A fundamental dichotomous classification for all physical systems is according to whether they are spinless or spinful. This is especially crucial for the study of symmetry-protected topological phases, as the two classes have distinct symmetry algeb
ra. As a prominent example, the spacetime inversion symmetry $PT$ satisfies $(PT)^2=pm 1$ for spinless/spinful systems, and each class features unique topological phases. Here, we reveal a possibility to switch the two fundamental classes via $mathbb{Z}_2$ projective representations. For $PT$ symmetry, this occurs when $P$ inverses the gauge transformation needed to recover the original $mathbb{Z}_2$ gauge connections under $P$. As a result, we can achieve topological phases originally unique for spinful systems in a spinless system, and vice versa. We explicitly demonstrate the claimed mechanism with several concrete models, such as Kramers degenerate bands and Kramers Majorana boundary modes in spinless systems, and real topological phases in spinful systems. Possible experimental realization of these models is discussed. Our work breaks a fundamental limitation on topological phases and opens an unprecedented possibility to realize intriguing topological phases in previously impossible systems.
Symmetry groups are projectively represented in quantum mechanics, and crystalline symmetries are fundamental in condensed matter physics. Here, we systematically present a unified theory of quantum mechanical space groups from two complementary aspe
cts. First, we provide a decomposition form for the space-group factor systems to characterize all quantum space groups. It consists of three factors, the factor system for the translation subgroup $L$, an in-homogeneous factor system for the point group $P$, and a factor connecting $L$ and $P$. The three factors satisfy three consistency equations, which are exactly solvable and can completely exhaust all factor systems for space groups. Second, since factors systems are classified by the second cohomology group, we show the (co)homology groups for space groups can be derived from Borels equivariant (co)homology theory, which leads to an algorithm that can compute all (co)homology groups for space groups. To demonstrate the general theory, we explicitly present quantum wallpaper groups with the $mathbb{Z}_2$ gauge group. Furthermore, as a primitive application, we find the time-reversal invariant quantum space groups with inversion symmetry can lead to a novel clifford band theory, where each band is fourfold degenerate to represent certain real Clifford algebras with topologically nontrivial pinor structures over the Brillouin zone. Our work serves as a foundation for exploring quantum mechanical space groups, and can find applications in spin liquids, unconventional superconductors, and artificial lattice systems, including cold atoms, photonic and phononic crystals, and even LC electric circuit networks.
Symmetry is fundamental to topological phases. In the presence of a gauge field, spatial symmetries will be projectively represented, which may alter their algebraic structure and generate novel topological phases. We show that the $mathbb{Z}_2$ proj
ectively represented translational symmetry operators adopt a distinct commutation relation, and become momentum dependent analogous to twofold nonsymmorphic symmetries. Combined with other internal or external symmetries, they give rise to many exotic band topology, such as the degeneracy over the whole boundary of the Brillouin zone, the single fourfold Dirac point pinned at the Brillouin zone corner, and the Kramers degeneracy at every momentum point. Intriguingly, the Dirac point criticality can be lifted by breaking one primitive translation, resulting in a topological insulator phase, where the edge bands have a M{o}bius twist. Our work opens a new arena of research for exploring topological phases protected by projectively represented space groups.
We present an exactly solvable spin-3/2 model defined on a pentacoordinated three-dimensional graphite lattice, which realizes a novel quantum spin liquid with second-order topology. The exact solutions are described by Majorana fermions coupled to a
background $mathbb{Z}_2$ gauge field, whose ground-state flux configuration gives rise to an emergent off-centered spacetime inversion symmetry. The symmetry protects topologically nontrivial band structures for the Majorana fermions, particularly including nodal-line semimetal phases with twofold topological charges: the second Stiefel-Whitney number and the quantized Berry phase. The former leads to rich topological phenomena on the system boundaries. There are two nodal-line semimetal phases hosting hinge Fermi arcs located on different hinges, and they are separated by a critical Dirac semimetal state with surface helical Fermi arcs. In addition, we show that rich symmetry/topology can be explored in our model by simply varying the lattice or interaction arrangement. As an example, we discuss how to achieve a topological gapped phase with surface Dirac points.
