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

Florian Gebhard1<EMAIL_ADDRESS>Kevin Bauerbach
1Fachbereich Physik, Philipps-Universität Marburg, 35032 Marburg, Germany  Örs
Legeza1,2,3<EMAIL_ADDRESS>2Strongly Correlated Systems Lendület
Research Group, Institute for Solid State Physics and Optics, MTA Wigner
Research Centre for Physics, P.O. Box 49, 1525 Budapest, Hungary  3Institute
for Advanced Study, Technical University of Munich, Lichtenbergstraße 2a,
85748 Garching, Germany

(September 9, 2023)

###### Abstract

We determine the ground-state phase diagram for the $1/r$-Hubbard model with
repulsive nearest-neighbor interaction at half band-filling using the density-
matrix renormalization group (DMRG) method. Due to the absence of Umklapp
scattering, the phase diagram displays finite regions for the three generic
phases, namely, a Luttinger liquid metal for weak interactions, a Mott-Hubbard
insulator for dominant Hubbard interactions, and a charge-density-wave
insulator for dominant nearest-neighbor interactions. Up to moderate
interactions strengths, the quantum phase transitions between the metallic and
insulating phases are continuous, i.e., the gap opens continuously as a
function of the interaction strength. We conclude that generic short-range
interactions do not change the nature of the Mott transition qualitatively.

##  I Overview

After a short introduction in Sect. I.1, we present in Sect. I.2 the generic
Mott-Hubbard phase diagram for extended Hubbard models without Umklapp
scattering, the central result of our work. The corresponding model and its
ground-state properties are discussed in the remainder of this work, as
outlined in Sect. I.3.

###  I.1 Introduction

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.

###  I.2 Phase diagram

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
$V$.

Figure 1: Phase diagram of the one-dimensional extended $1/r$-Hubbard model;
energies in units of the bandwidth, $W=1$. Dots: estimate for the critical
interaction, $\bar{U}_{\rm c}$, with error bounds; continuous lines: spline
interpolations through the dots as guide to the eye; dotted line: Hartree-Fock
(HF) result for the transition between metal and charge-density-wave
insulator.

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.

###  I.3 Outline

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.

##  II Hubbard model with linear dispersion

###  II.1 Hamiltonian

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

$\hat{H}=\hat{T}+U\hat{D}+V\hat{V}$ (1)

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

####  II.1.1 Kinetic energy

The kinetic energy describes the tunneling of electrons with spin
$\sigma=\uparrow,\downarrow$ along a ring with $L$ sites,

$\displaystyle\hat{T}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}l,m=1\\\ l\neq
m;\sigma\end{subarray}}^{L}t(l-m)\hat{c}_{l,\sigma}^{+}\hat{c}_{m,\sigma}^{\vphantom{+}}\;,$
(2) $\displaystyle t(r)$ $\displaystyle=$ $\displaystyle(-{\rm
i}t)\frac{(-1)^{r}}{d(r)}\;,$ $\displaystyle d(r)$ $\displaystyle=$
$\displaystyle\frac{L}{\pi}\sin\left(\frac{\pi r}{L}\right)\;.$ (3)

The creation and annihilation operators $\hat{c}_{l,\sigma}^{+}$,
$\hat{c}_{l,\sigma}^{\vphantom{+}}$ for an electron with spin
$\sigma=\uparrow,\downarrow$ on lattice site $l$ obey the usual anti-
commutation relations for fermions.

In Eq. (3), $d(l-m)$ is the chord distance between the sites $l$ and $m$ on a
ring. In the thermodynamic limit and for $|l-m|\ll L$ fixed, we have
$d(l-m)=(l-m)+{\cal O}(1/L^{2})$, and the electron transfer amplitude between
two sites decays inversely proportional to their distance (‘$1/r$-Hubbard
model’).

Since $L$ is even, we have anti-periodic electron transfer amplitudes because
$d(L+r)=-d(r)$. Therefore, we must choose anti-periodic boundary conditions

$\hat{c}_{L+l,\sigma}=-\hat{c}_{l,\sigma}$ (4)

for the operators, too. With these boundary conditions, the kinetic energy
operator is diagonal in Fourier space,

$\displaystyle\hat{C}_{k,\sigma}^{+}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L}}\sum_{l=1}^{L}e^{{\rm
i}kl}\hat{c}_{l,\sigma}^{+}\;,$ $\displaystyle\hat{c}_{l,\sigma}^{+}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{L}}\sum_{k}e^{-{\rm
i}kl}\hat{C}_{k,\sigma}^{+}\;,$ $\displaystyle k$ $\displaystyle=$
$\displaystyle\frac{(2m+1)\pi}{L}\;,\;m=-\frac{L}{2},\ldots,\frac{L}{2}-1\;,$
(5)

so that

$\hat{T}=\sum_{k,\sigma}\epsilon(k)\hat{C}_{k,\sigma}^{+}\hat{C}_{k,\sigma}^{\vphantom{+}}\;,\quad\epsilon(k)=tk\;.$
(6)

The dispersion relation of the $1/r$-Hubbard model is linear. We set

$t=\frac{1}{2\pi}$ (7)

so that the bandwidth is unity, $W\equiv 1$.

In this work, we focus on the case of a paramagnetic half-filled ground state
where we have the same number of electrons per spin species,
$N_{\uparrow}=N_{\downarrow}$, that equals half the number of lattice sites,
$N_{\sigma}=L/2$ ($\sigma=\uparrow,\downarrow$).

####  II.1.2 Coulomb interaction

The Coulomb interaction is parameterized by two terms in Eq. (1). The on-site
(Hubbard) interaction [3, 4, 5] acts locally between two electrons with
opposite spins,

$\hat{D}=\sum_{l=1}^{L}\hat{n}_{l,\uparrow}\hat{n}_{l,\downarrow}\;,\quad\hat{n}_{l,\sigma}=\hat{c}_{l,\sigma}^{+}\hat{c}_{l,\sigma}^{\vphantom{+}}\;,$
(8)

where $\hat{n}_{l,\sigma}$ counts the number of electrons with spin $\sigma$
on site $l$, and $\hat{n}_{l}=\hat{n}_{l,\uparrow}+\hat{n}_{l,\downarrow}$
counts the number of electrons on site $l$. The corresponding operators for
the total number of electrons with spin $\sigma=\uparrow,\downarrow$ are
denoted by $\hat{N}_{\sigma}=\sum_{l}\hat{n}_{l,\sigma}$, and
$\hat{N}=\hat{N}_{\uparrow}+\hat{N}_{\downarrow}$.

