Robustness of bipolaronic superconductivity to electron-density-phonon coupling

TL;DR

The paper addresses how simultaneous Holstein (local) and bond SSH (nonlocal) electron-phonon couplings influence bipolaron formation and superconductivity on a 2D square lattice. Using unbiased Diagrammatic Monte Carlo, it analyzes binding energy $\Delta_{\text{BP}}$, effective mass $m^*_{\text{BP}}$, and size $R^2_{\text{BP}}$, and estimates the superconducting transition temperature $T_c$; it finds a cooperative regime where moderate $g_H$ and $g_B$ enhance $T_c$ by forming compact, still-mobile bipolarons. The study also explores frequency asymmetry, notably $\omega_H/t=2\omega_B/t$, showing that higher Holstein frequencies broaden the cooperative window and can mitigate mass renormalization, thereby sustaining higher $T_c$ across wider parameter ranges. These results highlight how multimode phonon environments can be engineered to optimize high-$T_c$ bipolaronic superconductivity, offering concrete strategies for materials with multiple phonon branches.

Abstract

We study bipolaron formation and bipolaronic superconductivity on a square lattice, where electrons couple to both local Holstein phonons via on-site charge density and nonlocal bond Su-Schrieffer-Heeger phonons via modulation of hopping amplitudes. Using an unbiased Diagrammatic Monte Carlo method, we investigate how the interplay between these two types of electron-phonon coupling affects the bipolaron binding energy, effective mass, spatial extent (quantified by the mean-squared radius), and the superconducting transition temperature $T_c$. We find that, in some parameter space, the moderate Holstein coupling, though detrimental to $T_c$ when acting alone, can enhance superconductivity when combined with the bond SSH coupling by further compressing the bipolaron without significantly increasing its mass. Similarly, introducing bond SSH coupling into a Holstein bipolaron reduces its size while keeping the effective mass nearly unchanged, leading a higher $T_c$. These effects give rise to nonmonotonic behavior and reveal a cooperative regime in which both couplings work together to enhance superconductivity. We further examine phonon frequency asymmetry, particularly the case $ω_H/t = 2ω_B/t$, and show that in the deep adiabatic regime, adding Holstein coupling can even raise $T_c$ when combined with bond SSH coupling. These results highlight the distinct and complementary roles of local Holstein and non-local bond SSH electron-phonon couplings, and suggest strategies for optimizing high-$T_c$ superconductivity in systems with multiple phonon modes.

Figures (4)

• Figure 1: (a) Bipolaronic superconducting transition temperature $T_c$ (in units of $\omega_H$) as a function of Holstein coupling $g_H/t$ for various phonon frequencies, with $\omega_H/t = \omega_B/t$ and bond SSH coupling $g_B/t$ fixed at the optimal values identified for the pure bond SSH model PhysRevX.13.011010: $g_B/t = 1.25$ for $\omega_B/t = 1.0$ (black dots), $0.75$ for $0.5$ (red squares), $0.5477$ for $0.3$ (blue upward triangles), and $0.433$ for $0.2$ (green downward triangles). (b) $T_c$ (in units of $\omega_B$) as a function of bond SSH coupling $g_B/t$ for fixed Holstein coupling $g_H/t = 1.25$ and $\omega_H/t = \omega_B/t$. Results are shown for $\omega_B/t = 1.0$ (black downward triangles) and $0.5$ (red circles). The addition of bond SSH coupling significantly enhances $T_c$, revealing the cooperative role of nonlocal bond SSH electron-phonon coupling in Holstein bipolarons.
• Figure 2: Bipolaron properties at equal adiabaticity $\omega_B/t = \omega_H/t = 1.0$. (a) Binding energy $\Delta_{\text{BP}}$ (in units of $t$), and (b) effective mass $m^*_{\text{BP}}/m_0$ (green upward triangles, with $m_0 = 2m_e = 1/t$) and mean-squared radius $R^2_{\text{BP}}$ (orange downward triangles), plotted as functions of Holstein coupling $g_H/t$ at fixed bond SSH coupling $g_B/t = 1.25$. (c,d) Same quantities as in (a,b), now shown as functions of bond SSH coupling $g_B/t$ at fixed Holstein coupling $g_H/t = 1.25$ and $\omega_H/t=1.0$. Error bars, if not visible, are smaller than the symbol size.
• Figure 3: (a) Bipolaronic superconducting transition temperature $T_c$ (in units of $\omega_H$) as a function of Holstein coupling $g_H/t$ for various phonon frequencies, with $\omega_H/t = 2\omega_B/t$. The bond SSH coupling $g_B/t$ is fixed at the optimal values identified for the corresponding pure bond model: $g_B/t = 1.25$ for $\omega_B/t = 1.0$ (black dots), $0.75$ for $0.5$ (red squares), $0.5477$ for $0.3$ (blue upward triangles), and $0.433$ for $0.2$ (green downward triangles). (b) $T_c$ (in units of $\omega_B$) as a function of bond SSH coupling $g_B/t$ for fixed Holstein coupling $g_H/t = 1.25$ and $\omega_H/t = 2\omega_B/t$. Results are shown for $\omega_B/t = 1.0$ (black downward triangles) and $0.5$ (red circles). The addition of bond SSH coupling significantly enhances $T_c$, revealing the cooperative role of nonlocal electron-phonon couplings in Holstein bipolarons.
• Figure 4: Bipolaron properties at unequal adiabaticity $\omega_H/t = 2\omega_B/t$. (a) Binding energy $\Delta_{\text{BP}}$ (in units of $t$), and (b) effective mass $m^*_{\text{BP}}/m_0$ (green upward triangles, with $m_0 = 2m_e = 1/t$) and mean-squared radius $R^2_{\text{BP}}$ (orange downward triangles), plotted as functions of Holstein coupling $g_H/t$ at fixed bond SSH coupling $g_B/t = 0.5477$, for $\omega_H/t = 0.6$ and $\omega_B/t = 0.3$. (c,d) Same quantities as in (a,b), now shown as functions of Holstein coupling $g_H/t$ at fixed bond SSH coupling $g_B/t = 1.25$, for $\omega_H/t = 1.0$ and $\omega_B/t = 0.5$. Error bars, if not visible, are smaller than the symbol size.