Recursive Method for the Density of States in One Dimension

We derive a powerful yet simple method for analyzing the local density of states (DOS) in gapless one-dimensional fermionic systems, including extensions such as momentum dependent interaction parameters and hard-wall boundaries. We study the crossover of the local DOS from individual density waves to the well-known asymptotic power laws and identify characteristic signs of spin charge separation in possible STM experiments. For semi-infinite systems a closed analytic expression is found in terms of hypergeometric functions.

Figure 1

${\rho }_u(\omega ,x)$ in units of $|c_{x=L/2}{|}^2$ for $K_c=0.531$, $K_s=0.618$, and ${\upsilon }_c/{\upsilon }_s=1.618$ averaged over an energy interval of 0.3 in units of $\Delta {\omega }_c$.

Figure 2

Spatial Fourier transform of the local DOS with parameters as in Fig. 1 in arbitrary units. Inset: Cut at fixed energy.

Figure 3

The uniform spectral weights ${\rho }_u$ in units of $|c_x{|}^2$ near a boundary as a function of $\omega x/v_c$ for $K_c=0.531$, $K_s=1$, and ${\upsilon }_c/{\upsilon }_s=1.618$. The large number of spin and charge levels for $x=0.1L$ (blue dots) follow Eq. (11) (red line) after averaging.