# Octupole corner state in a three-dimensional topological circuit

August 26, 2020Topological phases of matter have been one of the research interests in the field of condensed matter physics due to its unique properties in designing fascinating materials possessing quantized invariants in both electronics and photonics systems, and have shown great potential in lasing, quantum computing platform, and robust signal transmission in optics, acoustic, and mechanical systems. While most of the research interests of topological insulators have focused on observation of protected nontrivial mode localized at the surface of a bulk material, recent emergence of higher-order topological insulators (HOTIs) has led to discoveries of topological boundary states with dimensions lower than that of the bulk by more than 1. These quantized higher order multipole corner states are localized at the intersection of edges of a square (2D, quadrupole moment) or cubic (3D, octupole moment) lattice, and are protected by specially designed spatial symmetries. So far, the study of HOTIs are mostly limited to 2D cases, and their corner states are either induced by the quadrupole moment or the 2D Zak phase of the bulk lattice.

In a new paper published in

*Light Science & Application*, a team of scientists, led by Professor Shuang Zhang from School of Physics and Astronomy, University of Birmingham, United Kingdom, Prof Tiejun Cui from State Key Laboratory of Millimeter Waves, Southeast University, Nanjing, China, Prof. Yuanjiang Xiang from School of Physics and Electronics, Hunan University, Changsha, China and co-workers have reported the experimental observation of 0D corner state in a three-dimensional (3D) topological circuit, which is built from a 3D cubic network of inductors and capacitors with deliberately designed values. They verify that such corner state is induced by the nontrivial octupole moment of the 3D circuit, and is topologically protected by three anticommuting reflection symmetries of the bulk lattice. This is achieved by engineering the dimerized coupling in each smallest loop (plaquette) in the circuit to have opposite sign to the other three, making this circuit a cubic lattice version of the famous Hofstadter model with π-flux per plaquette. 'This is critical for generating a synthetic magnetic π-flux threading the plaquette that finally gives the octuple corner state in the finite-sized system.' they emphasized.

The topological features of the circuit were analyzed from the band structures of the circuit with both infinite and finite boundary conditions. This was achieved by constructing the circuit Laplacian and circuit Hamiltonian of the circuit based on the Kirchhoff law. They found an isolated midgap mode in the band gap of the finite band structure, which is the octupole corner state that is localized at the corner of the cubic circuit. To verify their theoretical prediction, they fabricated a sample that comprises of 2.5×2.5×2.5 unit cells (5×5×5 nodes) using five layers of circuit board, and measured the impedance spectra between every adjacent circuit node using a vector network analyzer (VNA). A distinct peak was clearly identified from the impedance spectrum at one of the circuit corners at exactly the corner mode frequency (2.77MHz), which was confirmed to be the octupole corner state they expected. The experimental results were in good agreement with the theoretical calculations for the impedance spectra at all the circuit nodes. To theoretically confirm the topology of the corner state observed in the simulation and experiment, they calculated the topological invariant of the circuit through a series of procedures called the nested Wilson loops, and obtained a quantized value of 1/2 and 0, which correspond to the nontrivial and trivial states, respectively.

'Similar to the 1D edge state (2D surface state) in conventional 2D (3D) topological materials, which exhibits excellent immunity against defects and disorder, the 0D corner state in our HOTI circuit is also highly robust against certain types of disorder.' To evaluate the robustness of the octupole corner state, they provided the statistical distribution of the frequency of the corner state and the bandgap of the bulk from a number of disordered systems with different levels of variations in the circuit components. It was observed that the level of frequency shift of the corner mode is proportional to the randomness of the component variation, but its peak persists even at 20% circuit component variation. Further analyses were also performed to reveal the relationship between the bandgap and robustness of the corner state under different level of component disorder.

'The successful realization of octupole topological insulators paves the way for future investigations of higher-dimensional topological insulators possessing multipole moments without introducing synthetic dimensions, benefitting from the convenient electrical connections among nodes at arbitrary distances.' The authors also mentioned that this work may provide an experimental platform for further investigation of 3D higher-order topological circuit combined with non-Hermitian and nonlinear effects with the employment of active and nonlinear circuit devices such as operational amplifiers and varactor diodes.

-end-

Light Publishing Center, Changchun Institute of Optics, Fine Mechanics And Physics, Chinese Academy

## Related Topological Insulators Articles from Brightsurf:

Tunable THz radiation from 3D topological insulator

Wu's research group has been investigating a three-dimensional topological insulator of bismuth telluride (Bi2Te3) as a promising basis for an effective THz system.

