LECTURE ON THE PHYSICS OF HIGHLY CORRELATED ELECTRON SYSTEMS VII: Seventh Training Course in the Physics of Correlated Electron Systems and HighTc Superconductors

Polarons
View Description Hide DescriptionAn overview is presented of the theory of continuum (“Fröhlich‐Pekar”‐) polarons. Experimental manifestations of polarons are analysed and compared to theory. The emphasis is on the response characteristics of one‐ and many polarons: optical absorption, cyclotron resonance, photoluminescence, Raman scattering. I consider free polarons and polarons confined in space, polarons in 3D, 2D, 0D (quantum dots).

Multielectron bubbles in liquid helium: a spherical two‐dimensional electron system
View Description Hide DescriptionThe electron system with its many phases and realizations remains a fascinating subject of study. It has been extensively analysed in three, two, one and zero dimensions, as well as in confined nanoscopic and mesoscopic geometries generated with microfabrication techniques. An obvious next step in this investigation is to study the electron system in curved geometries. In particular we will focus in these lectures on the two‐dimensional spherical electron system, realized in for example multielectron bubbles (MEBs), buckyballs, and nanoshells.

The numerical approach to the correlated electron problem: quantum Monte Carlo methods
View Description Hide DescriptionWe present a review of the auxiliary field (i.e. determinantal) Quantum Monte Carlo method applied to various problems of correlated electron systems. The ground state projector method, the finite temperature approach as well as the Hirsch‐Fye impurity algorithm are described in details. It is shown how to apply those methods to a variety of models: Hubbard Hamiltonians, periodic Anderson model, Kondo lattice and impurity problems, as well as hard core bosons and the Heisenberg model. An introduction to the world‐line method with loop upgrades as well as an appendix on the Monte Carlo method is provided.

Basic Models in the Quantum Theory of Magnetism
View Description Hide DescriptionWe regard the Heisenberg model, the Hubbard model, the tJ‐model and the sd‐model as the basic models of the quantum theory of magnetism in solids. They can describe localized and itinerant magnets and strongly correlated electron systems. This review is devoted to analytical approaches for these models: diagrammatic techniques and the method of generating functional. The diagrammatic techniques are based on a generalization of the Wick theorem for spin and X operators. Peculiarities of such techniques for the basic models appear because the spin and X operators do not commute on a C‐value, but their commutator (anticommutator) is an operator itself. The method of generating functional is a generalization of the Kadanoff‐Baym approach, developed earlier for usual Fermi systems. The generating functional describes the interaction of a system with fluctuating fields, and different Green’s functions can be treated as variational derivatives with respect to these fields. Such approach allows to derive the equation of motion for the Green’s functions in each model in terms of functional derivatives. These equations help to find common features in the behavior of the basic models, particularly in finding the multiplicative structure of one‐particle Green’s functions. Iteration of the equations generates perturbation theory, which is compared with the diagrammatic techniques. Both approaches are applied to the calculation of the quasiparticle spectrum of the models and of collective excitations. A generalized random phase approximation (GRPA) is suggested for calculation of different dynamical susceptibilities. This approximation is developed in both approaches: the diagrammatic technique and the generating functional method.

How to detect unconventional density waves?
View Description Hide DescriptionWe report about our study on unconventional density waves (UDW) (i.e. density waves with wavevector dependent gap) in quasi‐one dimensional systems. Due to the zero average of the gap over the Fermi surface, these systems are not characterized by the periodic modulation of either the charge or spin density, and are referred to as systems with hidden‐order parameter. The quantities signaling the phase transition are determined. Depending on the explicit wavevector dependence of the gap, optical absorption is possible at low frequencies. In the presence of magnetic field, the different phases exhibit distinct behaviour, and the nonlinear electric response is strongly influenced by the applied field.

Mott domains of bosons and fermions confined in optical lattices
View Description Hide DescriptionWe study the ground state properties of the 1‐D boson and fermion Hubbard models in harmonic traps with an underlying lattice. The physics of these systems is found to be fundamentally different from the homogeneous case. Global quantities lose their meaning and Mott domains appear for a continuous range of incommensurate fillings. A local order parameter is defined to characterize the Mott insulating phases that coexist with the metallic ones. Finally, the phase diagrams show richer structures than in the homogeneous models.

