4D spinless topological insulator in a periodic electric circuit

TL;DR

This paper demonstrates how a 2D electric circuit can simulate a 4D spinless topological insulator, revealing topological boundary states and offering a platform for exploring higher-dimensional topological phenomena.

Contribution

The authors construct a 2D electric circuit model that simulates a 4D spinless topological insulator, highlighting the realization of higher-dimensional topological states in synthetic systems.

Findings

Resonance spectrum shows pairs of 3D Weyl boundary states

Boundary states are protected by classical time-reversal symmetry

Simulation of 4D topological band structure in a 2D circuit

Abstract

According to the mathematical classification of topological band structures, there exist a number of fascinating topological states in dimensions larger than three with exotic boundary phenomena and interesting topological responses. While these topological states are not accessible in condensed matter systems, recent works have shown that synthetic systems, such as photonic crystals or electric circuits, can realize higher-dimensional band structures. Here, we argue that the 4D spinless topological insulator, due to its symmetry properties, is particularly well suited to be implemented in these synthetic systems. We explicitly construct a 2D electric circuit lattice, whose resonance frequency spectrum simulate the 4D spinless topological insulator. We perform detailed numerical calculations of the circuit lattice and show that the resonance frequency spectrum exhibit pairs of 3D Weyl boundary states, a hallmark of the nontrivial topology. These pairs of 3D Weyl states with the same chirality are protected by classical time-reversal symmetry that squares to $+1$, which is inherent in the proposed circuit lattice. We also discuss how the simulated 4D topological band structure can be observed in experiments.

Equations

$$\mathcal{H}(\mathbf{k})=\sum _{a=0}^5{f}_a(\mathbf{k}){\gamma }_a.$$

$$C_2=\frac{3}{8{\pi }^3}\int d^{4}k\text{leavevmode}{ϵ}^{\mu \nu \lambda \rho \sigma }{\hat{f}}_{\mu }{\partial }_{k_1}{\hat{f}}_{\nu }{\partial }_{k_2}{\hat{f}}_{\lambda }{\partial }_{k_3}{\hat{f}}_{\rho }{\partial }_{k_4}{\hat{f}}_{\sigma },$$

$${\mathcal{H}}_s(\tilde{\mathbf{k}})=f_0(\tilde{\mathbf{k}}){\sigma }_0-f_3(\tilde{\mathbf{k}}){\sigma }_1+f_4(\tilde{\mathbf{k}}){\sigma }_2+f_5(\tilde{\mathbf{k}}){\sigma }_3.$$

Full text

Introduction

With the great success of topological band theory in condensed matter physics Hasan and Kane (2010); Bernevig and Hughes (2013); Qi and Zhang (2011); Chiu et al. (2016); Armitage et al. (2018); Burkov (2018), recent research has branched out to the study of topological bands in synthetic lattices, such as, photonic crystals Lu et al. (2014); Rechtsman et al. (2013); Yan et al. (2018); Ozawa et al. (2019), ultracold atomic gases Goldman et al. (2010); Sun et al. (2012); Jotzu et al. (2014); Aidelsburger et al. (2015); Goldman et al. (2016); Zhang et al. (2018), and electric circuit networks Ningyuan et al. (2015); Albert et al. (2015); Lee et al. (2018); Luo et al. (2018a); Lu et al. (2019); Goren et al. (2018); Hofmann et al. (2018); Imhof et al. (2018); Hadad et al. (2018); Ezawa (2018); Serra-Garcia et al. (2019); Luo et al. (2018b); Li et al. (2019); Haenel et al. (2019); Eisenberg et al. (2019); Zhu et al. (2019); Liu et al. (2019); Helbig et al. (2019). These synthetic lattices have several advantages compared to their condensed matter counterparts. One is the ability to precisely control and manipulate the band structure, another is the possibility to create lattices in dimensions greater than three. The celebrated ten-fold classification of topological materials Kitaev (2009); Schnyder et al. (2008); Ryu et al. (2010); Zhao and Wang (2013, 2014) predicts a number of interesting higher-dimensional topological states, including four-dimensional (4D) topological insulators Schnyder et al. (2008); Qi et al. (2008), 4D topological superconductors, and a 4D generalization of the integer quantum Hall effect Zhang and Hu (2001). These 4D topological states exhibit many interesting phenomena, e.g., quantized nonlinear responses Zhang and Hu (2001); Froehlich and Pedrini ; Qi et al. (2008); Ryu et al. (2012); Lohse et al. (2018); Price et al. (2015), topological charge pumping, and in-gap boundary modes with protected level crossings Kraus et al. (2013). Unfortunately, these 4D states cannot be realized in condensed matter systems, which are limited to three spatial dimensions. However, recent technological advances in photonics and cold atomic gases have allowed to synthetically engineer the 4D integer quantum Hall effect, using, e.g., internal degrees of freedom as additional effective dimensions Price et al. (2015); Price (2018); Ozawa et al. (2016); Lohse et al. (2018); Petrides et al. (2018); Zilberberg et al. (2018); Kraus et al. (2013). These experiments have revealed signatures of charge pumping and topological transport Lohse et al. (2018); Zilberberg et al. (2018). Apart from these works, there has been no other experimental investigation of the 4D integer quantum Hall effect, and likewise no other 4D topological state has yet been realized experimentally. Among the five 4D topological states of the ten-fold classification Kitaev (2009); Schnyder et al. (2008); Ryu et al. (2010), the spinless topological insulator, belonging to symmetry class AI, is particularly intriguing. Its energy bands are characterized by a 4D topological invariant, namely the second Chern number, which has the distinguishing property of taking on only even integer values Ryu et al. (2010). This invariant leads to topological transport responses in the 4D bulk Ryu et al. (2012) and to pairs of Weyl fermions of same chirality on the 3D boundary Zhao and Wang (2013, 2014). Hence, an experimental realization of the 4D spinless topological insulator could allow to simulate chiral lattice gauge theory of high-energy physics Jackiw and Rebbi (1976); Kaplan (1992); Ball (1989).

