Generalised Geometric Logic

This paper introduces a notion of generalised geometric logic. Connections of generalised geometric logic with L-topological system and L-topological space are established.

Nonlocal topological insulators: Deterministic aperiodic arrays supporting localized topological states protected by nonlocal symmetries

The properties of topological systems are inherently tied to their dimensionality. Indeed, higher-dimensional periodic systems exhibit topological phases not shared by their lower-dimensional counterparts. On the other hand, aperiodic arrays in lower-dimensional systems (e.g., the Harper model) have been successfully employed to emulate higher-dimensional physics. This raises a general question on the possibility of extended topological classification in lower dimensions, and whether the topological invariants of higher-dimensional periodic systems may assume a different meaning in their lower-dimensional aperiodic counterparts. Here, we demonstrate that, indeed, for a topological system in higher dimensions one can construct a one-dimensional (1D) deterministic aperiodic counterpart which retains its spectrum and topological characteristics. We consider a four-dimensional (4D) quantized hexadecapole higher-order topological insulator (HOTI) which supports topological corner modes. We apply the Lanczos transformation and map it onto an equivalent deterministic aperiodic 1D array (DAA) emulating 4D HOTI in 1D. We observe topological zero-energy zero-dimensional (0D) states of the DAA—the direct counterparts of corner states in 4D HOTI and the hallmark of the multipole topological phase, which is meaningless in lower dimensions. To explain this paradox, we show that higher-dimension invariant, the multipole polarization, retains its quantization in the DAA, yet changes its meaning by becoming a nonlocal correlator in the 1D system. By introducing nonlocal topological phases of DAAs, our discovery opens a direction in topological physics. It also unveils opportunities to engineer topological states in aperiodic systems and paves the path to application of resonances associates with such states protected by nonlocal symmetries.

Simulating Floquet topological phases in static systems

We show that scattering from the boundary of static, higher-order topological insulators (HOTIs) can be used to simulate the behavior of (time-periodic) Floquet topological insulators. We consider D-dimensional HOTIs with gapless corner states which are weakly probed by external waves in a scattering setup. We find that the unitary reflection matrix describing back-scattering from the boundary of the HOTI is topologically equivalent to a (D-1)-dimensional nontrivial Floquet operator. To characterize the topology of the reflection matrix, we introduce the concept of `nested' scattering matrices. Our results provide a route to engineer topological Floquet systems in the lab without the need for external driving. As benefit, the topological system does not suffer from decoherence and heating.

Particle-antiparticle duality and fractionalization of topological chiral solitons

Although a prototypical Su–Schrieffer–Heeger (SSH) soliton exhibits various important topological concepts including particle-antiparticle (PA) symmetry and fractional fermion charges, there have been only few advances in exploring such properties of topological solitons beyond the SSH model. Here, by considering a chirally extended double-Peierls-chain model, we demonstrate novel PA duality and fractional charge e/2 of topological chiral solitons even under the chiral symmetry breaking. This provides a counterexample to the belief that chiral symmetry is necessary for such PA relation and fractionalization of topological solitons in a time-reversal invariant topological system. Furthermore, we discover that topological chiral solitons are re-fractionalized into two subsolitons which also satisfy the PA duality. As a result, such dualities and fractionalizations support the topological $$\mathbb {Z}_4$$ Z 4 algebraic structures. Our findings will inspire researches seeking feasible and promising topological systems, which may lead to new practical applications such as solitronics.

A possible route towards dissipation-protected qubits using a multidimensional dark space and its symmetries

Quantum systems are always subject to interactions with an environment, typically resulting in decoherence and distortion of quantum correlations. It has been recently shown that a controlled interaction with the environment may actually help to create a state, dubbed as “dark”, which is immune to decoherence. To encode quantum information in the dark states, they need to span a space with a dimensionality larger than one, so different orthogonal states act as a computational basis. Here, we devise a symmetry-based conceptual framework to engineer such degenerate dark spaces (DDS), protected from decoherence by the environment. We illustrate this construction with a model protocol, inspired by the fractional quantum Hall effect, where the DDS basis is isomorphic to a set of degenerate Laughlin states. The long-time steady state of our driven-dissipative model exhibits thus all the characteristics of degenerate vacua of a unitary topological system.

Nonlinearity-induced photonic topological insulator

A hallmark feature of topological insulators is robust edge transport that is impervious to scattering at defects and lattice disorder. We demonstrate a topological system, using a photonic platform, in which the existence of the topological phase is brought about by optical nonlinearity. The lattice structure remains topologically trivial in the linear regime, but as the optical power is increased above a certain power threshold, the system is driven into the topologically nontrivial regime. This transition is marked by the transient emergence of a protected unidirectional transport channel along the edge of the structure. Our work studies topological properties of matter in the nonlinear regime, providing a possible route for the development of compact devices that harness topological features in an on-demand fashion.