Grassmann coherent states representation of the path integral: Evaluation of the generating function for spin systems

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.

Dynamics and Nonclassical Photon Statistics in the Interaction of Two-Level Spin Systems With a Two-Mode Cavity Field: A Generalized Jaynes-Cummings Model

“…2 to obtain the energy of the spin system restores the discrete translational invariance of the infinite lattice in the finite sample needed for the simulations. [40][41][42]49 The size of the spin chain is n = 14, and thus the manifold of excited states was thermally averaged over the 16383 configurations out of the vacuum state. In fact, the number of independent spin states of a given multiplicity, which exist for a system of n spins, denoted by f (n, S), can be derived in an analogical fashion.…”

“…49,50 In Eq. (1), n † , n are fermionic operators and the Ξ k are the associated eigenvalues, given by 43…”

Population inversion, temperature, and photon distributions of the generalized fermionic Ising ferromagnetic model: Path‐integral representation of the spin system

“…(2.1) reduces to standard Grassmann integrals after repeated use of Eq. (2.14) 25, 26. Thus, invoking the antiperiodic boundary condition performing an analytic continuation to Euclidean times through the Wick rotation, Δτ → − i Δτ, and the subsequent substitution Δτ/ℏ → β (≡ 1/ kT ), the imaginary time partition function of the spin system is obtained through the trace formula for fermions 25, 26, 28, …”

“…The diagonal character of the Hamiltonian (2.9) does not destroy any generality concerning the possible existence of nondiagonal quadratic forms of creation and annihilation operators. In fact, it should be addressed that if the Hamiltonian of the spin system involving quadratic forms of the fermion creation and annihilation operators is not diagonal, it has to be diagonalized before mounting it in the path integral 14, 25, 26. In any case, the invariance of the spin Hamiltonian is preserved through the Jordan–Wigner transformation, at least in those systems of free ends, such as the present model (for cyclic systems, there is a change of sign in the quadratic form of the operators involving the last and the first site of the chain).…”

“…The model system to be studied involves the interaction of an Ising ferromagnetic model with a single mode of a quantized cavity field in the rotating‐wave approximation; this interaction will be investigated through a coherent state path‐integral representation for the spin system constructed over anticommutating Grassmann variables. This technique, pioneered by Klauder 24, is particularly suited for describing quantum dynamics of spin systems 14, 24–26 and shown to lead to a partition function generated as a configuration expansion (e.g., clusters of one‐, two‐ spins) from which the exact energy of the spin system involving all possible interactions (generalized Ising model) can be obtained in the limit of low and high temperatures 14, 25, 26.…”