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arXiv:cond-mat/0108290 · Physical review. B, Condensed matter

Tomonaga-Luttinger parameters for quantum wires

Wolfgang Häusler, Lars Kecke, Allan H. MacDonald

Published Feb 7, 2002• 43 citations
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Abstract

The low-energy properties of a homogeneous one-dimensional electron system are completely specified by two Tomonaga-Luttinger parameters ${K}_{\ensuremath{\rho}}$ and ${v}_{\ensuremath{\sigma}}.$ In this paper we discuss microscopic estimates of the values of these parameters in semiconductor quantum wires that exploit their relationship to thermodynamic properties. Motivated by the recognized similarity between correlations in the ground state of a one-dimensional electron liquid and correlations in a Wigner crystal, we evaluate these thermodynamic quantities in a self-consistent Hartree-Fock approximation. According to our calculations, the Hartree-Fock approximation ground state is a Wigner crystal at all electron densities and has antiferromagnetic order that gradually evolves from spin-density wave to localized in character as the density is lowered. Our results for ${K}_{\ensuremath{\rho}}$ are in good agreement with weak-coupling perturbative estimates ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}$ at high densities, but deviate strongly at low densities, especially when the electron-electron interaction is screened at long distances. ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}\ensuremath{\sim}{n}^{1/2}$ vanishes at small carrier density n, whereas we conjecture that ${K}_{\ensuremath{\rho}}\ensuremath{\rightarrow}1/2$ when $\stackrel{\ensuremath{\rightarrow}}{n}0,$ implying that ${K}_{\ensuremath{\rho}}$ should pass through a minimum at an intermediate density. Observation of this nonmonotonic dependence could be used to measure the effective interaction range in a realistic semiconductor quantum wire geometry. In the spin sector we find that the spin velocity decreases with increasing interaction strength or decreasing n. Strong correlation effects make it difficult to obtain fully consistent estimates of ${v}_{\ensuremath{\sigma}}$ from Hartree-Fock calculations. We conjecture that ${v}_{\ensuremath{\sigma}}{/v}_{\mathrm{F}}\ensuremath{\propto}{n/V}_{0},$ where ${V}_{0}$ is the interaction strength, in the limit $\stackrel{\ensuremath{\rightarrow}}{n}0.$

