Generic Mott-Hubbard phase diagram for extended Hubbard models
without Umklapp scattering

The Mott transition is one of the long-standing problems in condensed-matter many-body physics [1, 2]. As formalized in the Hubbard model [3, 4, 5], an electronic system with a single-band of width $W$ and a purely local interaction of strength $U$ will be a metal for weak interactions, $W\ll U$, and an insulator for strong interactions, $U\gg W$. As argued by Mott early on [6], there must be a metal-to-insulator transition, generically at $U_{\rm c}\approx W$ when the two energy scales are comparable, irrespective of magnetic or charge order.

The quantitative analysis of a quantum phase transition in an interacting many-particle system is notoriously difficult. Concomitantly, analytical solutions are scarce even for the simplest models and in one spatial dimension [7, 2, 8]. Numerical approaches in finite dimensions are hampered by finite-size effects so that the calculation of ground-state quantities is also best performed for one-dimensional model systems. In one dimension, the numerical density-matrix renormalization group (DMRG) method provides accurate data for large enough systems with the order of hundred lattice sites and particles [9, 10, 11, 12, 13].

In some respects, one-dimensional systems behave qualitatively different from their three-dimensional counterparts. Most importantly, they generically display the perfect-nesting instability because the two Fermi points at half band-filling are connected by half a reciprocal lattice vector. Umklapp scattering turns the system insulating as soon as the (effective) interaction of the particles becomes finite [14]. Therefore, $U_{\rm c}=0^{+}$ is the generic situation [7, 2, 8], in contrast to Mott’s expectations. Correspondingly, the phase diagram for the one-dimensional Hubbard model does not contain a finite metallic region. When the Hubbard model is extended by the inclusion of a nearest-neighbor interaction, the ground-state phase diagram becomes more varied but one can only study quantum phase transitions between Mott-Hubbard, charge-density-wave (CDW) insulator, and bond-order-wave (BOW) insulator [15, 16, 17]. For more information on density waves in strongly correlated quantum chains, see Ref. [18].

To avoid Umklapp scattering at weak coupling, one can investigate models with only right-moving electrons that display only one Fermi point. A known example is the $1/r$-Hubbard model with its linear dispersion relation within the Brillouin zone [19, 20, 2]. Indeed, as indicated analytically [19, 21] and recently corroborated using DMRG [22], the critical interaction strength for the Mott transition is finite in the $1/r$-Hubbard model.

Therefore, we can study the competition of the metallic and insulating phases and the corresponding quantum phase transitions using the extended $1/r$-Hubbard model in one dimension. The resulting phase diagram should be generic in the sense that each phase covers a finite region in the ground-state phase diagram, as is expected for a three-dimensional system at half band-filling without Umklapp scattering.

The phase diagram in Fig. 1 depicts the central result of our work. It shows the generic Mott-Hubbard phase diagram for extended Hubbard models without Umklapp scattering. Derived for the special case of the extended $1/r$-Hubbard model, the phase diagram displays finite regions for the generic phases of an interacting electron system with a single half-filled band of width $W\equiv 1$ and with tunable local interaction $U$ and nearest-neighbor interaction $V$.

As can be argued using weak-coupling and strong-coupling perturbation theory, there should be a metallic phase at weak interactions, $U,V\ll W$, that becomes unstable against a Mott-Hubbard insulator for dominant Hubbard interaction, $U\gg V,W$, or against a charge-density-wave (CDW) insulator for dominant nearest-neighbor interactions, $V\gg U,W$. The critical interactions for the corresponding quantum phase transitions should be finite, the competing interactions being of the same order of magnitude.

Indeed, when the Coulomb interactions are dominant, $U,V\gg W$, the separation line between Mott-Hubbard insulator and CDW insulator should be $V=U/2$. The corresponding curve is included as a dashed line in Fig. 1. For large $U,V$, we find $V_{\rm c}(U)\gtrsim U/2$, with small deviations in favor of the Mott-Hubbard insulator. For this reason, we only show the phase diagram for $U\leq 1.6$. A bond-order wave might separate the two insulating phases, as is found in the one-dimensional extended Hubbard model [15, 16, 17]. Therefore, the line separating Mott-Hubbard insulator and charge-density-wave insulator should be taken as a guide to the eye only.

