A Two-dimensional Algebraic Quantum Liquid Produced by an Atomic Simulator of the Quantum Lifshitz Model

Bosons have a natural instinct to condense at zero temperature. It is a long-standing challenge to create a high-dimensional quantum liquid that does not exhibit long-range order at the ground state, as either extreme experimental parameters or sophisticated designs of microscopic Hamiltonian are required for suppressing the condensation. Here, we show that ultra cold atoms with synthetic spin-orbit coupling provide physicists a simple and practical scheme to produce a two-dimensional algebraic quantum liquid at the ground state. This quantum liquid arises at a critical Lifshitz point, where the single-particle ground state shrinks to a point from a circle in the momentum space, and many fundamental properties of two-dimensional bosons are changed in its proximity. Such an ideal simulator of the quantum Lifshitz model allows experimentalists to directly visualize and explore the deconfinement transition of topological excitations, an intriguing phenomenon that is difficult to access in other systems.Comment: 3 figure

Structure and Topology of Band Structures in the 1651 Magnetic Space Groups

The properties of electrons in magnetically ordered crystals are of interest both from the viewpoint of realizing novel topological phases, such as magnetic Weyl semimetals, and from the applications perspective of creating energy-efficient memories. A systematic study of symmetry and topology in magnetic materials has been challenging given that there are 1651 magnetic space groups (MSGs). Here, by using an efficient representation of allowed band structures, we obtain a systematic description of several basic properties of free electrons in all MSGs in three dimensions as well as in the 528 magnetic layer groups relevant to two dimensional magnetic materials. We compute constraints on electron fillings and band connectivity compatible with insulating behavior. Also, by contrasting with atomic insulators, we identify band topology entailed by the symmetry transformation of bands, as determined by the MSG alone. We give an application of our results to identifying topological semimetals arising in periodic arrangements of hedgehog-like magnetic textures.Comment: (9 + 34) pages; 3 figures; (2+19) tables; v2: close to published versio

Symmetry-based Indicators of Band Topology in the 230 Space Groups

The interplay between symmetry and topology leads to a rich variety of electronic topological phases, protecting states such as the topological insulators and Dirac semimetals. Previous results, like the Fu-Kane parity criterion for inversion-symmetric topological insulators, demonstrate that symmetry labels can sometimes unambiguously indicate underlying band topology. Here we develop a systematic approach to expose all such symmetry-based indicators of band topology in all the 230 space groups. This is achieved by first developing an efficient way to represent band structures in terms of elementary basis states, and then isolating the topological ones by removing the subset of atomic insulators, defined by the existence of localized symmetric Wannier functions. Aside from encompassing all earlier results on such indicators, including in particular the notion of filling-enforced quantum band insulators, our theory identifies symmetry settings with previously hidden forms of band topology, and can be applied to the search for topological materials.Comment: 9+21 pages; (2+1) figures, (4+20) tables; v2: references added; title changed; results for quasi-2D and 1D systems adde

Fragile topological phases in interacting systems

Topological phases of matter are defined by their nontrivial patterns of ground-state quantum entanglement, which is irremovable so long as the excitation gap and the protecting symmetries, if any, are maintained. Recent studies on noninteracting electrons in crystals have unveiled a peculiar variety of topological phases, which harbors nontrivial entanglement that can be dissolved simply by the the addition of entanglement-free, but charged, degrees of freedom. Such topological phases have a weaker sense of robustness than their conventional counterparts, and are therefore dubbed "fragile topological phases." In this work, we show that fragile topology is a general concept prevailing beyond systems of noninteracting electrons. Fragile topological phases can generally occur when a system has a $\mathrm{U}(1)$ charge conservation symmetry, such that only particles with one sign of the charge are physically allowed (e.g. electrons but not positrons). We demonstrate that fragile topological phases exist in interacting systems of both fermions and of bosons.Comment: 14 pages. Comments welcome; v2: several discussions are improve

Landau Level Degeneracy in Twisted Bilayer Graphene: Role of Symmetry Breaking

The degeneracy of Landau levels flanking charge neutrality in twisted bilayer graphene is known to change from eight-fold to four-fold when the twist angle is reduced to values near the magic angle of $\approx 1.05^\circ$. This degeneracy lifting has been reproduced in experiments by multiple groups, and is known to occur even in devices which do not harbor the correlated insulators and superconductors. We propose $C_3$ symmetry breaking as an explanation of such robust degeneracy lifting, and support our proposal by numerical results on the Landau level spectrum in near-magic-angle twisted bilayer graphene. Motivated by recent experiments, we further consider the effect of $C_2$ symmetry breaking on the Landau levels.Comment: 12 pages, 10 figure

Higher-dimensional Jordan-Wigner Transformation and Auxiliary Majorana Fermions

We discuss a scheme for performing Jordan-Wigner transformation for various lattice fermion systems in two and three dimensions which keeps internal and spatial symmetries manifest. The correspondence between fermionic and bosonic operators is established with the help of auxiliary Majorana fermions. The current construction is applicable to general lattices with even coordination number and an arbitrary number of fermion flavors. The approach is demonstrated on the single-orbital square, triangular and cubic lattices for spin-1/2 fermions. We also discuss the relation to some quantum spin liquid models.Comment: 16 pages, 9 figures; modify the appendix