Expressive Power in First Order Topology

A first order representation (fo.r.) in topology is an assignment of finitary relational structures of the same type to topological spaces in such a way that homeomorphic spaces get sent to isomorphic structures. We first define the notions one f.o.r. is at least as expressive as another relative to a class of spaces and one class of spaces is definable in another relative to an f.o.r. , and prove some general statements. Following this we compare some well-known classes of spaces and first order representations. A principal result is that if X and Y are two Tichonov spaces whose posets of zero-sets are elementarily equivalent then their respective rings of bounded continuous real-valued functions satisfy the same positiveuniversal sentences. The proof of this uses the technique of constructing ultraproducts as direct limits of products in a category theoretic setting

The order topology for a von Neumann algebra

The order topology $\tau_o(P)$ (resp. the sequential order topology $\tau_{os}(P)$) on a poset $P$ is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra $M$ we consider the following three posets: the self-adjoint part $M_{sa}$, the self-adjoint part of the unit ball $M_{sa}^1$, and the projection lattice $P(M)$. We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology to the other standard locally convex topologies on $M$, and relate the properties of the order topology to the underlying operator-algebraic structure of $M$

Two-dimensional higher-order topology in monolayer graphdiyne

Based on first-principles calculations and tight-binding model analysis, we propose monolayer graphdiyne as a candidate material for a two-dimensional higher-order topological insulator protected by inversion symmetry. Despite the absence of chiral symmetry, the higher-order topology of monolayer graphdiyne is manifested in the filling anomaly and charge accumulation at two corners. Although its low energy band structure can be properly described by the tight-binding Hamiltonian constructed by using only the $p_z$ orbital of each atom, the corresponding bulk band topology is trivial. The nontrivial bulk topology can be correctly captured only when the contribution from the core levels derived from $p_{x,y}$ and $s$ orbitals are included, which is further confirmed by the Wilson loop calculations. We also show that the higher-order band topology of a monolayer graphdyine gives rise to the nontrivial band topology of the corresponding three-dimensional material, ABC-stacked graphdiyne, which hosts monopole nodal lines and hinge states.Comment: 19 pages, 4 figures, new titl

Three-dimensional superconductors with hybrid higher order topology

We consider three dimensional superconductors in class DIII with a four-fold rotation axis and inversion symmetry. It is shown that such systems can exhibit higher order topology with helical Majorana hinge modes. In the case of even-parity superconductors we show that higher order topological superconductors can be obtained by adding a small pairing with the appropriate $C_4$ symmetry implementation to a topological insulator. We also show that a hybrid case is possible, where Majorana surface cones resulting from non-trivial strong topology coexist with helical hinge modes. We propose a bulk invariant detecting this hybrid scenario, and numerically analyse a tight binding model exhibiting both Majorana cones and hinge modes.Comment: Published versio

Higher-order Topology of Axion Insulator EuIn2_2As2_2

Based on first-principles calculations and symmetry analysis, we propose that EuIn$_2$As$_2$ is a long awaited axion insulator with antiferromagnetic (AFM) long range order. Characterized by the parity-based invariant $\mathbb Z_4=2$, the topological magneto-electric effect is quantized with $\theta=\pi$ in the bulk, with a band gap as large as 0.1 eV. When the staggered magnetic moment of the AFM phase is along $a/b$ axis, it's also a TCI phase. Gapless surface states emerge on (100), (010) and (001) surfaces, protected by mirror symmetries (nonzero mirror Chern numbers). When the magnetic moment is along $c$ axis, the (100) and (001) surfaces are gapped. As a consequence of a high-order topological insulator with $\mathbb Z_4=2$, the one-dimensional (1D) chiral state can exist on the hinge between those gapped surfaces. We have calculated both the topological surface states and hinge state in different phases of the system, respectively, which can be detected by ARPES or STM experiments

Solve[order/topology == quasi-metric/x, x]

AbstractIn the study of the semantics of programming languages, the qualitative framework using partially ordered sets and the quantitative framework using pseudo-metric spaces have existed separately for years. Smyth however noticed that both concepts can be unified by means of quasi-metric spaces.Recent literature concerning these “quantitative domains”, lacks the canonicity which is so typical for the relationship between topological techniques and theoretical computer science in the classical settings mentioned above. On the one hand, this yields the use of structures which could be considered “ad hoc” from a categorical point of view, such as continuity spaces by Flagg and Kopperman. On the other hand, this yields “incomplete structures”, which essentially belong to one of both classical settings, such as the generalized Scott topology by Bonsangue e.a.We shall discuss a natural generalization of the symbiosis between ordered sets and topology to an analogous relationship between quasi-metric spaces and approach spaces. Approach spaces seem to be an important tool in the study of certain aspects concerning quantitative domains