In our study we focus on the transitions between the metallic Luttinger liquid and the two insulating phases. We determine $V_{\rm c}(U)$ for fixed $0\leq v=V/U\leq 0.7$ with increment $\Delta v=0.1$, and for fixed $U=0.2$; for the meaning of the error bars in Fig. 1, see Sect. IV.

We note the following.

• –

  In the absence of a nearest-neighbor interaction, the Mott-Hubbard transition is known to occur at $U_{\rm c}(V=0)=1$ [19, 2] which is well reproduced using DMRG [22]. The repulsive nearest-neighbor interaction increases the critical interaction strength, i.e., the inclusion of the nearest-neighbor interaction stabilizes the metallic phase. Apparently, the additional repulsive nearest-neighbor interaction softens the two-particle scattering potential that is purely local in the bare Hubbard model.

  As a major result we find that the Mott transition remains continuous in the presence of a nearest-neighbor interaction. We presume that short-range interactions that decrease as a function of the particle distance will not fundamentally alter this behavior.

• –

  The transition from the Luttinger liquid metal to the charge-density-wave insulator is fairly common in the sense that even Hartree-Fock theory qualitatively reproduces the transition for not too large interactions.

  In Fig. 1, the corresponding Hartree-Fock prediction is shown as a dotted line. As usual, Hartree-Fock theory overestimates the stability of the ordered state and thus underestimates the critical interaction, $V_{\rm c,CDW}^{\rm HF}(U)<V_{\rm c,CDW}(U)$. Since the metallic phase extends well beyond the line $V=U/2$, there is no indication for a bond-order wave that might separate the Luttinger liquid and the charge-density-wave insulator.

We use a third-order spline interpolation through the data points to draw the phase transition lines in Fig. 1. The full lines depict continuous quantum phase transitions in the sense that the gaps open and close continuously at the same critical interaction when the transition is approached from the metallic and insulating sides, respectively.

The endpoint of both continuous lines where all three phases meet deserves spatial attention. Unsurprisingly, finite-size corrections are most severe in this region of phase space, and the study of the region around the tri-critical point is cumbersome and beyond the scope of our presentation.

Our work is organized as follows. In Sect. II we define the Hubbard model with long-range electron transfers and onsite and nearest-neighbor Coulomb interactions. We introduce the ground-state properties of interest, namely, the ground-state energy, the two-particle gap, the momentum distribution, and the density-density correlation function from which we determine the Luttinger parameter in the metallic phase and the CDW order parameter. In Sect. III we present results for the ground-state properties and discuss their finite-size dependencies and extrapolations to the thermodynamic limit where appropriate.

In Sect. IV we focus on the Mott-Hubbard transition in the presence of a nearest-neighbor interaction. We propose and discuss several methods to extract the critical interaction strength for the Mott transition based on the ground-state energy, the two-particle gap, the Luttinger parameter, and the structure factor whereby we study the Mott transition at fixed $v\equiv V/U$ in the range $0\leq v\leq 0.7$ (increment $\Delta v=0.1$) in units of the bandwidth, $W\equiv 1$. In addition, we address the Mott transition as a function of $V$ for fixed $U=0.2$ and $U=1.7$.

Short conclusions, Sect. V, close our presentation. The Hartree-Fock calculations for the CDW transition are collected in the appendix.

In this work, we address the $1/r$-Hubbard model [19, 2] with nearest-neighbor interactions

on a ring with $L$ sites ($L$: even). We discuss the kinetic energy and the Coulomb interaction terms separately.

We are interested in the metal-insulator transition at half band-filling where the metallic Luttinger liquid for weak interactions turns into a paramagnetic Mott insulator for large interactions at some finite value $U_{\rm c}(V)$ when $V$ is small enough, or to a CDW insulator for strong nearest-neighbor interactions. The metal-insulator transition can be inferred from the finite-size extrapolation of the ground-state energy and of the two-particle gap [22]. Alternatively, the Luttinger parameter [23] and the finite-size extrapolation of the structure factor at the Brillouin zone boundary permit to determine the critical interaction strength. Moreover, the charge-density-wave state can be monitored by the CDW order parameter. In this section, we also introduce the momentum distribution for finite systems that is also accessible via DMRG.

For $V=0$, the ground-state energy per site for finite system sizes is given by ($n=N/L$, $N$: even) [19, 2, 22]

with the abbreviation $e_{0}\equiv e_{0}(N,L;U,V=0)$.

