Quantum criticality of bandwidth-controlled Mott transition

Metallic states near the Mott insulator show a variety of quantum phases, including various magnetic, charge-ordered states and high-temperature superconductivity in various transition metal oxides and organic solids. The emergence of a variety of phases and their competitions are likely intimately associated with quantum transitions between the electron-correlation-driven Mott insulator and metals characterized by its criticality, and is related to many central questions of condensed matter. The quantum criticality is, however, not well understood when the transition is controlled by the bandwidth through physical parameters such as pressure. Here, we quantitatively estimate the universality class of the transition characterized by a comprehensive set of critical exponents by using a variational Monte Carlo method implemented as an open-source innovated quantum many-body solver, with the help of established scaling laws at a typical bandwidth-controlled Mott transition. The criticality indicates a weaker charge and density instability in contrast to the filling-controlled transition realized by carrier doping, implying a weaker instability to superconductivity as well. The present comprehensive clarification opens up a number of routes for quantitative experimental studies for complete understanding of elusive quantum Mott transition and nearby strange metal that cultivate future design of functionality.

Figure 1

Schematic phase diagram of Mott metal-insulator transition. The MQCP (red cross) is the quantum critical point between the metal and the Mott insulator and simultaneously the end point of the first order transition and the quantum critical line (green line). Finite-temperature critical point ($T=T_{\mathrm{c}}$) (dark blue curve) appears as the endpoint of the first order boundary (light blue shaded surface. $a$ and $b$ represent the control parameters and are given by a combination of $t_{\perp }$ and $U$ in the present case. In the bandwidth-control case in general, the electron filling is fixed at an odd integer in this whole $T\text{−}a\text{−}b$ parameter space. For details see Ref. [10].

Figure 2

(a) Ground-state phase diagram obtained by the present VMC calculation. The purple solid and broken lines with open circles indicate the first-order and continuous metal-insulator transition (MIT) boundaries, respectively. The green solid curve with open squares is for the antiferromagnetic transitions (AFT) (see Sec. 7d for the method to determine the MIT and see Appendix pp6 for the AFT). Red large circle depicts the MQCP. Error bars are determined by considering the errors of size extrapolations and statistical errors of Monte Carlo calculations for finite-size systems. Inset: Lattice structure used for the present study; $t\text{−}{t}_{\perp }\text{−}{t}^{\prime }$ Hubbard model with the nearest-neighbor intrachain (red bonds), interchain (blue bonds), and next-nearest-neighbor (broken black bonds) hoppings $t,t_{\perp }$ and $t^{\prime }=t_{\perp }/2$, respectively. We take $t$ as the energy unit.

(b) Critical exponents of MQCP estimated in this article.

Figure 3

(a) $t_{\perp }$-dependence of jumps of extrapolated double occupancy $\mathrm{\Delta }D$ with the choice of $t_{\perp }^{\mathrm{MQCP}}=0.38$ determined in Sec. 7c. Green curve represents the optimized fitting leading to $\beta =0.97\pm 0.05$. (b) $U$ dependence of extrapolated double occupancy $D$ at $t_{\perp }^{\mathrm{MQCP}}=0.38$. Green (purple) line represents the fittings to estimate ${\delta }_{\mathrm{I}}$ for $U>U^{\mathrm{MQCP}}$ (${\delta }_{\mathrm{M}}$ for $U<U^{\mathrm{MQCP}}$), which indicate ${\delta }_{\mathrm{I}}=0.98\pm 0.03$ and ${\delta }_{\mathrm{M}}=1.05\pm 0.04$.

Figure 4

The exponent $\nu z$ estimated from the charge gap ${\mathrm{\Delta }}_{\mathrm{c}}$ at MQCP. $U$ dependence of ${\mathrm{\Delta }}_{\mathrm{c}}$ in the linear plot (a) and the logarithmic scaling plot (b) for $28\times 28$ site around the MQCP ($t_{\perp }^{\mathrm{MQCP}}=0.38$ and $U^{\mathrm{MQCP}}=1.83$). Purple curve in panel (a) and green line in panel (b) show the same fitting by the estimate of the MQCP point and the critical exponents, indicating $\nu z=1.13\pm 0.19$.

Figure 5

(a) $t_{\perp }$-dependence of ${\chi }^2$ value for the fitting Eq. (18), which results in $t_{\perp }^{\mathrm{MQCP}}=0.38\pm 0.05$ and $\beta =0.97\pm 0.05$. (b) $U$ dependence of ${\chi }^2$ value for the fittings to combined Eq. (19) and Eq. (23), which results in $U^{\mathrm{MQCP}}=1.83\pm 0.03$, $\nu z=1.13\pm 0.19$, ${\delta }_{\mathrm{I}}=0.98\pm 0.03$, and ${\delta }_{\mathrm{M}}=1.05\pm 0.04$.

