Topological markers for a one-dimensional fermionic chain coupled to a single-mode cavity

This paper investigates how a single-mode photonic cavity modifies the topological phases of a one-dimensional Su-Schrieffer-Heeger chain in the off-resonant regime by deriving an effective interacting Hamiltonian and validating the resulting phase diagram through consistent agreement between winding number, bulk polarization, and edge correlation function markers.

The Gist

Imagine a long, narrow hallway made of stepping stones. This is our Su-Schrieffer-Heeger (SSH) chain, a model physicists use to understand how electrons move through certain materials.

In a normal hallway, the stones are arranged in pairs. Sometimes the gap between the two stones in a pair is small (easy to jump), and the gap to the next pair is large (hard to jump). Other times, it's the opposite.

• The Trivial Phase: If the "easy" jumps are inside the pairs, the hallway is boring. If you drop a ball (an electron) in the middle, it stays stuck there.
• The Topological Phase: If the "easy" jumps connect the pairs to each other, the hallway becomes "topological." This is a fancy way of saying the material has a special, protected property. If you drop a ball at the very end of the hallway, it gets stuck there, unable to move to the middle. These stuck balls are called edge states, and they are the "magic" of topological materials.

The New Twist: The Cavity Mirror

Now, imagine we put this entire hallway inside a giant, mirrored room (a cavity) that bounces light back and forth. The light isn't just shining on the stones; it's interacting with the electrons jumping between them.

Usually, when physicists study this, they try to track every single electron and every single photon (particle of light) at the same time. It's like trying to solve a puzzle where the pieces keep changing shape and multiplying. It's a nightmare of complexity.

The Authors' Trick: The "High-Frequency" Shortcut

The authors of this paper, Anna and Olesia, decided to take a shortcut. They assumed the light in the room is oscillating (wiggling) incredibly fast—so fast that the electrons can't really keep up with the individual wiggles.

Think of it like this: If you spin a fan blade super fast, it looks like a solid, blurry disk to your eye. You don't see the individual blades anymore; you just see a new, solid object.

By using a mathematical tool called High-Frequency Expansion, they "blurred" the light. Instead of tracking the light and electrons separately, they calculated what the hallway looks like to the electrons when the light is spinning that fast.

• The Result: The light disappears as a separate entity and reappears as a new set of rules for the electrons. The electrons now feel like they are jumping on a modified hallway where the gaps have changed size, and they are "talking" to each other over long distances through the light.

How Did They Check if the Magic Still Worked?

Now that they had this new, simplified "electron-only" hallway, they needed to check: Is it still topological? Does it still have those special stuck balls at the ends?

They used three different "detectors" (markers) to check:

1. The "Handshake" Test (Correlation Functions):
   They checked if the electron at the very left end of the hallway could "feel" the electron at the very right end. In a topological hallway, these two ends are mysteriously connected, even though they are far apart. They found that yes, the ends were still holding hands, proving the edge states were still there.

2. The "Spiral Map" (Winding Number):
   Imagine drawing a line that represents the electron's path as it moves through the material. In a normal hallway, the line just goes back and forth. In a topological hallway, the line makes a full circle (a spiral) around a central point.
   They calculated this "spiral" for their new light-modified hallway. They found that the line still made a full circle. The "topology" (the shape of the path) was preserved, even with the light involved.

3. The "Electric Balance" (Polarization):
   This is like checking if the hallway is perfectly balanced. In a topological state, the electric charge is shifted slightly to one side in a very specific, quantized way (like a scale that only tips to exactly "0" or exactly "1/2"). They measured this balance and found it matched the "Spiral Map" perfectly.

The Big Discovery

The most exciting part is that all three different detectors agreed perfectly.

• The "Handshake" showed the edge states existed.
• The "Spiral Map" showed the topology was there.
• The "Electric Balance" confirmed the phase transition.

They also discovered that the light acts like a tuning knob. By changing how strong the light is or how the stones are spaced, they could shift the point where the hallway changes from "boring" to "magical."

Why Does This Matter?

This paper is a bridge. It takes a super complicated problem (light + matter) and turns it into a simpler, well-understood problem (just matter, but with new rules).

It proves that even when you put a topological material in a box of light, you can still use the old, reliable tools we have for studying matter to understand what's happening. It's like realizing that even if you paint a picture of a mountain in neon colors, you can still use the same map to find the peak. This gives scientists a powerful new way to design materials that can be controlled by light, potentially leading to better computers or new types of sensors.

Technical Summary

1. Problem Statement

The paper addresses the challenge of characterizing topological phases in quantum materials when they are strongly coupled to cavity photons. While cavity quantum electrodynamics (cQED) offers a route to manipulate material properties (e.g., modifying superconductivity or topological insulators), standard topological invariants (like the winding number or Zak phase) are typically defined for non-interacting, free-fermion systems.

When a topological system (specifically the Su-Schrieffer-Heeger or SSH chain) is coupled to a cavity, the resulting light-matter Hamiltonian involves both fermionic and photonic operators. This creates an interacting many-body problem where:

• Standard single-particle Bloch band theory breaks down.
• Directly computing topological invariants for the full light-matter system is computationally expensive and conceptually complex.
• There is a need to determine if the "effective" interacting electronic system retains well-defined topological markers and how the cavity modifies the phase diagram.