DOI
10.1103/physrevb.65.085104

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The low-energy properties of a homogeneous one-dimensional electron system are completely specified by two Tomonaga-Luttinger parameters K\ensuremathρ{K}_{\ensuremath{\rho}}K\ensuremathρ​ and v\ensuremathσ.{v}_{\ensuremath{\sigma}}.v\ensuremathσ​. In this paper we discuss microscopic estimates of the values of these parameters in semiconductor quantum wires that exploit their relationship to thermodynamic properties. Motivated by the recognized similarity between correlations in the ground state of a one-dimensional electron liquid and correlations in a Wigner crystal, we evaluate these thermodynamic quantities in a self-consistent Hartree-Fock approximation. According to our calculations, the Hartree-Fock approximation ground state is a Wigner crystal at all electron densities and has antiferromagnetic order that gradually evolves from spin-density wave to localized in character as the density is lowered. Our results for K\ensuremathρ{K}_{\ensuremath{\rho}}K\ensuremathρ​ are in good agreement with weak-coupling perturbative estimates K\ensuremathρpert{K}_{\ensuremath{\rho}}^{\mathrm{pert}}K\ensuremathρpert​ at high densities, but deviate strongly at low densities, especially when the electron-electron interaction is screened at long distances. K\ensuremathρpert\ensuremath∼n1/2{K}_{\ensuremath{\rho}}^{\mathrm{pert}}\ensuremath{\sim}{n}^{1/2}K\ensuremathρpert​\ensuremath∼n1/2 vanishes at small carrier density n, whereas we conjecture that K\ensuremathρ\ensuremath→1/2{K}_{\ensuremath{\rho}}\ensuremath{\rightarrow}1/2K\ensuremathρ​\ensuremath→1/2 when n\ensuremath→0,\stackrel{\ensuremath{\rightarrow}}{n}0,n\ensuremath→0, implying that K\ensuremathρ{K}_{\ensuremath{\rho}}K\ensuremathρ​ should pass through a minimum at an intermediate density. Observation of this nonmonotonic dependence could be used to measure the effective interaction range in a realistic semiconductor quantum wire geometry. In the spin sector we find that the spin velocity decreases with increasing interaction strength or decreasing n. Strong correlation effects make it difficult to obtain fully consistent estimates of v\ensuremathσ{v}_{\ensuremath{\sigma}}v\ensuremathσ​ from Hartree-Fock calculations. We conjecture that v\ensuremathσ/vF\ensuremath∝n/V0,{v}_{\ensuremath{\sigma}}{/v}_{\mathrm{F}}\ensuremath{\propto}{n/V}_{0},v\ensuremathσ​/vF​\ensuremath∝n/V0​, where V0{V}_{0}V0​ is the interaction strength, in the limit n\ensuremath→0.\stackrel{\ensuremath{\rightarrow}}{n}0.n\ensuremath→0.
Low-energy properties of the homogeneous electron gas in one dimension are completely described by the group velocities of its charge (plasmon) and spin collective excitations. Because of the long range of the electron-electron interaction, the plasmon velocity is dominated by an electrostatic contribution and can be estimated accurately. We report on Quantum Monte Carlo simulations which demonstrate that the spin velocity is substantially decreased by interactions in semiconductor quantum wire realizations of the one-dimensional electron liquid.
We study the ground-state properties of ferromagnetic quasi-one-dimensional quantum wires using the quantum Monte Carlo (QMC) method for various wire widths bbb and density parameters rs{r}_{\text{s}}rs​. 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=2kFk=2{k}_{\text{F}}k=2kF​ 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=0.01b=0.01b=0.01 and b=0.5b=0.5b=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 b2{b}^{2}b2 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 bbb and ${r}_{\text{s}}\ensuremath{\le}1$. These are compared with the QMC results in the range of applicability of the theory.
It is demonstrated that the charge Tomonaga-Luttinger parameter KρK_ρKρ​ of quantum wires can be estimated accurately using the Hartree-Fock approximation if carried out self consistently. The dependence of KρK_ρKρ​ on the carrier density distinguishes different regimes of importance of correlations.
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 (rs{r}_{\text{s}}rs​) and interwire separations (ddd). 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)ln(d)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 rs{r}_{\text{s}}rs​ to a single monowire of density parameter rs/2{r}_{\text{s}}/2rs​/2 as ddd is reduced from infinity to zero. The MD reveals Tomonaga-Luttinger (TL) liquid behavior with a power-law nature near kF{k}_{\text{F}}kF​ even in the presence of an extra interwire interaction between the electrons in biwire systems. It is observed that when ddd is reduced the MD decreases for k<kFk<{k}_{\text{F}}k<kF​ and increases for k>kFk>{k}_{\text{F}}k>kF​, similar to its behavior with increasing rs{r}_{\text{s}}rs​. The TL liquid exponent is extracted by fitting the MD data near kF{k}_{\text{F}}kF​, 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.
We explore the role of electron correlation in quasi-one-dimensional quantum wires as the range of the interaction potential is changed and their thickness is varied by performing exact quantum Monte Carlo simulations at various electronic densities. In the case of unscreened interactions with a long-range 1/x1/x1/x tail there is a crossover from a liquid to a quasi-Wigner crystal state as the density decreases. When this interaction is screened, quasi-long-range order is prevented from forming, although a significant correlation with 4kF4{k}_{F}4kF​ periodicity is still present at low densities. At even lower electron concentration, exchange is suppressed and the electrons behave like spinless fermions. Finally, we study the effect of electron correlations in the double quantum wire experiment [Steinberg et al., Phys. Rev. B 73, 113307 (2006)] by introducing an accurate model for the screening in the experiment and explicitly including the finite length of the system in our simulations. We find that decreasing the electron density continuously drives the system from a liquid to a state with quite strong 4kF4{k}_{F}4kF​ correlations. This crossover takes place around $22\text{ }\ensuremath{\mu}{\text{m}}^{\ensuremath{-}1}$, near the density where the electron localization occurs in the experiment. The charge and spin velocities are also in good agreement with the experimental findings in the proximity of the crossover. We argue that correlation effects play an important role at the onset of the localization transition.
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