Loop unitary and phase band topological invariant in generic multi-band Chern insulators

This article generalizes the dynamical 3-winding number invariant of minimal two-band systems to generic multiband Chern insulators, proves its equivalence to the difference of Chern numbers between post- and pre-quench Hamiltonians, and reveals unique multiple fermion structures in the phase band that are inaccessible in two-band models.

The Gist

The Big Picture: Observing a Quantum System "Jumping"

Imagine a complex machine made of many gears (this represents a multiband Chern insulator, a type of quantum material). Normally, this machine is in a stable, quiet state.

In this paper, the authors investigate what happens when you suddenly push the machine (a "quench"). They immediately change the rules by which the gears interact. The machine does not simply stop; it begins to rotate and evolves over time.

The big question the authors ask is: Can we measure the "topology" (the shape or knot-like structure) of this machine solely by how it moves after being pushed?

The Problem: Too Many Gears

For simple machines with only two gears (two-band systems), scientists already knew how to do this. They could track the movement and count a number that revealed something about the hidden shape of the machine.

However, real materials are like machines with many gears (multiband systems). The mathematics for this is incredibly messy and complicated. The authors wanted to find out whether the same "counting trick" also works for these complex, multi-gear machines.

The Solution: The "Loop Unitary" and the "Phase Band"

To solve this, the authors used a mathematical tool called the Loop Unitary.

• The Analogy: Imagine taking a photo of the machine at the beginning and then another photo of it after it has evolved for a specific period. The "Loop Unitary" is like a video loop that connects the initial state with the final state and back again, thereby creating a closed circle in time and space.

They proved that if you count the "twists" and "turns" in this video loop (which they call the 3-Winding Number), you obtain a specific integer.

• The Result: This number is exactly equal to the difference between the "shape" of the machine before the push and the "shape" of the machine after the push. It works perfectly, even for machines with many gears.

The Surprise: "Gapless Fermions" as Defects

The most exciting part of the paper is how they visualized this number.

In the simple two-gear machines, the "twists" in the video loop appeared as single points where the gears briefly stopped turning smoothly. In physics, these are called Weyl Fermions (like tiny, massless particles).

The authors discovered that in these complex, multi-gear machines, the "twists" can manifest as multiple fermions.

• The Analogy: Imagine a street intersection.
  • In the simple case, a "defect" is a single car stuck at a red light (a two-way intersection).
  • In the new multi-gear case, the authors found a scenario where three streets meet at a single point, creating a "traffic jam" there. This is a triple fermion.

They showed that by pushing a specific three-gear machine, they could create a "jam" where three different energy paths meet at a single point in time and space. This is something that simply cannot happen in the simpler two-gear machines.

Why This Matters (According to the Paper)

1. It is universal: They proved that this method works for any number of gears (bands), not just the simple ones.
2. It is visual: Instead of just doing abstract mathematics, they showed that these "twists" look like specific defects (such as the three-way jam) in the "Phase Bands" (a map of the machine's movement).
3. It connects the static and the dynamic: They linked the static shape of the material (before the push) with the dynamic movement (after the push) using these defects.

Summary

The authors took a complex mathematical tool used for simple quantum systems and successfully improved it so that it works for complex, multi-layered systems. They proved that the "shape" of the system before and after a sudden change can be measured by counting the "twists" in its time evolution. Particularly noteworthy is their finding that these twists can manifest as complex, multi-way intersections (triple fermions) in the system's movement, a phenomenon previously unknown in these types of dynamic systems.

Technical Summary

Technical Summary: Loop Unitarity and Topological Invariance of Phase Bands in Generic Multi-Band Chern Insulators

Problem Statement
While dynamical topological invariants have been formulated for quench dynamics in topological phases, existing approaches are largely restricted to minimal models where Hamiltonians are expanded using Gamma matrices (typically two-band systems). In generic condensed matter scenarios involving multi-band Chern insulators, the mathematical complexity of generalized Bloch spheres and the non-abelian nature of the underlying algebras (e.g., $su(N)$) have hindered the formulation of explicit dynamical topological invariants. In particular, it remained unclear whether the loop unitarity formalism, which successfully describes two-band quench dynamics via a 3-winding number, could be generalized to multi-band systems to yield integer invariants corresponding to the difference of static Chern numbers. Furthermore, it was unknown whether dynamical invariants in multi-band systems would manifest as gapless fermions in phase bands, and if so, whether these could include multiple fermions (degeneracy > 2) that are absent in two-band models.

