Dependence of structure factor and correlation energy on the width of electron wires

The structure factor and correlation energy of a quantum wire of thickness b ≪ aB are studied in random phase approximation (RPA) and for the less investigated region rs < 1. Using the single-loop approximation, analytical expressions of the structure factor are obtained. The exact expressions for the exchange energy are also derived for a cylindrical and harmonic wire. The correlation energy in RPA is found to be represented by ϵc(b, rs) = α(rs)/b + β(rs) ln(b) + η(rs), for small b and high densities. For a pragmatic width of the wire, the correlation energy is in agreement with the quantum Monte Carlo simulation data.


Introduction
The motion of electrons confined in one spatial dimension gives rise to a variety of interesting phenomena with anomalous properties [1]. Recently quasi one-dimensional systems are experimentally realized in carbon nanotubes [2][3][4][5], semiconducting nanowires [6,7] and cold atomic gases [8][9][10], edge states in quantum hall liquid [11][12][13] and conducting molecules [14]. The electrons in one dimension do not obey the conventional Fermi-liquid theory, hence the prospect to observe non-Fermi-liquid features has given a large impetus to both theoretical and experimental research. An appropriate description of the one-dimensional (1D) homogeneous electron gas is provided by the low-energy theory based on an exactly solvable Tomonaga-Luttinger model [15][16][17]. The random phase approximation (RPA) is the correct theory for a homogeneous electron gas in the high-density limit i.e. at large electron densities n = 1/(2r s a B ), with a B being the effective Bohr radius and r s being the coupling parameter.
We model the interactions by a smoothed long-range Coulomb potential v(x) ∝ (x 2 + b 2 ) −1/2 , where b is a parameter related to the width of the wire. We also use a harmonic confinement potential. The true long-range character of the Coulomb potential has been studied by Schulz [18] and Fogler [19,20] using a different approximation than RPA in certain domains of (b, r s ). In fact, a considerable amount of theoretical and numerical work a e-mail: morawetz@fh-muenster.de has been done in this domain [21][22][23][24][25][26][27][28][29][30][31][32][33][34] using RPA, its generalized version and other methods, but still there is a need to understand the accurate parametrization of correlation energy for thin wires in the high-density limit. Therefore the calculation of the ground-state energy for thin wires in the high-density limit for realistic long-range Coulomb interactions is still an open problem for the 1D homogeneous electron gas.
Recently Lee and Drummond [35] have studied the ground state properties of the 1D electron liquid for an infinitely thin wire, and the harmonic wire using a quantum Monte Carlo (QMC) method, and provided a benchmark of the total energy data for a limited range of r s . Furthermore, the harmonic wire with transverse confinement has been investigated with a lattice regularized diffusion Monte Carlo (LRDMC) technique by Casula et al. [36], and by others [37][38][39].
Loos [40] has considered the high-density correlation energy for the 1D homogeneous electron gas using the conventional perturbation theory by taking the smoothed Coulomb potential described above in the limit b → 0 (infinitely thin wire). At b = 0 they have reported a correlation energy of −27.4 mHartree per electron at r s → 0. In their calculations the divergences in the integral for small b cancels out exactly. Although there is no direct comparison possible between results of Loos [40] and de Oliveira and Verri [32] with the RPA result due to use of different method of calculations, however, we mention that in RPA the correlation energy diverges for b → 0 and r s → 0. The exchange energy has been found to be independent of b (leaving the logarithmic thickness L which may be cancelled with the uniform charge background). Further, the agreement between the RPA result with QMC for the structure factor [41] ensures that the RPA should recover the normal Coulomb operator.
The purpose of the present paper is to study electron correlation effects in the interacting electron fluid described by RPA at high densities. The dependence of the structure factor and correlation energy on the wirewidth is analyzed in the domain of r s < 1 and b a B . In this respect it is noted that RPA is a good approximation in the high density limit r s → 0. We derive analytical expressions for the static structure factor in the high-density limit. The exact analytical expression for the exchange energy is also obtained for cylindrical and harmonic potentials. It is found that on the basis of theoretical deduction and a logical assumption, the correlation energy can be represented by the formula The paper is organized as follows. In Section 2, we calculate the static structure factor within RPA and using the first-order approximation. In Section 3, the RPA groundstate energy formula is given. Subsection 3.1 provides the exact analytical result of exchange energy for cylindrical and harmonic potentials for finite b and r s . The result for small-b limit is also given there. Subsection 3.2 describes the correlation energy partially by an analytical formula and partially through numerical calculation. Then the final result of the correlation energy and its parametrization is presented. In Section 4 the results are discussed.

