Self-consistent Projection Operator Research Articles

Self-consistent projection operator theory in nonlinear quantum optical systems: A case study on degenerate optical parametric oscillators

Nonlinear quantum optical systems are of paramount relevance for modern quantum technologies, as well as for the study of dissipative phase transitions. Their nonlinear nature makes their theoretical study very challenging and hence they have always served as great motivation to develop new techniques for the analysis of open quantum systems. In this article we apply the recently developed self-consistent projection operator theory to the degenerate optical parametric oscillator to exemplify its general applicability to quantum optical systems. We show that this theory provides an efficient method to calculate the full quantum state of each mode with high degree of accuracy, even at the critical point. It is equally successful in describing both the stationary limit and the dynamics, including regions of the parameter space where the numerical integration of the full problem is significantly less efficient. We further develop a Gaussian approach consistent with our theory, which yields sensibly better results than the previous Gaussian methods developed for this system, most notably standard linearization techniques.

Self-consistent projection operator theory for quantum many-body systems

We derive an exact equation of motion for the reduced density matrices of individual subsystems of quantum many-body systems of any lattice dimension and arbitrary system size. Our projection operator based theory yields a highly efficient analytical and numerical approach. Besides its practical use it provides a novel interpretation and systematic extension of mean-field approaches and an adaption of open quantum systems theory to settings where a dynamically evolving environment has to be taken into account. We show its high accuracy for two significant classes of complex quantum many-body dynamics, unitary evolutions of non-equilibrium states in closed and stationary states in driven-dissipative systems.

Nonlocal excitations and 1/8 singularity in cuprates

Momentum-dependent excitation spectra of the two-dimensional Hubbard model on a square lattice have been investigated at zero temperature on the basis of the full self-consistent projection operator method in order to clarify nonlocal effects of electron correlations on the spectra. Intersite antiferromagnetic correlations are found to cause shadow bands and enhance the Mott-Hubbard splittings near the half-filling. Furthermore, nonlocal excitations are shown to move the critical doping concentration δ h *, at which the singular quasiparticle peak is located just on the Fermi level, from δ h * = 0.153 (the single-site value) to δ h * = 0.123. The latter suggests the occurrence of an instability such as the stripe at δ h * = 1/8.

Full Self-Consistent Projection Operator Approach to Nonlocal Excitations in Solids

A self-consistent projection operator method for single-particle excitations is developed. It describes the nonlocal correlations on the basis of a projection technique to the retarded Green function and the off-diagonal effective medium. The theory takes into account long-range intersite correlations making use of an incremental cluster expansion in the medium. A generalized self-consistent coherent potential is derived. It yields the momentum-dependent excitation spectra with high resolution. Numerical studies for the Hubbard model on a simple cubic lattice at half filling show that the theory is applicable in a wide range of Coulomb interaction strength. In particular, it is found that the long-range antiferromagnetic correlations in the strong interaction regime cause shadow bands in the low-energy region and sub-peaks of the Mott-Hubbard bands.

Nonlocal Excitation Spectra in Two-Dimensional Doped Hubbard Model

Single-particle excitation spectra of the two-dimensional Hubbard model on the square lattice near half filling and at zero temperature are investigated on the basis of the self-consistent projection operator method. The method guarantees a high accuracy of the spectra with high energy and high momentum resolutions. It takes into account long-range intersite correlations as well as the strong on-site correlations. Effects of nonlocal excitations are clarified by comparing the results with those of the single-site approximation. The calculated spectra verify the quantum Monte-Carlo results for finite temperatures. The spectra at the Fermi level yield a hole-like Fermi surface in the underdoped region and an electron-like Fermi surface in the overdoped region. From a numerical analysis of the momentum dependent effective mass and self-energy, it is concluded that a marginal Fermi-liquid like state persists even at finite doping concentrations in the strongly correlated region because a van Hove singularity ...

Marginal Fermi liquid and kink structure of quasiparticles in cuprates

On the basis of the self-consistent projection operator method for nonlocal excitations, we show that a kink structure appears in the quasiparticle excitation spectrum of the two-dimensional Hubbard model at low doping concentrations. It is caused by a mixing between the quasiparticle state and the excitations with short-range antiferromagnetic order. The results explain the kink in high- T c cuprates.

Magnetism and electron correlations

Electron correlation effects on magnetism are discussed from a local point of view on the basis of the dynamical coherent potential approximation (CPA) which completely takes account of the dynamical spin and charge fluctuations within the single-site approximation. The dynamical CPA is equivalent to other single-site theories of strongly correlated electrons: the many-body CPA, the dynamical mean-field theory, and the projection operator method CPA. The method is shown to describe both the dynamical effects on magnetic properties and the many-body band structure on the same footing. The effects of orbital degeneracy and the realistic calculation scheme are discussed with emphasis on Hund's rule coupling. Two types of theories for non-local correlations, the molecular dynamics approach and the self-consistent projection operator approach, are presented. The effects of magnetic short-range order and non-local correlations are discussed with some numerical results.

A self-consistent projection-operator approach to the kondo-lattice model

The Kondo-lattice model incorporates the coupling of two subsystems of electrons. Conduction electrons interact with local moments, which are formed by d(f)-electrons of the considered material. We suggest the use of a projection-operator method, which can equally well be applied to Fermi and quantum-spin operators, for a combined descriptions of both subsystems. Working with a two-dimensional basis in the fermionic and in the bosonic sector, a closed system of correlation functions is obtained. We discuss the derived spin-wave spectrum, which is characterized by a two-pole structure.