Methodology
The authors generalize the loop unitarity operator and its associated homotopy invariant (3-winding number) from two-band to generic $N$-band Chern insulators.

1. Band Flattening: A specific band flattening procedure is defined for multi-band Hamiltonians ($H \to h = \sum_a \text{sgn}(E_a)P_a$), where $P_a$ are eigenprojectors. This ensures the $\pi$-periodicity of the loop unitarity operator $U_l(t) = e^{-i h t} e^{i h_0 t}$ in the time direction, a prerequisite for the invariant to be an integer.
2. Algebraic Proof: In contrast to previous proofs relying on Pauli matrices and abelian Chern-Simons actions, the authors apply a direct algebraic approach using projection operators. They decompose the 3-winding number $W_3[U_l]$ into contributions from the pre-quench and post-quench unitary evolutions separately.
3. Phase Band Formalism: The loop unitarity is expressed in its eigenbasis ($U_l = \sum_a e^{i\phi_a} P_a$), thereby defining "phase bands" $\phi_a(k,t)$. The authors derive an expression for $W_3$ as a sum of topological numbers assigned to each phase band, using the Berry curvature of the loop unitarity.
4. Construction of a Specific Model: To demonstrate the occurrence of multiple fermions, the authors construct a specific three-band quench model using $su(3)$ generators (generalized Gell-Mann matrices) and contrast this with the standard Qi-Wu-Zhang model.

Main Contributions and Results

• Generalization of the 3-Winding Number: The work proves that for generic multi-band Chern insulators, the homotopy invariant of the loop unitarity is an integer 3-winding number. Crucially, its value exactly corresponds to the difference between the post-quench and pre-quench static Chern numbers ($W_3[U_l] = C_f - C_i$).
• Decoupling of Phase Bands: A central result is that in the phase band representation, the contributions of different bands decouple. The total 3-winding number is a sum of integer topological numbers from individual phase bands. This enables the calculation of the invariant using eigenprojectors without requiring an expansion in specific matrix algebras (such as Pauli matrices), thereby making the result applicable to any Hermitian Hamiltonian.
• Manifestation as Gapless Fermions: It is shown that the 3-winding number manifests as defects (gapless points) in the phase bands within $(k, t)$ space. These defects are identified as chiral gapless fermions with topological charge.
• Occurrence of Multiple Fermions: By quenching a specific three-band model, the authors demonstrate the occurrence of a triple fermion (a defect with topological charge 2) in the phase band. This is a phenomenon impossible in two-band models, where only Weyl fermions (two-fold degeneracy) can occur. The authors identify a "0-defect" (where the phase is 0) as the location of this multiple fermion.
• Formalism for $su(N)$: The work provides explicit formulas for calculating the topological charge using the coherence vector and the structure relations of $su(N)$ algebras, facilitating the visualization of Berry curvature vector fields for $N > 2$.

Significance and Claims
The authors claim that this work clarifies two fundamental questions regarding the generalization of loop unitarity dynamics:

1. It confirms that the Wess-Zumino-Witten term (3-winding number) yields an integer invariant for generic multi-band systems, provided the correct band flattening is applied.
2. It establishes a generic correspondence between static gapped fermions and dynamical gapless fermions in phase bands, extending this relationship to multiple fermions.

The work positions this approach as a practical tool for describing quench dynamics in multi-band insulators, overcoming the limitations of minimal models. It highlights that defects in phase bands offer a new perspective relating gapped and gapless fermions across different dimensions, distinct from the traditional bulk-edge correspondence. The authors note that while the current work focuses on three-band models, higher-order fermions can occur in systems with more bands. They further suggest that the combination of coherence vector mapping (via time-of-flight) and defect measurement methods could enable the experimental detection of these phenomena in multi-band models. The work proposes no new experimental setups but rather provides the theoretical framework and invariants required to interpret such dynamics.