Compact Localized States in Electric Circuit Flatband Lattices

Flatbands (FBs) arise as completely degenerate energy bands in certain tight-binding lattices with macroscopic degeneracy [1, 2]. FB signals zero group velocity, suppressing transmission, and producing compact localized states (CLSs), i.e., eigenstates trapped in a strictly finite number of sites. The extreme sensitivity of FBs to perturbations leads to the emergence of diverse and intriguing phases as it lifts the macroscopic degeneracy, including ferromagnetism [3, 4, 5, 6, 7], superfluidity [8, 9, 10, 11, 12], localization-delocalization transition [13, 14, 15, 16, 17], many-body flatband localization [18, 19, 20, 21, 22], symmetry-breaking transitions [23, 24], and compact discrete breathers [25, 26], among others [3, 2, 27, 28].

With the above, the experimental realization of artificial FB lattices becomes a priority. The challenge is in fine-tuning to preserve the CLS over a long time. Several experimental attempts, typically over a short time, lacked crucial relative phase information for the CLS and sufficient spatial resolution. Examples are photonic lattices [29, 30, 31, 32, 24, 33], cold atoms [34, 35], polariton condensates [36, 37], electrical circuits [38, 39, 40, 41] and topological material [42], as well as magnonic [43] crystal lattices. Electrical circuits offer a particularly promising approach for in-depth exploration of FBs and CLS. They provide flexibility in various lattice geometries, feasibility of fine-tuning lattice parameters, and precise experimental control and measurement.

In this paper, we experimentally construct and characterize one-dimensional FB electrical lattices with discrete circuit elements. We excite FB CLS through local sinusoidal driving at the FB frequency, reporting results of two lattice structures: diamond and stub lattices. The diamond lattice contains orthogonal CLSs and exhibits CLS resonant modes when locally driven at the FB. Finally, we impart nonlinearity by replacing the capacitors with varactor diodes characterized by a voltage-dependent capacitance. Our findings reveal the continuation of CLSs into the highly nonlinear regime in the diamond lattice and the onset of nonlinear CLS instability when it becomes resonant with a dispersive band. On the other hand, a stub lattice features non-orthogonal CLSs and shows exponentially localized resonant modes by overlapping CLSs.

In the electrical-lattice context, vertices and edges of a tight-binding lattice represent capacitors and inductors, respectively. The lattices are examples of electrical transmission lines with nontrivial geometry, as depicted in Fig. 1 (a) and (b), where a diamond lattice is shown with two different hopping values denoted by black and red lines. The capacitance of each node is $C$, and the lattice incorporates inductors with different inductance, $L_{b}$ and $L_{r}$, shown as black and red, respectively. The inductors are assumed to have an effective serial resistance (ESR) $R$, while the capacitors are treated ideally.

The voltages at the three nodes of the $n$-th unit cell are denoted as $T_{n},U_{n},V_{n}$. We drive the lattice at $U_{m}$ in the $m$-th unit cell with a driving capacitor of capacitance $C_{d}\ll C$. From Kirchhoff’s current law at each node, the equations of motion for the voltages in the $n$-th unit cell at the linear level take the form:

At the driven site $U_{m}$, Eq. (II) gets a correction factor $1/(1+\gamma)$ and an additive driving force $A\sin(\omega_{d}t)$,

$\gamma=C_{d}/C$, where $C_{d}\ll C$, is an impurity artifact that appears as a result of driving which can be minimized to within the experimental tolerance of $\omega_{b}^{2}$. Note that $\omega_{b}^{2}\!\!=\!\!1/(L_{b}C)$ and $A=v_{d}\omega_{d}^{2}\gamma(1+\gamma)^{-1}$. $\alpha=L_{b}/L_{r}$ tunes the flatband, and $\beta=R/L$ accounts for dissipation. Then, the equation of motion is approximately written as follows,

This equation describes wave dynamics on a diamond lattice. Here, the wave, $\ket{\psi(t)}=\sum_{n,X}X_{n}\ket{X_{n}}$, is the voltage at each node, where $\ket{X_{n}}$ is real-space lattice site basis at sublattice $X\in\{T,U,V\}$, and $n\in\mathbb{Z}$. The matrix $H$ represents the coupling of the diamond lattice, given from the right side of Eq. (II). $\ket{\mathbf{F}(t)}$ is the driving term, yielding the last term of Eq. (2). Unlike the hoppings in tight-binding diamond lattices, inductors also add to the “onsite potential” (diagonal elements of $H$), which in turn guarantees $\omega^{2}$ is positive and breaks the lattice bipartitness and chirality, but not the flatband [44]. In the absence of driving, the Bloch waveform ansatz, $U_{n}=U(k)\exp\left[i(\omega t-kn)\right]$ (and similarly for $V_{n},T_{n}$) solves the eigenvalue problem associated with Eq. (II). It results in three bands - an optical band, a gapped flatband, and lower frequency acoustic band which extends down to zero frequencies:

