The surfaces of three-dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time-reversal
T or
U(1) charge conservation symmetry. Here, we discuss a possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then, the surface must develop topological order of a kind that can not be realized in a two-dimensional (2D) system with the same symmetries. We discuss candidate surface states, non-Abelian quantum Hall states which, when realized in 2D, have ${\ensuremath{\sigma}}_{xy}=\frac{1}{2}$ and hence break
T symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as
T-symmetric surface states. The corresponding 3D phases are confined, and have $\ensuremath{\theta}=\ensuremath{\pi}$ magnetoelectric response. Two candidate states have the same 12-particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their
T transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24-particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a
T-symmetric surface state.
Exploring 3D topological matter with spin-orbit-coupled fermions in optical lattices1 CHENGDONG HE, BO SONG, Hong Kong University of Science and Technology, SEN NIU, LONG ZHANG, Peking University, ZEJIAN REN, ENTONG ZHAO, Hong Kong University of Science and Technology, XIONG-JUN LIU, Peking University, GYU-BOONG JO, Hong Kong University of Science and Technology — Ultracold atoms offer a promising testbed for the experimental study of topological matter. Various topological models have been recently realized in 1D and 2D atomic systems. While 3D topological matter with rich physics remains unexplored due to experimental complexity. Here we report, for the first time, the realization of a 3D nodal line semimetal in optical lattices with SO coupled ultracold fermions. The 3D topological band structure is achieved by stacking 2D Dirac semimetal in the x-y plane along z direction in Raman-dressed optical lattices. kz resolved spin texture measurement technique based on emergent magnetic group symmetry has been developed to detect 3D topological state. Spin texture in a specific kz plane can be directly imaged with different Zeeman splitting. 3D nodal lines can be reconstructed with this pseudo-tomography method. Band inversion lines, the bulk counterparts of Fermi arc states, can be extracted from quench dynamics, which reconfirmed the realization of topological phase. Our detection technique can be broadly applied to characterizing 3D topological states with similar symmetries, which provides a new possibility to study exotic quantum physics in higher dimensions.
Systems with dipole moment conservation have been of recent interest, as they realize both novel quantum dynamics and exotic ground state phases. In this work, we study some generic properties of 1-D and 2-D dipole-conserving fermionic models at integer fillings. We find that a dipolar symmetry-breaking phase can result in a mean-field band insulator whose topological indices can strongly affect the low-energy physics of the dipolar Goldstone modes. We study the 2-D topological phase transition of the mean-field ground states in the presence of the Goldstone modes. The critical theory resembles the 2+1d quantum electrodynamics coupled to massless Dirac fermions with some crucial differences and shows a novel quantum critical point featuring a nontrivial dynamical exponent. We also discuss the analogous case of 1-D dipole-conserving models and the role of topological invariants.