# Wire width and density dependence of the crossover in the peak of the static
structure factor from $2k_{\text{F}}$ $\rightarrow$ $4k_{\text{F}}$ in one-
dimensional paramagnetic electron gases

Ankush Girdhar  Department of Physics, Dr. B. R. Ambedkar National Institute
of Technology, Jalandhar, Punjab-144027, India  Vinod Ashokan
<EMAIL_ADDRESS>Department of Physics, Dr. B. R. Ambedkar National
Institute of Technology, Jalandhar, Punjab-144027, India  Rajesh O. Sharma
Department of Physics, Indian Institute of Science, Bengaluru-560012, India.
N. D. Drummond  Department of Physics, Lancaster University, Lancaster LA1
4YB, United Kingdom  K. N. Pathak  Centre for Advanced Study in Physics,
Panjab University, Chandigarh-160014, India

###### Abstract

We use the variational quantum Monte Carlo (VMC) method to study the wire
width ($b$) and electron density ($r_{\text{s}}$) dependences of the ground-
state properties of quasi-one-dimensional paramagnetic electron fluids. The
onset of a quasi-Wigner crystal phase is known to depend on electron density,
and the crossover occurs in the low density regime. We study the effect of
wire width on the crossover of the dominant peak in the static structure
factor from $k=2k_{\text{F}}$ to $k=4k_{\text{F}}$. It is found that for a
fixed electron density, in the charge structure factor the crossover from the
dominant peak occurring at $2k_{\text{F}}$ to $4k_{\text{F}}$ occurs as the
wire width decreases. Our study suggests that the crossover is due to
interplay of both $r_{\text{s}}$ and $b<r_{\text{s}}$. The finite wire width
correlation effect is reflected in the peak height of the charge and spin
structure factors. We fit the dominant peaks of the charge and spin structure
factors assuming fit functions based on our finite wire width theory and clues
from bosonization, resulting in a good fit of the VMC data. The pronounced
peaks in the charge and spin structure factors at $4k_{\text{F}}$ and
$2k_{\text{F}}$, respectively, indicate the complete decoupling of the charge
and spin degrees of freedom. Furthermore, the wire width dependence of the
electron correlation energy and the Tomonaga-Luttinger parameter $K_{\rho}$ is
found to be significant.

††preprint: APS/123-QED

##  I Introduction

Many-body correlation effects in one-dimensional (1D) electron systems give
rise to fascinating properties with diverse technological applications [1, 2].
Advances in fabrication techniques have made the realization of extremely thin
wires feasible, although only a few aspect of theoretical predictions have yet
been experimentally tested. This is due to the experimental difficulties of
realizing ideal and controllable 1D systems. There are several features of 1D
systems, such as spin-charge separation [3], power-law behavior of correlation
functions and other physical quantities such as conductance [4], charge
fractionalization [5], Wigner crystallization [6, 7, 8, 9], etc., which make
them unique and interesting. The physics of 1D electron systems cannot be
explained using Landau’s Fermi liquid theory due to the fact that single-
particle excitation energies and their inverse lifetimes are of the same order
of magnitude.

The exactly solvable model by Tomonaga and Luttinger [10, 11, 12] describes
the low-energy spectrum of a 1D homogeneous electron gas (HEG) assuming short-
range interactions and a linear dispersion relation. The assumption of short-
range interactions is quite successfully used in studies of the properties of
various 1D structures such as quasi-one-dimensional conductors where screening
between adjacent chains leads to effective short-range interactions within
each chain [13]. However, the electrons in systems which do not have effective
screening, such as isolated metallic carbon nanotubes [8], interact via the
true long-range Coulomb potential $[V(x)=e^{2}/|x|]$. For the purpose of
theoretical studies of 1D electron systems, there are several choices of
confinement model available such as hard wall [14, 15], harmonic [16, 17], and
Coulombic [18]. In all these cases, the effective electron-electron
interaction is known to have a $1/|x|$ long range tail in the lowest subband
of the transverse motion. In this study, we consider electrons interacting
through an effective long-range Coulomb interaction using a harmonic
confinement model.