To discuss the influence of the extended nature of the Coulomb interaction, we
consider the case of pure nearest-neighbor interactions,

$\hat{V}=\sum_{l=1}^{L}(\hat{n}_{l}-1)(\hat{n}_{l+1}-1)\;,$ (9)

where we disregard the long-range parts of the Coulomb interaction for
distances $|l-m|\geq 2$. The model in Eq. (1) describes the ‘extended’
$1/r$-Hubbard model with on-site interaction $U$ and nearest-neighbor
interaction $V$.

As we shall show in this work, the Mott-Hubbard transition at half band-
filling remains continuous in the presence of short-range interactions. For
not too large interactions and for $V\lesssim U/2$, the model contains a
transition from the Luttinger-liquid metal to the Mott-Hubbard insulator. For
larger nearest-neighbor interactions, the model eventually describes
transitions from the metallic state to a charge-density-wave (CDW) insulator.
For strong interactions, $U\gg W$, the model contains a transition from the
Mott-Hubbard insulator to the CDW insulator around $V\approx U/2$.

We study several values for the ratio $v=V/U$, namely,
$v=0,0.1,0.3,0.4,0.5,0.6,0.7$ for weak to strong nearest-neighbor
interactions. Since we scan the value of $U$, we must limit the number of
values for $v$ to keep the numerical effort within bounds when we include
systems up to $L_{\rm max}=80$ lattice sites; when finite-size effects are
well behaved, e.g., for the ground-state energy, we limit our investigations
to $L=64$. Moreover, we scan $V$ for fixed $U=0.2$ and $U=1.7$ to study the
Mott transition as a function of the nearest-neighbor interaction.

####  II.1.3 Particle-hole symmetry

Under the particle-hole transformation

$\hat{c}_{l,\sigma}^{\vphantom{+}}\mapsto\hat{c}_{l,\sigma}^{+}\quad,\quad\hat{n}_{l,\sigma}\mapsto
1-\hat{n}_{l,\sigma}\;,$ (10)

the kinetic energy remains unchanged,

$\displaystyle\hat{T}$ $\displaystyle\mapsto$
$\displaystyle\sum_{\begin{subarray}{c}l,m=1\\\ l\neq
m;\sigma\end{subarray}}^{L}t(l-m)\hat{c}_{l,\sigma}^{\vphantom{+}}\hat{c}_{m,\sigma}^{+}$
(11) $\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}l,m=1\\\ l\neq
m;\sigma\end{subarray}}^{L}\left[-t(m-l)\right]\hat{c}_{l,\sigma}^{+}\hat{c}_{m,\sigma}^{\vphantom{+}}=\hat{T}$

because $t(-r)=-t(r)$. Furthermore,

$\hat{D}\mapsto\sum_{l=1}^{L}(1-\hat{n}_{l,\uparrow})(1-\hat{n}_{l,\downarrow})=\hat{D}-\hat{N}+L\;,$
(12)

and

$\hat{V}\mapsto\hat{V}\;.$ (13)

Therefore, $\hat{H}(N_{\uparrow},N_{\downarrow})$ has the same spectrum as
$\hat{H}(L-N_{\uparrow},L-N_{\downarrow})-U(2L-N)+LU$, where
$N=N_{\uparrow}+N_{\downarrow}$ is the particle number.

###  II.2 Ground-state properties

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.

####  II.2.1 Ground-state energy and two-particle gap

We denote the ground-state energy by

$E_{0}(N,L;U,V)=\langle\Psi_{0}|\hat{H}|\Psi_{0}\rangle$ (14)

for given particle number $N$, system size $L$, and interaction parameters
$U,V$. Here, $|\Psi_{0}\rangle$ is the normalized ground state of the
Hamiltonian (1). We are interested in the thermodynamic limit, $N,L\to\infty$
with $n=N/L$ fixed. We denote the ground-state energy per site and its
extrapolated value by

$\displaystyle e_{0}(N,L;U,V)$ $\displaystyle=$
$\displaystyle\frac{1}{L}E_{0}(N,L;U,V)\;,$ $\displaystyle e_{0}(n;U,V)$
$\displaystyle=$ $\displaystyle\lim_{L\to\infty}e_{0}(N,L;U,V)\;,$ (15)

respectively.

The two-particle gap is defined by

$\Delta_{2}(L;U,V)=\mu_{2}^{+}(L;U,V)-\mu_{2}^{-}(L;U,V)\;,$ (16)

where

$\displaystyle\mu_{2}^{-}(L;U,V)$ $\displaystyle=$ $\displaystyle
E_{0}(L,L;U,V)-E_{0}(L-2,L;U,V)\;,$ $\displaystyle\mu_{2}^{+}(L;U,V)$
$\displaystyle=$ $\displaystyle E_{0}(L+2,L;U,V)-E_{0}(L,L;U,V)$

are the chemical potentials for adding the last two particles to half filling
and the first two particles beyond half filling, respectively.

Due to particle-hole symmetry, we have

$\mu_{2}^{-}(L;U,V)=2U-\mu_{2}^{+}(L;U,V)$ (18)

so that

$\Delta_{2}(L;U,V)=2\mu_{2}^{+}(L;U,V)-2U$ (19)

and

$\Delta_{2}(U,V)=\lim_{L\to\infty}\Delta_{2}(L;U,V)$ (20)

in the thermodynamic limit. We always consider the spin symmetry sector
$S=S^{z}=0$. For this reason, we study the two-particle gap rather than the
single-particle gap.

The two added particles repel each other so that, in the thermodynamic limit,
they are infinitely separated from each other. Therefore, we have

$\Delta_{2}(U,V)=2\Delta_{1}(U,V)\;,$ (21)

where $\Delta_{1}(U,V)$ is the gap for single-particle excitations. For finite
systems, we expect the interaction energy

$e_{\rm R}(L;U,V)=\Delta_{2}(L;U,V)-2\Delta_{1}(L;U,V)={\cal O}(1/L)>0$ (22)

to be positive, of the order $1/L$. We verified that the interaction energy
vanishes in the thermodynamic limit for the case $V=0$ [22].

