One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems

This paper establishes a theoretical framework for one-dimensional nonsymmorphic Kramers degenerate systems with $\mathbb{Z}_2$ and $\mathbb{Z}_4$ topological classifications, extends winding-number invariants to these four-band models, and validates their topological phases and robust zero-energy modes through proposed topolectric circuit realizations and disorder analysis.

The Gist

Imagine you are a detective trying to solve a mystery about the hidden "personality" of materials. In the world of physics, some materials have a secret superpower: they can conduct electricity on their edges or surfaces while remaining insulating (non-conducting) in their interior. This is called topology.

Think of a material like a piece of dough. If you twist it into a donut shape, it has a hole in the middle. No matter how much you stretch or squish the dough, that hole remains. That "hole" is a topological feature. In quantum materials, these "holes" are mathematical patterns that protect special states of energy, often allowing electrons to flow without resistance.

This paper, by Max Tymczyszyn and Edward McCann, explores a very specific, tricky kind of topological material and shows how we can build a "toy version" of it using simple electrical circuits to study it.

Here is the breakdown in everyday language:

1. The Mystery: "Nonsymmorphic" Symmetry

Most topological materials are like a standard brick wall: the pattern repeats perfectly every time you move one brick's length. This is called "symmorphic."

But these researchers are looking at "nonsymmorphic" materials. Imagine a brick wall where, to see the pattern repeat, you have to move half a brick and flip the wall over. It's a more complex, "glued-together" pattern.

• The Problem: These complex patterns usually create a "Kramers degeneracy," which is a fancy way of saying that energy levels come in pairs that are locked together. This makes the math very hard (like trying to solve a Rubik's cube while wearing blindfolds).
• The Goal: They wanted to figure out how to count the "topological holes" in these complex, paired systems.

2. The New Tool: A "Winding Number" for Pairs

To solve the mystery, they invented a new way to count.

• The Old Way: Imagine walking around a circle. If you walk around once, you have a "winding number" of 1. If you walk twice, it's 2. This worked for simple materials.
• The New Way: Because these materials have paired energy levels (Kramers pairs), the path they walk isn't a simple circle; it's a more complex, open path that doesn't quite close up. They developed a formula to track this "open path" to get a new kind of count.
• The Result: They found two types of these materials:
  • Type A (The Z2 Model): Like a light switch (On/Off). It has two possible states.
  • Type B (The Z4 Model): This is the star of the show. It has four distinct states. Think of it like a four-way traffic light (Red, Yellow, Green, Blue) instead of just Red/Green. This is rare and special in 1D materials.

3. The Experiment: Building a "Topolectric" Circuit

Building these materials out of real atoms (like gold or silicon) is incredibly difficult and expensive. It's like trying to build a miniature city to test traffic laws.

Instead, the authors built a "Topolectric Circuit."

• The Analogy: Imagine replacing the atoms with tiny electronic components: capacitors (which store charge like a bucket) and inductors (which store magnetic energy like a flywheel).
• The Magic: When you connect these components in a specific pattern, the electricity flowing through them behaves exactly like the electrons in the complex atomic material.
• The Test: They measured the impedance (how hard it is for electricity to flow). If the circuit is in a "topological" state, the electricity gets stuck at the edges or at specific points inside, creating a huge spike in the measurement. It's like hearing a specific musical note ring out loudly only when the circuit is in the right "shape."

They successfully built circuits that mimicked both the "Light Switch" (Z2) and the "Four-Way Traffic Light" (Z4) models. The circuits confirmed their math: the electricity behaved exactly as predicted, ringing at the right frequencies.

4. The Twist: What Happens When Things Get Messy? (Disorder)

In the real world, nothing is perfect. Wires have tiny flaws, and components vary slightly. This is called "disorder." Usually, disorder destroys these delicate topological states, like a gust of wind blowing away a house of cards.

The researchers tested what happens when they added "noise" (random variations) to their Z4 circuit.

• The Surprise: They found that for the simplest version of the circuit (using only immediate neighbors), the special "zero-energy" states (the magic notes) were surprisingly stubborn. Even with noise, they stayed right at zero energy.
• Why? It wasn't because of the topological protection (the "shield"). It was an accident. The specific math of the simple circuit meant the noise didn't know how to "talk" to these special states. It was like trying to push a boulder with a feather; the feather (noise) just couldn't move the boulder (the state).
• The Catch: When they added more complex connections (longer-range links), this "accidental protection" vanished. The noise could finally push the states away from zero energy. This tells us that while these states are robust in simple models, they might be fragile in more complex, realistic materials.

