Abstract We study the properties of a dissipative two‐state fermionic system embedded in a phonon bath at zero temperature. In contrast to the traditional path‐integral and canonical transformation approaches, we propose a self‐consistent Green's function approach. Our results show that the mean field theory overestimates the polarization of the system. To justify our approach, we consider a simplest case in which there is only one phonon mode. This simple model can be ‘exactly’ diagonalized. We find that the results from the self‐consistent Green's function approach agree well with the quantum mechanical diagonalization results.
Abstract We study the properties of a dissipative two‐state fermionic system embedded in a phonon bath at zero temperature. In contrast to the traditional path‐integral and canonical transformation approaches, we propose a self‐consistent Green's function approach. Our results show that the mean field theory overestimates the polarization of the system. To justify our approach, we consider a simplest case in which there is only one phonon mode. This simple model can be ‘exactly’ diagonalized. We find that the results from the self‐consistent Green's function approach agree well with the quantum mechanical diagonalization results.
Abstract A general method for studying the properties of free and bound polarons at arbitrary strength of electron‐phonon coupling is given from a unified point of view by first performing a suitable canonical transformation of the electron and phonon field operators and subsequently applying the double‐time Green's function formalism to the transformed polaron‐phonon Hamiltonian. Calculating the one‐polaron Green's function in the lowest approximation and employing the variational principle for the free energy, a pair of coupled differential equations is derived for the effective wave function and a trial function as introduced by the canonical transformation. From these equations, the weakly excited polaron energy levels at low temperatures can be found in the entire range of coupling constants when external fields are present. In this paper, detailed calculations are confined to the optical‐polaron problem in the zero‐field case. Intermediate‐ and strong‐coupling formulas for temperature‐dependent polaron self‐energy and effective mass are given. The strong‐coupling regime appears to increase with temperature.
In this paper we study the dynamical properties of a dissipative two-state fermionic system embedded in a phonon bath at finite temperature. The non-equilibrium thermo field dynamics, a real time formalism of thermal quantum field theory, is employed to formulate time dependent energy renormalization and derive transport equations. The renormalization procedures derived in this paper replace the on-shell energy definitions in stationary situations and thus can be used in any situations. The procedure reduces to that of equilibrium situations when there is no time dependence. We show that in a system where there are many coupled stages (with multi-components) quantum fluctuations among these states cannot be eliminated so that the long time limit of non-equilibrium theory does not exactly reproduce an equilibrium theory. The procedures and conclusions obtained also apply to more general systems with multi-components.
We have studied an infinite Hubbard chain with a spin-orbit coupling term, submitted to a uniform magnetic field as well as local phonons. By means of a Lang-Firsov transformation, we show that an effective interacting fermion model emerges. Moreover, a self-consistent mean-field theory of this model, formulated in terms of thermal Green's functions, shows that a BCS term emerges, thus leading to a superconducting phase transition at zero temperature. We find analytical expressions for the phase boundary, that agree well with exact numerical diagonalization of the Hamiltonian.
We characterize the response of a Mott insulating system to a static electric field in terms of its conducting and spectral properties. Dissipation is included by a coupling to fermionic baths and to either optical or acoustic phonons. This paper extends and completes the analysis made in a previous work by the authors [Mazzocchi et al., Phys. Rev. B 106, 125123 (2022)]. In the present work, phonons are included diagrammatically within the Migdal approximation by also including self-consistency from the electronic feedback. The nonequilibrium steady state is addressed by means of the dynamical mean-field theory based on the nonequilibrium Green's function approach, while the so-called auxiliary master equation approach is employed as impurity solver. With optical phonons the self-consistency suppresses the steady-state current for field strengths comparable to the band gap with respect to the non-self-consistent case. This is due to the interaction of phonons with the hot electrons of the lattice which increases their temperature, thus providing a less effective relaxation channel for the current-induced Joule heat. In addition, in the case of optical phonons, the results are essentially independent of the temperature of the fermionic baths, as the latter is sensibly smaller than their characteristic frequency. On the other hand, with acoustic phonons the steady-state current is slightly suppressed by the self-consistent treatment only at field strengths close to half of the gap and especially at very small phonon frequency. Also, in this case, the results seem to slightly depend on the temperature of the fermionic baths.
The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kBT. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.
Abstract Green-function techniques are among the most versatile tools for investigating quantum many-body systems at finite temperature. Yet, standard textbook treatments often fragment the formalism—separating bosons and fermions, systems with massless and massive particles, or restricting proofs to special Hamiltonians—thereby obscuring the structural unity of the theory. This work develops a general, Hamiltonian-independent framework for both real- and imaginary-time Green-functions, valid for any combination of bosonic and fermionic degrees of freedom, including mixed systems such as interacting electron–phonon models. The key conceptual advance is the nature index η , which complements the conventional statistical sign ζ . While ζ encodes particle statistics, η distinguishes canonical quantization of systems with massless particles ( η = 0) from second quantization of systems with massive particles ( η = 1). This classification removes ambiguities in chemical-potential terms and unifies the treatment of photons, phonons, electrons, and their mixtures under a single notation. From this basis, we formulate generalized ensemble constructions, generalized Lehmann-representations, and a fluctuation-dissipation theorem valid for arbitrary Hamiltonians. Several results—such as the monotonicity of particle number in the chemical potential, an explicit formula for the quadratic contribution to the internal energy from the retarded Green-function, and the Hartree-contribution to the self-energy in electron–phonon systems—are either absent from, or only sparsely mentioned in, existing literature. We further introduce Green’s Chart, a visual and algebraic map that systematically connects all Green-function types through analytic continuation, Fourier-transforms, and Wick-rotations, enabling all to be generated from the Matsubara Green-function. Applications to both non-interacting and interacting models demonstrate the framework’s ability to compute densities of states, thermodynamic potentials, and spectral functions with a consistent formalism. The results provide a complete and reproducible toolkit for equilibrium many-body theory, consolidating scattered results into a unified structure suitable for both pedagogical and computational purposes.