####  II.2.2 Momentum distribution

We also study the spin-summed momentum distribution in the ground state at
half band-filling, $N=L$,

$\displaystyle n_{k}(L;U,V)$ $\displaystyle=$
$\displaystyle\langle\Psi_{0}|\hat{n}_{k,\uparrow}+\hat{n}_{k,\downarrow}|\Psi_{0}\rangle$
(23) $\displaystyle=$ $\displaystyle\sum_{l,m;\sigma}e^{{\rm
i}k(l-m)}P_{l,m;\sigma}$

with
$\hat{n}_{k,\sigma}=\hat{C}_{k,\sigma}^{+}\hat{C}_{k,\sigma}^{\vphantom{+}}$
and the single-particle density matrix
$P_{l,m;\sigma}=\langle\Psi_{0}|\hat{c}_{l,\sigma}^{+}\hat{c}_{m,\sigma}^{\vphantom{+}}|\Psi_{0}\rangle$.
Due to particle-hole symmetry we have

$n_{k}(L;U,V)=1-n_{-k}(L;U,V)$ (24)

at half band-filling. Therefore, it is sufficient to study wave numbers from
the interval $-\pi<k<0$.

In contrast to our previous work [22], the slope of the momentum distribution
at the band edge cannot be used to trace the Mott-Hubbard transition in the
extended $1/r$-Hubbard model because the bound state moves away from the band
edge for $V>0$.

####  II.2.3 Density-density correlation function and Luttinger parameter

Lastly, we address the density-density correlation function at half band-
filling, $N=L$,

$C^{\rm
NN}(r,L;U,V)=\frac{1}{L}\sum_{l=1}^{L}\bigl{(}\langle\hat{n}_{l+r}\hat{n}_{l}\rangle-\langle\hat{n}_{l+r}\rangle\langle\hat{n}_{l}\rangle\bigr{)}\;,$
(25)

where $\langle\ldots\rangle\equiv\langle\Psi_{0}|\ldots|\Psi_{0}\rangle$. The
limit $L\gg r\gg 1$ for $U,V\ll W$ is also accessible from field theory [24,
25, 26],

$C^{\rm NN}(r\gg 1;U,V)\sim-\frac{K(U,V)}{(\pi
r)^{2}}+\frac{A(U,V)(-1)^{r}}{r^{1+K}[\ln(r)]^{3/2}}+\ldots\,,$ (26)

where $A(U,V)$ is a constant that depends on the interaction but not on the
distance $r$.

We extract the Luttinger exponent $K(U,V)$ from the structure factor,

$\tilde{C}^{\rm NN}(q,L;U,V)=\sum_{r=0}^{L-1}e^{-{\rm i}qr}C^{\rm
NN}(r,L;U,V)\;,$ (27)

where the wave numbers are from momentum space, $q=(2\pi/L)m_{q}$,
$m_{q}=-L/2,-L/2+1,\ldots,L/2-1$. By construction, $\tilde{C}^{\rm
NN}(q=0,L;U,V)=0$ because the particle number is fixed, $N=L$ in the half-
filled ground state. In the thermodynamic limit, the structure factor
$\tilde{C}^{\rm NN}(q,L;U,V)$ remains of the order unity even in the CDW phase
because we subtract the contributions of the long-range order in the
definition (25).

The transition to a charge-density-wave insulator can be monitored from the
CDW order parameter. In this work, we do not study the standard CDW order
parameter,

$D(L;U,V)=\frac{1}{L}\left|\sum_{r=0}^{L-1}(-1)^{r}\left(\langle\hat{n}_{r}\rangle-1\right)\right|\leq
1\;.$ (28)

Instead, we include all short-range contributions and address

$N_{\pi}(L;U,V)=\frac{1}{L}\sum_{r=0}^{L-1}(-1)^{r}\frac{1}{L}\sum_{l=0}^{L-1}\left(\langle\hat{n}_{r+l}\hat{n}_{l}\rangle-1\right)\;.$
(29)

When the charges are distributed homogeneously, $\langle\hat{n}_{l}\rangle=1$,
we have $N_{\pi}(L;U,V)=\tilde{C}^{\rm NN}(\pi,L;U,V)/L$, and the order
parameter vanishes in the metallic phase. More generally, in the thermodynamic
limit we have $N_{\pi}(U,V)=(D(U,V))^{2}$. In the $1/r$-Hubbard model with its
long-range electron transfer, it is advantageous to analyze $N_{\pi}(L;U,V)$
to facilitate a reliable finite-size analysis.

When Eq. (26) is employed, it follows that the Luttinger parameter for finite
systems,

$K(L;U,V)=\frac{L}{2}\tilde{C}^{\rm NN}(2\pi/L,L;U,V)\;,$ (30)

can be used to calculate the Luttinger parameter in the thermodynamic limit,

$\displaystyle K(U,V)$ $\displaystyle=$
$\displaystyle\lim_{L\to\infty}K(L;U,V)$ (31) $\displaystyle=$
$\displaystyle\pi\lim_{q\to 0}\frac{\tilde{C}^{\rm NN}(q;U,V)}{q}\;,$

where we denote the structure factor in the thermodynamic limit by
$\tilde{C}^{\rm NN}(q;U,V)$. Using Eq. (31), the Luttinger exponent can be
calculated numerically with very good accuracy [27]. The Luttinger parameter
can be used to locate the metal-insulator transition in one spatial dimension.

##  III Ground-state properties

Before we investigate the Mott transition for the half-filled extended $1/r$
Hubbard model in more detail in the next section, we present DMRG results for
the ground-state energy, the two-particle gap, the momentum distribution, the
structure factor, and the CDW order parameter. For the numerical calculations
we employ a DMRG code that permits the treatment of arbitrary quantum system
with long-ranged complex interactions. It uses non-Abelian symmetries and
optimization protocols inherited from quantum information theory [28]. Further
technical details of the DMRG implementation can be found in Ref. [22]. Note
that our finite-size scaling analysis requires very accurate data. We obtain
those by imposing strict accuracy settings in our DMRG code, and by
restricting the largest system size to $L_{\rm max}=80$ to limit the
truncation errors.

