The Hubbard Model Within the Equations of Motion Approach

Advances in Physics, July^October 2004, Vol. 53, Nos. 5–6, 537–768

The Hubbard model within the equations of motion approach

F. Mancini and A. Avella Dipartimento di Fisica ‘E.R. Caianiello’ – Unita` INFM di Salerno Universita` degli Studi di Salerno, I-84081 Baronissi (SA), Italy

[Received 10 February 2004; Accepted in revised form 17 August 2004]

Abstract The Hubbard model plays a special role in condensed matter theory as it is considered to be the simplest Hamiltonian model one can write in order to describe anomalous physical properties of some class of real materials. Unfortunately, this model is not exactly solved except in some limits and therefore one should resort to analytical methods, like the Equations of Motion Approach, or to numerical techniques in order to attain a description of its relevant features in the whole range of physical parameters (interaction, ﬁlling and temperature). In this paper, the Composite Operator Method, which exploits the above mentioned analytical technique, is presented and systematically applied in order to get information about the behaviour of all relevant properties of the model (local, thermodynamic, single- and two-particle properties) in comparison with many other analytical techniques, the above cited known limits and numerical simulations. Within this approach, the Hubbard model is also shown to be capable of describing some anomalous behaviour of cuprate superconductors.

Contents page Introduction 538 1. The single-band Hubbard model 539 1.1. The model 539 1.2. Generalizations 541 1.3. Symmetries 542 1.4. Thermodynamic relations 545 1.5. Conservation laws 546 2. A formalism for highly interacting system 547 2.1. Physical systems and composite ﬁelds 548 2.2. Green’s functions (GF) and equations of motion formalism 553 2.3. GF properties, spectral representation, zero-frequency functions 556 2.4. The Composite Operator Method 563 2.5. Physical properties 566 2.6. Self-energy approximations 571 3. The paramagnetic solution of the single-band Hubbard model 577 3.1. The retarded Green’s function 577 3.2. local properties 583 3.3. The single-particle properties 591 3.4. Thermodynamics 602 3.5. The two-particle Green’s functions 618 3.6. The optical conductivity 641 3.7. The Knight shift and the relaxation rate 645 3.8. The Mott–Hubbard transition 648

Advances in Physics ISSN 0001–8732 print/ISSN 1460–6976 online # 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00018730412331303722 538 F. Mancini and A. Avella

4. The attractive Hubbard model 653 4.1. The self-consistent parameters and the chemical potential 653 4.2. The double occupancy 655 4.3. The energy spectra and density of states 656 4.4. The spin magnetic susceptibility 657 5. The Hubbard model with inter-site interaction 658 5.1. The Hamiltonian and field equations 659 5.2. The chemical potential 660 5.3. The phase diagram 663 5.4. The double occupancy, kinetic and internal energy 667 5.5. The Fermi surface and the DOS 669 6. The Hubbard model in presence of an external magnetic field 671 6.1. The exact solutions 672 6.2. The local properties 676 6.3. The internal energy and electronic specific heat 681 6.4. The metal–insulator transition in magnetic field 683 7. Spontaneously broken symmetry solutions 684 7.1. The ferromagnetic phase 684 7.2. The antiferromagnetic phase 690 7.3. The superconducting phase 705 8. The bosonic sector of the Hubbard model 716 8.1. The bosonic Green’s function 717 8.2. The self-consistent equations for the 1D system 720 8.3. The self-consistent equations for the 2D system 722 8.4. The charge and spin susceptibility 725 8.5. The correlation functions 727 9. The Hubbard model as minimal model for cuprate superconductors 730 9.1. Introduction 731 9.2. The van Hove scenario 732 9.3. The normal state 734 9.4. The superconducting state 747 Conclusions 748 Acknowledgements 749 Appendices 749 A. The d-dimensional cubic Bravais lattice 749 B. Non-interacting DOS 751 C. Energy spectra and spectral density matrices 752 D. Equivalence of basis 753 E. Green’s functions and spectral moments 753 F. Algebra of Hubbard operators 754 G. Representations for the Hubbard operators 755 H. Self-consistent equations at half filling 755 References 757

Introduction In the last few decades, the experimental and theoretical study of materials presenting anomalous properties of technological relevance (superconductivity, magneto-resistance, etc.) has raised more and more interest also thanks to the possibilities, just recently opened, to design and synthesize materials at the micro- and nanoscopical level and to characterize them with quite impressive resolutions. Unfortunately, the significant improvements of the experimental techniques do not correspond to significant enlightenment coming from the theoretical approaches. Many of the anomalous properties under analysis are due to the strength and The Hubbard model within the equations of motion approach 539 complexity of the correlations present in these systems. According to this, in order to attempt a microscopical description and to gain some comprehension of the physical properties observed in these materials, new methods of theoretical investigation must be formulated. In the last ten years, we have been developing a theoretical scheme, named the Composite Operator Method (COM), for strongly correlated systems mainly based on two concepts: the use of composite fields to describe the fundamental excitations dynamically generated in a complex system and the use of symmetry constraints to bind the dynamics to the correct Hilbert space. COM has shown to be capable of catching the physics and describing the features of many strongly interacting systems and, in particular, of the Hubbard model and of the materials mimed by it (e.g., the cuprate superconductors). The Hubbard model is recognized as the archetype for the strongly correlated class of Hamiltonian systems: competition of itinerancy and localization, spin and charge ordering, metal–insulator transition, complex phase diagram, etc. In this paper we wish to accomplish three main goals: first, to give a detailed presentation of the Composite Operator Method; second, to provide a quite exten- sive review of the physical properties of the Hubbard model in known exact limits (noninteracting, atomic and 1D), within the two-pole approximation and by means of numerical techniques (quantum Monte Carlo, Lanczos, Exact Diagonalization); third, to summarize the results (regarding local, thermodynamic, single-particle, two- particle and, in general, response properties) of the application of the COM to the Hubbard model within various boundary conditions, to compare them with the numerical simulations and to interpret the experimental data for the cuprate superconductors. The aim is to show the peculiarities and capabilities of this method as a general approach for strongly correlated systems and, in combination with the Hubbard model, its potentialities to describe, on the microscopical level, both relevant theoretical issues and anomalous experimental features of real materials. The Composite Operator Method tries to give global answers (i.e., in the whole space of model and physical parameters: interaction strength U, temperature T, filling n) in the quest for a deep comprehension of the Hubbard model properties. The model response is profoundly different according to the region of the latter space we decide to explore. Other approximations focus just on one region and usually give wrong results in the rest of the parameter space. We will see that the comparisons with both the numerical simulations and the experimental data show very good agreement on the whole parameter space. According to this, it is fully justified to adopt a very high degree of confidence in the results themselves and in the related microscopical interpretations of the features of the model and materials. Anyway, we are aware that the approximation used in this review, the two-pole approximation, misses relevant features at low temperature/frequencies. Then, it is worth noticing that this drawback is due to the approximation chosen and not to the COM itself. In fact, this latter allows the inclusion of a fully momentum- and frequency- dependent self-energy in the scheme and we are currently working in this direction.

1. The single-band Hubbard model 1.1. The model The Hubbard model [1–5] was ﬁrst introduced for describing the correlations of electrons in the narrow energy d-bands of transition metals. In particular, the