In the thermodynamic limit and at half band-filling, $n=1$, the ground-state energy per site becomes particularly simple,

The analytic expressions (III.1) and (33) are useful for a comparison with numerical data at $V=0$.

For finite $V$, we can use first-order perturbation theory for weak interactions, $U,V\ll 1$, to find

with $v=V/U$ in the thermodynamic limit and at half band-filling. Note that Eq. (34) holds for all $v$.

We display the ground-state energy per site at half band-filling, $e_{0}(L,L;U,V)$, as a function of the inverse system size ($L=8,16,24,32,48,64$) and various values of $U$ in Fig. 2a ($v=0.1$), Fig. 2b ($v=0.3$), and Fig. 2c ($v=0.5$). For the extrapolation to the thermodynamic limit, we use the algebraic fit function

where $e_{0}(n=1;U,V)$ denotes the numerical estimate for the ground-state energy density in the thermodynamic limit and $a_{0}(U,V)$ and $\gamma_{0}(U,V)$ are the two other fit parameters. This extrapolation scheme is appropriate for $V=0$ [22] because the ground-state energy per site scales with $(1/L)^{2}$ for $U\neq 1$ and with $(1/L)^{3/2}$ for $U=U_{\rm c}(V=0)=1$, as follows from Eq. (III.1). More generally, we assume for all $(U,V)$

These exponents apply for very large system sizes. We shall discuss the finite-size modifications in detail in Sect. IV.

The extrapolated ground-state energies are shown in Fig. 3 together with the exact result for $V=0$. For small interactions, the nearest-neighbor interaction in the particle-hole symmetric form decreases the ground-state energy because the Hartree contribution at half band-filling is subtracted in the definition of the interaction, and the Fock contribution is negative because of the exchange hole. Therefore, the linear term in the interaction $(U/4)(1-8v/\pi^{2})$, see Eq. (34), is smaller in the presence of a nearest-neighbor interaction.

At large interactions, the ground-state energy approaches zero, $\lim_{U\to\infty}e_{0}(n=1;U,V=vU)=0$, as long as the charge-density wave is absent. In the presence of a CDW, the ground-state energy is negative and proportional to $U$, $e_{0}(U\gg 1,V)=U(1/2-v)$.

For $V=0$ the two-particle gap is known exactly for all system sizes [19, 2, 22],

In the thermodynamic limit, we find

The gap opens linearly above the critical interaction strength, $U_{\rm c}(U,V=0)=1$. Eq. (37) shows that the finite-size data approach the value in the thermodynamic limit algebraically in $1/L$,

with $\gamma_{2}(U\neq U_{\rm c},V=0)=1$ and $\gamma_{2}(U=U_{\rm c},V=0)=1/2$.

More generally, we assume for all $(U,V)$

As for the ground-state energy, these exponents apply for very large system sizes. We shall discuss the finite-size modifications in more detail in Sect. IV.

In Fig. 4 we show the DMRG results for $\Delta_{2}(L;U,V)$ as a function of $1/L$ for various values for $U$ as a function of $1/L$ for $L=8,16,24,32,48,64$ for (a) $v=0.1$, (b) $v=0.3$, (c) $v=0.5$. The lines are fits to the algebraic function in Eq. (39). The fits in Fig. 4 are seen to agree very well with the data, showing a steep decrease of the finite-size gap as a function of inverse system size. This indicates that large system sizes are required to obtain reasonable gap extrapolations.

The extrapolated gaps becomes smaller as a function of $V$, i.e., the nearest-neighbor interaction reduces the tendency to form a Mott-Hubbard insulator. The extrapolated gaps $\Delta(U,V)$ are shown in Fig. 5 as a function of $U$ for $v=0$, $v=0.1$, $v=0.3$, and $v=0.5$. Apparently, the nearest-neighbor interaction not only shifts the critical interaction to higher values, it also reduces the size of the gap in the Mott insulating phase.

At first sight, the increase of the critical interaction is counter-intuitive because one might argue that an additional repulsive nearest-neighbor Coulomb interaction should favor the insulating state, not the metallic state. From a wave-mechanical viewpoint, however, the repulsive nearest-neighbor interaction softens the two-particle scattering potential. Figuratively speaking, particles that are scattered by the weaker nearest-neighbor interaction $V$ do not experience the stronger on-site interaction $U$. For a quantitative analysis, see Sect. IV.

