Approximation theory for Green's functions via the Lanczos algorithm

TL;DR

This work develops a theoretical framework for approximating Green's functions of chaotic many-body systems via continued fractions obtained from the Lanczos recursion, focusing on stitching the tail of the coefficients with a solvable sequence. Under the Operator Growth Hypothesis, the authors derive how the convergence rate of the stitching approximation depends on the decay of staggered subleading terms in the Lanczos coefficients and on the differentiability of the spectral function at the origin. They establish an explicit infinite-product relation linking the zero-frequency Green's function to Lanczos coefficients, provide upper bounds and uniform-convergence conditions along the imaginary axis, and apply these results to diffusion constants in the mixed-field Ising model. The analysis reveals two regimes (relevant vs irrelevant staggering) and shows that, in most physically interesting diffusive cases, the stitching error decays only as a power of log N, implying exponential-in-poly(1/ε) resource growth to achieve a target accuracy ε. Numerical results in d>1 and d=1 confirm the theory and illustrate when Meixner–Pollaczek stitching offers advantages over simpler constant-tail approaches.

Abstract

It is known that Green's functions can be expressed as continued fractions; the content at the $n$-th level of the fraction is encoded in a coefficient $b_n$, which can be recursively obtained using the Lanczos algorithm. We present a theory concerning errors in approximating Green's functions using continued fractions when only the first $N$ coefficients are known exactly. Our focus lies on the stitching approximation (also known as the recursion method), wherein truncated continued fractions are completed with a sequence of coefficients for which exact solutions are available. We assume a now standard conjecture about the growth of the Lanczos coefficients in chaotic many-body systems, and that the stitching approximation converges to the correct answer. Given these assumptions, we show that the rate of convergence of the stitching approximation to a Green's function depends strongly on the decay of staggered subleading terms in the Lanczos cofficients. Typically, the decay of the error term ranges from $1/\mathrm{poly}(N)$ in the best case to $1/\mathrm{poly}(\log N)$ in the worst case, depending on the differentiability of the spectral function at the origin. We present different variants of this error estimate for different asymptotic behaviours of the $b_n$, and we also conjecture a relationship between the asymptotic behavior of the $b_n$'s and the smoothness of the Green's function. Lastly, with the above assumptions, we prove a formula linking the spectral function's value at the origin to a product of continued fraction coefficients, which we then apply to estimate the diffusion constant in the mixed field Ising model.

Figures (5)

• Figure 1: Stitching approximation for irrelevant (a) and relevant (b) cases. For the irrelevant case there is an exact expression $|G(-\mathrm{i}0^{+})| = \pi^2/8$. However, the relevant case does not have a closed expression; therefore, we approximate $G(-\mathrm{i}0^{+})$ using a partial product (constant stitching) in Eq. \ref{['eqn: Product']} which yields $|G(-\mathrm{i}0^{+})| \approx 2.8071$. The error is predicted to scale as $N^{-2}$ for the irrelevant case and as $N^{-2/3}$ for the relevant case; these predictions agree with the fitting curves plotted in the insets.
• Figure 2: (a) Lanczos coefficients obtained by initialising the algorithm with a normalised current operator. The coefficients are expected to grow as $n/\log(n)$. However, resolving the logarithm and the sublinear corrections is challenging. (b) Diffusion constant as a function of the number of Lanczos coefficients $n$. We have rescaled it by a factor of two to compare it with the result quoted in Ref. PhysRevX.9.041017 which is the dashed line $2D \approx 3.35$.
• Figure 3: Truncation error as a function of the truncation level $N$. (a) The error for some values of $\eta$, which is a parameter in the recurrence coefficients of the Meixner–Pollaczek polynomials, at a fixed $z=-2\mathrm{i}$. The particular value of $\eta$ does not affect the convergence rate. (b) Plot of the error as a function of $N$ for different values of $\Im(z)|$, for fixed $\eta=2$. The larger the value of $|\Im(z)|$ the faster the approximation converges, independently of the value of $\eta$. (c) Fitting to determine the exponent $\beta(z)$ defined as $e_{N,t}(z) \sim \alpha(z) N^{\beta(z)}$. The analytical prediction from Eq. \ref{['eqn: Truncation error']} is $\beta(z) = -|\Im(z)|$. Therefore, we expect the fitted line to have a slope of $-1$ and a zero intercept. The fitted parameters $c_1$ and $c_2$ are indeed close to these values.
• Figure 4: Numerical evaluations of $G^{(1)}(0; 2n)$ for the different Lanczos coefficients listed in Table \ref{['tab: Prediction d=1']} ($d=1$) and Table \ref{['tab: Prediction d>1']} ($d>1$). The red annotations in each plot indicate the theoretical prediction for the scaling of $G^{(1)}(0; n)$. In all cases except Subfigure (b), $G^{(1)}(0;n)$ diverges as $n \to \infty$. To test the theoretical prediction in these cases, we present suitably rescaled quantities on the horizontal and vertical axes such that, if the prediction holds, the data should align along a straight line. For each of these cases, we also include the best fitting line for comparison.
• Figure 5: Numerical computations of $G^{(2)}(0; 2n)$ for the different Lanczos coefficients listed in Table \ref{['tab: Prediction d=1']} ($d=1$) and Table \ref{['tab: Prediction d>1']} ($d>1$). The red labels in each plot indicate the theoretical prediction for the scaling of $G^{(2)}(0; n)$. In all cases $G^{(2)}(0;n)$ is predicted to diverge as $n \to \infty$. To test the rate of divergence, we display rescaled quantities on the horizontal and vertical axes such that, if the prediction holds, the data should align along a straight line. In each case, a best-fit line is also shown for comparison.