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Physical review. B, Condensed matter

Fractional quantum numbers and electron correlations

W. Gasser

Published Jan 15, 1993• 2 citations
Publisher

Abstract

It will be shown that sequences of fractional quantum numbers (in this case occupation numbers) appear to be important for the instabilities of a one-dimensional Hubbard model in the strong-correlation limit (\ensuremath{\Delta}/U\ensuremath{\ll}1, but finite) if only the highest commensurability is observed and besides the 2${\mathit{k}}_{\mathit{F}}$ instability the 2${\mathit{lk}}_{\mathit{F}}$ instabilities are also taken into consideration. These sequences obey an odd-denominator rule. Further, it will be shown that an interacting two-dimensional electron gas in a strong external magnetic field and under the influence of a weak sinusoidal substrate potential may be described by a one-dimensional Hubbard model which also gives fractional quantum numbers (in this case fractional filling factors) again exhibiting an odd-denominator rule.

DOI
10.1103/physrevb.47.1324

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It will be shown that sequences of fractional quantum numbers (in this case occupation numbers) appear to be important for the instabilities of a one-dimensional Hubbard model in the strong-correlation limit (\ensuremath{\Delta}/U\ensuremath{\ll}1, but finite) if only the highest commensurability is observed and besides the 2kF{\mathit{k}}_{\mathit{F}}kF​ instability the 2lkF{\mathit{lk}}_{\mathit{F}}lkF​ instabilities are also taken into consideration. These sequences obey an odd-denominator rule. Further, it will be shown that an interacting two-dimensional electron gas in a strong external magnetic field and under the influence of a weak sinusoidal substrate potential may be described by a one-dimensional Hubbard model which also gives fractional quantum numbers (in this case fractional filling factors) again exhibiting an odd-denominator rule.
A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M > N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1/ q , q odd.. Within this model the complimentary wavefunction for v =1−1/ q is found explicitly, and extremely simple ground state wavefunctions for arbitrary odd-denominator filling factors are proposed.
We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.
The influence of spin and charge fluctuations on spectra of the two-dimensional fermionic Hubbard model is considered using the strong coupling diagram technique. Infinite sequences of diagrams containing ladder inserts, which describe the interaction of electrons with these fluctuations, are summed, and obtained equations are self-consistently solved for the ranges of Hubbard repulsions [Formula: see text], temperatures [Formula: see text] and electron concentrations [Formula: see text] with t the intersite hopping constant. For all considered U the system exhibits a transition to the long-range antiferromagnetic order at [Formula: see text]. At the same time no indication of charge ordering is observed. Obtained solutions agree satisfactorily with results of other approaches and obey moments sum rules. In the considered region of the U-T plane, the curve separating metallic solutions passes from [Formula: see text] at the highest temperatures to U = 2t at [Formula: see text] for half-filling. If only short-range fluctuations are allowed for the remaining part of this region is occupied by insulating solutions. Taking into account long-range fluctuations leads to strengthening of maxima tails, which transform a part of insulating solutions into bad-metal states. For low T, obtained results allow us to trace the gradual transition from the regime of strong correlations with the pronounced four-band structure and well-defined Mott gap for [Formula: see text] to the Slater regime of weak correlations with the spectral intensity having a dip along the boundary of the magnetic Brillouin zone due to an antiferromagnetic ordering for [Formula: see text]. For [Formula: see text] and [Formula: see text] doping leads to the occurrence of a pseudogap near the Fermi level, which is a consequence of the splitting out of a narrow band from a Hubbard subband. Obtained spectra feature waterfalls and Fermi arcs, which are similar to those observed in hole-doped cuprates.
Two-dimensional interacting electron systems become strongly correlated if the electrons are subject to a perpendicular high magnetic field. After introducing the physics of the quantum Hall regime the incompressible many-particle ground state and its excitations are studied in detail at fractional filling factors for spin-polarized electrons. The spin degree of freedom whose importance was shown in recent experiments is considered by studying the thermodynamics at filling factor one and near one.
Two-dimensional interacting electron systems become strongly correlated if the electrons are subject to a perpendicular high magnetic field. After introducing the physics of the quantum Hall regime the incompressible many-particle ground state and its excitations are studied in detail at fractional filling factors for spin-polarized electrons. The spin degree of freedom whose importance was shown in recent experiments is considered by studying the thermodynamics at filling factor one and near one.
The electronic system of the 1D Hubbard model is not stable due to Peierls instability; the correlations are strong even for the weak Coulomb interaction. The resulting strongly correlated state without Landau quisi--particle excitations is known as the Luttinger liquid. Critical exponents of a power--law dependence of correlation functions at low energies differ substantially for the Luttinger and Fermi liquids. In this paper we evaluate two critical exponents that define non--trivial behavior of the density of electronic states at low frequences and the momentum distribution of the occupation number nearby kFk_FkF​.