Symmetry plays a key role in modern physics, as manifested in the revolutionary topological classification of matter in the past decade. So far, we seem to have a complete theory of topological phases from internal symmetries as well as crystallograp
hic symmetry groups. However, an intrinsic element, i.e., the gauge symmetry in physical systems, has been overlooked in the current framework. Here, we show that the algebraic structure of crystal symmetries can be projectively enriched due to the gauge symmetry, which subsequently gives rise to new topological physics never witnessed under ordinary symmetries. We demonstrate the idea by theoretical analysis, numerical simulation, and experimental realization of a topological acoustic lattice with projective translation symmetries under a $Z_2$ gauge field, which exhibits unique features of rich topologies, including a single Dirac point, M{o}bius topological insulator and graphene-like semimetal phases on a rectangular lattice. Our work reveals the impact when gauge and crystal symmetries meet together with topology, and opens the door to a vast unexplored land of topological states by projective symmetries.
Response properties that are purely intrinsic to physical systems are of paramount importance in physics research, as they probe fundamental properties of band structures and allow quantitative calculation and comparison with experiment. For anomalou
s Hall transport in magnets, an intrinsic effect can appear at the second order to the applied electric field. We show that this intrinsic second-order anomalous Hall effect is associated with an intrinsic band geometric property -- the dipole moment of Berry-connection polarizability (BCP) in momentum space. The effect has scaling relation and symmetry constraints that are distinct from the previously studied extrinsic contributions. Particularly, in antiferromagnets with $mathcal{PT}$ symmetry, the intrinsic effect dominates. Combined with first-principles calculations, we demonstrate the first quantitative evaluation of the effect in the antiferromagnet Mn$_{2}$Au. We show that the BCP dipole and the resulting intrinsic second-order conductivity are pronounced around band near degeneracies. Importantly, the intrinsic response exhibits sensitive dependence on the N{e}el vector orientation with a $2pi$ periodicity, which offers a new route for electric detection of the magnetic order in $mathcal{PT}$-invariant antiferromagnets.
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.
The Haldane phase represents one of the most important symmetry protected states in modern physics. This state can be realized using spin-1 and spin-${1over 2}$ Heisenberg models and bosonic particles. Here we explore the emergent Haldane phase in an
alternating bond $mathbb{Z}_3$ parafermion chain, which is different from the previous proposals from fundamental statistics and symmetries. We show that this emergent phase can also be characterized by a modified long-range string order, as well as four-fold degeneracy in the ground state energies and entanglement spectra. This phase is protected by both the charge conjugate and parity symmetry, and the edge modes are shown to satisfy parafermionic statistics, in which braiding of the two edge modes yields a ${2pi over 3}$ phase. This model also supports rich phases, including topological ferromagnetic parafermion (FP) phase, trivial paramagnetic parafermion phase, classical dimer phase and gapless phase. The boundaries of the FP phase are shown to be gapless and critical with central charge $c = 4/5$. Even in the topological FP phase, it is also characterized by the long-range string order, thus we observe a drop of string order across the phase boundary between the FP phase and Haldane phase. These phenomena are quite general and this work opens a new way for finding exotic topological phases in $mathbb{Z}_k$ parafermion models.
Topological orders and associated topological protected excitations satisfying non-Abelian statistics have been widely explored in various platforms. The $mathbb{Z}_3$ parafermions are regarded as the most natural generation of the Majorana fermions
to realize these topological orders. Here we investigate the topological phase and emergent $mathbb{Z}_2$ spin phases in an extended parafermion chain. This model exhibits rich variety of phases, including not only topological ferromagnetic phase, which supports non-Abelian anyon excitation, but also spin-fluid, dimer and chiral phases from the emergent $mathbb{Z}_2$ spin model. We generalize the measurement tools in $mathbb{Z}_2$ spin models to fully characterize these phases in the extended parafermion model and map out the corresponding phase diagram. Surprisingly, we find that all the phase boundaries finally merge to a single supercritical point. In regarding of the rather generality of emergent phenomena in parafermion models, this approach opens a wide range of intriguing applications in investigating the exotic phases in other parafermion models.
