Classification Of Topological Phases Research Articles

Extrinsic and intrinsic nonlinear Hall effects across Berry-dipole transitions

Three-dimensional Hopf insulators are a class of topological phases beyond the tenfold-way classification. At the critical point of the transition between two distinct Hopf insulators with rotational symmetry, the band-touching points are point Berry dipoles with two opposite Berry charges overlapping in a mirror-symmetric way, and carry unique Berry curvature structures leading to a special quantization of Berry flux. Close to such Berry-dipole transitions, we find that the extrinsic and intrinsic nonlinear Hall conductivity tensors in the weakly doped regime are characterized by two universal functions of the ratio between doping level and bulk energy gap, and are directly proportional to the change in Hopf invariant across the transition. Our work suggests that the nonlinear Hall effects display a generalized-sense quantized behavior across Berry-dipole transitions, establishing a correspondence between nonlinear Hall effects and Hopf invariants.

Does causal dynamics imply local interactions?

We consider quantum systems with causal dynamics in discrete spacetimes, also known as quantum cellular automata (QCA). Due to time-discreteness this type of dynamics is not characterized by a Hamiltonian but by a one-time-step unitary. This can be written as the exponential of a Hamiltonian but in a highly non-unique way. We ask if any of the Hamiltonians generating a QCA unitary is local in some sense, and we obtain two very different answers. On one hand, we present an example of QCA for which all generating Hamiltonians are fully non-local, in the sense that interactions do not decay with the distance. We expect this result to have relevant consequences for the classification of topological phases in Floquet systems, given that this relies on the effective Hamiltonian. On the other hand, we show that all one-dimensional quasi-free fermionic QCAs have quasi-local generating Hamiltonians, with interactions decaying exponentially in the massive case and algebraically in the critical case. We also prove that some integrable systems do not have local, quasi-local nor low-weight constants of motion; a result that challenges the standard definition of integrability.

Interacting SPT phases are not morita invariant

The tenfold way provides a strong organizing principle for invertible topological phases of matter. Mathematically, it is intimately connected with K-theory via the fact that there exist exactly ten Morita classes of simple real superalgebras. This connection is physically unsurprising, since weakly interacting topological phases are classified by K-theory. We argue that when strong interactions are present, care has to be taken when formulating the exact ten symmetry groups present in the tenfold way table. We study this phenomenon in the example of class D by providing two possible mathematical interpretations of a class D symmetry. These two interpretations of class D result in Morita equivalent but different symmetry groups. As K-theory cannot distinguish Morita-equivalent protecting symmetry groups, the two approaches lead to the same classification of topological phases on the weakly interacting side. However, we show that these two different symmetry groups yield different interacting classifications in spacetime dimension 2+1. We use the approach to interacting topological phases using bordism groups, reducing the relevant classification problem to a spectral sequence computation.

Two-dimensional Stiefel-Whitney insulators in liganded Xenes

Two-dimensional (2D) Stiefel-Whitney insulator (SWI), which is characterized by the second Stiefel-Whitney class, is a class of topological phases with zero Berry curvature. As an intriguing topological state, it has been well studied in theory but seldom realized in realistic materials. Here we propose that a large class of liganded Xenes, i.e., hydrogenated and halogenated 2D group-IV honeycomb lattices, are 2D SWIs. The nontrivial topology of liganded Xenes is identified by the bulk topological invariant and the existence of protected corner states. Moreover, the large and tunable bandgap (up to 3.5 eV) of liganded Xenes will facilitate the experimental characterization of the 2D SWI phase. Our findings not only provide abundant realistic material candidates that are experimentally feasible but also draw more fundamental research interest towards the topological physics associated with Stiefel-Whitney class in the absence of Berry curvature.

Discovery of {hat{boldsymbol{C}}}_2 rotation anomaly in topological crystalline insulator SrPb

Topological crystalline insulators (TCIs) are insulating electronic states with nontrivial topology protected by crystalline symmetries. Recently, theory has proposed new classes of TCIs protected by rotation symmetries hat C_n, which have surface rotation anomaly evading the fermion doubling theorem, i.e., n instead of 2n Dirac cones on the surface preserving the rotation symmetry. Here, we report the first realization of the hat C_2 rotation anomaly in a binary compound SrPb. Our first-principles calculations reveal two massless Dirac fermions protected by the combination of time-reversal symmetry hat T and hat C_{2y} on the (010) surface. Using angle-resolved photoemission spectroscopy, we identify two Dirac surface states inside the bulk band gap of SrPb, confirming the hat C_2 rotation anomaly in the new classes of TCIs. The findings enrich the classification of topological phases, which pave the way for exploring exotic behavior of the new classes of TCIs.

