Mott physics and first-order transition between two metals in the normal-state phase diagram of the two-dimensional Hubbard model

For doped two-dimensional Mott insulators in their normal state, the challenge is to understand the evolution from a conventional metal at high doping to a strongly correlated metal near the Mott insulator at zero doping. To this end, we solve the cellular dynamical mean-field equations for the two-dimensional Hubbard model using a plaquette as the reference quantum impurity model and continuous-time quantum Monte Carlo method as impurity solver. The normal-state phase diagram as a function of interaction strength $U$, temperature $T$, and filling $n$ shows that, upon increasing $n$ toward the Mott insulator, there is a surface of first-order transition between two metals at nonzero doping. That surface ends at a finite temperature critical line originating at the half-filled Mott critical point. Associated with this transition, there is a maximum in scattering rate as well as thermodynamic signatures. These findings suggest a new scenario for the normal-state phase diagram of the high temperature superconductors. The criticality surmised in these systems can originate not from a T $=$ 0 quantum critical point, nor from the proximity of a long-range ordered phase, but from a low temperature transition between two types of metals at finite doping. The influence of Mott physics therefore extends well beyond half-filling.

Figure 1

Non-interacting local density of states ${\rho }_K\left(\omega \right)=-1/\pi \mathrm{Im}{G}_K\left(\omega \right)$ of the orbitals $K=(0,0),(\pi ,0),(\pi ,\pi )$ (solid, dashed and dot-dashed lines respectively).

Figure 2

Chemical potential $\mu$, interaction $U$, temperature $T$ phase diagram of the two-dimensional Hubbard model obtained by cellular DMFT. Because of particle-hole symmetry it is symmetric with respect to the $\mu =0$ plane. Cross-sections at constant $U$ are shown. Dark-gray (blue) shaded regions represent the coexistence of two phases. Light-gray (yellow) areas denote the onset of the Mott insulator state (MI), characterized by a plateau in the occupation at $n=1$. When these two regions overlap, a metal-insulator transition takes place (different shade of gray). Otherwise, the coexistence regions occur between two different metals. Projections on $T=0$, and $\mu =0$ planes are also shown (full lines and dashed lines respectively). Open dots mark the extrapolated $T=0$ values of $U_{c1}$ and $U_{c2}$. A critical line $T_{\mathrm{cr}}$ (dotted line) originates at the half-filled Mott critical endpoint $U_{\mathrm{MIT}}$ (full dot) and moves to progressively low temperatures and high doping as $U$ increases.

Figure 3

Phase diagram in the temperature $T$ versus chemical potential $\mu$ plane for different values of the interaction strength $U$. Each panel corresponds to a cross-section at constant $U$ of the previous figure. As before, light gray (yellow) area represents the Mott insulating phase (MI). Dark grey (blue) area represents the coexistence of two phases and is bounded by the spinodal lines ${\mu }_{c1}\left(T\right)$ and ${\mu }_{c2}\left(T\right)$. A star symbol marks the end of the coexistence region at finite temperature; by following this point as a function of $U$ gives rise to the dotted line $T_{\mathrm{cr}}$ of Fig. 2. Other symbols denote some of the points actually computed in our study. Three phases can be distinguished: Overdoped phase (open circles), Mott insulator (crosses), underdoped phase (filled squares). The former is metallic. In the latter some regions of the Brillouin zone are gapped and others gapless, and the system is compressible.

Figure 4

Occupation $n$ versus $\mu$ for several values of interaction strength $U$. The data shown are for temperatures $T/t=1/10$ (blue triangles), $1/25$ (green squares), $1/50$ (red circles), $1/100$ (black diamonds). When two solutions are found to coexist, the solutions obtained following the metallic and the insulating solution are indicated as full and open symbols respectively. ${\mu }_{c1}$ marks the vanishing of the underdoped phase. ${\mu }_{c2}$ signals the disappearance of the overdoped phase. The arrows indicate the values of ${\mu }_{c1}$ and ${\mu }_{c2}$ at the lowest temperature shown in the panel. The plateau in the occupation at $n=1$ signals the onset of the incompressible Mott state. Note that ${\mu }_{c1}$ in general occurs at finite values of doping. The inset in the panel for $U=6.2t$ shows the $n\left(\mu \right)$ curves for temperatures $T/t=1/40$ (orange down triangles), $1/50$ (red circles), $1/60$ (magenta left triangles), $1/64$ (violet right triangles), which are above the second-order critical temperature $T_{\mathrm{cr}}\sim 1/65t$ at which the two phases merge. Note the sigmoidal shape of the $n\left(\mu \right)$ curves as $T_{\mathrm{cr}}$ is approached from above.

