Motivated by the recent observation of evidence of ferromagnetism in correlated insulating states in systems with moir\'e superlattices, we study a two-orbital quantum antiferromagnetic model on the triangular lattice, where the two orbitals physically correspond to the two valleys of the original graphene sheet. For simplicity this model has a $\mathrm{SU}{(2)}^{s}\ensuremath{\bigotimes}\mathrm{SU}{(2)}^{v}$ symmetry, where the two
SU(2) symmetries correspond to the rotation within the spin and valley spaces, respectively. Through analytical argument, Schwinger boson analysis, and also density matrix renormalization group simulation, we find that even though all the couplings in the Hamiltonian are antiferromagnetic, there is still a region in the phase diagram with fully polarized ferromagnetic order. We argue that a Zeeman field can drive a metal-insulator transition in our picture, as was observed experimentally. We also construct spin liquids and topological ordered phases at various limits of this model. Then after doping this model with extra charge carriers, the system most likely becomes a spin-triplet/valley-singlet
d+id topological superconductor, as was predicted previously.
In the field of quantum magnetism, the advent of numerous spin-orbit assisted Mott insulating compounds, such as the family of Kitaev materials, has led to a growing interest in studying general spin models with nondiagonal interactions that do not retain the SU(2) invariance of the underlying spin degrees of freedom. However, the exchange frustration arising from these nondiagonal and often bond-directional interactions for two- and three-dimensional lattice geometries poses a serious challenge for numerical many-body simulation techniques. In this paper, we present an extended formulation of the pseudofermion functional renormalization group that is capable of capturing the physics of frustrated quantum magnets with generic (diagonal and off-diagonal) two-spin interaction terms. Based on a careful symmetry analysis of the underlying flow equations, we reveal that the computational complexity grows only moderately, as compared to models with only diagonal interaction terms. We apply the formalism to a kagome antiferromagnet which is augmented by general in-plane and out-of-plane Dzyaloshinskii-Moriya (DM) interactions, as argued to be present in the spin liquid candidate material herbertsmithite. We calculate the complete ground state phase diagram in the strength of in-plane and out-of-plane DM couplings, and discuss the extended stability of the spin liquid of the unperturbed kagome antiferromagnet in the presence of these couplings.
The pseudo-fermion functional renormalization group is generalized to treat spin Hamiltonians with finite magnetic fields, enabling its application to arbitrary spin lattice models with linear and bilinear terms in the spin operators. We discuss in detail an efficient numerical implementation of this approach making use of the system's symmetries. Particularly, we demonstrate that the inclusion of small symmetry breaking magnetic seed fields regularizes divergences of the susceptibility at magnetic phase transitions. This allows the investigation of spin models within magnetically ordered phases at
T=0 in the physical limit of vanishing renormalization group parameter
Λ. We explore the capabilities and limitations of this method extension for Heisenberg models on the square, honeycomb and triangular lattices. While the zero-field magnetizations of these systems are systematically overestimated, the types of magnetic orders are correctly captured, even if the local orientations of the seed field are chosen differently than the spin orientations of the realized magnetic order. Furthermore, the magnetization curve of the square lattice Heisenberg antiferromagnet shows good agreement with error controlled methods. In the future, the inclusion of magnetic fields in the pseudo-fermion functional renormalization group, which is also possible in three dimensional spin systems, will enable a variety of additional interesting applications such as the investigation of magnetization plateaus.
The pseudo-fermion functional renormalization group is generalized to treat spin Hamiltonians with finite magnetic fields, enabling its application to arbitrary spin lattice models with linear and bilinear terms in the spin operators. We discuss in detail an efficient numerical implementation of this approach making use of the system's symmetries. Particularly, we demonstrate that the inclusion of small symmetry-breaking magnetic seed fields regularizes divergences of the susceptibility at magnetic phase transitions. This allows the investigation of spin models within magnetically ordered phases at
T=0 in the physical limit of vanishing renormalization group parameter $\mathrm{\ensuremath{\Lambda}}$. We explore the capabilities and limitations of this method extension for Heisenberg models on the square, honeycomb, and triangular lattices. While the zero-field magnetizations of these systems are systematically overestimated, the types of magnetic orders are correctly captured, even if the local orientations of the seed field are chosen differently than the spin orientations of the realized magnetic order. Furthermore, the magnetization curve of the square lattice Heisenberg antiferromagnet shows good agreement with error controlled methods. In the future, the inclusion of magnetic fields in the pseudo-fermion functional renormalization group, which is also possible in three-dimensional spin systems, will enable a variety of additional interesting applications such as the investigation of magnetization plateaus.
Motivated by the recent proposal of realizing an SU(4) Hubbard model on triangular moire superlattices, we present a DMRG study of an
SU(4) spin model obtained in the limit of large repulsion for integer filling
νT=1,3. We retain terms in the
t/U expansion up to
O(U2t3) order, that generates nearest-neighbor exchange
J, as well as an additional three-site ring exchange term,
K, which is absent in the SU(2) S=1/2 case. For filling
νT=3, when increasing the three-site ring exchange term
K∼U2t3, we identify three different phases: a spin-symmetric crystal, an
SU(4)1 chiral spin liquid (CSL) and a decoupled one dimensional chain (DC) phase. The CSL phase exists at intermediate coupling:
U/t∈[11.3,22.9]. The sign of
K is crucial to stabilizing the CSL and the DC phase. For filling
νT=1 with the opposite sign of
K, the spin-symmetric crystal phase survives to very large
K. We propose to search for the CSL phase in moire bilayers. For example, in twisted AB stacked transition metal dichalcogenide (TMD) bilayers, the
SU(4) spin is formed by layer pseudospin combined with the real spin (locked to valley). The layer pseudospin carries an electric dipole moment in
z direction, thus the CSL is really a dipole-spin liquid, with quantum fluctuations in both the electric moment and magnetic moment . The CSL phase spontaneously breaks the time reversal symmetry and shows a quantum anomalous Hall effect in spin transport and dipole transport. Smoking gun evidence for the CSL could be obtained through measurement of the quantized dipole Hall effect in counter-flow transport.
Motivated by the realization of spin-valley Hubbard model on a triangular moir\'e superlattice in ABC trilayer graphene aligned with hexagon boron nitride (hBN) and possibly also in twisted transition metal dichalcogenide homobilayers, we study possible Mott insulating phases and pseudogap metals based on symmetry constraint and parton mean field theories. First, we show that the Luttinger constraint allows two distinct symmetric and featureless Fermi liquids when there is an intervalley Hund's term breaking
SU(4) spin rotation. Especially, there exists a symmetric and featureless ``pseudogap metal'' with small Fermi surfaces. Then, we suggest to search for such an unconventional metallic state by doping the Mott insulator at ${\ensuremath{\nu}}_{T}=2$. For this purpose, we study the ${\ensuremath{\nu}}_{T}=2$ Mott insulator using the
SO(6) Schwinger boson or Schwinger fermion parton. At the
SU(4)-symmetric point, we find two symmetric
Z2 spin liquids. With a large anti-intervalley Hund's term, a featureless Mott insulator is natural. Next, we show that doping the featureless Mott insulator or a
Z2 spin liquid can lead to featureless or orthogonal ``pseudogap metal'' with small Fermi surfaces proportional to the doping. Besides, we also provide one scenario for the evolution from pseudogap metal to the conventional Fermi liquid through an intermediate exotic ``deconfined metal'' phase. Last, we give brief comments on the possibility of
U(1) spinon Fermi surface state or
Z4 spin liquid at ${\ensuremath{\nu}}_{T}=1$.