Path integral of spin models

Grinberg, Horacio

doi:10.1016/s0375-9601(03)00464-x

Cited by 9 publications

(11 citation statements)

References 30 publications

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

Section: Cyclic Xy Spin Modelmentioning

confidence: 99%

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

Grinberg

2008

Int. J. Mod. Phys. B

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The interaction of a two-level XY n-spin system with a two-mode cavity field is investigated through a generalized Jaynes-Cummings model in the rotating wave approximation. The spontaneous decay of a spin level was treated by considering the interaction of the two-level spin system with the modes of the universe in the vacuum state. The different cases of interest, characterized in terms of a detuning parameter for each mode, which emerge from the nonvanishing of certain commutation relations between interaction picture Hamiltonians associated with each mode, were analytically implemented and numerically discussed for various values of the initial mean photon number and spin-photon coupling constants. Photon distribution, time evolution of the spin population inversion, as well as the statistical properties of the field leading to the possible production of nonclassical states, such as antibunched light and violations of the Cauchy-Schwartz inequality are examined for an excited initial state. It was assumed that the two modes are initially in coherent states and have the same photon distribution. The case of zero detuning of both modes was treated in terms of a linearization of the expansion of the time evolution operator, while in other three cases, the computations were conducted via second-and third-order Dyson perturbation expansion of the time evolution operator matrix elements for the excited and ground states respectively.

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

Section: Cyclic Xy Spin Modelmentioning

confidence: 99%

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

Grinberg

2008

Int. J. Mod. Phys. B

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“…Two different Grassmann coherent states are not orthogonal 26, and the overlap is given by After computing the Hamiltonian matrix elements for each time partition according to the multiple integral involving short‐time propagators in Eq. (2.1) reduces to standard Grassmann integrals after repeated use of Eq.…”

Section: Generating Function Of An Interacting Spin Systemmentioning

confidence: 99%

“…(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, …”

Section: Generating Function Of An Interacting Spin Systemmentioning

confidence: 99%

“…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).…”

Section: Generating Function Of An Interacting Spin Systemmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Grinberg¹

2006

Int J of Quantum Chemistry

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ABSTRACT:The quantum partition function and the emerging energy of a fermionic Ising ferromagnetic model involving all possible interactions (generalized Ising model) are obtained from an appropriate tracing of the analytic propagator path integral over Grassmann variable coherent nonorthogonal states in the imaginary time domain. The dynamics derived from the interaction of this system with a single-mode cavity field in the rotating wave approximation is investigated for nonresonant states within the framework of the Jaynes-Cummings two-level model consisting of the vacuum state and a thermally averaged manifold of excited states. Time evolution of the population inversion is computed in the nanosecond time scale, assuming that the initial coherent state of the field is given by a Poisson distribution. The limit of high temperatures characterizing the manifold of excited states becomes chaotic with rapid oscillations, whereas the ground state is described correctly in the thermodynamic limit by the vacuum state. A breakup is seen in the photon distribution into a series of peaks because of the detuning between the spin system and the field. However, this structure is smeared out, and the general shape is preserved in the computation emerging from the Laplace transform of the photon distribution.

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Interaction of a two‐level cyclic XY n‐spin model with a two‐mode cavity field in off‐resonant states

Grinberg¹

2007

Int J of Quantum Chemistry

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ABSTRACT:The interaction of the XY n-spin cyclic model with a two-mode cavity field in the rotating-wave approximation is investigated in the framework of a generalized Jaynes-Cummings two-level system consisting of the vacuum state and a thermally averaged manifold of excited sates. Computation of the energy of this manifold allows this interaction to be examined in off-resonant states. Time evolution of the population inversion, photon distribution, and temperature distribution for an excited initial state are computed via second-and third-order perturbation expansion of the time evolution operator matrix elements for the excited and ground states, respectively and for an ideal squeezed initial coherent state of the cavity field. It was assumed that the two modes have initially the same photon distribution. The pattern of the spin population inversion appears as a manifestation of multiple and complicated inerferences, which is mathematically reflected in a double discrete summation that appears in the calculation of the dynamics.

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Path integral of spin models

Cited by 9 publications

References 30 publications

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

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

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

Interaction of a two‐level cyclic XY n‐spin model with a two‐mode cavity field in off‐resonant states

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