Solving the dissipative case yields $s^{2}\!-\!i\beta s\!=\!\omega^{2}_{\mathrm{FB}/\mathrm{DB}}$,

with the dissipation time $\tau\!=\!2/\beta$ and the slightly shifted $\omega_{\mathrm{FB}/\mathrm{DB}}$ ($<\!1\%$ in our experiment) from Eq. (4). We assume only underdamped frequencies making the square root part always real, i.e. $\omega^{2}_{\mathrm{FB}/\mathrm{DB}}>\beta^{2}/4$. Hence, we set $s_{\mathrm{FB}/\mathrm{DB}}\approx\omega_{\mathrm{FB}/\mathrm{DB}}$ throughout the paper.

While the Bloch eigenvectors at FB are spatially extended, the degeneracy of the FB allows CLS eigenstates. For the diamond lattice, the CLS is given by $U_{n}=\delta_{nm}$, $V_{n}=-U_{n}$ and $T_{n}=0$ for any unit cell $n$ forming an orthogonal basis in the FB subspace, as we will see below in further detail in Fig. 2. A similar analysis is performed on the stub lattice (Fig. 3 (b)), and is discussed in Sec. V.

We analyze the impact of local driving on the flat-band analytically in this section, and we apply this result to the diamond lattice and stub lattice in the following sections.

To analyze the impact of local driving on the flatband, we employ the transfer function method [45], a powerful tool to understand the dynamic behavior of systems described by linear differential equations. In our case, the transfer function operator $G(\omega)$ is defined in the following:

The transfer function operator $G(\omega)$ encodes all information on the response to local sinusoidal driving. With zero initial condition, the response to the ideal sinusoidal driving is then given as:

The transfer function operator $G(\omega)$ can be easily calculated in the eigenbasis of $H$:

Note that we treat the frequency responses of the flat band and dispersive bands seperately, to isolate the flatband response, where $j\in\mathrm{DB}$ is the band index for the dispersive bands. If we assume ideal local sinusoidal driving at $Y_{m}$, we can write as

We assume local driving at $Y_{m}$, where $Y=T,U,V$ is a sublattice index. We neglect the dispersive term $G_{\mathrm{DB}}$, which is reasonable when $\omega_{d}\approx\omega_{\mathrm{FB}}$ and the dispersive bands are sufficiently far from the flat bands compared to the width of resonance peaks. We use Eqs. (7), (III),(9) and obtain the spatial profile

We express the projector in terms of the CLS basis,

where $i,j$ are lattice sites, and $S$ is the overlap matrix with elements defined as $S_{ij}\!=\!\innerproduct{\mathrm{CLS}_{i}}{\mathrm{CLS}_{j}}$. For flatbands supporting orthogonal CLSs, $S_{ij}=\delta_{ij}$ (Eq. (11)), the projector $P_{\mathrm{FB}}$ is expressed simply as

resulting in a compact projector [46]. The term compact implies the existence of an integer $l\leq|n-m|$, such that the matrix elements satisfy $\langle{X_{m}|P_{\mathrm{FB}}|Y_{n}}\rangle=0$, where $X,Y\!\in\!\{T,U,V\}$. For instance, for the diamond lattice (Sec. II), the CLS at the $i$-th unit cell is represented as

Then, the real-space representation of the projector is obtained from Eq. (10), and the CLS response of the diamond lattice at $U_{n}$ is obtained as

which is identical for the $V_{n}$ sites. For the parameters in the experimental setup, we have $\gamma=0.015$, $v_{d}=1\textrm{ V}$, $\omega_{d}=\omega_{\mathrm{FB}}=2\pi\times 429\textrm{ kHz}$, and $\beta=R/L_{b}=49356\textrm{/sec}$. The maximum CLS response is given by $0.4\textrm{ V}$, which is in excellent agreement with the experiment and simulation within 10 % (Fig. 2 (b) and (c)).

