Krylov-space anatomy and spread complexity of a disordered quantum spin chain

This paper demonstrates that Krylov spread complexity serves as a sharp diagnostic tool to distinguish between ergodic and many-body localized phases in disordered spin chains, revealing that long-time complexity scales linearly with system size in the ergodic phase but grows sublinearly with a stretched-exponential decay profile in the localized phase.

The Gist

Imagine you have a giant, chaotic library filled with every possible arrangement of a million books. This library represents the "Fock space" of a quantum system—a place where every possible state of a quantum chain of spins (tiny magnets) exists.

Now, imagine you drop a single book onto a specific shelf at the start. As time passes, that book doesn't just sit there; it magically copies itself and spreads out to other shelves, influenced by the rules of the library (the Hamiltonian).

This paper asks a simple but profound question: How far does that book spread, and how "complex" is its journey?

The authors investigate this in two different types of libraries:

1. The "Ergodic" Library: A chaotic, well-connected library where the book eventually spreads to almost every shelf.
2. The "MBL" (Many-Body Localized) Library: A library with broken aisles and locked doors, where the book gets stuck in a small corner and never really explores the rest.

Here is the breakdown of their findings using everyday analogies:

1. The Magic Map: The Krylov Chain

Usually, looking at this library is like trying to navigate a 3D maze that keeps changing shape. It's incredibly hard to measure how far the book has traveled.

The authors use a clever trick called the Krylov basis. Think of this as taking that messy, multi-dimensional library and flattening it out into a single, long, straight hallway.

• The start of the hallway is where you dropped the book.
• The end of the hallway is the furthest possible point.
• As the book spreads, it moves down this hallway.

This "Krylov hallway" is special because it is the most efficient way to measure the book's journey. It strips away all the unnecessary complexity and gives you a direct ruler to measure "spread complexity."

2. The Two Types of Journeys

The Ergodic Phase (The Chaotic Library)

In this phase, the library is fully connected.

• The Journey: If you drop the book, it spreads out rapidly. By the time a long time has passed, the book is effectively present on every single shelf in the hallway.
• The Measurement: The "complexity" (how far it spread) grows linearly with the size of the library. If the library doubles in size, the spread doubles.
• The Analogy: It's like dropping a drop of ink in a glass of water. Eventually, the ink is evenly distributed everywhere. The system is "thermalized" or "ergodic"—it has forgotten where it started and explored everything.

The MBL Phase (The Broken Library)

In this phase, the disorder (randomness) is so strong that the library is full of dead ends and locked doors.

• The Journey: The book tries to spread, but it gets stuck. Even after a very long time, it only occupies a tiny fraction of the hallway.
• The Measurement: The complexity grows sub-linearly. If the library doubles in size, the spread doesn't double; it grows much slower (like the square root of the size).
• The Analogy: It's like dropping that ink in a block of jelly. The ink spreads a little bit, but it never reaches the edges. The system is "localized"—it remembers where it started and refuses to explore the rest of the universe.

3. The Shape of the Spread

The authors looked closely at how the book spreads in the MBL (jelly) phase.

• They found that the probability of finding the book at a certain distance follows a stretched exponential decay.
• The Metaphor: Imagine a crowd of people running away from a starting line. In the chaotic phase, they run at similar speeds and spread out evenly. In the localized phase, most people stop almost immediately, but a few "lucky" runners sprint very far ahead. However, the number of these lucky runners drops off very quickly. The "tail" of the distribution is long but thin, meaning rare, long-distance jumps happen, but they are very uncommon.

4. The "Rare Resonance" Secret

One of the most interesting findings is about who is doing the spreading in the localized phase.

• In the chaotic phase, everyone contributes equally to the spread.
• In the localized phase, the spread is dominated by a tiny, vanishingly small fraction of the "runners" (quantum states).
• The Metaphor: Imagine a marathon where 99.9% of runners stop after 10 meters. But a few "super-runners" (rare resonances) manage to run for miles. Even though they are a tiny fraction of the total crowd, they are the only ones who determine how far the group has spread. The authors found that these "super-runners" are rare events hidden in the tails of the distribution.

