Since the discovery of superconductivity and correlated insulator at fractional electron fillings in the twisted bilayer graphene, most theoretical efforts have been focused on describing this system in terms of an effective extended Hubbard model. However, it was recognized that an exact tight binding model on the Moiré superlattice which captures all the subtleties of the bands can be exceedingly complicated. Here we pursue an alternative framework of coupled wires to describe the system based on the observation that the lattice relaxation effect is strong at small twist angle, which substantially enlarges the AB and BA stacking domains. Under an out-of-plane electric field which can have multiple origins, the low energy physics of the system is dominated by interconnected wires with (approximately) gapless 1d conducting quantum valley hall domain wall states. We demonstrate that the Coulomb interaction likely renders the wires a U (2)2 (1 + 1)d conformal field theory with a tunable Luttinger parameter for the charge U (1) sector. Spin triplet and singlet Cooper pair operator both have quasi-long range order in this CFT. The junction between the wires at the AA stacking islands can lead to either a two dimensional superconductor, or an insulator. PACS numbers:Surprising correlated physics such as superconductivity and correlated insulator at fractional electron fillings away from charge neutrality has been discovered in different systems with Moiré superlattices [1-4], which motivated a series of active theoretical studies . These systems have narrow electron bandwidth near charge neutrality [28][29][30][31], hence interaction effects are significantly enhanced. In several systems that are microscopically different, for example, (1) the heterostructure of trilayer graphene (TLG) and hexagonal boron nitride (hBN), and (2) twisted bilayer graphene (TBG), (3) twisted double bilayer graphene (TDBG) [32], insulating behavior was observed at commensurate fractional fillings away from the charge neutral point [1,2,4]; superconductivity has been observed in all these systems near the insulator phases [3,4,32,33].A consensus of the mechanism for the observed insulator and superconductor has not yet been reached. A minimal triangular lattice extended Hubbard model [5] at least describes the TLG/hBN heterostructure and twisted double bilayer graphene with certain out-of-plane electric field (displacement field) [6,[34][35][36], since in these cases there is no symmetry protected band touching below the fermi energy, and the isolated narrow band has trivial topology. This minimal model would then naturally predict either a spin-triplet [5] or spin-singlet d + id topological superconductor [7], depending on the sign of the Hund's coupling. Signatures of spin triplet pairing predicted in Ref. [5] was recently found in TDBG [32], though further experiments are demanded to determine the exact pairing symmetry.On the contrary, for one of the systems, i.t. the TBG, it was recognized that a standard tight binding model on the sup...

We propose a lattice model for strongly interacting electrons with the potential to explain the main phenomenology of the strange metal phase in the cuprate high temperature superconductors. Our model is motivated by the recently developed "tetrahedron" rank-3 tensor model that mimics much of the physics of the better-known Sachdev-Ye-Kitaev (SYK) model. Our electron model has the following advantageous properties: 1. it only needs one orbital per site on the square lattice; 2. it does not require any quenched random interaction; 3. it has local interactions and respects all the symmetries of the system; 4. the soluble limit of this model has a longitudinal DC resistivity that scales linearly with temperature within a finite temperature window; 5. again the soluble limit of this model has a fermion pairing instability in the infrared, which can lead to either superconductivity or a "pseudogap" phase. The linear−T longitudinal resistivity and the pairing instability originate from the generic scaling feature of the SYK model and the tetrahedron tensor model. PACS numbers:arXiv:1802.04293v2 [cond-mat.str-el]

One dimensional (1d) interacting systems with local Hamiltonians can be studied with various well-developed analytical methods. Recently novel 1d physics was found numerically in systems with either spatially nonlocal interactions, or at the 1d boundary of 2d quantum critical points, and the critical fluctuation in the bulk also yields effective nonlocal interactions at the boundary. This work studies the edge states at the 1d boundary of 2d strongly interacting symmetry protected topological (SPT) states, when the bulk is driven to a disorder-order phase transition. We will take the 2d Affleck-Kennedy-Lieb-Tasaki (AKLT) state as an example, which is a SPT state protected by the SO(3) spin symmetry and spatial translation. We found that the original (1+1)d boundary conformal field theory of the AKLT state is unstable due to coupling to the boundary avatar of the bulk quantum critical fluctuations. When the bulk is fixed at the quantum critical point, within the accuracy of our expansion method, we find that by tuning one parameter at the boundary, there is a generic direct transition between the long range antiferromagnetic Néel order and the valence bond solid (VBS) order. This transition is very similar to the Néel-VBS transition recently found in numerical simulation of a spin-1/2 chain with nonlocal spatial interactions. Connections between our analytical studies and recent numerical results concerning the edge states of the 2d AKLT-like state at a bulk quantum phase transition will also be discussed.

We study the concept of 'categorical symmetry' introduced recently, which in the most basic sense refers to a pair of dual symmetries, such as the Ising symmetries of the 1d quantum Ising model and its self-dual counterpart. In this manuscript we study discrete categorical symmetry at higher-dimensional critical points and gapless phases. At these selected gapless states of matter, we can evaluate the behavior of categorical symmetries analytically. We analyze the categorical symmetry at the following examples of criticality: (i) (2 + 1)d Lifshitz critical point of a quantum Ising system; (ii) (3 + 1)d photon phase as an intermediate gapless phase between the topological order and the confined phase of 3d Z 2 quantum gauge theory; (iii) 2d and 3d examples of systems with both categorical symmetries (either zero-form or one-form categorical symmetries) and subsystem symmetries. We demonstrate that at some of these gapless states of matter the categorical symmetries have very different behavior from the nearby gapped phases.

Motivated by the recent observation of evidences of ferromagnetism in correlated insulating states in systems with Moiré 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 SU(2) s ⊗ SU(2) v symmetry, where the two SU(2) symmetries correspond to the rotation within the spin and valley space respectively. Through analytical argument, Schwinger boson analysis and also DMRG 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 metalinsulator 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 spin-triplet/valley-singlet d + id topological superconductor as was predicted previously.PACS numbers: tration. This state Eq. 11 is always the eigenstate of Eq. 1 because a fully polarized ferromagnetic spin state is the eigenstate of operator σ i · σ j on every link < i, j >. Then in the limit of J v J J s > 0, this eigenstate |Ψ FM is also the ground state, because intuitively on every link the spin σ i will see a background "effective" ferromagnetic coupling