Metal-insulator transition and antiferromagnetism in the generalized Hubbard model: Treatment of correlation effects

The ground state for the half-filled $t-t^{\prime }$ Hubbard model is treated within the Hartree-Fock approximation and the slave-boson approach including correlations. The criterium for the metal-insulator transition in the Slater scenario is formulated using an analytical free-energy expansion in the next-nearest-neighbor transfer integral $t^{\prime }$ and in direct antiferromagnetic gap $\mathrm{\Delta }$. The correlation effects are generally demonstrated to favor the first-order transition. For a square lattice with a strong van Hove singularity, accidental close degeneracy of AFM and paramagnetic phases is analytically found in a wide parameter region. As a result, there exists an interval of $t^{\prime }$ values for which the metal-insulator transition is of the first order due to the existence of the van Hove singularity. This interval is very sensitive to model parameters (direct exchange integral) or external parameters. For the simple and body-centered cubic lattices, the transition from the insulator AFM state with increasing $t^{\prime }$ occurs to the phase of an AFM metal and is a second-order transition, which is followed by a transition to a PM metal. These results are quantitatively modified when taking into account the intersite Heisenberg interaction, which can induce first-order transitions. A comparison with the Monte Carlo results is performed.

Figure 1

Schematic phase diagram of MIT in the ground state in $\tau -U$ terms within the Slater scenario. In the vicinity of $U=0, \tau =0$, the AFM order is stable and its treatment within HFA is well justified. The region of AFM metal between AFM insulator and paramagnetic regions is present for some lattices only. Away from close vicinity of $U=0, \tau =0$, other phases may present.

Figure 2

Density of states for (a) square at $\tau =0$, 0.1, and 0.2; (b) sc at $\tau =0,0.075,\text{and}0.15$; (c) bcc lattice at $\tau =0,0.1$, and 0.2. The positions of the Fermi level corresponding to half-filling are shown by vertical dashed lines with corresponding colors. Electron energy is counted from the van Hove singularity level. The Fermi level $E_{\mathrm{F}}$ corresponding to half-filling is shown by dashed lines.

Figure 3

$E_{\mathrm{F}}/\tau$ as a function of $\tau$ for the square, sc, and bcc lattices.

Figure 4

Ground state phase diagram of $\tau -{\mathrm{\Delta }}_{*}$ within AFM phase for the square (a), sc (b), and bcc (c) lattices. (a) In the white region $E_{1,\text{max}}^{\mathrm{sq}}/z^2=-{\mathrm{\Delta }}_{*}$, in the dark-gray region $E_{1,\text{max}}^{\mathrm{sq}}/z^2=-4\tau -{\mathrm{\Delta }}_{*}\sqrt{\frac{1-2\tau }{1+2\tau }}$, in the light-gray region $E_{1,\text{max}}^{\mathrm{sq}}/z^2=4\tau -\sqrt{16+{\mathrm{\Delta }}_{*}^2}$. $E_{2,\text{min}}^{\mathrm{sq}}/z^2=-4\tau +{\mathrm{\Delta }}_{*}$ The breakpoint of the MIT line is $\tau =1/\sqrt{2},{\mathrm{\Delta }}_{*}=\sqrt{2}$. (b) White region $E_{1,\max }^{\mathrm{sc}}/z^2=-{\mathrm{\Delta }}_{*}$, at ${\mathrm{\Delta }}_{*}<3{\tau }^{-1}/2-6\tau$, blue region $E_{1,\max }^{\mathrm{sc}}/z^2=12\tau -\sqrt{36+{\mathrm{\Delta }}_{*}^2}$ otherwise; everywhere $E_{2,\min }^{\mathrm{sc}}/z^2=-4\tau +{\mathrm{\Delta }}_{*}$. (c) White region $E_{1,\max }^{\mathrm{bcc}}/z^2=8\tau -{\mathrm{\Delta }}_{*}$ at ${\mathrm{\Delta }}_{*}<8{\tau }^{-1}-2\tau$, blue region $E_{1,\max }^{\mathrm{bcc}}/z^2=12\tau -\sqrt{64+{\mathrm{\Delta }}_{*}^2}$; everywhere $E_{2,\min }^{\mathrm{bcc}}/z^2={\mathrm{\Delta }}_{*}$.

Figure 5

(a) $\delta F_{\mathrm{AFM}}^{\mathrm{HFA}}({\mathrm{\Delta }}_{\mathrm{MIT}}\left(\tau \right))$ (solid line) and $\delta F_{\mathrm{PM}}^{\mathrm{HFA}}\left(\tau \right)$ (dashed line) within HFA; (b) $\delta F_{\mathrm{AFM}}({\mathrm{\Delta }}_{\mathrm{MIT}}\left(\tau \right))$ (solid line) and $\delta F_{\mathrm{PM}}\left(\tau \right)$ (dashed line) within SBA. The square, sc, and bcc lattices are considered.