1D interacting electron systems have been studied using several techniques
such as density functional theory (DFT) [18], exact diagonalization [19], the
random phase approximation (RPA) [20, 21, 22, 23, 16, 24], density matrix
renormalization group (DMRG) [25, 26], quantum Monte Carlo (QMC) [27, 28, 29,
30, 31], lattice regularized diffusion Monte Carlo (LRDMC) [32, 33] and
bosonization [34]. The latter was used by Schulz in an interesting paper [34]
studying a 1D electron gas with long range interactions assuming a linearized
kinetic energy dispersion. The major finding of the paper was an extremely
slowly decaying $4k_{\text{F}}$ component in the charge-charge correlations,
signifying the presence of a one-dimensional Wigner crystal state even for the
weakest long-range Coulomb interaction. However, it is not clear how the
finite width of a wire and the coupling strength will affect the Wigner
crystallization. In fact, one of the purposes of this paper is to find
possible answers to this question. At the same time, in the spin correlation
function, a finite $2k_{\text{F}}$ peak was obtained whereas the
$4k_{\text{F}}$ peak was absent.

The experimental observation of a Wigner crystal is a tricky task since it
requires extreme conditions such as low temperatures and low densities [9]. In
addition, we show in this paper that it also requires small wire widths.
Examining such a fragile crystal requires the use of noninvasive imaging
probes. In case of 1D infinite systems, true long-range order is forbidden due
to thermal and quantum fluctuations. However, in finite systems, physicists
have studied 1D Wigner crystals and produced crystalline correlations due to
quasi-long-range order. Introducing long-range Coulomb interactions results in
a $4k_{\text{F}}$ oscillation in charge correlations whose decay rate is
slower than any power law [34]. The experimental realization of long-range
interactions was made possible in a recent work by Shapir et al. [8], where a
few electrons ($<10$) were housed in a carbon nanotube. A second nanotube was
used as a scanning probe to measure the spatial charge distribution of
electrons in the first nanotube. The measured charge densities were found to
exhibit the features of a quasi-Wigner crystal. There is no critical density
associated with the onset of the 1D Wigner crystal.

Our aim in this work is to study the effects of confinement and electron
correlations on the ground-state properties of a paramagnetic harmonic wire.
Unlike our previous work in Ref. 31 we investigate the emergence of quasi-
Wigner crystal phases at low density and we examine crossover in the position
of the dominant peak in the static structure factor for various wire widths at
low density. We employ VMC technique to calculate ground-state properties of
the system under consideration. The paper is structured as follows. The model
for confinement of electrons in 1D is described in Sec. II. The energies for
the paramagnetic harmonic wire are presented in Sec. III.1. The dependence of
the $2k_{\text{F}}\rightarrow 4k_{\text{F}}$ crossover on wire width is
discussed in detail in Sec. III.2. In Sec. III.3, we present MDs and Tomonaga-
Luttinger (TL) parameters $K_{\rho}$ extracted from the MD data around $k\sim
k_{\text{F}}$ as a function of wire width. Our conclusions are drawn in Sec.
IV. Hartree atomic units ($\hbar=|e|=m_{\text{e}}=4\pi\epsilon_{0}=1$) are
used throughout this paper.

##  II Model and calculation details

The form of Hamiltonian used for simulating a 1D electron fluid is

$\displaystyle\hat{H}=-\frac{1}{2}\sum_{i=1}^{N}\frac{\partial^{2}}{\partial
x_{i}^{2}}+\sum_{i<j}\tilde{V}(x_{ij})+\frac{N}{2}V_{\rm Mad},$ (1)

where $\tilde{V}(x_{ij})$ and $V_{\rm Mad}$ describe the Ewald interaction and
Madelung energy, respectively. The Ewald-like interaction for a harmonic wire
in a periodic cell of length $L$ is [35, 29]

$\displaystyle\tilde{V}(x_{ij})$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{m=-\infty}\bigg{[}\frac{\sqrt{\pi}}{2b}e^{{(x_{ij}-mL)}^{2}/{(4b)}^{2}}{\rm
erfc}\bigg{(}\frac{|x_{ij}-mL|}{2b}\bigg{)}$ (2) $\displaystyle\hskip
40.00006pt{}-\frac{1}{|x_{ij}-mL|}{\rm
erf}\bigg{(}\frac{|x_{ij}-mL|}{2b}\bigg{)}\bigg{]}$
$\displaystyle{}+\frac{2}{L}\sum^{\infty}_{n=1}{\rm
E}_{1}[{(bGn)}^{2}]\cos(Gnx_{ij}),$

where $G=2\pi/L$, $b$ is the wire width, and ${\rm E}_{1}$ is the exponential
integral function.