###  III.1 Ground-state energy

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

$\displaystyle e_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4}n(n-1)+\frac{U}{4}n$
$\displaystyle-\frac{1}{2L}\sum_{r=0}^{(N/2)-1}\sqrt{1+U^{2}-4U(2r+1-L/2)/L}$

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,

$\displaystyle e_{0}(n=1;U\leq 1,V=0)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}+\frac{U}{4}-\frac{U^{2}}{12}\;,$ $\displaystyle
e_{0}(n=1;U\geq 1,V=0)$ $\displaystyle=$ $\displaystyle-\frac{1}{12U}\;.$ (33)

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

(a)

(b)

(c)

Figure 2: Ground-state energy per lattice site at half band-filling,
$e_{0}(L,L;U,V)$, for the extended $1/r$-Hubbard model as a function of $1/L$
for $L=8,16,24,32,48,64$ and various values for $U$ for (a) $v=0.1$, (b)
$v=0.3$, (c) $v=0.5$. The continuous lines are fits to the algebraic fit
function (35). The intercept of the extrapolation curves with the ordinate
defines the extrapolation estimate $e_{0}(n=1;U,V)$ in the thermodynamic
limit.

$e_{0}^{\rm
PT}(U,V)=-\frac{1}{4}+\frac{U}{4}\left(1-\frac{8v}{\pi^{2}}\right)+\mathcal{O}(U^{2})$
(34)

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

$e_{0}(L,L;U,V)=e_{0}(n=1;U,V)+a_{0}(U,V)\left(\frac{1}{L}\right)^{\gamma_{0}(U,V)}\;,$
(35)

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
$(U,V)$

$\gamma_{0}(U,V)=\left\\{\begin{array}[]{ccl}2&\hbox{for}&U\neq U_{\rm
c}(V)\\\\[6.0pt] \displaystyle\frac{3}{2}&\hbox{for}&U=U_{\rm
c}(V)\end{array}\right.\;.$ (36)

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.

Figure 3: Ground-state energy per lattice site at half band-filling in the
thermodynamic limit, $e_{0}(n=1;U,V)$, for the extended $1/r$-Hubbard model
from the extrapolation to the thermodynamic limit in Fig. 2. The dashed lines
represent first-order order perturbation theory for $v=V/U=0.3,0.5,0.7$, see
Eq. (34). The continuous line is the exact result for $V=0$,
$e_{0}(n=1;U,V=0)$, see Eq. (33).

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 $U$, $e_{0}(U\gg 1,V)=U(1/2-v)$.

###  III.2 Two-particle gap

(a)

(b)

(c)

Figure 4: Two-particle gap $\Delta_{2}(L;U,V)$ for the extended $1/r$-Hubbard
model as a function of inverse system size for $L=8,16,24,32,48,64$ and
various values for $U$ for (a) $v=0.1$, (b) $v=0.3$, (c) $v=0.5$. The
continuous lines are fits to the algebraic fit function (39). The intercept of
the extrapolation curves with the ordinate defines the extrapolation estimate
$\Delta_{2}(U,V)$ for the two-particle gap.

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

$\Delta_{2}(L;U\geq 1,V=0)=U-1+\frac{2}{L}+\sqrt{(U-1)^{2}+\frac{4U}{L}}\;.$
(37)

In the thermodynamic limit, we find

$\Delta_{2}(U\geq 1,V=0)=2(U-1)\;.$ (38)

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 $1/L$,

$\Delta_{2}(L;U,V)=\Delta_{2}(U,V)+a_{2}(U,V)\left(\frac{1}{L}\right)^{\gamma_{2}(U,V)}$
(39)

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 $(U,V)$

$\gamma_{2}(U,V)=\left\\{\begin{array}[]{ccl}1&\hbox{for}&U\neq U_{\rm
c}(V)\\\\[6.0pt] \displaystyle\frac{1}{2}&\hbox{for}&U=U_{\rm
c}(V)\end{array}\right.\;.$ (40)

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.

Figure 5: Two-particle gap $\Delta_{2}(U,V)$ for the extended $1/r$-Hubbard
model as a function of $U$ for $v=0.1$ (red dots), $v=0.3$ (green dots),
$v=0.5$ (purple dots), extrapolated from finite-size data with up to $L=64$
sites. The continuous line is the exact result in the thermodynamic limit for
$V=0$, $\Delta_{2}(U,V=0)=2(U-1)$, see Eq. (38).

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

$C(L;U,V)=\frac{e_{0}(L,L;U,V)-e_{0}(L,L;U,V=0)}{V}\;.$ (41)

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

$\lim_{V\to 0}C(L;U,V)=C^{\rm NN}(r=1,L;U,V=0)\;,$ (42)

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.

###  III.3 Momentum distribution

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 $U,V>0$.

(a)
(b)
(c)

Figure 6: Momentum distribution $n_{k}(L;U,V)$ from DMRG at half band-filling
for the extended $1/r$-Hubbard model for $L=64$ sites and for various values
for $U$ for $v=0.1$, $v=0.3$, and $v=0.5$ (from top to bottom).

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.

###  III.4 Structure factor and CDW order parameter

(a)
(b)
(c)

Figure 7: Structure factor $\tilde{C}^{\rm NN}(q,L;U,V)$ for the extended
$1/r$-Hubbard model for $L=16,64$ below (left) and above (right) the Mott
transition for (a) $v=0.1$, (b) $v=0.3$, (c) $v=0.5$.

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,

$\tilde{C}^{\rm NN}(q,n=1;U=0,V=0)=\frac{|q|}{\pi}\;,$ (43)

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.

(a)

(b)

Figure 8: (a) CDW order parameter $N_{\pi}(L;U,V)$ for the extended half-
filled $1/r$-Hubbard model as a function of $1/L$ ($L\leq 80$) for $v=0.7$ and
various $U$-values. Lines are a second-order polynomial fit in $1/L$, see Eq.
(44); (b) Extrapolated CDW order parameter $N_{\pi}(U,V=0.7U)$ as a function
of $U$. The line is an algebraic fit to the data in the vicinity of the CDW
transition, see Eq. (45), with $U_{\rm c}(v=0.7)=0.6$, $N_{0}=1$ and
$2\nu=0.3$.

