Topological Floquet Green's function zeros

This paper investigates topological Green's function zeros in interacting Floquet systems, specifically Kitaev-like chains in symmetry class BDI, by deriving new topological invariants that account for both bulk and edge zeros (which can exist even without interactions) and proposing a digital quantum emulator circuit to detect these boundary features.

The Gist

Imagine you are trying to understand the behavior of a complex crowd of people (electrons) in a city. Usually, physicists use a map called the Green's function to predict where people will go. Think of this map as a weather forecast for the crowd: it tells you where the "storms" (poles) are likely to happen. In a calm, normal city, the map only shows storms.

However, in some very chaotic or "strongly interacting" cities, the map starts showing holes (zeros) where the weather is perfectly calm, even though the crowd is wild. These "holes" aren't empty; they are special features that tell us the city has a hidden, robust structure. This is the concept of Topological Green's Function Zeros.

Now, imagine this city isn't static; it's a Floquet system. This means the city's rules change rhythmically, like a DJ spinning a record that repeats every few seconds. The city goes through a cycle of "day" and "night" (or different phases of a drive) over and over again.

This paper, written by Elio J. König and Aditi Mitra, explores what happens when you combine these two ideas: Topological Zeros in a Rhythmically Changing City.

Here is the breakdown of their discovery using simple analogies:

1. The "Magic" of the Rhythm (Floquet Systems)

In a normal, static city (equilibrium), if you have a crowd of people who don't talk to each other (free electrons), your weather map can only show storms. It cannot show "holes" (zeros). You need a lot of chaos (interactions) to create a hole.

But in this Floquet city, the rhythm itself changes the rules. The authors discovered that even if the people in the crowd don't talk to each other at all (free fermions), the rhythmic spinning of the DJ (the Floquet drive) creates holes in the map automatically. It's as if the rhythm of the music creates "quiet zones" in the crowd just by virtue of the beat. This is a new phenomenon unique to time-driven systems.

2. The "Symmetric Mass Generation" (The Interaction)

The paper then asks: What happens if we add real interactions? Specifically, they look at a special type of interaction called Symmetric Mass Generation (SMG).

Imagine the crowd is made of pairs of dancers. In a normal topological city, these dancers are stuck at the edges of the city, creating a "protected" boundary that is hard to break. SMG is like a magical handshake that allows these edge dancers to pair up and disappear into the crowd, making the edge look "trivial" or empty.

Usually, when the edge dancers disappear, the whole city loses its special topological identity. But here is the twist: The topological identity doesn't vanish; it just moves.

3. The "Ghost" in the Machine (The Zeros)

Even though the edge dancers (the physical particles) have disappeared due to the interaction, the Green's function zeros (the "holes" in the map) remain!

• Before Interaction: The map has storms (poles) at the edge.
• After Interaction: The storms disappear, but the "holes" (zeros) appear at the edge.

The authors show that these "holes" act as the new guardians of the city's topology. They prove that you can count these holes to determine if the city is still topologically special, even if the physical particles at the edge are gone. It's like a building that looks empty, but if you look at the shadows it casts (the zeros), you can still tell it's a cathedral.

4. The Digital Quantum Emulator (The Experiment)

The paper isn't just math; it proposes a way to test this on a Quantum Computer (specifically, a NISQ device, which is a noisy, early-stage quantum computer).

They designed a "circuit" (a recipe of quantum gates) that acts like the rhythmic city.

• The Setup: They use qubits (quantum bits) to represent the dancers.
• The Interaction: They program a specific interaction (the Fidkowski-Kitaev interaction) that forces the dancers to pair up and vanish from the edge.
• The Measurement: They don't measure the dancers directly (which would be hard). Instead, they measure the autocorrelation (how the system remembers its past state).

The Analogy: Imagine you want to know if a room is empty. Instead of looking inside, you clap your hands and listen to the echo.