Figure 6

The difference of $\mathrm{\Delta }{F}_{\mathrm{MIT}}\left(\tau \right)$ within SBA (solid lines) and HFA approximations (dashed lines).

Figure 7

(a) The free-energy difference $\mathrm{\Delta }{F}_{\mathrm{MIT}}\left(\tau \right)$ on the MIT line for square (SQ), simple cubic (sc), and body centered cubic (bcc) lattices within SBA in the presence of $J_{\mathbf{Q}}=0$ (solid lines), $J_{\mathbf{Q}}=+z_{\mathrm{A}}^2$ (dashed lines), $J_{\mathbf{Q}}=-z_{\mathrm{A}}^2$ (dotted lines). (b) The phase diagram demonstrating the MIT type in the $J-\tau$ variables for square (SQ), simple cubic (sc), and body centered cubic (bcc) lattices, $J$ being the exchange integral between the nearest neighbors. The negative sign of $J$ corresponds to AFM, positive—to FM ordering. To the left of the curves, a first-order transition takes place, to the right—the second-order transition.

Figure 8

$\zeta$ plot of $\phi (m,\zeta ), {\phi }_J(m,\zeta ), v(m,\zeta )$, and $v_J(m,\zeta )$ at $m=0.3$ and $m=0$.

Figure 9

Critical $U$ at MIT line within the AFM phase in units of $U_{\mathrm{BR}}$ for square, sc, and bcc lattices from numerical solution of SBA equations (solid lines), numerical solution of HFA equations (dashed lines) and asymptotic solution (74) appropriate in HFA (dotted lines). Two latter lines practically coincide. Analytical solution (75) is also shown by dot-dashed lines.

Figure 10

The dependence of $\tau$ as a function of $1/w_0\left({\tilde{E}}_{\mathrm{F}}^{\mathrm{sq}}\right)$, Eq. (76), is shown by solid line, a simple approximation (79) by dashed line.

Figure 11

Free-energy expansion parameters for the square lattice. (a) The plot of difference of $\tau$-expansion coefficients of the free energy in AFM $[f_{2,\mathrm{AFM}}^{\mathrm{sq}}\left(w_0\right),f_{2,\mathrm{AFM}}^{\mathrm{sq},\mathrm{eff}}\left(w_0\right)$, see Eqs. (88) and (89)] and PM $[f_{2,\mathrm{PM}}^{\mathrm{sq}}\left(w_0\right),f_{2,\mathrm{PM}}^{\mathrm{sq},\mathrm{eff}}\left(w_0\right)$, see Eqs. (82) and (83)] phases as a function of $\tau$. (b) The plot of coefficients of $\tau$ expansion $f_{2,\mathrm{AFM}}^{\mathrm{sq}}\left(w_0\right), f_{2,\mathrm{AFM}}^{\mathrm{sq},\mathrm{eff}}\left(w_0\right), f_{2,\mathrm{PM}}^{\mathrm{sq}}\left(w_0\right), f_{2,\mathrm{PM}}^{\mathrm{sq},\mathrm{eff}}\left(w_0\right)$ and $f_{2,\mathrm{AFM}}^{\mathrm{sq},\mathrm{corr}}\left(w_0\right)$ with account of correlation correction $f_{4,\mathrm{AFM}}^{\mathrm{sq},\mathrm{eff}}\to f_{4,\mathrm{AFM}}^{\mathrm{sq},\mathrm{corr}}$, see Eq. (93), as a function of $w_0^{-1}$.

Figure 12

Plots of different contributions to $\mathrm{\Delta }{F}_{\mathrm{MIT}}$ (in units of ${10}^{-4}t$) at $J_{\mathbf{Q}}=0.5z_{\mathrm{A}}^2$ for the square lattice as a function of $\tau$. Two way of calculations is used: Eq. (61), its terms being shown by solid lines, and Eq. (65), its terms being shown by dashed lines. The following contributions are shown: $\mathrm{\Delta }{F}_0$ corresponds to first term (originating from HFA), $\mathrm{\Delta }{F}_{\mathrm{c}}$ to second term (corresponding to many-electron corrections to HFA), $\mathrm{\Delta }{F}_J$ to third term (originating from many-electron corrections induced by exchange). $\mathrm{\Delta }{F}_{\mathrm{MIT}}^{\mathrm{HFA},\mathrm{lead}}\left(\tau \right)$ is shown by orange dotted line, $\mathrm{\Delta }{F}_{\mathrm{MIT}}^{\mathrm{HFA},\mathrm{lead}}\left(\tau \right)$ is shown by red dotted line, $\mathrm{\Delta }{F}_{\mathrm{MIT}}^{\mathrm{sq},\mathrm{HFA}}$ is shown by dash-dot-dotted line.