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),

$N_{\pi}(L;U,V)=N_{\pi}(U,V)+\frac{N_{1}(U,V)}{L}+\frac{N_{2}(U,V)}{L^{2}}\;.$
(44)

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

$N_{\pi}(U,V)=N_{0}\left[U-U_{\rm c}(V)\right]^{2\nu}\;,$ (45)

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 $\nu\leq 1/2$.

##  IV Mott transition

In this section we determine the critical value for the Mott transition in the
extended $1/r$-Hubbard model. We investigate the two-particle gap, the ground-
state energy, the Luttinger parameter, and the structure factor at the
Brillouin zone boundary to locate the critical interaction strength $U_{\rm
c}(V)$. The Mott transition remains continuous for all $V/U$.

###  IV.1 Two-particle gap

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.

Figure 9: Extrapolated gap exponent $\gamma_{2}(U)=\gamma_{2}(U,V=0)$ using
the analytical expression of the two-particle gap in Eq. (37). Various system
sizes are used, ranging from
$L=8,16,24,32,48,64,80,96,128,256,512,1024,2048,4096$. Figure 10: Exponent
$\gamma_{2}(U,V)$ for the two-particle gap in the extended $1/r$-Hubbard model
as a function of $U$ for various values of $v=V/U$, based on system sizes
$16\leq L\leq 80$. The minimum of the curve determines $U_{\rm c,gap}(V)$.

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 $v\to 0.5$.

$V/U$ | $U_{\rm c,gap}(V)$ | $U_{\rm c,gs}(V)$ | $U_{\rm c,LL}(V)$ | $U_{\rm c,sf}(V)$ | $\overline{U}_{\rm c}(V)$
---|---|---|---|---|---
0 | 1.009 | 1.000 | 1.033 | 0.965 | 1.002
0.1 | 1.024 | 1.022 | 1.056 | 0.984 | 1.021
0.2 | 1.055 | 1.056 | 1.090 | 1.018 | 1.055
0.3 | 1.109 | 1.116 | 1.144 | 1.075 | 1.111
0.4 | 1.202 | 1.221 | 1.243 | 1.175 | 1.210
0.5 | 1.425 | 1.500 | 1.540 | 1.456 | 1.480
0.6 | 0.828 | 0.838 | 0.883 | 0.876 | 0.856
0.7 | 0.587 | 0.600 | 0.616 | 0.611 | 0.604
Table 1: Critical interaction strengths for the extended $1/r$-Hubbard model,
as obtained from the two-particle gap, the ground-state energy, the Luttinger
parameter, and the structure factor for systems with $16\leq L\leq 80$ lattice
sites. For $V=0$, the exact result in the thermodynamic is known [19], $U_{\rm
c}(V=0)=1$.

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.

Figure 11: Exponent $\gamma_{0}(U,V)$ for the ground-state energy of the
extended $1/r$-Hubbard model as a function of $U$ for various values of
$v=V/U$, based on system sizes $16\leq L\leq 80$. The minimum of the curve
determines $U_{\rm c,gs}(V)$.

###  IV.2 Ground-state energy

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.

###  IV.3 Luttinger parameter

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

$K(U_{\rm c}(V),v)=\frac{1}{2}$ (46)

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

Figure 12: 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 with $K_{\rm c}=1/2$ determines $U_{\rm c}(V)$.

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

$K(U,V)=a+b\tanh(c+dU)$ (47)

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 $U_{\rm c}(V)$.

Figure 13: Structure factor $\tilde{C}_{\pi}(L;U,V)$ at $q=\pi$ as a function
of $1/L$ for various values of $U$ for the extended $1/r$-Hubbard model with
nearest-neighbor interaction $V=0.3U$ for system sizes $L=8,16,24,32,48,64$.
Lines are a second-order polynomial extrapolation to the thermodynamic limit,
see Eq. (48).

###  IV.4 Structure factor and CDW order parameter

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

$\tilde{C}_{\pi}(L;U,V)=\tilde{C}_{\pi}(U,V)+\frac{C_{1}(U,V)}{L}+\frac{C_{2}(U,V)}{L^{2}}\;,$
(48)

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
$L=8,16,24,32,48,64$.

(a)

(b)

Figure 14: (a) Finite-size coefficient $C_{1}(U,V)$ of the structure factor as
a function of $U$ for the extended $1/r$-Hubbard model for $v=0.1$ and $v=0.3$
(inset: $v=0.5$). (b) Finite-size coefficient $N_{1}(U,V=0.7U)$ for the CDW
order parameter, see Eqs. (29) and (44). Lines are fitted Fano resonance
curves, see Eq. (50).

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

$C_{1}(U_{\rm c,sf}(V),V)=0\;.$ (49)

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,

$C_{1}^{\rm Fano}(U,V)=a(V)+b(V)\frac{[q_{\rm F}(V)\Gamma(V)+U-U_{\rm
c}(V)]^{2}}{[\Gamma(V)]^{2}+[U-U_{\rm c}(V)]^{2}}\;.$ (50)

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 $U_{\rm c}(V)$.

###  IV.5 Critical interactions for fixed interaction ratios

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.

###  IV.6 Transitions at fixed Hubbard interaction

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

####  IV.6.1 Transition from Luttinger liquid to CDW insulator

At $U=0.2$, we find a transition from the Luttinger liquid metal to the CDW
insulator at $V_{\rm c}(U=0.2)=0.29\pm 0.01$. The analysis follows the route
outlined in the previous subsections, and will not be repeated here. We
increase $V$ in steps of $\Delta V=0.02$ around the transition.

Using the coefficient $\gamma_{0}$ from the ground-state energy, see Sect.
IV.2, we find $V_{\rm c,gs}(U=0.2)=0.286$, the coefficient $\gamma_{2}$ from
the two-particle gap in Sect. IV.1 leads to $V_{\rm c,gap}(U=0.2)=0.280$, and
the Luttinger parameter of Sect. IV.3 leads to $V_{\rm c,LL}(U=0.2)=0.298$,
almost identical to the values from the structure factor, see Sect. IV.4. This
leads to the average value quoted above.

