Symbolic recursion method for strongly correlated fermions in two and three dimensions

Abstract

We present a symbolic implementation of recursion method for the dynamics of strongly correlated fermions on one-, two- and three-dimensional lattices. Focusing on two paradigmatic models, interacting spinless fermions and the Hubbard model, we first directly confirm that the universal operator growth hypothesis holds for interacting fermionic systems, manifested by the linear growth of Lanczos coefficients. Equipped with symbolically computed Lanczos coefficients and knowledge of their asymptotics, we are able to compute infinite-temperature autocorrelation functions up to times long enough for thermalization to occur. In turn, the knowledge of autocorrelation functions unlocks transport properties. We compute with high precision the charge diffusion constant over a broad range of interaction strengths, $V$. Surprisingly, we observe that these results are well described by a simple functional dependence $\sim 1/V^2$ universally valid both for small and large $V$. All results are obtained directly in the thermodynamic limit. Our results highlight the promise of symbolic computational paradigm where the most costly step is performed once and outputs symbolic results that can further be used multiple times to easily compute physical quantities for specific values of model parameters.

Figures (3)

• Figure 1: Conceptual scheme of the symbolic--computational workflow: the computationally expensive symbolic evaluation of the $n$-fold commutators $[H(V),A]^{(n)}$ is carried out once, while the resulting symbolic moments $\mu_{2n}(V)$ are subsequently reused multiple times to compute correlation functions $C(t)$ at specific values of the parameter $V$.
• Figure 2: Dynamics of infinite-temperature autocorrelation function (\ref{['infC']}) computed using the recursion method. The results are valid in thermodynamic limit. Top row: spinless-fermion $t$-$V$ model (\ref{['HamTV']}). Bottom row: Hubbard model (\ref{['HamHub']}). Curves of different colors correspond to different values of the interaction parameter $V=2,4$, while being obtained from the same set of symbolically computed moments. Gray dashed lines indicate rigorous upper and lower bounds (\ref{['bounds']}) obtained from the Taylor expansion. Green dots are the exact diagonalization results for a periodic chains of $L=12$ and $L=7$ sites in panels (a) and (d) correspondingly. Insets show the Lanczos coefficients and their linear extrapolation.
• Figure 3: Diffusion constant in the infinite-temperature limit at half-filling, $\langle n_{i\sigma}\rangle=1/2$, as a function of the interaction strength. Top row: spinless-fermion $t$-$V$ model (\ref{['HamTV']}). Bottom row: Hubbard model (\ref{['HamHub']}). Solid blue -- the results of the recursion method, with the width of the curve indicating the estimated uncertainty. Dashed red -- the $\kappa/V^2$ fit. For the 2D Hubbard model, this fit coincides with the perturbative result of refKovacevic_2025_Toward, see supplement for details.