Competition between phase separation and spin density wave or charge density wave order: Role of long-range interactions

Recent studies of pairing and charge order in materials such as FeSe, ${\mathrm{SrTiO}}_3$, and 2H-${\mathrm{NbSe}}_2$ have suggested that momentum dependence of the electron-phonon coupling plays an important role in their properties. Initial attempts to study Hamiltonians which either do not include or else truncate the range of Coulomb repulsion have noted that the resulting spatial nonlocality of the electron-phonon interaction leads to a dominant tendency to phase separation. Here we present quantum Monte Carlo results for such models in which we incorporate both on-site and intersite electron-electron interactions. We show that these can stabilize phases in which the density is homogeneous and determine the associated phase boundaries. As a consequence, the physics of momentum dependent electron-phonon coupling can be determined outside of the trivial phase separated regime.

Figure 1

(a) Phase diagram of the HEH in the $U\text{−}\xi$ plane at fixed $V=0$ and $\omega =0.5,{\lambda }_0=1.0$. The lattice size is $N=32$ and inverse temperature $\beta =26$. The procedure for determining the positions of the phase boundaries is described in subsequent figures and discussion, and in the right-hand panel. (b) Snapshots of electron density $(n_i=n_{i,\uparrow }+n_{i,\downarrow })$ profiles. For $\xi =0.5,U=3$, the particles are clumped in one region of the lattice—the system exhibits PS. For $\xi =0.2,U=7$, the total density is uniform, as expected for an SDW where spin density oscillates, but charge density is constant. The three data sets at $\xi =0.2$ with $U=1,2,3$ are in the CDW phase. The order parameter (size of charge oscillation) decreases as $U$ grows and the phase boundary to the SDW is approached.

Figure 2

(a) Electron kinetic energy as a function of on-site repulsion $U$ for fixed intersite repulsion, $V=0.5$, and different values of electron-phonon interaction screening length $\xi$. (b) Double occupancy as a function of $U$. A sharp drop in $\mathcal{D}$ and a sharp increase in the magnitude of ${\mathcal{K}}_{\text{el}}$ indicate the entry into the SDW phase from the large $\xi$ PS region as $U$ increases. Structure at smaller $\xi$ is associated with the CDW-SDW boundary [see text and Fig. 3]. (c) Charge susceptibility as a function of $U$. ${\chi }_{\mathrm{charge}}$ is big in the weak $U$ CDW phase, and drops at the SDW boundary. (d) The spin susceptibility shows the opposite trend: ${\chi }_{\mathrm{spin}}$ is small at weak $U$, but increases as the SDW phase is entered at large $U$. The lattice size is $N=32$ and inverse temperature $\beta =26$. The phonon frequency is fixed at $\omega =0.5$ and electron-phonon coupling strength $\lambda =1.0$.

Figure 3

(a) Phase diagram of the HEH model in the $U\text{−}\xi$ plane for $V=0.5$. Compared to Fig. 1 the CDW-PS line is pushed out from $\xi \approx 0.38$ at $V=0$ to $\xi \approx 0.8$ here. In the Holstein limit, CDW correlations are stabilized by an amount $2V$ against SDW. The uncertainty in the precise nature of the phase where the CDW, SDW, and PS regions meet is indicated by the shaded region. See text. (b) Finite-size effect on the normalized charge structure factor $S_{\mathrm{charge}}\left(\pi \right)/N$ at $\xi =0.1$. The magnitude of $S_{\mathrm{charge}}\left(\pi \right)$ is proportional to the size of lattice, indicating the existence of true long-range order in the CDW phase. Therefore, the normalized charge structure factor $S_{\mathrm{charge}}\left(\pi \right)/N$ is $N$ independent in the CDW phase. $S_{\mathrm{charge}}\left(\pi \right)/N$ changes abruptly at the same $U/t$ value for all $N$. (c) Electron kinetic energy ${\mathcal{K}}_{\text{el}}$ (per site) for different lattice sizes at $\xi =0.1$. The position of the cusp is N independent.

Figure 4

Phase diagram of the HEH model in the $U\text{−}\xi$ plane for $V=1.0$ (solid lines). The dashed and dotted lines indicate the phase boundary (PB) of the CDW region $V=0$ and 0.5, with the SDW boundaries of Figs. 1 and 3 suppressed for clarity. As the intersite $V$ increases, the charge density wave is stabilized to longer-range electron-phonon interaction range $\xi$. At $\xi =0$ the CDW-SDW phase boundary moves upward approximately by $2V$ [30]. As with the preceding figure, the uncertainty in the precise nature of the phase where the CDW, SDW, and PS regions meet is indicated by the shaded region.

Figure 5

(a) Phase diagram of the HEH model in the $V\text{−}\xi$ plane for $U=3.0$. The on-site repulsion is too weak to produce SDW. (b) Phase diagram of the HEH model in the $V\text{−}\xi$ plane for $U=8.0$. Here the on-site repulsion results in the robust SDW phase and the combination of $U$ and $V$ eliminates all phase separation.

Figure 6

For the HEH model at $U=4$ and several values of $\xi$ we show as functions of $V$ the (a) electron kinetic energy, (b) the double occupancy, (c) the charge susceptibility, (d) the spin susceptibility. The number of sites $N=32$ and inverse temperature $\beta =26$. Phonon frequency $\omega =0.5$ and electron-phonon coupling strength ${\lambda }_0=1.0$.

Figure 7

Phase diagram of the HEH model in the $V\text{−}\xi$ plane for $U=4.0$ inferred from Fig. 6. The on-site repulsion is now big enough to give SDW sandwiched in between PS at large $\xi$ and CDW at large $V$.