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INTERACTING FERMIONS ON A LATTICE

Thierry Giamarchi

in Quantum Physics in One Dimension

This chapter examines the canonical microscopic models used to study interacting fermions. Most of these models are quite general and are not special to one dimension. These models try to implement ... More

This chapter examines the canonical microscopic models used to study interacting fermions. Most of these models are quite general and are not special to one dimension. These models try to implement the essential ingredients for studying a strongly interacting electronic system: the kinetic energy, the interaction, and the lattice in order to be able to study effects such as the Mott transition. Single chain fermionic systems are considered, and transport in fermionic systems and the Mott transition are discussed. In addition, the relaxation of the momentum due to scattering on the lattice is analyzed.Less

Fermi and Fermi–Bose Hubbard models: methods of treatment

Maciej Lewenstein, Anna Sanpera, and Verònica Ahufinger

in Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems

This chapter presents various methods specific for fermionic systems and Fermi–Bose mixtures. It starts by discussing a lattice version of the BCS theory and then reviews BCS-BEC crossover theory for ... More

This chapter presents various methods specific for fermionic systems and Fermi–Bose mixtures. It starts by discussing a lattice version of the BCS theory and then reviews BCS-BEC crossover theory for trapped Fermi gases close to the Feshbach resonance. The chapter analyses Fermi Hubbard models in the strongly correlated regime and t-J limit, and introduces the Gutzwiller projection method and the slave boson approach. It discusses in detail, in particular in the context of Fermi-Bose mixtures, effective low energy Hamiltonians and their derivation.Less

Models of Strongly Interacting Quantum Matter

Hans-Peter Eckle

in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

This chapter introduces a select number of models of strongly interacting quantum many-particle physics and examines their basic properties. These models represent Bosonic and Fermionic systems as ... More

This chapter introduces a select number of models of strongly interacting quantum many-particle physics and examines their basic properties. These models represent Bosonic and Fermionic systems as well as systems where magnetic moments, i.e. spins, interact. The main selection criterion has been the existence of a variant of the model that is quantum integrable using Bethe ansatz methods. After studying the Bose fluid, the Landau Fermi liquid, and the one-dimensional concept of the Luttinger liquid, it reviews some of the major models of condensed matter theory, including the Hubbard model describing itinerant magnetism, the Heisenberg model describing localized magnetism, and the Kondo model describing the interaction of a magnetic impurity and band electrons. It also presents the Rabi model and some of its descendants in order to describe the interaction of light and quantum matter in quantum optics.Less

Numerical Studies of Metal—Insulator Transitions in Disordered Hubbard Models

S. Chiesa, R.T. Scalettar, P.J.H. Denteneer, P. Chakraborty, T. Paiva, and S. Story

in Conductor-Insulator Quantum Phase Transitions

This chapter reviews Determinant Quantum Monte Carlo studies of the disordered Hubbard Hamiltonian. The effect of the interplay of interactions and a variety of realizations of the randomness, ... More

This chapter reviews Determinant Quantum Monte Carlo studies of the disordered Hubbard Hamiltonian. The effect of the interplay of interactions and a variety of realizations of the randomness, including spatially varying hopping and chemical potentials, on the conductivity, magnetism, Mott gap formation, and s-wave superconductivity is evaluated. Advances in algorithms and in computer hardware have made possible an order of magnitude increase in system sizes over the last several years, and optical lattice emulators of the Hubbard Hamiltonian are providing a new experimental realization of disordered and interacting fermions. This suggests the possibility of new efforts in this field.Less

The ground state construction of the two-dimensional Hubbard model on the honeycomb lattice

Giuliani Alessandro

in Quantum Theory from Small to Large Scales: Lecture Notes of the Les Houches Summer School: Volume 95, August 2010

The fermionic Hubbard model on the two-dimensional hexagonal lattice at half-filling is the simplest model for undoped single layer graphene with screened interactions. It is remarkable that, in the ... More

The fermionic Hubbard model on the two-dimensional hexagonal lattice at half-filling is the simplest model for undoped single layer graphene with screened interactions. It is remarkable that, in the weak coupling regime, the ground state thermodynamic functions and the equilibrium correlations can be fully constructed (in the form of convergent resummed series in the coupling strength) by rigorous renormalization group methods. This chapter presents a self-contained proof of the analyticity of the specific ground state energy and it reviews the relevant multiscale cluster expansion methods, which are of growing importance in the condensed matter community and, in particular, in the study of interaction effects in graphene.Less

Quantum simulators of condensed matter

Maciej Lewenstein, Anna Sanpera, and Verònica Ahufinger

in Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems

This chapter presents an operational definition of quantum simulators, and discusses what kind of Hubbard models and spin models can, in principle, be simulated with ultracold atoms.