Figure 5

Real and imaginary parts of the cluster Green’s function $G_K\left(i{\omega }_n\right)$ at $U=7.0t$ and low temperature $T/t=1/100$ for several dopings. Left panels show the results for the orbital $K=(0,0)$, central panels for $K=(\pi ,0)$, and right panels for $K=(\pi ,\pi )$. The finite (zero) value of the imaginary part of the cluster Green’s function at ${\omega }_n\to 0$ indicates the metallic (insulating) character of the solution. Accordingly, three regimes can be distinguished: Metal (circles), insulator (crosses), and a strong cluster momentum differentiation, with orbital $K=(\pi ,\pi )$ gapped and $K=(0,0)$, $(\pi ,0)$ gapless. The first and the last behavior characterize respectively the overdoped and the under-doped phases. The dopings on the figure correspond to $\mu =-2.0,-1.5,-1.3,-1.2,-1.15,-1.05,-0.2,\mathrm{and}0.0$.

Figure 6

Extrapolated zero frequency value of the imaginary part of the cluster Green’s function, −Im$G_K(\omega \to 0)$, as a function of doping $\delta =1-n$, for several values of the interaction strength $U$. For each figure, the upper panel shows cluster momentum $K=(\pi ,0)$ (circles) and the lower panel displays $K=(\pi ,\pi )$ (triangles). Note the difference of scale of the y-axis. The data shown are for temperatures $T/t=1/10$ (blue dotted), $1/25$ (green dashed), $1/50$ (red solid), and $1/100$ (black dot-dashed). When two solutions are found to coexist, the solutions obtained following the metallic and the insulating solution are indicated as full and open symbols respectively. This observable measures the density of states at the Fermi energy averaged in a coarse-grained cluster momentum region. Doping of the Mott state occurs gradually in cluster momentum, with the metallization that starts first in the orbital $K=(\pi ,0)$ and $(0,0)$ whereas $K=(\pi ,\pi )$ remains insulating. This behavior characterizes the underdoped phase and is highlighted by the gray (orange) background for the lowest temperature displayed. The transition between the underdoped and overdoped phase can be first-order or it can be a crossover depending on $U$ and $T$. In the latter case, for concreteness we define the boundary, illustrated in the figure for $T/t=1/50$, by -Im$G_{(\pi ,\pi )}(\omega \to 0)=0.05$.

Figure 7

Re$G_K(\omega \to 0)$ as a function of doping $\delta$, for several values of the interaction strength $U$. Circles indicate the data for the $K=(\pi ,0)$ orbital, triangles for the $K=(\pi ,\pi )$. The data shown are for temperatures $T/t=1/10$ (blue dotted), $1/25$ (green dashed), $1/50$ (red circles), and $1/100$ (black dot-dashed). When two solutions are found to coexist, the solutions obtained following the metallic and the insulating solution are indicated as full and open symbols respectively. The assignment of symbols is the same as in Fig. 6. This observable measures the low frequency asymmetry of the orbitals. The $(\pi ,0)$ orbital crosses zero at a characteristic doping close to the underdoped-overdoped transition.

Figure 8

Real and imaginary part of the cluster self-energy ${\Sigma }_K\left(i{\omega }_n\right)$ at $U=7.0t$ and the low temperature $T/t=1/100$ for several dopings. Left panels show the results for the orbital $K=(0,0)$, central panels for $K=(\pi ,0)$, and right panels for $K=(\pi ,\pi )$. The assignment of symbols is the same as in Fig. 5.

