Local spectral weight of a Luttinger liquid: Effects from edges and impurities

We calculate the finite-temperature local spectral weight (LSW) of a Luttinger liquid with an “open” (hard-wall) boundary. Close to the boundary the LSW exhibits characteristic oscillations indicative of spin-charge separation. The line shape of the LSW is also found to have a Fano-like asymmetry, a feature originating from the interplay between electron-electron interaction and scattering off the boundary. Our results can be used to predict how edges and impurities influence scanning tunneling microscopy (STM) of one-dimensional electron systems at low temperatures and voltage bias. Applications to STM on single-walled carbon nanotubes are discussed.

Figure 1

(Color online) Branch points and branch cuts for $G_{LL}(t,x,x;\beta )$. The phases are different in different regions defined by branch points.

Figure 2

Local spectral weight $A(\omega ,x;\beta )$ for $K_c=0.7$, $v_c∕v_F\approx 1.43$, $v_c∕v_s\approx 3$ (a), and $K_c=0.9$, $v_c∕v_F\approx 1.11$, $v_c∕v_s\approx 1.26$ (b) with $T∕k_F{v}_F=2.6\times {10}^{-6}$, $x=10a$, $n_e=0.97$.

Figure 3

(Color online) $A(\omega ,x;\beta )$ for $K_c=0.7$, $v_c∕v_F\approx 1.43$, $v_c∕v_s\approx 3$, $T∕k_F{v}_F=2.6\times {10}^{-6}$, $n_e=0.97$, shown for even and odd sublattices.

Figure 4

(Color online) $A(\omega ,x;\beta )$ for $K_c=0.7$, $v_c∕v_F\approx 1.43$, $v_c∕v_s\approx 3$, $T∕k_F{v}_F=2.6\times {10}^{-4}$, $n_e=0.97$.

Figure 5

(Color online) $A(\omega ,x;\beta )$ for $T∕k_F{v}_F=2.6\times {10}^{-6}$, $n_e=0.97$, with $K_c=0.7$, $v_c∕v_F\approx 1.43$, $v_c∕v_s\approx 3$ (a) and $K_c=0.9$, $v_c∕v_F\approx 1.11$, $v_c∕v_s\approx 1.26$ (b). The color coding is the same as in Fig. 4.

Figure 6

$A(\omega ,x;\beta )$ for $T∕k_F{v}_F=2.6\times {10}^{-6}$, $n_e=0.97$, with $x=50a$, $K_c=0.7$, $v_c∕v_F\approx 1.43$, $v_c∕v_s={\lambda }_c∕{\lambda }_s\approx 3$ (a) and $x=100a$, $K_c=0.9$, $v_c∕v_F\approx 1.11$, $v_c∕v_s={\lambda }_c∕{\lambda }_s\approx 1.26$ (b).

Figure 7

(Color online) Fourier transform of the LSW for $K_c=0.7$, $v_c∕v_F\approx 1.43$, $v_c∕v_s\approx 3$, $T∕k_F{v}_F=2.6\times {10}^{-4}$, $n_e=0.97$ (a) and the same graph with charge (solid lines) and spin (dashed lines) wave dispersions superimposed (b).

Figure 8

(Color online) $A(\omega ,x;\beta )$ for $K_c=0.7$, $v_c∕v_F\approx 1.43$, $v_c∕v_s\approx 3$, $x=10a$, $n_e=0.97$ at different temperatures.

Figure 9

(Color online) $A(\omega ,x;\beta )$ for $T∕k_F{v}_F=2.6\times {10}^{-2}$, $K_c=0.7$, $v_c∕v_F\approx 1.43$, $v_c∕v_s\approx 3$, $n_e=0.97$ (a) and cut at $x=200a$ (b).

Figure 10

$dI∕dV(V,x;\beta )$ for $T∕k_F{v}_F=2.6\times {10}^{-3}$, $n_e=0.97$, with $x=50a$, $K_c=0.7$, $v_c∕v_F\approx 1.43$, $v_c∕v_s={\lambda }_c∕{\lambda }_s\approx 3$ (a) and $x=100a$, $K_c=0.9$, $v_c∕v_F\approx 1.11$, $v_c∕v_s={\lambda }_c∕{\lambda }_s\approx 1.26$ (b).