Strongly Correlated Systems, Coherence and Entanglement

The following sections are included:

• Preface

• Contents

We review developments concerning the effect of correlations on the electronic properties of one-dimensional systems, focusing our analysis on the one-dimensional Hubbard model. We consider methods used to describe the exotic properties of these systems, ranging from bosonization associated with the Tomonaga and Luttinger liquid behavior, to the Bethe ansatz solution, referring to all energy scales of solvable quantum problems and the pseudoparticle description. We use that description to study the model energy spectrum and the low-energy quantities. In the ensuing companion chapter we discuss the relation of the electronic operators to these quantum objects.

This chapter follows its companion, chapter 1. Here we review different methods based on the Bethe ansatz solution of the one-dimensional Hubbard model, in order to study quantities related to charge transport and the momentum dependent conductivity. Moreover, we report recent developments on finite-energy dynamical properties. This is achieved by introducing new entities called pseudofermions which are basically free, in the sense that their energies are additive, and where the effect of the interactions appears through phase shifts that are absorbed by their discrete momentum values. The resulting pseudofermion dynamical theory enables the evaluation of matrix elements between energy eigen-states and hence the derivation of finite energy expressions for the one- and two-electron correlation and spectral functions. Comparison with experimental results is also discussed.

We address the doping evolution of the low energy electronic structure of high-temperature superconducting copper-oxide compounds, as described by the tt′t″ J model. Following experimental evidence for well defined quasiparticles in the normal state of these doped Mott insulators, we use a new slave-particle basis that includes electron-like operators, namely, the doped-carrier basis, and extensively discuss the mean-field electron spectral function of the tt′t″ J model. We show that the above mean-field theory reproduces many aspects of the non-trivial microscopic single electron dynamics probed by angle-resolved photoemission experiments in hole and electron doped cuprates; these include: the emergence of spectral peaks inside the Mott gap upon doping away from half-filling; the differentiation between the nodal and antinodal regions of momentum space, which displays distinct properties in the hole and electron doped regimes; the low energy spectral weight arcs, whose length increases with doping; the nodal dispersion kink, which is sharper in the underdoped regime; the strong dispersion renormalization, which renders the dispersion close to (0,π) and (π, 0) surprisingly flat. We further argue that measured angle-resolved photoemission spectral dispersions, together with the associated spectral weight intensity, impose strong constraints on the character of coexisting short-range correlations. The agreement between our results and experimental data supports that the two predominant local spin correlations in cuprate superconductors are: (i) d-wave singlet pairing correlations, and (ii) staggered moment correlations.

In this chapter we revise the basic physics of a single layer and a double layer of graphene. In both cases, and starting from a tight-binding description, we show how to construct the effective continuum models. Also for the single layer and for the bilayer, both with zigzag edges, we discuss the existence of a zero energy band made of surface states localized near the edges of the sample. The spectrum of the single layer in the presence of a magnetic field is studied and its relation with the half-odd integer quantum Hall effect is discussed. For the bilayer the electronic bulk properties are studied and the unconventional quantum Hall effect is addressed. The concept of a biased bilayer is introduced together with its energy spectrum.

We review the theory and main physical mechanisms of the anomalous Hall effect in magnetic metals and semiconductors. Recently proposed mechanisms of the chirality-induced Hall effect, topological Hall effect, and the intrinsic mechanism of the anomalous Hall effect are reviewed. In the latter case, we discuss in more details the problem of defects and scattering from impurities. Our consideration is mostly based on the original articles of the authors.

In this contribution, we review some of our recent results on the nonequilibrium properties of the spin S = 1/2 Heisenberg ferromagnet. We consider the situation when the system is coupled to a phonon heat bath and/or in the presence of an external time-dependent magnetic field. The problem is studied by means of a path integral approach using the Majorana fermion representation for the spin operators. In particular, we consider the relaxation of the magnetization in the case when the magnetic field suddenly changes its direction. Another important case considered in this paper is the process of spinodal decomposition, or magnetic domain growth after the temperature is lowered below the critical value. We compare some of our results with the corresponding results in the case of classical spin models and some other models, and discuss possible applications of the results.

We present a review of our recent work in extending the successful dynamical mean-field theory from the equilibrium case to nonequilibrium cases. In particular, we focus on the problem of turning on a spatially uniform, but possibly time varying, electric field (neglecting all magnetic field effects). We show how to work with a manifestly gauge-invariant formalism, and compare numerical calculations from a transient-response formalism to different types of approximate treatments, including the semiclassical Boltzmann equation and perturbation theory in the interaction. In this review, we solve the nonequilibrium problem for the Falicov-Kimball model, which is the simplest many-body model and the easiest problem to illustrate the nonequilibrium behavior in both diffusive metals and Mott insulators. Due to space restrictions, we assume the reader already has some familiarity both with the Kadanoff-Baym-Keldysh nonequilibrium formalism and with equilibrium dynamical mean-field theory; we provide a guide to the literature where additional details can be found.

