The topological classification of insulators and superconductors, the "ten-fold way", is grounded on fermionic many-body symmetries and has had a dramatic impact on many fields of physics. Therefore, it seems equally important to investigate a similar approach for bosons as tightly analogous to the fermionic prototype as possible. There are, however, several obstacles coming from the fundamental physical differences between fermions and bosons. Here, we propose a theory of free boson topology (topological classification and bulk-boundary correspondence) protected by bosonic many-body symmetry operations, namely, squeezing transformations, particle number, and bosonic time reversal. We identify two symmetry classes that are topologically non-trivial in one dimension. They include key models like the bosonic Kitaev chain, protected by a squeezing symmetry within our framework, and the celebrated bosonic SSH model, protected by a squeezing symmetry and particle number. To provide a robust experimental platform for testing our theory, we introduce a new quantum meta-material: photo-magnonic crystals. They are remarkable for their experimental flexibility and natural affinity for displaying band topological physics at microwave frequencies. We engineer a many-body symmetry-protected topological photo-magnonic chain with boundary modes mandated by a Pfaffian invariant. Using an electromagnetic finite-element modelling, we simulate its reflection and transmission and identify experimental signatures of its boundary modes. The experimental tuning of the crystal to its symmetry-protected topological phase is also addressed. Our modelling of the photo-magnonic chain provides a thorough blueprint for its experimental realisation and the unambiguous observation of its exotic physics.
The topological classification of insulators and superconductors, the "ten-fold way", is grounded on fermionic many-body symmetries and has had a dramatic impact on many fields of physics. Therefore, it seems equally important to investigate a similar approach for bosons as tightly analogous to the fermionic prototype as possible. There are, however, several obstacles coming from the fundamental physical differences between fermions and bosons. Here, we propose a theory of free boson topology (topological classification and bulk-boundary correspondence) protected by bosonic many-body symmetry operations, namely, squeezing transformations, particle number, and bosonic time reversal. We identify two symmetry classes that are topologically non-trivial in one dimension. They include key models like the bosonic Kitaev chain, protected by a squeezing symmetry within our framework, and the celebrated bosonic SSH model, protected by a squeezing symmetry and particle number. To provide a robust experimental platform for testing our theory, we introduce a new quantum meta-material: photo-magnonic crystals. They are remarkable for their experimental flexibility and natural affinity for displaying band topological physics at microwave frequencies. We engineer a many-body symmetry-protected topological photo-magnonic chain with boundary modes mandated by a Pfaffian invariant. Using an electromagnetic finite-element modelling, we simulate its reflection and transmission and identify experimental signatures of its boundary modes. The experimental tuning of the crystal to its symmetry-protected topological phase is also addressed. Our modelling of the photo-magnonic chain provides a thorough blueprint for its experimental realisation and the unambiguous observation of its exotic physics.
Tight bosonic analogs of free-fermionic symmetry-protected topological phases, and their associated edge-localized excitations, have long evaded the grasp of condensed-matter and AMO physics. In this paper, building on our initial exploration [Phys. Rev. Lett. 127, 245701 (2021)], we identify a broad class of quadratic bosonic systems subject to Markovian dissipation that realize tight bosonic analogs of the Majorana and Dirac edge modes characteristic of topological superconductors and insulators, respectively. To this end, we establish a general framework for topological metastability for these systems, by leveraging pseudospectral theory as the appropriate mathematical tool for capturing the nonnormality of the Lindbladian generator. The resulting dynamical paradigm, which is characterized by both a sharp separation between transient and asymptotic dynamics and a nontrivial topological invariant, is shown to host edge-localized modes, which we dub Majorana and Dirac bosons. Generically, such modes consist of one conserved mode and a canonically conjugate generator of an approximate phase-space translation symmetry of the dynamics. The general theory is exemplified through several representative models exhibiting the full range of exotic boundary physics that topologically metastable systems can engender. In particular, we explore the extent to which Noether's theorem is violated in this dissipative setting and the way in which certain symmetries can nontrivially modify the edge modes. Notably, we also demonstrate the possibility of anomalous parity dynamics for a bosonic cat state prepared in a topologically metastable system, whereby an equal distribution between even and odd parity sectors is sustained over a long transient. For both Majorana and Dirac bosons, observable multitime signatures in the form of anomalously long-lived quantum correlations and divergent zero-frequency power spectral peaks are proposed and discussed in detail. Our results point to a paradigm for symmetry-protected topological physics in free bosons, embedded deeply in the long-lived transient regimes of metastable dynamics.
