We outline a method for rigorously calculating the time-dependent as well as the static thermal properties of localized spin systems from those of an ordinary many-boson system. Our method retains the advantages of both the Holstein-Primakoff and the Dyson-Maleev transformations without having their main disadvantages. The spin operators and the boson Hamiltonian are all finite series in the boson creation and annihilation operators. Our boson Hamiltonian is Hermitian. This method establishes a rigorous correspondence between the thermal properties of the spin-\textonehalf{} isotropic Heisenberg model and those of a hardcore boson system with only two-body interactions.
We outline a method for rigorously calculating the time-dependent as well as the static thermal properties of localized spin systems from those of an ordinary many-boson system. Our method retains the advantages of both the Holstein-Primakoff and the Dyson-Maleev transformations without having their main disadvantages. The spin operators and the boson Hamiltonian are all finite series in the boson creation and annihilation operators. Our boson Hamiltonian is Hermitian. This method establishes a rigorous correspondence between the thermal properties of the spin-\textonehalf{} isotropic Heisenberg model and those of a hardcore boson system with only two-body interactions.
Within the formalism of thermo field dynamics, the boson-fermion Holstein-Primakoff transformation is constructed for the case of nonzero temperatures. For the example of the Lipkin model, the transformation in question is used to construct a thermal Hamiltonian in the form of an expansion in the parameter $${1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right. \kern-\nulldelimiterspace} {\sqrt N }}$$ , where N is the number of particles in the system. The temperature dependence of quasiparticle (fermion) and collective-excitation (boson) energies is calculated to terms of order 1/N.
We formulate the thermofield dynamics for time-dependent systems by combining the Liouville-von Neumann equation, its invariant operators, and the basic notions of thermofield dynamics. The new formulation is applied to time-dependent bosons and fermions by using the time-dependent annihilation and creation operators that satisfy the Liouville-von Neumann equation. It is shown that the thermal state is the time- and temperature-dependent vacuum state and a general formula is derived to calculate the thermal expectation value of operators.
We present a systematic numerical iteration approach to study the evolution properties of the spin-boson systems, which works well in whole coupling regime. This approach involves the evaluation of a set of coefficients for the formal expansion of the time-dependent Schrödinger equation ∣t⟩=e−iH^t∣t=0⟩ by expanding the initial state ∣t=0⟩ in Fock space. The main advantage of this method is that this set of coefficients is unique for the Hamiltonian studied, which allows one to calculate the time evolution based on the different initial states. To complement our numerical calculations, the method is applied to the Buck-Sukumar model. Furthermore, we pointed out that, when the ground state energy of the model is unbounded and no ground state exists in a certain parameter space, the unstable time evolution of the physical quantities is the natural results. Furthermore, we test the performance of the numerical method to the Hamiltonian that use anti-Hermitian terms for modeling open quantum systems.
Given a time-independent Hamiltonian H , one can construct a time-dependent Hamiltonian Ht by means of the gauge transformation Ht=UtHUt†−iUt∂tUt† . Here Ut is the unitary transformation that relates the solutions of the corresponding Schrödinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as physical. We formulate sufficient conditions on Ut ensuring that Ht is physical as long as H is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians Ht and H such that the driven many-body dynamics governed by Ht can be reduced to the quench dynamics due to the time-independent H . We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to a time-independent system without a magnetic field [F. Yan, L. Yang, and B. Li, Phys. Lett. A 251, 289–293; 259, 207–211 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where H=Ht is time-dependent but dynamically integrable.
In order to construct time-dependent pseudo-bosonic coherent states, first, we extend the non-Hermitian integrals of motion method to cases where the quantum systems are described by time-dependent non-Hermitian Hamiltonians; second, we introduce a pseudo-bosonic annihilation operator defined as a time-dependent non-Hermitian linear invariant. The pseudo-bosons operators are a pseudo-Hermitian extension of usual boson operators. In fact, they are obtained from the modification of usual boson commutation relations where the annihilation and creation operators are related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. Thus, the pseudo-bosonic coherent states are just obtained as eigenstates of the pseudo-bosonic annihilation operator. As an illustration, we study the time-dependent non-Hermitian Swanson Hamiltonian and we compare the obtained results with those in the literature (Swanson model but time-independent).
In order to develop a previously proposed method to investigate thermal effects in quantum many-particle systems, basic ideas are formulated in more general systems obeying su(2)-algebra. It is shown that Hamiltonians of a single orbit shell model with pairing correlation, the Lipkin model, a symmetric rigid rotor model and an asymmetric rigid rotor model have almost the same structure, that is algebraic structure of su(1, 1), by the use of the Holstein-Primakoff boson representation for su(2)-algebra and so-called phase-space doubling introduced to study thermal phenomena. It is also discussed that the interaction term between intrinsic system and external environment approximately represents the interesting physical process in each system at finite temperature.