Structure factor
The structure factor S(q) is now calculated within the RPA and its first-order version, where it is possible to obtain the result analytically. The RPA density response function χ(q, ω) is given by [42], where V (q) is the Fourier transform of the inter-electronic interaction potential. For harmonically trapped electron wires, and for cylindrical wires [26], it is given respectively by , where E 1 is the exponential integral and K 0 the modified Bessel function of 2nd kind.
The static structure factor is defined through the fluctuation-dissipation theorem as where χ (q, ω) is the imaginary part of the density response function (1). The integral in (2) can be re-written using the contour integration method [1] as, where n = (k F g s )/π is the linear electron number density, g s is the spin degeneracy factor and k F is the Fermi wave vector. Using the high-density expansion where, the structure factor (3) can be calculated for r s → 0 using (4) and (5) The zeroth-order static structure factor is easily calculated The first-order correction to the structure factor can be obtained by substituting χ 0 (q, iω) in the second term of (4), and than using it in (3) The resulting integral can be performed analytically and the result for q < 2k F reads Similarly, for q > 2k F one obtains Here and in the following we use V (q) = v(q)e 2 /4π 0 . In the limit of small q, q around 2k F and large q, the S 1 (q) takes the simpler forms where Λ(z) = (8 ln(2) − 2|z| + 2|z| ln |z| − 3 4 |z| 2 ) and z = q−2k F k F . It can be easily seen that for harmonic wires the interaction potential approaches with the Euler Gamma constant γ.
For a cylindrical potential the corresponding results are, for bq → ∞ . (11) Both potentials behave similarly at the small bq limit, but at large bq they differ. Substituting values of v(q) from (10) in (9), the corresponding leading term agrees with Fogler [20].
To see the effect of thickness b of the wire we calculate the structure factor from (3) by using (1) and (5), for r s < 1 and plot them in Figure 1. It is seen from Figure 1a that as b decreases, the structure factor S(q) also decreases. A similar trend is also obtained for other r s . To see the validity of the first-order structure factor we plot in Figure 1b for r s = 0.1, S 0 (q) + S 1 (q) and RPA for b = 0.025. These are compared with diffusion quantum Monte Carlo simulation [43] for an infinitely thin wire, as implemented in CASINO code [44]. All three curves match perfectly. It is noted that in the high density limit (r s 1), the structure factor is more or less the one of a non-interacting gas of electrons. This is due to the fact that in this region, the kinetic energy is the dominant contribution to the total energy leading the system to behave as a gas of non-interacting electrons as conjectured by Fogler [20].

Ground state energy
The ground-state energy can be obtained by the density-density response function in conjunction with the fluctuation-dissipation theorem as [1], It further simplifies into a sum of kinetic energy of the noninteracting gas with the exchange energy and the residual energy (i.e. correlation energy) as, where

Exchange energy
In this section the exchange energy for a cylindrical as well as for a harmonic electron wire is obtained analytically, by integrating (14). Specifically for cylindrical wire it turns out to be, where K n (x) is nth order modified Bessel function of 2nd kind, and L n (x) is modified Struve function [45]. Similarly, the exchange energy can also be obtained for a harmonic wire of finite thickness given as and Γ 0, 4b 2 k 2 F are the Meijer G function [46] and the incomplete gamma function, respectively. As it is difficult to visualize the former function, it is desirable to write it in its integral form For thin harmonic wires b a B the exchange energy can be simplified to be, where ψ (0) (1/2) and ψ (0) (3/2) are polygamma functions [45]. We now use the simpler expansion of the polygamma function as 2 ψ (0) (1/2) = −2γ − 2 ln(4) and 2 ψ (0) (3/2) = 4 − 2γ − 2 ln(4) in the above equation. Equations (16) and (17) can also be written for a polarized gas by defining k F ↑(↓) = k F (1 ± p), N ↑(↓) = N (1 ± p)/2 and k F = π/(2g s r s a B ). Explicitly for thin cylindrical wires b a B , the exchange energy per particle can be obtained by expanding equation (16) as Similarly for harmonic wires, equation (19) gives where L = ln(a B /b). It is noted that equations (16) and (17) are new results and for special cases they reduce to (20) and (21). It is worth mentioning that the logarithmic thickness of the wire is defined by L −1 . For polarized (g s = 1 and p = 1) and unpolarized fluids (g s = 2 and p = 0), the exchange energy of a cylindrical wire is obtained respectively to be and These are in agreement with Fogler's results [19].

