Universal Growth of Krylov Complexity Across a Quantum Phase Transition

This paper establishes that across second-order quantum phase transitions, the growth of Krylov complexity follows universal power-law scaling identical to the Kibble-Zurek defect density, with the full complexity distribution becoming asymptotically Gaussian, as demonstrated analytically in the transverse field Ising model and numerically in long-range Kitaev models.

The Gist

Imagine you are watching a complex dance performance. In a calm, slow dance, the dancers move in perfect sync, and you can easily predict where everyone will be next. But what happens if you suddenly speed up the music? The dancers might stumble, bump into each other, and create a chaotic mess.

In the world of quantum physics, this "dance" is the evolution of a system of particles. The paper you are asking about investigates what happens when we force a quantum system to change its state quickly—specifically, when it crosses a "phase transition" (like water turning instantly into ice, but for quantum particles).

Here is the breakdown of their discovery using simple analogies:

1. The Problem: Measuring the "Mess"

When a quantum system changes, it becomes hard to describe. Physicists use a mathematical tool called Krylov Complexity to measure how "spread out" or "complicated" the system has become.

• The Analogy: Imagine a single drop of ink falling into a glass of water.
  • Low Complexity: The ink is still a tight drop.
  • High Complexity: The ink has spread out, mixing with every part of the water.
  • The paper asks: If we push the system through a critical change quickly, how does this "ink" spread?

2. The Tool: The "Diabatic Magnus" Map

To study this, the authors invented a new way to look at the system. They used a method called the diabatic Magnus expansion.

• The Analogy: Imagine trying to track a chaotic crowd. Instead of watching every individual person, you map the crowd onto a simple, one-dimensional hallway.
  • In this hallway, the "complexity" is just the average distance the crowd has moved from the starting door.
  • This map strips away the confusing background noise (the slow, predictable parts of the dance) and focuses only on the chaotic, non-adiabatic "stumbles" caused by the speed of the change.

3. The Discovery: The "Universal Law" of Chaos

The researchers tested this on a famous model called the Transverse Field Ising Model (think of it as a row of tiny magnets that can flip up or down). They found something surprising:

The "Defect" Connection:
When you cool a material too fast, it forms cracks or "defects" (like ice forming too fast and getting bubbles inside). Physicists already knew that the number of these defects follows a specific rule based on how fast you cool the system (the Kibble-Zurek mechanism).

• The Paper's Claim: They discovered that the complexity of the system follows the exact same rule as the number of defects.
  • If you double the speed of the change, the number of defects goes up by a specific power.
  • The "spread" of the complexity goes up by that exact same power.
  • It's as if the "messiness" of the dance is perfectly synchronized with the number of "stumbles" the dancers make.

4. The Shape of the Chaos: The Bell Curve

Usually, when things get chaotic, the results are unpredictable and lopsided. However, the authors found that in this specific "fast change" regime, the distribution of complexity becomes perfectly Gaussian (a Bell Curve).

• The Analogy: Imagine rolling a die. If you roll it once, the result is random. If you roll it a million times and average the results, you get a predictable, smooth bell curve.
• The paper shows that even though the quantum system is complex, the "spread" of its complexity behaves like that average of a million dice rolls. All the different "layers" of complexity (the average, the variance, the skewness) scale up together in a uniform way.

5. Does This Apply Everywhere?

The authors didn't just stop at the magnet model. They tested their theory on Long-Range Kitaev Models (a more complex system where particles talk to each other over long distances).

• The Result: Even in these more complicated systems, the same rules applied. Whether the particles were close neighbors or far apart, the complexity still grew according to the universal laws of the phase transition.

Summary

In short, this paper says:
When you push a quantum system through a critical change quickly, the "complexity" of its evolution doesn't just grow randomly. It grows in a universal, predictable pattern that is mathematically identical to how physical defects (like cracks in ice) form. Furthermore, this complexity settles into a smooth, predictable "Bell Curve" shape, proving that even in the chaos of a quantum phase transition, there is a deep, underlying order.

The authors provide the mathematical "blueprint" (the Magnus operator and Krylov space) that proves this connection exists, showing that the "messiness" of quantum evolution is governed by the same laws that govern the formation of defects in the universe.

Technical Summary

Technical Summary: Universal Growth of Krylov Complexity Across a Quantum Phase Transition

Problem Statement
The characterization of operator growth and Krylov complexity in time-dependent quantum systems remains a significant challenge, particularly for nonequilibrium processes such as quantum quenches and annealing protocols. While Krylov subspace methods have proven effective for time-independent systems in quantifying complexity, operator spreading, and chaos, existing frameworks for time-dependent settings (e.g., those employing nonlocal-in-time Floquet operators) are often difficult to apply to many-body systems. Furthermore, it remains an open question whether the growth of quantum complexity exhibits universal scaling behavior analogous to the Kibble-Zurek (KZ) mechanism, which describes defect formation in systems driven across second-order quantum phase transitions (QPTs) at finite rates.

