arXiv:2112.12064 · Physical review. B./Physical review. B

Electron correlation and confinement effects in quasi-one-dimensional quantum wires at high density

Ankush Girdhar, Vinod Ashokan, N. D. Drummond, Klaus Morawetz, K. N. Pathak

Published Mar 30, 2022• 9 citations

Abstract

We study the ground-state properties of ferromagnetic quasi-one-dimensional quantum wires using the quantum Monte Carlo (QMC) method for various wire widths $b$ and density parameters ${r}_{\text{s}}$. The correlation energy, pair-correlation function, static structure factor, and momentum density are calculated at high density. It is observed that the peak in the static structure factor at $k=2{k}_{\text{F}}$ grows as the wire width decreases. We obtain the Tomonaga-Luttinger liquid parameter ${K}_{\ensuremath{\rho}}$ from the momentum density. It is found that ${K}_{\ensuremath{\rho}}$ increases by about 10% between wire widths $b=0.01$ and $b=0.5$. We also obtain ground-state properties of finite-thickness wires theoretically using the first-order random phase approximation (RPA) with exchange and self-energy contributions, which is exact in the high-density limit. Analytical expressions for the static structure factor and correlation energy are derived for $b\ensuremath{\ll}{r}_{\text{s}}<1$. It is found that the correlation energy varies as ${b}^{2}$ for $b\ensuremath{\ll}{r}_{\text{s}}$ from its value for an infinitely thin wire. It is observed that the correlation energy depends significantly on the wire model used (harmonic versus cylindrical confinement). The first-order RPA expressions for the structure factor, pair-correlation function, and correlation energy are numerically evaluated for several values of $b$ and ${r}_{\text{s}}\ensuremath{\le}1$. These are compared with the QMC results in the range of applicability of the theory.

DOI: 10.1103/physrevb.105.115140

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arXiv:2112.12064 · Physical review. B./Physical review. B

Electron correlation and confinement effects in quasi-one-dimensional quantum wires at high density

Ankush Girdhar, Vinod Ashokan, N. D. Drummond, Klaus Morawetz, K. N. Pathak

Published Mar 30, 2022• 9 citations

arXiv HTML PDF

Abstract

We study the ground-state properties of ferromagnetic quasi-one-dimensional quantum wires using the quantum Monte Carlo (QMC) method for various wire widths

b

and density parameters

{r}_{\text{s}}

. The correlation energy, pair-correlation function, static structure factor, and momentum density are calculated at high density. It is observed that the peak in the static structure factor at

k=2{k}_{\text{F}}

grows as the wire width decreases. We obtain the Tomonaga-Luttinger liquid parameter

{K}_{\ensuremath{\rho}}

from the momentum density. It is found that

{K}_{\ensuremath{\rho}}

increases by about 10% between wire widths

b=0.01

and

b=0.5

. We also obtain ground-state properties of finite-thickness wires theoretically using the first-order random phase approximation (RPA) with exchange and self-energy contributions, which is exact in the high-density limit. Analytical expressions for the static structure factor and correlation energy are derived for

b\ensuremath{\ll}{r}_{\text{s}}<1

. It is found that the correlation energy varies as

{b}^{2}

for

b\ensuremath{\ll}{r}_{\text{s}}

from its value for an infinitely thin wire. It is observed that the correlation energy depends significantly on the wire model used (harmonic versus cylindrical confinement). The first-order RPA expressions for the structure factor, pair-correlation function, and correlation energy are numerically evaluated for several values of

b

and

{r}_{\text{s}}\ensuremath{\le}1

. These are compared with the QMC results in the range of applicability of the theory.

