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arXiv:2204.09691 · Physical review. B./Physical review. B

Twisted bilayer U(1) Dirac spin liquids

Zhu-Xi Luo, Urban F. P. Seifert, Leon Balents

Published Oct 28, 2022• 10 citations
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Abstract

When two layers of two-dimensional materials are assembled with a relative twist, moir\'e patterns arise, inducing a tremendous wealth of exotic phenomena. In this work, we consider twisting two triangular lattices hosting Dirac quantum spin liquids. A single decoupled layer is described by compact quantum electrodynamics in $2+1$ dimensions (${\mathrm{QED}}_{3}$) with an emergent $\mathrm{U}(1)$ gauge field, which is assumed to flow to a strongly interacting fixed point in the IR with conformal symmetry. We use recent results for the quantum numbers of monopole operators, which tunnel $2\ensuremath{\pi}$ fluxes of the compact gauge field. It is found that, in the bilayer system, interlayer monopole tunneling is a symmetry-allowed relevant perturbation which induces an (ordering) instability. We show using perturbation theory that upon twisting the two layers the system remains unstable under the interlayer interaction, but any finite twist angle softens this instability compared to the untwisted case. To analyze the resulting phase induced by the (twisted) interlayer tunneling, we use ``conformal mean-field theory'', which reduces the interacting bilayer system to two copies of ${\mathrm{QED}}_{3}$ coupled to background fields which are to be determined self-consistently. In the weak-coupling regime, where the interlayer coupling is weak compared to the energy scale set by the moir\'e lattice constant, we solve the self-consistency equations perturbatively. In the limit of strongly coupled layers, a local scaling approximation is used, and we find that the magnetically ordered state exhibits a lattice of magnetic vortices, with the lattice constant tunable through the twisting angle.

DOI
10.1103/physrevb.106.144437

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•Aug 27, 2020•1 citation
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A complicating factor in the realization and observation of quantum spin liquids in materials is the ubiquitous presence of other degrees of freedom, in particular lattice distortion modes (phonons). These provide additional routes for relieving magnetic frustration, thereby possibly destabilizing spin-liquid ground states. In this work, we focus on triangular-lattice Heisenberg antiferromagnets, where recent numerical evidence suggests the presence of an extended U(1) Dirac spin liquid phase which is described by compact quantum electrodynamics in 2+1 dimensions (QED3_33​), featuring gapless spinons and monopoles as gauge excitations, and believed to flow to a strongly-coupled fixed point with conformal symmetry. Using complementary perturbation theory and scaling arguments, we show that a symmetry-allowed coupling between (classical) finite-wavevector lattice distortions and monopole operators of the U(1) Dirac spin liquid generally induces a spin-Peierls instability towards a (confining) 12-site valence-bond solid state. We support our theoretical analysis with state-of-the-art density matrix renormalization group simulations. Away from the limit of static distortions, we demonstrate that the phonon energy gap establishes a parameter regime where the spin liquid is expected to be stable, and show that the monopole-lattice coupling leads to softening of the phonon in analogy to the Kohn anomaly. We discuss the applicability of our results to similar systems, in particular the Dirac spin liquid on the Kagome lattice.
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In recent years, there have been numerous examples of twisted bilayer systems that host remarkable physical properties that are not found in their untwisted counterparts. Motivated by this, we study the properties of twisted bilayers of the Kitaev honeycomb model in the Abelian spin-liquid phase. We show that for strong, short-ranged, interlayer interactions, a superlattice of non-Abelian defects forms in the twisted bilayer system. These non-Abelian defects are wormhole-like genons that allow anyons from one layer to tunnel to the other layer. We find that when a magnetic field is applied to the system, the low-energy dynamics of the twisted bilayer system can be mapped onto four quantum Ising models arising from the degrees of freedom localized on the genon defects. At small twist angles the Ising models are in a trivial paramagnetic phase, and at large twist angles they are in a ferromagnetic phase.
Moiré related physics in twisted bilayer two-dimensional (2D) materials has attracted widespread interest in condensed matter physics. Simulation of moiré related physics in cold atom platform is expected to outperform the 2D materials thanks to its advantage of higher tunablility. Here, we demonstrate that, the cold atom platform enables a new mechanism of moiré lattice formation, induced by interlayer interaction with intrinsic "dynamical" character, in contrast to conventional moiré lattice induced by "static" ways such as single-particle interlayer tunneling. Specifically, we consider a twisted bilayer Bose-Hubbard model with vanishing interlayer tunneling, and the bilayer is solely coupled through interlayer interaction that originates from contact interaction of atoms. We find that this system hosts a plethora of novel phases unique to this dynamical lattice, including a variety of Mott insulator (MI) and superfluid (SF) phases either preserving or breaking moiré lattice symmetry, phases with one layer in SF and the other in MI, "interlocked" MI, and self-localized phases at commensurate twist angles, which exhibits the characteristics of Bose glass and quasi-many-body localization in the absence of (quasi)disorder or quasicrystalline lattices. Our prediction can be readily observed in current experimental setup of twisted bilayer optical lattices, opening up new avenues for exploring the rich physics of interaction induced moiré systems in cold atoms.
The U(1)U(1)U(1) Dirac spin liquid might realize an exotic phase of matter whose low-energy properties are described by quantum electrodynamics in 2+12+12+1 dimensions, where gapless modes exist but spinons and gauge fields are strongly coupled. Its existence was proposed in frustrated Heisenberg models in the presence of frustrating superexchange interactions by the (Abrikosov) fermionic representation of the spin operators [Wen, Phys. Rev. B 65, 165113 (2002)], supplemented by the Gutzwiller projection. Here, we construct charge-QQQ monopole excitations in the Heisenberg model on the triangular lattice with nearest-neighbor (J1{J}_{1}J1​) and next-nearest-neighbor (J2{J}_{2}J2​) couplings. In the highly frustrated regime, singlet and triplet monopoles with Q=1Q=1Q=1 become gapless in the thermodynamic limit; in addition, the energies for generic QQQ agree with field-theoretical predictions, obtained for a large number of gapless fermion modes. Finally, we consider localized gauge excitations, in which magnetic $\ensuremath{\pi}$ fluxes are concentrated in the triangular plaquettes (in analogy with Z2{\mathbb{Z}}_{2}Z2​ visons), showing that these kinds of states do not play a relevant role at low energies. All our findings lend support to a stable U(1)U(1)U(1) Dirac spin liquid in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice.
It has been recently demonstrated [Song et al., Nature Comm. 10, 4254 (2019), Iqbal et al., Phys. Rev. B 93, 144411 (2016)] that the staggered π\piπ-flux Dirac spin liquid phase on the non-bipartite triangular lattice may be stable in the clean limit. However, quenched disorder plays a crucial role in determining whether such a phase can be experimentally viable. The effective low-energy description of Dirac spin liquids in (2+1)(2+1)(2+1) dimensions is given by the compact quantum electrodynamics (cQED2+1\rm cQED_{2+1}cQED2+1​) which admits monopoles. It is already known that generic quenched random perturbations to the non-compact version of QED2+1\rm QED_{2+1}QED2+1​ (where monopoles are absent) lead to strong-coupling instabilities. In this work we study cQED2+1\rm cQED_{2+1}cQED2+1​ in the presence of a class of time-reversal invariant quenched disorder perturbations. We show that in this model, random non-abelian vector potentials make the symmetry-allowed monopole operators more relevant. The disorder induced under-screening of monopoles thus generically makes the gapless spin liquid phase fragile.