Session N17: Topology in New Settings: Non-Hermitian and Beyond

Analysis of the bulk-boundary correspondence in nonlinear eigenvalue problems by the auxiliary eigenvalues

Topological band theory has been extensively studied in these fifteen years [1,2]. So far, the platforms of the topological band theory have been extended including classical systems [3]. Furthermore, recent studies have been attempting to extend the topological band theory to systems that involve nonlinearity of eigenvectors [4]. This is because the nonlinearity is common in nature and those extension is significant for the development of the topological band theory. On the other hand, few studies focusing on the interplay between the topology and the nonlinearity of eigenvalues, which is another type of the nonlinearity.

In this talk, we address the bulk-boundary correspondence (BBC) [5] in systems with the nonlinearity of eigenvalues. Specifically, we introduce auxiliary eigenvalues of the matrix pencil and establish the BBC between the bulk topology of auxiliary eigenstates and physical boundary states [6]. We demonstrate the existence of the BBC in insulators and semimetals by analyzing models with nonlinearities of eigenvalues.

 

[1] M. Z. Hasan and C. L. Kane, RMP 82, 3045 (2010).

[2] X. -L. Qi and S. -C. Zhang, RMP 83, 1057 (2011).

[3] F. D. M. Haldane and S. Raghu, PRL 100, 013904 (2008). 

[4] K. Sone, Y. Ashida, and T. Sagawa, PRR 4, 023211 (2022).

[5] Y. Hatsugai, PRL 71, 3697 (1993).

[6] T. Isobe, T. Yoshida, and Y. Hatsugai, arXiv:2310.12577 (2023).   

Floquet non-Abelian topological insulator and multifold bulk-edge correspondence

Topological phases characterized by non-Abelian charges are beyond the scope of the paradigmatic tenfold way and have gained increasing attention recently. Here we investigate topological insulators with multiple tangled gaps in Floquet settings and identify uncharted Floquet non-Abelian topological insulators without any static or Abelian analog. We demonstrate that the bulkedge correspondence is multifold and follows the multiplication rule of the quaternion group Q8. The same quaternion charge corresponds to several distinct edge-state configurations that are fully determined by phase-band singularities of the time evolution. In the anomalous non-Abelian phase, edge states appear in all bandgaps despite trivial quaternion charge. Furthermore, we uncover an exotic swap effect—the emergence of interface modes with swapped driving, which is a signature of the non-Abelian dynamics and absent in Floquet Abelian systems. Our work, for the first time, presents Floquet

topological insulators characterized by non-Abelian charges and opens up exciting possibilities for exploring the rich and uncharted territory of nonequilibrium topological phases.   

Symmetry-Protected Locally Confined Eigenstates Oscillation in Topolectrical Heterosystem

We demonstrate a controllable novel local confinement of voltage oscillations in a topolectrical (TE) heterosystem constructed with typical RLC circuit components by exploiting the underlying symmetries of its admittance spectrum. The system comprises of TE Su-Schrieffer-Heeger chain with different inter- and intra-unit cell coupling capacitances in which alternating onsite gain and loss are introduced through resistors coupled to operational amplifiers in the non-Hermitian (NH) half of the chain while the Hermitian part remains resistor-free. By adjusting the resistance in the NH segment, the admittance spectrum of the system can be made to exhibit parity-time (PT) symmetry in which all the admittance eigenvalues of the entire chain are real, and the corresponding voltage eigenstates are delocalized throughout all the circuit nodes. In contrast, when the NH segment is tuned to an anti-PT symmetric configuration, the eigenmodes with imaginary eigenvalues are localized exclusively within the NH segments while those with real eigenvalues are localized within only the Hermitian segment. This remarkable phenomenon represents a novel symmetry-protected and controlled confinement of voltage oscillations in TE heterojunctions, which is of practical utility in topological electronics.   

Topological Triviality of Strictly Local Projectors

The topological properties of a Bloch band are closely related to the localization properties of wavefunctions spanning it. It has been shown that if a set of compactly supported Wannier-type functions spans a band or a set of bands, then the band(s) are necessarily topologically trivial. This situation arises, for example, when a band is described by a strictly local projector. We show that if the occupied (gapped) subspace of a single-particle Hamiltonian is described by a projector that is strictly local, then the system is topologically trivial. We do not rely on the assumption of lattice translational invariance.   

