We show that anisotropic spin interactions do not merely break spin-space group (SSG) symmetries, but instead them through cohomology invariants, yielding symmetry classes beyond subgroups of O ( 3 ) × Isom ( R 3 ) . This requires redefining the spin-only group S 0 in terms of proper spin rotations. Based on this unitary S 0 , we formulate a twisted SSG (tSSG) theory that captures the complete set of spin-space symmetries. We then study a spin-1 model with tSSG symmetry using linear flavor wave theory and find Z 2 topological quadrupolar excitations defined on a spin Brillouin Klein bottle. Specifically, the quadrupolar excitations possess a momentum-space glide-reflection symmetry and the edge states exhibit a nonlocal momentum twist. These results establish the symmetry language required for interacting spin systems, whether realized in magnetic materials or on programmable quantum simulators, and open a route to unconventional magnetism.
The quasi-particle excitations in Weyl semimetals, known as Weyl fermions, are usually forced to emerge in charge-conjugate pairs by the Nielsen--Ninomiya theorem. When the Brillouin zone is non-orientable, this constraint is replaced by a
Z2 charge cancellation, as a result of the chirality becoming ill-defined on such manifolds; this results in configurations with seemingly non-zero total chirality. Here, we set out to explain this behaviour from a purely topological perspective, and provide a classification of non-orientable Weyl semimetal topology in terms of exact sequences of twisted (co)homology groups. This leads to several discoveries of direct physical importance: in particular, we recover the
Z2 charge cancellation in a coordinate-independent way, allowing meaningful limits to be set on its physical interpretation. A detailed discussion is provided on a specific Klein bottle-like topology induced by a momentum-space glide symmetry, including a full review of the insulating and semimetallic invariants of the system and a classification of the surface states on the non-orientable boundary. Beyond this, we provide a complete survey of all possible non-orientable Brillouin zones and their associated invariants, and extend our formalism into the realm of non-Hermitian topological physics and inversion-symmetric Weyl semimetals. Our work exemplifies the vast potential of fundamental mathematical descriptions to not only aid the corresponding physical intuition, but also predict novel and hitherto overlooked phenomena of great relevance throughout the physics research forefront.
Crystallographic groups are conventionally studied in real space to characterize crystal symmetries. Recent work has recognized that when these symmetries are realized projectively, momentum space inherently accommodates nonsymmorphic symmetries, thereby evoking the concept of \textit{momentum-space crystallographic groups} (MCGs). Here, we reveal that the cohomology of MCGs encodes fundamental data of crystalline topological band structures. Specifically, the collection of second cohomology groups,
H2(ΓF,Z), for all MCGs
ΓF, provides an exhaustive classification of Abelian crystalline topological insulators, serving as an effective approximation to the full crystalline topological classification. Meanwhile, the third cohomology groups
H3(ΓF,Z) across all MCGs exhaustively classify all possible twistings of point-group actions on the Brillouin torus, essential data for twisted equivariant K-theory. Furthermore, we establish the isomorphism
Hn+1(ΓF,Z)≅Hn(ΓF,F(RFd,U(1))) for $ n\ge 1$, where
F(RFd,U(1)) denotes the space of continuous
U(1)-valued functions on the
dD momentum space
RFd. The case
n=1 yields a complete set of topological invariants formulated in purely algebraic terms, which differs fundamentally from the conventional formulation in terms of differential forms. The case
n=2, analogously, provides a fully algebraic description for all such twistings. Thus, the cohomological theory of MCGs serves as a key technical framework for analyzing crystalline topological phases within the general setting of projective symmetry.
Topological spin liquids can be described by topological gauge theories with global symmetry. Due to the presence of both nontrivial bulk deconfined gauge fluxes and global symmetry, topological spin liquids are examples of the so-called ``symmetry enriched topological phases'' (SETs). In this paper, we find that, in some twisted versions of topological gauge theories (with discrete Abelian gauge group
Gg), implementing a global symmetry (denoted by
Gs) is anomalous although symmetry charge carried by topological pointlike excitations is normally fractionalized and classified by the second cohomology group. To demonstrate the anomaly, we fully gauge the global symmetry, rendering a new gauge theory that is not gauge invariant. Therefore, the SET order of the ground state is anomalous, which cannot exist in the three-dimensional system alone. Such an anomalous state construction generalizes the ``2D surface topological order'' to three dimensions. A concrete example with ${G}_{g}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ and
Gs=Z2 is calculated.
A projective symmetry group (PSG) has been regarded as a classification theory of spin liquids.However, it does not include a symmetry-protected topological order of fermionic spinon excitations, and thus the classification of gapped spin liquids is incomplete.We demonstrate the classification beyond PSG by utilizing the Kitaev model on the square-octagon lattice, where two gapped spin liquids are distinguished by a topological Z 2 invariant.This Z 2 invariant can be defined solely by the time-reversal and translation symmetries on condition that the time-reversal symmetry is implemented projectively.Thus, it is a hidden class of topological Kitaev spin liquids with helical edge states, which has been ignored for a long time.This suggests that there exists an unknown classification scheme of gapped spin liquids beyond PSG.
We revisit the fermionic parton approach to
S=1/2 quantum spin liquids with
SU(2) spin-rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin-liquid states with
S=1 and
S=0 Majorana fermion excitations coupled to a deconfined
Z2 gauge field. The crucial observation leading us to this result is that, like space group and time-reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous
SU(2) spin and gauge rotations. We show that there are only two realizations of spin rotations acting on fermionic partons: the familiar naive realization where spin rotation is not accompanied by any gauge transformation, and a single type of projective realization. We discuss the PSG classification for states with projective spin rotations. To illustrate these results, we show that there are four such PSGs on the two-dimensional square lattice. We study the properties of the corresponding states, finding that one---with gapless Fermi points---is a stable phase beyond mean-field theory. In this phase, depending on parameters, a small Zeeman magnetic field can open a partial gap for the Majorana fermion excitations. Moreover, there are nearby gapped phases supporting
Z2 vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such
S=1 Majorana spin liquids may occur in models and in real systems.
We study quantum many-body systems in the presence of an exotic antiunitary translation or inversion symmetry involving time reversal. Based on a symmetry-twisting method and spectrum robustness, we propose that a half-integer spin chain that respects any of these two antiunitary crystalline symmetries in addition to the discrete Z_{2}×Z_{2} global spin-rotation symmetry must either be gapless or possess degenerate ground states. This explains the gaplessness of a class of chiral spin models not indicated by the Lieb-Schultz-Mattis theorem and its known extensions. Moreover, we present symmetry classes with minimal sets of generators that give nontrivial Lieb-Schultz-Mattis-type constraints, argued by the bulk-boundary correspondence in 2D symmetry-protected topological phases as well as lattice homotopy. Our results for detecting the ingappability of 1D quantum magnets from the interplay between spin-rotation symmetries and magnetic space groups are applicable to systems with a broader class of spin interactions, including Dzyaloshinskii-Moriya and triple-product interactions.