We construct an electrical-circuit diamond lattice of $N=5$ unit cells, with periodic boundary conditions. The lattice includes $3N=15$ capacitors, each with a capacitance of $C=1\pm 0.01~{}\mathrm{nF}$; the driving capacitor of $C_{d}=15~{}\mathrm{pF}$ yields $\gamma=0.015$. Two inductors are employed with $L_{b}=466~{}\mu\mathrm{H}$ and $L_{r}=674~{}\mu\mathrm{H}$, within a 1% tolerance. The main sources of inductor dissipation include ferrite cores and coil-wire resistance, diminishing the quality factor $Q=\omega L/R_{\mathrm{eff}}$. The $Q$ of an inductor remains essentially constant while the effective serial resistance (ESR) varies according to the resonant frequency.

With $Q=55$ at $232~{}\mathrm{kHz}$, we estimate ESR $R_{\mathrm{eff}}\approx 23~{}\Omega$ for the $L_{b}$ inductor at $f_{\mathrm{FB}}$ [47]. A $10~{}\mathrm{k}\Omega$ resistor is placed in parallel with the lattice to suppress a DC voltage component. We then obtain the band structure using Eq. (4) (red (dispersive) and blue (FB) lines in Fig. 2 (a)). The flatband frequency $f_{\mathrm{FB}}=429~{}\mathrm{kHz}$ is in the spectral gap between the two dispersive bands. The flatness of this band has been ascertained experimentally by measuring its frequency using both a spatially uniform and a staggered driver.

To experimentally probe the flatband and its CLS, we supply energy locally via a sinusoidal voltage input from a signal generator (Agilent 33220A function/sweep generator), see Fig. 1. The measurement results are displayed in Fig. 2. We also monitor the response voltage at all lattice sites, which corresponds to $(U_{0},V_{0},T_{0},\ldots,U_{4},V_{4},T_{4})$, simultaneously with a 16-channel data acquisition system (NI PXI-1033 with NI 6133 cards) at a $2.5~{}\mathrm{MHz}$ sampling rate. The driving voltage is injected at site 3 ($U_{1}$). While we show a continuous band structure, there is only $N=5$ resonance modes per band, hence $\lceil N/2\rceil=3$ peaks per band due to two-fold degeneracy.

Let us turn to the impact of local driving frequency $f_{d}$. We sweep $200$ – $600~{}\mathrm{kHz}$ frequency range within $25~{}\mathrm{ms}$ using a function generator. The steady-state amplitude responses at site 4 ($V_{1}$) were measured with an oscilloscope (no DAQ card), shown as the black trace in Fig. 2 (a). The flatband frequency prediction is accurately matched. This resonance peak strength depends on dissipation, driving voltage, and amplitude of resonant eigenvector at $U_{m}$. We observe the largest peak reaching $0.4~{}\mathrm{V}$ at $429~{}\mathrm{kHz}$. Two other prominent modes in the acoustic branch are also visible at $k=2\pi/5,4\pi/5$. We now tune the function generator to the frequencies of the observed resonance peaks. Fig. 2 (1) – (5) display spatial patterns at various drive frequencies once they reach a steady state.

At $f_{d}=429~{}\mathrm{kHz}$, corresponding to the largest resonance peak on the flatband, the associated CLS is predicted to sit at $U_{n},V_{n}$ of a single unit cell, with their excitations out of phase. Figs. 2 (b) and (c) compare experiment and numerics, respectively. In Fig. 2 (b), the red trace depicts the response of the driven site, whereas the blue depicts that for the other CLS site. The traces show voltage-time profiles for all 15 sites within the time interval $(0,2/f_{d})$. We observe precise out-of-phase behavior, causing destructive interference at neighboring bottleneck sites $T_{n},T_{n-1}$. However, small leakage to the rest of the sites is evident due to experimental imperfections and dissipation, which broadens the dispersive resonance peaks – yet, this does not adversely affect the CLS (even in the nonlinear regime, as we will see), as its frequency is detuned from any other eigenmodes. Note that driving at a site in the CLS is essential for its generation.

In Fig. 2 (c), the corresponding numerical simulation demonstrates excellent agreement with the experiment. In the simulation, for $\beta=0$, all sites other than the CLS sites approach to near-zero voltages, resulting in a true CLS localized on the two CLS sites, oscillating at the flatband frequency of $429~{}\mathrm{kHz}$. Note that the experimental frequency is shifted down slightly to $401~{}\mathrm{kHz}$ due to small parasitic capacitances associated with the measurement apparatus (ribbon cables and DAQ board).

