Observation of novel topological states in hyperbolic lattices

Hyperbolic Non-Abelian Semimetal

We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in a negatively curved plane. Because of their distinct translation group structure, such lattices are associated with a high-dimensional reciprocal space. In addition, they support non-Abelian Bloch states which, unlike conventional Bloch states, acquire a matrix-valued Bloch factor under lattice translations. Combining diverse numerical and analytical approaches, we uncover an unconventional scaling in the density of states at low energies, and illuminate a nodal manifold of codimension five in the reciprocal space. The nodal manifold is topologically protected by a nonzero second Chern number, reminiscent of the characterization of Weyl nodes by the first Chern number.

Higher-order topological phases in crystalline and non-crystalline systems: a review

In recent years, higher-order topological phases have attracted great interest in various fields of physics. These phases have protected boundary states at lower-dimensional boundaries than the conventional first-order topological phases due to the higher-order bulk-boundary correspondence. In this review, we summarize current research progress on higher-order topological phases in both crystalline and non-crystalline systems. We firstly introduce prototypical models of higher-order topological phases in crystals and their topological characterizations. We then discuss effects of quenched disorder on higher-order topology and demonstrate disorder-induced higher-order topological insulators. We also review the theoretical studies on higher-order topological insulators in amorphous systems without any crystalline symmetry and higher-order topological phases in non-periodic lattices including quasicrystals, hyperbolic lattices, and fractals, which have no crystalline counterparts. We conclude the review by a summary of experimental realizations of higher-order topological phases and discussions on potential directions for future study.

Anomalous and Chern topological waves in hyperbolic networks

Hyperbolic lattices are a new type of synthetic materials based on regular tessellations in non-Euclidean spaces with constant negative curvature. While so far, there has been several theoretical investigations of hyperbolic topological media, experimental work has been limited to time-reversal invariant systems made of coupled discrete resonances, leaving the more interesting case of robust, unidirectional edge wave transport completely unobserved. Here, we report a non-reciprocal hyperbolic network that exhibits both Chern and anomalous chiral edge modes, and implement it on a planar microwave platform. We experimentally evidence the unidirectional character of the topological edge modes by direct field mapping. We demonstrate the topological origin of these hyperbolic chiral edge modes by an explicit topological invariant measurement, performed from external probes. Our work extends the reach of topological wave physics by allowing for backscattering-immune transport in materials with synthetic non-Euclidean behavior.

Observation of a Higher-Order End Topological Insulator in a Real Projective Lattice

The modern theory of quantized polarization has recently extended from 1D dipole moment to multipole moment, leading to the development from conventional topological insulators (TIs) to higher-order TIs, i.e., from the bulk polarization as primary topological index, to the fractional corner charge as secondary topological index. The authors here extend this development by theoretically discovering a higher-order end TI (HOETI) in a real projective lattice and experimentally verifying the prediction using topolectric circuits. A HOETI realizes a dipole-symmetry-protected phase in a higher-dimensional space (conventionally in one dimension), which manifests as 0D topologically protected end states and a fractional end charge. The discovered bulk-end correspondence reveals that the fractional end charge, which is proportional to the bulk topological invariant, can serve as a generic bulk probe of higher-order topology. The authors identify the HOETI experimentally by the presence of localized end states and a fractional end charge. The results demonstrate the existence of fractional charges in non-Euclidean manifolds and open new avenues for understanding the interplay between topological obstructions in real and momentum space.

Hyperbolic photonic topological insulators

Topological photonics provides a new degree of freedom to robustly control electromagnetic fields. To date, most of established topological states in photonics have been employed in Euclidean space. Motivated by unique properties of hyperbolic lattices, which are regular tessellations in non-Euclidean space with a constant negative curvature, the boundary-dominated hyperbolic topological states have been proposed. However, limited by highly crowded boundary resonators and complicated site couplings, the hyperbolic topological insulator has only been experimentally constructed in electric circuits. How to achieve hyperbolic photonic topological insulators is still an open question. Here, we report the experimental realization of hyperbolic photonic topological insulators using coupled ring resonators on silicon chips. Boundary-dominated one-way edge states with pseudospin-dependent propagation directions have been observed. Furthermore, the robustness of edge states in hyperbolic photonic topological insulators is also verified. Our findings have potential applications in the field of designing high-efficient topological photonic devices with enhanced boundary responses.

