I was given an interesting paper to read by a theorist colleague who works in the area of non-Hermitian systems, in response to my question asking for an example of a real-world system where non-Hermitian phenomena can be observed.
The article describes a 1D lattice of LC resonators that are coupled via opamp voltage followers.
The paper (as do others on non-Hermetian circuit lattices (cf. Kaifa Luo et al. (2018) Topological Nodal States in Circuit Lattice) uses the concept of "circuit Laplacian" and "circuit Hamiltonian" without really explaining what those mean, and I've never been exposed to these concepts. So I'd like to ask for help pointing me to some basic references that explain how to obtain these.
Question: How does one go about writing the Laplacian and Hamiltonian of an electrical circuit containing active and passive components?
ABSTRACT: The conventional bulk-boundary correspondence directly connects the number of topological edge states in a finite system with the topological invariant in the bulk band structure with periodic boundary condition (PBC). However, recent studies show that this principle fails in certain non-Hermitian systems with broken reciprocity, which stems from the non-Hermitian skin effect (NHSE) in the finite system where most of the eigenstates decay exponentially from the system boundary. In this work, we experimentally demonstrate a 1D non-Hermitian topological circuit with broken reciprocity by utilizing the unidirectional coupling feature of the voltage follower module. The topological edge state is observed at the boundary of an open circuit through an impedance spectra measurement between adjacent circuit nodes. We confirm the inapplicability of the conventional bulk-boundary correspondence by comparing the circuit Laplacian between the periodic boundary condition (PBC) and open boundary condition (OBC). Instead, a recently proposed non-Bloch bulk-boundary condition based on a non-Bloch winding number faithfully predicts the number of topological edge states.
Figure 1 Schematic and circuit diagram of the nonreciprocal non-Hermitian circuit. (a) Schematic of the nonreciprocal non-Hermitian model in the electronic system. The dashed line outlines a single unit cell. (b) Circuit implementation of the nonreciprocal non-Hermitian model. (c) Circuit diagram of the voltage follower module. (d) Real PCB layout of one unit cell of the nonreciprocal non-Hermitian circuit.
Figure 3 Experimental and numerical results for the topological edge mode. (a) Fabricated sample of the finite circuit chain containing 9.5 unit cells (19 nodes), with circuit parameters C0 = 470 pF, C1 = 470 pF, C2 = 1000 pF, C2 = 1000 pF, C3 = C4 = 680 pF, and L0 = 47 uH, corresponding to t1 = 1.72, t2 = 2.85, and γ1 = γ2 = 1.45. (b) Sorted eigenfrequencies of the finite circuit chain. Red circle represents the topological edge mode. (c) Imaginary part of the eigenvalue spectra of J(ω) for the finite circuit chain. The isolated curve represents the edge mode. (d, e) Experimentally measured and numerically calculated impedance spectra of the finite circuit chain, respectively. Red curve indicates the impedance measured across the leftmost coupling capacitors. (f, g) Experimentally measured and numerically calculated impedance distributions of the finite circuit chain. Note that the impedance spectra in all experiments and simulations are measured across the each adjacent node (i.e., across each coupling capacitor).
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This post imported from StackExchange Physics at 2025-04-08 17:13 (UTC), posted by SE-user uhoh
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