Doping-driven Mott transition in the one-band Hubbard model

A powerful diagrammatic impurity solver is shown to permit a systematic study of the doping-driven Mott transition in a one-band Hubbard model within the framework of single-site dynamical mean-field theory. At small dopings and large interaction strengths, we are able to access low enough temperatures that a reliable extrapolation to temperature $T=0$ may be performed, and ground state energies of insulating and metallic states may be compared. We find that the $T=0$ doping-driven transition is of second order and is characterized by an interaction-strength dependent electronic compressibility, which vanishes at the critical interaction strength of the half filled model. Over wide parameter ranges, the compressibility is substantially reduced relative to the noninteracting system. The metal-insulator transition is characterized by the appearance of in-gap states, but these are relevant only for very low dopings of less than 3%.

Figure 1

(Color online) Sketch of the phase diagram of the Hubbard model in the single-site DMFT approximation (semicircular density of states of bandwidth $4t$) with magnetic order suppressed by averaging over spin. Top panel: the thick red lines indicate the surface in the space of temperature, interaction strength, and chemical potential, where a first-order metal-insulator transition occurs. The thin black lines delimit the coexistence region (hashed for cuts across the $\mu ={\mu }_1$ and $U=6t$ planes) where both metallic and insulating solutions to the DMFT equations exist. Lower panel: $T=0$ phase diagram and spectroscopic gap $\Delta$.

Figure 2

Doping per spin, $0.5-n$, as a function of chemical potential for $\beta t=400$ and indicated values of $U∕t$. At this temperature, the transition at half-filling $(\mu ={\mu }_1=U∕2)$ occurs at $U_c\left(T\right)\approx 5.65$. For larger interactions, a gap opens and shifting the chemical potential induces a first-order metal-insulator transition. Near the critical point $U_{c2}$, $n-0.5\sim {(\mu -{\mu }_1)}^2$, but as one moves away from the critical point, the onset of doping becomes linear in $\mu$.

Figure 3

Doping-vs-$\mu$ for $U∕t=6$ (upper panel) and 6.5 (lower panel) and $\beta t=100$, 200, and 400. For fixed $\mu$, the doping increases with decreasing temperature. The vertical lines indicate the positions of the first-order phase transition, as estimated from the crossing of the thermodynamic potential curves (dashed, $\beta t=200$; solid, $\beta t=400$). The solid triangles show the densities obtained from the thermodynamic potential using $2n=-\partial \Omega ∕\partial \mu$ (up-triangles, $\beta t=200$; down-triangles, $\beta t=400$).

Figure 4

Quasiparticle weights estimated using $Z\approx 1∕(1-\mathfrak{I}m\Sigma \left({\omega }_0\right)∕{\omega }_0)$, evaluated at the lowest Matsubara frequency ${\omega }_0$, plotted as a function of doping per spin. The data points connected by dashed lines correspond to $\beta t=400$ and approximation (10). The solid dots show an estimate for $Z$ which is based on a two-parameter fitting function.

Figure 5

Solid points: thermodynamic potential difference between the metallic and insulating solutions for the indicated temperatures, $U=6t$ (upper panel) and $U=6.5t$ (lower panel). Open symbols: “thermodynamic potential” computed by neglecting the $1∕2\gamma T^2$ term in Eq. (9).

Figure 6

Close-up view of the small doping results for $U\gtrsim U_{c2}$, showing the essentially linear onset of doping, which becomes more pronounced as one moves away from the critical point. The lines show parabolic fits to the data points, which were extrapolated to $T\to 0$ $(U∕t=6,6.5)$ or can be considered indistinguishable from that limit $(U∕t=8)$. We assume that these curves correspond to $n(T=0,\mu )$.

Figure 7

Energy difference between the metallic and insulating solutions at $T=0$, extrapolated from the data for $\beta t=200$ and $\beta t=400$, respectively. The lines show the result obtained from the $n(T=0,\mu )$, assuming a second-order transition.

Figure 8

Open circles: critical chemical potential ${\mu }_{c2}$ determined by the extrapolation of the density to temperature $T=0$. Crosses: position of the spinodal point ${\mu }_{c2}(\beta t=400)$ (where the metallic solution ceases to exist) as a function of $U$. The stars indicate the size of the spectroscopic gap at half-filling $\Delta$ as determined by the analytic continuation of insulating Green’s functions for $\beta t=40$, and the solid line gives a rough estimate for this gap, which assumes a semicircular density of states.

Figure 9

$\partial n∕\partial \mu$ normalized to the noninteracting value ${(\partial n∕\partial \mu )}_0\approx 1∕\pi t$, as a function of density per spin, $0.5-n$. For $U$ close to $U_{c2}$, the compressibility is substantially reduced relative to the noninteracting system.