• If the room is full of people (poles), the echo is loud and complex.
• If the room is empty but has a special shape (zeros), the echo has a specific, silent pattern.

The paper shows that by measuring this "echo" on a quantum computer, you can detect the Topological Zeros. This proves that even on a noisy, imperfect quantum computer, we can detect these deep, hidden topological features.

Summary of the Big Picture

• Old Idea: Topology is defined by where the particles (storms) are.
• New Idea: Topology can also be defined by where the particles aren't (the zeros), especially in rhythmic, time-driven systems.
• The Breakthrough: Even when interactions make the physical edge particles disappear, the "zeros" stay behind, preserving the topological fingerprint.
• The Application: We can build a digital quantum circuit to simulate this and "hear" the zeros, proving that these abstract mathematical concepts are real and observable.

In short, the paper tells us that in the quantum world, what is missing (zeros) can be just as important as what is present (poles), and rhythmic driving can create these "missing" features out of thin air.

Technical Summary

1. Problem Statement

The paper addresses the characterization of interacting topological phases in non-equilibrium (Floquet) systems, specifically focusing on Green's function zeros.

• Context: While topological invariants based on Green's function poles (quasiparticle excitations) are well-established for equilibrium systems, the role of Green's function zeros (vanishing spectral weight) in interacting systems is a growing area of interest. In equilibrium, zeros typically require strong interactions (e.g., Symmetric Mass Generation, SMG) to appear.
• Gap: There is a lack of understanding regarding how these zeros behave in periodically driven (Floquet) systems, particularly in the presence of interactions. Furthermore, existing topological invariants for Floquet systems often rely on free-fermion classifications, which fail to capture the physics of strongly correlated interacting phases.
• Goal: To develop a framework for defining and calculating Floquet Green's-function-based topological invariants for interacting systems (Symmetry Class BDI) and to demonstrate how these invariants capture topological features via Green's function zeros, even in the absence of interactions in the Floquet setting.

2. Methodology

The authors employ a combination of analytical field theory, perturbation theory, and digital quantum simulation concepts:

• Model System: They study 1D interacting Kitaev-like Floquet chains (equivalent to transverse-field Ising circuits) with Majorana fermions. The system belongs to Symmetry Class BDI (possessing time-reversal, particle-hole, and chiral symmetries).
• Green's Function Formalism:
  • They define discrete-time retarded and advanced Green's functions, $G_R(n)$ and $\tilde{G}_R(n)$, associated with two distinct Floquet cycles: one starting at $t=0$ and another shifted by half a period ($t=T_F/2$).
  • They introduce two topological invariants, $N_1$ and $\tilde{N}_1$, defined as integrals over momentum space involving the Green's function and its derivative:
    $$N_1 = \int \frac{dp}{4\pi i} \text{tr}[\Gamma G_R^{-1} \partial_p G_R]_{\Omega=0}$$
    (and similarly for $\tilde{N}_1$).
• Interaction Mechanism (SMG): They introduce Fidkowski-Kitaev (FK) interactions to induce Symmetric Mass Generation (SMG). This mechanism gaps out edge Majorana zero modes ($\pi$ modes) without breaking symmetry, effectively "trivializing" the boundary while preserving bulk topology in a specific sense.
• Analytical Calculation:
  • Free Limit: They derive exact Green's functions for the non-interacting Floquet chain, showing that free Floquet systems naturally possess Green's function zeros (unlike continuum equilibrium systems).
  • Interacting Limit: Using Floquet perturbation theory near special points in the phase diagram (where the system decays into FK quantum dots), they analytically calculate bulk and edge Green's functions in the presence of interactions.
• Quantum Emulation: They propose a digital quantum circuit implementation using Jordan-Wigner transformation to simulate the FK interaction and measure the relevant observables (edge autocorrelations) on NISQ devices.