Summary

This paper is a tour de force of theoretical physics and experimental engineering:

1. The Theory: They figured out how to mathematically describe complex, paired topological materials that have four distinct states (Z4).
2. The Simulation: They built a "toy" version using electrical circuits to prove their math works in the real world.
3. The Discovery: They found that while these special states are robust against some types of messiness, it's often just a lucky accident of the simple model, not a fundamental law.

In a nutshell: They took a confusing, complex quantum puzzle, solved the math, built a LEGO version of it with wires and batteries, and discovered that the "magic" parts of the puzzle are surprisingly tough to break—unless you make the puzzle slightly more complicated.

Technical Summary

1. Problem Statement

The paper addresses the theoretical characterization and experimental realization of nonsymmorphic topological phases in one-dimensional (1D) systems with Kramers degeneracy (four-band systems).

• Context: While topological insulators and superconductors are well-studied, most 1D models (like the Su-Schrieffer-Heeger (SSH) or Kitaev chains) utilize two bands and symmorphic symmetries.
• Gap: There is a lack of comprehensive models and experimental platforms for 1D systems where:
  1. The system possesses Kramers degeneracy (requiring at least four bands).
  2. The protecting symmetries are nonsymmorphic (involving fractional lattice translations, e.g., half-unit cell shifts).
  3. The resulting topological invariants are non-trivial (specifically $Z_2$ and $Z_4$ classifications) and distinct from standard Majorana classifications.
• Challenge: Calculating topological invariants for these systems is difficult because standard winding number formulas (often closed loops) do not directly apply to nonsymmorphic systems with Kramers degeneracy, which exhibit open-path trajectories in the complex plane. Furthermore, realizing these complex symmetry requirements in physical materials is challenging, motivating the use of topolectric circuits as a simulation platform.

2. Methodology

The authors employ a three-pronged approach combining theoretical modeling, topological invariant calculation, and circuit simulation.

A. Theoretical Modeling

They construct two representative four-band tight-binding models:

1. Nonsymmorphic AII Class:
   • Symmetries: Symmorphic Time-Reversal Symmetry (TRS, $T^2=-1$) combined with nonsymmorphic Charge-Conjugation (C) and Chiral (S) symmetries.
   • Structure: Two coupled Charge-Density-Wave (CDW) chains acting as TRS partners.
2. Nonsymmorphic D Class:
   • Symmetries: Symmorphic Charge-Conjugation symmetry combined with nonsymmorphic TRS and Chiral symmetries.
   • Structure: A Bogoliubov-de Gennes (BdG) model representing an electron chain coupled to a hole chain with alternating superconducting phases.

B. Topological Invariant Calculation

• Generalized Winding Number: The authors extend the open-path winding number formulation (previously used for non-Kramers nonsymmorphic systems) to Kramers-degenerate four-band systems.
• Q-Matrix Formalism: They utilize chiral symmetry to define a $Q$-matrix, $Q(k) = U^\dagger H(k) U$, reducing the Hamiltonian to an off-diagonal block $q(k)$.
• Trajectory Analysis: Instead of a closed loop, they analyze the trajectory of the determinant of $q(k)$, denoted as $E_q(k)$, in the complex plane across the Brillouin zone.
  • Due to nonsymmorphic symmetry, $E_q(k)$ exhibits $4\pi/a$ periodicity, resulting in open paths rather than closed loops.
  • Topological phase transitions occur when these paths cross the origin.

C. Topolectric Circuit Realization

• Mapping: They map the tight-binding Hamiltonians to RLC (Resistor-Inductor-Capacitor) circuits.
• Impedance Measurement: The circuit Laplacian $J(\omega)$ is related to the Hamiltonian $H$ via $J = i\omega H$. Topological edge states or solitons are detected by measuring the two-point impedance $Z_{mn}(\omega)$. A divergence in impedance indicates a zero-energy mode.
• Components:
  • SSH/CDW: Used as a baseline comparison using staggered capacitors and inductors.
  • AII/Z4 Models: Implemented using two coupled channels (electron/hole or TRS partners) connected by Phase-Control Units (PCUs) containing operational amplifiers to simulate complex phases and non-Hermitian-like directional coupling without breaking Hermiticity.