Figure 9

Effective chemical potential ${\mu }_{\mathrm{eff}}^K=\mu -Re{\Sigma }_K(\omega \to 0)$ renormalized by the electronic correlation versus $\delta$. Data are shown for the cluster momenta $K=(\pi ,0)$ (circles) and $K=(\pi ,\pi )$ (triangles) and are obtained for temperatures $T/t=1/10$ (blue dotted line), $1/25$ (green dashed line), $1/50$ (red solid line), $1/100$ (black dot-dashed line). Note that ${\mu }_{\mathrm{eff}}^{(\pi ,0)}$ crosses zero at a characteristic doping close to the OD-UD transition, or more precisely at the spinodal ${\mu }_{\mathrm{c}2}(U,T)$ and its high temperature crossover. The hatched grey lines indicate the region where ${\mu }_{\mathrm{eff}}^{(\pi ,\pi )}$ exceeds the noninteracting bandwidth of the $(\pi ,\pi )$ orbital.

Figure 10

Extrapolated zero-frequency value of the imaginary part of the cluster self-energy, −Im${\Sigma }_K(\omega \to 0)$ as a function of doping $\delta$ and for several values of the interaction $U$. For each figure, the upper panel shows cluster momentum $K=(\pi ,0)$ (circles) and the lower panel displays $K=(\pi ,\pi )$ (triangles). This quantity is proportional to the scattering rate ${\Gamma }_K$. The data shown are for temperatures $T/t=1/10$ (blue dotted line), $1/25$ (green dashed line), $1/50$ (red solid line), and $1/100$ (black dot-dashed line). −Im${\Sigma }_{(\pi ,0)}(\omega \to 0)$ is peaked close to the spinodal surface ${\mu }_{c1}(U,T)$ of the first-order transition and to its high temperature crossover. That peak reaches its overall maximum in the 3D phase diagram at the Mott end point $U_{\mathrm{MIT}}\approx 5.95$ and $\delta =0$ [see Figs. 10c and 10d]. As the temperature increases, the value of -Im${\Sigma }_{(\pi ,0)}(\omega \to 0)$ at its maximum increases as it does its width in doping. The peak in −Im ${\Sigma }_{(\pi ,0)}(\omega \to 0)$ then disappears at a characteristic temperature which is progressively higher as $U$ increases.

Figure 11

Statistical weight $P$ of the following cluster eigenstates: the singlet $S$ $|N=4;S=0;K=(0,0)\rangle$, triplet $T$ $|N=4;S=1;K=(\pi ,\pi )\rangle$, and doublet $D$ $|N=3;S=1/2;K=(\pi ,0)\rangle$, where $N$, $S$, and $K$ are the number of electrons, the total spin, and the cluster momentum of the cluster eigenstate. The remnant statistical weight is summed up in the histogram $R$. Data are obtained at $U=12.0t$ and shown at $T/1=1/10$ (left) and $1/50$ (right). The value of doping is (a) $\delta =0$, representative of the Mott insulating phase, (b) $\delta =0.02$, corresponding to the UD phase, and (c) $\delta =0.25$, in, the OD phase.

Figure 12

Statistical weight $P$ of the following cluster eigenstates as a function of doping $\delta$, for several values of $U$: The singlet $|N=4,S=0,K=(0,0)\rangle$, the triplet $|N=4,S=1,K=(\pi ,\pi )\rangle$, and the doublet $|N=3,S=1/2,K=(\pi ,0)\rangle$ (black squares, red triangles, and blue circles respectively) where $N$, $S$ and $K$ are the number of electrons, the total spin, and the cluster momentum of the cluster eigenstate. The data shown are for temperatures $T/t=1/10,1/25,1/50$ (dotted, dashed, and solid line respectively). The statistical weight $P$ of the half-filled singlet crosses $1/2$ (green solid horizontal line) roughly at the transition between the UD and OD phase, suggesting competing singlet spin versus other spins and charge fluctuations at the transition.