Electron repulsion

Enric Canadell, Marie-Liesse Doublet, and Christophe Iung

in Orbital Approach to the Electronic Structure of Solids

This chapter shows that the repulsion between correlated electrons can take different mathematical forms depending on the Hamiltonian and wavefunction chosen to describe the n-interacting particles ... More

This chapter shows that the repulsion between correlated electrons can take different mathematical forms depending on the Hamiltonian and wavefunction chosen to describe the n-interacting particles system. It accounts for the electronic repulsion Uee , firstly in an effective manner, starting from the extended Hückel model (Uee = 0) and going to the Hubbard (Uee ≠ 0) and Heisenberg (Uee → +∞) models; and secondly in a more quantitative manner using the exact electronic Hamiltonian of the many-body problem, i.e., a system of n interacting particles.Less

EXCITONIC INSULATORS, ELECTRON–HOLE LIQUIDS AND METAL–INSULATOR TRANSITIONS

T. Maurice Rice

in Problems of Condensed Matter Physics: Quantum coherence phenomena in electron-hole and coupled matter-light systems

This chapter describes metal-insulator transitions in disorder-free crystals. It reviews excitonic insulators, electron-hole liquids, and metal-insulator transitions due to band crossing. The ... More

This chapter describes metal-insulator transitions in disorder-free crystals. It reviews excitonic insulators, electron-hole liquids, and metal-insulator transitions due to band crossing. The realization of exciton (Bose-Einstein) condensate in the high-density limit is illustrated by experiments on a quantum Hall bilayer. In the second part of the chapter, the Mott transition from a metal to a Coulomb localized insulator is discussed in terms of the one band Hubbard model.Less

Cuprate superconductivity

A.J. Leggett

in Quantum Liquids: Bose condensation and Cooper pairing in condensed-matter systems

Starting with an account of the chemical composition, crystalline structure, and phase diagram of the high-temperature (cuprate) superconductors, this chapter reviews the principal experimental ... More

Starting with an account of the chemical composition, crystalline structure, and phase diagram of the high-temperature (cuprate) superconductors, this chapter reviews the principal experimental properties of the optimally doped normal phase, the superconducting phase, and the so-called “pseudogap” region of the phase diagram, and some general comments made on the implications of the experimental data. The question is then raised: what do we know for sure about cuprate superconductivity in the absence of a specific microscopic model? And some answers are attempted. Next, various ideas which may be important in understanding these systems are reviewed. Finally, some novel consequences of the type of pairing realized in the cuprates are explored.Less

Ultracold spinor atomic gases

Maciej Lewenstein, Anna Sanpera, and Verònica Ahufinger

in Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems

This chapter focuses on ultracold gases of atoms that possess internal spin structure. First, it considers spinor interactions, and then turns to trapped spinor Bose–Einstein condensates. In ... More

This chapter focuses on ultracold gases of atoms that possess internal spin structure. First, it considers spinor interactions, and then turns to trapped spinor Bose–Einstein condensates. In particular, the chapter discusses various possible mean field phases, spin textures, and topological defects. Next, it describes bosonic spinor gases in optical lattices, effective Hamiltonians, and possible quantum phases. Finally, spinor Fermi gases in optical lattices for spins larger than ½ are discussed.Less

Ultracold atoms in optical lattices

Immanuel Bloch

in Many-Body Physics with Ultracold Gases: Lecture Notes of the Les Houches Summer School: Volume 94, July 2010

This chapter provides an introduction to the field of strong correlation physics with ultracold atoms in optical lattices. After a basic introduction to the single-particle band structure and lattice ... More

This chapter provides an introduction to the field of strong correlation physics with ultracold atoms in optical lattices. After a basic introduction to the single-particle band structure and lattice configurations, the effect of strong interactions on the Hubbard model is discussed. Detection methods are introduced, which allow us to reveal in-trap density and (quasi)-momentum distributions, as well as correlations between particles on the lattice. The fundamental phases of the bosonic and fermionic Hubbard model are discussed. Superexchange spin-spin interactions that form the basis of quantum magnetism are introduced and the current status on observing such magnetic phenomena is highlighted. Finally, novel possibilities for detecting single-site and single-atom resolved quantum gases are outlined.Less

Electronic Systems

Hans-Peter Eckle

in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ... More

The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ansatz. Electronic systems are the most relevant examples for condensed matter physics. Prominent electronic many-particle systems in one dimension solvable by a nested Bethe ansatz are the one-dimensional δ‎-Fermi gas, the one-dimensional Hubbard model, and the Kondo model. The major difference to the Bethe ansatz for one component systems is a second, spin, eigenvalue problem, which has the same form in all cases and is solvable by a second Bethe ansatz, e.g. an algebraic Bethe ansatz. A quantum dot tuned to Kondo resonance and coupled to an isolated metallic ring presents an application of the coupled sets of Bethe ansatz equations of the nested Bethe ansatz.Less