It is tempting to apply the elegant proposal of the Real Space Renormalization Group (RSRG) method to the treatment of heavy fermion lattices. The method proceeds through the definition of blocks and a truncation of the Hilbert space to products of the lowest eigenstates of the blocks. As such the method is not accurate enough. A great improvement is obtained when one defines effective interactions (instead of the bare ones) between the products of the selected block eigenstates. These effective interactions are obtained from the exact treatment of pairs or trimers of blocks through the effective Hamiltonian theory of Bloch. If the blocks and the effective inter-block Hamiltonian maintain an isomorphism between the block lattice and the starting one, the exponential scale change may be iterated to convergence. The qualitative and quantitative potentialities of the method are illustrated on a series of one-dimensional (1D) and two-dimensional (2D) spin lattices (non frustrated and frustrated), concerning the cohesive energies, the localization of quantum phase transitions and the spin gaps.

A review is given on the theory of Ising spin glasses in a magnetic field. We consider the nature of the low-temperature phase, below the freezing transition, and present the “replica symmetry breaking” and the “droplet” pictures, which have been proposed to describe spin glass behavior. The spin glass transitions in zero and nonzero magnetic field are analyzed within the renormalization group, which indicates that there is no Almeida-Thouless transition, i.e., no spin glass transition occurs in a finite magnetic field.

Ab initio calculations combined with the effective Hamiltonian theory of Bloch provide a rational way to determine model Hamiltonians. The embedded cluster approach is the most reliable method of extraction of effective interactions for the study of highly correlated material. In the specific case of half-doped man-ganites, several model Hamiltonians can be considered to reproduce the local physics generated by the interactions between the magnetic sites according to the position of the doping holes. While a double exchange mechanism takes place between the Mn sites if the holes are localized on the metals, a purely magnetic Heisenberg Hamiltonian should be considered for a localization of the holes on the bridging oxygens. For intermediate situations in which both elements share the doping holes, a truncated Hubbard model which treats variationaly double exchange and Heisenberg configurations seems to be the most appropriate. This model can be mapped on both simpler double exchange and Heisenberg Hamiltonians. The analytical spectrum of the Heisenberg model in the case of two metals bridged by a magnetic oxygen is identical (except for one state) to the double exchange one, for a peculiar relation between the electronic interactions of the two models. Finally, the most appropriate hamiltonians is a refined double exchange model which combines the Anderson-Hazegawa and the Girerd-Papaefthymiou antiferromagnetic contributions.

Some materials, most noticeably the manganites, show, in certain composition ranges, concomitant para-ferromagnetic and metal-insulator transitions. It was precisely in the context of the experimental discovery of this remarkable correlation between transport and magnetism in the case of the manganites, that Zener proposed a magnetic exchange mechanism, double exchange, in which charge transport and magnetic correlations are closely inter-dependent.

In this article we review studies which addressed the conditions under which a simple double exchange model can present a Anderson-like metal-insulator transition when it orders ferromagnetically. We present arguments that show that intrinsic disorder in some calcium doped manganites is much higher than has generally been admitted in the literature. Nevertheless, such a model is a dramatic over simplification in the case of the manganites, in which orbital degeneracy, antiferromagnetic interactions, and strong electron-lattice coupling play an important part in the physics.

We also review some recent work on another class of compounds showing very large magnetoresistance at the Curie temperature, the europium hexaborides, in which is is argued that a variety of optical, transport and magnetic properties can be well understood as a manifestation of Anderson localization in the context of a double exchange model, without the complicating factors that are present in the manganites.

We review briefly the problem of electron transport in magnetic nanowires with thin domain walls. Transmission of electrons in such structures is associated with charge and spin currents leading to the occurrence of a spin torque that acts on the domain wall. Experimentally, the properties of such structures are manifested as a large magnetoresistance, current-induced motion of the domain wall, generation of spin currents, etc. The effect of electron interactions on the scattering from a sharp domain wall is also considered in more details. Using a renormalization group approach for the interactions, we obtain scaling equations for the scattering amplitudes. The RG equations obtained are independent of the single-particle model for the domain wall. We describe the nature of the zero temperature fixed points. For repulsive interactions, the wall reflects all incident electrons at the fixed points. However, the interactions determine whether this reflection is accompanied by spin reversal or not. In one of the fixed points the wall flips the spin of all incident electrons, generating a finite spin current without an associated charge current. It is also shown that the RG flow affects short walls more quickly than long walls, implying that correlations have a more important effect on short walls.

