Phenomenological theory of the pseudogap state

An ansatz is proposed for the coherent part of the single particle Green’s function in a doped resonant valence bond (RVB) state by analogy with the form derived by Konik and co-workers for a doped spin liquid formed by an array of two-leg Hubbard ladders near half-filling. The parameters of the RVB state are taken from the renormalized mean field theory of Zhang and co-workers for underdoped cuprates. The ansatz shows good agreement with recent angle resolved photoemission on underdoped cuprates and resolves an apparent disagreement with the Luttinger sum rule. The transition in the normal state from a doped RVB spin liquid to a standard Landau Fermi liquid, that occurs in the renormalized mean field theory, appears as a quantum critical point characterized by a change in the analytic form of the Green’s function. A $d$-wave superconducting dome surrounding this quantum critical point is introduced phenomenologically. Results are also presented for the Drude weight and tunneling density of states as functions of the hole density.

Figure 1

(Color online) Contours on which $G(\mathit{k},0)$ changes sign at various hole concentrations $x$ are shown in (a)–(e). In the shaded area $G(\mathit{k},0)>0$, satisfying the Luttinger sum rule. In the normal pseudogap phase, the line connecting $(\pi ,0)-(0,\pi )$ is the Luttinger surface of zeroes and the pockets in the thick line represent the infinities of $G(\mathit{k},0)$. The values of the parameters used here are given in Fig. 2. The evolution of the contours of infinities in $G(\mathit{k},0)$ is illustrated in (f).

Figure 2

(Color online) The values of the parameters $t\left(x\right)$, $t^{\prime }\left(x\right)$, $t^{″}\left(x\right)$, ${\mu }_p$, and $\Delta \left(x\right)$ that enter in Eqs. (8, 9) used in the present calculations $(\chi =0.338,J∕t_0=1∕3)$. Results presented in the text are for a choice of $t_0^{\prime }∕t_0=-0.3$, $t_0^{″}∕t_0=0.2$, estimated from the calculated band structure of ${\mathrm{Ca}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ (Ref. 23).

Figure 3

(Color online) The spectral weight distribution $z\left(\mathit{k}\right)∕g_t$ around the hole quasiparticle pockets in the normal pseudogap phase.

Figure 4

Quasiparticle dispersion in the normal pseudogap phase [Eq. (7)] along some high symmetry directions. The thickness of the lines is proportional to the spectral weight $z\left(\mathit{k}\right)∕g_t$ of the quasiparticle.

Figure 5

(Color online) Comparison between our theory and some recent ARPES experiments on ${\mathrm{Ca}}_{2-x}\mathrm{Na}\mathrm{Cu}{\mathrm{O}}_2\mathrm{Cl}$ by Shen et al. (Ref. 3). The experimental results are replotted in panels (A)–(F). Panels (A)–(C) show the distribution of spectral weight in the Brillouin zone within a $\pm 10\mathrm{meV}$ window around the Fermi level. The open/closed circles in panels (D)–(F) are detected by this experiment to determine the Fermi surface. The pockets in panels (A)–(I) show the infinities of $G(\mathit{k},0)$ in the normal pseudogap phase. The “Fermi surfaces” (blue and black curves) shown in panels (D)–(I) are defined by the minimum distance from the lower quasiparticle band to the Fermi level in our calculation along radial directions centered at $(\pi ,\pi )$. The thickness of the curve in panels (D)–(I) represents the spectral weight $z\left(\mathit{k}\right)∕g_t$ of the quasiparticle.

Figure 6

(Color online) Panels (a)–(d) show the dispersion of the lower quasiparticle band $E\left(\mathit{k}\right)$ on the “Fermi surface” and on the hole pockets for various dopings. In panel (e) we show the angular dependence of $\mid E\left(\mathit{k}\right)\mid$ on the “Fermi surface.” $\varphi$ is defined in the inset of panel (e).

Figure 7

(Color online) Nodal Fermi wave vector $k_F$, “antinodal Fermi wave vector” $k_F^A$, Fermi velocity $v_F$ (with values $t_0=0.5\mathrm{eV}$, lattice constant $a=4.0\mathrm{\AA }$). The experimental data in panels (a) and (b) for nodal and “antinodal Fermi wave vector” are for ${\mathrm{Ca}}_{2-x}{\mathrm{Na}}_x\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ (Ref. 3) while the experimental results in panel (c) are for ${\mathrm{Ca}}_{2-x}{\mathrm{Na}}_x\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$, ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$, and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ (Ref. 28).

Figure 8

(Color online) Density of states at the Fermi level $N_T\left(0\right)$ and the Drude weight $D_{\mu \mu }$ in the normal pseudogap phase.

Figure 9

Density of states $N_T\left(\omega \right)$ in the normal pseudogap phase. The zero energy values $N_T\left(0\right)$ are shown in Fig. 8a.

Figure 10

Regular and anomalous Green’s functions $G(\mathit{k},\omega )$, $F^{†}(\mathit{k},\omega )$ which appear in the coupled Eqs. (14, 15).

Figure 11

The phenomenological form of the superconducting gap that follows from a parabolic relation for $T_c\left(x\right)$: ${\Delta }_S\left(x\right)∕t_0=0.07[1-82.6{(x-0.2)}^2]$.

Figure 12

(Color online) The dispersion [panels (a)–(c)] and weight [panel (d)] on the Luttinger surface with nonzero superconducting gap. There are two bands below the Fermi level, and shown here is the one closer to Fermi level. Around $(\pi ,0)$ and $(0,\pi )$, there is a substantial part of the spectral weight located at the lower band, only a small part remains on the band closer to the Fermi level as indicated in Fig. 13.

Figure 13

Quasiparticle dispersion $E\left(\mathit{k}\right)$ in the superconducting state [Eq. (10)] along high symmetry directions. The thickness of the lines is proportional to the spectral weight $z\left(\mathit{k}\right)∕g_t$.

Figure 14

Density of states $N_T\left(\omega \right)$ in the presence of a nonzero superconducting gap. The additional peaks closest to Fermi level are determined by the superconducting gap which opens on the hole pockets for $x<x_c$. For $x⩾x_c$ the peaks come from the antinodal directions for $(\pm \pi ,0)$, $(0,\pm \pi )$.