Due to the absence of perfect nesting in the dispersion relation, it requires
a finite interaction strength $V$ to stabilize the CDW phase even at $U=0$.
Qualitatively, Hartree-Fock theory leads to the same result. Hartree-Fock
theory systematically overestimates the stability of the CDW phase and thus
underestimates $V_{\rm c}(U)$, see Fig. 1. The analytical approach can be
improved by including second-order corrections to Hartree-Fock theory, see,
e.g., Refs. [29, 23]. This is beyond the purpose of our present analysis.

####  IV.6.2 Transition from Mott-Hubbard to CDW insulator

At $U=1.7$, not included in the phase diagram in Fig. 1, we have a brief look
at the transition from the Mott-Hubbard insulator to the CDW insulator. The
results for the two-particle gap are shown in Fig. 15. They are corroborated
by the behavior of the order parameter quantities $C_{\pi}$ and $N_{\pi}$. The
analysis of the parameters $\gamma_{0}$ and $\gamma_{2}$ lead to
quantitatively identical but less accurate results.

For the one-dimensional extended Hubbard model it is known that the critical
interaction is larger than $U/2$. For the extended $1/r$-Hubbard model we also
find $V_{\rm c}=0.87\pm 0.01>1.7/2=0.85$ for the onset of the CDW. When
expressed in units of the bandwidth, the offset of $\delta_{\rm c}(U)=V_{\rm
c}(U)-U/2$ agrees almost quantitatively with the value obtained from DMRG and
QMC calculations for the one-dimensional extended Hubbard model, $\delta_{\rm
c}(U=1.7)\approx 0.02$, see Ref. [30].

The shift $\delta_{\rm c}(U)$ can be determined analytically using higher-
order strong-coupling perturbation [31]. Unfortunately, this program cannot be
carried out for the extended $1/r$-Hubbard model because the exact ground
state is not known for the effective spin model which is a linear combination
of the Heisenberg model with nearest-neighbor interaction and the Haldane-
Shastry model with $1/r^{2}$-exchange interaction [32, 33]. A variational
strong-coupling approach that employs the Baeriswyl wave function [34, 35] can
neither be carried out analytically because it requires the evaluation of
$\langle\hat{T}^{3}\rangle$ in the Gutzwiller-projected Fermi sea.

(a)

(b)

Figure 15: (a) Two-particle gap for the extended $1/r$-Hubbard model at
$U=1.7$ for various values of $V$ as a function of $1/L$ for
$L=8,16,32,64,80$. (b) Extrapolated two-particle gap as a function of $V$.

In the one-dimensional extended Hubbard model, there is a bond-order-wave
phase below a critical interaction strength $U_{\rm tri}$ that separates the
Mott-Hubbard and CDW insulators [15, 16, 17]. For $U>U_{\rm tri}$, the
transition from the Mott-Hubbard insulator to the CDW insulator is
discontinuous. As can be seen from Fig. 15, we also find indications for the
existence of a bond-order-wave phase. The charge gap of the Mott-Hubbard
insulator closes around $V_{\rm c}(U=1.7)\approx 0.87$ and reopens beyond
$V_{\rm c}(U=1.7)$ with a small value. The extrapolation of the gap remains
linear as a function of $1/L$ even at $V=0.88$, as seen in Fig. 15a. For
larger values of $V$, the gap drastically increases and the extrapolation
displays a $1/L^{2}$ behavior for large $L$. The same behavior of the gap was
observed for the one-dimensional extended Hubbard model at $U=2W$ [16, 17]
where it was numerically shown in detail that as a function of $V$ the Mott-
Hubbard insulator gives in to a bond-order-wave insulator before the CDW phase
eventually takes over.

Further investigations are necessary to corroborate the existence of a bond-
order-wave phase in the vicinity of the CDW transition also for the extended
$1/r$-Hubbard model. Note, however that we do not expect a bond-order wave as
an intermediate phase for small interactions because in the extended
$1/r$-Hubbard model the metallic Luttinger liquid overrides a conceivable
bond-order wave.

##  V Conclusions

In this work we applied the density-matrix renormalization group (DMRG) method
to the half-filled extended $1/r$-Hubbard model where the electron transfer
amplitudes decay proportional to the inverse chord distance of two lattice
sites on a ring. The model describes a linear dispersion within the Brillouin
zone and thus provides an ideal case to study the Mott-Hubbard transition
because it lacks Umklapp scattering. Therefore, the metal-to-insulator
transitions occur at finite interaction strengths. Consequently, all generic
phases, namely Luttinger-liquid metal, Mott-Hubbard insulator, and charge-
density-wave insulator, occupy a finite region in the $(U,V)$ ground-state
phase diagram, see Fig. 1.

Mapping the quantum phase transition boundaries for the specific model is one
of the main achievements of this work. To this end, we use DMRG data for up to
$L=80$ sites to calculate the ground-state energy, the two-particle gap, the
momentum distribution, the Luttinger parameter, and the structure factor. The
finite-size behavior of the ground-state energy, of the two-particle gap, and
of the structure factor permit to determine the critical interaction
parameters for the instability of the Luttinger liquid metal against the Mott-
Hubbard insulator and the charge-density-wave insulator, respectively.
Moreover, we monitor the Luttinger parameter that also signals the breakdown
of the Luttinger liquid metal at a metal-to-insulator transition. We tested
the validity of our analysis against exact results for $V=0$ for which
analytic results for the ground-state energy and the gap exist for all
interactions $U$ and system sizes $L$.

The phase diagram in Fig. 1 shows that the nearest-neighbor interaction and
the Hubbard interaction counteract each other. On the one hand, the Mott
transition shifts to larger values, i.e., a weak to moderate nearest-neighbor
interaction stabilizes the Luttinger liquid metal. Apparently, the two-
particle scattering interaction becomes smoother in position space and renders
the total interaction less effective. On the other hand, as can readily be
understood from classical considerations, the Hubbard interaction opposes the
formation of a charge-density wave because, by definition, a CDW augments the
particle density on the same lattice site.