When $v=V/U$ is small, the change in the critical interaction strength is also small, and one might think of using perturbation theory around the bare $1/r$-Hubbard model. To test this idea, we consider

In the limit $V\to 0$, leading-order perturbation theory gives

where $C^{\rm NN}(r=1,L;U,V=0)$ is the nearest-neighbor density-density correlation function at half band-filling for the bare $1/r$-Hubbard model at finite system sizes $L$, see Eq. (25). As an example, for $v=0.3$ and $U\lesssim 0.7$, we find that $C^{\rm NN}(r=1,L;U,V=0)$ agrees fairly well with $C(L;U,V)$ from Eq. (41). Around the Mott transition, however, the corrections become sizable, more noticeably for larger systems. Therefore, low-order perturbation theory around the limit $V=0$ cannot be used to determine the critical interaction strength $U_{\rm c}(V)$ reliably.

In Fig. 6 we show the momentum distribution from DMRG at half band-filling for $L=64$ sites for various values of $U$ and $v=0.1$, $v=0.3$, and $v=0.5$ (from top to bottom). For small interactions, the momentum distribution resembles that of a Fermi liquid with all states $-\pi<k<0$ occupied and all states $0<k<\pi$ empty. For small $U$, low-energy scattering processes are limited to the vicinity of the sole Fermi point $k_{\rm F}=0$. Indeed, in the field-theoretical limit, $U,V\ll 1$, the model reduces to a bare $g_{4}$-model of only right-moving particles [24]. This ‘non-interacting Luttinger liquid’ displays a jump discontinuity at $k_{\rm F}$.

However, the $1/r$-Hubbard model is defined on a lattice and the bandwidth is finite. Consequently, the second Fermi point at $k_{{\rm F},2}=-\pi$ starts to play a role when $U$ becomes large, of the order of half the bandwidth. States near $k_{{\rm F},2}$ are depleted more quickly as a function of $U$ than those deeper in the Brillouin zone. Therefore, as seen in Fig. 6, the momentum distribution develops a maximum around $k=-\pi/2$, with a corresponding minimum around $k=\pi/2$.

These considerations show that the Luttinger parameter must deviate from unity, $K(U,V)<1$, for all $(U,V)$, even though corrections to unity are (exponentially) small for $U,V\ll 1$. Therefore, the momentum distribution is a continuous function in the (extended) $1/r$-Hubbard model for all $U,V>0$.

In contrast to the case $V=0$ [22], there is no Fano resonance discernible in the slope of the momentum distribution at $k=-\pi$ as the slope is always positive at $k=-\pi$. This indicates that the bound state for $V=0$ moves away from the band edge for finite $V>0$ and thus cannot be detected in the momentum distribution. Consequently, we cannot use the resonance to locate the metal-insulator transition in the extended $1/r$-Hubbard model.

Lastly, we show the structure factor from DMRG in Fig. 7 for $v=0.1$, $v=0.3$, and $v=0.5$ (from top to bottom) for the extended $1/r$-Hubbard model at system sizes $L=16,64$ below (left) and above (right) the Mott transition. It is seen that the finite-size effects are fairly small but larger systems permit a much better resolution in momentum space. In comparison with the exact result for the non-interacting system,

we see that the local interaction reduces the charge fluctuations. This is expected because the suppression of double occupancies likewise reduces the number of holes and the charges are more homogeneously distributed in the system. Therefore, the charge correlations become smaller when we compare the left and right figure in the same row.

The nearest-neighbor interaction counters the effect of the Hubbard interaction because nearest-neighbor pairs of a double occupancy and a hole are energetically favorable. Therefore, the charge correlations increase when we go from top to bottom in the left/right row, even though $U$ also increases from top to bottom.

When the nearest-neighbor interaction increases beyond a certain threshold value $V_{\rm c}(U)$, the ground state displays charge-density-wave order. In Fig. 8(a) we show the charge-density-wave order parameter $N_{\pi}(L;U,V=0.7U)$, see Eq. (29), as a function of $1/L$ for various values of $U$, and the extrapolated result $N_{\pi}(U,V=0.7U)$ into the thermodynamic limit using a second-order polynomial fit in Fig. 8(b),

Apparently, the CDW order parameter is continuous over the CDW transition. Close to the transition, $U\gtrsim U_{\rm c}(V)$,

where $\nu$ is the critical exponent for the CDW order parameter $D(U,V)$. Note that we pass the CDW transition for a fixed ratio $v=U/V$.

