Gutzwiller magnetic phase diagram of the cuprates

A general constructive procedure is presented for analyzing magnetic instabilities in two-dimensional materials, in terms of (predominantly) double nesting, and applied to Hartree-Fock plus random-phase approximation $\left(\text{HF}+\text{RPA}\right)$ and Gutzwiller approximation plus RPA calculations of the Hubbard model. Applied to the cuprates, it is found that competing magnetic interactions are present only for hole doping, between half filling and the Van Hove singularity. While $\text{HF}+\text{RPA}$ instabilities are present at all dopings (for sufficiently large Hubbard $U$), in a Gutzwiller approximation they are restricted to a doping range close to the range of relevance for the physical cuprates. The same model would hold for charge instabilities, except that the interaction is more likely to be $q$ dependent.

Figure 1

(Color online) (a) Susceptibility ${\chi }_0^{\prime }\left(\omega =0\right)$ for LSCO as a function of $\mathbf{q}$ for a series of dopings from $x_{\text{VHS}}=0.207$ (blue dashed line, top curve) to $x=-0.99$ (brown dashed line, bottom curve). [Light gray curves (blue online) are for hole doping, $x>0$, others for electron doping, $x\le 0$.] Note the evolution from a plateau near $\left(\pi ,\pi \right)$ (upper curves, blue and red online) to one near $\left(\pi ,0\right)$ (lower curves, green online) to one near $\Gamma$ (lowest curves, brown online). Band parameters appropriate to LSCO(2) (Table ). (b) Same as (a), except for hole dopings $x=0.207$ (red dashed line) to $x=0.99$ (brown dashed line). Note similar evolution of plateaus. [(c) and (d)] Similar to (a) and (b), except for Bi-2201 (2 in Table ). Note overall similarity, except for curvature near $\left(\pi ,\pi \right)$. Dashed curves correspond to $x=\left(\text{c}\right)$: $-0.99$ (brown) or 0.43 (blue) and (d): 0.44 (red) or 0.99 (brown). Curves are generally spaced by $\Delta x=0.05$, except for (1) higher density near points of rapid change (e.g., the VHS), (2) $\Delta x=0.1$ near top and bottom of band, and (3) end points at $x=\pm 0.99$.

Figure 2

(Color online) (a) Holelike Fermi surface (solid) shifted by the double nesting vector $\mathbf{q}=\left(0.38,1.62\right)\pi /a$ and folded back into the first BZ (dashed). (b) Electronlike Fermi surface (solid) shifted by the double nesting vector $\mathbf{q}=\left(0.58,1\right)\pi /a$ and folded back into the first BZ (dashed). The (double) nesting points are marked by a solid dot. [(c) and (d)] Construction of nesting curves (dashed) for (c) holelike and (d) electronlike FS’s (solid lines) from condition Eq. (2). The FS is shown in the first BZ whereas the nesting curves are folded into the momentum space $0\le q_{x/y}\le 2\pi$ as defined by the dashed square. The arrows indicate scattering processes which lead to double nesting as explained in the text. The dotted lines in (c) correspond to the boundary of the magnetic BZ in the first and enlarged zone. [(a) and (c)] Parameters for NCCO, $x=-0.15$ and [(b) and (d)] Parameters for LSCO(1), $x=0.41$.

Figure 3

(Color online) (a) Nesting maps $q=2k_F$ for LSCO, dispersion 1, and $x=0.41$ (red curves), 0.62 (blue), and 0.79 (green). Corresponding susceptibility ${\chi }_0^{\prime }\left(\omega =0\right)$ as a function of $q$ for a series of hole dopings from (b) $E_F\left[x\right]=-0.16\left[0.41\right]$, (c) $-0.3\left[0.62\right]$], and (d) $-0.5\text{eV}\left[0.79\right]$. In all figures, whites are largest $\chi$’s, blues are smallest.

Figure 4

(Color online) Susceptibility ${\chi }_0^{\prime }\left(\omega =0\right)$ as a function of $q$ with superposed nesting maps $q=2k_F$ for NCCO, for a series of electron dopings $\left(x<0\right)$ from (a) $x\left[E_F\right]=-0.26\left[0.10\right]$, (b) $-0.37\left[0.20\right]$, (c) $-0.52\left[0.35\right]$, and (d) $-0.65\left[0.50\text{eV}\right]$.

Figure 5

(Color online) Susceptibility maps for [(a)–(d)] Bi2201, [(e)–(h)] LSCO(1), and [(i)–(l)] LSCO(2) at a series of increasing dopings. (a) $x\left[E_F\right]=0.12\left[-0.20\right]$, (b) $0.20\left[-0.25\right]$, (c) $0.31\left[-0.30\right]$, (d) $0.40\simeq x_{\text{VHS}}\left[0.33\right]$; (e) $0.19\left[-0.12\right]$, (f) $0.23\left[-0.13\right]$, (g) $0.29\left[-0.14\right]$, (h) $\left(x_{\text{VHS}}\simeq 0.33\right)$, $0.37\left[-0.15\right]$; (i) $0.0\left[-0.05\right]$, (j) $0.14\left[-0.10\right]$, (k) $0.20\simeq x_{\text{VHS}}\left[-0.111\right]$, and (l) $0.21\left[-0.1112\text{eV}\right]$.

Figure 6

(Color online) (a) Susceptibility map for LSCO(1) (h) with nesting curves superposed. Red curves: $\mathbf{q}=2{\mathbf{k}}_F$ nesting for both $\Gamma$-centered barrel and $\left(\pi ,0\right)$, $\left(0,\pi \right)$-centered pockets. Blue $\text{lines}=\text{barrel-pocket}$ nesting and green $\text{line}=\text{interpocket}$ nesting. (b) Corresponding Fermi-surface map showing four $\mathbf{q}$-shifted pockets, labeled as in (a).

Figure 7

(Color online) Susceptibility maps (a) ${\chi }_{11}^0$, (b) ${\chi }_{12}^0$, and (c) ${\chi }_{22}^0$, along with derived quantities (d) ${\overline{E}}_1$ and (e) ${\overline{E}}_2$, as a function of $\mathbf{q}$ for Bi2201 with $x=0.15$.

Figure 8

(Color online) (a) Variation in $U_{\text{GA}}$ with $U$ for $x=0.01$. (b) Variation in $U_{\text{GA}}$ with $x$ for $U=2U_{BR}$. $U_{\text{GA}}$ varies with $\mathbf{q}$, and the figure shows $\mathbf{q}=\Gamma$ (red solid line), $\left(\pi ,0\right)$ (blue dashed line), $\left(\pi /2,\pi /2\right)$ (brown dotted line), and $\left(\pi ,\pi \right)$ (green dot-dashed line). The $\mathbf{q}$ points along the zone diagonal, $\left(\pi ,0\right)$ and $\left(\pi /2,\pi /2\right)$ have nearly identical behavior.

Figure 9

(Color online) Gutzwiller magnetic phase diagrams for (a) LSCO(2) and (b) Bi2201(2). Dashed lines indicate metastable states—extensions of the condition $U_{Gutz}\chi =1$ for one phase beyond the point where another phase has become the dominant instability.