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The last decade has seen a large increase in the number of electronic-structure calculations that involve adding a Hubbard term to the local density approximation band-structure Hamiltonian. The Hubbard term is then solved either at the mean-field level or with sophisticated many-body techniques such as dynamical mean field theory. We review the physics underlying these approaches and discuss their strengths and weaknesses in terms of the larger issues of electronic structure that they involve. In particular, we argue that the common assumptions made to justify such calculations are inconsistent with what the calculations actually do. Although many of these calculations are often treated as essentially first-principles calculations, in fact, we argue that they should be viewed from an entirely different point of view, viz., as phenomenological many-body corrections to band-structure theory. Alternatively, they may also be considered to be just a more complex Hubbard model than the simple one- or few-band models traditionally used in many-body theories of solids.
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The repulsive fermionic Hubbard model is a typical model describing correlated electronic systems. Although it is a simple model with only a kinetic term and a local interaction term, their competition generates rich phases. When the interaction part is significant, usually in many strongly correlated, flat or narrow band systems, lots of novel correlated phases may emerge. One way to understand the possible correlated phases is to go beyond finite interaction and solve the infinite-$U$ Hubbard model. Solving infinite-$U$ Hubbard model is usually extremely hard, and a large-scale unbiased numerical study is still missing. In this Letter, we propose a projection approach, such that a controllable quantum Monte Carlo (QMC) simulation on infinite-$U$ Hubbard model may be done at some integer fillings where either it is sign problem free or surprisingly has an algebraic sign structure -- a power law dependence of average sign on system size. We demonstrate our scheme on the infinite-$U$ $SU(2N)$ fermionic Hubbard model on both square and honeycomb lattice at half-filling, where it is sign problem free, and suggest possible correlated ground states. The method can be generalized to study certain extended Hubbard models applying to cluster Mott insulators or 2D Morie systems, among one of them at certain non-half integer filling, the sign has an algebraic behavior such that it can be numerically solved within a polynomial time. Further, our projection scheme can also be generalized to implement the Gutzwiller projection to spin basis such that $SU(2N)$ quantum spin models and Kondo lattice models may be studied in the framework of fermionic QMC simulations.
Two-dimensional Hubbard model is very important in condensed matter physics. However it has not been resolved though it has been proposed for more than 50 years. We give several methods to construct eigenstates of the model that are independent of the on-site interaction strength $U$.
The Hubbard model, which augments independent-electron band theory with a single parameter to describe electron-electron correlations, is widely regarded to be the `standard model of condensed matter physics. The model has been remarkably successful at addressing a range of correlation effects in solids, but beyond one dimension its solution is intractable. Much current research aims, therefore, at finding appropriate approximations to the Hubbard model phase diagram. Here we take the new approach of using ab initio electronic structure methods to design a material whose Hamiltonian is that of the single-band Hubbard model. Solution of the Hubbard model will then be available through measurement of the materials properties. After identifying an appropriate crystal class and several appropriate chemistries, we use density functional theory and dynamical mean-field theory to screen for the desired electronic band structure and metal-insulator transition. We then explore the most promising candidates for structural stability and suitability for doping and propose specific materials for subsequent synthesis. Finally, we identify a regime -- that should manifest in our bespoke material -- in which the single-band Hubbard model on a triangular lattice exhibits exotic d-wave superconductivity.
We study the flat-band ferromagnetic phase of a topological Hubbard model within a bosonization formalism and, in particular, determine the spin-wave excitation spectrum. We consider a square lattice Hubbard model at 1/4-filling whose free-electron term is the pi-flux model with topologically nontrivial and nearly flat energy bands. The electron spin is introduced such that the model either explicitly breaks time-reversal symmetry (correlated flat-band Chern insulator) or is invariant under time-reversal symmetry (correlated flat-band $Z_2$ topological insulator). We generalize for flat-band Chern and topological insulators the bosonization formalism [Phys. Rev. B 71, 045339 (2005)] previously developed for the two-dimensional electron gas in a uniform and perpendicular magnetic field at filling factor u=1. We show that, within the bosonization scheme, the topological Hubbard model is mapped into an effective interacting boson model. We consider the boson model at the harmonic approximation and show that, for the correlated Chern insulator, the spin-wave excitation spectrum is gapless while, for the correlated topological insulator, gapped. We briefly comment on the possible effects of the boson-boson (spin-wave--spin-wave) coupling.
Using a self-consistent Hartree-Fock approximation we investigate the relative stability of various stripe phases in the extended $t$-$t$-$U$ Hubbard model. One finds that a negative ratio of next- to nearest-neighbor hopping $t/t<0$ expells holes from antiferromagnetic domains and reinforces the stripe order. Therefore the half-filled stripes not only accommodate holes but also redistribute them so that the kinetic energy is gained, and these stripes take over in the regime of $t/tsimeq -0.3$ appropriate for YBa$_2$Cu$_3$O$_{6+delta}$.
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