Krylov Complexity in Periodically Driven CFTs and Critical Fermions

This paper investigates Krylov complexity in periodically driven conformal field theories and their critical fermion lattice realizations, revealing that while both square-wave and sinusoidal drives exhibit similar Krylov growth patterns in the heating and non-heating phases, their underlying spectral and graph signatures differ significantly, pointing to distinct mechanisms governing the phase transitions.

The Gist

Imagine a quantum system as a vast, intricate dance floor where particles are constantly moving and interacting. In a normal, calm situation, these dancers might move in a predictable, rhythmic pattern. But what happens if you start shaking the floor rhythmically, like a DJ changing the beat? This is the world of periodically driven systems.

This paper explores what happens when you shake two specific types of quantum "dance floors":

1. Conformal Field Theories (CFTs): Highly abstract, perfect mathematical models of quantum physics.
2. Critical Fermions: A more concrete, "lattice" version of the same physics, like a grid of atoms on a computer chip.

The researchers are trying to measure how "complex" the dance becomes over time. They use a tool called Krylov Complexity. Think of this as a "complexity meter" that tracks how far a simple starting move spreads out into a chaotic, tangled mess of interactions.

The Two Types of Shaking (Driving Protocols)

The paper tests two different ways of shaking the floor:

1. The Square-Wave Drive: Imagine turning the music on and off instantly. One moment the floor is still, the next it's shaking violently, then still, then shaking. It's a choppy, abrupt rhythm.
2. The Continuous Sinusoidal Drive: Imagine a smooth, rolling wave. The shaking gradually increases and decreases in a smooth, sine-wave pattern. It's a gentle, flowing rhythm.

The Two Outcomes: Heating vs. Non-Heating

When you shake these systems, they fall into one of two distinct moods:

• The Heating Phase (The Chaotic Party): The system absorbs energy endlessly. The dancers get more and more frantic, spreading out across the entire floor until they are completely scrambled. The system effectively reaches a state of "infinite temperature" where all order is lost.
• The Non-Heating Phase (The Organized Rehearsal): The system absorbs energy but stays bounded. The dancers move in a coordinated, oscillating pattern. They don't get lost; they stay within a specific, repeating loop.

What the "Complexity Meter" Reveals

The authors used their "complexity meter" (Krylov complexity) and a specific set of numbers called Arnoldi coefficients to see how the system behaves in these two phases.

• In the Heating Phase: The complexity meter shoots up. The Arnoldi coefficients (which measure how much the system jumps to a new, more complex state) rapidly approach 1.
  • Analogy: Imagine a ball rolling down a steep hill. It keeps picking up speed and moving forward without stopping. The system is constantly exploring new, more complex states.
• In the Non-Heating Phase: The complexity meter wiggles. The coefficients oscillate (go up and down) but never settle at 1.
  • Analogy: Imagine a pendulum swinging back and forth. It moves, but it keeps returning to the same spots. The system is stuck in a loop, never fully escaping its initial structure.

The Big Surprise: The Lattice vs. The Theory

Here is where the paper gets interesting. The researchers found that while the abstract math (CFT) and the concrete computer simulation (Lattice) agreed on the basic behavior (chaotic vs. organized), they disagreed on why and how the transition happened.

1. The Square-Wave Drive (The Choppy Rhythm):

• The Math: The system behaves like a chaotic random matrix.
• The Lattice: When they looked at the "spectral statistics" (the spacing between energy levels), it looked like a chaotic crowd (Wigner-Dyson statistics) in the heating phase and a quiet, orderly crowd (Poisson statistics) in the non-heating phase.
• The Graph: If you draw a map of how particles move, the map is directed (like a one-way street). The flow is messy and asymmetric.

2. The Continuous Drive (The Smooth Rhythm):

• The Math: Similar chaotic vs. organized behavior.
• The Lattice: Surprisingly, the energy levels did not look like the standard chaotic or orderly crowds. They were in a weird middle ground.
• The Graph: The map of particle movement was undirected (like a two-way street). The researchers could see the "connectivity" of the system change clearly. In the non-heating phase, the whole network was one big connected cluster. In the heating phase, it split into two isolated islands.

The Takeaway

The paper concludes that even though two different ways of shaking a system (choppy vs. smooth) might look similar when you just measure "how complex it gets," the underlying machinery is totally different.

• The choppy drive creates a system that behaves like a classic chaotic randomizer, with one-way traffic flows.
• The smooth drive creates a system that retains more local structure, with two-way traffic flows and a different kind of spectral signature.

Essentially, the "how" of the driving matters just as much as the "what." You can't just look at the final complexity; you have to look at the hidden structure of the dance to understand the difference between a smooth wave and a sudden jolt.