In this Chapter we present a brief overview of the physics of an ideal ultra cold quantum gas formed within a harmonic trapping potential focusing especially on its coherence properties. Recently it has become experimentally feasible to study these particle correlations. We include a short introduction to the relevant experimental techniques and their limitations. Measurements are typically carried out on atomic clouds released from the trap and left to expand under the influence of gravity. We model the appropriate ballistic expansion for a non-interacting atomic cloud and derive the corresponding one-body and two-body correlation functions. We conclude by summarizing recent measurements of the second order particle correlations present within a falling cloud of metastable Helium atoms close to the Bose-Einstein condensation point.

The hamiltonian describing the physics of atomic Bose gases is a many-body hamiltonian with a confining potential and two-particle repulsive interactions. Dealing with such a hamiltonian is difficult, even for dilute gases, at the level of mean-field theory, because of the very repulsive nature of the potential. It becomes even more difficult when the atomic density is increased and many-particle correlations need to be considered. In this article we overview the way in which density functional theory deals with both these problems.

A kinetic approach to cold atoms and Bose-Einstein condensates is explored. This approach is based on the Wigner transformation, which allows for a classical phase space representation of a quantum system. Wave kinetic equations exactly equivalent to the Gross Pitaevskii equation are considered, and various approximations are discussed. In the quasi-classical limit, we obtain the particle number conservation equation. Several different examples of application of this method are given. They include, self-phase modulation of a BE condensate, modulational instability and wakefield generation by a cold atom beam in a thermal background, and kinetic dispersion relation of Bogoliubov oscillations with collisionless Landau damping.

We present a review of the temperature-magnetic field phase diagram of homogeneous and inhomogeneous superconductivity in the case of a clean quasi-two-dimensional superconductor with singular density of states. For transverse magnetic field, the superconducting pairing susceptibility K_T (r) displays anomalous short-range behavior which leads to positive curvature in the upper critical field. The Pauli limit (H_p) is strongly enhanced and a huge metastability region appears when the magnetic field is applied parallely to the conducting planes. A non-uniform superconducting FFLO state is not favored by the presence of the van Hove singularity.

The revision is made of Green function methods that describe the dynamics of electronic quasiparticles in disordered superconducting systems with d-wave symmetry of order parameter. Various types of impurity perturbations are analyzed within the simplest T-matrix approximation. The extension of the common self-consistent T-matrix approximation (SCTMA) to the so-called group expansions in clusters of interacting impurity centers is discussed and hence the validity criteria for SCTMA are established. A special attention is payed to the formation of impurity resonance states and localized states near the characteristic points of energy spectrum, corresponding to nodal points on the Fermi surface.

The BCS theory of ³P₀ quark-antiquark condensation is given. A method to evaluate overlap kernels is discussed. An introduction to the derivation of Salpeter equations and its relation with RGM equations is presented. The mechanism for the spontaneous breakdown of chiral symmetry and its relation with issues like the pion Goldstne boson, chiral restoration, scalar versus vectorial confinement and hadronic decay are also treated.

We introduce and define the concept of quantum entanglement and its properties. We concentrate our discussion in the case of bipartite entanglement in a pure state, the Schmidt decomposition and the quantification of entanglement by the von Neumann entropy. We then review the Matrix Product State and Projected Entangled-Pair States representations of entangled states and their application to the classical simulation of many-body quantum systems, showing how these novel techniques allow us to extend the study of low-energy pure states to systems with periodic boundary conditions, as well as to obtain low-energy states and simulate the time evolution of d-dimensional systems at zero temperature and one-dimensional systems at finite temperature.

We review recent studies on the role of entanglement in many-body quantum systems at zero temperature, focusing on quantum phase transitions (QPT). We start by introducing the entanglement entropy and the concurrence, and then discuss how these measures of quantum correlations behave in one-, low- and infinite-dimensional spin and electron systems in (first- and second-order) phase transitions, giving us new insight into the correlations present in those systems, new order parameters and, in some cases, even new QPT points. Finally, we introduce the idea of the quantum computer, which in practice will be a many-body quantum system. We present the adiabatic model of quantum computation and discuss its connections to quantum phase transitions, as well as the role of entanglement in both phenomena.

We review recent results on the existence and use of entanglement in macroscopic systems at finite temperatures. Entanglement represents a particular quantum form of correlations and we explore the relevance of these correlations on thermodynamic properties of macroscopic systems, as well as its survival at finite temperatures. Various lower bounds for the entanglement temperature (the temperature below which entanglement surely exists in a system) are discussed and it is shown that, in some cases, macroscopic entanglement can exist even at room temperatures. Furthermore, the relevant correlations determining the macroscopic behavior of some systems are present due to entanglement only, as we show in the example of high-temperature superconductivity. Entanglement can also affect various thermodynamic properties, such as magnetic susceptibility, internal energy and pressure. Motivated by the fact that entanglement is a crucial resource for quantum information processing, we also discuss how to extract it from macroscopic systems, namely from spin-chains. Finally, we illustrate the generation and manipulation of entanglement in a macroscopic system of non-interacting hopping bosons.

The following sections are included:

• Subject Index