Quantum Chaos Diagnostics for non-Hermitian Systems from Bi-Lanczos Krylov Dynamics

This paper demonstrates that Krylov complexity, computed via the bi-Lanczos algorithm, serves as a robust and universal diagnostic for distinguishing chaotic from integrable phases in non-Hermitian systems, aligning with complex spectral statistics across various models and symmetry classes.

The Gist

Imagine you are trying to understand how a complex machine works. In the world of quantum physics, scientists have a special tool called Krylov Complexity (let's call it the "Complexity Meter") that measures how quickly a system gets messy and unpredictable.

For a long time, this tool only worked well for "closed" systems—like a perfect, sealed box where nothing enters or leaves. In these sealed boxes, if the machine is chaotic (wild and unpredictable), the Complexity Meter shows a big, sharp spike. If the machine is integrable (orderly and predictable), the meter stays flat. This spike has been a reliable way to tell chaos from order.

The Problem: The Leaky Box
But real-world quantum systems aren't sealed boxes. They are "open systems," interacting with their environment (like a leaky box). In physics, we describe these using non-Hermitian math. This is tricky because the numbers involved are complex (involving imaginary numbers), and the usual rules of symmetry break down.

When scientists tried to use the old Complexity Meter on these "leaky boxes," it failed. The needle wouldn't move correctly, and they couldn't tell if the system was chaotic or orderly. It was like trying to measure the speed of a car using a broken speedometer.

The Solution: The Two-Handed Dance
The authors of this paper invented a new way to use the Complexity Meter. They realized that in a leaky box, you can't just look at the system from one side. You need to look at it from two sides simultaneously, like a dance between a Left Hand and a Right Hand.

They used a mathematical technique called the Bi-Lanczos Algorithm.

• The Old Way: Imagine trying to build a tower with blocks using only your right hand. In a chaotic, leaky environment, the tower keeps falling over.
• The New Way: Now, imagine you have a partner. You use your right hand to place a block, and your partner uses their left hand to steady it. They constantly check each other's work to make sure the tower stays upright. This "two-handed" approach (bi-orthogonal) allows them to build a stable tower even in the messy, leaky environment.

What They Found
Using this new "two-handed" method, they tested two famous models:

1. The Non-Hermitian SYK Model: A complex system of interacting particles.
2. Random Matrices: A mathematical way to simulate pure randomness.

The Results:

• In Chaotic Systems: Just like in the sealed boxes, the Complexity Meter showed a huge, distinct spike. The system was wild, and the meter knew it.
• In Orderly Systems: The meter stayed flat. No spike. The system was calm.

This was a huge discovery. It proved that the "Complexity Spike" is a universal sign of chaos, even in open, leaky systems. It works just as well as other advanced tests (like looking at the spacing of the system's energy levels).

A Surprising Pattern
While building their "two-handed" tower, they noticed a strange, beautiful pattern. The strength of the blocks they placed (the mathematical coefficients) followed a specific rule: the "on-site" blocks and the "hopping" blocks always balanced each other out in a fixed ratio (roughly 1 to 1.4).

Think of it like a tightrope walker. Whether they are walking on a calm day (integrable) or a windy day (chaotic), they always adjust their balance pole in the exact same way to stay upright. This suggests that deep down, all these chaotic quantum systems share a hidden, universal geometry.

Why This Matters
This paper gives scientists a reliable new tool to study open quantum systems. These are systems that interact with the real world, which is crucial for:

• Understanding how quantum computers lose information (decoherence).
• Studying biological systems that are never truly isolated.
• Designing new materials that behave in specific ways when exposed to their environment.

In short, the authors fixed a broken tool, taught it how to dance with two hands, and proved that it can still spot chaos anywhere, even in the messiest, leakiest quantum systems imaginable.

Technical Summary

1. Problem Statement

The study of quantum chaos in Hermitian (closed) systems is well-established, with Krylov Complexity (KC) emerging as a powerful diagnostic tool. In Hermitian systems, KC quantifies the growth of an operator or state within the Krylov subspace generated by time evolution. It reliably distinguishes chaotic phases (characterized by a distinct early-time peak) from integrable phases (lacking such a peak), aligning with traditional probes like spectral statistics and Out-of-Time-Order Correlators (OTOCs).

However, non-Hermitian systems, which model open quantum systems coupled to environments, present significant challenges:

• Complex Eigenvalues: The spectrum is complex, rendering standard real-spectrum diagnostics insufficient.
• Biorthogonality: Eigenstates of non-Hermitian Hamiltonians are not orthogonal in the standard sense; they require a biorthogonal basis (left and right eigenvectors).
• Failure of Naive Extensions: Standard extensions of KC using Singular Value Decomposition (SVD) or standard Lanczos algorithms fail to distinguish chaos from integrability in non-Hermitian settings due to the loss of orthogonality and the complex nature of the evolution.

The Core Question: Can Krylov Complexity remain a universal and reliable diagnostic of quantum chaos in non-Hermitian (open) systems?

2. Methodology

The authors propose a rigorous framework based on the bi-Lanczos algorithm to compute KC in non-Hermitian systems.