Why Does This Matter?

This paper is important because it provides a new, very clear way to tell the difference between a "normal" quantum system and a "frozen" (localized) one.

• Old Way: Look at entanglement or energy levels (very abstract and hard to visualize).
• New Way (This Paper): Look at the "Krylov Hallway." If the state fills the hallway, it's chaotic. If it stays in a corner, it's localized.

The authors show that the Krylov space acts like a microscope that simplifies the complex, high-dimensional world of quantum mechanics into a simple 1D line, making it much easier to see the fundamental differences between order and chaos in quantum matter.

In a nutshell: The paper proves that by reorganizing the quantum library into a single hallway, we can clearly see that in a chaotic world, everything mixes together, but in a disordered, localized world, everything stays stuck in its own little corner, driven only by a few rare, lucky exceptions.

Technical Summary

1. Problem Statement

The paper investigates the fundamental differences between the ergodic and Many-Body Localized (MBL) phases in disordered, interacting quantum systems. While traditional characterizations rely on real-space observables (eigenstate thermalization hypothesis), entanglement entropy (volume vs. area law), and Fock-space properties (inverse participation ratios), these measures can sometimes be subtle or difficult to interpret dynamically.

The authors address the question: How does the "complexity" of a quantum state, defined by its spread over the Hilbert space, distinguish between ergodic and MBL phases? Specifically, they focus on the Krylov spread complexity, a basis-optimized measure of complexity derived from the time-evolution of an initial state. They aim to characterize the "anatomy" of quantum states within the Krylov basis to reveal structural signatures of the MBL phase that are distinct from the ergodic phase.

2. Methodology

Model System

The authors study a disordered, tilted-field Ising (TFI) spin-1/2 chain defined by the Hamiltonian:
$$H = \sum_{i=1}^{L-1} J_i \sigma_i^z \sigma_{i+1}^z + \sum_{i=1}^L h_i \sigma_i^z + \Gamma \sum_{i=1}^L \sigma_i^x$$
where $J_i$ and $h_i$ are random variables representing disorder. The system exhibits an ergodic phase at weak disorder ($W < W_c \approx 3.7$) and an MBL phase at strong disorder ($W > W_c$).

Krylov Basis Construction

The core methodology involves constructing the Krylov basis $\{|k_n\rangle\}$ generated by the Hamiltonian $H$ acting on an initial state $|k_0\rangle$ (chosen as a specific $\sigma^z$-product state near the middle of the spectrum).

• The basis is generated via the Lanczos recursion relation.
• In this basis, the Hamiltonian takes a tridiagonal form (Eq. 4), effectively mapping the many-body problem onto a one-dimensional, nearest-neighbor, correlated tight-binding chain of length $N_H = 2^L$ (the Fock-space dimension).
• Unlike the high-dimensional, complex Fock-space graph, the Krylov chain offers a simple, ordered 1D geometry with a clear notion of distance.

Spread Complexity Definition

The Krylov spread complexity $S_K(t)$ is defined as the first moment of the probability distribution of the state on the Krylov chain:
$$S_K(t) = \sum_{n=0}^{N_H-1} n |c_n(t)|^2$$
where $c_n(t) = \langle k_n | \psi_t \rangle$. The authors focus on the infinite-time limit $S_{K,\infty}$, defined as the time average of $S_K(t)$ as $t \to \infty$. This quantity serves as a basis-optimized measure of how far the state has spread in the Hilbert space.

3. Key Contributions

1. Basis-Optimized Complexity Measure: The paper rigorously applies the concept that the Krylov basis minimizes the spread complexity of a time-evolving state, providing a "bona fide" measure of complexity that is independent of arbitrary basis choices.
2. Scaling Laws for Infinite-Time Complexity: The authors establish distinct scaling behaviors for $S_{K,\infty}$ in the two phases:
   • Ergodic Phase: $S_{K,\infty} \propto N_H$ (Linear scaling).
   • MBL Phase: $S_{K,\infty} \propto N_H^\alpha$ with $\alpha < 1$ (Sublinear scaling).
3. Stretched-Exponential Decay Profile: They discover that in the MBL phase, the probability profile of the infinite-time state along the Krylov chain, $\langle \Lambda_n \rangle$, follows a stretched-exponential decay rather than a simple exponential or power law.
4. Large-Deviation Analysis: The paper introduces a large-deviation analysis of the contributions of individual eigenstates to the total spread complexity, revealing that the MBL complexity is dominated by a vanishing fraction of "rare" eigenstates with anomalously high complexity.

