Matrix-product-state-based band-Lanczos solver for quantum cluster approaches

TL;DR

It is found that only when treating large cluster sizes, observables can be extrapolated to the thermodynamic limit, which is demonstrated at the example of the staggered magnetization.

Abstract

We present matrix-product state (MPS) based band Lanczos method as solver for quantum cluster methods such as the variational cluster approximation. While a naïve implementation of MPS as cluster solver would barely improve its range of applicability, we show that our approach makes it possible to treat cluster geometries well beyond the reach of exact diagonalization methods. The key modifications we introduce are a continuous energy truncation combined with a convergence criterion that is more robust against approximation errors introduced by the MPS representation and provides a bound to deviations in the resulting Green's function. The potential of the resulting cluster solver is demonstrated by computing the self-energy functional for the single-band Hubbard model at half filling in the strongly correlated regime, on different cluster geometries. Here, we find that only when treating large cluster sizes, observables can be extrapolated to the thermodynamic limit, which we demonstrate at the example of the staggered magnetization. Treating clusters sizes with up to $6\times 6$ sites we obtain significant improvement over the extrapolation accessible with exact diagonalization solvers when comparing to quantum Monte Carlo results. Finally, we illustrate the applicability of the MPS cluster solver to more complex models by calculating spectral properties as relevant for the electron-doped cuprate CaCuO$_2$.

Figures (12)

• Figure 1: fig:e-trunc:sketch Schematic representation of the energy truncation. Overlaps $\epsilon_n=\langle\tilde{E}_n\vert\psi\rangle<\delta$ are neglected when constructing the sublattice projection (marked red). fig:e-trunc:dos Effect of the truncation threshold on the (lattice) density of states $N(\omega)$ for a $4\times4$ cluster in the phase, see \ref{['sec:VCA']} for details. The results for $\delta=10^{-2},10^{-3}$ are shifted for clarity.
• Figure 2: Comparison of convergence critera. The blue curve (right $y$- axis) shows the lowest eigenvalue $E^D$ of the effective Hamiltonian in the Krylov subspace as a function of the dimension $D$ for a $4\times 6$- site Hubbard cluster at $U=8t$ and a staggered magnetic field $h_z=0.09t$. The conventional convergence criterion $\delta E = E^{D+1} - E^{D}$ is given by the red line (left $y$- axis), exhibiting strong fluctuations over several orders of magnitude. In contrast, oscillations in the suggested Hochbruck- Lubich critertion $\epsilon^D$ shown by the green line are significantly smaller, overlaying a systematic decrease with respect to the number of iterations.
• Figure 3: (A) Illustration of the geometries of different clusters with $N=24$ sites labeled according to Betts et al.Betts1996. Red arrows indicate the superlattice vectors that tile the lattice with the respective cluster, dashed lines indicate inter- cluster hopping terms and the Néel-type sublattice structure is indicated by different tones of grey. The clusters differ in their squareness ${\sigma}$, geometrical imperfection $J$ and bipartite imperfection $I_B$, see (B). (C) Points in the Brillouin zone that are multiples of reciprocal superlattice vectors, which are indicated by arrows. The corresponding reduced (superlattice) Brillouin zones are marked by parallelograms. An illustration of the corresponding mapping is shown in \ref{['app:1D']}.
• Figure 4: The $\Omega$ as a function of staggered Weiss field strength for a $2\times4$ cluster. Stationary points of $\Omega$ are indicated by arrows. Dots denote results obtained with the- based solver, in the inset shown for different maximal bond dimension $\chi^{\mathrm{max}}$ around the minimum of $\Omega$; the results obtained with an solver are shown as a line.
• Figure 5: Impact of the cluster geometry on the . $\Omega$ is calculated with $\chi^{\mathrm{max}}=1024$ for different 24-site clusters shown in \ref{['fig:BettsSketch']}. Minima of $\Omega$ are indicated by arrows, lines are only a guide to the eye. The inset shows $\Omega$ for two different mappings of the 24s0 cluster, '$4\times6$' and '$6\times4$', calculated with $\chi^{\mathrm{max}}=1024, 2048$.