Boundary states of 4D topological matter: Emergence and full 3D-imaging of nodal Seifert surfaces

Abstract

With its boundary tracing out a link or knot in 3D, the Seifert surface is a 2D surface of core importance to topological classification. We propose the first-ever experimentally realistic setup where Seifert surfaces emerge as the boundary states of 4D topological matter. Unlike ordinary real space knots that exist in polymers, biomolecules and everyday life, our knots and their Seifert surfaces exist as momentum space nodal structures, where topological linkages have profound effects on optical and transport phenomena. Realized with 4D circuit lattices, our nodal Seifert systems are freed from symmetry constraints and readily tunable due to the dimension and distance agnostic nature of circuit connections. Importantly, their Seifert surfaces manifest as very pronounced impedance peaks in their 3D-imaging via impedance measurements, and are directly related to knot invariants like the Alexander polynomial and knot Signature. This work thus unleashes the great potential of Seifert surfaces as sophisticated yet accessible mathematical tools in the study of exotic band structures.

19 pages, 14 figures

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Cited by in corpus (27)

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• Experimental Observation of Higher-Order Topological Anderson Insulators
• Circuit Implementation of a Four-Dimensional Topological Insulator
• Topolectric circuits: Theory and construction
• Topolectrical-circuit realization of 4D hexadecapole insulator
• Quantized classical response from spectral winding topology
• Dimensional transmutation from non-Hermiticity
• Zoology of non-Hermitian spectra and their graph topology
• Classification of Exceptional Nodal Topologies Protected by $\mathcal{P}\mathcal{T}$ Symmetry
• Stabilizing multiple topological fermions on a quantum computer
• Impedance responses and size-dependent resonances in topolectrical circuits via the method of images
• What's knot to like? Observation of a linked loop quantum state
• Enhanced higher harmonic generation from nodal topology
• Anomalous fractal scaling in two-dimensional electric networks
• Activating non-Hermitian skin modes by parity-time symmetry breaking
• The Bulk-boundary Correspondence in Non-Hermitian Hopf-link Exceptional Line Semimetals
• Moiré circuits: engineering magic-angle behaviors
• Topological corner modes in a brick lattice with nonsymmorphic symmetry
• Generalization of Benalcazar-Bernevig-Hughes model to arbitrary dimensions
• Engineering topological states and quantum-inspired information processing using classical circuits
• Superfluid Vortices in Four Spatial Dimensions
• Anisotropic non-linear optical response of nodal loop materials
• Observation of non-local impedance response in a passive electrical circuit
• Topological Electromagnetic Effects and Higher Second Chern Numbers in Four-Dimensional Gapped Phases
• Curved vortex surfaces in four-dimensional superfluids: II. Equal-frequency double rotations
• Curved vortex surfaces in four-dimensional superfluids: I. Unequal-frequency double rotations