In magic angle twisted bilayer graphene (TBG), electron-electron interactions play a central role, resulting in correlated insulating states at certain integer fillings. Identifying the nature of these insulators is a central question, and it is potentially linked to the relatively high-temperature superconductivity observed in the same devices. Here, we address this question using a combination of analytical strong-coupling arguments and a comprehensive Hartree-Fock numerical calculation, which includes the effect of remote bands. The ground state we obtain at charge neutrality is an unusual ordered state, which we call the Kramers intervalley-coherent (K-IVC) insulator. In its simplest form, the K-IVC order exhibits a pattern of alternating circulating currents that triples the graphene unit cell, leading to an "orbital magnetization density wave." Although translation and time-reversal symmetry are broken, a combined "Kramers" timereversal symmetry is preserved. Our analytic arguments are built on first identifying an approximate Uð4Þ × Uð4Þ symmetry, resulting from the remarkable properties of the TBG band structure, which helps select a low-energy manifold of states that are further split to favor the K-IVC state. This low-energy manifold is also found in the Hartree-Fock numerical calculation. We show that symmetry-lowering perturbations can stabilize other insulators and the semimetallic state, and we discuss the ground state at half-filling and give a comparison with experiments.

We use a lowest Landau level model to study the recent observation of an anomalous Hall effect in twisted bilayer graphene. This effective model is rooted in the occurrence of Chern bands which arise due to the coupling between the graphene device and its encapsulating substrate. Our model exhibits a phase transition from a spin-valley polarized insulator to a partial or fully valley unpolarized metal as the bandwidth is increased relative to the interaction strength, consistent with experimental observations. In sharp contrast to standard quantum Hall ferromagnetism, the Chern number structure of the flat bands precludes an instability to an inter-valley coherent phase, but allows for an excitonic vortex lattice at large interaction anisotropy.Moiré graphene systems are a class of simple van der Waals heterostructures [1] hosting interaction driven lowenergy physics, making them an exciting platform to advance our understanding of strongly correlated quantum matter. In twisted bilayer graphene (TBG) with a small twist angle between adjacent layers, interaction effects are enhanced by van Hove singularities coming from 8 bands around charge neutrality in the Moiré-or mini-Brillouin zone (mBZ) with a very small bandwidth [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. A confirmation of the important role played by interactions in these mBZ flat bands was provided in Ref. [22] and Refs. [23][24][25][26][27], where interaction-dominated gaps were observed when 2 or 6 (filling ν = −2, 2) of the 8 flat bands in TBG are filled. Also in ABC stacked trilayer graphene Moiré systems Mott insulating behavior has been reported [28]. Interestingly, at densities near some of these Mott insulators the system becomes superconducting [25,29].Recent experiments indicate that certain magic angle graphene devices have large resistance peaks at ν = 0, 3, with the latter featuring an anomalous Hall (AH) effect detected via hysteresis in the Hall conductance as a function of the out-of-plane magnetic field [30]. The Hall conductance is of order e 2 /h but not yet quantized. Some have detected an meV-scale gap at charge neutrality, and a hysteretic behaviour of the Hall conductance with applied field at ν = −1 [31]. In this work we discuss how the breaking of the 180-degree rotational symmetry (C 2z ) by a partially aligned hexagonal boron-nitride (h-BN) substrate could explain these observations. A variety of works [32][33][34][35][36][37][38] have found that h-BN opens up a band gap at the Dirac points of monolayer graphene whose magnitude depends on the graphene / h-BN alignment angle, reaching ∆ AB ∼ 17meV [38] to ∼ 30meV [36, 37] at perfect alignment. Notably, even in seemingly unaligned devices with little or no observable h-BN induced Moiré potential, band gaps of several meV are still observed [37,38]. In TBG, the substrate can likewise gap-out the flat band Dirac points at the K ± points of the mBZ, splitting the bands as 8 = 4 + 4 to create a gap at charge * N.B. and S.C. contributed equally to this w...

Quantum tensor network states and more particularly projected entangledpair states provide a natural framework for representing ground states of gapped, topologically ordered systems. The defining feature of these representations is that topological order is a consequence of the symmetry of the underlying tensors in terms of matrix product operators. In this paper, we present a systematic study of those matrix product operators, and show how this relates entanglement properties of projected entangled-pair states to the formalism of fusion tensor categories. From the matrix product operators we construct a C * -algebra and find that topological sectors can be identified with the central idempotents of this algebra. This allows us to construct projected entangled-pair states containing an arbitrary number of anyons. Properties such as topological spin, the S matrix, fusion and braiding relations can readily be extracted from the idempotents. As the matrix product operator symmetries are acting purely on the virtual level of the tensor network, the ensuing Wilson loops are not fattened when perturbing the system, and this opens up the possibility of simulating topological theories away from renormalization group fixed points. We illustrate the general formalism for the special cases of discrete gauge theories and string-net models.1 arXiv:1511.08090v2 [cond-mat.str-el]

