⚛️ quantum physics

Realizing Next-Nearest-Neighbor Coupling and Peierls Phase in Circuits

This paper designs and experimentally validates trimerized circuits implementing non-Hermitian Su-Schrieffer-Heeger models with next-nearest-neighbor coupling and Peierls phases, demonstrating their effectiveness as an adjustable platform for investigating topological states and their dynamic responses.

Original authors: Xin-Sheng Zhang, Wen-Jie Xu, Hang-Yu Gu, Chuan-Xun Du, Yong-Long Wang

Published 2026-07-07

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CC BY 4.0

Original authors: Xin-Sheng Zhang, Wen-Jie Xu, Hang-Yu Gu, Chuan-Xun Du, Yong-Long Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a long chain of dominoes, but instead of just falling over, they are connected by springs and wires. In the world of physics, scientists use these chains to study "topological states"—special conditions where energy or signals get stuck at the very ends of the chain, behaving differently than in the middle. This paper is about building a specific kind of electrical chain (using resistors, inductors, and capacitors) to see how we can control where these signals get stuck.

Here is the story of their experiment, broken down into simple steps:

1. The Basic Setup: The "Trimer" Chain

The researchers started with a basic design called a "trimerized" chain. Think of this as a repeating pattern of three distinct stations: A, B, and C.

Station A is a "leaky" station (it loses energy, like a bucket with a hole).
Station C is a "boosted" station (it gains energy, like a bucket being filled from a hose).
Station B is the neutral middle ground.

In this basic setup, Station A and Station C are connected to their neighbors (B) by wires. The researchers found that because of the difference between the "leaky" and "boosted" stations, signals naturally wanted to hide at the very ends of the chain. This is the "topological state" they were looking for.

2. Adding a Shortcut: The "Next-Nearest-Neighbor"

Next, they asked: What if we add a shortcut?
In the original chain, A only talks to B, and B talks to C. But in the new design, they added a direct wire connecting A and C directly, skipping B.

The Analogy: Imagine you are walking down a hallway. Usually, you walk from Room 1 to Room 2, then to Room 3. The new shortcut is a secret tunnel that lets you jump from Room 1 straight to Room 3.
The Result: This shortcut changed the rules of the game. It didn't just tweak the system; it created a new "step" in the behavior of the chain. Instead of the signals just being at the ends or nowhere, the system could now show three different types of behavior (0, 1, or 2 pairs of signals at the ends) depending on how you adjusted the connections. It made the system much more flexible.

3. The Magic Twist: The "Peierls Phase"

Finally, they added a "Peierls Phase." This sounds complicated, but think of it as a one-way turnstile or a traffic light on that new shortcut tunnel.

The Analogy: Imagine the shortcut tunnel between Room 1 and Room 3. Without the phase, you can walk through it normally. With the Peierls phase, the tunnel has a magical property: if you walk from 1 to 3, you feel a gentle push; if you walk from 3 to 1, you feel a pull. It introduces a "direction" or a "twist" to the connection.
The Result: This twist didn't change how strong the connection was, but it changed how the signals interfered with each other.
- When the twist was set one way, the "upper" signal (the one at the top of the energy scale) was strong and stuck at the edge, while the "lower" signal faded away.
- When they turned the twist to the other side, the opposite happened: the lower signal became strong and stuck, while the upper one faded.
- At a middle setting, both signals behaved similarly.

4. Building and Testing the Machine

The team didn't just do math on a computer; they built a real physical circuit board (PCB) with actual electronic components.

They used a computer program (LTspice) to simulate how the circuit would behave, which matched their math perfectly.
They then built the board, plugged in a signal generator, and measured the voltage at different points.
The Proof: When they turned the "Peierls Phase" knob (by adjusting a specific circuit component), they watched the signals move. Just like their theory predicted, they could make the signal jump from the top edge to the bottom edge, or make it disappear entirely, simply by changing this phase.

The Bottom Line

The paper claims that by building these specific electrical circuits, they created a simple, adjustable platform to study complex physics.

The "Shortcut" (NNN coupling) gave them more control over how many signals could exist at the edges.
The "Twist" (Peierls phase) gave them a dial to swap which signal was strong and which was weak.

They conclude that these circuits are a great "playground" for physicists to test new ideas about how energy behaves in strange, non-standard environments, without needing expensive or complex quantum computers. It's a way to turn abstract math into a tangible, tunable machine.

Technical Summary: Realizing Next-Nearest-Neighbor Coupling and Peierls Phase in Circuits

Problem Statement
In discrete lattice systems, Next-Nearest-Neighbor (NNN) coupling and the Peierls phase are critical for extending coupling access and introducing gauge-phase modulation. While these elements are essential for describing band structures, gap opening, and the emergence of new topological phases in models like the Su-Schrieffer-Heeger (SSH) lattice, Haldane model, and Kane-Mele lattices, their experimental investigation faces significant challenges. In traditional platforms such as photonic systems, waveguide arrays, and atomic lattices, the parameters governing NNN coupling and Peierls phases are often difficult to tune dynamically. This limitation restricts the ability to fully explore the controllability of topological states and the effects of synthetic gauge fields. There is a need for an alternative platform with highly adjustable parameters to investigate these mechanisms effectively.