We have considered a 1D HEG in a simulation cell of length $L=2Nr_{\text{s}}$
subject to periodic boundary conditions, with $N$ being the total number of
electrons in the periodic cell. We keep $N^{\uparrow}=N^{\downarrow}$ odd to
avoid shell degeneracy effects. A Slater-Jastrow-backflow wave function was
used in the VMC calculations [36, 37]. The orbitals in the Slater determinant
were chosen to be plane waves with wave vectors up to
$k_{\text{F}}=\pi/(4r_{\text{s}})$ for the paramagnetic systems. The VMC
method as implemented in the casino code [38] was used to calculate ground-
state expectation values. More simulation details can be found in our previous
works [29, 30, 31].

In a 1D electron gas with an electron-electron interaction potential that is
appropriate for a finite wire of width $b$, opposite-spin electrons have a
small but nonzero contact probability density. By contrast, in an infinitely
thin 1D electron gas, the contact probability density is zero for both same-
spin and opposite-spin electrons. QMC calculations for 1D electron gases with
finite-width interactions are challenging because the width introduces
features on the small length scale $b$ into the wave function near coalescence
points. Typically $b$ will be much smaller than other length scales in the
problem, such as the density parameter $r_{\text{s}}$.

One issue that affects diffusion quantum Monte Carlo calculations of finite-
width wires is ergodicity; to explore all of configuration space, opposite-
spin electrons have to be able to pass one another. However the “quasinodes”
at opposite-spin coalescence points lead to a very low (but nonzero)
probability of electrons passing each other, resulting in very long
correlation times. This is much less of an issue in VMC, where there is no
need to work in the limit of small time steps, so that electron moves past
coalescence points are permitted. A second issue is simply the challenge of
creating and optimizing a trial wave function that contains features on the
small length scale $b$ as well as the large length scale $L$ of the simulation
cell.

##  III Results and discussion

###  III.1 Energies

Figure 1:  Correlation energy per particle ($E_{\text{c}}$) as a function of
wire width $b$ for $r_{\text{s}}$ = 0.5, 1, and 2 (top to bottom).

The ground-state energy per electron for the harmonic wire is calculated for
$r_{\text{s}}=0.5$, 1, 2, and 5 and for $b=0.1$, 0.3, 0.5, 0.7, 1, 2 a.u. We
calculate the energies for $N=30$, 50, 74, 98 electrons and then extrapolating
them to obtain the thermodynamic limit. The ground state energy is found to
scale with system size as $E(N)=E_{\infty}+BN^{-2}$ [29] where $B$ and
$E_{\infty}$ are fitting parameters. We also calculate the correlation energy
per electron using the ground-state energy. These energies are tabulated in
Table 1 of Appendix A. The wire width dependence of correlation energy per
particle for different values of density parameter $r_{\text{s}}$ is drawn in
Fig. 1.

###  III.2 Structure and correlation function

Figure 2:  Charge structure factor $S_{\rho\rho}(k)$ plotted against
$k/k_{\text{F}}$ (a)–(c) and charge-charge pair correlation function plotted
against $r/r_{\text{s}}$ (d)–(f) for $r_{\text{s}}=0.5$, 2, and 5 for various
$b$ values with $N=98$ electrons. The insets in (b) and (c) show the peak at
$k=4k_{\text{F}}$ in greater detail. Figure 3:  Spin structure factor
$S_{\sigma\sigma}(k)$ plotted against $k/k_{\text{F}}$ (a)–(c) and spin-spin
pair correlation function plotted against $r/r_{\text{s}}$ (d)–(f) for
$r_{\text{s}}=0.5$, 1, and 2 for various $b$ values with $N=98$ electrons. The
insets (a), (b), and (c) show the peak at $k=2k_{\text{F}}$ in greater detail.

Figure 4:  (Top) Charge SSF peak height at $k=4k_{\text{F}}$ as a function of
wire width $b$ for $r_{\text{s}}=0.5$, 1, and 2 for $N=98$ electrons. The peak
heights at $k=4k_{\text{F}}$ are fitted by Eq. (7) and the fits are shown by
solid lines, whereas the $2k_{\text{F}}$ peak heights are joined by dashed
lines. (Bottom) Height of spin SSF plotted against $b$ for $r_{\text{s}}=0.5$,
1, and 2 for $N=98$ electrons and fitted by Eq. (III.2).