Competition between Vertical and Diagonal Stripes in the Hartree‐Fock Approximation
View Description Hide DescriptionUsing the Hartree‐Fock approximation we analyze the properties and stability of the filled (one doped hole per stripe site) vertical stripes (VS) and diagonal stripes (DS), found in the two‐dimensional Hubbard model at two representative doping levels x = 1/8 and x = 1/6, by varying the ratio of the on‐site Coulomb repulsion U to the hopping element t. In the weak coupling regime of U ∼ 3.5t the stability of VS is best explained by the solitonic mechanism which leads to the kinetic energy gain due to the hopping perpendicular to the stripes, while DS have a lower potential energy which explains their stability in the regime of large U. The results obtained beyond the Hubbard model are shortly reviewed as well.

Pseudogaps from classical fluctuations of the order parameter
View Description Hide DescriptionThe problem of pseudo‐gap formation in an electronic system, induced by the fluctuations of the order parameter is revisited. We make the observation that a large class of current theories are theoretically equivalent to averaging the Free energy of the pseudo‐gap system over quenched‐disordered distribution of the order parameter. To illustrate how our approach works we examine the interplay of pseudogap, superconductivity and disorder.

Properties of a correlated nanoscopic chain close to the metal‐insulator transition
View Description Hide DescriptionWe review our recent results concerning the electronic properties of a nanoscopic system with inclusion of long‐range Coulomb interactions that are obtained by combining exact diagonalization in the Fock space with an ab initio method (EDABI). The system evolution is discussed as a function of lattice parameter R. A transformation from a nanometal to localized spin system is observed. With the help of a finite‐size scaling we determine the discontinuity of the momentum distribution at the Fermi points. Our results imply Fermi‐liquid‐like behavior for R ≲ 3a _{0} (a _{0} is the Bohr radius). The quasiparticle mass is almost divergent at the localization threshold, where the particle distribution of the Fermi‐Dirac type gets smeared out. The charge‐gap evolution is compared with that of the Slater gap. Also, the single‐particle spectral function and the corresponding density of states are presented. The analysis is performed using 1s‐like Gaussian‐type STO‐3G basis.

Scaling, Entanglement, and Quantum Phase Transitions
View Description Hide DescriptionWe study the entanglement near a quantum phase transition in a class of exactly solvable one‐dimensional quantum magnets, the anisotropic XY models. The critical behaviour (scaling) is reflected by the derivatives of the bipartite entanglent, quantified by the concurrence. However, we found a profound difference between classical correlation and non‐local quantum correlation: The range of the concurrence is found to behave in a non‐universal way, being extremely short‐ranged for the Ising model. This range tends to infinity towards the isotropic XY model.

Structural, Transport and Magnetic Properties of the Doped Magnetic Superconductor Ru‐1212
View Description Hide DescriptionSingle phase materials of RuSr_{2}Gd_{0.5}Eu_{0.5} Cu _{2}O_{8} (Ru‐1212Gd_{0.5}Eu_{0.5}) and Ru(Sr_{1−x } Na _{ x })_{2}GdCu_{2}O_{8} (0 ⩽ x ⩽ 0.10) have been successfully synthesized in oxygen at 1065 °C. DC‐resistivity and magnetic susceptibility measurements show a decreases of the superconducting transition temperature T_{c} from 46 K in the pure Gd compound to 41 K in Ru‐1212Gd_{0.5}Eu_{0.5}, whereas the magnetic ordering temperature T_{M} does not change. The influence of the monovalent substitution Na ^{+} for Sr^{2+} on the properties of Ru‐1212Gd is investigated and compared with previous work. Finally, the effects of Cu substituted RuSr_{2}Gd(Cu_{1−x }M_{ x })_{2}O_{8} compounds, where M is Co, Ni and Ga for 0 ⩽ x ⩽ 0.03 are reported and compared with similar doping experiments in other high‐T_{c} superconductors.

Space‐Group Approach to the Wavefunction of a Cooper Pair and Its Application to High‐T_{ c } Superconductors
View Description Hide DescriptionThe general two‐electron wavefunctions obeying the Pauli exclusion principle and Anderson criteria for Cooper pairs are constructed for D_{2h } and D_{4h } symmetries. It is shown that in axial symmetry groups the Blount theorem is violated and lines of nodes of triplet superconducting order parameter are required by the symmetry. Application of the results to high −T_{ c } superconductors is discussed.

Two‐Dimensional t‐J Model in a Staggered Field
View Description Hide DescriptionThe two‐dimensional t‐J model in a staggered field is studied by exact diagonalization up to 20 sites. For the low‐hole‐density region and a realistic value of J/t, it is found that the presence of a staggered field strengthens the attraction between two holes. With increasing field, the d_{ x 2−y 2 }‐wave superconducting correlations are enhanced while the extended‐s‐wave ones hardly change. This implies that coexistence of the d_{ x 2−y 2 }‐wave superconducting order and the commensurate antiferromagnetic order occurs in a staggered field.