Interaction-range effects for fermions in one dimension

Experiments on quasi-one-dimensional systems such as quantum wires and metallic chains on surfaces suggest the existence of electron-electron interactions of substantial range and hence physics beyond the Hubbard model. We therefore investigate one-dimensional, quarter-filled chains with a Coulomb potential with variable screening length by quantum Monte Carlo methods and exact diagonalization. The Luttinger liquid interaction parameter $K_{\rho }$ decreases with increasing interaction strength and range. Experimentally observed values close to 1/4 require strong interactions and/or large screening lengths. As predicted by bosonization, we find a metal-insulator transition at $K_{\rho }=1/4$. Upon increasing the screening length, the charge and spin correlation functions reveal the crossover from dominant $2k_{\mathrm{F}}$ spin correlations to dominant $4k_{\mathrm{F}}$ charge correlations, and a strong enhancement of the charge velocity. In the metallic phase, the signatures of spin-charge separation in the single-particle spectrum, spinon and holon bands, remain robust even for rather long-ranged interactions. The charge-density-wave state exhibits backfolded shadow bands.

Figure 1

Finite-size scaling of the rescaled density structure factor $\pi S_{\rho }\left(q_1\right)/q_1$, with $q_1=2\pi /L$, for $V/t=3$, $n=0.5$, and $\xi =0.1$, 0.5, 1, 2, 3, 4, 5, 10, 20 (top to bottom). Lines are linear fits, and the extrapolated value in the thermodynamic limit $L\to \infty$ defines the Luttinger liquid parameter $K_{\rho }$. The extrapolation has been carried out for all data points shown in Fig. 2. The system sizes were $L=32,44,60$ for $n=0.5$ and $L=60,100,140$ for $n=0.1$. The temperature was $\beta t=2L$ for $V/t=1$ and $\beta t=L$ for $V/t=3,6,9$.

Figure 2

Luttinger liquid parameter $K_{\rho }$ as a function of screening length $\xi$. Points represent values obtained from a finite-size scaling (see Fig. 1). Lines are guides to the eye. (a) Results at quarter filling $n=0.5$. (b) Fixed $V/t=3$ and different fillings $n$. The phases are a Luttinger liquid (LL) for $K_{\rho }>n^2$ and a charge-density-wave insulator (CDW) for $K_{\rho }<n^2$.

Figure 3

Charge [(a), (b)] and spin [(c), (d)] structure factors $S_{\rho /\sigma }\left(q\right)$ for different values of the screening length $\xi$, and $\beta t=L=44$. The key in (a) applies to (a)–(d). (e) Rescaled charge structure factor for $V/t=3$ ($\beta t=L=60$), revealing the asymptotic approach to the small-$q$ behavior of the WC, $S_{\rho }\left(q\right)\sim q/{\left|\ln q\right|}^{1/2}$. Here and in all subsequent figures, results are for quarter filling $n=0.5$.

Figure 4

Density-density correlations in real space (symbols). Here, $V/t=5$, $\beta t=L=84$, $n=0.5$ and (a) $\xi =10$, (b) $\xi =20$. Lines are fits to the LL result for $\langle n_x{n}_0\rangle$ with fitting parameters $A_1$, $A_2$ [see Eq. (9)] and $K_{\rho }$ determined from a (linear) finite-size extrapolation based on $L=44,84$. The fitting interval was (a) $[15:35]$, (b) $[25:45]$.

Figure 5

Finite-size scaling of the amplitude of $4k_{\mathrm{F}}$ charge correlations at fixed $\xi =10$. The results reveal the absence of long-range order for $V/t=3$, and long-range $4k_{\mathrm{F}}$ charge correlations for $V/t=9$. The case $V/t=6$ is on the metallic side but very close to the critical point, and we find a small but finite extrapolated value. The lines are linear fits. Here, $\beta t=L$ and $n=0.5$.

Figure 6

(a) Dynamical charge and (b) spin structure factor for $V/t=1$ and $\xi =10$. Results were obtained with the projective CTQMC method (Ref. 34) using $L=28$, $\theta t=15$, and $n=0.5$. Dashed lines indicate the velocity of long-wavelength charge and spin excitations.

Figure 7

Dynamical charge [(a)–(c)] and spin [(d)–(f)] structure factor from simulations in the SSE representation for $V/t=6$ and different screening lengths $\xi$. Results are for $n=0.5$, $L=44$, and $\beta t=2L$. The dashed lines indicate the velocity at long wavelengths.

Figure 8

(a) Dynamical charge and (b) spin structure factor for $V/t=9$ and $\xi =10$, corresponding to the insulating CDW phase. Results were obtained in the SSE representation using $n=0.5$, $L=44$, and $\beta t=2L$.

Figure 9

Single-particle spectral function $A(k,\omega -\mu )$ for $n=0.5$, $V/t=1$, and $\xi =10$ from exact diagonalization with $L=20$. We used an artificial broadening of $0.05t$. The curves marked spinon, holon, and shadow are explained in the text.

Figure 10

Single-particle spectral function $A(k,\omega -\mu )$ for $n=0.5$, $V/t=6$, and different screening lengths $\xi$ from exact diagonalization with $L=20$. Insets: full energy range, revealing the upper Hubbard band.

Figure 11

As in Fig. 9 but for $V/t=9$ and $\xi =10$, corresponding to the insulating CDW phase (see Fig. 2). The inset shows a logarithmic density plot of the spectrum, revealing backfolded shadow bands related to the $4k_{\mathrm{F}}$ charge order.