We calculate the VMC charge and spin static structure factors (SSFs), which
are defined as

$\displaystyle S_{\rho\rho}(k)$ $\displaystyle=$ $\displaystyle
S_{\uparrow\uparrow}+S_{\uparrow\downarrow}$ (3) $\displaystyle
S_{\sigma\sigma}(k)$ $\displaystyle=$ $\displaystyle
S_{\uparrow\uparrow}-S_{\uparrow\downarrow},$ (4)

respectively, for a paramagnetic quantum wire. Here $S_{\uparrow\uparrow}$ and
$S_{\uparrow\downarrow}$ are parallel-spin and antiparallel-spin SSFs. The SSF
is a crucial property of electron fluids that is used to look into the
structural properties of the electron system in reciprocal space. The charge
SSF, also known as the total SSF, describes correlations between electron
pairs irrespective of their spins. The spin PCF describes the difference
between pair correlations for parallel- and antiparallel-spin electron pairs.
To study the charge-charge and spin-spin correlation functions in real space,
we define alternative quantities, namely charge and spin pair correlation
functions (PCFs), as

$\displaystyle g_{\rho\rho}$ $\displaystyle=$
$\displaystyle\frac{g_{\uparrow\uparrow}+g_{\uparrow\downarrow}}{2}$ (5)
$\displaystyle g_{\sigma\sigma}$ $\displaystyle=$
$\displaystyle\frac{g_{\uparrow\uparrow}-g_{\uparrow\downarrow}}{2},$ (6)

respectively. Here $g_{\uparrow\uparrow}$ and $g_{\uparrow\downarrow}$ are
parallel-spin and antiparallel-spin PCFs, respectively. A peak at
$k=4k_{\text{F}}$ in the charge structure factor corresponds to slowly
decaying oscillations of period $2r_{\text{s}}$, which is the average
interparticle spacing. This is a signature of crossover to a ordered phase
known as a Wigner crystal. For the case of ferromagnetic infinitely thin
wires, the singularity at $k=4k_{\text{F}}$ is absent for densities
$r_{\text{s}}<15$ a.u., but a small feature at $4k_{\text{F}}$ starts to
develop at $r_{\text{s}}\approx 15$ [29]. The effect of singularity at
$4k_{\text{F}}$ has been observed recently [39] for the case of electron-
electron biwires, where the parallel wire effectively provides an extra spin
degree of freedom to electrons. In the case of a paramagnetic quasi-quantum
wire, it has been verified in previous works [27, 28] that the dominant
$4k_{\text{F}}$ peak height grows sublinearly with system size $N$ at low
density, which is a signature of a quasi-Wigner crystal. The adjective ‘quasi’
has been used to emphasize the absence of true long-range order due to quantum
fluctuations. Further, it has been found that for any fixed wire width, this
crossover occurs as one decreases the electron density (i.e., low density with
large values of $r_{\text{s}}$). In this work, we have observed a similar
correlation effect by decreasing the wire width at fixed coupling parameter
$r_{\text{s}}$.

The charge structure factor $S_{\rho\rho}(k)$ is plotted for several values of
$b$ and $r_{\text{s}}$ for $N=98$ electrons in Fig. 2. In Fig. 2(a),
$S_{\rho\rho}(k)$ shows a peak at $k=2k_{\text{F}}$ and no peak at
$4k_{\text{F}}$ is present. This leads to oscillations of period
$4r_{\text{s}}$ in charge-charge PCF, as can be seen in Fig. 2(d). It is
evident from Fig. 2(b) that as $b$ decreases for $r_{\text{s}}=2$, the
$4k_{\text{F}}$ peak grows and the $2k_{\text{F}}$ peak is suppressed,
indicating the crossover ($2k_{\text{F}}\rightarrow 4k_{\text{F}}$) to an
ordered phase. This crossover leads to a change in the period of the
oscillations in the charge-charge PCF from $4r_{\text{s}}$ to $2r_{\text{s}}$,
as apparent in Fig. 2(e). Further, it can be noted from Fig. 2(c) that the
$2k_{\text{F}}$ peak is absent, but decreasing $b$ leads to an increase in the
height of the $4k_{\text{F}}$ peak. This is also reflected in a more
structured behavior of $g_{\rho\rho}$, with oscillation period $2r_{\text{s}}$
in Fig. 2(f).