3. Key Contributions

1. Floquet Green's Function Zeros in Free Systems: The authors demonstrate a fundamental distinction between continuum and Floquet time evolution: Free Floquet systems inherently possess Green's function zeros (interposed between poles in the extended quasienergy zone), whereas free continuum systems do not.
2. New Topological Invariants: They define two robust topological invariants ($N_1, \tilde{N}_1$) based on Green's functions that remain valid for interacting systems. These invariants reduce to linear combinations of established free-fermion Floquet invariants ($\nu_0, \nu_\pi$) when interactions are turned off.
3. Bulk-Boundary Correspondence for Zeros: They establish a generalized bulk-boundary correspondence where the topological invariant is determined entirely by the topological bands of Green's function zeros in the bulk, rather than poles. In the interacting SMG phase, the bulk poles disappear (becoming trivial), but the bulk zeros remain topological and carry the invariant.
4. Edge Zeros in SMG: They show that while SMG gaps out the edge Majorana modes (removing spectral weight at $\Omega=0, \pi$), it leaves behind topological edge zeros in the Green's function at these frequencies.
5. NISQ Implementation: They provide a concrete circuit design for a gate-based quantum emulator to observe these phenomena, identifying specific observables (edge autocorrelations) that encode the topological zeros.

4. Key Results

• Free Fermion Phase Diagram: The free Floquet Kitaev chain is characterized by integers $(\nu_0, \nu_\pi)$ representing the number of Majorana zero and $\pi$ modes. The authors show that $N_1 = \nu_0 + \nu_\pi$ and $\tilde{N}_1 = \nu_0 - \nu_\pi$.
• Interaction Effects (SMG):
  • When FK interactions are applied to specific phases (e.g., 8 chains with $(\nu_0, \nu_\pi) = (8,0)$), the edge modes are gapped out.
  • Edge Green's Function: The edge Green's function develops a zero at $\Omega=0$ (or $\pi$) even though the spectral weight (poles) is shifted away. The zero is pinned by symmetry.
  • Bulk Green's Function: In the interacting bulk, the topological bands of poles vanish, but topological bands of zeros emerge. These zero bands have an "opposite" Hamiltonian structure compared to the non-interacting pole bands.
• Invariant Stability: The calculated invariants $N_1$ and $\tilde{N}_1$ for the interacting system remain identical to those of the corresponding non-interacting system in the same parameter regime. This confirms that the invariants are robust against interactions that induce SMG.
• Circuit Implementation: The paper details how to map the Majorana model to qubits. The FK interaction (a 4-body term) can be implemented using generalized Toffoli gates or specific sequences of two-qubit gates. The edge autocorrelation function $\langle Z_1(n) Z_1(0) \rangle$ is shown to directly relate to the retarded Green's function $G_R(n)$, allowing experimental detection of the zeros.

5. Significance

• Theoretical Advancement: This work extends the theory of topological Green's function zeros from equilibrium to the non-equilibrium Floquet regime. It resolves the question of how topological invariants are defined when the bulk spectrum is trivialized by interactions (SMG), showing that zeros, not poles, carry the topological information in such phases.
• Experimental Relevance: By focusing on NISQ devices, the paper bridges the gap between abstract many-body theory and near-term experimental realization. It provides a concrete protocol to detect topological features (zeros) that are difficult to observe in solid-state materials due to disorder and decoherence.
• New Diagnostic Tool: The identification of Green's function zeros as a primary diagnostic for interacting Floquet SPT phases offers a new tool for classifying quantum matter, particularly in regimes where traditional gap-closing criteria or free-fermion invariants fail.
• Symmetry Protection: The results reinforce the concept that Symmetric Mass Generation is a robust mechanism for trivializing boundary states while preserving bulk topology, but with the unique signature of persistent Green's function zeros.

In summary, König and Mitra provide a comprehensive framework for understanding and detecting topological order in interacting, periodically driven systems, highlighting the crucial role of Green's function zeros as the carriers of topological information in the presence of strong correlations.