• Bi-Lanczos Algorithm:

  • Instead of a single orthonormal basis, the algorithm generates two sets of bases: a right basis $\{|q_n\rangle\}$ and a left basis $\{|p_n\rangle\}$.
  • These satisfy the bi-orthonormal condition: $\langle p_n | q_m \rangle = \delta_{nm}$.
  • The algorithm produces three sequences of Lanczos coefficients: $\{a_n, b_n, c_n\}$, which encode the system's dynamics.
  • The non-Hermitian Hamiltonian $H$ is represented as a tridiagonal matrix in this basis, where $a_n$ are diagonal elements, $b_n$ are superdiagonal, and $c_n$ are subdiagonal.

• Krylov Complexity Definition:

  • The time-evolved state is expanded in the biorthogonal bases: $|\Psi(t)\rangle = \sum \Phi^p_n(t)|p_n\rangle = \sum \Phi^q_n(t)|q_n\rangle$.
  • KC is defined as: $C(t) \equiv \sum_n n |\tilde{\Phi}^{p*}_n(t) \tilde{\Phi}^q_n(t)|$, where tildes denote dynamically normalized coefficients ensuring probability conservation ($P(t)=1$).

• Initial State:

  • The authors utilize the Thermofield Double (TFD) state (a maximally entangled state at infinite temperature) as the seed for the bi-Lanczos recursion. This choice is motivated by its success in Hermitian systems for generating a characteristic chaos peak.

• Models Studied:

  1. Non-Hermitian Sachdev-Ye-Kitaev (nHSYK) Model: A model of $N$ Majorana fermions with random $q$-body interactions, including a non-Hermitian term ($iM$).
  2. Non-Hermitian Random Matrix Ensembles: Specifically focusing on Class A (Ginibre Unitary Ensemble, GinUE) and other symmetry classes (AI$^\dagger$, AII$^\dagger$).

• Benchmarking:

  • Results are compared against Complex Level Spacing Statistics (Poisson vs. Ginibre distributions) and Complex Spacing Ratios (CSR).

3. Key Contributions

1. Algorithmic Implementation: The paper provides a robust numerical implementation of the bi-Lanczos algorithm for non-Hermitian systems, addressing numerical instabilities through complete bi-orthogonalization (Gram-Schmidt procedure).
2. Validation of KC in Open Systems: It demonstrates that KC, when computed via bi-Lanczos, successfully distinguishes between chaotic and integrable regimes in non-Hermitian systems, a capability previously unproven.
3. Discovery of Universal Relations: The authors identify a universal relation among the Lanczos coefficients in non-Hermitian systems:
   $$\frac{1}{\sqrt{2}}|a_n| \approx |b_n| = c_n$$
   This relation holds for both chaotic and integrable cases, suggesting a "balanced tight-binding" structure in the Krylov chain that is intrinsic to the bi-orthogonal construction.
4. Universality Across Symmetry Classes: The findings are validated across multiple non-Hermitian symmetry classes (A, AI$^\dagger$, AII$^\dagger$) and various parameters ($N, q$), establishing KC as a universal probe beyond the Hermitian realm.

4. Key Results

• Chaos Signatures (e.g., nHSYK with $q=4$):

  • KC Dynamics: Exhibits a distinct, pronounced early-time peak followed by saturation. This peak is the hallmark of quantum chaos.
  • Spectral Statistics: The complex eigenvalue spacing distribution follows the Ginibre (GinUE) distribution.
  • CSR: Shows strong angular anisotropy (cubic radial repulsion), with $\langle \cos \theta \rangle \approx 0.22$.

• Integrability Signatures (e.g., nHSYK with $q=2$):

  • KC Dynamics: No peak is observed; the complexity grows monotonically or saturates without the characteristic overshoot.
  • Spectral Statistics: Follows a 2D Poisson distribution.
  • CSR: Isotropic distribution with $\langle \cos \theta \rangle \approx 0.003$.

• Robustness:

  • The results are insensitive to the specific choice of the TFD basis (constructed from $H$, $H^\dagger$, or a mix), confirming the universality of the diagnostic.
  • The Lanczos coefficient relation $(1/\sqrt{2})|a_n| \approx |b_n| = c_n$ is observed universally across all tested models and symmetry classes, regardless of whether the system is chaotic or integrable.

5. Significance

• New Diagnostic Tool: This work establishes bi-Lanczos Krylov Complexity as a reliable, universal tool for diagnosing quantum chaos in open, dissipative, and non-Hermitian systems, filling a critical gap in the literature.
• Theoretical Unification: It bridges the gap between Hermitian chaos diagnostics and non-Hermitian physics, showing that the "peak" signature of chaos persists even when the Hamiltonian is non-Hermitian, provided the correct biorthogonal formalism is used.
• Physical Insight: The discovery of the universal Lanczos coefficient relation suggests that diverse non-Hermitian Hamiltonians may share a common underlying geometric structure in their Krylov subspaces, governed by the constraints of bi-orthogonality.
• Future Directions: The paper opens avenues for exploring the transition from Hermitian to non-Hermitian regimes, connecting these theoretical probes to experimental platforms (e.g., photonic or cold atom systems) that emulate non-Hermitian Hamiltonians, and extending the framework to other Krylov-space tools like Krylov entropy and metric.

In summary, the paper successfully adapts the Krylov complexity framework to the non-Hermitian domain, proving its efficacy as a universal chaos indicator and revealing new structural properties of non-Hermitian dynamics.