In contrast to the ‘standard’ extended Hubbard model in one dimension, the
absence of Umklapp scattering and the competition of both interactions leads
to an extended metallic region in the phase diagram. The extrapolations
suggest that there is a tri-critical point where all three phases touch. It
will be interesting to analyze this region in phase space with higher
accuracy, i.e., more data points in the $(U,V)$ parameter space close to
$(U_{\rm tri},V_{\rm tri})\approx(1.5,0.75)$, and larger system sizes, $L>80$.
Moreover, a conceivable bond-order wave above the tri-critical point between
Mott insulator and charge-density-wave insulator should be investigated in
more detail. These tasks are left for future studies.

###### Acknowledgements.

Ö.L. has been supported by the Hungarian National Research, Development, and
Innovation Office (NKFIH) through Grant No. K134983, and TKP2021-NVA by the
Quantum Information National Laboratory of Hungary. Ö.L. also acknowledges
financial support from the Hans Fischer Senior Fellowship program funded by
the Technical University of Munich – Institute for Advanced Study and from the
Center for Scalable and Predictive methods for Excitation and Correlated
phenomena (SPEC), which is funded as part of the Computational Chemical
Sciences Program by the U.S. Department of Energy (DOE), Office of Science,
Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences,
and Biosciences at Pacific Northwest National Laboratory.

##  Appendix Hartree-Fock theory

###  A.1 CDW Hartree-Fock Hamiltonian

In Hartree-Fock theory, we decouple the two-particle interaction as follows,

$\displaystyle\hat{D}^{\rm HF}=\hat{D}^{\rm H}$ $\displaystyle=$
$\displaystyle\sum_{l}\langle\hat{n}_{l,\uparrow}\rangle\hat{n}_{l,\downarrow}+\hat{n}_{l,\uparrow}\langle\hat{n}_{l,\downarrow}\rangle-\langle\hat{n}_{l,\uparrow}\rangle\langle\hat{n}_{l,\downarrow}\rangle\;,$
$\displaystyle\hat{V}^{\rm HF}$ $\displaystyle=$ $\displaystyle\hat{V}^{\rm
H}+\hat{V}^{\rm F}\;,$ (52) $\displaystyle\hat{V}^{\rm H}$ $\displaystyle=$
$\displaystyle\sum_{l}\left(\langle\hat{n}_{l}\rangle-1\right)\left(\hat{n}_{l+1}-1\right)$
(53)
$\displaystyle\hphantom{\sum_{l}}+\left(\hat{n}_{l}-1\right)\left(\langle\hat{n}_{l+1}\rangle-1\right)$
$\displaystyle\hphantom{\sum_{l}}-\left(\langle\hat{n}_{l}\rangle-1\right)\left(\langle\hat{n}_{l+1}\rangle-1\right)\;,$
$\displaystyle\hat{V}^{\rm F}$ $\displaystyle=$
$\displaystyle\sum_{l,\sigma}\langle\hat{c}_{l,\sigma}^{+}\hat{c}_{l+1,\sigma}^{\vphantom{+}}\rangle\hat{c}_{l,\sigma}^{\vphantom{+}}\hat{c}_{l+1,\sigma}^{+}$
(54)
$\displaystyle\hphantom{\sum_{l,\sigma}}+\hat{c}_{l,\sigma}^{+}\hat{c}_{l+1,\sigma}^{\vphantom{+}}\langle\hat{c}_{l,\sigma}^{\vphantom{+}}\hat{c}_{l+1,\sigma}^{+}\rangle$
$\displaystyle\hphantom{\sum_{l}}-\langle\hat{c}_{l,\sigma}^{+}\hat{c}_{l+1,\sigma}^{\vphantom{+}}\rangle\langle\hat{c}_{l,\sigma}^{\vphantom{+}}\hat{c}_{l+1,\sigma}^{+}\rangle\;.$

Here, where $\langle\hat{A}\rangle$ denotes the ground-state expectation value
of the operator $\hat{A}$,

$\langle\hat{A}\rangle\equiv\langle\Phi_{0}|\hat{A}|\Phi_{0}\rangle$ (55)

with $|\Phi_{0}\rangle$ as the ground state of the Hartree-Fock Hamiltonian
$\hat{H}^{\rm HF}$, see below. We make the CDW Ansatz for the order parameter

$\langle\hat{n}_{l,\sigma}\rangle=\frac{1}{2}\left(1+(-1)^{l}\Delta\right)$
(56)

with the real CDW parameter $\Delta\geq 0$, and introduce the abbreviation

$B=\langle\hat{c}_{l,\sigma}^{+}\hat{c}_{l+1,\sigma}^{\vphantom{+}}\rangle={\rm
i}b\;.$ (57)

Particle-hole symmetry implies that $B$ is purely complex at half band-
filling, i.e., $b$ is real. Note that we disregard a possible bond-order wave
(BOW) by assuming that $B$ does not alternate from site to site.

With these abbreviations, we can rewrite the Hartree-Fock interaction at half
band-filling as

$\displaystyle\hat{D}^{\rm H}$ $\displaystyle=$
$\displaystyle\frac{L}{4}\left(1-\Delta^{2}\right)+\frac{\Delta}{2}\sum_{l,\sigma}(-1)^{l}\hat{n}_{l,\sigma}\;,$
(58) $\displaystyle\hat{V}^{\rm H}$ $\displaystyle=$ $\displaystyle
L\Delta^{2}-2\Delta\sum_{l,\sigma}(-1)^{l}\hat{n}_{l,\sigma}\;,$ (59)
$\displaystyle\hat{V}^{\rm F}$ $\displaystyle=$ $\displaystyle 2Lb^{2}+{\rm
i}b\sum_{l,\sigma}\left[\hat{c}_{l,\sigma}^{+}\hat{c}_{l+1,\sigma}^{\vphantom{+}}-\hat{c}_{l+1,\sigma}^{+}\hat{c}_{l,\sigma}^{\vphantom{+}}\right].\;$
(60)

The resulting single-particle problem defines the Hartree-Fock Hamiltonian for
a possible CDW ground state

$\hat{H}^{\rm HF}=\hat{T}+U\hat{D}^{\rm H}+V\left(\hat{V}^{\rm H}+\hat{V}^{\rm
F}\right)\;.$ (61)

It has to be solved self-consistently, i.e., $\Delta$ must be chosen such that
the ground state fulfills Eq. (56).