The spin structure factor $S_{\sigma\sigma}(k)$ is plotted for several values
of $b$ and $r_{\text{s}}$ for $N=98$ electrons in Fig. 3.
$S_{\sigma\sigma}(k)$ is found to show a peak at $k=2k_{\text{F}}$, which
grows as $b$ decreases for all the densities considered [see Figs. 3(a)–(c)].
This is followed by the spin-spin PCF in Figs. 3(d)–(f), where the amplitude
of oscillations of period $4r_{\text{s}}$ is enhanced as $b$ decreases. It can
also be noted from Figs. 3(a)–(c) that the peak height increases with
increasing $r_{\text{s}}$ for a given value of $b$. In the present work, we
have observed complete decoupling of the charge and spin degrees of freedom in
the SSFs. For example, in $r_{\text{s}}=2$, there is a crossover from
$2k_{\text{F}}\rightarrow 4k_{\text{F}}$ behavior, with complete disappearance
of $2k_{\text{F}}$ peak at $b=0.5$, in the charge SSF [Fig. 2(b)], while for
the same values of $b$ and $r_{\text{s}}$ the spin SSF shows a peak at
$k=2k_{\text{F}}$ [see Fig. 3(c)]. The spin and charge excitations in the SSFs
manifest the complete separation of spin and charge of electrons due to strong
Coulomb correlation effects induced by the small transverse confinement width.

The crossover observed is between (i) a state in which opposite spin electrons
can easily slide past each other (large $b/r_{\text{s}}$; in this case
oscillations in both charge and spin PCFs have period $4r_{\text{s}}$, because
correlation is only important for same-spin electrons and negligible for
opposite-spin pairs) and (ii) a state in which opposite spin electrons cannot
slide past each other (small $b/r_{\text{s}}$; in this case the system is like
a floating antiferromagnetic crystal lattice; the spin PCF has period
$4r_{\text{s}}$, whereas the charge PCF has period $2r_{\text{s}}$). The
“fluid to crystal” crossover can be visualized in terms of the behavior of
interlocking lattices of spin-up electrons and spin-down electrons, with the
crossover occurring at the point at which the lattices start being able to
slide past each other.

On the basis of our finite wire width theory [31], the SSF is a function of
$b/r_{\text{s}}$. In view of this, and clues from bosonization theory [34,
25], we propose that the dependence of the peak heights
$S_{\rho\rho}(4k_{\text{F}},b)$ and $S_{\sigma\sigma}(2k_{\text{F}},b)$ on
wire width can be represented as

$S_{\rho\rho}(4k_{\text{F}},b)=a_{0}\frac{L\;r_{\text{s}}}{b}\exp\left(-4c\sqrt{\ln\frac{L\;r_{\text{s}}}{b}}\right)+a_{1},$
(7)

and

$\displaystyle S_{\sigma\sigma}(2k_{\text{F}},b)=$ $\displaystyle-
a_{2}\left(\sqrt{\ln\frac{L\;r_{\text{s}}}{b}}+\frac{1}{c}\right)\exp\left(-c\sqrt{\ln\frac{L\;r_{\text{s}}}{b}}\right)$
$\displaystyle+a_{3}$ (8)

where $a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$, and $c$ are fitting parameters and
$L=2Nr_{\text{s}}$.

Finite wire width effects are studied in Fig. 4. The $4k_{\text{F}}$ and
$2k_{\text{F}}$ peak heights in the charge SSF are plotted against wire width
in Fig. 4 (top) for $r_{\text{s}}=0.5$, 1, and 2. The dependence of the peak
heights on $b$ is found to be significant for $r_{\text{s}}=1$ and 2, and for
these coupling parameters both the $4k_{\text{F}}$ and $2k_{\text{F}}$ peaks
are present. On the other hand, for $r_{\text{s}}=0.5$, there is no
$4k_{\text{F}}$ peak present for any of the wire widths considered and even
the $2k_{\text{F}}$ peak height depends only weakly on $b$ (at least until
$b\ll r_{\text{s}}$). The $4k_{\text{F}}$ peak heights have been fitted to Eq.
(7), which was deduced on the basis of bosonization and which shows a good
representation of the numerically calculated data. The wire width dependence
of the $2k_{\text{F}}$ peak height in the spin SSF is plotted in Fig. 4
(bottom) for $r_{\text{s}}=0.5$, 1, and 2. It is observed that the peak height
in the spin SSF depends on both $r_{\text{s}}$ and $b$. It is also observed
that for a fixed value of electron coupling, the amplitude of the peak height
at $k=2k_{\text{F}}$ in the spin SSF strongly depends on the confinement
width. Further, the $2k_{\text{F}}$ peak heights are well fitted by Eq.
(III.2).

