Mott transition and electronic excitation of the Gutzwiller wavefunction

The Mott transition is usually considered as resulting from the divergence of the effective mass of the quasiparticle in the Fermi-liquid theory; the dispersion relation around the Fermi level is considered to become flat toward the Mott transition. Here, to clarify the characterization of the Mott transition under the assumption of a Fermi-liquid-like ground state, the electron-addition excitation from the Gutzwiller wavefunction in the $t\text{−}J$ model is investigated on a chain, ladder, square lattice, and bilayer square lattice in the single-mode approximation using a Monte Carlo method. The numerical results demonstrate that an electronic mode that is continuously deformed from a noninteracting band at zero electron density loses its spectral weight and gradually disappears toward the Mott transition. It exhibits essentially the magnetic dispersion relation shifted by the Fermi momentum in the small-doping limit as indicated by recent studies for the Hubbard and $t\text{−}J$ models, even if the ground state is assumed to be a Fermi-liquid-like state exhibiting gradual disappearance of the quasiparticle weight. This implies that, rather than as the divergence of the effective mass or disappearance of the carrier density that is expected in conventional single-particle pictures, the Mott transition can be better understood as freezing of the charge degrees of freedom while the spin degrees of freedom remain active, even if the ground state is like a Fermi liquid.

Figure 1

Spectral function on chain [(a)–(c)], ladder at $k_{\perp }=0,\pi$ [(d)–(f)], plane [(g)–(i)], and bilayer at $k_{\perp }=0,\pi$ [(j)–(l)]. (a), (d), (g), (j) $A(\mathit{k},\omega )t$ at $n=0$ on chain [(a)], ladder [(d)], plane [(g)], and bilayer [(j)] [Eqs. (14) and (15)]. (b), (e), (h), (k) $A^{\mathrm{s}}(\mathit{k},\omega )t$ for $\omega >0$ on chain at $n=0.95$ [(b)], ladder at $n=0.95$ [(e)], plane at $n=0.905$ [(h)], and bilayer at $n=0.95$ [(k)], where the dispersion relation and spectral weight by cubic spline interpolation in Fig. 2 are used. (c), (f) $A(\mathit{k},\omega )t$ on chain at $n=0.95$ [(c)] and ladder at $n=0.95$ [(f)] obtained using the non-Abelian DDMRG method with 240 density-matrix eigenstates on a 120-site cluster [8, 12]. (i) $A(\mathit{k},\omega )t$ at $n=0.905$ on plane obtained using CPT with $4\times 4$-site clusters [9]. (l) $A(\mathit{k},\omega )t$ at $n=0.95$ on bilayer in the effective theory near half filling for $t_{\perp }\gg t$ and $J_{\perp }\gg J$ [Eqs. (12) and (13)] [12]. The green lines represent the Fermi level ($\omega =0$). Gaussian broadening with a standard deviation of $0.1t$ is used.

Figure 2

Dispersion relation $\omega =\varepsilon \left(\mathit{k}\right)$ [(a)–(f)] and spectral weight $W\left(\mathit{k}\right)$ [(g)–(l)] for $\omega >0$ on chain [(a), (g)], ladder at $k_{\perp }=0$ [(b), (h)] and $\pi$ [(c), (i)], plane [(d), (j)], and bilayer at $k_{\perp }=0$ [(e), (k)] and $\pi$ [(f), (l)]. The red diamonds denote the Monte Carlo results. The blue lines indicate the cubic spline interpolation. For the chain and ladder [(a)–(c), (g)–(i)], $n\approx 0.017$, 0.083, 0.150, 0.217, 0.283, 0.350, 0.417, 0.483, 0.550, 0.617, 0.683, 0.750, 0.817, 0.850, 0.883, 0.917, 0.950, and 0.983 from above. For the plane [(d), (j)], $n=0.005$, 0.105, 0.225, 0.245, 0.305, 0.405, 0.505, 0.605, 0.705, 0.745, 0.825, 0.845, 0.885, and 0.905 from above. For the bilayer [(e), (f), (k), (l)], $n=0.01$, 0.05, 0.09, 0.13, 0.21, 0.25, 0.29, 0.37, 0.41, 0.59, 0.63, 0.71, 0.75, 0.79, 0.87, 0.91, 0.95, and 0.99 from above.

Figure 3

Characteristic energies and spectral weights as a function of electron density. (a) $\varepsilon \left(\pi \right)/t$, (b) $W\left(\pi \right)$, and (c) $Z$ on chain. (d) $\varepsilon \left(\mathit{\pi }\right)/t$, (e) $W\left(\mathit{\pi }\right)$, and (f) $Z$ for $k_x=k_y$ on plane. (g) ${\varepsilon }_{\mathrm{c}}/t$ (solid blue circles) and $\mathrm{\Delta }\varepsilon /t$ (solid purple triangles), (h) $W(\pi ,\pi )$, and (i) $Z$ on ladder. (j) ${\varepsilon }_{\mathrm{c}}/t$ (solid blue circles) and $\mathrm{\Delta }\varepsilon /t$ (solid purple triangles), (k) $W(\mathit{\pi },\pi )$, and (l) $Z$ for $k_x=k_y$ on bilayer. The solid blue diamonds, solid blue circles, and solid purple triangles indicate the Monte Carlo results. The red curve in (c) represents the exact result for the Gutzwiller wavefunction on the chain [22, 23]. The magenta curves are guides for the eye. In (a) and (d), the open red diamonds at $n=1$ indicate $e_{1\mathrm{D}}(\pi /2)/t$ with $v_{1\mathrm{D}}=\pi J/2$ [17] [(a)] and $e_{2\mathrm{D}}(\mathit{\pi }/2)/t$ with $v_{2\mathrm{D}}=1.18\sqrt{2}J$ [19] [(d)]. In (g) and (j), the open red circles at $n=1$ indicate $e_{\mathrm{c}}/t$ on the ladder [(g)] and bilayer [(j)], and the open red triangles at $n=1$ indicate $\mathrm{\Delta }e/t$ on the ladder [(g)] and bilayer [(j)].