Figure 6

Fermi surfaces of the model for $U=0$ at half filling. Green, purple, and light-blue curves indicate the Fermi surface at $t_{\perp }=1,0.62$, and 0.3, respectively.

Figure 7

Examples of charge gaps ${\mathrm{\Delta }}_{\mathrm{c}}$ for $28\times 28$ sites in the cases of $t_{\perp }=0.05$ (a), 0.1 (b), 0.2 (c) and 0.3 (d). Thin dashed blue lines are the finite-size gaps in the noninteracting case.

Figure 8

An example of charge gap estimate by Maxwell rule. (a) Ground-state energy as a function of carrier density $n$ at $t_{\perp }=t_{\perp }^{\mathrm{MQCP}}=0.38$ and $U=2.5$ for $28\times 28$-site system. (b) Chemical potential $\mu$ vs $n$ for the same case as panel (a). The Horizontal solid lines in both electron- and hole-doped regions represent the lines used for the Maxwell's construction, where the horizontal line is drawn to make the areas of the two shaded regions surrounded by the line and the curve equal. For the curve and line adjacent to $n=0$, we count the area of the shaded domain surrounded by the curve, the horizontal line and the $n=0$ vertical line. Then the two crossing points of the line and the curve determines the two points of the phase separation for each electron and hole side as assured by the thermodynamic stability. The difference in $\mu$ between electron and hole doped sides (the difference of $\mu$ between the two horizontal lines) gives the charge gap.

Figure 9

Momentum distribution functions $n\left(\vec{k}\right)$ for $t_{\perp }=$ 0.5, 0.7 and 1.0 around the metal-insulator transition points. Panels (a), (c), (e) indicate the metallic phase with sharp Fermi surface, while panels (b), (d), (f) indicate the insulating phase without the Fermi surface.

Figure 10

Ground-state energy as a function of carrier density $n$ (a) and $U$ (b), which support the absence of the singular behavior.

Figure 11

Correlation-ratio plot for $20\times 20$, $24\times 24,$ and $28\times 28$ sites in the cases of (a) $t_{\perp }=0.05$ and (b) $t_{\perp }=0.1$. The crossing points suggest $U^{\mathrm{AF}}\sim 1.48$ and 1.73, respectively.

Figure 12

Determination of the antiferromagnetic transition along $t_{\perp }=t_{\perp }^{\mathrm{MQCP}}=0.38$. Analysis by correlation-ratio method by using $20\times 20$, $24\times 24,$ and $28\times 28$ lattices is shown in wide region in panel (a) and the zoom-in plot in panel(b).

Figure 13

Finite-size scaling of $S(\pi ,\pi )$ by using scaling relation (F3). The data collapse to a single scaling curve is found for $\nu =0.52\pm 0.02$, and $\eta +z=1.9\pm 0.1$ with $U^{\mathrm{AF}}=1.85\pm 0.02$.

Figure 14

Spin structure factor at MQCP.

Figure 15

Spatial correlation of double-occupancy fluctuation in $x$ direction defined in Eq. (26). Error bars are those of statistical errors in the Monte Carlo sampling. The fitting curve is obtained by using the form $Q\left(\vec{r}\right)\propto 1/{\left|r\right|}^{z+\eta }+{\sum }_{n\ne (0,0)}\left[\right(1/|r+L\vec{n}{|}^{z+\eta })-(1/|L\vec{n}{|}^{z+\eta }\left)\right]$ to fit the data at $r\ge 3$ to estimate the asymptotic form at long distance. It suggests more or less isotropic power-law decay with $z+\eta =3.3\pm 0.8$ by taking account of the combined errors of the fitting and the Monte Carlo sampling.

Figure 16

Size extrapolation of the double occupancy at $t_{\perp }^{\mathrm{MQCP}}=0.38$ by using Eq. (H1). The VMC data of the double occupancy are plotted for the metallic phase ($U=1.5,1.6,1.7$, and 1.8) (a) and for the insulating phase ($U=1.9,2.0,2.1$, and 2.2) (b) by using the lattices with $L=20,24,$ and 28.

Figure 17

Comparison of ground-state energy between the VMC and AFQMC for $L=4,8$, and 12 at $t=1,U=1.8$, and $t_{\perp }=0.4$, which is close to the MQCP.

Figure 18

Averaged sign in AFQMC as a function of the effective imaginary time $\theta t$ at the same parameter as Fig. 17.