It should be noted from Fig. 4 that the SSF of the finite-width paramagnetic
wire is not expected to converge to that of a ferromagnetic infinitely thin
wire in the limit $b\rightarrow 0$, because the latter has only parallel spin
correlations whereas the former has both parallel and antiparallel spin
correlations. Further, the quasi-long-range order in ferromagnetic and
paramagnetic electron gases is represented by dominant peaks in the charge
structure factor $S_{\rho\rho}(k)$ at $k=2k_{\text{F}}$ and $k=4k_{\text{F}}$,
respectively, when $b$ is small. The dominant peaks in $S_{\rho\rho}(k)$ for
these two cases correspond to the same $2r_{\text{s}}$ oscillation period in
the charge-charge PCF. A nonmonotonic shift of the dominant peak from
$4k_{\text{F}}$ to $2k_{\text{F}}$ for $b\rightarrow 0$ in a paramagnetic wire
is not therefore expected. This behavior is consistent with Eq. (7), the model
formula obtained from the bosonization prediction [34, 25]. It is known that
the Lieb-Mattis theorem is not applicable for infinitely thin wires, and in
fact the ferromagnetic and paramagnetic states are energetically degenerate in
this limit [29]. However, physical observables which do not commute with the
Hamiltonian, such as the SSF, need not be same for infinitely thin
paramagnetic and ferromagnetic wires.

Figure 5:  MD against $k/k_{\text{F}}$ for different values of wire width $b$
and $r_{\text{s}}$ for $N=98$ electrons.

###  III.3 Momentum density and TL parameters

TL theory suggests that the 1D MD has a peculiar power-law behavior, which is
continuous at $k_{\text{F}}$ with a singular first-order derivative and takes
the form $n(k)=n(k_{\text{F}})+A[{\rm
sgn}(k-k_{\text{F}})]|k-k_{\text{F}}|^{\alpha}$ close to $k=k_{\text{F}}$,
where $n(k_{\text{F}})$, $A$, and $\alpha$ are constants [11, 40]. The method
used to estimate the interaction exponent $\alpha$ is discussed in great
detail in our previous works [29, 30, 31].

In Fig. 5, the MD is plotted for $N=98$ electrons at several values of $b$ and
$r_{\text{s}}$. The effect of wire width on the behavior of the MD is clearly
visible in the figure. It can be seen that as $b$ decreases or $r_{\text{s}}$
increases, the power-law behavior becomes more appreciable. It can also be
noted that the value of $n(k=0)$ decreases from 1 and the value of $n(k)$
increases for $k>k_{\text{F}}$ as $b$ decreases or $r_{\text{s}}$ increases.

We also compare our QMC MD data with bosonization results [34, 25], finding a
good fit of the jump in the MD at $k=k_{\text{F}}$ at finite size by the
asymptotic scaling function predicted by bosonization theory:

$\Delta n(k_{\text{F}})\sim Le^{-c{(\ln L)}^{3/2}}.$ (9)

Figure 6 shows that Eq. (9) provides quite a good numerical representation of
the system-size scaling of the jump in the MD at $k=k_{\text{F}}$.

The TL parameter $K_{\rho}$ for repulsive interactions ($K_{\rho}<1$) and
exponent $\alpha$ are plotted against $b$ for $r_{\text{s}}=0.5$, 1, and 2 in
Fig. 7. It has been observed that as $b$ decreases or $r_{\text{s}}$
increases, $K_{\rho}$ decreases and the interacting electrons become more
strongly correlated. Hence it is found that $K_{\rho}$ significantly depends
on the confinement width $b$ and $r_{\text{s}}$.