DOI: 10.1103/physrevb.105.115140

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Dynamical response theory is employed to investigate the effects of various transverse confinements on electron correlations in the ground state of a ferromagnetic one-dimensional quantum wire for different wire widths b and density parameters r s . In the regime of a thin quantum wire, electrons are treated as a one-dimensional gas under different confinement models via effective electron-electron interaction potentials. Using the first-order random phase approximation (FRPA) including self-energy and exchange contribution, which provides the ground-state structure beyond the random phase approximation, we numerically compute the structure factor, pair-correlation function, correlation energy, and ground-state energy for various values of b and r s . Our results reveal that the correlation energy depends on the choice of confinement model. For the ultrathin wire ( b → 0 ) in the high-density limit, we find that the correlation energy for transverse confinement models V 1 ( q ) (harmonic), V 2 ( q ) (cylindrical), and V 5 ( q ) (harmonic-delta) approaches ε c ( r s ) = − π 2 / 360 ∼ − 0.02741 a.u., which agrees with the exact results in this limit [P.-F. Loos, ; V. Ashokan, , ]. This clearly illustrates that for at least these three confinement potentials, the one-dimensional Coulomb potential can be regularized at interparticle distance x = 0 to yield the same correlation energy. In contrast, other confinement potentials, V 3 ( q ) (infinite square well), V 4 ( q ) (infinite square-infinite triangular well), and V 6 ( q ) (infinite square-delta well), do not approach the same high-density limit; instead, the correlation energy tends to ε c ∼ − 0.03002 a.u. for these potentials. The percentage difference in correlation energy between the confinement models V 1 ( q ) , V 2 ( q ) , V 5 ( q ) and V 3 ( q ) ,Dynamical response theory is employed to investigate the effects of various transverse confinements on electron correlations in the ground state of a ferromagnetic one-dimensional quantum wire for different wire widths b and density parameters r s . In the regime of a thin quantum wire, electrons are treated as a one-dimensional gas under different confinement models via effective electron-electron interaction potentials. Using the first-order random phase approximation (FRPA) including self-energy and exchange contribution, which provides the ground-state structure beyond the random phase approximation, we numerically compute the structure factor, pair-correlation function, correlation energy, and ground-state energy for various values of b and r s . Our results reveal that the correlation energy depends on the choice of confinement model. For the ultrathin wire ( b → 0 ) in the high-density limit, we find that the correlation energy for transverse confinement models V 1 ( q ) (harmonic), V 2 ( q ) (cylindrical), and V 5 ( q ) (harmonic-delta) approaches ε c ( r s ) = − π 2 / 360 ∼ − 0.02741 a.u., which agrees with the exact results in this limit [P.-F. Loos, ; V. Ashokan, , ]. This clearly illustrates that for at least these three confinement potentials, the one-dimensional Coulomb potential can be regularized at interparticle distance x = 0 to yield the same correlation energy. In contrast, other confinement potentials, V 3 ( q ) (infinite square well), V 4 ( q ) (infinite square-infinite triangular well), and V 6 ( q ) (infinite square-delta well), do not approach the same high-density limit; instead, the correlation energy tends to ε c ∼ − 0.03002 a.u. for these potentials. The percentage difference in correlation energy between the confinement models V 1 ( q ) , V 2 ( q ) , V 5 ( q ) and V 3 ( q ) ,

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Wigner crystallization in quasi-one-dimensional quantum wire

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Abstract We study the ground-state properties of quasi-one-dimensional paramagnetic electron fluid using variational quantum Monte Carlo method. The electrons are transversally confined using the harmonically regularized Coulomb potential with long range order. In this work we have investigated the crossover from liquid to crystalline phase in the finite width paramagnetic quantum wire. The calculated pair correlation function shows the oscillations of period $2r_\text {s} a_0$ 2 r s a 0 , where $a_0$ a 0 is the Bohr’s radius and $r_\text {s}$ r s is Wigner Seitz radius or electron-density parameter which corresponds to a $k=4k_\text {F}$ k = 4 k F peak with $k_\text {F}$ k F being Fermi wave vector in the static structure factor at finite electron density. It is a signature of the onset of Wigner crystallization. It is found that the cross-over from liquid to a quasi-Wigner phase is due to enhancement of the electron correlation effects as $r_\text {s}$ r s increases or as the wire width b decreases.Abstract We study the ground-state properties of quasi-one-dimensional paramagnetic electron fluid using variational quantum Monte Carlo method. The electrons are transversally confined using the harmonically regularized Coulomb potential with long range order. In this work we have investigated the crossover from liquid to crystalline phase in the finite width paramagnetic quantum wire. The calculated pair correlation function shows the oscillations of period $$2r_\text {s} a_0$$ 2 r s a 0 , where $$a_0$$ a 0 is the Bohr’s radius and $$r_\text {s}$$ r s is Wigner Seitz radius or electron-density parameter which corresponds to a $$k=4k_\text {F}$$ k = 4 k F peak with $$k_\text {F}$$ k F being Fermi wave vector in the static structure factor at finite electron density. It is a signature of the onset of Wigner crystallization. It is found that the cross-over from liquid to a quasi-Wigner phase is due to enhancement of the electron correlation effects as $$r_\text {s}$$ r s increases or as the wire width b decreases.

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We present calculations of the energy, pair-correlation function (PCF), static structure factor (SSF), and momentum density (MD) for the one-dimensional electron gas using the quantum Monte Carlo method. We are able to resolve peaks in the SSF at even-integer multiples of the Fermi wave vector, which grow as the coupling is increased. Our MD results show an increase in the effective Fermi wave vector as the interaction strength is raised in the paramagnetic harmonic wire; this appears to be a result of the vanishing difference between the wave functions of the paramagnetic and ferromagnetic systems. We have extracted the Luttinger liquid exponent from our MDs by fitting to data around ${k}_{\mathrm{F}}$ , finding good agreement between the exponent of the ferromagnetic infinitely thin wire and the ferromagnetic harmonic wire.We present calculations of the energy, pair-correlation function (PCF), static structure factor (SSF), and momentum density (MD) for the one-dimensional electron gas using the quantum Monte Carlo method. We are able to resolve peaks in the SSF at even-integer multiples of the Fermi wave vector, which grow as the coupling is increased. Our MD results show an increase in the effective Fermi wave vector as the interaction strength is raised in the paramagnetic harmonic wire; this appears to be a result of the vanishing difference between the wave functions of the paramagnetic and ferromagnetic systems. We have extracted the Luttinger liquid exponent from our MDs by fitting to data around ${k}_{\mathrm{F}}$, finding good agreement between the exponent of the ferromagnetic infinitely thin wire and the ferromagnetic harmonic wire.