Methodology
The authors propose and implement a non-Hermitian one-dimensional trimerized circuit based on the SSH model using Resistor-Inductor-Capacitor (RLC) networks. The study proceeds through three progressive stages of circuit design and analysis:

Theoretical Modeling:
- Initial Model: A trimerized unit cell containing three sublattice sites (A: loss, B: neutral/grounded, C: gain) is constructed. The circuit Laplacian is derived, serving as an effective Hamiltonian to describe the system's spectrum and topological invariants.
- Enhanced Model: An NNN coupling channel is introduced by connecting the loss node (A) and gain node (C) within the same unit cell via an additional capacitor. This modifies the circuit Laplacian to include a coupling strength parameter $\nu$ .
- Reenhanced Model: A Peierls phase ( $\phi$ ) is incorporated into the NNN coupling channel using a nonreciprocal phase-shift circuit. This introduces a direction-dependent complex phase ( $e^{\pm i\phi}$ ) to the coupling term without altering its amplitude.
Numerical Simulation:
- The authors calculate the eigenfrequency spectra and the total Zak phase for circuits consisting of 8 unit cells.
- They analyze the evolution of topological states as a function of the intercell coupling parameter ( $\theta$ ), NNN strength ( $\nu$ ), and Peierls phase ( $\phi$ ).
- Stability intervals ( $\Delta\theta_{\pm}$ ) and Inverse Participation Ratios (IPRs) for upper and lower edge states are computed to quantify localization strengths.
- LTspice simulations are performed to verify amplitude-frequency responses, using the difference between boundary and bulk node responses as an edge-state indicator.
Experimental Realization:
- A Printed Circuit Board (PCB) is fabricated containing 10 unit cells.
- The circuit utilizes tunable capacitor arrays for coupling, operational amplifiers to simulate gain and loss (negative resistance), and a specific module to introduce the Peierls phase.
- Sinusoidal excitation signals are applied to terminal nodes, and steady-state voltage responses are measured using an oscilloscope.
- Negative resistors are integrated to compensate for inductor losses and maintain edge-state responses.

Key Contributions

Circuit Design: The paper successfully designs and fabricates a trimerized RLC circuit that realizes non-Hermitian SSH models with tunable NNN coupling and Peierls phases.
Topological Evolution: The study demonstrates that introducing NNN coupling transforms the topological phase sequence from a simple two-sector response ( $0 \to 2 \to 0$ ) in the initial model to a stepwise sequence ( $0 \to 1 \to 2 \to 1 \to 0$ ) in the enhanced model.
Phase-Dependent Control: The research establishes that the Peierls phase acts as a synthetic gauge field that redistributes the stability intervals and localization strengths of upper and lower edge states in a complementary manner. Specifically, increasing the phase can widen the stability region and enhance localization for one edge state while narrowing it for the other.
Experimental Verification: The theoretical predictions regarding phase-dependent shifts in response intervals and the modulation of boundary-voltage localization are verified through both numerical simulations and physical measurements on the fabricated PCB.

Results

Spectral Analysis: Numerical results show that the non-Hermitian parameter $\gamma$ induces PT-symmetry breaking, leading to exceptional points where eigenfrequencies and eigenstates degenerate.
Zak Phase Transitions: The introduction of NNN coupling ( $\nu \neq 0$ ) creates intermediate topological regions with Zak phases of 1, which are absent in the initial model.
Peierls Phase Modulation:
- At $\phi = 0^\circ$ , the lower edge state exhibits a wider stability interval and stronger localization compared to the upper edge state.
- At $\phi = 90^\circ$ , the stability intervals and localization strengths of both edge states become equivalent.
- At $\phi = 180^\circ$ , the trend reverses, with the upper edge state showing a wider interval and stronger localization.
Experimental Data: Measured voltage profiles confirm the theoretical predictions. For instance, at $g_2 = 1.5$ , increasing $\phi$ from $0^\circ$ to $180^\circ$ causes the boundary voltage of the upper edge state to decrease (weakened localization) while the lower edge state's boundary voltage increases (enhanced localization).

Significance
The paper concludes that trimerized circuits provide an adjustable and simple platform for investigating new topological physical states. By utilizing the abundant elements and flexible connections of RLC circuits, the authors demonstrate a practical route to reconfigurable spectral and topological control. The ability to introduce NNN coupling and Peierls phases allows for the manipulation of topological responses without creating additional spectral frameworks, but rather by regulating existing edge-state branches. This work supports the development of programmable routes for magnetic-like topological control in non-Hermitian topolectrical circuits, offering a viable alternative to systems where parameter tuning is difficult. The results are presented as a foundation for further investigation into topological states using circuit-based platforms compatible with integrated-circuit technology.