Bootstrap Embedding for Interacting Electrons in Phonon Coherent-state Mean Field

Abstract

We develop a fermi-bose bootstrap embedding (fb-BE) framework for the ground state of interacting elec- trons coupled to phonon mean field. The method combines bootstrap embedding for correlated electrons with a self-consistent coherent-state mean-field treatment for phonons. This method models the interacting electron-phonon problem as a system of correlated electrons traveling in a self-consistently specified potential landscape, allowing for efficient treatment of large lattice systems. Convergence of the methods for frag- ment size and total system size are demonstrated for one-dimensional Hubbard-Holstein model for up to 350 sites. Finite-size scaling is performed to extrapolate to infinite system size. Benchmarking against density matrix renormalization group for small 8-site system at half- and quarter-filling shows orders-of-magnitude runtime advantage. The comparison further reveals that the method performs best in regimes dominated by localization, such as the Mott insulating phase and the strong-coupling tiny polaron regime, where the local embedding ansatz is still valid. However, due to the mean-field treatment for phonons, we find limitations of our methods in the weakly coupled delocalized region and at the Peierls transition, where quantum phonon fluctuations and long-range kinetic correlations become substantial.

Figures (7)

• Figure 1: Self-Consistent Fermi-Bose Bootstrap Embedding (fb-BE)
• Figure 1: Convergence of the outer Hubbard--Holstein self-consistency loop, quantified by the Euclidean norm of the phonon displacement update $\|\Delta \alpha^{(k)}\|_2$ as a function of outer iteration. For the $L=8$ half-filled system at $U=5$, $g=0.5$ (blue circles), linear mixing of $\alpha$ produces monotonic and approximately exponential convergence. For the larger system ($L=350$, $U=5$, $g=0.1$; orange squares), the direct (unmixed) update converges in two iterations, with the residual dropping below $10^{-6}$ after the first correction.
• Figure 2: Finite-size scaling of the ground-state energy per site at at different $U=$ values. The energy density $E(L,U)$ is plotted versus inverse system size $1/L$. The dashed line is a linear fit of the form $E(L,U)=E_\infty(U)+a(U)/L$, whose intercept yields the infinite-size-limit value $E_\infty$.
• Figure 3: Ground-state energy per site $E(L,U)$ as a function of interaction strength $U$ for $L=150$--350 together with the extrapolated infinite-size result $E_\infty(U)$ (upper panel). Finite-size deviation $\Delta E(L,U)=E(L,U)-E_\infty(U)$, highlighting the systematic suppression of size effects with increasing $L$ (lower panel).
• Figure 4: CPU time (total runtime) comparison between DMRG and fb-BE for an 8-site Hubbard--Holstein chain at half filling ($g=0.4$).