The Hubbard model is reformulated in terms of different ``colored'' fermion species for the electrons or holes at different lattice sites. Antiferromagnetic ordering or d-wave superconductivity can then be described in terms of translationally invariant expectation values for colored composite scalar fields. A suitable mean field approximation for the two-dimensional colored Hubbard model shows indeed phases with antiferromagnetic ordering or d-wave superconductivity at low temperature. At low enough temperature the transition to the antiferromagnetic phase is of first order. The present formulation also allows an easy extension to more complicated microscopic interactions.
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including
d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For
d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum
q=0 and
q=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.
We begin by scrutinizing a recent proposal that presents an alternate description of the half-filled Landau level in terms of massless Dirac fermions. In Chapter \\ref{chap:HFLL}, we examine the possibility of pairing of these Dirac fermions by numerically solving the coupled Eliashberg equations unlike a related previous calculation (Wang and Chakravarty, 2016). In addition, vertex corrections are calculated to be zero from the Ward identity. We find that pairing is possible in non-zero angular momentum channels; only differences are minor numerical shifts. As before, the pairing leads to the gapped Pfaffian and anti-Pfaffian states. However, in our approximation scheme, pairing is not possible in the putative particle–hole symmetric state for
l=0 angular momentum. The specific heat at low temperatures of a system of massless Dirac fermions interacting with a transverse gauge field, expected to be relevant for the half-filled Landau level, is calculated. Using the Luttinger formula, it is found to be
proptoTlnT in the leading low temperature limit, due to the exchange of transverse gauge bosons. The result agrees with the corresponding one in the nonrelativistic composite fermion theory of Halperin, Lee and Read of the half-filled Landau level.The rest of the thesis concerns the cuprate high-
Tc superconductors (``cuprates''). Conventional wisdom says that beyond the superconducting ``dome'' in cuprates, the material behaves as a Fermi liquid. However, this picture does not help explain the disappearance of the superconducting order parameter, and there are some anomalous measurements that cannot be explained by a Fermi liquid phase. It was proposed by Kopp, Ghosal, and Chakravarty that there is a \\emph{ferromagnetic} phase at zero temperature beyond the superconducting dome, and that fluctuations of the ferromagnetic order parameter compete with the superconducting order parameter and work to suppress the superconducting transition temperature \\cite{Kopp:2007fn}. In Chapter \\ref{chap:ferro} we summarize the experimental evidence for ferromagnetic fluctuations in the overdoped cuprates, and present several calculations supporting the existence of a ferromagnetic ground state in the 2D single-band Hubbard model, which model is thought to provide an adequate description of the cuprate superconductors.Another region of interest in the cuprate phase diagram is the pseudogap phase. It is unclear which of a host of competing order parameters is responsible for the behavior in this phase, such as the recent observation of an anomalous thermal Hall conductance in the cuprate La
2−xSr
xCuO
4 \\cite{Grissonnanche:2019gx}. One promising candidate is the
d-density wave state. In Chapter \\ref{chap:DMI}, we investigate the effect that density wave states have on the localized spins of a square lattice. We derive the effective Dzyaloshinskii-Moriya (DM) interaction from first principles and study its effects on both ferromagnetic and antiferromagnetic backgrounds. We find that topologically nontrivial density wave states can induce stable DM interactions among the localized spins of the lattice when an external magnetic field is present. Furthermore, these density wave-induced DM vectors point along the external magnetic field's direction--implying that they break time-reversal and spin rotation symmetries in the same manner. Due to these symmetry considerations alone we find that the underlying magnon excitations cannot induce any thermal Hall effect. Utilizing a Holstein-Primakoff substitution about a mean-field ground state expansion we calculate the topological density wave corrections to magnetic ground state energy, spin canting angles, and the dispersion of the magnons for both the ferromagnetic and antiferromagnetic cases.
The mechanism that leads to high‐temperature superconductivity in cuprates remains an open question despite intense study for nearly two decades. Here, we introduce an SU(4) model for cuprate systems having many similarities to dynamical symmetries known to play an important role in nuclear structure physics and in elementary particle physics. Analytical solutions in three dynamical symmetry limits of this model are found: an SO(4) limit associated with antiferromagnetic order; an SU(2) limit that may be interpreted as a d‐wave pairing condensate; and an SO(5) limit that may be interpreted as a doorway state between the antiferromagnetic order and the superconducting order. It is demonstrated that with a slightly broken SO(5) but under constraint of the parent SU(4) symmetry, the model is capable of describing the rich physics that is crucial in explaining why cuprate systems that are antiferromagnetic Mott insulators at half filling become superconductors through hole doping.
To date, the Hubbard model and its strong coupling limit, the t-J model, serve as the canonical model for strongly correlated electron systems in solids. Approximating the Coulomb interaction by only the on-site term (Hubbard U-term), however, may not be sufficient to describe the essential physics of interacting electron systems. We develop a more complete model in which all the next leading order terms besides the on-site term are retained. Moreover, we discuss how the inclusion of these neglected interaction terms in the Hubbard model changes the t-J model.