KRYLOV-BOGOLIUBOV-MITROPOLSKY AVERAGING USED TO CONSTRUCT EFFECTIVE HAMILTONIANS IN THE THEORY OF STRONGLY CORRELATED ELECTRON SYSTEMS

A. P. Saǐko

Published Aug 16, 2016• 1 citation

Abstract

We show that the Krylov-Bogoliubov-Mitropolsky averaging in the canonical formulation can be used as a method for constructing effective Hamiltonians in the theory of strongly correlated electron systems. As an example, we consider the transition from the Hamiltonians of the Hubbard and Anderson models to the respective Hamiltonians of the t-J and Kondo models. This is a very general method, has several advantages over other methods, and can be used to solve a wide range of problems in the physics of correlated systems.

Related papers

A Theory for Magnetic Fluctuations in Strongly Correlated Electron-Systems

Torben Jabben

Strongly correlated electron systems show a rich variety of astonishing physical phenomena. However, the strong interactions make the theoretical description of these systems a highly non-trivial task. Even the simplest models cannot be solved exactly, and one has to reside to approximative solutions, which often cover only certain aspects of the physics contained in these models. In recent years progress has been made in the theoretical description of correlated lattice systems through the mapping of lattice models onto effective impurity models. The most prominent example is the mapping of the Hubbard model onto an effective single impurity Anderson model (SIAM). Within this description both models feature the prominent Kondo effect, which leads to the emergence of low energy quasiparticles. The occurrence of such excitations is a result of the dynamical screening of local magnetic moments and the corresponding formation of a low temperature Fermi liquid phase. On the other hand these theories usually fail in the description of non Fermi liquid behavior, which is for example observed in the normal state of high temperature superconductors and some Heavy-Fermion systems. Strong nonlocal fluctuations drive the dynamics of these systems. In the theoretical treatment, due to the mapping of the lattice problem to an effective single impurity problem, these fluctuations are not adequately described.\\\\ In this thesis a new self-consistent approach for the inclusion of spatial correlations is presented for the Hubbard model. In contrast to existing quantum cluster theories in this field, the approach allows for the simultaneous description of short and long range pair correlations. Due to the intimate connection of this new lattice theory with the two impurity Anderson model (TIAM) a thorough numerical investigation of the TIAM is undertaken in this thesis. The competition of the Kondo effect with the RKKY interactions is studied in great detail. Furthermore two new solvers for the TIAM are introduced, which extend existing non crossing approximations for the SIAM.

Jan 1, 20100 citations

Non-Linear Canonical Methods in Strongly Correlated Electron Systems

Matteo Bazzanella

In this thesis some new ideas to perform the analysis of Strongly Correlated Electronic Systems (SCES) are developed. In particular the use of non-linear canonical transformations is considered thoroughly. Using such transformations it is possible, in some circumstances, to simplify the quantum problem redefining the fermionic degrees of freedom used to describe the system. To understand and use effectively these non-linear transformations it is convenient to work in the Majorana fermion representation, i.e., to represent the quantum mechanical operators in terms of Majorana fermions. These objects can be imagined as algebraic constituents of the fermionic degrees of freedom. In a fermionic system, different equivalent sets of (emergent) Majorana fermions can be used to build the fermionic operators that characterize the system. The non-linear transformations can be seen as a way to mix these equivalent sets. Thanks to this insight, it becomes possible to determine the full structure of the group of canonical transformations and to identify an advantageous framework, which allows their use in the study of a generic SCES system. To test these statements the Hubbard and the Kondo lattice models were intensively studied making use of non-linear canonical transformations, obtaining interesting results in both cases. For example, in the Hubbard model a free fermion mean-field description of the paramagnetic Mott insulator was identified, while in the Kondo lattice it was possible to describe already at mean-field level the spin-selective Kondo insulating phase, consistently (from a quantitative and qualitative point of view) with the known numerical results. Moreover the method elaborated for the study of the Hubbard model is suitable for a systematic generalization to other situations and shows great room for improvement. These results prove that, thanks to the redefinition of the degrees of freedom used in the analysis of the system, it becomes possible to obtain quite non-trivial results already at mean-field level, or to consider very involved (but meaningful) correlated quantum states via simple variational trial states. This will potentially permit a more judicious and profitable choice of the fundamental degrees of freedom, allowing for an improvement of the efficiency of the analytical and numerical techniques used in the analysis of many SCES systems.