Besides these interesting properties, the 4D spinless topological insulator has the advantage that it can be realized easily and in a robust manner in bosonic synthetic or classical systems, such as photonic lattices or periodic electric circuits. This is because such systems naturally exhibit a time-reversal symmetry that squares to $+1$, which is the protecting symmetry of the 4D spinless topological insulator in class AI. Hence, there is no need to introduce artificial gauge fields or to engineer fine-tuned intra-unit-cell degrees of freedom for the simulation of additional symmetries. The time-reversal symmetry also guarantees that the first Chern numbers vanish, such that the topological responses originate purely from the second Chern number.

Motivated by these considerations, we propose in this paper an experimental realization of the 4D spinless topological insulator in a periodic electric circuit composed of inductors (L), capacitors (C), and operational amplifiers. By using a mapping between circuit Laplacians and single-particle Hamiltonians, we explicitly construct an circuit lattice, whose resonance frequency spectrum is identical to a 4D spinless topological insulator in class AI. We perform detailed numerical simulations of the resonance frequency spectrum for various boundary conditions. For open boundary conditions we observe pairs of 3D Weyl cones that traverse a gap in the resonance frequency spectrum. Since the LC circuit lattice is non-dissipative, it has a built-in time-reversal symmetry of class AI, which leads to a strong and robust protection of the 3D Weyl boundary states. Even though the proposed circuit lattice realizes a 4D state, it can readily be implemented on a 2D circuit board or integrated-circuit wafer by projecting the 4D hyperlattice onto the 2D plane. The crossings of the projected lattice links can be avoided by using a bridge structure for the wiring. The predicted pairs of Weyl modes can be experimentally observed using frequency-dependent measurements.

4D spinless topological insulator

We start by discussing a minimal model for the 4D topological insulator in class AI and its boundary Weyl modes. A minimal model can be constructed from a four-band Hamiltonian of the form,

[TABLE]