###  A.2 Diagonalization of the Hartree-Fock Hamiltonian

In the CDW phase, the Hartree-Fock Hamiltonian is identical for both spin
species, $\hat{H}^{\rm HF}=\sum_{\sigma}\hat{H}_{\sigma}^{\rm HF}$. Dropping
the spin index we must diagonalize

$\displaystyle\hat{H}_{\rm sf}$ $\displaystyle=$
$\displaystyle\sum_{k}\epsilon(k)\hat{C}_{k}^{+}\hat{C}_{k}^{\vphantom{+}}+\left(\frac{U}{2}-2V\right)\Delta\sum_{l}(-1)^{l}\hat{n}_{l}$
(62) $\displaystyle+{\rm
i}b\sum_{l}\left[\hat{c}_{l}^{+}\hat{c}_{l+1}^{\vphantom{+}}-\hat{c}_{l+1}^{+}\hat{c}_{l}^{\vphantom{+}}\right]+C$

for spinless fermions (‘sf’), where
$C=UL/8(1-\Delta^{2})+LV\Delta^{2}/2+LVb^{2}$. In momentum space, the
Hamiltonian reads

$\displaystyle\hat{H}_{\rm sf}$ $\displaystyle=$ $\displaystyle
C+\sideset{}{{}^{\prime}}{\sum}_{k}\Bigl{[}\left(\epsilon(k)+b(k)\right)\hat{C}_{k}^{+}\hat{C}_{k}^{\vphantom{+}}$
$\displaystyle\hphantom{\sideset{}{{}^{\prime}}{\sum}_{k}\Bigl{[}}+\left(\epsilon(k+\pi)-b(k)\right)\hat{C}_{k+\pi}^{+}\hat{C}_{k+\pi}^{\vphantom{+}}\Bigr{]}$
$\displaystyle+\left(\frac{U}{2}-2V\right)\Delta\sideset{}{{}^{\prime}}{\sum}_{k}\left(\hat{C}_{k}^{+}\hat{C}_{k+\pi}^{\vphantom{+}}-\hat{C}_{k+\pi}^{+}\hat{C}_{k}^{\vphantom{+}}\right)\,,$

where the prime on the sum indicates the $k$-space region $-\pi<k<0$ and
$b(k)=-2bV\sin(k)\geq 0$.

We diagonalize $\hat{H}_{\rm sf}$ with the help of the linear transformation

$\displaystyle\hat{C}_{k}$ $\displaystyle=$ $\displaystyle
c_{k}\hat{\alpha}_{k}-s_{k}\hat{\beta}_{k}\;,$ $\displaystyle\hat{C}_{k+\pi}$
$\displaystyle=$ $\displaystyle s_{k}\hat{\alpha}_{k}+c_{k}\hat{\beta}_{k}\;,$
(64)

where we abbreviate $c_{k}\equiv\cos(\varphi_{k})$ and
$s_{k}=\sin(\varphi_{k})$. The mixed terms in $\hat{H}_{\rm sf}$ vanish when
we demand

$\tan(2\varphi_{k})=-\frac{(2V-U/2)\Delta}{b(k)+(\epsilon(k)+\epsilon(k+\pi))/2}\geq
0\;.$ (65)

The diagonal terms result in

$\hat{H}_{\rm
sf}=\sideset{}{{}^{\prime}}{\sum}_{k}E_{\alpha}(k)\hat{\alpha}_{k}^{+}\hat{\alpha}_{k}^{\vphantom{+}}+E_{\beta}(k)\hat{\beta}_{k}^{+}\hat{\beta}_{k}^{\vphantom{+}}+C$
(66)

with

$\displaystyle E_{\begin{subarray}{c}\alpha\\\\[1.0pt]
\beta\end{subarray}}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\epsilon(k)+\epsilon(k+\pi)\right)\mp s(k)$
$\displaystyle s(k)$ $\displaystyle=$
$\displaystyle\sqrt{\left[2V-\frac{U}{2}\right]^{2}\Delta^{2}+\left[b(k)+\frac{\epsilon(k)-\epsilon(k+\pi)}{2}\right]^{2}}$

so that $E_{\alpha}(k)<E_{\beta}(k)$ for all $-\pi<k<0$. Therefore, the ground
state contains only $\alpha$-particles,

$|\Phi_{0}\rangle=\prod_{-\pi<k<0}\hat{\alpha}_{k,\sigma}^{+}|{\rm
vac}\rangle\;,$ (68)

where we re-introduced the spin index.

###  A.3 Self-consistency equations and CDW transition

The self-consistency equations (56) and (57) become

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\Delta\int_{-\pi}^{0}\frac{{\rm d}k}{\pi}\frac{2V-U/2}{s(k)}\;,$
(69) $\displaystyle b$ $\displaystyle=$
$\displaystyle-\int_{-\pi}^{0}\frac{{\rm
d}k}{2\pi}\frac{\sin(k)\left[b(k)+(\epsilon(k)-\epsilon(k+\pi))/2\right]}{s(k)}$

in the thermodynamic limit. The set {$\Delta=0$, $b=-1/\pi$} provides the
solution for the Fermi sea of non-interacting particles.

Within Hartree-Fock theory, the CDW transition is continuous. We seek a
solution for $\Delta=0^{+}$ and $b=-1/\pi$ so that $V_{\rm c}(U)$ must obey
the equation

$\frac{1}{2V_{\rm c}(U)-U/2}=\int_{0}^{\pi}\frac{{\rm
d}k}{\pi}\frac{1}{1/4+2V_{\rm c}(U)\sin(k)/\pi}\;.$ (70)

Using Mathematica [36] and with the abbreviation $a_{\rm c}=8V_{\rm c}/\pi$ we
find

$\frac{1}{a_{\rm
c}-2U/\pi}=\frac{\pi}{\sqrt{1-a_{c}^{2}}}-\frac{2}{\sqrt{1-a_{c}^{2}}}\arctan\left(\frac{a_{c}}{\sqrt{1-a_{c}^{2}}}\right)\;.$
(71)

This equation must be solved numerically for given $U$. For example, at $U=0$
we find $a_{c}\approx 0.394235$ so that $V_{\rm c}(U=0)\approx 0.154816$ in
Hartree-Fock theory. The resulting curve $V_{\rm c}^{\rm HF}(U)$ is shown in
Fig. 1.

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