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An $SU(4)$ chiral spin liquid and quantized dipole Hall effect in moir\'e bilayers

Ya-Hui Zhang, D. N. Sheng, Ashvin Vishwanath

Published Mar 17, 2021• 0 citations
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Abstract

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 $\nu_T=1,3$. We retain terms in the $t/U$ expansion up to $O(\frac{t^3}{U^2})$ 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 $\nu_T=3$, when increasing the three-site ring exchange term $K \sim \frac{t^3}{U^2}$, 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: $U/t \in [11.3,\,22.9]$. The sign of $K$ is crucial to stabilizing the CSL and the DC phase. For filling $\nu_T=1$ 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.

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As one of the most intriguing states of matter, the chiral spin liquid (CSL) has attracted much scientific interest while its existence and mechanism in crystalline strongly correlated systems remain hotly debated. On the other hand, strong correlation driven emergent phenomena can be realized in twisted transition metal dichalcogenide bilayers with a tremendously tunable large length scale providing a new platform for the emergence of CSLs. We focus on a strongly correlated model relevant to heterobilayer WSe2/MoSe2{\text{WSe}}_{2}/{\text{MoSe}}_{2}WSe2​/MoSe2​ and investigate the Mott insulating phase at half filling under an out-of-plane magnetic field. Considering both its orbital and spin Zeeman effects we identify three conventionally ordered phases including a ${120}^{\ensuremath{\circ}}$ N\'eel phase, a stripe phase, and an up-up-down phase. For intermediate fields an emergent quantum spin liquid phase is identified with partial spin polarization. We further characterize its topological nature as the $\ensuremath{\nu}=1/2$ Laughlin CSL through the topological entanglement spectrum and quantized spin pumping under spin flux insertion. In addition, we map out the quantum phase diagram for different twisted angles in an experimentally accessible parameter regime.
Twisted transition metal dichalcogenides offer an exceptionally tunable moiré platform for studying correlation physics beyond conventional condensed matter systems. In particular, the intriguing interplay between the displacement field and the twist angle remains to be fully resolved. In this paper, we use large-scale density matrix renormalization group simulations to study the minimal moiré Hubbard model on a triangular lattice at half-filling, where the displacement field effect is captured by a spin-dependent staggered flux. We find that the displacement field significantly enriches the triangular Hubbard phase diagram in several qualitative ways. It rapidly destabilizes the chiral spin liquid phase beyond a narrow weak-field regime, induces pronounced chiral correlations in the strong-coupling 120∘120^\circ120∘-antiferromagnetic phase, and stabilizes incommensurate spin-density wave phases at weaker coupling. We further find signatures of a continuous transition between the chiral spin liquid and chiral antiferromagnetic phases at a finite displacement field, potentially driven by spinon condensation. Our results uncover rich displacement-field-driven many-body physics and provide useful guidance for future experiments in moiré superlattice systems.
At n=3/4n=3/4n=3/4 filling of the moiré flat band, transition metal dichalcogenide moiré bilayers will develop kagome charge order. We derive an effective spin model for the resulting localized spins and find that its further neighbor spin interactions can be much less suppressed than the corresponding electron hopping strength. Using density matrix renormalization group simulations, we study its phase diagram and, for realistic model parameters relevant for WSe2_22​/WS2_22​, we show that this material can realize the exotic chiral spin liquid phase and the highly debated kagome spin liquid. Our work thus demonstrates that the frustration and strong interactions present in TMD heterobilayers provide an exciting platform to study spin liquid physics.
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)\text{SU}(4)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)\text{SO}(6)SO(6) Schwinger boson or Schwinger fermion parton. At the SU(4)\text{SU}(4)SU(4)-symmetric point, we find two symmetric Z2{Z}_{2}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{Z}_{2}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)\text{U}(1)U(1) spinon Fermi surface state or Z4{Z}_{4}Z4​ spin liquid at ${\ensuremath{\nu}}_{T}=1$.
Motivated by experimental studies that have found signatures of a quantum spin liquid phase in organic crystals whose structure is well described by the two-dimensional triangular lattice, we study the Hubbard model on this lattice at half filling using the infinite-system density matrix renormalization group (iDMRG) method. On infinite cylinders with finite circumference, we identify an intermediate phase between observed metallic behavior at low interaction strength and Mott insulating spin-ordered behavior at strong interactions. Chiral ordering from spontaneous breaking of time-reversal symmetry, a fractionally quantized spin Hall response, and characteristic level statistics in the entanglement spectrum in the intermediate phase provide strong evidence for the existence of a chiral spin liquid in the full two-dimensional limit of the model.
A chiral spin liquid (CSL) phase has been recently reported in the Hubbard model on a triangular lattice. It emerges in an intermediate coupling regime at half-filling, which is sandwiched between a ${120}^{\ensuremath{\circ}}$ antiferromagnetic (AFM) phase and a metallic phase as a function of on-site repulsion UUU. In this work, we examine the mechanism of the CSL and complex phase diagram via analytic analysis and numerical density matrix renormalization group (DMRG) method. First, we identify an exact Berry-phase-like sign structure in the partition function of the model at arbitrary UUU, which is originated from the Fermi sign structure at U=0U=0U=0.The spin and charge degrees of freedom are generally entangled via a singular phase string in such a many-body sign structure. In the large-UUU limit, the suppression of the charge fluctuation at half-filling can render the phase string ineffective, resulting in the AFM order. However, if one precisely switches off such a phase string in the DMRG calculation, the ${120}^{\ensuremath{\circ}}$ AFM is shown to survive all the way down to a much weaker UUU without the emergence of the CSL and metallic phases. It indicates that the phase-string sign structure plays the key role to mediate the mutual interaction between the charge and spin fluctuations, which results in the CSL and metallic phases at finite UUU. General implications for the Mott physics in the Hubbard model will also be discussed.