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Wire-width and electron-density dependence of the crossover in the peak of the static structure factor from 2kF → 4kF in one-dimensional paramagnetic electron gases

Ankush Girdhar, Vinod Ashokan, R. Sharma +2 more

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 $k=2{k}_{\text{F}}$ to $k=4{k}_{\text{F}}$ . It is found that, for a fixed electron density, in the charge structure factor the crossover from the dominant peak occurring at $2{k}_{\text{F}}$ to $4{k}_{\text{F}}$ occurs as the wire width decreases. Our study suggests that the crossover is due to the 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 $4{k}_{\text{F}}$ and $2{k}_{\text{F}}$ , 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 ${K}_{\ensuremath{\rho}}$ is found to be significant.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 $k=2{k}_{\text{F}}$ to $k=4{k}_{\text{F}}$. It is found that, for a fixed electron density, in the charge structure factor the crossover from the dominant peak occurring at $2{k}_{\text{F}}$ to $4{k}_{\text{F}}$ occurs as the wire width decreases. Our study suggests that the crossover is due to the 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 $4{k}_{\text{F}}$ and $2{k}_{\text{F}}$, 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 ${K}_{\ensuremath{\rho}}$ is found to be significant.

arXiv:2212.02225Physical review. B./Physical review. BMar 15, 20234 citations

Ground-state properties of electron-electron biwire systems

R. Sharma, N. D. Drummond, Vinod Ashokan +2 more

The correlation between electrons in different quantum wires is expected to affect the electronic properties of quantum electron-electron biwire systems. Here, we use the variational Monte Carlo method to study the ground-state properties of parallel, infinitely thin electron-electron biwires for several electron densities ( ${r}_{\text{s}}$ ) and interwire separations ( $d$ ). Specifically, the ground-state energy, the correlation energy, the interaction energy, the pair-correlation function (PCF), the static structure factor (SSF), and the momentum distribution (MD) function are calculated. We find that the interaction energy increases as $ln(d)$ for $d\ensuremath{\rightarrow}0$ and it decreases as ${d}^{\ensuremath{-}2}$ when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. The PCF shows oscillatory behavior at all densities considered here. As two parallel wires approach each other, interwire correlations increase while intrawire correlations decrease as evidenced by the behavior of the PCF, SSF, and MD. The system evolves from two monowires of density parameter ${r}_{\text{s}}$ to a single monowire of density parameter ${r}_{\text{s}}/2$ as $d$ is reduced from infinity to zero. The MD reveals Tomonaga-Luttinger (TL) liquid behavior with a power-law nature near ${k}_{\text{F}}$ even in the presence of an extra interwire interaction between the electrons in biwire systems. It is observed that when $d$ is reduced the MD decreases for $k<{k}_{\text{F}}$ and increases for $k>{k}_{\text{F}}$ , similar to its behavior with increasing ${r}_{\text{s}}$ . The TL liquid exponent is extracted by fitting the MD data near ${k}_{\text{F}}$ , from which the TL liquid interaction parameter ${K}_{\ensuremath{\rho}}$ is calculated. The value of the TL parameter is found to be in agreement with that of a single wire for large separation between the two wires.The correlation between electrons in different quantum wires is expected to affect the electronic properties of quantum electron-electron biwire systems. Here, we use the variational Monte Carlo method to study the ground-state properties of parallel, infinitely thin electron-electron biwires for several electron densities (${r}_{\text{s}}$) and interwire separations ($d$). Specifically, the ground-state energy, the correlation energy, the interaction energy, the pair-correlation function (PCF), the static structure factor (SSF), and the momentum distribution (MD) function are calculated. We find that the interaction energy increases as $ln(d)$ for $d\ensuremath{\rightarrow}0$ and it decreases as ${d}^{\ensuremath{-}2}$ when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. The PCF shows oscillatory behavior at all densities considered here. As two parallel wires approach each other, interwire correlations increase while intrawire correlations decrease as evidenced by the behavior of the PCF, SSF, and MD. The system evolves from two monowires of density parameter ${r}_{\text{s}}$ to a single monowire of density parameter ${r}_{\text{s}}/2$ as $d$ is reduced from infinity to zero. The MD reveals Tomonaga-Luttinger (TL) liquid behavior with a power-law nature near ${k}_{\text{F}}$ even in the presence of an extra interwire interaction between the electrons in biwire systems. It is observed that when $d$ is reduced the MD decreases for $k<{k}_{\text{F}}$ and increases for $k>{k}_{\text{F}}$, similar to its behavior with increasing ${r}_{\text{s}}$. The TL liquid exponent is extracted by fitting the MD data near ${k}_{\text{F}}$, from which the TL liquid interaction parameter ${K}_{\ensuremath{\rho}}$ is calculated. The value of the TL parameter is found to be in agreement with that of a single wire for large separation between the two wires.

