Localized-basis formulation of interacting Hamiltonians in flat topological bands: coherent states and coherent-like states for fractional physics

This paper proposes a unified framework for describing fractional quantum Hall systems and fractional Chern insulators by extending the concept of coherent states to Chern bands via "coherent-like states," enabling the construction of a localized-basis Hamiltonian that exhibits topological degeneracy and zero-energy ground states across both systems.

The Gist

Imagine you are trying to organize a massive, chaotic dance floor where the dancers are electrons. In most materials, these dancers move in predictable patterns, like a grid of people marching in step. But in topological materials (the subject of this paper), the rules of the dance floor are weird. The electrons are forced to move in loops or swirls that are protected by the very shape of the material's "energy landscape."

The problem is: How do you describe these swirling dancers using simple, local steps?

The Core Problem: The "Impossible" Grid

In normal materials, physicists use something called Wannier functions. Think of these as "personal bubbles" for each dancer. If you know where a dancer's bubble is, you know exactly where they are. You can map the whole dance floor by tiling these bubbles together.

However, in topological materials (like Quantum Hall systems or Chern insulators), the "dance floor" is twisted. It's mathematically impossible to create these perfect, non-overlapping bubbles without breaking the rules of the dance (symmetry). The bubbles would have to stretch infinitely or overlap in a way that makes them useless for calculations.

The Solution: The "Overlapping Spotlight"

The author, Nobuyuki Okuma, proposes a new way to look at the dance floor. Instead of trying to force the dancers into non-overlapping bubbles, he suggests using Coherent States.

The Analogy:
Imagine trying to take a photo of a spinning top. You can't take a sharp, single snapshot of exactly where every part of the top is at once. Instead, you use a flashlight that sweeps across the room.

• The flashlight beam is localized (it shines on a specific spot).
• But the beams overlap. If you shine a light on the left side of the room, it spills a little onto the center. If you shine it on the center, it spills onto the right.
• This creates an overcomplete basis: You have more light beams than strictly necessary to cover the room, but because they overlap, you can describe the spinning top perfectly from any angle.

In this paper, Okuma creates a mathematical "flashlight" (called a Coherent-like State) that works for these twisted topological dance floors. These "flashlights" are localized in space but overlap with their neighbors, allowing physicists to describe the system without breaking the topological rules.

The New "Interaction" Game

Once you have these overlapping flashlights, you can ask: "What happens if the dancers bump into each other?"

In the real world, electrons repel each other (they don't like to be in the same spot). Okuma builds a Hamiltonian (a mathematical rulebook for energy) based on these overlapping flashlights.

• The Rule: If two dancers are in the same "flashlight beam," they pay a penalty (energy cost).
• The Result: Because the beams overlap in a very specific, topological way, the dancers arrange themselves into a highly ordered, "fractional" state.

Why is this cool?
This single rulebook works for two very different scenarios:

1. Quantum Hall Effect: Electrons swirling in a strong magnetic field (like water going down a drain).
2. Fractional Chern Insulators: Electrons swirling in a material without a magnetic field, just because of the material's internal structure.

Okuma shows that by using his "overlapping flashlight" method, you can write one single equation that describes both phenomena. It's like discovering that the same set of dance moves works for both a ballroom waltz and a breakdance battle, provided you look at them through the right lens.

The "Magic" Outcome: Topological Degeneracy

When the author tested this new rulebook on a computer, he found something magical.

• In a normal system, if you have a group of dancers, there is usually just one best way for them to arrange themselves (the ground state).
• In these topological systems, the dancers can arrange themselves in three different ways that are all equally perfect and have the same energy.
• This is called Topological Degeneracy. It's the "fingerprint" of a fractional quantum state. It means the system is robust; you can't easily mess it up by shaking the dance floor slightly.

Extending to "Double" Dancers (Z2 Insulators)

The paper also touches on Z2 Topological Insulators.

• Imagine the dancers come in pairs (Kramers pairs) that are linked by a special symmetry (time-reversal).
• Okuma shows that his "flashlight" method can be adapted to handle these pairs naturally. Instead of one beam per dancer, you get a pair of beams that are perfectly synchronized. This opens the door to studying even more complex, "strongly correlated" topological phases.

Summary in Plain English

1. The Problem: We couldn't easily describe "twisted" topological materials using standard local tools because the math didn't fit.
2. The Fix: The author invented a new tool called "Coherent-like states," which are like overlapping spotlights that cover the material perfectly without breaking its rules.
3. The Discovery: By using these spotlights to define how electrons repel each other, he created a simple model that perfectly predicts the strange, fractional behavior of these materials.
4. The Impact: This unifies our understanding of magnetic quantum Hall systems and non-magnetic topological insulators, giving scientists a powerful new way to design materials that could revolutionize quantum computing.

In short, Okuma gave us a new pair of glasses that lets us see the hidden, swirling order of electrons in topological materials, proving that even in a chaotic, twisted world, there is a beautiful, unified structure waiting to be found.

Technical Summary

1. Problem Statement

The study of topological phases, particularly the Fractional Quantum Hall Effect (FQHE) and Fractional Chern Insulators (FCIs), faces a fundamental theoretical challenge: the inability to construct exponentially localized Wannier functions in topological bands while preserving the system's symmetries.

• Context: In the Integer Quantum Hall Effect (IQHE) and FQHE, the lowest Landau level (LLL) allows for the definition of coherent states (eigenstates of annihilation operators) which are exponentially localized in real space. These states form an overcomplete basis, enabling a real-space description of interactions.
• Gap: In Chern insulators (lattice analogs of IQHE) and other topological bands, standard Wannier functions cannot be constructed due to non-trivial topology. While "coherent-like states" were previously proposed, a unified framework to define interacting Hamiltonians for both continuum (FQHE) and lattice (FCI) systems using these localized bases was lacking.
• Goal: The author aims to construct a unified framework where interacting Hamiltonians for FQHE and FCIs are formulated using localized bases (coherent and coherent-like states), allowing for a direct comparison and understanding of fractional physics in both systems.