We conduct a similar analysis on a stub lattice. In Fig. 3 (b) and (c), we illustrate the circuit representation and schematic of the stub lattice. In the stub case, only a single type of inductor is required (we use $L_{b}$). The equations of motion for the stub lattice are

The driven site is $A_{m}$ and is modified similarly to Eq. (2). The eigenfrequencies for the stub lattice for $\beta=0$ are computed as

Similar to the diamond chain the onsite potentials break chirality (but, again, not the flatband [44]), and in addition here also make both dispersive bands optical-like, with a finite gap to zero frequencies. The blue and red traces in Fig. 3 (a) plot Eq. (17); the black trace along the right vertical axis depicts the experimental spectrum obtained by sweeping the frequency of the driver at a CLS site. Note that the largest experimental response at this CLS site is registered at the predicted flat-band frequency. The zone-center acoustic mode occurs at a nonzero frequency and is registered in the spectrum. At $\omega_{\mathrm{FB}}$, the CLS labeled with $i$ occupies the $i$-th and $(i+1)$-th unit cells:

As a result of the overlap between the closest CLSs, they form a nonorthogonal basis. Then, the impact of local driving on the stub flatband exhibits distinct behavior compared to the diamond case, leading to $S_{ij}\neq 0$ for $i\neq j$. Given that the stub CLSs only overlap with their nearest CLSs, as defined by Eq. (18)), the overlap matrix $S_{ij}$ in Eq. (11)) is determined as

where the overlap between neighboring CLSs is denoted as $\sigma$. $S$ is a tridiagonal matrix with translational symmetry, thus its inverse can be readily obtained in the momentum basis,

The integration is solved in the complex plane using Cauchy’s integral formula with substitution, $\omega\!=\!\exp(ik)$ and $d\omega\!=\!i\omega dk$ [48]. Here, a characteristic localization length $\xi$ is obtained as,

Therefore, local driving induces exponential localization around the driven site, not an excited CLS mode. We have $\sigma=1/3$ and thus $\xi\approx 1.03$ for the CLS in Eq. (18).

To verify the theoretical prediction, we introduce a driver at site 6 (indicated in red arrow) as depicted in Fig. 3 (b), it is expected to induce partial excitation in both CLSs which share the site 6. This situation experimentally shown in Fig. 3 (d), using a driver frequency of $312~{}\mathrm{kHz}$ with a driving amplitude of $11~{}\mathrm{V}$. The corresponding result is shown in the numerical simulation of Fig. 3 (e). The voltage-time profiles of all 15 sites are presented for two periods. When site 6 (red) is driven, neighboring CLSs also undergo excitation (depicted in yellow and blue). This phenomenon arises due to the shared sites 5 and 8 with adjacent CLSs. It is important to note that slight inhomogeneities result in uneven excitation amplitudes between the two neighboring CLSs.

We extend our studies and investigate the impact of nonlinearity on the CLS in the diamond chain both experimentally and numerically. It has been predicted that linear homogeneous CLSs (absolute values of all nonzero amplitudes are equal) can be extended into the nonlinear regime as families of compact periodic orbits or compact discrete breathers in the presence of a suitable symmetric nonlinearity [26]. In order to create a symmetric hard-type nonlinearity, we substitute the capacitors with varactors – diode pairs oriented in opposite directions [49]. The varactor configuration is illustrated in the inset of Fig. 4 (a). The additional ground-connected resistor ($100~{}\mathrm{k}\Omega$) is needed to prevent a DC charge buildup. It breaks the symmetry, but the effect is weak for the chosen large resistance value due to the small current actually flowing through it. The diodes ensure a symmetric nonlinear current-voltage characteristics. Consequently, the term $\omega^{2}_{b}$ in Eq. (II), becomes a nonlinear symmetric function of the voltage $U_{n}$, at sites $U_{n}$ (and similarly for $V_{n},T_{n}$).

In order to experimentally demonstrate nonlinear CLS, we employ these diodes and a $680~{}\mu\text{H}$ inductor to build an RF-resonator. We drive it using a sweep generator and a linear capacitor, recording the resulting resonance curves. In Fig. 4 (a), a low driving amplitude yields a symmetric peak (black). As we increase the driving amplitude (red), the curve shifts to higher frequencies. At $v_{d}=4\text{ V}$, a significant bistability window emerges, (340 – 500 kHz), with hysteresis evident in the up and down sweeps (blue and green, depicting only peak-to-peak amplitude for visual clarity).

