We explore ordering tendencies in weakly coupled Hubbard and t-J models on the two-dimensional square lattice for fermions with N flavors. Such systems may be realized using ultracold atoms with several hyperfine states in optical lattices. The analysis is based on a functional renormalization group approach. In the repulsive Hubbard model for N > 6 and half band filling staggered flux order is found as the dominant instability, in agreement with the large-N limit. In the t-J model without occupancy constraint this type of ground state is suggested already for N > 2. In the attractive case and odd N a full Fermi surface coexists with a superfluid in the ground state. We describe how these unconventional paired states can be identified in the density response of the fermions.
Abstract Ultracold atoms have attracted enormous interest recently, and Cooper pairing between fermionic alkali atoms may have been observed experimentally. Besides the demonstration of phenomena that we are acquainted with from solid state physics, ultracold atoms possibly exhibit interesting many‐body ground states that do not have straightforward counterparts in solid state physics. The hyperfine degree of freedom of the atoms in combination with optical lattices may allow for the approximate realization of fermionic SU(N) Hubbard models where N counts the number of trapped hyperfine states (‘flavors’). We analyze the ground states of this model on the two‐dimensional square lattice using one‐loop renormalization group and meanfield techniques. In the repulsive case for N > 6 and half band filling staggered flux order is found as the dominant instability, in agreement with the large‐N limit. In the attractive case and odd N a full Fermi surface coexists with a superfluid in the ground state. We describe how these unconventional paired states can be identified in the density response of the fermions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
To gain insight into the behavior of the Hubbard model, we define a
SU(n) invariant generalization of the Hubbard-Heisenberg model and, in the large-
n limit, solve it in one dimension and in two dimensions on a square lattice. In one dimension the ground state is completely dimerized near half filling. We show that this behavior agrees with a renormalization-group solution of the one-dimensional
SU(n) Hubbard model. In two spatial dimensions we find several different ground states depending on the size of the hopping term
t, the doping $\ensuremath{\delta}$, and the biquadratic spin interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. In particular, the undimerized "flux" or "
s+id" phase is the ground state at half filling for sufficiently large
t or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. We study the electronic and spin excitations of the various phases and comment on the relevance of the large-
n problem to the high-
Tc superconductors.
The advent of ultracold alkaline-earth atoms in optical lattices has established a platform for investigating correlated quantum matter with SU(
N) symmetry, offering highly tunable model parameters that allow experiments to access phenomena that are unavailable in conventional materials. Understanding the ground-state physics of SU(
N) Fermi-Hubbard models away from the Heisenberg limit and from the spin-flavor balanced setting is important, as examining the flavor imbalance reveals new physics in Fermi-Hubbard models and shows how SU(
N) phases react to practical experimental imperfections in optical lattices. In this study, mean-field phase diagrams are presented for the unit-density SU(3) Fermi-Hubbard model at two sets of flavor densities,
(31−δ,31+δ,31) and
(41−δ,41+δ,21), with the flavor imbalance introduced as
δ. Novel phases are identified at moderate interaction strengths for both densities and their robustness is investigated in the presence of flavor imbalance. Furthermore, we provide microscopic explanations of the phases found and their stability. Analysis of thermal ensembles of random mean-field solutions indicate that, at temperatures accessible in state-of-the-art cold atom experiments, some spin orders are hard for conventional scattering or local observable measurements to detect, but can be more accessible with quantum gas microscopy in optical lattice experiments. This work also shows that nesting and Mottness, intertwined in the usual SU(2) Hubbard model in stark contrast to generic materials, can be tuned in the SU(3) model and play distinct roles. The resulting phase diagrams not only deepen our understanding of SU(
N) Fermi-Hubbard models but also inform future experimental search for new phases.
We study Hubbard models for ultracold bosonic or fermionic atoms loaded into an optical lattice. The atoms carry a high spin F> 1/2, and interact on site via strong repulsive Van der Waals forces. Making convenient rearrangements of the interaction terms, and exploiting their symmetry properties, we derive low-energy effective models with nearest-neighbor interactions, and their properties. We apply our method to F =3/2, and 5/2 fermions on two-dimensional square lattice at quarter, and 1/6 fillings, respectively, and investigate mean-field equations for repulsive couplings. We find for F =3/2 fermions that the plaquette state appearing in the highly symmetric SU (4) case does not require fine tuning, and is stable in an extended region of the phase diagram. This phase competes with an SU (2) flux state, that is always suppressed for repulsive interactions in the absence of external magnetic field. The SU (2) flux state has, however, lower energy than the plaquette phase, and stabilizes in the presence of weak applied magnetic field. For F =5/2 fermions a similar SU (2) plaquette phase is found to be the ground state without external magnetic field.
We study the phase diagram of the extended Hubbard model on a two-dimensional square lattice, including on-site (
U) and nearest-neighbor (
V) interactions, at weak couplings. We show that the charge-density wave phase that is known to occur at half filling when
4V>U gives way to a
dxy-wave superconducting instability away from half filling, when the Fermi surface is not perfectly nested, and for sufficiently large repulsive
V and a range of on-site repulsive interaction
U. In addition, when nesting is further suppressed and in the presence of a nearest-neighbor attraction, a triplet time-reversal breaking
(px+ipy)-wave pairing instability emerges, competing with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing state that is known to dominate at fillings just slightly away from half. At even smaller fillings, where the Fermi surface no longer presents any nesting, the
(px+ipy)-wave superconducting phase dominates in the whole regime of on-site repulsions and nearest-neighbor attractions, while
dxy pairing occurs in the presence of on-site attraction. Our results suggest that zero-energy Majorana fermions can be realized on a square lattice in the presence of a magnetic field. For a system of cold fermionic atoms on a two-dimensional square optical lattice, both an on-site repulsion and a nearest-neighbor attraction would be required, in addition to rotation of the system to create vortices. We discuss possible ways of experimentally engineering the required interaction terms in a cold atom system.