The Anderson transition - a view from Krylov space

TL;DR

The paper develops a Krylov-space approach to the Anderson localization transition, constructing local integrals of motion as zero modes of the Krylov Liouvillian generated from a local seed operator. By analyzing the Lanczos coefficients, it connects LIOM localization to dimerization on a semi-infinite Krylov chain, revealing three regimes: negligible finite-size decay in the weakly disordered (extended) phase, power-law decay at criticality with α ≈ 0.29, and stretched-exponential decay in the strongly localized phase with exponent γ = 1/(2d). The results provide a unifying framework linking operator growth, Krylov complexity, and localization, and suggest pathways to study interacting systems and mobility edges within Krylov space. Overall, the work shows that edge-like Krylov LIOMs capture essential aspects of the Anderson transition and offers a quantitative, dimension-aware description of LIOM decay patterns.

Abstract

The Krylov subspace expansion is a workhorse method for sparse numerics that has been increasingly explored as source of physical insight into many-body dynamics in recent years. In this work we revisit the venerable Anderson model of localization in dimensions $d=1, 2, 3, 4$ to construct local integrals of motion (LIOM) in Krylov space. These appear as zero eigenvalue edge states of an effective hopping problem in the Krylov superoperator subspace and can be analytically constructed given the Lanczos coefficients. We exploit this idea, focusing on $d=3$, to study the manifestation of the disorder driven Anderson transition in the anatomy of LIOMs. We find that the increasing complexity of the Krylov operators results in a suppression of the fluctuations of the Lanczos coefficients. As such, one can study the phenomenology of the integrals of motion in the disorder averaged Krylov chain. We find edge states localized on vanishing fraction of Krylov space (of dimension $D_K=V^2$ for cubes of volume $V$), both in localized and extended phases. Importantly, in the localized phase, disorder induces powerlaw decaying dimerization in the (Krylov) hopping problem, producing stretched exponential decay of the LIOMs in Krylov space with a stretching exponent $1/2d$. Metallic LIOMs are completely delocalized albeit across only $\propto \sqrt{D_K}$ states. Critical LIOMs exhibit powerlaw decay with an exponent matching the expected value of $0.29$.

Figures (7)

• Figure 1: Normalized variance of the Lanczos coefficients for the three-dimensional Anderson model at $W=4,16,40$. The inset shows an anti-diagonal cut of the correlation matrix $\rho_{ij}={\rm cov}[b_{i},b_j]/\sigma(b_i)\sigma(b_j)$ which we denote $\rho_{i,N+1-i}$ and which is sharply peaked only at the diagonal index ($i=0$). Different color shades indicate different system sizes: from lightest to darkest are $L=14,18,22$. All results have been obtained from $N_s = 1000$ samples.
• Figure 2: Disorder averaged Lanczos coefficients of the three-dimensional Anderson model over a range of different disorders $W$, both above and below the critical disorder (in order from top to bottom are $W=40,30,22,20,18,16,14,12,10,4$). All results are obtained for a system size of $L=22$ with $N_s=1000$ samples.
• Figure 3: Parameters of the dimerization model, expression (\ref{['eq:dimerization_model']}), fitted by a least squares procedure as a function of disorder (lighter dots show the confidence interval at $10\%$ significance). The critical point of the model, corresponding to $\gamma=0$, occurs at $W_c \approx 16$ consistent with the accepted value $W_c\simeq16.5$. The black dotted line marks the transition point $\gamma=0$ and the grey dotted line marks the localized prediction $\gamma=1/2d$. In the inset, the grey dotted line marks the predicted value in the extended regime: $\epsilon=L^{-d}$
• Figure 4: Zero-mode probability density in Krylov space for $W=4$ in the extended regime. The dot-dashed line represents the geometric average of the zero-modes constructed in each sample whereas the colored lines are obtained from the sample-averaged $b_n$: the two are in complete agreement. There is an extremely small exponential decay which vanishes as a function of increasing $L$. The inset shows the disorder averaged dimerization of the Lanczos coefficients for different system sizes $L=14,18,22,23$, revealing its inverse scaling with the volume roughly as $\simeq 6.3L^{-3}$. There are $N_s=1000$ samples for all sizes except $L=23$ for which there are $N_s=500$.
• Figure 5: The zero-mode probability density for the three-dimensional Anderson model in Krylov space close to the critical point ($W=16$) is shown to decay as a powerlaw $n^{-2\alpha}$ with an exponent $\alpha$ approaching $\alpha\approx0.29$ (shown by the dotted line). The dot-dashed line represents the geometric average of the zero-modes constructed in each sample whereas the colored lines are obtained from the sample-averaged $b_n$: the two are in complete agreement.