Krylov Complexity: Flat bands and Carroll breaking deformations

This paper investigates the state dynamics of flat band systems invariant under supertranslation symmetries by utilizing Krylov complexity to analyze quenches induced by Carroll-breaking perturbations, revealing how this measure resolves phase-dependent resilience and exhibits UV/IR mixing in continuum Carroll scalar field theories.

The Gist

The Big Picture: A World Where Nothing Moves

Imagine a crowded dance floor where, under normal rules, people can walk around, bump into each other, and spread out. This is how most quantum systems work: energy moves, and information spreads.

But this paper studies a very strange, special kind of dance floor called a "Flat Band" system. In this world, the "dance floor" is designed so perfectly that no one can move. If you place a dancer (a particle) in a spot, they stay there forever. They are "frozen" in place.

In physics, this happens because of a hidden symmetry called Carrollian symmetry (named after a fictional character who moves very slowly). In this state, the system is "ultra-local," meaning every spot on the floor is completely disconnected from its neighbors. It's like having a room full of people in soundproof glass boxes; no matter what happens in one box, the others don't know about it.

The Theoretical Study: Breaking the Freeze

The researchers wanted to see what happens if they "break" this perfect freeze. They introduced a tiny nudge (a perturbation) that allows the dancers to finally take a step.

They asked: How quickly does the system explore new patterns once the freeze is broken?

To measure this, they used a tool called Krylov Complexity. Think of this as a "spreadometer." It doesn't just count how many people moved; it measures how much the entire pattern of the dance floor has changed and how deeply the system has explored all its possible arrangements.

The Two Types of Dancers

The system has two main "phases" or types of dance floors, and they react very differently to the nudge:

1. The "Vanilla" Phase (The Uniform Start)

• The Setup: Everyone is frozen in a very specific, simple, and uniform pattern. Imagine that every single dancer is sitting in the exact same kind of local frozen box. It is a simple, unique arrangement where everyone is identical in their isolation.
• The Reaction: As soon as the "freeze" is weakly broken, this simple arrangement starts exploring new dance patterns efficiently. The complexity (the spread) grows rapidly because the system quickly finds new ways to arrange itself.
• The Analogy: Imagine a group of people standing in perfect, identical rows. The moment the music starts, they all have the same simple instructions to move, so they quickly begin to fill the floor with new formations. The "Vanilla" state is efficient at starting to move.

2. The "Exotic" Phase (The Resilient Puzzle)

• The Setup: This is a much more complex state. There are millions of different ways to arrange the dancers that all look the same energy-wise. It's a giant, degenerate puzzle.
• The Reaction: Here, the result depends entirely on which specific puzzle piece you start with.
  • The "Frozen" Pieces: Some specific arrangements are so perfectly aligned that even when the nudge is applied, they don't move at all. They are "immune" to the breaking of the symmetry.
  • The "Fast" Pieces: Other arrangements have "active links" (places where a dancer is right next to an empty spot on the same side). These start moving very fast, spreading out even quicker than the "Vanilla" phase.
  • The "Middle" Pieces: Some arrangements move at a moderate pace.
• The Analogy: Imagine a giant jigsaw puzzle. If you pick up a piece that fits perfectly into a locked slot, it won't budge. But if you pick up a piece that is hanging off the edge, it will fall immediately. The "Exotic" phase is a mix of locked pieces and loose pieces.

The "Active Link" Secret

The researchers discovered a simple rule to predict how fast a state will spread, but it applies specifically to a special family of exotic frozen arrangements. They called it the "Active Link" count.

• Imagine the dancers are on a ladder. An "active link" exists if a dancer is standing on a rung next to an empty rung of the same color.
• Zero Active Links: The state is frozen. It doesn't care about the nudge.
• Many Active Links: The state is ready to run. It spreads complexity very quickly.

This rule allows them to predict exactly how "brittle" or "resilient" a specific quantum state is within this exotic family, just by looking at its pattern. The author offers a helpful analogy: it is like a group of people who start enjoying jazz music and break into a swing dance (high active links, fast movement), versus another group who find it boring and barely move (zero active links, frozen). The active-link count distinguishes between these groups.

The Continuum Analogy: The Smooth Field

To provide a complementary perspective, the researchers also looked at a continuous field theory (like a smooth sheet of rubber instead of a grid of rungs).

• They found that when they tried to make this smooth sheet move, the math got weird. The speed at which things spread depended heavily on the tiniest, most microscopic details (the "ultraviolet" scale).
• The Analogy: It's like trying to measure the smoothness of a beach by looking at individual grains of sand. In this "Carroll" world, the behavior of the whole system is dictated entirely by the smallest, invisible grains. This is called UV/IR mixing—a fancy way of saying the tiny stuff controls the big stuff. This smooth-field version complements the grid-based story, showing that the sensitivity to microscopic details is a robust feature of these systems.

The Conclusion

The paper concludes that Krylov Complexity is a powerful new tool for understanding these quantum systems.

1. It reveals that "flat band" systems are not just static; they have hidden layers of resilience.
2. It shows that some quantum states are naturally protected against chaos (the "frozen" exotic states), while others are incredibly efficient at exploring new patterns.
3. It indicates that in these special systems, the way a state spreads is determined by its local geometry (the "active links") and its sensitivity to the tiniest microscopic details.

In short: The researchers found that in a world where nothing usually moves, the way things start to move depends entirely on how they were arranged before the movement started. Some arrangements are locked tight; others are ready to efficiently explore new patterns.

