In the tensor network representation, a deformed Z2 topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller (AT) model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the AT model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate of pairs of logarithmically confined electric charges and magnetic fluxes, and the scaling behavior of various anyon correlations can be exactly derived, revealing the effective interaction between anyons and their condensation. Deformations away from the self-duality drive the Coulomb state into either the gapped Higgs phase or confining phase.Introduction.-The toric code model proposed by Kitaev[1] is a prototypical model realizing the Z 2 intrinsic topological phase of matter with anyonic excitations. It is interesting and fundamentally important to consider the possible topological phase transitions out of the toric code phase, because such phase transitions are beyond the conventional Ginzburg-Landau paradigm for the symmetry breaking phases. From the perspective of lattice gauge theory, it has been known that there exists the Higgs/confinement transition, where the electric charge is condensed/confined accompanied by the confinement/condensation of magnetic flux due to electricmagnetic duality [2][3][4][5][6][7]. However, there is a long-standing puzzle: what is the nature of the phase transition along the self-dual line and how the Higgs and confinement transition lines merge into the self-dual phase transition point [6,8]. Should there be a tricritical point, it would go beyond the anyon condensation scenario[9], because the electric charge and magnetic flux are not allowed to simultaneously condense.In this Letter, we shall resolve this puzzle and provide new insight into the nature of this topological phase transition. Instead of solving a Hamiltonian with tuning parameters, we propose a deformed topological wave-function interpolating from the nontrivial to trivial phases in the tensor network representation [10][11][12], which provides a clearer scope into the essential physics of abelian anyonic excitations [13][14][15][16][17]. In this scheme, the usual pure Higgs/confinement transition of the toric code [14,18,19] has a special path, where the deformed wave-function can be exactly mapped to a twodimensional (2D) classical Ising model. The topological phase transition is associated with the 2D Ising phase transition, drawing the striking topological-symmetrybreaking correspondence [20].Further deformation of the toric code wave functions can span a generalized phase diagram [21][22][23], where the perturbed Higgs and confinement transitions were gener-ically obtained by the symmetry breaking pattern and long-range-...
Recent experiment has shown that the ABC-stacked trilayer graphene-boron nitride Moire super-lattice at half-filling is a Mott insulator. Based on symmetry analysis and effective band structure calculation, we propose a valley-contrasting chiral tight-binding model with local Coulomb interaction to describe this Moire super-lattice system. By matching the positions of van Hove points in the low-energy effective bands, the valley-contrasting staggered flux per triangle is determined around π/2. When the valence band is half-filled, the Fermi surfaces are found to be perfectly nested between the two valleys. Such an effect can induce an inter-valley spiral order with a gap in the charge excitations, indicating that the Mott insulating behavior observed in the trilayer graphene-boron nitride Moire super-lattice results predominantly from the inter-valley scattering.
Different from the spin-1 Haldane gapped phase, we propose a novel SO(3) spin-1 matrix product state (MPS), whose parent Hamiltonian includes three-site spin interactions. From the entanglement spectrum of a single block with l sites, an enlarged SU(3) symmetry is identified in the edge states, which are conjugate to each other for the l = even block but identical for the l = odd block. By blocking this novel state, the blocked MPS explicitly displays the SU(3) symmetry with two distinct structures. Under a symmetric bulk bipartition with a sufficient large block length l = even, the entanglement Hamiltonian (EH) of the reduced system characterizes a spontaneous dimerized phase with two-fold degeneracy. However, for the block length l = odd, the corresponding EH represents an SU(3) quantum critical point with delocalized edge quasiparticles, and the critical field theory is described by the SU(3) level-1 Wess-Zumino-Witten conformal field theory. Topological phases of matter have become one of the most important subjects in physics, because their lowenergy excitations have potential use for fault-tolerant quantum computation. Symmetry protected topological (SPT) phases belong to a new type of topological phases with robust gapless edge excitations [1][2][3]. Without breaking the protecting symmetry or closing the energy gap, these SPT phases can not be continuously connected to the trivial phase. A topological quantum critical point (TQCP) exists to separate an SPT phase from its adjacent trivial phase, and the corresponding critical theory is beyond the Landau-Ginzburg-Wilson paradigm [4,5]. The simplest example of SPT phases is the Haldane gapped phase of the antiferromagnetic Heisenberg spin-1 chain [6]. Recently it has been shown that the symmetric bulk bipartition of the SO(3) symmetric Affleck-Kennedy-Lieb-Tasaki (AKLT) wave function for the Haldane phase [7] is an effective way to create an array of fractionalized spin-1/2 edge spins in the bulk subsystem, and the corresponding bulk entanglement spectrum (ES) represents a TQCP described by the SU(2) level-1 Wess-Zumino-Witten (WZW) theory with spinon excitations [8,10]. The TQCP is argued to characterize the quantum critical state between the Haldane gapped phase and its adjacent trivial phase [8][9][10].It is well-known that the quantum spin-1 chain may exhibit an SU(3) symmetry. The SU(3) symmetric matrix product state (MPS) has been constructed [11][12][13][14][15][16]. The on-site physical space is spanned by the SU(3) adjoint representation 8 consisting of a fundamental representation 3 (quarks) and a conjugate representation 3 (antiquarks), while the adjacent lattice sites are connected by the SU(3) quark-antiquark singlet bonds. Different from the Haldane gapped phase, the zero-energy states localized on the edges are conjugate to each other. The natural question arises whether a distinct spin-1 MPS can be found so that the symmetric bulk bipartition leads to the SU(3) quantum criticality for the reduced bulk system. According to the complete class...
Motivated by the recent observations of nodeless superconductivity in the monolayer CuO2 grown on the Bi2Sr2CaCu2O 8+δ substrates, we study the two-dimensional superconducting (SC) phases described by the two-dimensional t-J model in proximity to an antiferromagnetic (AF) insulator. We found that (i) the nodal d-wave SC state can be driven via a continuous transition into a nodeless d-wave pairing state by the proximity induced AF field. (ii) The energetically favorable pairing states in the strong field regime have extended s-wave symmetry and can be nodal or nodeless. (iii) Between the pure d-wave and s-wave paired phases, there emerge two topologically distinct SC phases with (s+id) symmetry, i.e., the weak and strong pairing phases, and the weak pairing phase is found to be a Z2 topological superconductor protected by valley symmetry, exhibiting robust gapless non-chiral edge modes. These findings strongly suggest that the high-Tc superconductors in proximity to antiferromagnets can realize fully gapped symmetry protected topological SC.
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