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Lecture-4 Slides (PDF, 16 Course Outline: Feb. 15-19, 2021 ZOOM: https://uzh.zoom.us/j/98416350390?pwd=WEhkaTBkMzdjL21BYjZSdGMyZUdNUT09 Monday Tuesday Wednesday Thursday Friday Room ZOOM ZOOM ZOOM ZOOM ZOOM 10-10h45 Lecture 1 Lecture 4 Lecture 7 Lecture 10 Lecture 13 Johan Johan Johan Marc Johan Fermi-liquids Strongly Supercond. Magnetism Anomalous Correlated Hall effect Insulators 11-11h45 Lecture 2 Lecture 5 Lecture 8 Lecture 11 Lecture 14 Marc Marc Marc Marc Johan Kondo-physics Quantum Supercond. Skyrmions Thermal Hall Phase Effect and Transitions Conductivity Lunch Break Lunch Break Lunch Break Lunch Break Lunch Break 13h30- Lecture 3 Lecture 6 Lecture 9 Lecture 12 Lecture 15 14h15 Marc Johan Johan Marc Johan Heavy Fermions Non-Fermi Nematicity Spin Liquids Charge Order liquids Exercise Class Exercise Class 14h30-16 14h30-16 Metal insulator transitions (MIT) Scientific RepoRts | 6:23652 | DOI: 10.1038/srep23652 MIT applications: Smart windows Fathers of Insulators Rudolf Peierls Nevill Francis Mott Jun Kondo Nobel Prize 1977 Background information: Material classification Band Peierls Mott Kondo …. Background information: Band structure Band Gap Band insulator Metal Fathers of Insulators Rudolf Peierls Nevill Francis Mott Jun Kondo Crystal structure -> Band structure Peierls Transition Half-filling 1d Chains http://galileo.phys.virGinia.edu/classes/752.mf1i.sprinG03/PeierlsTrans.htm BechGaard Salts CRYSTAL STRUCTURE beta-co-(BEDT-TTF)2I3 BechGaard Salts RESISTIVITY J. Phys.: Condens. Matter 25 (2013) 343201 Fathers of Insulators Rudolf Peierls Nevill Francis Mott Jun Kondo Kinetic Energy Dominates Potential EnerGy Dominates 1042 Imada, Fujimori, and Tokura: Metal-insulator transitions the single-particle (carrier) number, as in the transition MIT in correlated metals has been most thorough and to a band insulator. For example, this takes place in the systematic in d-electron systems, namely, transition- transition from an antiferromagnetic metal to an antifer- metal compounds. Many examples will be reviewed in romagnetic Mott insulator, where the folding of the Bril- this article. In d-electron systems, orbital degeneracy is louin zone due to the superstructure of the magnetic an important and unavoidable source of complicated be- periodicity creates a completely filled lower band. Car- havior. For example, under the cubic crystal-field sym- riers are doped into small pockets of Fermi surface metry of the lattice, any of the threefold degenerate t2g whose Fermi volume vanishes at the MIT. bands, dxy , dyz , and dzx as well as twofold degenerate From this heuristic argument, one can see at least two eg bands, dx22y2 and d3z22r2, can be located near the distinct routes to the Mott insulator when one ap- Fermi level, depending on transition-metal ion, lattice proaches the MIT point from the metallic side, namely, structure, composition, dimensionality, and so on. In ad- the mass-diverging type and carrier-number-vanishing dition to strong spin fluctuations, effects of orbital fluc- type. The diversity of anomalous features of metallic tuations and orbital symmetry breaking play important phases near both types of MIT is a central subject of this roles in many d-electron systems, as discussed in Secs. review. Mass enhancement or carrier-number reduction II.H and IV. The orbital correlations are frequently as well as more complicated features have indeed been strongly coupled with spin correlations through the observed experimentally. The experiments were exam- usual relativistic spin-orbit coupling as well as through ined from various, more or less independently devel- orbital-dependent exchange interactions and quan- oped theoretical approaches, as detailed in Sec. II. In drupole interactions. An example of this orbital effect particular, anomalous features of correlated metals near known as the double-exchange mechanism is seen in Mn the Mott insulator appear more clearly when the MIT is oxides (Sec. IV.F), where strong Hund’s-rule coupling continuous. Theoretically, this continuous MIT has been between the eg and t2g orbitals triggers a transition be- a subject of recent intensive studies in which unusual tween the Mott insulator with antiferromagnetic order Mott-insulator: Filling conditionmetallic properties are understood from various critical and the ferromagnetic metal. Colossal negative magne- fluctuations near the quantum critical point of the MIT. toresistance near the transition to this ferromagnetic A prototype of theoretical understanding for the tran- metal phase has been intensively studied recently. Filling and Energeticssition between the Mott insulator and metals was Another aspect of orbital degeneracy is the overlap or achieved by using simplified lattice fermion models, in the closeness of the d band and the p band of ligand particular, in the celebrated Hubbard model (Anderson, atoms which bridge the elements in transition-metal 1959; Hubbard, 1963, 1964a, 1964b; Kanamori, 1963). compounds. For example, as clarified in Secs. II.A and The Hubbard model considers only electrons in a single III.A, in the transition-metal oxides, the oxygen 2ps band.Hubbard Model Its Hamiltonian in a second-quantized form is level becomes close to that of the partially filled 3d band given by near the Fermi level for heavier transition-metal ele- ments such as Ni and Cu. Then the charge gap of the 5 1 2mN, (1.1a) HH Ht HU Mott insulator cannot be accounted for solely with d KINETIC ENERGY † electrons, but p-electron degrees of freedom have also t52t( ~ciscjs1H.c.!, (1.1b) to be considered. In fact, when we could regard the H ^ij& Hubbard model as a description of a d band only, the POTENTIAL ENERGY 1 1 charge excitation gap is formed between a singly occu- U5U ~ni 2 2 !