To make use of Eq. (45), the critical interaction $U_{\rm c}(V)$ must be known. In addition, the region of validity of Eq. (45) is unknown a priori. Typically, one has to study system parameters close to the transition to obtain a reliable estimate for $\nu$. Therefore, very large system sizes might be necessary to reach the scaling limit, and we have to be satisfied with the result from Fig. 8(b) that the CDW transition at $v=0.7$ is continuous with exponent $\nu\leq 1/2$.

In our previous work [22], we showed that the exponent $\gamma_{2}(U)=\gamma_{2}(U,V=0)$ sensitively depends on $U$ in the vicinity of the Mott-Hubbard transition, and the critical interaction for the $1/r$-Hubbard model, $U_{\rm c}(V=0)=1$, was obtained with an accuracy of one per mil.

To illustrate this result for the bare $1/r$-Hubbard model, in Fig. 9 we show the extrapolated gap exponent $\gamma_{2}(U)\equiv\gamma_{2}(U,V=0)$ using the analytic expression (37) for various combinations of system sizes in the range $L=8,16,24,32,48,64,80,96,128,256,512,1024,2048,4096$.

The minimal value for $\gamma_{2}(U)$ depends on the selected range of system sizes. The gap exponent in the thermodynamic limit, see Eq. (40), cannot be reproduced from finite-size studies but it is approached systematically with increasing system size. Furthermore, it can be seen from Fig. 9 that the inclusion of smaller system sizes such as $L=8,16$ leads to stronger deviations so that the smallest system sizes should be discarded. Note, however, that the position of the minimum and thus the critical interaction strength are very well reproduced in all cases. Therefore, the minimum of $\gamma_{2}(U,V)$ permits to locate the Mott transition $U_{\rm c}(V)$ fairly accurately.

In Fig. 10 we display the exponent $\gamma_{2}(U,V)$, as obtained from the fit of the finite-size data in the range $16\leq L\leq 80$ to the algebraic function in Eq. (39). Also shown in the figure are the quartic fits around the minima which lead to the critical interactions $U_{\rm c,gap}(V)$ listed in table 1. Note that the curves flatten out for increasing $v$ so that it becomes more difficult to determine accurately the minima for $v\to 0.5$.

The comparison with the exact value for $V=0$ shows that the gap exponent $\gamma_{2}(U,V)$ provides a fairly accurate estimate for the critical interaction. The same accuracy can be obtained when using the ground-state energy exponent $\gamma_{0}(U,V)$, as we shall show next.

As seen from Eq. (36), the $1/L$ corrections to the ground-state energy density also permit to locate the Mott transition in the extended $1/r$-Hubbard model, in the same way as the two-particle gap. In Fig. 11 we show the exponent $\gamma_{0}(U,V)$, as obtained from the fit of the finite-size data in the range $16\leq L\leq 80$ to the algebraic function in Eq. (35). Also shown in the figure are the quartic fits around the minima which lead to the critical interactions $U_{\rm c,gs}(V)$ listed in table 1.

The critical interaction strengths obtained from the minima of $\gamma_{0}(U,V)$ very well agree with the exact result at $V=0$ and with the values obtained from the gap exponent $\gamma_{2}(U,V)$ with deviations in the low percentage range. Therefore, we can be confident that we found reliable estimates for the critical interaction strength for the Mott transition.

As an alternative to locate the Mott transition, we monitor the Luttinger parameter and determine $U_{\rm c}(U,V)$ from the condition [24]

for fixed ratios $v=V/U$, see also Ref. [22].

In Fig. 12 we show the Luttinger parameter $K(L;U,V)$ from DMRG for the extended $1/r$-Hubbard model with nearest-neighbor interaction $V=0.3U$ as a function of $U$ for system sizes $L=8,16,24,32,48,64$ including a second-order polynomial extrapolation to the thermodynamic limit. The intersection of the extrapolation into the thermodynamic limit with $K_{\rm c}=1/2$ determines $U_{\rm c}(V)$. To obtain a reliable estimate for the intersection we can either use the two data points closest to the transition and perform a linear interpolation, in this case $U=1.1$ and $U=1.2$. Alternatively, we use a four-parameter fit of the whole data set that employs the information that the Luttinger parameter deviates from unity by exponentially small terms for $U,V\to 0$. Thus, we use

to fit the extrapolated data for finite values of $U$ to a continuous curve which is parameterized by $a,b,c,d$ that depend on $v$. Then, we solve Eq. (46) for $U_{\rm c,LL}(V)$. The results are also listed in table 1.