We model the nonlinearity with the following ansatz:

where $g=0.1$ characterizes the hard-type nonlinearity, and $\omega_{b0}=\omega_{b}(0)$. This particular choice of nonlinearity provides a good fit to our experimental data at strong driver voltage, see the blue curves in Fig. 4 (a), and Fig. 4 (g), (h). At weak voltage, $gU_{n}^{2}\!\ll\!1$, the model simplifies to a quadratic form, $\omega^{2}_{b}\to\omega^{2}_{b0}(1+gU^{2}_{n})$, capturing the essence of nonlinearity at lower voltages. In Appendix. A, we conduct a stability analysis following Ref. [26] for the undriven nonlinear diamond lattice, for quadratic, and logarithmic nonlinearities.

In the nonlinear diamond lattice, we simulate frequency sweeps starting near the flat band – see Fig. 4 (c) – (e). When reaching the optical band-edge for this lattice of 100 unit cells, 735 kHz, the signal either drops abruptly to zero or (at higher amplitudes) continues with additional noise. The black (red) curves represent the response at a CLS (non-CLS) site. Notably, panel (c) is essentially unchanged over a range of driver amplitudes ($A$ up to $0.8\text{ V}$), and upon entering the optical band ($A\geq 0.9\text{ V}$), the response at the adjacent non-CLS site jumps up discontinuously.

In the experiment, panels (f) – (h), we observe similar behavior in a smaller lattice ($N=5$). These panels show the response again for different driving amplitudes (top $1\text{ V}$, middle $2.25\text{ V}$, bottom $2.75\text{ V}$). The two optical modes show up prominently in the red curve. More importantly, when the driving frequency reaches the first of these modes at $800\text{ kHz}$, it either drops or continues with enhanced noise.

Fig. 4 (i) – (n) display the spatio-temporal profiles at stationary resonant states at multiple driving amplitudes, both experimentally and numerically. We start in the nearly linear regime at $578^{*}\text{ kHz}$ (equivalent to $630\text{ kHz}$, shift due to DAQ/ribbon cable), and raise the frequency to $650^{*}\text{ kHz}$ ($708\text{ kHz}$) with a stronger driver. This procedure does not disrupt the CLS. When extending deeper into the nonlinear regime, the emergence of harmonics in the CLS spectrum (see insets) underscores the nonlinear nature of the stationary CLS. Lowering the amplitude while maintaining $650\text{ kHz}$ destroys the CLS due to the nonlinearity-induced bistability and hysteresis.

Exploiting the strong nonlinearity of the varactors, we aim to shift the nonlinear CLS into the optical band, in line with theoretical predictions [25, 26]. To achieve this, we must chirp the driver frequency up as we enter high-frequency regions due to the system’s hysteresis. In Fig. 4 (k), as the local driving frequency sweeps on the second mode of the optical band $735^{*}\text{ kHz}$ ($801\text{ kHz}$), we observe the CLS transitioning to a pattern where energy oscillates between the CLS sites in an alternating (“zig-zag”) fashion and partially “leaks”, i.e., radiates into the rest of the lattice. This observation is consistent with simulations in panel (n), which exhibit a similar instability pattern. We conclude that a nonlinear CLS in this electrical diamond lattice displays a special instability pattern, yet highly localized when its resonance frequency intersects the linear optical band, in contrast to Ref. 26, where instability destroys the CLS.

Using complex electrical circuits, we constructed 1D flatband lattices and observed resonant modes through local sinusoidal driving. In a diamond lattice, we find that the driving at the flatband frequency excites a CLS. In the stub lattice the lack of orthogonality of neighbouring CLSs precludes the observation of individual CLSs, leading instead to resonance modes with exponentially localized spatial profiles. Finally, we found that CLSs persist in the diamond lattice when nonlinearity is introduced by replacing the capacitors with varactor diodes exhibiting symmetrical nonlinearity of capacitance, but that it either disintegrates entirely or becomes unstable when it is nonlinearly shifted into resonance with a dispersive band.

The clarity, as well as the qualitative and quantitative correspondence between theory, numerics and computations affirms the particular relevance of such linear, and, especially through our work, nonlinear electrical lattices as a fruitful platform for exploring flat bands and CLSs, including for relevant applications such as, e.g., targeted energy transfer [50]. Naturally, numerous open questions emerge from the present work, including, e.g., whether a CLS can be distilled suitably in stub lattices, and whether more complex quasi-one-dimensional [51], but also two-dimensional structures [32] can also be engineered. In such settings the fate of linear and, perhaps especially, nonlinear flatband states remains an appealing and challenging task for future considerations.