Technical Summary

Technical Summary: Krylov Complexity in Periodically Driven CFTs and Critical Fermions

Problem Statement
This work investigates the dynamics of quantum states and operators in periodically driven (Floquet) systems, specifically focusing on (1+1)-dimensional conformal field theories (CFTs) and their lattice realizations via critical free fermions. The central problem is to characterize the distinct dynamical phases—heating (where the system continuously absorbs energy and approaches an infinite-temperature state) and non-heating (where energy absorption remains bounded and return amplitudes oscillate)—using Krylov complexity (K-complexity). While previous studies have utilized diagnostics such as entanglement entropy, return probability, and quasienergy spectra, this paper applies the Krylov construction to probe operator growth and the transition between integrable and chaotic regimes in driven settings.

Methodology
The authors employ two distinct driving protocols:

1. Square-wave drive: The system alternates between a uniform Hamiltonian ($H_0$) and a sine-square deformed (SSD) Hamiltonian ($H_{SSD}$) in discrete time steps.
2. Continuous sinusoidal drive: The system is driven by a time-dependent Hamiltonian with a continuous sinusoidal modulation.

The study proceeds in two parallel tracks:

• Analytical CFT Approach: Using the $SL(2, \mathbb{R})$ sector of the CFT, the authors map the time evolution to Möbius transformations. They construct the Krylov basis by orthonormalizing the sequence of states generated by repeated applications of the Floquet operator $U_F$. The complexity is quantified via the Arnoldi coefficients ($h_{n,n-1}$), which represent the sub-diagonal elements of the Floquet operator in the Krylov basis.
• Lattice Simulations: The CFT dynamics are realized on a one-dimensional lattice of spinless critical fermions. The authors compute the many-body return amplitude, the Krylov complexity of the correlation matrix, and spectral statistics (level-spacing ratios). Additionally, they construct a stochastic transition matrix $W_{ij} = |(U_F)_{ij}|^2$ to analyze the graph structure of the dynamics using the Fiedler value (algebraic connectivity).

Key Contributions and Results

1. Behavior of Arnoldi Coefficients:

   • Heating Phase: In both CFT and lattice simulations, the Arnoldi coefficients $h_{n,n-1}$ approach unity exponentially as the number of Floquet cycles $n$ increases. This indicates rapid spreading of the state into higher-complexity Krylov vectors, characteristic of chaotic dynamics.
   • Non-heating Phase: The coefficients exhibit oscillatory behavior and do not converge to unity. The state remains largely confined within a lower-dimensional subspace of the Krylov basis, reflecting non-ergodic dynamics.
   • Analytical Approximations: The authors derive analytical expressions for the autocorrelation function $G_n$ and the resulting Arnoldi coefficients in the heating phase, showing excellent agreement with exact CFT computations.

2. Lattice vs. CFT Agreement:

   • For smaller values of $n$, the lattice simulations of critical fermions show quantitative agreement with CFT predictions for both the return amplitude and Arnoldi coefficients.
   • Discrepancies arise at late times due to finite-size effects, where the Krylov chain saturates.

3. Spectral Statistics and Phase Transitions:

   • Square-wave Drive: The transition between phases is sharply captured by the mean adjacent level-spacing ratio $\langle r \rangle$. In the heating phase, $\langle r \rangle$ fluctuates around the Wigner-Dyson value ($\approx 0.53$), indicating chaotic spectral correlations. In the non-heating phase, it fluctuates around the Poisson value ($\approx 0.386$), indicating integrable-like behavior.
   • Continuous Drive: The spectral statistics differ markedly. In the heating phase, $\langle r \rangle$ shows erratic fluctuations, while in the non-heating phase, it varies smoothly but does not saturate to either the Wigner-Dyson or Poisson limits. This suggests that the effective lattice dynamics in the continuous drive do not conform to standard random matrix theory classifications in either phase.

4. Graph-Theoretic Signatures:

   • Continuous Drive: The transition matrix $W$ is effectively symmetric, defining an undirected graph. The connectivity of this graph undergoes a structural reorganization across the phase transition: the non-heating phase is dominated by a single large connected cluster, while the heating phase fragments into two dominant clusters. The Fiedler value serves as a sensitive diagnostic for this transition.
   • Square-wave Drive: The transition matrix is intrinsically asymmetric (directed graph). Consequently, graph connectivity measures (like the Fiedler value) fail to capture the heating transition, highlighting that the mechanism governing the phase transition differs fundamentally between the two drive types.

Significance and Claims
The paper claims that Krylov complexity, specifically through the behavior of Arnoldi coefficients, provides a robust and intuitive framework for distinguishing heating and non-heating phases in driven CFTs and critical fermions. The work highlights a crucial finding: while the Krylov growth (behavior of $h_{n,n-1}$) appears similar for both square-wave and continuous drives on the CFT side, their lattice realizations exhibit fundamentally different spectral and graph-theoretic signatures.

The authors conclude that the mechanism governing the transition between heating and non-heating phases is distinct for discrete versus continuous drives. The square-wave drive leads to a transition characterized by a shift from Poisson to Wigner-Dyson statistics and a directed graph structure, whereas the continuous drive exhibits non-generic spectral statistics and a reorganization of undirected graph connectivity. This underscores the importance of the specific driving protocol in determining the nature of nonequilibrium criticality and ergodicity in quantum many-body systems.