Figure 6:  Jump in the MD at $k=k_{\text{F}}$ at finite size, plotted against
system size and fitted by $\Delta n(k_{\text{F}})=aLe^{-c{(\ln L)}^{3/2}}$.
The fitted parameter values are $a=1.03(2)$ a.u. and $c=0.731(2)$ for
$r_{\text{s}}=1$ and $b=1$; and $a=0.93(5)$ a.u. and $c=0.777(7)$ for
$r_{\text{s}}=2$ and $b=2$. Figure 7:  TL parameter $K_{\rho}$ against wire
width $b$ for different values of the density parameter $r_{\text{s}}$ as
indicated in the plot. The TL parameter is obtained by fitting MD data to
appropriate fitting functions to obtain the exponent $\alpha$ and $K_{\rho}$.
The exponent $\alpha$ is plotted against $b$ in the inset.

##  IV Conclusions

We have studied the wire width dependence of the ground-state properties of
quasi-1D quantum wires for several electron densities. We analyze the effects
of adjusting the wire width on the charge structure factor, where a crossover
from a peak at $2k_{\text{F}}$ to a peak at $4k_{\text{F}}$ was observed as
$b$ decreases. The crossover indicates the onset of a quasi-Wigner crystalline
state in the 1D electron fluid. The PCF is also found to show a more
structured behavior as $b$ decreases, which again suggests a crossover to a
quasi-Wigner crystal phase. We find the crossover does not appear for all
values of $r_{\text{s}}$ and $b$, a result at variance with the bosonization
findings of Schulz in Ref. [34]. We also fit the dominant peaks of the charge
and spin structure factors assuming fitting functions on the basis of our
finite wire width theory and clues from bosonization, giving a good fit of the
VMC data. The crossover occurs only for those density and wire width values
which bring strong correlations in the system.

The correlation energy as a function of $b$ and $r_{\text{s}}$ is represented
by a formula which fits our VMC data perfectly. The MD $n(k)$ is found to show
a power law behavior. We find that $n(k)$ at $k=0$ reduces from $1$, whereas
$n(k)$ increases beyond $k_{\text{F}}$ as $b$ decreases or $r_{\text{s}}$
increases. We find the TL liquid exponent $\alpha$ by fitting the MD data to
an appropriate fitting function near $k_{\text{F}}$. The TL parameter
$K_{\rho}$ is found to depend significantly on the wire width at a constant
$r_{\text{s}}$.

###### Acknowledgements.

One of the authors (A.G.) would like to thank MHRD for financial support.
K.N.P. acknowledges the National Academy of Sciences of India, Prayagraj for
partial financial assistance. R.S. acknowledges SERB for providing NPDF grant
PDF/2021/000546. V.A. acknowledges financial support from DST-SERB Grant No.
EEQ/2019/000528, and the National PARAM Supercomputing Facility at C-DAC Pune
for computational resources.

## Appendix A

The thermodynamic limit of the VMC ground state energy per electron
$E_{\infty}$ and the correlation energy per electron $E_{\text{c}}$ are
calculated by extrapolation from calculations performed at several different
values of system size $N$ at different values of $b$ and $r_{\text{s}}$; these
are tabulated in Table 1.

