Krylov Reciprocity Metric in Non-Hermitian Systems

• Krylov Reciprocity Metric is a quantitative diagnostic that measures phase coherence between forward and backward hopping amplitudes in Krylov-space to detect chaos transitions.
• It employs the bi-Lanczos algorithm to construct bi-orthonormal bases, with phase angle computations indicating the degree of hopping reciprocity.
• The metric reveals critical signatures at chaos transitions, aligning with established diagnostics like entanglement entropy variance and level-spacing statistics.

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 (Zhou et al., 27 Jan 2025).

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: $$i\, d\varphi_n/dt = b_n \varphi_{n-1} + a_n \varphi_n + c_{n+1} \varphi_{n+1}$$ (Zhou et al., 27 Jan 2025).

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 $R_K^{(d)}=+1$ (Zhou et al., 27 Jan 2025).

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 (Zhou et al., 27 Jan 2025).

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 $W_\gamma \approx 1.6$) (Zhou et al., 27 Jan 2025).

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 (Chen et al., 2024). 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 (Zhou et al., 27 Jan 2025, Chen et al., 2024, Craps et al., 19 Nov 2025).