Non-Bloch band theory of sub-symmetry-protected topological phases

A generic feature of symmetry-protected topological (SPT) phases of matter is the bulkboundary correspondence (BBC) which connects the concept of bulk topology to the emergence of robust boundary states. In recent years, non-Hermitian systems have shown unconventional properties and phenomena such as exceptional points, non-Hermitian skin effect, and many more in different research fields without Hermitian analog. Therefore, the topological Bloch band theory with the notion of the Brillouin zone (BZ) has been extended to the non-Bloch band theory with the notion of the generalized Brillouin zone (GBZ) defined by generalized momenta which can take complex values. The non-Bloch band theory has successfully proven that non-Hermitian systems show two types of modified BBC: (i) complex eigenvalue topology of the bulk leads to non-Hermitian skin effect, where all bulk states localize at one boundary of the system, and (ii) the eigenstate topology in the GBZ leads to the conventional topological boundary modes.

In our recent work [1], we reported a new type of BBC in non-Hermitian systems, which originates from the intrinsic topology of the GBZ. Topologically non-trivial GBZ appears due to general boundary conditions that break the translation symmetry of the system locally. In our case, the topological phase transition is characterized by the generalized momentum touching of GBZ, which accompanies the emergence of exceptional points.

In this seminar, I will discuss a simple extension of our work to Hermitian systems. Firstly, we found that the intrinsic topology of GBZ successfully characterizes the SPT phases. In this case, the topological phase transition of GBZ is characterized by the generalized momenta touching of GBZ, which accompanies the emergence of band touching points. Moreover, the non-Bloch band theory provides a natural topological invariant characterizing the sub-symmetry-protected topological phases that a conventional Bloch topological invariant (winding number) fails to characterize.

Reference: [1] Sonu Verma and Moon Jip Park, arXiv:2305.08584.   

Particle-Hole Symmetry Groups and Topological Quantum Chemistry

With the recent completion of Topological Quantum Chemistry (TQC) for magnetic symmetry groups (SGs), efforts have now shifted towards generalizing TQC to other ordered states, including weakly spin-orbit-coupled magnets (via spin space groups) and weak-coupling superconductors (SCs). However, to construct on equal footing TQC- (Wannier-) based characterizations of topological states in all of the classes of the tenfold way with additional crystal symmetries, it is necessary to formulate notions of elementary band representations (EBRs) and topological invariants that are agnostic as to the strength of SC pairing and sublattice, orbital, and spin-orbit interactions. We here accomplish this for crystals in Classes AIII and A with additional combinations of crystal and unitary particle-hole (chiral) symmetry S by introducing a new notion of S-SGs. We compute the EBRs, real-space invariants, and bulk-boundary correspondences for all nontrivial bands in the 1D and 2D S-SGs. We relate our findings to quantum anomalies and recent discussions of fermion doubling theorems, highlighting cases where non-on-site S symmetry in nonsymmorphic S-SGs leads to frequently overlooked gauge-invariance issues in the conventional 1D winding number.   

Delicate Topology of Luttinger Semimetal

Recent advances in delicate topology have expanded the classification of topological bands, but its presence in solid-state materials remains elusive. Here we show that delicate topology naturally emerges in the Luttinger-Kohn model that describes many semiconductors and semimetals. In particular, the Luttinger semimetal is found to be a quantum critical point leading to a quantized jump of an integer-valued delicate topological invariant. Away from this criticality, we have identified new types of electronic insulators and semimetals with intertwined stable and delicate topologies. They all carry gapless surface states that transform anomalously under rotation symmetry. Our work provides a starting point for exploring delicate topological phenomena in quantum materials.   

PT-symmetric Non-Hermitian Hopf metal

Hopf insulator represents a class of three-dimensional topological insulators protected by the non-trivial Hopf bundle structure of the eigenstates. In this letter, we propose the generalization of the Hopf bundle in the non-Hermitian systems. While the Hopf invariant is not a stable topological index due to the additional non-Hermitian degree of freedom, we show that the $mathcal{PT}$-symmetry stabilizes the Hopf invariant even in the presence of the non-Hermiticity. In sharp contrast to the Hopf insulator phase in the Hermitian counterpart, we discover an interesting result that the non-Hermitian Hopf bundle exhibits the topologically protected band degeneracy, which is characterized by the surface of exceptional points. In addition to the topological degeneracy, this Hopf metal phase has non-trivial bulk-boundary correspondence, which manifests as the drumhead surface state in the open boundary condition. Finally, we show that, by breaking $mathcal{PT}$-symmetry, the nodal surface deforms into the knotted exceptional lines. Our discovery of the Hopf metal phase, for the first time, shows the existence of the non-Hermitian topological phase beyond the known topological classification methods such as $K$-theory and symmetry indicator.   

