Tangent space Krylov computation of real-frequency spectral functions: Influence of density-assisted hopping on 2D Mott physics

TL;DR

The paper introduces TaSK, a tangent-space Krylov method that efficiently computes real-frequency spectral functions on top of ground-state MPS from DMRG by projecting dynamics to the tangent space of the GS manifold. By enforcing a tangent-space vector-space structure, TaSK avoids the orthogonality problems of traditional Lanczos in MPS, enabling accurate spectral weights and pole positions via a Lanczos series on the projected Hamiltonian H^{1⊥}. Benchmarking on Haldane–Shastry and Heisenberg chains validates TaSK against analytical and Bethe-ansatz results, while the application to a 2D Hubbard model on a cylinder with density-assisted hopping (DAH) reveals particle-hole asymmetries in the Mott insulator and a nonlocal modification of the self-energy pole dispersion, including emergent next-nearest-neighbor terms. The findings highlight robust, scalable access to dynamical correlations in challenging systems and point to DAH-induced dynamical effects that could influence cuprate-like physics and potential superconducting tendencies.

Abstract

We present a tangent-space Krylov (TaSK) method for efficient computation of zero-temperature real-frequency spectral functions on top of ground state (GS) matrix product states (MPS) obtained from the Density Matrix Renormalization Group. It relies on projecting resolvents to the tangent space of the GS-MPS, where they can be efficiently represented using Krylov space techniques. This allows for a direct computation of spectral weights and their corresponding positions on the real-frequency axis. We demonstrate the accuracy and efficiency of the TaSK approach by showcasing spectral data for various models. These include the 1D Haldane-Shastry and Heisenberg models as benchmarks. As an interesting application, we study the Hubbard model on a cylinder at half-filling, augmented by a density-assisted hopping (DAH) term. We find that DAH leads to particle-hole asymmetric single-particle mobilities and lifetimes in the resulting Mott insulator, and identify the responsible scattering processes. Further, we find that DAH influences the dispersion of Green's function zeros beyond its range, which has a frustrating effect on the Mott insulator studied here.

Figures (10)

• Figure 1: Dynamical structure factor of the ${\hbox{\small$\mathscr{L}$}} = 40$ Haldane-Shastry model at $k=\pi$ and $\mathcal{N}_{\mathrm{kr}}=30$. (a) Comparison between analytical values (blue dots) and the TaSK result (black circles) for $D^* = 700$. Horizontal bars depict $\delta_\alpha$ as error estimate for $\omega_\alpha$. Inset: zoom to high frequencies. (b) Convergence trend across data sets with increasing $D^*$, relative to $\delta_\alpha$. Dashed lines indicate the analytical values from (a).
• Figure 2: Dynamical structure factor of the Heisenberg chain at $k = \pi$. The dashed line indicates the exact Bethe ansatz result for the full $2+4$-spinon contribution from Ref. Caux2006.
• Figure 3: Comparison of $A_\mathbf{k}(\omega)$ and $-\mathrm{Im}\Sigma_\mathbf{k}(\omega)$ for the Hubbard model on a $10\times 4$ cylinder, computed via TaSK $\mathrm{(a,b)}$ without and $\mathrm{(c,d)}$ with DAH, and (e,f) computed by treating DAH simplistically via Eq. \ref{['eq:SE_correction_term']}. Solid gray lines in (b,d,f) mark the self-energy pole where it becomes too sharp for the frequency resolution of our grid. Dashed gray lines indicate $\omega = \pm U/2$. Inset in (b): purple lines show the Brillouin zone paths $\Gamma \to X$ and $X\to\Pi$. (g,h,i) Line cuts of $A_\mathbf{k}(\omega)$ for $\mathbf{k}=\Gamma,X,\Pi$.
• Figure S-1: Convergence of $\omega_\alpha$ with the number of Krylov steps $\mathcal{N}_{\mathrm{kr}}$ for (a) the Haldane-Shastry model on a ring of size ${\hbox{\small$\mathscr{L}$}} = 40$ with $D^* = 700$ and (b) the Heisenberg chain of length ${\hbox{\small$\mathscr{L}$}} = 64$ with $D^* = 512$. The color scale indicates spectral weights $S_\alpha$, represented on a logarithmic scale. Spectral weights $\leq 10^{-4}$ are clipped to the minimum color value to highlight dominant contributions to the DSF.
• Figure S-2: Discrete spectral data for the dynamical structure factor of the ${\hbox{\small$\mathscr{L}$}} = 64$-site Heisenberg chain at $k=\pi$. Blue dots are computed via TaSK, black circles by targeting low-energy states with DMRG. We obtain the TaSK data in Fig. \ref{['fig:HeisenbergPowerLaw']} by binning the data shown here, as described near the end of Sec. \ref{['sec:Convergence_Spins']}.