4. Detailed Results

A. Scaling of Infinite-Time Complexity ($S_{K,\infty}$)

• Ergodic Phase: The distribution of $S_{K,\infty}$ over disorder realizations is Gaussian and narrow. The mean scales linearly with the Fock-space dimension: $\langle S_{K,\infty} \rangle \approx N_H / 2$. This indicates the state spreads over a finite fraction of the entire Krylov chain.
• MBL Phase: The distribution becomes broad and follows an exponential form ($P(s) \sim e^{-s}$). The mean scales sublinearly: $\langle S_{K,\infty} \rangle \sim N_H^\alpha$ where $\alpha < 1$ (e.g., $\alpha \approx 0.6$ for strong disorder). This implies the state occupies only a vanishing fraction of the Krylov chain as system size increases.

B. Anatomy of the State Profile ($\Lambda_n$)

The quantity $\Lambda_n$ represents the long-time probability of finding the system on the $n$-th Krylov orbital.

• Ergodic: $\Lambda_n$ is roughly flat (constant) across the chain, consistent with delocalization.
• MBL: $\Lambda_n$ decays systematically with $n$. The decay is not a simple exponential but a stretched exponential:
  $$\langle \Lambda_n \rangle \sim N_H^{-\alpha} \exp\left[ -c \left( \frac{n}{N_H^\alpha} \right)^\gamma \right]$$
  Numerical data suggests a stretch exponent $\gamma \approx 1/2$ for accessible system sizes, though a phenomenological theory suggests it may asymptotically approach $\gamma = 1/3$. This reflects a broad distribution of decay length scales associated with different eigenstates.

C. Phenomenological Theory

The authors propose a theory where the stretched-exponential profile arises from the interplay of:

1. Exponential decay of state amplitudes for individual eigenstates on the Krylov chain.
2. A broad (but not heavy-tailed) distribution of these decay length scales ($\xi$) across eigenstates and disorder realizations.
   The theory successfully reproduces the observed stretched-exponential decay and the exponential distribution of the total complexity.

D. Statistics of Eigenstate Contributions (Large-Deviation Analysis)

The total complexity is a sum over eigenstate contributions: $S_{K,\infty} = \sum_E S_{K,|E\rangle}$.

• Ergodic: Contributions come from almost all eigenstates (finite fraction of the spectrum). The entropy density $\Sigma(x)$ at the saddle point equals $\ln 2$.
• MBL: The sum is dominated by a vanishing fraction of eigenstates (though exponentially large in number) that lie in the tails of the distribution of eigenstate complexities. These are "rare resonant" states. The entropy density at the saddle point satisfies $\Sigma(x^*) < \ln 2$, indicating multifractality and the dominance of rare events.

5. Significance

• New Diagnostic for MBL: The Krylov spread complexity provides a sharp, quantitative distinction between ergodic and MBL phases, scaling linearly vs. sublinearly with system size.
• Insight into MBL Structure: The analysis reveals that the MBL phase is not just "localized" in a simple sense but involves a complex interplay of rare resonances. The stretched-exponential profile suggests a hierarchy of length scales in the Krylov space.
• Conceptual Clarity: By mapping the high-dimensional Fock space to a 1D correlated chain, the Krylov approach simplifies the visualization of quantum state spreading, making the distinction between "extended" (ergodic) and "localized" (MBL) states more transparent.
• Rare Events: The work highlights the critical role of rare, high-complexity eigenstates in determining the macroscopic properties of the MBL phase, offering a new perspective on the "avalanche" instability and rare thermal bubbles in MBL systems.

In conclusion, the paper establishes Krylov space as a powerful framework for analyzing quantum complexity, demonstrating that the long-time spread of a state serves as a robust order parameter for the many-body localization transition.