Luttinger parameters of interacting fermions in one dimension at high energies

Interactions between electrons in one dimension are fully described at low energies by only a few parameters of the Tomonaga-Luttinger model, which is based on linearization of the spectrum. We consider a model of spinless fermions with short-range interaction via the Bethe-Ansatz technique and show that a Luttinger parameter emerges in an observable beyond the low-energy limit. A distinct feature of the spectral function, the edge that marks the lowest possible excitation energy for a given momentum, is parabolic for arbitrary momenta and the prefactor is a function of the Luttinger parameter, $K$.

Figure 1

The numerical evaluation of the Luttinger parameter $K$ as a function of next-neighbor interaction strength $V$ using Eqs. (3), (5), and (8), full ellipses; $L=100$ and $n=11$. We compare it with the bound on $K$ at infinite interaction strength from Eq. (9), thick dashed line. Small and larger $V$ asymptotes are $K=1-2n/L\times V/t$ and $K=K\left(\infty \right)+2n/(L-n)\times t/V$, thin dashed lines.

Figure 2

(a) We show the main result of this paper, that the edge of the support of the spectral function satisfies Eq. (10). Numerical results at intermediate coupling ($V=1.9t$) are shown as open circles and compared to the analytical result, Eq. (10), shown as a full line for $L=400$ and $n=39$. The asymptotes in the weak-coupling limit from Eq. (6) and the strong-coupling limit from Eq. (7) are shown as thin and thick dashed lines, respectively. (b) Sketch of the sets of quasimomenta, using classification Eq. (6), that correspond to the edge states; parameter $\Delta P$ corresponds to different momenta $k$.

Figure 3

The numerical solutions of Bethe equations, Eq. (5) of the main text, for $k_j$ as a function of interaction strengths $V$ for $n=12$ particles and $L=100$, full lines. The thin dashed lines at large $V$ correspond to asymptotes from Eq. (7) of the main text. The states are classified according to Eq. (6) of the main text: (a) ${\lambda }_j=\{-5,-4,-3,-2,-1,0,1,2,3,4,10\}$; (b) ${\lambda }_j=\left\{-6,-4,-3,-2,-1,0,1,2,3,4,9\right\}$; (c) ${\lambda }_j=\left\{-7,-6,-3,-2,-1,0,1,2,3,4,12\right\}$; (d) ${\lambda }_j=\left\{-5,-4,-3,-2,-1,0,1,2,3,6,32,33\right\}$. A bound state forms out of a pair of quasimomenta with $k_j>/\pi /2$ above a finite value of $V$. The thick dashed line marks the value of $k=\pi /2$, and the inset is the imaginary parts of all quasimomenta $k_j$.

Figure 4

Spectrum of the eigenstates, Eq. (3) of the main text, that are involved in the form factor in Eq. (2) of the main text for the ground state with $n=19$ particles, $L=200$, and $V\ll t$, dots. Large dots are the states at the edge. The insets are sketches of sets of quasimomenta that correspond to the edge states using the classification Eq. (6) of the main text. The positive half planes $E-E_0>\mu$ are the states with an extra added particle: (a) momenta are from $-3k_F$ to $k_F$, the energies at the edge at $-3k_F$ and $-k_F$ correspond to chemical potentials ${\mu }_{-1}$ and ${\mu }_0$; (b) momenta are from $k_F$ to $5k_F$, the energies at the edge at $k_F$, $3k_F$, and $5k_F$ correspond to chemical potentials ${\mu }_1$, ${\mu }_2$, and ${\mu }_3$. The negative half planes $E_0-E<\mu$ are the states with one particle removed: (c) momenta are from $-3k_F$ to $k_F$; and (d) momenta are from $k_F$ to $5k_F$.