Here $f_{a}(\bm{k})$ are real functions of the 4D quasi-momentum $\bm{k}=(k_{1},k_{2},k_{3},k_{4})$, $\gamma_{0}=\bm{1}_{4\times 4}$, and $\gamma_{i}$ (with $i=1,2,\cdots,5$) are five $4\times 4$ gamma matrices, which satisfy the Clifford algebra $\{\gamma_{i},\gamma_{j}\}=2\delta_{ij}$ and act on the spinors $\Psi^{\dagger}=(\psi_{a}^{\dagger},\psi_{b}^{\dagger},\psi_{c}^{\dagger},\psi_{d}^{\dagger})$. For concreteness we choose the following representation for the gamma matrices: $\gamma_{1,2,3}=\tau_{1,2,3}\otimes\rho_{1}$, $\gamma_{4}=\tau_{0}\otimes\rho_{2}$, and $\gamma_{5}=\tau_{0}\otimes\rho_{3}$, with $\tau_{\alpha}$ and $\rho_{\alpha}$ two sets of the Pauli matrices. Time-reversal symmetry acts on $\mathcal{H}(\bm{k})$ as $\mathcal{H}^{*}(\bm{k})=\mathcal{H}(-\bm{k})$, which implies that $f_{0,1,3,5}$ ($f_{2,4}$) are even (odd) functions of $\bm{k}$. With this condition, one possible choice for $f_{i}$ that yields a finite second Chern number is: $f_{0}(\bm{k})=\epsilon-t\cos(k_{2}+k_{3})$, $f_{1}(\bm{k})=-t(1+\cos k_{1}+\cos k_{2})$, $f_{2}(\bm{k})=t(\sin k_{1}+\sin k_{2})$, $f_{3}(\bm{k})=-t(1+\cos k_{3}+\cos k_{4})$, $f_{4}(\bm{k})=t(\sin k_{3}+\sin k_{4})$, and $f_{5}(\bm{k})=m-t\cos(k_{2}+k_{3})$, similar to a previous model introduced in a general context Price (2018). Since the term $f_{0}(\bm{k})$ only affects the global energy at each $\bm{k}$, rather than the topological property as indicated by $E(\bm{k})=f_{0}(\mathbf{k})\pm(\sum_{a=1}^{5}f_{a}^{2}(\bm{k}))^{1/2}$, we choose its form only for the convenience of the realization of the 4D topological electric circuit.

The topology of the gapped 4D class AI system can be characterized by the second Chern number. For the Dirac model, the second Chern number can be nicely simplified as the winding number of $\hat{\bm{f}}=\bm{f}/|\bm{f}|$ from the 4D Brillouin zone (BZ) to the 4D unit sphere $S^{4}$ Qi et al. (2008)

[TABLE]

where $\epsilon^{\mu\nu\lambda\rho\sigma}$ is the rank-5 Levi-Civita symbol with $\mu,\nu,\lambda,\rho,\sigma=1,2,\cdots,5$, and repeated indices are summed over. Straightforward calculation gives that $C_{2}=-2$ if $-t/2<m<t$, and otherwise $C_{2}=0$ as shown in Fig.1 a, for which a detailed derivation can be found in the supplemental materials.

According to the general theory of bulk-boundary correspondence of topological insulators, a nontrivial second Chern number leads to boundary Weyl fermions. We consider a 3D boundary perpendicular to the $\bm{r}_{1}$-axis, putting the semi-infinite system in the region with $r_{1}>0$. For simplicity we set $m=0$ and the system is in the topologically nontrivial phase with $C_{2}=-2$. For the Dirac model (1) the boundary effective Hamiltonian can be derived analytically as Mong and Shivamoggi (2011)

[TABLE]

Here $\sigma_{\alpha}$ acts in the sub-lattices $c$ and $d$, $f_{0,3,4,5}$ are functions defined in (1), depending only on $\tilde{\bm{k}}\equiv(k_{2},k_{3},k_{4})$, where $k_{2}\in(2\pi/3,4\pi/3)$ and $k_{3,4}\in[0,2\pi]$. From the boundary spectrum $E(\tilde{\bm{k}})=f_{0}\pm(\sum_{a=3}^{5}f_{a}^{2})^{1/2}$, it is easy to obtain that there are two Weyl points located at $w_{1,2}=\pm 2\pi(5/12,1/3,-1/3)$ in the 3D boundary BZ as shown in Fig.1 c-d. As the Weyl points are located at generic momenta, they have anisotropic dispersion relations due to the lack of rotational symmetry. The $\gamma_{0}$ term in Eq. (1) leads to the unwanted $\sigma_{0}$ term in Eq. (3), which tilts the boundary Weyl points. But, we have to make a trade off between the simplicity of the model and the magnitude of the term. Since the two Weyl points are related by time-reversal symmetry, they have the same chirality, right-handedness as shown in the supplementary materials. More details are given in the supplementary materials. Generically the boundary states decay exponentially towards the bulk, which will be confirmed by our simulation results in Fig.3.