~ni 2 2 !, (1.1c) H (i " # pied d band (the so-called lower Hubbard band) and a doubly occupied (with spin up and down) d band (the half and so-called upper Hubbard band). However, if the ps level - fillinG becomes closer, the character of the minimum charge N[ nis , (1.1d) excitation gap changes to that of a gap between a singly (is M. Imada et al., occupied d band with fully occupied p band and a singly where the creation (annihilation) of the single-band occupied d band with a ps hole. This kind of insulator, Rev. Mod. Phys. 70, 1040 (1998) † electron at site i with spin s is denoted by cis(cis) with which was clarified by Zaanen, Sawatzky, and Allen † nis being the number operator nis[ciscis . In this sim- (1985), is now called a charge-transfer (CT) insulator as plification, various realistic complexities are ignored, as contrasted to the former case, the Mott-Hubbard (MH) we shall see in Sec. II.A. However, at the same time, insulator, which we discuss in detail in Secs. II.A and low-energy and low-temperature properties are often III.A, and the distinction is indeed observed in high- well described even after this simplification since only a energy spectroscopy. Correspondingly, compounds that small number of bands (sometimes just one band) are have the MH insulating phase are called MH com- crossing the Fermi level and have to do with low-energy pounds while those with the CT insulating phase are excitations. The parameters of the simplified models in called CT compounds. The term ‘‘Mott insulator’’ is this case should be taken as effective values derived used in this review in a broad sense which covers both from renormalized bands near the Fermi level. types. Recent achievements in the field of strongly cor- One of the most drastic simplification in the Hubbard related electrons, especially in d-electron systems, have model is to consider only electrons in a single orbit, say brought us closer to understanding more complicated the s orbit. In contrast, the experimental study of the situations in which there is an interplay between orbital Rev. Mod. Phys., Vol. 70, No. 4, October 1998 1042 Imada, Fujimori, and Tokura: Metal-insulator transitions the single-particle (carrier) number, as in the transition MIT in correlated metals has been most thorough and to a band insulator. For example, this takes place in the systematic in d-electron systems, namely, transition- transition from an antiferromagnetic metal to an antifer- metal compounds. Many examples will be reviewed in romagnetic Mott insulator, where the folding of the Bril- this article. In d-electron systems, orbital degeneracy is louin zone due to the superstructure of the magnetic an important and unavoidable source of complicated be- periodicity creates a completely filled lower band. Car- havior. For example, under the cubic crystal-field sym- riers are doped into small pockets of Fermi surface metry of the lattice, any of the threefold degenerate t2g whose Fermi volume vanishes at the MIT. bands, dxy , dyz , and dzx as well as twofold degenerate From this heuristic argument, one can see at least two eg bands, dx22y2 and d3z22r2, can be located near the distinct routes to the Mott insulator when one ap- Fermi level, depending on transition-metal ion, lattice proaches the MIT point from the metallic side, namely, structure, composition, dimensionality, and so on. In ad- the mass-diverging type and carrier-number-vanishing dition to strong spin fluctuations, effects of orbital fluc- type. The diversity of anomalous features of metallic tuations and orbital symmetry breaking play important phases near both types of MIT is a central subject of this roles in many d-electron systems, as discussed in Secs. review. Mass enhancement or carrier-number reduction II.H and IV. The orbital correlations are frequently as well as more complicated features have indeed been strongly coupled with spin correlations through the observed experimentally. The experiments were exam- usual relativistic spin-orbit coupling as well as through ined from various, more or less independently devel- orbital-dependent exchange interactions and quan- oped theoretical approaches, as detailed in Sec. II. In drupole interactions. An example of this orbital effect particular, anomalous features of correlated metals near known as the double-exchange mechanism is seen in Mn the Mott insulator appear more clearly when the MIT is oxides (Sec.
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