arXiv:2104.00494Physical review. B./Physical review. BJul 26, 20218 citations

We study the ground-state properties of ferromagnetic quasi-one-dimensional quantum wires using the quantum Monte Carlo (QMC) method for various wire widths

b

and density parameters

{r}_{\text{s}}

. The correlation energy, pair-correlation function, static structure factor, and momentum density are calculated at high density. It is observed that the peak in the static structure factor at

k=2{k}_{\text{F}}

grows as the wire width decreases. We obtain the Tomonaga-Luttinger liquid parameter

{K}_{\ensuremath{\rho}}

from the momentum density. It is found that

{K}_{\ensuremath{\rho}}

increases by about 10% between wire widths

b=0.01

and

b=0.5

b\ensuremath{\ll}{r}_{\text{s}}<1

. It is found that the correlation energy varies as

{b}^{2}

for

b\ensuremath{\ll}{r}_{\text{s}}

b

and

{r}_{\text{s}}\ensuremath{\le}1

. These are compared with the QMC results in the range of applicability of the theory.

{r}_{s}

. The extracted exponent $\ensuremath{\alpha}$ of the momentum density for $k\ensuremath{\sim}{k}_{F}$ is used to determine the Tomonaga-Luttinger parameter ${K}_{\ensuremath{\rho}}$ as a function of

{r}_{s}

in the high-density regime. We find that the simulated static structure factor and pair correlation function for infinitely thin wires agree with our recent high-density theory [K. Morawetz et al., Phys. Rev. B 97, 155147 (2018)].

2r_\text {s} a_0

2 r s a 0 , where

a_0

a 0 is the Bohr’s radius and

r_\text {s}

r s is Wigner Seitz radius or electron-density parameter which corresponds to a

k=4k_\text {F}

k = 4 k F peak with

k_\text {F}

k F being Fermi wave vector in the static structure factor at finite electron density. It is a signature of the onset of Wigner crystallization. It is found that the cross-over from liquid to a quasi-Wigner phase is due to enhancement of the electron correlation effects as

r_\text {s}

r s increases or as the wire width b decreases.

{k}_{\mathrm{F}}

, finding good agreement between the exponent of the ferromagnetic infinitely thin wire and the ferromagnetic harmonic wire.

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=2{k}_{\text{F}}

k=4{k}_{\text{F}}

. It is found that, for a fixed electron density, in the charge structure factor the crossover from the dominant peak occurring at

2{k}_{\text{F}}

4{k}_{\text{F}}

occurs as the wire width decreases. Our study suggests that the crossover is due to the 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

4{k}_{\text{F}}

and

2{k}_{\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 ${K}_{\ensuremath{\rho}}$ is found to be significant.

{r}_{\text{s}}

) and interwire separations (

d

). Specifically, the ground-state energy, the correlation energy, the interaction energy, the pair-correlation function (PCF), the static structure factor (SSF), and the momentum distribution (MD) function are calculated. We find that the interaction energy increases as

ln(d)

for $d\ensuremath{\rightarrow}0$ and it decreases as ${d}^{\ensuremath{-}2}$ when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. The PCF shows oscillatory behavior at all densities considered here. As two parallel wires approach each other, interwire correlations increase while intrawire correlations decrease as evidenced by the behavior of the PCF, SSF, and MD. The system evolves from two monowires of density parameter

{r}_{\text{s}}

to a single monowire of density parameter

{r}_{\text{s}}/2

d

is reduced from infinity to zero. The MD reveals Tomonaga-Luttinger (TL) liquid behavior with a power-law nature near

{k}_{\text{F}}

even in the presence of an extra interwire interaction between the electrons in biwire systems. It is observed that when

d

is reduced the MD decreases for

k<{k}_{\text{F}}

and increases for

k>{k}_{\text{F}}

, similar to its behavior with increasing

{r}_{\text{s}}

. The TL liquid exponent is extracted by fitting the MD data near

{k}_{\text{F}}

, from which the TL liquid interaction parameter ${K}_{\ensuremath{\rho}}$ is calculated. The value of the TL parameter is found to be in agreement with that of a single wire for large separation between the two wires.