Alternatively, we could have solved Eq. (46) for each system size, and extrapolated the resulting system-size dependent critical interactions strengths to the thermodynamic limit. Since the results deviate more strongly from the exact value for $V=0$, we refrain from pursuing this approach further.

As seen from table 1, the critical value from Luttinger parameter systematically overestimate the correct interaction strengths by some three percent. A similar effect was found for the charge-density-wave transition in a one-dimensional model for spinless fermions with nearest-neighbor interactions (‘$t$-$V$-model’) [23]. Apparently, much larger systems are required to overcome this systematic error. In this work, we do not apply correction factors for a better fit but use the critical interaction strengths $U_{\rm c,LL}(V)$ as an upper bound to the exact value $U_{\rm c}(V)$.

For the $1/r$-Hubbard model, the finite-size corrections to the structure factor $\tilde{C}_{\pi}(U,V)\equiv\tilde{C}(\pi;U,V)$,

and the CDW order parameter $N_{\pi}(L;U,V)$, see Eq. (44), permit to locate the critical interaction strength. In Fig. 13 we show the structure factor for $v=0.3$ and various values of $U$ as a function of inverse system size for $L=8,16,24,32,48,64$.

As can be seen from the figure, the coefficient in $1/L$ changes its sign at the critical interaction strength,

To see this more clearly, in Fig. 14(a) we show the coefficient $C_{1}(U,V)$ as a function of $U$ for $v=0.1$, $v=0.3$, and $v=0.5$, and fit the data to a Fano resonance,

Analogously, we find the critical interaction strengths in the CDW phase from the $1/L$ corrections to the CDW order parameter (29), see Eq. (44), in Fig. 14(b).

As in our study of the $1/r$-Hubbard model [22], a bound state that interacts with the continuum shows up in physical quantities and thus contributes a Fano resonance to various physical quantities, with weight of the order $1/L$. Using the Fano resonance formula and the conditions $C_{1}(U_{\rm c,sf},V)=0=N_{1}(U_{\rm c,sf},V)$, the $1/L$-corrections of the structure factor and the CDW order parameter provide the estimate $U_{\rm c,sf}(V)$ for the critical interaction. The resulting data are listed for various $v$ in table 1.

The critical interaction strength $U_{\rm c,sf}(V)$ systematically underestimates the exact value for the Mott transition by a few percent. Together with the critical interaction strength from the Luttinger parameter $U_{\rm c,LL}(V)$ we thus can set tight limits to $U_{\rm c}(V)$.

In table 1 we collect the results for the critical interaction strengths $U_{\rm c}(V)$ obtained from the analysis of the two-particle gap, the ground-state energy, the Luttinger parameter, and the structure factor for $v=V/U=0,0.1,0.2,0.3,0.4,0.5,0.6,0.7$, as obtained from the previous four subsections. We observe that

• –

  the arithmetic average of the four values, $\overline{U}_{\rm c}$, reproduces the exact result at $V=0$ with an accuracy of a few per mil;

• –

  the values for $U_{\rm c,gs}(V)$ are close to the average for all $V$, with a deviation below two percent. Therefore, the ground-state exponent alone provides a reliable estimate for $U_{\rm c}(V)$ in all cases;

• –

  the estimates $U_{\rm c,LL}(V)$, using the Luttinger parameter, and $U_{\rm c,sf}$, using the structure factor, respectively, systematically overestimate and underestimate the critical interaction strength for the transition from the Luttinger liquid to the Mott-Hubbard insulator. Therefore, they provide natural bounds to $U_{\rm c}(V)$ for $v\leq 0.5$;

• –

  the transitions to the CDW insulator at $v=0.6,0.7$ can be determined fairly accurately from all four approaches individually.

In Fig. 1, we connect the data points for $\overline{U}_{\rm c}(V)$ using a third-order spline interpolation. Error bars at the data points result from the overestimates and underestimates listed in table 1. In Fig. 1 we also include the results from the analysis for the Mott transition between the Luttinger liquid and the CDW insulator at fixed $U=0.2$, as we discuss next.

Lastly, we study the metal-to-insulator transition at fixed Hubbard interaction $U$ as a function of $V$, namely for $U=0.2$ and $U=1.7$.