3. Key Contributions

A. New Topological Classification

• $Z_2$ Invariant (AII Class): They demonstrate that nonsymmorphic AII systems support a $Z_2$ topological index. Unlike symmorphic AII (which is trivial), the nonsymmorphic C and S symmetries induce non-trivial topology.
• $Z_4$ Invariant (D Class): They identify a unique $Z_4$ topological classification for nonsymmorphic D-class superconductors. This is distinct from the standard $Z_2$ Majorana classification, offering four distinct topological phases ($N^{NS}_D = 1, 2, 3, 4$).
  • The phases are distinguished by transitions at $k=0$ (Majorana-like) and at non-zero $k$ values (nonsymmorphic specific).

B. Generalized Winding Number Method

The paper provides a robust method to calculate invariants for Kramers-degenerate nonsymmorphic systems by analyzing open-path trajectories in the complex plane, overcoming the limitations of standard closed-loop winding numbers.

C. Topolectric Circuit Prototypes

• They successfully design and simulate circuits for both the AII and $Z_4$ models.
• The circuits reproduce the predicted phase boundaries and impedance signatures (peaks at resonant frequencies) corresponding to zero-energy modes.
• They demonstrate that solitons (domain walls between topological phases) host zero-energy states detectable via impedance measurements.

D. Disorder Analysis and Emergent Robustness

• Finding: In the minimal nearest-neighbor $Z_4$ model, domain-wall zero modes (solitons) exhibit channel-selective robustness against disorder.
  • Disorder in the superconducting phase ($\phi_s$) does not shift the soliton energy from zero (first-order coupling is zero).
  • Disorder in the hopping parameter ($v$) does shift the energy.
• Mechanism: This is an accidental emergent property of the minimal Hamiltonian where the disorder terms do not couple to the soliton subspace at first order.
• Lifting: Introducing longer-range parameters (e.g., $p$-wave pairing $\Delta_p$ or next-nearest-neighbor hopping $t_{AA}$) breaks this emergent constraint, causing the soliton energy to shift away from zero under disorder.

4. Key Results

• Phase Diagrams: The authors derived explicit phase diagrams for the $Z_4$ model, showing four distinct phases separated by transitions dependent on chemical potential ($\mu$), hopping ($v$), pairing ($\Delta_s$), and phase ($\phi_s$).
• Impedance Signatures:
  • SSH vs. CDW: Confirmed that CDW circuits (nonsymmorphic) host soliton modes rather than edge modes, with impedance peaks localized at the soliton center.
  • AII & $Z_4$: Simulations showed distinct impedance peaks at the resonant frequency for solitons located at phase boundaries. Smooth solitons produced significantly sharper and larger impedance peaks compared to atomically sharp solitons.
• Disorder Robustness:
  • Numerical simulations of the Density of States (DOS) confirmed that solitons in the minimal model remain pinned at zero energy under specific disorder channels (e.g., phase disorder).
  • Adding long-range terms ($\Delta_p$) lifted this pinning, confirming the accidental nature of the robustness in the minimal model.

5. Significance

• Theoretical Advancement: The work expands the "ten-fold way" classification of topological matter by rigorously treating nonsymmorphic symmetries in Kramers-degenerate 1D systems, introducing a unique $Z_4$ classification.
• Experimental Feasibility: By proposing topolectric circuits, the paper offers a practical route to experimentally realize and probe these complex topological phases. Electrical circuits allow for precise tuning of parameters and direct measurement of topological signatures (impedance) that are difficult to access in solid-state nanowires or magnetic chains.
• Insight into Disorder: The discovery of channel-selective robustness in soliton states highlights a subtle interplay between symmetry breaking and emergent constraints in minimal models. This suggests that while nonsymmorphic symmetries are fragile to disorder, specific topological features can persist due to the specific structure of the Hamiltonian, a nuance critical for designing robust topological devices.
• Soliton Physics: The paper reinforces the role of solitons as topological defects in nonsymmorphic systems, showing they can host protected zero modes even in the absence of edge states.

In summary, this paper bridges the gap between abstract nonsymmorphic topological theory and experimental realization, providing a comprehensive framework for understanding, calculating, and detecting $Z_2$ and $Z_4$ topological phases in 1D Kramers-degenerate systems using topolectric circuits.