Non-Abelian Topological Bound States in the Continuum

Bound states in the continuum (BICs), which are spatially localized states with energies lying in the continuum of extended modes, have been widely investigated in both quantum and classical systems. Recently, the combination of topological band theory with BICs has led to the creation of topological BICs that exhibit extraordinary robustness against disorder. However, the previously proposed topological BICs are only limited in systems with Abelian gauge fields. Whether non-Abelian gauge fields can induce topological BICs and how to experimentally explore these phenomena remains unresolved. Here, we report the theoretical and experimental realization of non-Abelian topological BICs, which are generated by the interplay between two inseparable pseudospins and can coexist in each pseudospin subspace. This unique characteristic necessitates non-Abelian couplings that lack any Abelian counterparts. Furthermore, the non-Abelian couplings can also offer a new avenue for constructing topological subspace-induced BICs at bulk dislocations. Those exotic phenomena are observed by non-Abelian topolectrical circuits. Our results establish the connection between topological BICs and non-Abelian gauge fields, and serve as the catalyst for future investigations on non-Abelian topological BICs across different platforms.

Non-Abelian Hyperbolic Band Theory from Supercells

Wave functions on periodic lattices are commonly described by Bloch band theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic lattices support non-Abelian Bloch states that have so far eluded analytical treatments. By adapting the solid-state-physics notions of supercells and zone folding, we devise a method for the systematic construction of non-Abelian Bloch states. The method applies Abelian band theory to sequences of supercells, recursively built as symmetric aggregates of smaller cells, and enables a rapidly convergent computation of bulk spectra and eigenstates for both gapless and gapped tight-binding models. Our supercell method provides an efficient means of approximating the thermodynamic limit and marks a pivotal step toward a complete band-theoretic characterization of hyperbolic lattices.

Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations

Tessellations of the hyperbolic spaces by regular polygons support discrete quantum and classical models with unique spectral and topological characteristics. Resolving the true bulk spectra and the thermodynamic response functions of these models requires converging periodic boundary conditions and our Letter delivers a practical and rigorous solution for this open problem on generic {p,q}-tessellations. This enables us to identify the true spectral gaps of bulk Hamiltonians and construct all but one topological models that deliver the topological gaps predicted by the K theory of the lattices. We demonstrate the emergence of the expected topological spectral flows whenever two such bulk models are deformed into each other and prove the emergence of topological channels whenever a soft physical interface is created between different topological classes of Hamiltonians.

Non-Abelian Inverse Anderson Transitions

Inverse Anderson transitions, where the flat-band localization is destroyed by disorder, have been wildly investigated in quantum and classical systems in the presence of Abelian gauge fields. Here, we report the first investigation on inverse Anderson transitions in the system with non-Abelian gauge fields. It is found that pseudospin-dependent localized and delocalized eigenstates coexist in the disordered non-Abelian Aharonov-Bohm cage, making inverse Anderson transitions depend on the relative phase of two internal pseudospins. Such an exotic phenomenon induced by the interplay between non-Abelian gauge fields and disorder has no Abelian analogy. Furthermore, we theoretically design and experimentally fabricate non-Abelian Aharonov-Bohm topolectrical circuits to observe the non-Abelian inverse Anderson transition. Through the direct measurements of frequency-dependent impedance responses and voltage dynamics, the pseudospin-dependent non-Abelian inverse Anderson transitions are observed. Our results establish the connection between inverse Anderson transitions and non-Abelian gauge fields, and thus comprise a new insight on the fundamental aspects of localization in disordered non-Abelian flat-band systems.

Non-Hermitian Topolectrical Circuit Sensor with High Sensitivity

Electronic sensors play important roles in various applications, such as industry and environmental monitoring, biomedical sample ingredient analysis, wireless networks and so on. However, the sensitivity and robustness of current schemes are often limited by the low quality-factors of resonators and fabrication disorders. Hence, exploring new mechanisms of the electronic sensor with a high-level sensitivity and a strong robustness is of great significance. Here, a new way to design electronic sensors with superior performances based on exotic properties of non-Hermitian topological physics is proposed. Owing to the extreme boundary-sensitivity of non-Hermitian topological zero modes, the frequency shift induced by boundary perturbations can show an exponential growth trend with respect to the size of non-Hermitian topolectrical circuit sensors. Moreover, such an exponential growth sensitivity is also robust against disorders of circuit elements. Using designed non-Hermitian topolectrical circuit sensors, the ultrasensitive identification of the distance, rotation angle, and liquid level is further experimentally verified with the designed capacitive devices. The proposed non-Hermitian topolectrical circuit sensors can possess a wide range of applications in ultrasensitive environmental monitoring and show an exciting prospect for next-generation sensing technologies.

