Properties of Holons in the Quantum Dimer Model

I introduce a doped two-dimensional quantum dimer model describing a doped Mott insulator and retaining the original Fermi statistics of the electrons. This model shows a rich phase diagram including a $d$-wave hole-pair unconventional superconductor at small enough doping and a bosonic superfluid at large doping. The hole kinetic energy is shown to favor binding of topological defects to the bare fermionic holons turning them into bosons, in agreement with arguments based on resonating valence bond wave function. Results are discussed in the context of cuprate superconductors.

Figure 1

Dimer flip (a) and holon hopping (b) in the QDM.

Figure 2

Phase diagram of the doped QDMs versus doping ($x$) and $V/|J|$ (a),(c) or $|J/t|$ (b),(d). The Frobenius [(a),(b), left] and non-Frobenius [(c),(d), right] models are compared. The $x=0$ insulator has VBC order. The symbols in (a),(b) [(d)] are obtained from the data of Fig. 3 [Fig. 4].

Figure 3

GS energy (per site) versus doping calculated on a $6\times 6$ cluster. Full (open) symbols are used for the non-Frobenius (Frobenius) QDM labeled as $J<0$ ($J>0$). The linear behavior at $x\to 0$ has been subtracted for convenience. (a) $V=0$ and different values of $|t|$; (b) $|t/J|=0.5$ and different values of $V$ (all in units of $|J|=1$). Insets: Mean slopes between $x=0$ and doping $x$ plotted versus $x$. For $J>0$, a polynomial fit gives a minimum at $x_c$ providing the range [$0,x_c$] of PS (Maxwell construction) reported in Figs. 2a, 2b. For $J<0$ no minimum at finite $x$ is seen.

Figure 4

(a) Columnar [i.e., $\mathbf{q}=(\pi ,0)$] VBC structure factor (normalized to its $x=0$ value) versus $|t/J|$. (b) Fermionic and bosonic signs (as defined in text) versus $|t/J|$. (a),(b) correspond to the non-Frobenius doped QDM with $V=0$ (circles) and $V/|t|=0.2$ (lozenges) at $x=1/6$ (computed on a $6\times 6$ cluster). In (a) additional data for the Frobenius case are shown for comparison. Vertical dotted segments give the approximate separations between the phases discussed in the text.

Figure 5

(a)–(f) Spectral functions associated with the propagation of a $\mathbf{q}=(0,0)$ $d$-wave pair of holes versus frequency $\omega$ (computed on a periodic $8\times 8$ cluster) for fixed $V=0$ (left) or $|t/J|=1$ (right). The weight of the hole-pair QP peak (at the lowest energy edge) is plotted versus $|J/t|$ (g) and $V/|J|$ (h). Comparison with data on a $6\times 6$ cluster in (g) suggests weak finite-size effects. Data for the $t\mathrm{\text{−}}J$ model (stars) from Ref. 16 are also shown for comparison in (g). Note that “string resonances” can be seen above the QP peak in (a)–(f).