Tight-binding model and circuit lattice

We now proceed to address the realization of the above 4D topological states by constructing a realistic electric-circuit in a practical way. For this purpose, it is more convenient to write the model Hamiltonian (1) in real space, that reads $H=\sum_{\alpha,\beta,i,s}t_{\alpha\beta}(R^{s}_{\alpha\beta})c_{\alpha}^{+}(i+R^{s}_{\alpha\beta})c_{\beta}(i)$, where $\alpha,\beta$ label nodes in each unit cell and $i$ labels the unit cells. $R_{\alpha\beta}^{s}$ are hopping vectors, which can be obtained by the inverse Fourier transform of (1), and are listed as: $R_{ad}^{s}=R_{bc}^{s}=(0,0,0,0),(1,0,0,0),(0,1,0,0)$ and $R^{s}_{ab}=-R^{s}_{cd}=(0,0,0,0),(0,0,1,0),(0,0,0,1)$ with $s=1,2,3$, respectively, and $R^{s}_{aa}=R^{s}_{cc}=(0,1,1,0),(0,-1,-1,0)$ with $s=1,2$, respectively. Here, each number in the parentheses is in the unit of the corresponding lattice constant for the 4D hypercube lattice. The hopping amplitudes are assumed to be $t_{ab}=t_{bc}=t_{ad}=t_{aa}=t_{cc}=-t_{cd}=-t$, where $t$ is a real constant so that time-reversal symmetry is preserved. Exchanging the order of the subscripts, the amplitudes $t_{\alpha\beta}$ are unchanged while the vectors $R_{\alpha\beta}$ are reversed. The above 4D tight-binding model can be mapped to a 4D circuit lattice as detailed in the supplementary materials. As the property of a circuit lattice depends only on the connection relations among its nodes, regardless of the shape of circuit lattice, one can project the 4D circuit lattice onto a 2D plane to obtain an equivalent 2D circuit lattice as shown in Fig.2, preserving the property of the circuit. In more detail, the circuit in Fig.2 is constructed by the following two steps. First, the sub-circuits in the $\bm{r}_{3}$-$\bm{r}_{4}$ planes, with the Born-von Karman periodic boundary conditions, are constructed as shown in Fig.2 a. The nodes and lines in Fig.2 a are detailed in Fig.2 b-d. Then, arrange the sub-circuits on the $\bm{r}_{1}$ and $\bm{r}_{2}$ plane, and connect the nodes between sub-circuits with capacitors to realize the connections in the $\bm{r}_{1}$-$\bm{r}_{2}$ planes as shown in Fig.2 e. By the two steps, we have constructed a 2D circuit that is genuinely equivalent to the 4D circuit, since the connections of nodes in the two circuits have a one-to-one correspondence. According to the Kirchhoff current law, it is easy to check that the current equations for the circuit in Fig.2 e lead to a Hamiltonian with exactly the same form as Eq. (1). Now the functions $f_{a}(\bm{k})$ have the parameters concretely specified in terms of capacitance values as $t=C$, $m=(C_{a0}-C_{b0}+2C)/2$ and $\epsilon=(C_{a0}+C_{b0})/2+7C$ as detailed in the supplementary materials. If the capacitance values satisfy $C_{a0}+2C=C_{b0}$, namely $m=0$, the circuit is in a topologically nontrivial phase with the second Chern number $C_{2}=-2$.

Simulation results

In order to extract the resonance frequency spectrum of the circuit lattice, we perform the time-domain transient simulations to obtain the voltage $v(t,\bm{R},\alpha)$ on each node as a function of time. Here, $\bm{R}$ is the unit cell label, $\alpha=a,b,c,d$ is the index for the nodes in each unit cell, and $t$ is the time. Taking periodical boundary conditions in $\bm{r}_{1,2,3,4}$ directions, respectively, and performing the Fourier transform, the voltage $v(\omega,\bm{k},\alpha)$ can be obtained in the momentum $\bm{k}$ and frequency $\omega$ space. The band-structure-like dispersions are obtained by plotting $|v(\omega,\bm{k},\alpha)|$. As introduced in the supplementary materials, the eigenvalue $\varepsilon$ of the tight-binding model corresponds to the resonance frequency $\omega$ of the circuit lattice, with the relation $\varepsilon=1/(\omega^{2}L)$. Therefore, the vertical axis in Fig.3 are plotted as $1/(\omega^{2}L)$, so as to compare with the eigenvalues of the Hamiltonian (1). In Fig.3 a and b, it is easy to see that the simulation results (gray points) are in good agreement with the bulk band dispersions (red dashed line) obtained from the model Hamiltonian (1).