Kink Structure in the Quasiparticle Band of Doped Hubbard Systems

By making use of the self-consistent projection operator method with high-momentum and high-energy resolutions, we find a kink structure in the quasiparticle excitation spectrum of the two-dimensional Hubbard model in the underdoped regime. The kink is caused by a mixing between the quasiparticle state and excitations with short-range antiferromagnetic order. We suggest that this might be the origin of the strong concentration dependence of the 'kink' found in La_{2-x}Sr_{x}CuO_{4} (x=0.03-0.07).

Marginal Fermi Liquid Theory in the Hubbard Model

We find marginal-Fermi-liquid- (MFL) like behavior in the Hubbard model on a square lattice for a range of hole doping and on-site interaction parameter U. Thereby we use a self-consistent projection operator method. It enables us to compute the momentum and frequency dependence of the single-particle excitations with high resolution. The Fermi surface is found to be holelike in the underdoped regime and electronlike in the overdoped regime. Our calculations concern normal state properties of the system. When a comparison is possible, we find consistency with finite temperature quantum Monte Carlo results. We also find a discontinuous change with doping concentration from a MFL to a Fermi-liquid behavior resulting from a collapse of the lower Hubbard band. This renders Luttinger's theorem inapplicable in the underdoped regime.

Self-consistent projection operator approach to nonlocal excitation spectra

Self-consistent projection operator approach to nonlocal excitation spectra

Self-consistent projection operator approach to excitation spectra: Role of correlated wave function

A recently proposed combination of a projection operator method with the coherent potential approximation for the computation of the excitation spectra of solids is further extended. In particular, the effect of electron correlations on the matrix elements of the self-energy are investigated. This is done with the help of a Gutzwiller-type variational wave function. Numerical calculations have been performed for a half-filled band of a hypercubic lattice in infinite dimensions. They show that for strong electron correlations the higher-order dynamical corrections in an renormalized perturbation expansion are screened except in the low-energy regime. This provides for a justification of earlier work, where a Hartree-Fock approximation with a cutoff parameter was used for the computation of the static matrix elements.

Self-consistent projection operator method for the single-particle excitation spectrum

A new projection operator method combined with the coherent potential approximation is presented. The theory interpolates between the strong- and the weak-Coulomb interaction limits, and allows us to calculate the momentum dependence of the self-energy. Numerical results for the case of infinite dimensions indicate that it can describe quantitatively the quasi-particle properties of a system.

APPLICATION OF THE SEKE SELF-CONSISTENT PROJECTION OPERATOR METHOD IN THE CASE OF A DISCRETIZED ATOM-FIELD INTERACTION MODEL

The relaxation dynamics in the case of a discretized atom-field interaction model, being of great importance in quantum optics, is investigated numerically by an algorithm based on Seke's self-consistent projection operator method. The transition from reversible to irreversible dynamics in the continuous limit is simulated numerically. The dependence of the time evolution of atomic nondecay probability on the number of discrete-field modes, coupled to the atom, is discussed.

The Seke self-consistent projection-operator approach for the calculation of quantum-mechanical eigenvalues and eigenstates

Originally, the Seke self-consistent projection-operator method has been developed for treating non-Markovian time evolution of probability amplitudes of a relevant set of state vectors. In the so-called Born approximation the method leads automatically to an Hamiltonian restricted to a subspace and thus enables the construction of effective Hamiltonians. In the present paper, in order to explain the efficiency of Seke's method its algebraic operator structure is analyzed and a new successive approximation technique for the calculation of eigenstates and eigenvalues of an arbitrary quantum-mechanical system is developed. Since, unlike most perturbative techniques, in the present approach a well-defined effective (restricted) Hamiltonian in each order of the applied approximation exists, the self-consistency of all obtained results is self-evident. Finally, to illustrate the efficiency of the developed formalism, the optical polaron model is investigated.

A novel complete treatment of the time-evolution of a spontaneously decaying atomic state in relativistic QED

A new method, which, for the first time to our knowledge, makes a complete treatment of a spontaneously decaying atomic state in relativistic QED possible, is presented. The method is based on a self-consistent projection-operator technique, developed recently by the author. A new physical real-transition interaction picture, in which the unobservable interaction of the free electron with the vacuum radiation field is eliminated, is introduced. This leads automatically to the familiar finite relativistic Lamb shift, without using the concept of the mass renormalization. Moreover, in this picture a new real-transition Hamiltonian is derived. Finally, by applying the method to the relativistic Lyman-σ spontaneous emission, explicit analytic results for the non-Markovian time evolution and the radiative line shape including the correct frequency (Lamb) shift are obtained.

Novel Technique for Quantum-Mechanical Eigenstate and Eigenvalue Calculations based on Seke's Self-Consistent Projection-Operator Method

Seke's self-consistent projection-operator method has been developed for deriving non-Markovian equations of motion for probability amplitudes of a relevant set of state vectors. This method, in a Born-like approximation, leads automatically to an Hamiltonian restricted to a subspace and thus enables the construction of effective Hamiltonians. In the present paper, in order to explain the efficiency of Seke's method in particular applications, its algebraic operator structure is analyzed and a new successive approximation technique for the calculation of eigenstates and eigenvalues of an arbitrary quantum-mechanical system is developed. Unlike most perturbative techniques, in the present case each order of the approximation determines its own effective (approximating) Hamiltonian ensuring self-consistency and formal exactness of all results in the corresponding approximation order.