Nov 14, 20140 citations

SYMMETRIES IN THE PHYSICS OF STRONGLY CORRELATED ELECTRONIC SYSTEMS

Mancini, Avella

Strongly correlated electron systems require the development of new theoretical schemes in order to describe their unusual and unexpected properties.The usual perturbation schemes are inadequate and new concepts must be introduced.In our scheme of calculations, the Composite Operator Method is possible to recover through a self-consistent calculation, a series of fundamental symmetries by choosing a suitable Hilbert space.

arXiv:cond-mat/9807019Condensed Matter PhysicsJan 1, 19986 citations

A NEW INTEGRABLE MODEL OF STRONGLY CORRELATED ELECTRONS WITH HUBBARD-LIKE INTERACTION

Xiangyu Ge

A new completely integrable model of strongly correlated electrons is proposed which describes two competitive interactions: one is the correlated one-particle hopping, the other is the Hubbard-like interaction. The integrability follows from the fact that the Hamiltonian is derivable from a one-parameter family of commuting transfer matrices. The Bethe ansatz equations are derived by algebraic Bethe ansatz method.

Modern Physics Letters BMar 20, 20010 citations

Theory of strongly correlated electron systems: Hubbard–Anderson models from an exact Hamiltonian, and perturbation theory near the atomic limit within a nonorthogonal basis set

I. Sandalov, Börje Johansson, Olle Eriksson

Abstract The theory of correlated electron systems is formulated in a form that allows the use as a reference point a density functional theory in the local density approximation (LDA DFT) for solids or molecules. The theory is constructed in two steps. As a first step, the total Hamiltonian is transformed into a correlated form. To elucidate the microscopic origin of the parameters of the periodic Hubbard–Anderson model (PHAM) the terms of the full Hamiltonian that have the operator structure of PHAM are separated. It is found that the matrix element of mixing interaction includes ion configuration‐ and number of particles‐dependent contributions from the Coulomb interaction. In a second step the diagram technique (DT) is developed by means of generalization of the Baym–Kadanoff method for correlated systems. The advantages of the method are that: (1) A nonorthogonal basis can be used, in particular the one generated by LDA DFT; (2) the equations for Green's functions (GFs) for the Fermi and Bose types of quasiparticles can be formulated in the form of a closed system of functional equations. The latter allows us to avoid the question of the nonunique decoupling procedure existing in previous versions of the DT and perform the expansion in terms of dressed GFs. Although the expressions for all interactions depend on the overlap matrix, it is shown that the theory is formally equivalent to one with orthogonal states with redefined interactions. When the PHAM is treated from the atomic‐limit side the vertexes are generated by kinematic interactions. The latter arise due to nontrivial commutation relations between X‐operators and come from the mixing, hopping, and overlap of states. The equations for GFs are derived within the nonorthogonal basis set in Hubbard‐I, one‐loop and random‐phase approximations with respect to kinematic interactions. The self‐consistent equation for “Hubbard Us” is derived. The technique developed is general, in particular its “bosonic” part can be used for description of spin systems with arbitrary anisotropy, systems with orbital ordering, or ordering of multipoles. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 113–143, 2003

arXiv:cond-mat/0011259International Journal of Quantum ChemistryJan 1, 200328 citations

Electronic Dynamics of the Anderson Model. The Many-Body Approach

A. L. Kuzemsky

A review of electronic dynamics of single-impurity and many-impurity Anderson models is contained in this report. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the non-perturbative technique for the thermodynamic Green functions. The method of the irreducible Green functions is used as the basic tool. This irreducible Green functions method allows one to describe the quasiparticle spectra with damping of the strongly correlated electron systems in a very general and natural way and to construct the relevant dynamical solution in a self-consistent way on the level of Dyson equation without decoupling the chain of the equations of motion for the Green functions. The subject matter includes the improved interpolating solution of the Anderson model. It was shown that an interpolating approximation, which simultaneously reproduces the weak-coupling limit up to second order in the interaction strength U and the strong-coupling limit up to second order in the hybridization V (and thus also fulfils the atomic limit) can be formulated self-consistently. This approach offers a new way for the systematic construction of approximate interpolation dynamical solutions of strongly correlated electron systems.

arXiv:1503.00983Mar 3, 20150 citations