---
title: Krylov Reciprocity Metric in Non-Hermitian Systems
url: https://www.emergentmind.com/topics/krylov-reciprocity-metric
type: topic
---

# Krylov Reciprocity Metric in Non-Hermitian Systems

The Krylov Reciprocity Metric is a quantitative diagnostic that probes the reciprocal nature of hopping amplitudes in Krylov-space tridiagonalizations, providing a sharp, computationally efficient method to identify the transition between chaotic and non-chaotic dynamics in non-Hermitian quantum many-body systems. It is formulated by measuring the phase coherence between forward and backward hopping elements in the bi-Lanczos algorithm, and has been shown to yield critical signatures of chaos–non-chaos transitions, in agreement with established diagnostics such as entanglement entropy variance and level-spacing statistics [2501.15982].

## 1. Krylov Basis Construction and Bi-Lanczos Recursion

In the context of non-Hermitian quantum systems, the Krylov basis is constructed via the bi-Lanczos algorithm, which generates bi-orthonormal sets of right and left vectors, $\{ |A_n\rangle \}$ and $\{ \langle B_n| \}$, starting from an initial state $| \psi_0 \rangle$. These bases satisfy $\langle B_n | A_m \rangle = \delta_{nm}$. The recurrence proceeds as follows:

- Initialize $|A_0\rangle = |\psi_0\rangle$, $\langle B_0| = \langle \psi_0|$, with $b_0 = c_0 = 0$.
- For $n = 0, 1, 2, \ldots$:
  - Compute $a_n = \langle B_n | H | A_n \rangle$.
  - Update right basis:
    $r_{n+1} = H |A_n\rangle - a_n |A_n\rangle - b_n |A_{n-1}\rangle$,
    $b_{n+1} = \| r_{n+1} \|$,
    $|A_{n+1}\rangle = r_{n+1}/b_{n+1}$.
  - Update left basis:
    $\ell_{n+1} = \langle B_n | H - a_n - c_n \langle B_{n-1}|$,
    $c_{n+1} = \| \ell_{n+1} \|$,
    $\langle B_{n+1}| = \ell_{n+1} / c_{n+1}$.

In Hermitian cases ($H^\dagger = H$), $b_n = c_n$; generally, for non-Hermitian $H$, $b_n$ and $c_n$ are independent complex numbers. The time-dependent state in Krylov space evolves according to a tight-binding-like equation: $i\, d\varphi_n/dt = b_n \varphi_{n-1} + a_n \varphi_n + c_{n+1} \varphi_{n+1}$ [2501.15982].

## 2. Formal Definition of the Krylov Reciprocity Metric

The Krylov Reciprocity Metric detects the degree to which hopping in the Krylov chain is reciprocal. At each site $n$, define
$\theta_n = \mathrm{Arg}(b_n c_n)$,
so that $\cos\,\theta_n = +1$ for perfect reciprocity and $-1$ for maximal non-reciprocity. The global reciprocity metric is then defined as the mean over the first $d$ bonds:
$$
R_K^{(d)} = \frac{1}{d} \sum_{n=1}^d \cos\,\theta_n,
$$
where empirically $d=4,5,6$ yields robust results for chains of length $L > d$.

This metric directly quantifies the symmetry breaking between forward ($b_n$) and backward ($c_n$) hoppings, which is a generic phenomenon in non-Hermitian quantum systems. In the Hermitian limit, $b_n c_n$ is real and positive, so $R_K^{(d)}=+1$ [2501.15982].

## 3. Computation Procedure and Statistical Protocol

To calculate $R_K^{(d)}$, one generates disorder configurations (e.g., for a non-Hermitian quantum spin chain with parameters $\Delta_j$, $\gamma_j$ drawn from specified intervals), constructs the Hamiltonian, and applies the bi-Lanczos algorithm up to $n=d$:

- Record Lanczos coefficients $\{b_n, c_n\}$ for $n=1...d$.
- Compute $\theta_n = \mathrm{Arg}(b_n c_n)$ and $R_K^{(d)} = \frac{1}{d} \sum_{n=1}^d \cos\,\theta_n$.
- Repeat over many ($\sim 10^3$) independent disorder realizations.
- Obtain the disorder-averaged metric $\langle R_K^{(d)}(W_{\gamma}, L) \rangle$.

This protocol provides a notably inexpensive probe, requiring only a limited number of bi-Lanczos steps per realization [2501.15982].

## 4. Role as a Chaos–Non-Chaos Diagnostic

The Krylov Reciprocity Metric exhibits sharp sensitivity to dynamical phase transitions:

- **Chaotic regime (small $W_\gamma$):** $R_K^{(d)}\approx+1$, reflecting reciprocal hopping.
- **Intermediate disorder ($W_{L} < W_\gamma < W_C$):** $R_K^{(d)}$ begins to decrease but remains positive.
- **At critical disorder $W_C$:** $R_K^{(d)}$ crosses zero, indicating that even the short-range Krylov hops become non-reciprocal—marking the transition to non-chaotic dynamics.

Finite-size scaling collapses of $R_K^{(d)}$ across system sizes reveal a critical point $W_C = 1.647$ and scaling exponent $\alpha \approx 0.820$. This crossing matches, to within $1\%$, the transitions determined by entanglement entropy variance ($W_C' \approx 1.763$) and complex level-spacing ratios (crossover at $W_\gamma \approx 1.6$) [2501.15982].

## 5. Comparison with Other Krylov Metrics and Reciprocity Notions

The Krylov Reciprocity Metric $R_K^{(d)}$ is distinct from other recent Krylov metrics, such as the $K_{mn}$ matrix introduced to measure the effective dimension of Krylov space through operator size and out-of-time-order correlator (OTOC) expansions [2404.08207]. While $K_{mn}$ provides criteria for fast scrambling and chaotic growth in Hermitian systems,
- $R_K^{(d)}$ specifically detects the breakdown of reciprocity in non-Hermitian hopping amplitudes and is directly sensitive to non-Hermitian effects.

In the Hermitian limit, $R_K^{(d)}$ is trivial ($=1$), whereas $K_{mn}$ further captures the operator growth structure, including nonchaotic systems with exponentially growing Krylov complexity. These metrics are complementary: $K_{mn}$ provides a refined understanding of many-body chaos in Hermitian models, while $R_K^{(d)}$ excels as a non-Hermitian chaos diagnostic.

## 6. Theoretical and Practical Significance

The Krylov Reciprocity Metric provides a sharply-defined, easily computable indicator of dynamical phase transitions in non-Hermitian systems. Its critical crossing point identifies the chaos–non-chaos boundary with high precision and low computational overhead, using only the first few Lanczos hops and finite-size scaling.

Because $R_K^{(d)}$ does not require long-time evolution or high moments of distributed operator dynamics, it is robust and applicable to a wide range of non-Hermitian settings. Its agreement with conventional chaos diagnostics, such as entanglement entropy fluctuations and level-spacing statistics, reinforces its reliability.

A plausible implication is that similar reciprocity-based metrics could be adapted for broader classes of non-Hermitian or open quantum systems, exploiting the complex structure of Krylov recurrences to reveal phase structures inaccessible to Hermitian diagnostics [2501.15982][2404.08207][2511.15799].

Source: https://www.emergentmind.com/topics/krylov-reciprocity-metric