The repulsive Hubbard model has been immensely useful in understanding strongly correlated electron systems, and serves as the paradigmatic model of the field. Despite its simplicity, it exhibits a strikingly rich phenomenology which is reminiscent of that observed in quantum materials. Nevertheless, much of its phase diagram remains controversial. Here, we review a subset of what is known about the Hubbard model, based on exact results or controlled approximate solutions in various limits, for which there is a suitable small parameter. Our primary focus is on the ground state properties of the system on various lattices in two spatial dimensions, although both lower and higher dimensions are discussed as well. Finally, we highlight some of the important outstanding open questions. 

The Hubbard Model

We study the spin liquid candidate of the spin-1/2 J_{1}-J_{2} Heisenberg antiferromagnet on the triangular lattice by means of density matrix renormalization group (DMRG) simulations. By applying an external Aharonov-Bohm flux insertion in an infinitely long cylinder, we find unambiguous evidence for gapless U(1) Dirac spin liquid behavior. The flux insertion overcomes the finite size restriction for energy gaps and clearly shows gapless behavior at the expected wave vectors. Using the DMRG transfer matrix, the low-lying excitation spectrum can be extracted, which shows characteristic Dirac cone structures of both spinon-bilinear and monopole excitations. Finally, we confirm that the entanglement entropy follows the predicted universal response under the flux insertion.

Dirac Spin Liquid on the Spin- 1 / 2 Triangular Heisenberg Antiferromagnet

Magnetic susceptibility, NMR, muon spin relaxation, and inelastic neutron scattering measurements show that kapellasite, Cu3Zn(OH)6Cl2, a geometrically frustrated spin-1/2 kagome antiferromagnet polymorphic with herbertsmithite, is a gapless spin liquid showing unusual dynamic short-range correlations of noncoplanar cuboc2 type which persist down to 20 mK. The Hamiltonian is determined from a fit of a high-temperature series expansion to bulk susceptibility data and possesses competing exchange interactions. The magnetic specific heat calculated from these exchange couplings is in good agreement with experiment. The temperature dependence of the magnetic structure factor and the muon relaxation rate are calculated in a Schwinger-boson approach and compared to experimental results.

https://absimage.aps.org/image/MAR13/MWS_MAR13-2012-008221.pdf

Kapellasite: a kagome quantum spin liquid with competing interactions

At strong repulsion, the triangular-lattice Hubbard model is described by s=1/2 spins with nearest-neighbor antiferromagnetic Heisenberg interactions and exhibits conventional 120° order. Using the infinite density matrix renormalization group and exact diagonalization, we study the effect of the additional four-spin interactions naturally generated from the underlying Mott-insulator physics of electrons as the repulsion decreases. Although these interactions have historically been connected with a gapless ground state with emergent spinon Fermi surface, we find that, at physically relevant parameters, they stabilize a chiral spin liquid (CSL) of Kalmeyer-Laughlin (KL) type, clarifying observations in recent studies of the Hubbard model. We then present a self-consistent solution based on a mean-field rewriting of the interaction to obtain a Hamiltonian with similarities to the parent Hamiltonian of the KL state, providing a physical understanding for the origin of the CSL.

/pdf/four-spin-terms-and-the-origin-of-the-chiral-spin-liquid-in-2x2rtxiqn9.pdf

Four-Spin Terms and the Origin of the Chiral Spin Liquid in Mott Insulators on the Triangular Lattice.

We propose a moiré bilayer as a platform where exotic quantum phases can be stabilized and electrically detected. Moiré bilayers consist of two separate moiré superlattice layers coupled through the interlayer Coulomb repulsion. In the small distance limit, an SU(4) spin can be formed by combining layer pseudospin and the real spin. As a concrete example, we study an SU(4) spin model on triangular lattice in the fundamental representation. By tuning a three-site ring exchange term K∼(t^{3}/U^{2}), we find the SU(4) symmetric crystallized phase and an SU(4)_{1} chiral spin liquid at the balanced filling. We also predict two different exciton supersolid phases with interlayer coherence at imbalanced filling under displacement field. Especially, the system can simulate an SU(2) Bose-Einstein condensation by injecting interlayer excitons into the magnetically ordered Mott insulator at the layer polarized limit. Smoking gun evidences of these phases can be obtained by measuring the pseudospin transport in the counterflow channel. 

SU(4) Chiral Spin Liquid, Exciton Supersolid, and Electric Detection in Moiré Bilayers

We study the interplay of competing interactions in spin-$\frac{1}{2}$ triangular Heisenberg model through tuning the first- (${J}_{1}$), second- (${J}_{2}$), and third-neighbor (${J}_{3}$) couplings. Based on a large-scale density-matrix renormalization group calculation, we identify a quantum phase diagram of the system and discover a gapless chiral spin-liquid (CSL) phase in the intermediate ${J}_{2}$ and ${J}_{3}$ regime. This CSL state spontaneously breaks time-reversal symmetry with finite scalar chiral order, and it has gapless excitations implied by a vanishing spin triplet gap and a finite central charge on the cylinder. Moreover, the central charge grows rapidly with the cylinder circumference, indicating emergent spinon Fermi surfaces. To understand the numerical results we propose a parton mean-field spin-liquid state, the $U(1)$ staggered flux state, which breaks time-reversal symmetry with chiral edge modes by adding a Chern insulator mass to Dirac spinons in the $U(1)$ Dirac spin liquid. This state also breaks lattice rotational symmetries and possesses two spinon Fermi surfaces driven by nonzero ${J}_{2}$ and ${J}_{3}$, which naturally explains the numerical results. This realizes an example of a gapless CSL state with coexisting spinon Fermi surfaces and chiral edge states, demonstrating the rich family of interesting quantum phases emergent from competing interactions in triangular-lattice magnets.

/pdf/chiral-spin-liquid-with-spinon-fermi-surfaces-in-the-spin-1-omcx1sdp7s.pdf

Chiral spin liquid with spinon Fermi surfaces in the spin-1/2 triangular Heisenberg model

Gapless Chiral Spin Liquid Phase in Spin-$1/2$ Triangular Heisenberg Model

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Chiral spin liquid with spinon Fermi surfaces in the spin-1/2 triangular Heisenberg model

Gapless Chiral Spin Liquid Phase in Spin-$1/2$ Triangular Heisenberg Model