2. Methodology

The paper employs a combination of topological band theory, operator algebra, and exact diagonalization.

• Lattice Vortex Functions: The author defines a "lattice vortex function" $Z$, a generalization of the complex coordinate $z = x+iy$ used in the continuum. In a lattice model, $Z$ is a diagonal matrix involving position operators and projection operators onto sublattices.
• Projected Vortex Operators: The vortex function is projected onto a specific Chern band ($P Z P$). In momentum space, these projections act as differential operators ($\hat{z}_k, \hat{z}^\dagger_k$) analogous to ladder operators in the LLL.
• Exact Zero Modes: The core of the construction relies on finding the exact zero modes of the projected operator $P Z^\dagger P$.
  • According to the Atiyah–Singer index theorem, if the Chern number is $C$, there are at least $C$ zero modes.
  • For $C=1$, a unique zero mode wavefunction $a_0(k)$ exists in momentum space.
• Coherent-like States: These zero modes are used to define coherent-like states $|\zeta_R\rangle$ localized at position $R$. These states are eigenstates of the projected annihilation operator.
• Overcomplete Basis Construction:
  • To treat all momenta equally and cover the Brillouin zone, the author introduces a shift vector $\delta$ within the unit cell.
  • This creates a set of states $|\zeta_{R+\delta}\rangle$ forming an overcomplete basis (similar to the von Neumann lattice in the LLL).
  • The identity operator is decomposed using this basis: $\hat{P} = \sum_{R, \delta} f(\delta) |\zeta_{R+\delta}\rangle\langle\zeta_{R+\delta}|$.
• Interaction Hamiltonian: A minimal repulsive interaction Hamiltonian is defined on this basis:
  $$H = \sum_{R, \delta} U(\delta) n_{R,\delta,0} n_{R,\delta,1}$$
  where $n$ is the number operator for local orbitals defined by the coherent-like states. This Hamiltonian is mapped to momentum space, replacing standard Bloch states with the coefficients of the coherent-like states.
• Extension to $Z_2$ Insulators: The framework is extended to $Z_2$ topological insulators, where time-reversal symmetry leads to Kramers-degenerate coherent-like pairs.

3. Key Contributions

1. Unified Formalism: The paper provides a single mathematical framework to describe both FQHE (continuum) and FCIs (lattice). The difference between the two systems is encoded solely in the functional form of the wave packet $a_0(k)$ in momentum space, while the interaction Hamiltonian structure remains identical.
2. Definition of Coherent-like States: It rigorously defines coherent-like states for Chern bands using the zero modes of projected vortex functions, establishing them as a valid substitute for Wannier functions in topological systems.
3. Proof of Zero-Energy Ground States (FQHE): By relating the constructed Hamiltonian to Haldane's pseudopotential models, the author proves that in the lowest Landau level, the Hamiltonian possesses exact zero-energy ground states with topological degeneracy (3-fold degeneracy for $\nu=1/3$).
4. Numerical Verification (FCI): The paper performs exact diagonalization on representative Chern insulator models (Checkerboard lattice and Qi-Wu-Zhang model). It demonstrates that the Hamiltonian exhibits nearly threefold-degenerate gapped ground states, confirming the emergence of FCI physics.
5. $Z_2$ Extension: It proposes a method to define coherent-like states in $Z_2$ topological insulators that preserve Kramers degeneracy, opening a path to study fractional phases in time-reversal invariant systems.

4. Results

• Analytical Results:
  • The coherent-like states are shown to be exponentially localized in real space.
  • The Hamiltonian defined on these states is shown to be frustration-free in the LLL limit, guaranteeing exact zero-energy ground states.
  • The scattering matrix elements in the momentum-space representation of the Hamiltonian exhibit a symmetry that mimics continuous translational invariance, which is crucial for stabilizing fractional states.
• Numerical Results:
  • Checkerboard Model: For filling $\nu=1/3$, the system shows a clear gap with three nearly degenerate ground states at specific total momenta, characteristic of the FQHE.
  • QWZ Model: Similar topological degeneracy is observed, though the stability depends on the interaction details (specifically the position dependence $U(\delta)$).
  • Stability: The results suggest that while the topological degeneracy is robust, the energy gap and the exact nature of the ground state can be sensitive to the microscopic details of the interaction and the band geometry (dispersion).

5. Significance

• Theoretical Unification: This work bridges the gap between continuum quantum Hall physics and lattice Chern insulator physics. It demonstrates that the "fractional" nature of these phases arises from the non-orthogonality and localization properties of the basis states, rather than the specific details of the magnetic field or lattice.
• Model for FCIs: It offers a concrete, minimal Hamiltonian model for FCIs that can be used to study strongly correlated topological phases without relying on complex, ad-hoc interaction terms.
• Experimental Relevance: With recent experimental signatures of FCIs in moiré materials (e.g., twisted $MoTe_2$), this framework provides a theoretical tool to interpret these findings and design materials with ideal flat bands and specific interaction profiles.
• Future Directions: The extension to $Z_2$ insulators suggests that this localized basis approach can be applied to a broader class of topological materials, potentially leading to the discovery of new fractional topological phases in time-reversal invariant systems.

In summary, Okuma successfully constructs a localized-basis formulation that unifies the description of fractional quantum Hall effects and fractional Chern insulators, proving that the essential physics of fractionalization can be captured by a simple repulsive interaction defined on coherent-like states.