Figure 13

Extrapolated zero-frequency value of the imaginary part of the cluster hybridization function, −Im${\Delta }_K(\omega \to 0)$, for cluster momentum $K=(\pi ,0)$ (circles), and $K=(\pi ,\pi )$ (triangles) as a function of doping, for different values of the interaction strength $U$. The data shown are for temperatures $T/t=1/10$ (blue dotted line) $1/25$ (green dashed line), $1/50$ (full red line), and $1/100$ (black dot-dashed line). This observable measures the density of states of the conduction bath at the Fermi level averaged in the coarse grained cluster momentum region. Note the weakly cluster momentum dependence in the underdoped phase, to be contrasted with the strong cluster momentum differentiation of both the cluster Green’s function and self-energy.

Figure 14

Local spin susceptibility ${\chi }_0$ versus $\delta$ for several values of $U$. The data shown are for temperatures $T/t=1/10,1/25,\mathrm{and}1/50$ (blue dotted, green dashed, red solid line respectively). In the underdoped phase ${\chi }_0$ is approximately linear with doping, as highlighted by the solid cyan line, substantiating the idea that the doped carriers propagate among short-range singlets formed by the superexchange interaction.

Figure 15

Phase diagram temperature $T$ versus doping $\delta$ for $U=6.2t$, representative of the regime $U>U_{\mathrm{MIT}}$. The star symbols denotes the second-order finite temperature critical point, with coordinates $(\delta ,T)\approx (0.05,1/65)$. Blue lines with circles identify the spinodals ${\delta }_{c1}$ (full line) and ${\delta }_{c2}$ (dashed line) where the UD and OD phases respectively disappear. Dotted blue line with circles denotes the high-temperature crossover of the transition measured by a maximum in the compressibility ${(dn/d\mu )}_T$. The red lines with squares indicate the vanishing of Re$G_{(\pi ,0)}(\omega \to 0)$ (solid line), of ${\mu }_{\mathrm{eff}}^{(\pi ,0)}$ (dashed line), and the crossing of the isotherms $n\left(\mu \right)$ (dot-dashed line), and they are early warning indicators of the spinodal ${\delta }_{c2}\left(T\right)$. The green lines with triangles are precursors of the spinodal ${\delta }_{c1}\left(T\right)$. Solid line marks the maximum of the scattering rate ${\Gamma }_{(\pi ,0)}=-\mathrm{Im}{\Sigma }_{(\pi ,0)}(\omega \to 0)$. The strong momentum differentiation found in Im$G_K(\omega \to 0)$ disappears at the dot-dashed line. When that differentiation occurs as a crossover, we defined the boundary by −Im$G_{(\pi ,\pi )}(\omega \to 0)=0.05$. The dot-dashed line corresponds to the doping below which the probability of the half filled singlet state becomes larger than 1/2. Below the critical point the metallic state separates into two different metallic states, the UD and the OD phase, with a hysteretic change in the occupation (the jumps in the occupation define hatched region). The lines above the critical point are crossover lines. The Mott insulator (MI) exists only on the $\delta =0$ line.

Figure 16

Characteristic dopings versus $U$ that signal the UD to OD transition, measured at the temperature $T/t=1/50$. The blue line with circles denotes the maximum of the compressibility ${(dn/d\mu )}_T$. The red full line with squares indicates the vanishing of Re$G_{(\pi ,0)}(\omega \to 0)$. Green line with triangles denotes the maximum of the scattering rate ${\Gamma }_{(\pi ,0)}=$ –Im${\Sigma }_{(\pi ,0)}(\omega \to 0)$. Data points are displayed for the constant temperature $T/t=1/50$ and thus they measure the features of the three-dimensional phase diagram at that temperature. The overall maximum (as a function of $U$, $T$ and $\delta$) of the compressibility follows the critical line $T_{\mathrm{cr}}$ that originates at $U_{\mathrm{MIT}}$, with coordinates $(U,\delta ,T)\approx (5.95,0,1/12)$ indicated by the vertical arrow. The overall maximum as a function of $U$, $T$, and $\delta$ of the scattering rate ${\Gamma }_{(\pi ,0)}$ follows the ${\delta }_{\mathrm{c}1}(U,T)$ surface and peaks at $U_{\mathrm{MIT}}$. The zero of the extrapolated value of real part of the $(\pi ,0)$ cluster Green’s function at $\omega =0$ bifurcates at the line $U_{c2}\left(T\right)$.