Topology of non-Hermitian systems from the exceptional point

We discuss classifications of multiple arbitrary-order exceptional points by invoking the permutation group and its conjugacy classes. We classify topological structures of Riemann surfaces generated by multiple states around multiple arbitrary-order exceptional points, using the permutation properties of stroboscopic encircling exceptional points. The results are realized in non-Hermitian effective Hamiltonian based on Jordan normal forms and fully desymmetrized optical microcavities. Additionally, we reveal the relation between the spectral topology originating from complex eigenvalues in non-Hermitian systems and wavefunction topology related to the additional geometric phases. Finally, we discuss the topology of one-dimensional multi-bands systems based on exceptional points.    

Universal Platform of point-gap topological phases from topological materials

Whereas point-gap topological phases are responsible for exceptional phenomena intrinsic to non-Hermitian systems, their realization in quantum materials is still elusive. Here we propose a simple and universal platform of point-gap topological phases constructed from Hermitian topological insulators and superconductors. We show that (d-1)-dimensional point-gap topological phases are realized by making a boundary in d-dimensional topological insulators and superconductors dissipative. A crucial observation of the proposal is that adding a decay constant to boundary modes in d-dimensional topological insulators and superconductors is topologically equivalent to attaching a (d-1)-dimensional point-gap topological phase to the boundary. We furthermore establish the proposal from the extended version of the Nielsen-Ninomiya theorem, relating dissipative gapless modes to point-gap topological numbers. From the bulk-boundary correspondence of the point-gap topological phases, the resultant point-gap topological phases exhibit exceptional boundary states or in-gap higher-order non-Hermitian skin effects.   

Quantum corrections to the magnetoconductivity of surface states in three-dimensional topological insulators

The interplay between quantum interference, electron-electron interaction (EEI), and disorder is one of the central themes of condensed matter physics. Such interplay can cause high-order magnetoconductance (MC) corrections in semiconductors with weak spin-orbit coupling (SOC). However, it remains unexplored how the magnetotransport properties are modified by the high-order quantum corrections in the electron systems of symplectic symmetry class, which include topological insulators (TIs), Weyl semimetals, graphene with negligible intervalley scattering, and semiconductors with strong SOC. Here, we extend the theory of quantum conductance corrections to two-dimensional (2D) electron systems with the symplectic symmetry, and study experimentally such physics with dual-gated TI devices in which the transport is dominated by highly tunable surface states. We find that the MC can be enhanced significantly by the second-order interference and the EEI effects, in contrast to the suppression of MC for the systems with orthogonal symmetry. Our work reveals that detailed MC analysis can provide deep insights into the complex electronic processes in TIs, such as the screening and dephasing effects of localized charge puddles, as well as the related particle-hole asymmetry.   

A variant of bulk-boundary correspondence in topological phases in AZ+I classification

Topological insulators usually have gapless boundary states due to the well-known bulk-boundary correspondence. For this to hold true there is an implicit necessary condition: the symmetry of the bulk must be maintained locally even near the boundary. This condition is violated for symmetries that involve spatial inversion operations, such as parity-time-reversal (PT) symmetry. For two-dimensional PT-symmetric topological insulators, bulk-edge correspondence in the usual sense does not generally hold for the reasons mentioned above. In the previous work, we showed that by using entanglement spectra instead of ordinary edge spectra, a variant of "bulk boundary correspondence" holds even for PT symmetric topological insulators [1].

 

In this presentation, we reveal the extension of such "bulk-boundary correspondence" to topological phases of the AZ+I classification, which is an extension of the AZ classification by replacing time reversal T and particle-hole symmetry C with PT and PC, respectively. We focus on the fact that the entanglement Hamiltonian always has an anti-symmetry Ξ. By combining the PT and PC operations with Ξ, we show that symmetries of the AZ+I classification are recovered in the entanglement Hamiltonian. As a result, the entanglement spectrum for topological insulators in the AZ+I classification becomes gapless, and a variant of “bulk-boundary correspondence” holds. 

 

[1] R. Takahashi and T. Ozawa, Phys. Rev. B 108, 075129 (2023).   

Non-Hermitian skin effect enforced by nonsymmorphic symmetries

Crystal symmetries play an essential role in band structures of non-Hermitian Hamiltonian.  In this talk, we propose a non-Hermitian skin effect (NHSE) enforced by nonsymmorphic symmetries [1]. We show that the NHSE inevitably occurs if a two-dimensional non-Hermitian system satisfies conditions derived from the nonsymmorphic symmetry of the doubled Hermitian Hamiltonian. This NHSE occurs in symmetry classes with and without time-reversal symmetry. The NHSE enforced by nonsymmorphic symmetries always occurs simultaneously with the closing of the point gap at zero energy. We also show that such a NHSE can occur in specific three-dimensional space groups with nonsymmorphic symmetries. 

[1] Y. Tanaka, R. Takahashi, and R. Okugawa, arXiv:2306.08923 (2023).