9781003411666-p

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In the path integral representation of quantum amplitudes ( chapter 16 ) the fields are regarded as classical functions. Matrix elements of time-ordered products of bosonic operators could be satisfactorily represented (see the discussion following ( 16.79 )). But something new is needed to represent, for example, the time-ordered product of two fermionic operators: there must be a sign difference between the two orderings, since the fermionic operators anticommute . Thus it seems that to represent amplitudes involving fermionic operators by path integrals we must think in terms of ‘classical’ anticommuting variables.
A variation to the usual formulation of Grassmann representation path integrals is presented. Time-indexed anticommuting partners are introduced for each Grassmann coherent state variable and a general method for handling the effect of these introduced Grassmann partners is also developed. These Grassmann partners carry the nilpotency and anticommutivity of the Grassmann coherent state variables into the propagator and allow the propagator to be written as a path integral. Two examples are introduced in which this variation is shown to yield exact results. In particular, exact results are demonstratated for a spin-1/2 in a time-dependent magnetic field, and for a spin-boson system. The stationary path approximation is then shown to be exact for each example.
Abstract An interacting spin system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a discrete time formalism are used in combination with a basis set involving Grassmann variables coherent states to get a many‐body analytic propagator. The generating function thus obtained leads, after an adequate tracing over Grassmann variables in the imaginary time domain, to the partition function. A spin 1/2 Hamiltonian involving the whole set of interactions is considered. Fermion operators satisfying the standard anticommutation relations are constructed from the raising and lowering spin operators via the Jordan–Wigner transformation. The partition function obtained is more general than the partition function of the traditional Ising model involving only first‐neighbor interactions. Computations were performed assuming that the coupling as a function of the distance can be reasonably well represented by an Airy function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002
Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory.