Quantum Monte Carlo study of an anharmonic Holstein model

We study the effects of anharmonicity on the physics of the Holstein model, which describes the coupling of itinerant fermions and localized quantum phonons, by introducing a quartic term in the phonon potential energy. We find that the presence of this anharmonic term reduces the extent of the charge density wave (CDW) phase at half-filling as well as the transition temperature to this phase. Doping away from half-filling, we observe a first-order phase transition between the CDW and a homogeneous phase which is also present in the harmonic model. In addition, we study the evolution of the superconducting susceptibility in the doped region and show that anharmonicity can enhance the superconducting response.

Figure 1

Behavior of the average density, $\langle n_i\rangle$, and phonon displacement, $|\langle x_i\rangle |$, as functions of the position, $i$, along one axis in the square lattice in the homogeneous and charge density wave (CDW) phases. In CDW phase at half-filling, there is symmetry breaking and two alternate values of $\langle n_i\rangle$ and $\langle x_i\rangle$ are observed. Out of half-filling, in an homogeneous phase, $\langle n_i\rangle$ and $\langle x_i\rangle$ are independent of the position.

Figure 2

Density as a function of $\mu$ for different values of ${\omega }_4$. $L=6, g=1, \beta =20$, and $\omega =0.5$. We find a reduction of the charge density gap at half-filling when ${\omega }_4$ is increased. The apparent gaps away from half-filling are shell effects. A simulation at $L=8, {\omega }_4=0.0025$ shows that half-filled plateau is not affected by finite size effects, while the plateaux off of half-filling are reduced for larger sizes.

Figure 3

Phase diagrams of the system obtained from $L=6$ simulations at $\beta =20$ and $g=1$, for (a) $\omega =0.5$ and (b) $\omega =1$ and (c) comparison of the charge gaps in these cases. The anharmonic parameter ${\omega }_4$ ranges from 0 to 0.01. The area enclosed by blue curves is the incompressible charge density wave (CDW) phase at half-filling, while the rest of the phase diagram corresponds to compressible phases that should become superconducting at low temperatures. The red triangles mark the half-filled chemical potential inferred from the approximate theory of Appendix pp1. The width of the CDW phase is strongly reduced due to the anharmonic effects for (a) $\omega =0.5$. For (b) $\omega =1$, the charge gap is relatively unaffected by the anharmonicity ${\omega }_4$ in the range shown, although (c) it is generally smaller for $\omega =1$ than for $\omega =0.5$.

Figure 4

The density $\rho$ and charge density wave (CDW) structure factor $S_{\mathrm{cdw}}$ (rescaled for better visibility) for an $L=10$ system, focusing on the CDW plateau. Notice that, for $\beta =10$, the density does not yet show a plateau at half-filling; $\beta =20$ is necessary for the system to display the ground state behavior and exhibit the CDW gap. The structure factor $S_{\mathrm{cdw}}$ displays an abrupt change of values when the system is doped away from half-filling.

Figure 5

Average value $\langle x_i\rangle$ (top) and double occupancy (bottom) as functions of $L^{-1}$ for different ${\omega }_4$ in half-filled systems. $|\langle x_i\rangle |$ is reduced as ${\omega }_4$ increases, although it always extrapolates to nonzero values. For the harmonic case ${\omega }_4=0, \langle x_i\rangle =-4$. The double occupancy is also reduced but always extrapolates to values larger than $\langle n_{i\uparrow }\rangle \langle n_{i\downarrow }\rangle =0.25$.

Figure 6

Structure factor $S_{\mathrm{cdw}}$ as a function of size $L$ for different values of ${\omega }_4$ at half-filling. For all ${\omega }_4$, the linear extrapolation of $S_{\mathrm{cdw}}$ to $L\to \infty$ is nonzero. The linear extrapolation is based on fits for the data with $L\ge 8$.

Figure 7

Finite size analysis for $\omega =0.5$ and ${\omega }_4=0.001$. (top) Structure factor for different sizes as a function of temperature $T$. (bottom) Rescaled structure factor as a function of the rescaled reduced temperature. The critical temperature $T_c=0.24\pm 0.01$ is chosen to obtain the best possible collapse between the different curves.

Figure 8

Charge density wave (CDW) critical temperature $T_c$ at half-filling as a function of anharmonicity ${\omega }_4$ for $\omega =0.5, t=1$, and $g=1$. With sizes up to $L=12$, the finite-size analysis was only feasible for ${\omega }_4\le 0.005$.

Figure 9

Density $\rho$ and structure factor $S_{\mathrm{cdw}}$ as functions of the chemical potential for $\omega =1$ and several sizes at $\beta =16$. An abrupt change of the density is found when the system is doped away from half-filling, which also corresponds to the disappearance of charge density wave (CDW) order.

Figure 10

Density $\rho$ as a function of the chemical potential for $\omega =1$ and several values of ${\omega }_4$ at $\beta =16$. The abrupt change of the density found when the system is doped is present for all values of ${\omega }_4$.

Figure 11

Density $\rho$, energy per site $E/L^2$, and grand potential per site $E/L^2-\mu \rho$ as functions of $\mu$ for different initial conditions of the simulations [charge density wave (CDW) start or homogeneous start]. We observe hysteresis with an intermediate region where two different phases coexist.

Figure 12

Density $\rho$ and s-wave pairing susceptibility ${\chi }_{\mathrm{s}}$ as functions of $\beta$ for $\omega =0.5, {\omega }_4=0.001$, and $g=t=1$. $\mu =-3.1$ corresponds to the half-filled system, $\mu =-4$ to a lower density $\rho \simeq 0.25$, and $\mu =-2.5$ to $\rho >1.5$.

Figure 13

The s-wave pairing susceptibility ${\chi }_{\mathrm{s}}$ as a function of density $\rho$ for $\omega =0.5, g=t=1$, different values of ${\omega }_4$, and $\beta =8,10$. The susceptibility increases as ${\omega }_4$ or $\beta$ increases.

Figure 14

The s-wave pairing susceptibility ${\chi }_{\mathrm{s}}$ as a function of density $\rho$ for $\omega =1, g=t=1$, different values of ${\omega }_4$, and $\beta =8,10$. The susceptibility is not sensitive to changes in ${\omega }_4$ but increases with $\beta$.

Figure 15

For ${\omega }_4=0.0075, \omega =0.5$, and $L=6$ to $L=16$, rescaled data for $S_{\mathrm{cdw}}$ do not cross each other, precluding the use of finite-size scaling. This is probably due to larger finite-size scaling corrections.