2. Methodology

The authors employ a multi-faceted approach combining analytical expansions and numerical exact diagonalization (ED):

• Model System: A 1D SSH chain (dimerized with alternating hoppings $v$ and $w$) coupled to a single-mode photonic cavity with frequency $\omega_c$. The coupling is introduced via the Peierls substitution, dressing the hopping terms with a phase dependent on the vector potential and the intra-cell distance $d$.
• High-Frequency Expansion (HFE): To simplify the problem, the authors apply a high-frequency expansion valid in the off-resonant regime ($\omega_c$ is the largest energy scale). This expansion is non-perturbative in the light-matter coupling strength $g$.
  • The expansion yields an effective, purely fermionic Hamiltonian ($H_{\text{eff}}$) acting within the zero-photon subspace.
  • $H_{\text{eff}}$ contains renormalized hopping amplitudes ($v_{\text{eff}}, w_{\text{eff}}$) and long-range, cavity-mediated interactions (non-local correlations between hopping terms).
• Topological Markers: Instead of treating the full light-matter system, the authors analyze $H_{\text{eff}}$ using three distinct markers adapted for interacting systems:
  1. Edge Correlation Functions: Calculating two-point correlations $\langle c^\dagger_{1,A} c_{j,\alpha} \rangle$ in an open chain to detect edge states.
  2. Winding Number: Using the single-particle Green's function $G(k, \Omega)$ at zero frequency. They verify that a generalized chiral symmetry holds for $H_{\text{eff}}$, allowing the definition of a quantized winding number $\nu$.
  3. Bulk Electric Polarization: Using Resta's many-body formula for polarization $P$ on a chain with periodic boundary conditions, relying on inversion symmetry.

3. Key Contributions

• Derivation of an Effective Interacting Hamiltonian: The authors explicitly derive the effective Hamiltonian up to the first order in $1/\omega_c$, demonstrating that cavity coupling induces long-range interactions that cannot be captured by simple mean-field renormalization of hoppings.
• Validation of Chiral Symmetry: They prove that despite the presence of cavity-mediated interactions, the effective Hamiltonian retains a generalized chiral symmetry. This ensures that the winding number remains a valid, quantized topological invariant for the interacting system.
• Comparative Analysis of Markers: The study provides a rigorous comparison between three different topological markers (edge correlations, winding number, and polarization) in the context of cavity-coupled systems, showing they yield consistent results.
• Benchmarking Approaches: The work contrasts their exact diagonalization of the interacting effective Hamiltonian with previous mean-field (Hartree-Fock) treatments, highlighting the importance of interaction terms in the phase diagram.

4. Key Results

• Agreement of Markers: There is excellent agreement between the winding number ($\nu$) and the bulk polarization ($P$) approaches.
  • Trivial Phase: $\nu = 0$, $P = 0$ (mod 1).
  • Topological Phase: $\nu = 1$, $P = 1/2$ (mod 1).
  • The transition points identified by both methods coincide numerically.
• Edge State Verification: Calculations of the two-point correlation function on open chains confirm the presence of edge states in the topological phase. The correlation function shows non-local correlations between the two edges, and the energy splitting between the ground state and the first excited state decreases exponentially with system size, characteristic of topological edge modes.
• Phase Diagram Shifts: The cavity coupling shifts the critical point of the topological phase transition ($w/v$ ratio).
  • The shift depends on the coupling strength $g$, the cavity frequency $\omega_c$, the system size $L$, and the intra-cell distance $d$.
  • Specifically, when $d=0.5$ (symmetric unit cell), the phase transition point is unaffected by the cavity coupling because the Peierls phases factorize. For $d \neq 0.5$, the transition point shifts significantly.
• Validity of HFE: The results obtained from the effective Hamiltonian ($H_{\text{eff}}$) show very good agreement with calculations performed on the full light-matter Hamiltonian, validating the high-frequency expansion for studying topological properties in this regime.
• Finite Size Effects: While the study uses small system sizes ($L \leq 7$) due to computational constraints of ED, the results suggest that finite-size effects do not significantly alter the topological classification (winding number), consistent with previous literature on interacting 1D systems.

5. Significance

• Alternative Perspective: The paper offers a complementary approach to studying light-matter topological phases. Instead of treating the full light-matter entanglement directly, it maps the problem to an interacting electronic system where established topological tools can be applied.
• Beyond Mean-Field: By including the interaction terms generated by the HFE, the authors demonstrate that mean-field approximations (which often ignore these long-range correlations) may fail to capture the correct phase boundaries, especially at strong coupling.
• Framework for Future Work: The methodology provides a robust framework for extending topological markers to other light-matter systems. The authors suggest that these techniques can be scaled up using numerical methods like Density Matrix Renormalization Group (DMRG) to study larger systems.
• Experimental Relevance: The findings are relevant for ongoing experiments in cavity quantum materials, particularly those involving ultracold atoms, graphene nanoribbons, or superconducting circuits where topological phases are manipulated via cavity coupling.

In summary, the paper successfully establishes that topological invariants remain well-defined and quantized in cavity-coupled SSH chains when treated as effective interacting fermionic systems, providing a clear roadmap for characterizing and manipulating topological phases in the strong light-matter coupling regime.