Operator growth and Krylov Complexity in Bose-Hubbard Model

TL;DR

This work studies operator growth and Krylov complexity in a three-site Bose-Hubbard system, a quantum many-body model that shows chaotic and integrable aspects depending on the interaction strength. The authors implement a non-perturbative Lanczos procedure with full orthogonalization to build the Krylov basis and extract Lanczos coefficients for a discrete quartic bosonic Hamiltonian. They find that finite interactions drive the Krylov space to saturate its upper bound and produce ramp-then-plateau Lanczos coefficients, consistent with chaotic dynamics, while integrable limits show much smaller Krylov dimensions. Krylov complexity exhibits three distinct time-scale behavior—a rapid scrambling, a linear growth stage, and late-time saturation—validating Krylov-geometry as a diagnostic of chaos versus integrability and illustrating the method’s non-perturbative applicability to bosonic lattice systems.

Abstract

We study Krylov complexity of a one-dimensional Bosonic system, the celebrated Bose-Hubbard Model. The Bose-Hubbard Hamiltonian consists of interacting bosons on a lattice, describing ultra-cold atoms. Apart from showing superfluid-Mott insulator phase transition, the model also exhibits both chaotic and integrable (mixed) dynamics depending on the value of the interaction parameter. We focus on the three-site Bose Hubbard Model (with different particle numbers), which is known to be highly mixed. We use the Lanczos algorithm to find the Lanczos coefficients and the Krylov basis. The orthonormal Krylov basis captures the operator growth for a system with a given Hamiltonian. However, the Lanczos algorithm needs to be modified for our case due to the instabilities instilled by the piling up of computational errors. Next, we compute the Krylov complexity and its early and late-time behaviour. Our results capture the chaotic and integrable nature of the system. Our paper takes the first step to use the Lanczos algorithm non-perturbatively for a discrete quartic bosonic Hamiltonian without depending on the auto-correlation method.

Figures (5)

• Figure 1: Lanczos sequence for different configurations of M, and N. Fig.(\ref{['fig:Lanczos_2.5']}) shows the Lanczos sequence for N=3, and M=3, Fig.(\ref{['fig:Lanczos_3,5']}) for N=5, and M=3, and Fig.(\ref{['fig:fig:Lanczos_3,7']}) for N=7, and M=3 respectively for the values of $\Lambda=0.1, 2.5, 6.6$. Fig.(\ref{['fig:Lanczos_3,10']}) plots the Lanczos coefficients $b_{n}$ for $\Lambda=2.5$, N=10, and M=3. For all the cases, the Lanczos coefficients $b_{n}$ initially grows linearly with n up to $n\sim \ln{D}$, and then slowly decrease to zero with a slope of order $\sim -\frac{1}{D^{2}}\sim -\frac{1}{K}$. The Lanczos sequence $\{b_{n}\}$ smoothens as we increase the system size.
• Figure 2: The above figure plots the variance $\sigma^2$ as a function of the interaction parameter $\Lambda$ for $(N,M)= (3,3),(3,5)$. It can be seen from the plot that as $\Lambda$ becomes finite starting from $\Lambda=0$, the variance decreases and becomes close to zero for $\Lambda=1$. This behaviour of the variance points towards the transition from integrable to chaotic.
• Figure 3: Early time plot ( scrambling) of Krylov complexity $C_{K}(t)$ for $\Lambda=2.5$ with time t. The magenta, blue, and green line represents plot of $C_{K}(t)$ for $\{N=3, M=3, t_{s}=0.36\}$, $\{N=5, M=3, t_{s}=0.48\}$, $\{N=7, M=3, t_{s}=0.55\}$ respectively.
• Figure 5: Lanczos sequence Fig.(\ref{['fig:diffopLan_2.5']}) and K-complexity Fig.(\ref{['fig:diffopCom_2.5']}) for a different choice of initial operator \ref{['iniopdiff']}. Fig.(\ref{['fig:diffopLan_2.5']}) shows the Lanczos sequence for $N=5$, and $M=3$. One can easily see and relate to the previous results and conclude that the Lanczos coefficients $b_{n}$ initially grow linearly with n up to $n\sim \ln{D}$, and then slowly decrease to zero with a slope of order $\sim -\frac{1}{D^{2}}\sim -\frac{1}{K}$. Fig.(\ref{['fig:diffopCom_2.5']}) plots K-complexity at all time scales for $\Lambda=2.5$ with $N=5$, and $M=3$. The linear growth and the saturation can be observed at times of $\mathcal{O}(D)$, and at $\mathcal{O}(D^{2})$ respectively. For $N=5$, and $M=3$, $D=441$ and $K\sim 421$.
• Figure 6: Plot of K-complexity at all different scales for $\Lambda=0.1$ with $N=5$, and $M=3$. Numerical fits are included at both the early time plot in Fig.(\ref{['fig:numexpfit']}) near the scrambling time $t_{s}=0.48$ and at times $\sim \mathcal{O}(D)$ in Fig.(\ref{['fig:numlinfit']}) respectively. Fig.(\ref{['fig:numexpfit']}) shows the exponential growth of K-complexity at early times ($t\sim \log(ln D)$). The non-linear fit function is $C_{K}(t)=0.0379053\,\exp{(0.49635\,t)}$ with $r^{2}=0.989944$ denoted by red dashed lines. Fig.(\ref{['fig:numlinfit']}) shows the linear growth of K-complexity at times of $\mathcal{O}(D)$.The linear fit function is given by $C_{K}(t)=-2.92944 + 1.85039\,t$ with $r^{2}=0.992016$ denoted by red line.