Superconductivity and charge-density-wave in the Holstein model on the Penrose Lattice

Abstract

The exotic quantum states emerging in the quasicrystal (QC) have attracted extensive interest because of various properties absent in the crystal. In this paper, we systematically study the Holstein model at half filling on a prototypical structure of QC, namely rhombic Penrose lattice, aiming at investigating the superconductivity (SC) and other intertwined ordering arising from the interplay between quasiperiodicity and electron-phonon ({\it e}-ph) interaction. Through unbiased sign-problem-free determinant quantum Monte Carlo simulations, we reveal the salient features of the ground-state phase diagram. Distinct from the results on bipartite periodic lattices at half filling, SC is dominant in a large parameter regime on the Penrose lattice. When {\it e}-ph coupling is sufficiently strong, charge-density-wave order appears and strongly suppresses the SC. The strongest SC emerges at intermediate {\it e}-ph coupling strength and pronounced pairing fluctuation exists above the SC transition temperature. The strong pairing originates from the cooperative effects of unique lattice structure and macroscopically degenerate confined states at Fermi energy which uniquely exist on the Penrose lattice. Moreover, we demonstrate the forbidden ladders substantially suppress the phase coherence of SC. Our unbiased numerical results suggest that Penrose lattice is a potential platform to realize strong SC pairing, providing a promising avenue to searching for relatively high-$T_c$ SC dominantly induced by {\it e}-ph coupling.

Figures (5)

• Figure 1: (a) The different cluster regions (green, magenta, blue) and forbidden ladders (white) on the Penrose lattice. We have not shown the whole part of green cluster, which is connected to the edge of the largest forbidden ladder shown in this figure. The two red dots are the typical positions used in Fig. \ref{['spectrum']}. (b) Single-particle energy spectrum and DOS on the Penrose lattice with $13926$ sites. Inset is a magnified graph for energy spectrum around half-filling. (c) Schematic phase diagram of the Holstein model on the Penrose lattice. When the EPC is weak $g<g_{\rm CDW}$, the ground state is SC. When $g>g_{\rm SC}$, the ground state is CDW state. There is an SC-CDW coexistence phase between $g_{\rm CDW}$ and $g_{\rm SC}$.
• Figure 2: The inverse of SC susceptibilities versus temperature $T$ for different EPC strength $g$ with $N=201$. The phonon frequency is (a). $\Omega=1$ and (b). $\Omega=4$. The solid lines in (a) and (b) are the power law fit of susceptibilities. The fit function is $\chi_{\rm SC}^{-1}(T)=a+bT^c$. Insets in (a) and (b) present the whole range of temperatures simulated.
• Figure 3: The results of LDOS with varying temperature calculated at (a) the center of blue cluster and (b) the edge of forbidden ladder (shown in Fig. \ref{['penrose']}(a)). The parameters are fixed as $\Omega = 4$ and $g=4$. The dash lines indicate the position of zero for each curve of LDOS.
• Figure 4: The CDW structure factor versus temperature $T$ for different EPC strengths $g$ with $N=201$. The phonon frequencies are (a) $\Omega=1$ and (b) $\Omega=4$ respectively.
• Figure 5: The spatial pattern of (a) nearest neighbor hopping $T_{ij}$ and (b) nearest neighbor SC correlations $\Delta_i\Delta_j$ for $g=1$ with $\Omega=1$. The color bar on the right codes the value from the minimum to the maximum. (c), (d) show all the minimum and maximum values of $T_{ij}$ and $\Delta_i\Delta_j$ for different EPC strengths with $\Omega=1$ respectively. The minimum values of $T_{ij}$ and $\Delta_i\Delta_j$ are both on the forbidden ladders.