Next we study the surface states by assuming open boundary conditions in the $\bm{r}_{1}$ direction and periodic boundary conditions in the$\bm{r}_{2,3,4}$ directions. The technical details are provided in the supplementary materials. The pulse voltage source is connected to the (1,1,1,1) cell on the bottom edge, and thereby the voltage $v(t,\bm{R},\alpha)$ are obtained for a slab geometry with nine layers in the $\bm{r}_{1}$ direction. We then carry out the Fourier transforms for $\bm{r}_{2,3,4}$ and $t$, which gives $v(\omega,R_{1},\tilde{\bm{k}},\alpha)$, where $R_{1}$ is the lattice index in the $\bm{r}_{1}$ direction and $\tilde{\bm{k}}=(k_{2},k_{3},k_{4})$. The corresponding boundary band structures for the voltage intensity along a number of selected $k$ lines crossing one of two Weyl points, $w_{1}$, are listed in Fig.3 c-h. We now briefly introduce these figures, while more information can be found in the figure caption. For all of them, the data from simulation and analytic solutions from the model Hamiltonian are plotted by gray dots and dashed red lines, respectively, for comparison. In Fig.3 c, the data from simulation shows that the surface Weyl states appear in the gap of the band structure, in good agreement with the results obtained from the model Hamiltonian. To reveal the local nature of the topological boundary states, the intensity of voltage for the fifth layer to the top layers is depicted in Fig.3 d, where the surface Weyl states disappear, because the pulse source on the bottom boundary cannot excite the Weyl surface states located on the top boundary. Furthermore, according to the model Hamiltonian, the boundary states are contributed only by the node-$c$ and $d$ components, and have vanishing $a$ and $b$ components, which is confirmed by comparing the simulation result in Fig.3 f with that in Fig.3 e. The intensity of voltage for the bottom four layers is clearly visible only for the component of the $c$ and $d$ nodes plotted in Fig.3 f, while the component of the $a$ and $b$ nodes plotted in Fig.3 e is too weak to be seen. Finally, the band structures of the Weyl states along the $k_{3}$ and $k_{4}$ directions are depicted in Fig.3 g and h, respectively. The linear dispersion relations in the vicinity of the point $w_{1}$ w.r.t. all boundary momentum components $\tilde{\bm{k}}$ show that the point $w_{1}$ is indeed a Weyl point.

Conclusions

In summary, 4D topological states exhibit many interesting phenomena that are markedly different from lower-dimensional topological phases. Unfortunately, they cannot be realized in condensed-matter materials, which are limited to three spatial dimensions. In this article we have shown that periodic electric circuits, composed of inductors, capacitors, and operational amplifiers, provide a realistic and ideal platform to create higher-dimensional topological states in the laboratory. We have explicitly constructed an electric circuit lattice that realizes the 4D spinless topological insulator. By projecting onto two dimensions, this 4D circuit lattice can readily be implemented on a printed circuit board or an integrated-circuit wafer. In this way, the higher dimensions of the 4D circuit lattice are faithfully realized through long-ranged lattice connectivity, rather than by internal degrees of freedom. Furthermore, the circuit implementation of higher-dimensional topological states has the advantage of being highly controllable and easily reconfigurable. This allows, for example, to investigate topological phase transitions, non-Hermitian phenomena, and the effects of nonlinear couplings Serra-Garcia et al. (2019). Using detailed numerical simulations, we have shown that the resonance frequency spectrum of our circuit lattice exhibits a pair of 3D Weyl boundary states, which is the hallmark of nontrivial topology.

Our work opens up the possibility of realizing topological phases in arbitrarily high dimensions, for example the 5D topological Weyl state Lian and Zhang (2016), or the 6D chiral topological superconductors Kitaev (2009); Schnyder et al. (2008); Ryu et al. (2010). Even topological states on non-orientable surfaces of any dimension could be realized, such as, topological phases on Möbius strips Ningyuan et al. (2015), Klein bottles, or real projective planes. Other interesting directions for future research concern the study of quantum effects and interactions in higher-dimensional topological states. The former could be simulated by use of periodic Josephson junction arrays Tsomokos et al. (2010); Cosmic et al. (2018). Moreover, the nonlinearity effect could be achieved by bringing the electronic device into a nonlinear region. We hope that our work will stimulate further investigations along these lines.

Acknowledgements

We thank Sheng Chang and Dong Zhang for fruitful discussion. This work was supported by the National Key Research and Development Program of China (No.2017YFA0303402, No.2017YFA0304700), the National Natural Science Foundation of China (No.11874048, No.11674077), and the GRF (HKU 173057/17P) of Hong Kong. The numerical calculations in this work have been done on the supercomputing system in the Supercomputing Center of Wuhan University.