Pseudogap crossover in the electron-phonon system

Thermodynamic properties of the square-lattice Holstein model of the electron-phonon problem with phonon frequencies small compared to the bare Fermi energy are obtained using Monte Carlo methods, a strong-coupling (bipolaronic) expansion, and a weak-coupling Migdal-Eliashberg approach. Already at elevated temperatures where the charge-density wave (CDW) and superconducting (SC) correlations are very short range, a crossover occurs as a function of increasing electron-phonon coupling, ${\lambda }_0$, from a normal metallic regime to a pseudogap regime. At sufficiently low $T$, a SC phase is found for small ${\lambda }_0$ and a commensurate insulating CDW phase for large ${\lambda }_0$.

Figure 1

Schematic phase diagram of the electron-phonon problem with small $w\equiv \hslash {\omega }_0/E_F^{\left(0\right)}$. Depending on details of the band structure and the electron density, additional CDW phases can arise (including metallic ones at intermediate ${\lambda }_0$).

Figure 2

Phase diagram of the Holstein model in the $M\to \infty$ limit with $t^{\prime }/t=-0.3$. Chemical potential has been chosen such that $n=0.8$ for $T=0.25t$. The inset shows the phase diagram extended to higher temperatures where, for ${\lambda }_0\gtrsim 1, T^{★}$ becomes equal to half the bipolaron binding energy, $V/2$, as expected from the strong-coupling limit. See text for additional details.

Figure 3

Density $n$ as a function of $T$ for various coupling strengths. The dashed line shows the evolution of the density for noninteracting electrons. Arrows indicate $T_{\mathrm{cdw}}$ inferred from finite-size scaling. Linear system size is $L=12$.

Figure 4

DOS as a function of energy relative to $E_F$ at various $T$ (a) for ${\lambda }_0=0.54$ and (b) for ${\lambda }_0=2.43$. The dashed vertical lines indicate the bipolaron binding energy $\pm V$. Linear system size is $L=12$. Delta functions are broadened with $\eta \approx 10/L^2$. $T^{★}$ is identified as the highest temperature at which there is a local minimum at $\omega =0$. (c) The spin susceptibility as a function of $T$ for various ${\lambda }_0$; the arrows indicated $T^{★}$ inferred from the DOS. Linear system size is $L=10$.

Figure 5

Temperature averaged DOS at the Fermi energy ${\rho }_{\beta }$ at $T=0.25t$, normalized by the zero-temperature, noninteracting DOS ${\rho }^{\left(0\right)}$. At this elevated temperature ${\rho }_{\beta }<{\rho }^{\left(0\right)}$ even for ${\lambda }_0=0$. The breakdown of ME theory occurs for ${\lambda }_0\approx 0.5$. Linear system size is $L=12$.