Gutzwiller magnetic phase diagram of the undoped $t\text{−}{t}^{\prime }\text{−}U$ Hubbard model

We calculate the magnetic phase diagram of the half-filled $t\text{−}{t}^{\prime }\text{−}U$ Hubbard model as a function of $t^{\prime }$ and $U$, within the Gutzwiller approximation plus random-phase approximation. As $U$ increases, the system first crosses over to one of a wide variety of incommensurate phases, whose origin is clarified in terms of double nesting. We evaluate the stability regime of the incommensurate phases by allowing for symmetry breaking with regard to the formation of spin spirals, and find a crossover to commensurate phases as $U$ increases and a full gap opens. The results are compared with a variety of other recent calculations, and in general good agreement is found. For parameters appropriate to the cuprates, double occupancy should be only mildly suppressed in the absence of magnetic order, inconsistent with a strong-coupling scenario.

Figure 1

(Color online) Phase diagram, as a function of $U$ and $t^{\prime }$, showing $U_{\text{BR}}$ [upper solid line (red)] and calculations from Ref. 1 (green lines). Shown also are $\text{GA}+\text{RPA}$ calculations ${\tilde{U}}_{\text{GA}}$ in which the momentum of the instability is restricted to be $\left(\pi ,\pi \right)$ (dashed blue lines) or $\left(\pi ,0\right)$ (dot-dashed blue lines).

Figure 2

(Color online) (a) Phase diagram obtained including the regime of incommensurate phases, comparing $U_{\text{HF}}$ (brown line with symbols) and $U_{\text{GA}}$ (dark blue line). Shown also is the line of first-order transitions to commensurate $\left(\pi ,\pi \right)$ or $\left(\pi ,0\right)$ order (light green solid line) and the $\left(\pi ,\pi \right)$ to $\left(\pi ,0\right)$ crossover line (light green dashed line). For ease in comparisons, the TBPS crossover line (green dotted line) from Fig. 1 is reproduced. To illustrate the dominant $q$ vector [Fig. 2b], the $\text{HF}+\text{RPA}$ points are color coded and numbered consecutively, although only some of the numbers are indicated. (b) Position of the dominant susceptibility peak in $\text{HF}+\text{RPA}$; color code and numbers match the values in (a). Note that some extra, metastable points are included, not shown in (a). Blue dotted line traces evolution of stable points. (c) $n_d/n_{d0}$ as a function of $t^{\prime }$, assuming a constant $U=8t$ (red line) or $10t$ (blue line). Here $n_d$ is the double occupancy and $n_{d0}=0.25$ is its uncorrelated value. Symbols indicate values of $n_d/n_{d0}$ estimated from experimental dispersion renormalization $Z_{disp}$ as a function of $t^{\prime }$, from Ref. 13. Letters refer to $\text{L}={\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$; $\text{N}={\text{Nd}}_{2-x}{\text{Ce}}_x{\text{CuO}}_{4\pm \delta }$; $\text{B}={\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$.

Figure 3

(Color online) (a) Plot of bare susceptibility ${\chi }_0$ in the first Brillouin zone, for $t^{\prime }=-0.55$, at half filling. (b) Similar plot with nesting curve (solid line) and its folded replicas (dashed lines). (c) and (d) Similar plots for $t^{\prime }=-0.73$, with additional susceptibility peaks and nesting curves, as discussed in text.

Figure 4

(Color online) Fermi surfaces (red and blue lines) and the $q$-shifted versions, illustrating two (competing) examples of double nesting. Data are for $t^{\prime }=-0.73t$, $x=0$, as in Figs. 3c, 3d (arrows). Labeled dots denote the double nesting points $A_1$, $A_2$, $B_1$, and $B_2$.