Grassmann coherent states for spin systems

Sanjiv Shresta

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.

K. Berrada, M. El Baz, Y. Hassouni

Pablo G. O. Anicich, Horacio Grinberg

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

Monique Combescure, Didier Robert

Sumiyoshi Abe

The adiabatic evolution of Pauli spin in a slowly varying external magnetic field is studied by using the fermionic coherent state defined by the Grassmann-variant unitary displacement from the stable adiabatic vacuum. This manifests the interrelation between Berry's phase and Hannay's angle in a peculiar manner. The gauge field strength giving rise to the geometrical phase is found to be that of a monopole of a Grassmann charge. We quantize this charge based on the Bohr-Sommerfeld-type rule for Grassmannian systems and then correctly obtain Berry's phase.