Square root topological lattice models such as the non-Hermitian 1D Su-Schrieffer-Heeger (SSH) model [1] and the 2D honeycomb lattice [2, 3] have recently been introduced and realised in several platforms [4, 6]. The recipe for the square root construction has paved the way for 2n root models [7], in which the transition from higher to lower roots can be readily made due to the arborescent structure of the resultant lattice by taking the n-th power of the corresponding Hamiltonian ((Hroot)n) and subtracting a constant energy. The present method is only applicable for the square root operation. Here, we introduce a new procedure which enables us to realise any arbitrary n-th root of the lattice, including prime number roots, by simply adding n-1 new sites between two parental sites such that the new directional couplings create a closed loop in either the clockwise or anti-clockwise directions (Fig. 1a). While the Hermiticity is preserved in square root operations, the non-reciprocal couplings in the n-th root method require breaking the Hermiticity so that the arborescent construction can be followed to form (Hroot)n as the direct sum of Hparental and Hresidual where Hparental is the original lattice Hamiltonian, which is topologically non-trivial [8], and Hresidual represents the direct sum of all the residual parts. This form allows us to transition between roots (Fig. 1b) without the need for a constant energy shift in every operation (Fig. 2). Every root operation results in a multiplication of the energy band structure, which depends on the degree of the root. This multiplication in energy also results in the multiplication of the topological states. In contrast to conventional TIs, the n-th root version of the topological lattices possesses multiple topological mid-gap states with finite energy as well as non-Hermitian phenomenon such as the non-Hermitian skin effect, which may be useful for various potential applications such as magnetic sensing due to the extensive sensitivity and robustness of the topological states.

![](https://assets.underline.io/uploads/markdown_image/1/image/7985dfc21e709bda9aa9472f8edf38c1.jpg)

Illustration of systematic shifting between (SSH)1/n lattices.

![](https://assets.underline.io/uploads/markdown_image/1/image/c228455f335d23670adca71013893745.jpg)

Real energy spectrum as a function of site index of cubic root (3n) SSH model with 1 unit cell, the transition from n=3 to n=0.

N-th Root Topological Lattices

Topolectrical (TE) circuits have not only bridged the seemingly different disciplines of electrical circuits and topological condensed matter but also allowed the realization of highly tunable lattice systems with desired topological properties based on the parameters of the corresponding TE circuit lattices [1, 2]. The frequency dependence of the electrical inter-node couplings (i.e., the admittance of capacitors and inductors) suggests that the circuits can be switched between the insulating, semimetallic, and Chern insulator phases by varying the frequency of the alternating signal [3]. Here, we propose a TE circuit lattice that exhibits this switching within a single circuit set-up. We consider a 2-dimensional (2D) TE circuit containing two types of nodes A and B (Fig. 1a) with complex nearest-neighbor couplings (combination of capacitive and resistive) realized via negative impedance converters (NICs) (Fig. 1b) and capacitive next-nearest neighbor (NNN) couplings. The critical frequency of the circuit depends only on the grounding inductor and NNN capacitor (Fig. 1c). In the absence of resistive coupling, the TE circuit respects time-reversal symmetry (TRS) and hosts insulating and semimetallic phases when the frequency is less than, and greater or equal to the critical frequency, respectively. Furthermore, the 2D Weyl points in the semimetallic phase are connected by topological edge states in the admittance spectra (Fig. 2b,e). However, a non-zero Rd breaks TRS and a bulk bandgap appears. Chiral edge states emerge under open boundary conditions and the circuit exhibits the quantum anomalous Hall effect at ω &gt; ωc (Fig. 2f). Moreover, the Chern number of the system changes signs when the frequency crosses the critical value.

![](https://assets.underline.io/uploads/markdown_image/1/image/335a98dd245cf287b314d00c26f10710.jpg)

a. Schematic of frequency-dependent Chern circuit. b. NIC and grounding connections. c. System Hamiltonian and critical frequency.

![](https://assets.underline.io/uploads/markdown_image/1/image/c73aa987cfe48dc4e49ba3714769b396.jpg)

a-f. Admittance dispersion of a finite system at different frequencies. The top and bottom rows correspond to zero and non-zero resistive couplings, respectively. Common parameters: C1=1.5 mF, C2=1 mF and Lm=1 nH.

AC Frequency Modulated Phase Transitions in Topolectrical Circuits

d-dimensional higher-order topological systems (TS) possess boundary states on (d−n)-dimensional interfaces where n≥2 [1, 2]. Here, we demonstrate a highly tunable 3-D second-order Dirac semimetal (SODSM) phase on a topolectrical (TE) circuit lattice comprising inductors and capacitors (Fig. 1) [3]. We first construct a 2-D second-order TS on the x−y plane with four types of sublattice nodes (Fig. 1a). Electrical couplings with negative signs are achieved via a negative impedance converter (NIC) (Fig. 1d). To induce SODSMs, we extend the TE lattice along the z-direction by stacking the circuit layers in the x−y plane via the interlayer capacitor Cz, which modifies the effective intercell hopping and hence the locations of the gapless points. The presence of both inversion and time-reversal symmetries ensures the existence of pairs of gapless points. The admittance band dispersion at resonant frequency hosts a pair of Dirac points (DPs) along the kz axis under periodic boundary conditions (PBC) along all directions (Fig. 2a). Moreover, two first-order DPs connected by Fermi arc surface edge states emerge under open-boundary conditions (OBC) in the x-direction and PBC in the y and z directions. Interestingly, second-order flat hinge-like states appear in the admittance bandgap and connect the two DPs in the SODSM when PBC is imposed in the z-direction and OBC in the x and y directions (Fig. 2c). Furthermore, the density of states (DOS) vanishes in the vicinity of the DPs. The DPs can be traced via the impedance readout between two terminal nodes. Large impedance peaks exist only near the locations of the DPs, and the impedance falls sharply to nearly zero at other points with finite DOS (Fig. 2d).

![](https://assets.underline.io/uploads/markdown_image/1/image/9ea35d73e7f491a4a7072356e5aa838a.jpg)

Schematic of SODSM TE circuit. Cross-section of TE circuit on a. x-y plane, b. x-z plane, c. y-z plane. d. Schematic of NIC using an op-amp. e. Grounding mechanisms.

![](https://assets.underline.io/uploads/markdown_image/1/image/f92e487050814f6b673b3db6dcaf22ef.jpg)

a. Admittance dispersion under PBC in all directions. b. First-order DPs under PBC in y and z directions and OBC in the z-direction. c. SODSM under PBC in z-direction and OBC in the x and y directions. d. Impedance reading of SODSM.

A Circuit Lattice Representation of a Second-Order Dirac Semimetal

**Session chairs:** Eleanor Clements, National Institute of Standards and Technology and Roberto Zivieri University of Ferrara



AOB - Fundamental Properties and Cooperative Phenomena II

technical paper

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This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

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Haydar Sahin

3

3

SHORT BIO

Presentations

N-th Root Topological Lattices

AC Frequency Modulated Phase Transitions in Topolectrical Circuits

A Circuit Lattice Representation of a Second-Order Dirac Semimetal

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