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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 64 , Issue 5 : 257011(2021) https://doi.org/10.1007/s11433-021-1675-0

Observation of antichiral edge states in a circuit lattice

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  • ReceivedJan 24, 2021
  • AcceptedJan 29, 2021
  • PublishedMar 24, 2021

PACS numbers

Abstract


Funded by

the National Natural Science Foundation of China(Grant,Nos.,11874274,12004425)

the Natural Science Foundation of Jiangsu Province(Grant,Nos.,BK20170058,BK20200630)

and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD)


Acknowledgment

We thank You Wang, Qiang Wang and Udvas Chattopadhyay from Nanyang Technological University for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11874274, and 12004425), the Natural Science Foundation of Jiangsu Province (Grant Nos. BK20170058, and BK20200630), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). YiDong Chong was supported by the Singapore MOE Academic Research Fund Tier 3 (Grant No. MOE2016-T3-1-006).


Supplement

Supporting Information

The supporting information is available online at http://phys.scichina.com and https://link.springer.com/journal/11433/. The supporting materials are published as submitted, without typesetting or editing. The responsibility for scientific accuracy and content remains entirely with the authors.


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  • Figure 1

    (Color online) Setup of the circuit. (a) Schematic of a modified Haldane mode. Each unit cell consists of sites A and B, with each site containing a pair of inductors. NN and NNN hoppings are indicated by black lines and green arrows. (b) Schematic of the braided capacitive couplings between inductors on different sites for NN hoppings (with zero phase) and NNN hoppings (with phase π/2), respectively. (c) Band diagram of a semi-infinite circuit lattice consisting of a strip 20 unit cells wide and infinitely long, with zigzag boundaries and circuit parameters L = 3.3 mH, C1 = 330 pF, and C2 = 33 pF. (d) Intensity distributions at wavenumber π/a for the two degenerate states at frequency 113.63 kHz (u2 and u3) and two representative bulk states at 105.08 and 124.28 kHz (u1 and u4). The antichiral edge states are localized to different edges but have the same group velocity.

  • Figure 2

    (Color online) Experimental characterization of bulk and edge states. (a) Photograph of a circuit corresponding to a lattice with periodic boundaries in x and zigzag boundaries in y. The 20 sites in the sample are explicitly numbered. Each site has an X or Y inductor (black cylinders), with NN and NNN hoppings implemented via capacitors (yellow components). (b) Experimentally measured voltage amplitudes. For the red curve, driving coils are placed on the X inductors at sites 2 and 12, and the pickup coil is placed on the Y inductor at site 1; the excitation and measurement thus occur along one edge of the effectively semi-infinite strip, and the peak at 111.1 kHz (red star) is close to the predicted eigenfrequency of the antichiral edge states. For the black curve, driving coils are placed on the X inductors at sites 1 and 4/6/8 (results are averaged over the three driving configurations), and the pickup coil is placed on the Y inductor at site 17; this serves as a probe of the spatially-averaged density of states, and the lack of a dip in the response shows the lack of a bulk band gap. (c) Circuit simulation results corresponding to (b). For the strip geometry, the response peaks at frequency 113.64 kHz, close to the experimental peak. (d) Voltage amplitudes measured at different sites, showing strong edge localization at 111.1 kHz (red), and no edge localization at frequency 117.6 kHz (blue). The driving coils are placed on the X inductors at sites 2 and 12, and the pickup coils are placed on the Y inductors at different sites.

  • Figure 3

    (Color online) Propagation of antichiral edge states in a finite lattice. (a) Photograph of the circuit boards implementing a 128-site lattice with open boundary conditions (left panel), and the schematic of the lattice (right panel). (b) Experimental results showing the voltage amplitude distribution at 112.7 kHz, produced by two driven coils with 90° phase difference (so as to excite spin-up states) placed on inductors X and Y at a corner site (marked by a black star). Red arrows indicate the expected propagation directions of the edge and bulk states. (c) Experimentally measured relative dwell times along the top edge (pink) and bottom edge (blue). The slopes of the linear least-squares fits correspond to group velocities of −0.045 sites/μs (top edge) and −0.042 sites/μs (bottom edge). (d) Relative dwell times obtained from corresponding circuit simulations. The slopes of the linear least-squares fits correspond to group velocities of −0.047 sites/μs (top edge) and −0.045 sites/μs (bottom edge).

  • Figure 4

    (Color online) Propagation of antichiral edge states in a Möbius strip. (a) Schematic of the Möbius strip circuit. Twisted electrical connections are applied to the left and right boundaries of the physical sample, so that the upper edge of the strip continues to the lower edge and vice versa. (b) Experimental results showing the voltage amplitude distribution at 116.4 kHz with spin up excitation. Red arrows indicate the propagation directions of the edge and bulk states. (c) Experimentally measured relative dwell times along the top edge (red) and bottom edge (blue). The slopes of the linear least-squares fits correspond to group velocities of −0.039 sites/μs (top edge) and −0.044 sites/μs (bottom edge).

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