Topological fermions as excitations from multi-degenerate Fermi points have been attracting increasing interests in condensed matter physics. They are characterized by topological charges, and magnetic fields are usually applied in experiments for th
eir detection. Here we present an index theorem that reveals the intrinsic connection between the topological charge of a Fermi point and the in-gap modes in the Landau band structure. The proof is based on mapping fermions under magnetic fields to a topological insulator whose topological number is exactly the topological charge of the Fermi point. Our work lays a solid foundation for the study of intriguing magneto-response effects of topological fermions.
The concept of topological fermions, including Weyl and Dirac fermions, stems from the quantum Hall state induced by a magnetic field, but the definitions and classifications of topological fermions are formulated without using magnetic field. It is
unclear whether and how the topological information of topological fermions can be probed once their eigen spectrum is completely rebuilt by a strong magnetic field. In this work, we provide an answer via mapping Landau levels (bands) of topological fermions in $d$ dimensions to the spectrum of a $(d-1)$-dimensional lattice model. The resultant Landau lattice may correspond to a topological insulator, and its topological property can be determined by real-space topological invariants. Accordingly, each zero-energy Landau level (band) inherits the topological stability from the corresponding topological boundary state of the Landau lattice. The theory is demonstrated in detail by transforming 2D Dirac fermions under magnetic fields to the Su-Schrieffer-Heeger models in class AIII, and 3D Weyl fermions to the Chern insulators in class A.
Recently Weyl fermions have attracted increasing interest in condensed matter physics due to their rich phenomenology originated from their nontrivial monopole charges. Here we present a theory of real Dirac points that can be understood as real mono
poles in momentum space, serving as a real generalization of Weyl fermions with the reality being endowed by the $PT$ symmetry. The real counterparts of topological features of Weyl semimetals, such as Nielsen-Ninomiya no-go theorem, $2$D sub topological insulators and Fermi arcs, are studied in the $PT$ symmetric Dirac semimetals, and the underlying reality-dependent topological structures are discussed. In particular, we construct a minimal model of the real Dirac semimetals based on recently proposed cold atom experiments and quantum materials about $PT$ symmetric Dirac nodal line semimetals.
We show that for two-band systems nonsymmorphic symmetries may enforce the existence of band crossings in the bulk, which realize Fermi surfaces of reduced dimensionality. We find that these unavoidable crossings originate from the momentum dependenc
e of the nonsymmorphic symmetry, which puts strong restrictions on the global structure of the band configurations. Three different types of nonsymmorphic symmetries are considered: (i) a unitary nonsymmorphic symmetry, (ii) a nonsymmorphic magnetic symmetry, and (iii) a nonsymmorphic symmetry combined with inversion. For nonsymmorphic symmetries of the latter two types, the band crossings are located at high-symmetry points of the Brillouin zone, with their exact positions being determined by the algebra of the symmetry operators. To characterize these band degeneracies we introduce a emph{global} topological charge and show that it is of $mathbb{Z}_2$ type, which is in contrast to the emph{local} topological charge of Fermi points in, say, Weyl semimetals. To illustrate these concepts, we discuss the $pi$-flux state as well as the SSH model at its critical point and show that these two models fit nicely into our general framework of nonsymmorphic two-band systems.
As PT and CP symmetries are fundamental in physics, we establish a unified topological theory of PT and CP invariant metals and nodal superconductors, based on the mathematically rigorous $KO$ theory. Representative models are constructed for all non
trivial topological cases in dimensions $d=1,2$, and $3$, with their exotic physical meanings being elucidated in detail. Intriguingly, it is found that the topological charges of Fermi surfaces in the bulk determine an exotic direction-dependent distribution of topological subgap modes on the boundaries. Furthermore, by constructing an exact bulk-boundary correspondence, we show that the topological Fermi points of the PT and CP invariant classes can appear as gapless modes on the boundary of topological insulators with a certain type of anisotropic crystalline symmetry.
We report two theoretical discoveries for $mathbb{Z}_2$-topological metals and semimetals. It is shown first that any dimensional $mathbb{Z}_2$ Fermi surface is topologically equivalent to a Fermi point. Then the famous conventional no-go theorem, wh
ich was merely proven before for $mathbb{Z}$ Fermi points in a periodic system without any discrete symmetry, is generalized to that the total topological charge is zero for all cases. Most remarkably, we find and prove an unconventional strong no-go theorem: all $mathbb{Z}_2$ Fermi points have the same topological charge $ u_{mathbb{Z}_2} =1$ or $0$ for periodic systems. Moreover, we also establish all six topological types of $mathbb{Z}_2$ models for realistic physical dimensions.