We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by noninteracting Bloch and Bogoliubov--de Gennes Hamiltonians that support additional order-two spatial symmetry, besides any of 10 classes of symmetries defined by time-reversal symmetry and particle-hole symmetry. The additional order-two spatial symmetry we consider is general and it includes ${\mathbit{Z}}_{2}$ global symmetry, mirror reflection, twofold rotation, inversion, and their magnetic point group symmetries. We find that the topological periodic table shows a periodicity in the number of flipped coordinates under the order-two spatial symmetry, in addition to the Bott periodicity in the space dimensions. Various symmetry-protected topological phases and gapless modes will be identified and discussed in a unified framework. We also present topological classification of symmetry-protected Fermi points. The bulk classification and the surface Fermi point classification provide a realization of the bulk-boundary correspondence in terms of the K theory.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T2=1 time-reversal Z2T and fermion-number-parity Z2f symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge c=1 gapless edge excitations, if the symmetry is not broken.
Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.
Crystalline symmetry and time-reversal symmetry are commonly present in real superconducting materials. However, the topological classification of systems respecting these symmetries, particularly for interacting fermions, remains incomplete. In this work, we systematically classify time-reversal symmetry-protected crystalline topological superconductors in two-dimensional interacting fermionic systems using an explicit real-space construction. Among the resulting phases, we identify intrinsically interacting fermionic topological superconductors, i.e., phases that cannot be realized in either free-fermion or interacting bosonic systems. For spinless fermions with protecting symmetry group C4×Z2T or D4×Z2T (plus fermion parity), the intrinsic sector has a Z4 classification. The corresponding root phases generating this Z4 classification admit a transparent real-space construction in terms of decorated 1D blocks. These blocks are 1D fermionic symmetry-protected topological (FSPT) phases, realizable as double Majorana chains. We further find the corresponding Z4 spinless intrinsic phases for wallpaper groups p4, p4m, and p4g. We also find an additional Z2 intrinsically interacting phase for spinless fermions with wallpaper group pm, which is absent with the corresponding point-group symmetry alone. Moreover, these intrinsic phases naturally give rise to higher-order FSPT phases that support corner zero modes. Finally, we verify the crystalline equivalence principle for generic 2D interacting FSPT systems with both crystalline and internal symmetries.
We explore one-dimensional fermionic symmetry-protected topological (SPT) phases related by the crystalline equivalence principle. In particular, we study charge-conserving many-body topological phases of fermions protected respectively by chiral and reflection symmetries. While the classifications of the two crystalline-equivalent SPT phases are identical, their topological properties and phase structures can be very different, depending on the microscopic details. Specifically, we consider certain extensions of the Su-Schrieffer-Heeger model, with and without interactions, that preserve both chiral and reflection symmetries, and explicitly compute the many-body topological invariants based on the systems' ground states. The phase structures determined by these topological invariants align perfectly with the many-body spectra of deformations among the models. As expected, gapped deformations exist only when all the topological invariants remain unchanged. Moreover, we show that decomposable systems -- those that can be decomposed into local and decoupled subsystems -- can be topologically characterized by real-space quantum numbers directly associated with the symmetries. For reflection-symmetric systems, these quantum numbers are related to the many-body topological invariants via a bulk-center correspondence, which can be justified using the Atiyah-Hirzebruch spectral sequence in generalized homology theory. Finally, we discuss the role of transition symmetry in the many-body topologies of these SPT phases.