Correlation energy
The integration over the coupling constant λ is easily done in (15) and the correlation energy becomes The above equation can be written further as, where The first term c1 can be integrated analytically for the cylindrical potential, Equation (28) is further simplified for an infinitely thin wire b → 0 for any finite r s as For a given r s the above equation has a functional dependence on b (in atomic unit) as where α 1 (r s ), β 1 (r s ) and η 1 (r s ) can be read from (29). Equation (27) cannot be integrated analytically, therefore we solve it numerically. Anticipating that the correlation energy c for b → 0 and r s → 0 turns out to be a constant, c2 may also be represented by (30) but with the coefficients α 2 (r s ), β 2 (r s ) and η 2 (r s ). Therefore we represent c2 by, and fit it to the numerical result. The coefficients α, β, η for c2 and c are given in Table 1, for r s =0.1, 0.01 and 0.001. Note that the coefficients for c1 are analytically known. Also the same formula as for c2 is assumed for c . To estimates the accuracy of the fit with the numerical calculation, we have provided the statistical analysis with different methods: χ 2 , R 2 adjusted for the number of model parameters, AdjustedRSquared (ARS), Akaike information criterion (AIC), Bayesian information criterion (BIC) and coefficient of determination R 2 (RSq). The fitted parameters by the statistical analysis in Table 1 reflect the quality of the function c2 (b, r s ) and c (b, r s ). The correlation energies per particle c1 (lower curve) and c2 (upper curve) are plotted in Figure 2a, as obtained analytically and numerically, and are shown as green continuous curve and green dots, respectively. The fitted c2 from representation (31) is shown by the black continuous line. It is clearly seen that there is a perfect fit of c2 , as also inferred above from the statistical analysis. It is seen that there is no cancellation between the two curves for r s = 0.1. The resulting sum is plotted for the same r s in Figure 2b. Total correlation energy for r s = 0.01 and r s = 0.001 are also plotted in Figures 2c and 2d respectively. These figures show that there is no indication that the correlation energy approaching a constant value for very small r s for an infinitely thin wire.
For a pragmatic width of the wire, the correlation energy for a polarized fluid is reported in Table 2. The correlation energy at high densities r s ≤ 0.1 and b = 0.1, is in agreement with the quantum Monte Carlo simulation [36,37] for polarized fluids.
To check the consistency of our result of the correlation energy for b → 0 and r s < 1, we plot it in Figure 3 as a function of r s for small values of b. It is seen that as b decreases, the correlation energy increases,  which is consistent with our previous results given in Figures 2b-2d. In Figure 4 the total ground-state energy is presented with different wire thicknesses as a function of r s . As b decreases, the ground-state energy increases. For the range r s 1 there are no QMC data available to compare the ground-state energy for an infinitely thin wire. Our calculation is suited for long-range interactions whereas the Fogler calculation deals with the short-range interaction.

Summary
In this paper we have calculated the dependence of the ground-state structure factor and the correlation energy on the thickness of an electron wire as a function of r s . The structure factor is calculated in the single-loop approximation of RPA. The electron-electron interactions are modeled by a cylindrical and a harmonic potential. We find an agreement with the variational calculation of Fogler [20]. The structure factor has also been compared for b = 0.025 and r s = 0.1 with the QMC data [43]. For first-order corrections in the interaction, the RPA results and the QMC data match perfectly, indicating that for small thickness and for high densities, the electron gas behaves as a gas of non-interacting particles though highly correlated, which is clear from the correlation energy calculations. In this sense Fogler [19,20] calls it a Coulomb-Tonks gas. We have also obtained new analytical expressions for the exchange energy of both cylindrical and harmonic electron wires. In the small-thickness limit the expressions simplify considerably and are more or less the same for both wires. This has been also worked out for polarized gases, from which the paramagnetic and ferromagnetic phases can easily be obtained. The exchange energy for a fully polarized gas agrees with Fogler [19].
In the present paper the total correlation energy in RPA is found to be fitted by with the parameters given explicitly. This correlation energy is the sum of two terms which only partially cancel. The first term is calculated analytically exactly by the expression (32) where the values of α, β and η are precisely known. The second term has been calculated numerically. It perfectly fits with the expression of (32) but with different parameters. This findings clearly indicate that the correlation energy is diverging in the limit of b → 0 and r s → 0. Further, the correlation energy as a function of r s for various b, again points out that the correlation energy increases as b decreases for r s → 0. The Coulomb correlations are enhanced, and the interacting electron gas behaves structure-less in the ultra-thin and high-density domain of L −1 r s 1 like a strongly-interacting electron gas named Coulomb-Tonks gas [19,20]. Further, we find that the correlation energy does not approach a constant value for an infinitely thin wire and r s → 0 within the RPA.