We develop the formalism of fermionic matrix product states (fMPS) and show how irreducible fMPS fall in two different classes, related to the different types of simple Z2 graded algebras, which are physically distinguished by the absence or presence of Majorana edge modes. The local structure of fMPS with Majorana edge modes also implies that there is always a two-fold degeneracy in the entanglement spectrum. Using the fMPS formalism we make explicit the correspondence between the Z8 classification of time-reversal invariant spinless superconductors and the modulo 8 periodicity in the representation theory of real Clifford algebras. Studying fMPS with general on-site unitary and anti-unitary symmetries allows us to define invariants that label symmetry-protected phases of interacting fermions. The behavior of these invariants under stacking of fMPS is derived, which reveals the group structure of such interacting phases. We also consider spatial symmetries and show how the invariant phase factor in the partition function of reflection symmetric phases on an unorientable manifold appears in the fMPS framework. CONTENTS

Projected entangled pair states (PEPS) provide a natural ansatz for the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We develop a framework to describe on-site symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, in terms of virtual symmetries of the local tensors expressed as a set of matrix product operators (MPOs) labeled by distinct group elements. These MPOs describe the possibly anomalous symmetry of the edge theory, whose local degrees of freedom are concretely identified in a PEPS. A classification of SPT phases is obtained by studying the obstructions to continuously deforming one set of MPOs into another, recovering the results derived for fixed-point models [X. Chen et al., Phys. Rev. B 87, 155114 (2013)] [1]. Our formalism accommodates perturbations away from fixed point models, opening the possibility of studying phase transitions between different SPT phases. We also demonstrate that applying the recently developed quantum state gauging procedure to a SPT PEPS yields a PEPS with topological order determined by the initial symmetry MPOs. The MPO framework thus unifies the different approaches to classifying SPT phases, via fixed-points models, boundary anomalies, or gauging the symmetry, into the single problem of classifying inequivalent sets of matrix product operator symmetries that are defined purely in terms of a PEPS.

Topological solitons, a class of stable nonlinear excitations, appear in diverse domains as in the Skyrme model of nuclear forces. Here, we argue that similar excitations play an important role in a remarkable material obtained on stacking and twisting two sheets of graphene. Close to a magic twist angle, insulating behavior is observed, which gives way to superconductivity on doping. Here, we propose a unifying description of both observations. A symmetry breaking condensate leads to the ordered insulator, while topological solitons in the condensate—skyrmions—are shown to be charge 2e bosons. Condensation of skyrmions leads to a superconductor, whose physical properties we calculate. More generally, we show how topological textures can mitigate Coulomb repulsion and provide a previously unexplored route to superconductivity. Our mechanism not only clarifies why several other moiré materials do not show superconductivity but also points to unexplored platforms where robust superconductivity is anticipated.

We study the entanglement structure of lattice gauge theories from the local operational point of view, and, similar to Soni and Trivedi (arXiv:1510.07455), we show that the usual entanglement entropy for a spatial bipartition can be written as the sum of an undistillable gauge part and of another part corresponding to the LOCC distillable entanglement, which is obtained by depolarizing the local superselection sectors. We demonstrate that the distillable entanglement is zero for pure abelian gauge theories in the weak coupling limit, while it is in general nonzero for the nonabelian case. We also consider gauge theories with matter, and show in a perturbative approach how area laws -including a topological correction -emerge for the distillable entanglement.Introduction-The concept of quantum entanglement is at the center of current day physics. While it used to be confined to the realms of quantum information theory, it is now appearing in many other fields. It is used for instance to characterize new phases of matter with topological order [1,2], and it also plays a crucial role as guiding principle in the classical simulation of general quantum many body systems [3][4][5]. Furthermore in the context of the holographic principle and quantum gravity there appears to be an intricate connection between entanglement and geometry [6,7]. At the same time entanglement lies at the root of the firewall paradox in the quantum behavior of black holes [8,9].It is hence highly desirable to gain a better understanding of the entanglement structure of quantum field theories. However, as first pointed out in [10], for gauge theories the concept of entanglement is obscured by their intrinsic non-locality. More formally, the Hilbert space for locally distinct regions is not of tensor product form, which renders the usual rules for computing the entanglement not applicable. While a number of tools have been developed to cope with this computational problem [10][11][12][13][14][15][16][17][18], the physical meaning of entanglement in gauge theories was still unclear.In this letter, we put forward the operational view on entanglement: for us, the physical entanglement is identified as the accessible one. That is, the asymptotic number of EPR-pairs E

We study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry induced domain walls. A close connection to the theory of graded unitary fusion categories is established. Tensor network representations of the topological defect superselection sectors are constructed for all domain walls. The emergent symmetry-enriched topological order is extracted from these representations, including the symmetry action on the underlying anyons. Dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory, are analyzed and the relationship between topological orders on either side of the transition is derived. Several examples are worked through explicitly.