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

Abstract

We investigate the anatomy and complexity of quantum states in Krylov space, in the ergodic and many-body localised (MBL) phases of a disordered, interacting spin chain. The Krylov basis generated by the Hamiltonian from an initial state provides a representation in which the spread of the time-evolving state constitutes a basis-optimised measure of complexity. We show that the long-time Krylov spread complexity sharply distinguishes the two phases. In the ergodic phase, the infinite-time complexity scales linearly with the Fock-space dimension, indicating that the state spreads over a finite fraction of the Krylov chain. By contrast, it grows sublinearly in the MBL phase, implying that the long-time state occupies only a vanishing fraction of the chain. Further, the profile of the infinite-time state along the Krylov chain exhibits a stretched-exponential decay in the MBL phase. This behaviour reflects a broad distribution of decay lengthscales, associated with different eigenstates contributing to the long-time state. Consistently, a large-deviation analysis of the statistics of eigenstate spread complexities shows that while the ergodic phase receives contributions from almost all eigenstates, the complexity in the MBL phase is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.

Figures (11)

• Figure 1: In the Krylov-space basis, the many-body Hamiltonian takes the form of a nearest-neighbour tight-binding chain (Eq. \ref{['eq:H-krylov-tridiag']}), of length $N_\mathcal{H}=2^L$, with the orbitals $\ket{k_n}$ labelled $n=0,1,\cdots,N_\mathcal{H}-1$. On the Fock-space graph (illustrated for $L=6$), the Krylov $\ket{k_0}$ orbital is identified with the top apical site $\ket{I_{0}}$. For $n\leq L$, $\ket{k_n}$ is supported on Fock-space sites which lie within Hamming distance $r\leq n$, as illustrated by the red-shaded connection between $\ket{k_2}$ and all Fock-space sites with $r\leq 2$. By contrast, for $L+1\leq n\leq N_\mathcal{H}-1$, $\ket{k_n}$ is in general supported on the entire Fock-space graph, as indicated by the grey-shaded connections.
• Figure 2: Numerical results for the effective size of the wave function on the Fock-space graph corresponding to the $n^{\rm th}$ Krylov orbital, as quantified by ${\cal R}_n$ defined in Eq. \ref{['eq:Rn']} (with system sizes $L$ indicated). Panels (a) and (b) with $W=1$ and $W=10$ exemplify, respectively, the ergodic and MBL regimes. The results conform to the scaling form in Eq. \ref{['eq:Rn-scaling']}. The (thin) shaded region around the data points reflects the statistical errors over disorder realisations.
• Figure 3: Distribution $P_{s}(s)$ of the infinite-time spread complexity rescaled by its mean, $s\equiv S_{K,\infty}/\braket{{S_{K,\infty}}}$, over disorder realisations in both the ergodic [panels (a), (b)] and MBL [panels (c), (d)] regimes, for different $L$. In the ergodic phase the distribution clearly approaches a Gaussian, whereas in the MBL phase it is an exponential distribution, as indicated by the blue dashed lines denoting $e^{-s}$.
• Figure 4: (a) Disorder averaged $\braket{S_{K,\infty}}$ as a function of $N_\mathcal{H}$ on logarithmic scales, for different disorder strengths $W$. The straight line fits show that $\braket{S_{K,\infty}}\sim N_\mathcal{H}^\alpha$ with $\alpha = 1$ in the ergodic phase and $\alpha<1$ in the MBL phase, as indicated explicitly next to fits. (b) Scaling with $N_\mathcal{H}$ of the standard deviation of ${S_{K,\infty}}$ relative to its mean, $\sigma_s \equiv {\rm std}[{S_{K,\infty}}]/\braket{{S_{K,\infty}}}$. The scaling exponents $\varsigma$, defined via $\sigma_s\sim N_\mathcal{H}^{-\varsigma}$, are mentioned explicitly next to the fits. In the deep ergodic regime $\varsigma=1/2$, whereas in the MBL regime $\varsigma\simeq 0$.
• Figure 5: Profile of the disorder-averaged $\braket{\Lambda_n}$ (defined in Eq. \ref{['eq:Lambda-n-E']}) on the Krylov chain, for different $L$. Panels (a) and (b) correspond to the ergodic phase for $W=1,2$, while panels (c) and (d) correspond to the MBL phase for $W=8, 10$.