Technical Summary

Technical Summary: Krylov Complexity in Flat Bands and Carroll Breaking Deformations

Problem Statement
Flat electronic bands, characterized by dispersionless energy spectra and Compact Localised States (CLS), are central to modern condensed matter physics, hosting phenomena like unconventional superconductivity and topological order. Recent work has established that flat-band lattice models admit an emergent infinite-dimensional algebra of conserved charges identifiable with Carrollian supertranslations, implying an ultra-local dynamics where time evolution is generated independently at each spatial site. While the static properties and correlation functions of these systems under Carroll-breaking perturbations (which restore finite band dispersion) have been studied, a comprehensive understanding of how these perturbations affect the dynamical exploration of the many-body Hilbert space remains lacking. Specifically, it is unknown how the ultra-locality of CLS-based states and the macroscopic degeneracy of "Exotic" Carroll phases manifest in Krylov (spread) complexity when supertranslation symmetry is weakly broken.

Methodology
The authors investigate this problem using Krylov complexity (or Spread Complexity) as a diagnostic tool. The methodology involves:

1. Model Construction: Utilizing a Fermionic ladder Hamiltonian (Creutz ladder type) where all bands are flat (ABF scenario) at the point where intra- and inter-chain hopping amplitudes are equal ($t_1 = t_2$). The system is augmented by supertranslation-preserving on-site density-density interactions.
2. Symmetry Breaking: Introducing a controlled perturbation by detuning the hopping amplitudes ($t_1 = \tau + \Delta, t_2 = \tau - \Delta$), which breaks the supertranslation symmetry and induces non-local hopping between CLS modes.
3. Quench Protocols: Analyzing the time evolution of initial states prepared in distinct phases of the unperturbed Hamiltonian ($H_C$) under the perturbed Hamiltonian ($H = H_C + H_\Delta$). Two primary classes of initial states are examined:
   • Vanilla Phase: The unique CLS-product ground state (e.g., all $\beta$ modes filled).
   • Exotic Phase: States from the macroscopically degenerate ground-state manifold, including domain-wall configurations and translation-symmetrised states.
4. Krylov Construction: Constructing the orthonormal Krylov basis via repeated action of the Hamiltonian on the initial state. The complexity $C_K(t)$ is defined as the first moment of the probability distribution in this basis, quantifying the spread of the state.
5. Analytical and Numerical Probes:
   • Deriving an "Active Link" invariant ($A$) to quantify the susceptibility of specific initial states in the degenerate manifold to the perturbation.
   • Performing exact numerical diagonalization in fixed particle-number and rung-parity sectors to compute Lanczos coefficients and Krylov complexity.
   • Complementing lattice results with a continuum Carroll scalar field theory deformed by a spatial gradient term to analyze Ultraviolet (UV) sensitivity.

Key Contributions and Results

• Differential Krylov Dynamics: The study reveals a sharp dichotomy in how different phases respond to Carroll-breaking perturbations.
  • Vanilla Phases: The unique CLS-product ground states are extremely "brittle." Upon turning on the perturbation, they exhibit rapid Krylov growth, quickly exploring a large fraction of the Hilbert space.
  • Exotic Phases: The response is highly state-dependent due to the macroscopic degeneracy. The authors introduce the Active Link invariant ($A$), which counts the number of active bonds (transitions between filled and empty rungs on the same parity sublattice) in the initial state configuration.
    • States with $A=0$ (e.g., specific Charge Density Wave configurations) remain frozen (Krylov complexity growth is zero) because the perturbation cannot act on them at first order.
    • States with intermediate $A$ (e.g., domain walls) show moderate growth.
    • States with maximal $A$ (e.g., period-four patterns) exhibit the fastest initial growth, even surpassing the growth rate of the brittle Vanilla phase.
• Active Link Invariant: The paper establishes that for the degenerate manifold, the initial curvature of Krylov complexity is directly proportional to the active link count ($b_1^2 \propto A$). This provides a basis-independent ordering principle for the dynamical fragility of states within the same degenerate ground-state manifold.
• Late-Time Behavior: While early-time growth is dictated by the active link structure, late-time behavior involves oscillations around a mean value dependent on the system size and parameters. The study identifies a "finite-size anti-resonance" on the exact critical line where off-site particle-hole states become degenerate, leading to coherent oscillations and partial revivals.
• Continuum Analogue and UV/IR Mixing: In a continuum Carroll scalar field theory, breaking the symmetry via a gradient deformation leads to Krylov growth where the leading coefficient is dominated by ultraviolet modes ($\Lambda^5$ in 1D). This demonstrates a form of UV/IR mixing, where an infrared observable (early-time complexity growth) is entirely determined by microscopic UV details. This mirrors the lattice result where the "active link" count (a discrete analogue of momentum summation) dictates the growth rate.

Significance
The paper claims that Krylov complexity serves as a powerful, dynamical tool for probing Carroll symmetry protection and destruction in flat-band systems. Its significance lies in:

1. Extending Analysis: Moving beyond static correlation functions and Loschmidt echoes to characterize how extensively quantum states explore the Hilbert space under time evolution.
2. Resolving Phase Resilience: Sharply resolving the phase-dependent resilience of Carrollian sectors. It demonstrates that while the "Vanilla" phase is fragile, the "Exotic" phase possesses an emergent resilience where specific symmetry-symmetrised states can remain localized (frozen) even when the global symmetry is broken.
3. Universality: Suggesting that the sensitivity of ultra-local states to gradient deformations (manifested as UV sensitivity in continuum or active-link dependence on lattices) is a generic hallmark of Carrollian systems.
4. Diagnostic Power: Providing a quantitative, basis-independent measure of the fragility of vanilla phases and the high initial-state dependence of degenerate vacuum manifolds in exotic Carroll phases.

The work does not propose new experimental setups but rather offers a theoretical framework for interpreting the dynamics of flat-band systems, particularly in the context of emergent spacetime symmetries and their breaking.