Table 1:  VMC energies in a.u. per electron [$E(N)$] for $N=30$, 50, 74, and 98 for a paramagnetic harmonic wire. $E_{\infty}$ and $E_{\text{c}}$ denote the ground-state energy per electron extrapolated to the thermodynamic limit and the correlation energy per electron respectively. ($r_{\text{s}},b$) | $E(30)$ | $E(50)$ | $E(74)$ | $E(98)$ | $E_{\infty}$ | $E_{\text{c}}$
---|---|---|---|---|---|---
(0.5, 0.1) |  $-$0.937394(8) |  $-$0.935599(7) |  $-$0.935042(5) |  $-$0.934791(5) |  $-$0.93455(2) |  $-$0.14477(2)
(0.5, 0.3) |  $-$0.336230(3) |  $-$0.334478(2) |  $-$0.333931(2) |  $-$0.333725(2) |  $-$0.333473(7) |  $-$0.032178(7)
(0.5, 0.5) |  $-$0.127807(2) |  $-$0.126094(1) |  $-$0.1255490(9) |  $-$0.1253469(9) |  $-$0.125100(9) |  $-$0.014176(9)
(0.5, 0.7) |  $-$0.0145572(9) |  $-$0.0128579(7) |  $-$0.0123206(6) |  $-$0.0121214(5) |  $-$0.011877(8) |  $-$0.008008(8)
(0.5, 1.0) | 0.0853158(5) | 0.0869805(4) | 0.0875067(3) | 0.0877014(3) | 0.087942(8) |  $-$0.004262(8)
(0.5, 2.0) | 0.225663(2) | 0.2272236(1) | 0.22772541(9) | 0.22791182(8) | 0.228151(5) |  $-$0.001176(5)
(1.0, 0.1) |  $-$0.899453(5) |  $-$0.898910(4) |  $-$0.898695(3) |  $-$0.898505(3) |  $-$0.89847(6) |  $-$0.23295(6)
(1.0, 0.3) |  $-$0.478440(2) |  $-$0.477901(2) |  $-$0.477725(2) |  $-$0.477671(1) |  $-$0.477589(4) |  $-$0.074459(4)
(1.0, 0.5) |  $-$0.328023(1) |  $-$0.327485(1) |  $-$0.327323(1) |  $-$0.3272540(9) |  $-$0.327179(4) |  $-$0.036280(4)
(1.0, 0.7) |  $-$0.245460(1) |  $-$0.2449252(8) |  $-$0.2447628(7) |  $-$0.2446985(6) |  $-$0.244622(2) |  $-$0.021498(2)
(1.0, 1.0) |  $-$0.1710782(7) |  $-$0.1705563(5) |  $-$0.1703899(4) |  $-$0.1703292(4) |  $-$0.170254(3) |  $-$0.011984(3)
(1.0, 2.0) |  $-$0.0610819(3) |  $-$0.0605826(2) |  $-$0.0604240(2) |  $-$0.0603641(1) |  $-$0.060292(3) |  $-$0.003585(3)
(2.0, 0.1) |  $-$0.687882(6) |  $-$0.687592(7) |  $-$0.687429(6) |  $-$0.687309(2) |  $-$0.68726(3) |  $-$0.24309(3)
(2.0, 0.3) |  $-$0.428893(1) |  $-$0.428709(1) |  $-$0.4286485(9) |  $-$0.4285848(9) |  $-$0.42858(2) |  $-$0.11954(2)
(2.0, 0.5) |  $-$0.321088(1) |  $-$0.3209185(8) |  $-$0.3208670(7) |  $-$0.3208426(6) |  $-$0.320819(2) |  $-$0.072522(2)
(2.0, 0.7) |  $-$0.2583233(8) |  $-$0.2581581(7) |  $-$0.2581048(6) |  $-$0.2580852(5) |  $-$0.258061(1) |  $-$0.048212(1)
(2.0, 1.0) |  $-$0.2006410(7) |  $-$0.2004732(5) |  $-$0.2004218(5) |  $-$0.2004024(4) |  $-$0.2003781(3) |  $-$0.0292266(3)
(2.0, 2.0) |  $-$0.1146782(3) |  $-$0.1145145(2) |  $-$0.1144626(2) |  $-$0.1144422(2) |  $-$0.114419(1) |  $-$0.009582(1)
(5.0, 0.1) |  $-$0.393883(2) |  $-$0.393808(3) |  $-$0.393718(2) |  $-$0.3935256(8) |  $-$0.39352(6) |  $-$0.16401(6)
(5.0, 0.3) |  $-$0.284488(3) |  $-$0.283437(9) |  $-$0.284425(1) |  $-$0.2843393(4) |  $-$0.28433(5) |  $-$0.10955(5)
(5.0, 0.5) |  $-$0.234260(4) |  $-$0.234304(6) |  $-$0.234296(8) |  $-$0.2342142(4) |  $-$0.23420(3) |  $-$0.08464(3)
(5.0, 0.7) |  $-$0.201875(3) |  $-$0.2019512(7) |  $-$0.2019229(7) |  $-$0.2019149(2) |  $-$0.201911(9) |  $-$0.068787(9)
(5.0, 1.0) |  $-$0.168710(3) |  $-$0.1687108(6) |  $-$0.1686853(6) |  $-$0.1686792(2) |  $-$0.168670(4) |  $-$0.052681(4)
(5.0, 2.0) |  $-$0.1102775(8) |  $-$0.1102383(7) |  $-$0.1102273(5) |  $-$0.1102212(1) |  $-$0.1102150(4) |  $-$0.0260165(4)

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