Hyperbolic band topology with non-trivial second Chern numbers

Topological band theory establishes a standardized framework for classifying different types of topological matters. Recent investigations have shown that hyperbolic lattices in non-Euclidean space can also be characterized by hyperbolic Bloch theorem. This theory promotes the investigation of hyperbolic band topology, where hyperbolic topological band insulators protected by first Chern numbers have been proposed. Here, we report a new finding on the construction of hyperbolic topological band insulators with a vanished first Chern number but a non-trivial second Chern number. Our model possesses the non-abelian translational symmetry of {8,8} hyperbolic tiling. By engineering intercell couplings and onsite potentials of sublattices in each unit cell, the non-trivial bandgaps with quantized second Chern numbers can appear. In experiments, we fabricate two types of finite hyperbolic circuit networks with periodic boundary conditions and partially open boundary conditions to detect hyperbolic topological band insulators. Our work suggests a new way to engineer hyperbolic topological states with higher-order topological invariants.

Hyperbolic matter in electrical circuits with tunable complex phases

Curved spaces play a fundamental role in many areas of modern physics, from cosmological length scales to subatomic structures related to quantum information and quantum gravity. In tabletop experiments, negatively curved spaces can be simulated with hyperbolic lattices. Here we introduce and experimentally realize hyperbolic matter as a paradigm for topological states through topolectrical circuit networks relying on a complex-phase circuit element. The experiment is based on hyperbolic band theory that we confirm here in an unprecedented numerical survey of finite hyperbolic lattices. We implement hyperbolic graphene as an example of topologically nontrivial hyperbolic matter. Our work sets the stage to realize more complex forms of hyperbolic matter to challenge our established theories of physics in curved space, while the tunable complex-phase element developed here can be a key ingredient for future experimental simulation of various Hamiltonians with topological ground states.

Realization of a Hopf Insulator in Circuit Systems

Three-dimensional (3D) two-band Hopf insulators are a paradigmatic example of topological phases beyond the topological classifications based on powerful methods like K theory and symmetry indicators. Since this class of topological insulating phases was theoretically proposed in 2008, they have attracted significant interest owing to their conceptual novelty, connection to knot theory, and many fascinating physical properties. However, because their realization requires special forms of long-range spin-orbit coupling, they have not been achieved in any 3D system yet. Here, we report the first experimental realization of the long-sought-after Hopf insulator in a 3D circuit system. To implement the Hopf insulator, we construct basic pseudospin modules and connection modules that can realize 2×2-matrix elements and then design the circuit network according to a tight-binding Hopf insulator Hamiltonian constructed by the Hopf map. By simulating the band structure of the designed circuit network and calculating the Hopf invariant, we find that the circuit realizes a Hopf insulator with Hopf invariant equaling 4. Experimentally, we measure the band structure of a printed circuit board and find the observed properties of the bulk bands and topological surface states are in good agreement with the theoretical predictions, verifying the bulk-boundary correspondence of the Hopf insulator. Our scheme brings the experimental study of Hopf insulators to reality and opens the door to the implementation of more unexplored topological phases beyond the known topological classifications.

Hyperbolic Topological Band Insulators

Recently, hyperbolic lattices that tile the negatively curved hyperbolic plane emerged as a new paradigm of synthetic matter, and their energy levels were characterized by a band structure in a four- (or higher-) dimensional momentum space. To explore the uncharted topological aspects arising in hyperbolic band theory, we here introduce elementary models of hyperbolic topological band insulators: the hyperbolic Haldane model and the hyperbolic Kane-Mele model; both obtained by replacing the hexagonal cells of their Euclidean counterparts by octagons. Their nontrivial topology is revealed by computing topological invariants in both position and momentum space. The bulk-boundary correspondence is evidenced by comparing bulk and boundary density of states, by modeling propagation of edge excitations, and by their robustness against disorder.