Knotting semimetals in topological electrical circuits

Scientists created exotic states of matter using electrical circuit enhanced by machine-learning algorithm

Penn engineers create helical topological exciton-polaritons

Researchers at the University of Pennsylvania's School of Engineering and Applied Science are the first to create an even more exotic form of the exciton-polariton, one which has a defined quantum spin that is locked to its direction of motion.

Bridging the gap between the magnetic and electronic properties of topological insulators

Scientists at Tokyo Institute of Technology shed light on the relationship between the magnetic properties of topological insulators and their electronic band structure.

Topological superconducting phase protected by 1D local magnetic symmetries

Scientists from China and USA classified 1D gapped topological superconducting quantum wires with local magnetic symmetries (LMSs), in which the time-reversal symmetry is broken but its combinations with certain crystalline symmetries, such as MxT, C2zT, C4zT, and C6zT, are preserved.

Octupole corner state in a three-dimensional topological circuit

Higher-order topological insulators featuring quantized bulk polarizations and zero-dimensional corner states are attracting increasing interest due to their strong mode confinement.

Quantum simulation for 3D chiral topological phase

Professor Liu at PKU, Professor Du and Professor Wang at USTC build up a quantum simulator using nitrogen-vacancy center to investigate a three-dimensional (3D) chiral topological insulator which was not realized in solid state system, and demonstrate a complete study of both the bulk and surface topological physics by quantum quenches.

Photonic amorphous topological insulator

The current understanding of topological insulators and their classical wave analogues, such as photonic topological insulators, is mainly based on topological band theory.

Recent advances in 2D, 3D and higher-order topological photonics

A research team from South Korea and the USA has provided a comprehensive review covering the recent progress in topological photonics, a recently emerging branch of photonics.

Synthetic dimensions enable a new way to construct higher-order topological insulators

Higher-order topological insulators (HOTIs) are a new phase of matter predicted in 2017, involving complicated high-dimensional structures which show signature physical effects called ''corner modes.'' Now, scientists have proposed a recipe to construct such HOTIs and observe corner modes for photons in simpler, lower-dimensional structures by harnessing an emerging concept called ''synthetic dimensions.'' This construction allows flexible tuning of the topological behavior and opens avenues for even more exotic phases of photons in very high dimensions.

Read More: Topological Insulators News and Topological Insulators Current Events

Wu's research group has been investigating a three-dimensional topological insulator of bismuth telluride (Bi2Te3) as a promising basis for an effective THz system.

Knotting semimetals in topological electrical circuits

Scientists created exotic states of matter using electrical circuit enhanced by machine-learning algorithm

Penn engineers create helical topological exciton-polaritons

Researchers at the University of Pennsylvania's School of Engineering and Applied Science are the first to create an even more exotic form of the exciton-polariton, one which has a defined quantum spin that is locked to its direction of motion.

Bridging the gap between the magnetic and electronic properties of topological insulators

Scientists at Tokyo Institute of Technology shed light on the relationship between the magnetic properties of topological insulators and their electronic band structure.

Topological superconducting phase protected by 1D local magnetic symmetries

Scientists from China and USA classified 1D gapped topological superconducting quantum wires with local magnetic symmetries (LMSs), in which the time-reversal symmetry is broken but its combinations with certain crystalline symmetries, such as MxT, C2zT, C4zT, and C6zT, are preserved.

Octupole corner state in a three-dimensional topological circuit

Higher-order topological insulators featuring quantized bulk polarizations and zero-dimensional corner states are attracting increasing interest due to their strong mode confinement.

Quantum simulation for 3D chiral topological phase

Professor Liu at PKU, Professor Du and Professor Wang at USTC build up a quantum simulator using nitrogen-vacancy center to investigate a three-dimensional (3D) chiral topological insulator which was not realized in solid state system, and demonstrate a complete study of both the bulk and surface topological physics by quantum quenches.

Photonic amorphous topological insulator

The current understanding of topological insulators and their classical wave analogues, such as photonic topological insulators, is mainly based on topological band theory.

Recent advances in 2D, 3D and higher-order topological photonics

A research team from South Korea and the USA has provided a comprehensive review covering the recent progress in topological photonics, a recently emerging branch of photonics.

Synthetic dimensions enable a new way to construct higher-order topological insulators

Higher-order topological insulators (HOTIs) are a new phase of matter predicted in 2017, involving complicated high-dimensional structures which show signature physical effects called ''corner modes.'' Now, scientists have proposed a recipe to construct such HOTIs and observe corner modes for photons in simpler, lower-dimensional structures by harnessing an emerging concept called ''synthetic dimensions.'' This construction allows flexible tuning of the topological behavior and opens avenues for even more exotic phases of photons in very high dimensions.

Read More: Topological Insulators News and Topological Insulators Current Events

Brightsurf.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com.