First-order metal-insulator transitions in the extended Hubbard model due to self-consistent screening of the effective interaction

While the Hubbard model is the standard model to study Mott metal-insulator transitions, it is still unclear to what extent it can describe metal-insulator transitions in real solids, where nonlocal Coulomb interactions are always present. By using a variational principle, we clarify this issue for short- and long-range nonlocal Coulomb interactions for half-filled systems on bipartite lattices. We find that repulsive nonlocal interactions generally stabilize the Fermi-liquid regime. The metal-insulator phase boundary is shifted to larger interaction strengths to leading order linearly with nonlocal interactions. Importantly, nonlocal interactions can raise the order of the metal-insulator transition. We present a detailed analysis of how the dimension and geometry of the lattice as well as the temperature determine the critical nonlocal interaction leading to a first-order transition: for systems in more than two dimensions with nonzero density of states at the Fermi energy the critical nonlocal interaction is arbitrarily small; otherwise, it is finite.

Figure 1

Schematic phase diagram of the extended Hubbard model: (a) The continuous (solid) and first-order (dotted) metal-insulator transition lines touch at $V_c\left(U\right)$. A coexistence region surrounds the first-order transition. For large $V/U$ a CDW phase occurs. (b) The CDW phase can mask the first-order MIT.

Figure 2

Effective screening factor $\alpha$ as defined above Eq. (4) for NN (top panels) and L (bottom panels) interaction. Left (right) panels show the case of the honeycomb (square) lattice. We show results from low (red) to high (cyan) inverse temperatures $\beta t$.

Figure 3

Double occupancy for (a) the honeycomb ($\beta t=10$) and (b) square lattice ($\beta t=20$) for $V/t=0$ (black line), 0.6 (dashed red line), 1.2 (dotted green line), and 1.8 (thin blue line) for the case of NN interaction. The inset in (b) shows a close-up of the rectangle with the thermodynamically unstable states (dotted line), coexistence region (shaded area), and double occupancy by Maxwell construction (dashed line).

Figure 4

(a) Dependence of $max\left[\right|\partial \alpha /\partial \tilde{U}\left|\right]$ on $\beta t$ for the square lattice with linear fit (red line). (b) Corresponding $V_c\left(T\right)=-{[\partial \alpha /\partial \tilde{U}]}^{-1}$. Results for $\mathrm{NN}$ and $\mathrm{L}$ coincide within error bars. The fading black curve is a weak-coupling estimate of the critical $V$ for the CDW valid for small $V$.

Figure 5

DMFT results for the cubic (left panels) and diamond (right panels) lattices. From top to bottom the panels show staggered magnetization $m$ and ${\partial }_{\tilde{U}}\langle n_0{n}_0\rangle$.

Figure 6

Finite $\mathrm{\Delta }\tau$ extrapolation for the nearest-neighbor charge correlator on the square lattice for $\beta t=10$ and $L=10$. Left: linear fit (blue line) to raw data on $\mathrm{\Delta }{\tau }^{-2}$ (black dots) resulting in extrapolated value at $\mathrm{\Delta }\tau \to 0$ (blue dot) for $\tilde{U}/t=1.0$. Right: $\tilde{U}$-dependent results for the extrapolation.

Figure 7

Finite-$L$ extrapolation for the nearest-neighbor charge correlator on the square lattice for $\beta t=10$ and $\mathrm{\Delta }\tau =0.2$. Left: linear fit (blue line) to raw data on $L^{-2}$ (black dots) resulting in the extrapolated value at $\mathrm{\Delta }\tau \to 0$ (blue dot) for $\tilde{U}/t=1.0$. Right: $\tilde{U}$-dependent results for the extrapolation.

Figure 8

Raw data and smoothed data for on-site and nearest-neighbor charge correlators ($\langle n_0{n}_0\rangle , \langle n_0{n}_1\rangle$; top panels) and their derivatives with respect to $\tilde{U}$ (bottom panels) for the square lattice with $L=12, \mathrm{\Delta }\tau =0.2$, and $\beta t=4.0$. We show smoothing results of different fit windows, $w=1.0$ and $w=0.4$, with cubic polynomials. Where no error bars are visible, they are hidden behind the markers.

Figure 9

Same as Fig. 8, but for $L=12, \mathrm{\Delta }\tau =0.05$, and $\beta t=20.0$.

Figure 10

DMFT results for the $d=\infty$ hypercubic (left panels) and hyperdiamond (right panels) lattices. From top to bottom the panels show staggered magnetization $m$ and ${\partial }_{\tilde{U}}\langle n_0{n}_0\rangle$.

Figure 11

Results for the derivative of the (a) local and (b) nearest-neighbor charge correlators with respect to $\tilde{U}$ and (c) ${\alpha }_{\text{NN}}$ as defined in the main text from the DQMC simulation of the cubic lattice at $\beta t=10$ for lattices with linear sizes $L=4$ (solid blue line), $L=6$ (dashed yellow line), $L=8$ (dash-dotted green line), and $L=10$ (dotted red line). Results for the local correlator in the thermodynamic limit $L\to \infty$ in the DMFT approximation are presented as purple crosses in (a).