Methodology
The authors introduce an analytical framework based on the diabatic Magnus expansion to describe the growth of Krylov complexity in driven quantum systems.

1. Diabatic Magnus Operator: To isolate non-adiabatic effects, the time-evolution operator $U(t)$ is mapped relative to the instantaneous ground state $|GS(t)\rangle$ via the diabatic operator $U(t) = U(t)U_{ad}(t)^\dagger$. The dynamics are governed by the diabatic Magnus operator $\Omega(t) = i \log(U(t))$, such that $|\psi(t)\rangle = e^{-i\Omega(t)}|GS(t)\rangle$.
2. Krylov Subspace Construction: A time-dependent Lanczos algorithm is employed to generate the Krylov basis $\{|K_n(t)\rangle\}$ from $\Omega(t)$. The quantum state is expanded in this basis, and the Krylov complexity is defined as the average position of the state in the Krylov lattice, $K_1 = \sum n |\phi_n(t)|^2$, where $\phi_n$ are the occupation amplitudes.
3. Model Systems:
   • Transverse-Field Ising Model (TFIM): Used as a primary testbed. The system is mapped to independent two-level systems (TLSs) in momentum space. The authors derive exact expressions for the Magnus operator elements and the resulting Krylov wavefunction in the slow-driving (KZ) regime.
   • Long-Range Kitaev Models (LRKM): Used to generalize findings to systems with long-range hopping and pairing interactions, testing the robustness of the scaling arguments against different critical exponents and interaction ranges.

Key Contributions and Results

• Exact Link in TFIM: For the TFIM driven across a critical point, the authors establish an exact link between Krylov complexity growth and the KZ defect scaling.

  • Lanczos Coefficients: The off-diagonal Lanczos coefficients $b_n$ and diagonal coefficients $a_n$ exhibit universal scaling behaviors: $b_n \sim L^{1/2}\tau^{-1/4}\sqrt{n}$ and $a_n \sim L\tau^{-1/2}$, where $L$ is system size and $\tau$ is the driving time.
  • Poissonian Statistics: The Krylov wavefunction components follow a Poissonian distribution in the leading order. Consequently, all cumulants of the Krylov complexity ($K_q$) become identical at the leading order, scaling as $K_q \approx 2CL\tau^{-1/2}$.
  • Defect Correlation: This scaling matches the universal power law of defect density ($n \sim \tau^{-1/2}$) predicted by the KZ mechanism for the TFIM ($d=1, \nu=1, z=1$).
  • Gaussian Limit: As the parameter $L\tau^{-1/2}$ becomes large, the central limit theorem applies to the underlying Poissonian statistics, causing the full distribution of Krylov complexity to asymptotically converge to a Gaussian distribution.

• General Scaling Argument: The authors extend these results to arbitrary $d$-dimensional quantum critical systems with free-fermion representations and a $(d-D)$-dimensional gapless critical surface.

  • They derive general scaling laws for the Lanczos coefficients and complexity cumulants: $K_q \sim L^{d-D}\tau^{-\alpha(d-D)}$, where $\alpha$ is a dynamical exponent determined by the excitation amplitudes.
  • This framework accommodates systems where standard KZ scaling is modified, such as long-range systems.

• Numerical Verification in LRKM: The general predictions are numerically demonstrated in Long-Range Kitaev Models with both short-range and long-range pairing interactions.

  • In regimes where the excitation density scales as $\tau^{-1/2}$ (short-range pairing) or $\tau^{-0.625}$ (long-range pairing), the complexity cumulants and Lanczos coefficients precisely follow the predicted power laws.
  • Histograms of the complexity confirm the transition to Gaussian statistics in the KZ scaling regime, with distributions shifting toward the origin as the driving rate slows (adiabatic limit).

• Time Evolution Dynamics: The study reveals that while the complexity exhibits universal growth near the critical point, it displays pronounced oscillatory behavior at intermediate times and in the symmetry-broken phase. These oscillations eventually dampen, and the system converges to an asymptotic value governed by universal power laws, distinct from the non-universal, model-specific details that dominate far from the critical point.

Significance
The paper claims to provide the first exact and versatile analytical framework for describing Krylov complexity growth in time-dependent settings, specifically bridging the gap between quantum complexity and nonequilibrium universality. By demonstrating that all cumulants of Krylov complexity scale identically to topological defect densities and converge to a universal Gaussian distribution, the work establishes a direct connection between the computational hardness of describing quantum evolution and the fundamental statistical mechanics of phase transitions. The results suggest that Krylov complexity is a robust observable for characterizing the dynamics of quantum phase transitions, offering a potential avenue for experimental verification of nonequilibrium universality classes in quantum simulators.