Ultracold dipolar gases

Maciej Lewenstein, Anna Sanpera, and Verònica Ahufinger

in Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems

This chapter examines ultracold dipolar gases. First, it discusses the general properties of dipole–dipole interaction and explains how ultracold dipolar systems can be realized in experiments. The ... More

This chapter examines ultracold dipolar gases. First, it discusses the general properties of dipole–dipole interaction and explains how ultracold dipolar systems can be realized in experiments. The chapter reviews the properties of ultracold trapped dipolar Bose gases, pointing out the appearance of the roton minimum in the excitation spectrum. It then moves on to dipolar gases in a lattice, and investigates the corresponding extended Hubbard models. Dipolar bosons in a 2D optical lattice are discussed within the Gutzwiller approximation in more detail, with an emphasis on the appearance of multiple metastable states. Next, the chapter considers quantum Monte Carlo studies of dipolar gases, and discusses some further dipolar effects, which can be realized with cold atoms.Less

Optical lattices and quantum simulation

Massimo Inguscio and Leonardo Fallani

in Atomic Physics: Precise Measurements and Ultracold Matter

This chapter extends the investigation of ultracold atoms in optical lattices to the emerging field of ‘quantum simulation’, in which atoms are used to experimentally realize basic condensed-matter ... More

This chapter extends the investigation of ultracold atoms in optical lattices to the emerging field of ‘quantum simulation’, in which atoms are used to experimentally realize basic condensed-matter models to precisely investigate their properties and their quantum phase transitions in an ultimately clean setting, where decoherence or unwanted interactions with the environment can be avoided. The chapter discusses the superfluid/metal-to-insulator transition exhibited by strongly-interacting atoms realizing Hubbard models, as well as the Anderson localization determining the suppression of transport for atoms moving in a disordered potential. It concludes with an illustration of the most recent developments in the field, discussing novel experimental techniques and important frontiers in this research.Less

Related Theories

Tuck C. Choy

in Effective Medium Theory: Principles and Applications

In this chapter the analogies between EMT with CPA, ATA and RPA are established. Analogies are made with the mean field concept in statistical mechanics and its more subtle variations in areas like ... More

In this chapter the analogies between EMT with CPA, ATA and RPA are established. Analogies are made with the mean field concept in statistical mechanics and its more subtle variations in areas like fluids or in the quantum case such as Hartree–Fock and the modern density functional theories (DFT). The theoretical similarities are emphasized through their fundamental variational principle treatment. Use of Feynman diagrams, relation of EMT with the localization of light, DFT and the classical theory of liquids will stretch the reader’s imagination and perspectives of EMT. The relevance to the quantum Hall effect and electrolytes is noted. The Hubbard model concludes the chapter and can be treated approximately using either CPA or DFT approaches. Landau’s Fermi liquid theory is mentioned.Less

Atomic Bose fluids in optical lattices

Nathan Gemelke and Cheng Chin

in Novel Superfluids: Volume 2

The physics of bosonic atoms in optical lattices is described. Starting from the basic experimental tools available to create such systems, the chapter introduces a simple model (known as the ... More

The physics of bosonic atoms in optical lattices is described. Starting from the basic experimental tools available to create such systems, the chapter introduces a simple model (known as the Bose—Hubbard model) for such a system. A mean-field type picture is developed to determine the basic character of the superfluid and Mott-insulating phases, and the physical conditions under which each should arise. A number of experimental methods are outlined to detect the properties of the gas. The fact that two thermodynamic ground states exist at zero temperature with different types of order implies the existence of a quantum phase transition. The behavior of the gas near this transition is described using an entirely different framework—quantum criticality—formed around the concepts of scaling symmetry and the renormalization group.Less

Making second quantization work

Tom Lancaster and Stephen J. Blundell

in Quantum Field Theory for the Gifted Amateur

This chapter considers how to build single-particle operators out of creation and annihilation operators, and this already gives enough information to discuss the tight-binding model of solid state ... More

This chapter considers how to build single-particle operators out of creation and annihilation operators, and this already gives enough information to discuss the tight-binding model of solid state physics and the Hubbard model.Less

Ultracold gases in optical lattices: basic concepts

Maciej Lewenstein, Anna Sanpera, and Verònica Ahufinger

in Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems

This chapter describes basic methods to realise optical potentials and optical lattices, listing in detail what can be controlled in ultracold atomic systems. It revisits the properties of ... More

This chapter describes basic methods to realise optical potentials and optical lattices, listing in detail what can be controlled in ultracold atomic systems. It revisits the properties of non-interacting particles in periodic lattices: band structure, Bloch functions, and Wannier states. The chapter derives the Hubbard model in the tight binding approximation, and discusses Bose–Einstein condensates in optical lattices in a weak interacting limit, and in a strongly correlated regime.Less