M-theoretic genesis of topological phases

We present a novel M-theoretic approach of constructing and classifying anyonic topological phases of matter, by establishing a correspondence between (2+1)d topological field theories and non-hyperbolic 3-manifolds. In this construction, the topological phases emerge as macroscopic world-volume theories of M5-branes wrapped around certain types of non-hyperbolic 3-manifolds. We devise a systematic algorithm for identifying the emergent topological phases from topological data of the internal wrapped 3-manifolds. As a benchmark of our approach, we reproduce all the known unitary bosonic topological orders up to rank 4. Remarkably, our construction is not restricted to an unitary bosonic theory but it can also generate fermionic and/or non-unitary anyon models in an equivalent fashion. Hence, we pave a new route toward the classification of topological phases of matter.

Classification of topological phases with finite internal symmetries in all dimensions

We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.

Photonic quadrupole topological phases

The topological phases of matter are characterized using the Berry phase, a geometrical phase associated with the energy-momentum band structure. The quantization of the Berry phase and the associated wavefunction polarization manifest as remarkably robust physical observables, such as quantized Hall conductivity and disorder-insensitive photonic transport1–5. Recently, a novel class of topological phases, called higher-order topological phases, were proposed by generalizing the fundamental relationship between the Berry phase and quantized polarization, from dipole to multipole moments6–8. Here, we demonstrate photonic realization of the quantized quadrupole topological phase, using silicon photonics. In our two-dimensional second-order topological phase, we show that the quantization of the bulk quadrupole moment manifests as topologically robust zero-dimensional corner states. We contrast these topological states against topologically trivial corner states in a system without bulk quadrupole moment, where we observe no robustness. Our photonic platform could enable the development of robust on-chip classical and quantum optical devices with higher-order topological protection. A higher-order topological phase is demonstrated, manifesting as robust, topologically protected corner modes in a silicon photonics system.

Classification of topological phases in periodically driven interacting systems

We consider topological phases in periodically driven (Floquet) systems exhibiting many-body localization, protected by a symmetry $G$. We argue for a general correspondence between such phases and topological phases of undriven systems protected by symmetry $\mathbb{Z} \rtimes G$, where the additional $\mathbb{Z}$ accounts for the discrete time translation symmetry. Thus, for example, the bosonic phases in $d$ spatial dimensions without intrinsic topological order (SPT phases) are classified by the cohomology group $H^{d+1}(\mathbb{Z} \rtimes G, \mathrm{U}(1))$. For unitary symmetries, we interpret the additional resulting Floquet phases in terms of the lower-dimensional SPT phases that are pumped to the boundary during one time step. These results also imply the existence of novel symmetry-enriched topological (SET) orders protected solely by the periodicity of the drive.

Chiral magnetic and vortical effects in higher dimensions at weak coupling

Chiral Magnetic Effect and Chiral Vortical Effect are parity odd transport phenomena originating from chiral anomaly, and have generalizations to all even dimensional space-time higher than four dimensions. We attempt to compute the associated P-odd retarded response functions in the weak coupling limit of chiral fermion theory in all even dimensions, using the diagrammatic technique of real-time perturbation theory. We also clarify the necessary Kubo formula relating the computed P-odd retarded correlation functions and the associated anomalous transport coefficients. We speculate on the 8-fold classification of topological phases.

Universal topological phase of two-dimensional stabilizer codes

Topological phases can be defined in terms of local equivalence: two systems are in the same topological phase if it is possible to transform one into the other by a local reorganization of its degrees of freedom. The classification of topological phases therefore amounts to the classification of long-range entanglement. Such local transformation could result, for instance, from the adiabatic continuation of one system's Hamiltonian to the other. Here, we use this definition to study the topological phase of translationally invariant stabilizer codes in two spatial dimensions, and show that they all belong to one universal phase. We do this by constructing an explicit mapping from any such code to a number of copies of Kitaev's code. Some of our results extend to some two-dimensional (2D) subsystem codes, including topological subsystem codes. Error correction benefits from the corresponding local mappings. In particular, it enables us to use decoding algorithm developed for Kitaev's code to decode any 2D stabilizer code and subsystem code.

Topological phases of fermions in one dimension

In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of one-dimensional systems. We focus on the time-reversal-invariant Majorana chain (BDI symmetry class). While the band classification yields an integer topological index $k$, it is known that phases characterized by values of $k$ in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half chains. We generalize these results to the classification of all one-dimensional gapped phases of fermionic systems with possible antiunitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.