Spectral-edge mode in interacting one-dimensional systems

A continuum of excitations in interacting one-dimensional systems is bounded from below by a spectral edge that marks the lowest possible excitation energy for a given momentum. We analyze short-range interactions between Fermi particles and between Bose particles (with and without spin) using Bethe-ansatz techniques and find that the dispersions of the corresponding spectral edge modes are close to a parabola in all cases. Based on this emergent phenomenon we propose an empirical model of a free, nonrelativistic particle with an effective mass identified at low energies as the bare electron mass renormalized by the dimensionless Luttinger parameter $K$ (or $K_{\sigma }$ for particles with spin). The relevance of the Luttinger parameters beyond the low-energy limit provides a more robust method for extracting them experimentally using a much wider range of data from the bottom of the one-dimensional band to the Fermi energy. The empirical model of the spectral edge mode complements the mobile impurity model to give a description of the excitations in proximity of the edge at arbitrary momenta in terms of only the low-energy parameters and the bare electron mass. Within such a framework, for example, exponents of the spectral function are expressed explicitly in terms of only a few Luttinger parameters.

Figure 1

A schematic representation of a tunneling process into a one-dimensional system for a particle with fixed momentum $k$ and energy $E$ that is described by the spectral function. Excitations of the system are (a) density waves or [(a) and (b)] spin wave for particles with spin.

Figure 2

A set of quasimomenta from Eq. (7) that corresponds to the edge mode of the spectral function for Fermi particles without spin. The momentum of each many-particle state is given by $k=-k_F+\Delta P$.

Figure 3

Parametrization of many-body states for fermions with spin using the $U=\infty$ limit in Eqs. (14) and (15). Chargelike excitations correspond to different sets of $I_j$ and spinlike excitations correspond to different sets of $J_j$.

Figure 4

Velocities of the collective modes for fermions with spins at low energies as a function of the interaction parameters ${\gamma }_{YG}$ from Eq. (18). The red line corresponds to the holon branch, the green line corresponds to the spinon branch, and the blue dashed line marks a quantum of momentum $v_1=2\pi /\left(mL\right)$; $L=400$, $N=40$, ${\gamma }_{YG}=3.44mU$. Inset: Degree of double degeneracy [see the definition in Eq. (19)] for the ground state in Fig. 3 with $\Delta k=\Delta \lambda =0$ as a function of the interaction parameter ${\gamma }_{YG}$.

Figure 5

Dispersion of the spectral edge mode (extension of the spinon branch to high energies) for fermions with spins for different values of the interaction parameter ${\gamma }_{YG}=0,1,6.88$; $L=400$, $N=40$. The blue triangles, green squares, and red ellipses are the numerical solutions of Eqs. (12) and (13); the solid black lines are the best parabolic fits by Eq. (1). Inset: Difference between the slope of the parabolic dispersion at $E_F$, which is given by the effective mass $m^{*}$, and the velocity of spin waves at low energies, which is obtained directly from Eq. (20) [$\left(v_{\sigma }-k_F/m^{*}\right)/v_{\sigma }$] as a function of the number of particles $N$ for ${\gamma }_{YG}=6.88$. The solid black line is the $1/N$ fit, $a+b/N$, that gives $a=0.22$ [29].

Figure 6

The sound velocity of the collective modes for spinless bosons $v$ at low energies as a function of the interaction parameter ${\gamma }_{LL}$ from Eq. (26) (red line) and the quantum of momentum $v_1=2\pi /\left(mL\right)$ (blue dashed line); $L=200$, $N=20$, and ${\gamma }_L=0.2mU$.

Figure 7

Dispersion of the spectral edge mode for spinless bosons for different values of the interaction parameter ${\gamma }_L=0.5,1,\infty$; $L=200$, $N=20$. The blue triangles, green squares, and red ellipses are the numerical solutions of Eq. (25), the solid black lines are the best parabolic fits by Eq. (1). Inset: Difference between the slope of the parabolic dispersion at $E_F$, which is given by the effective mass $m^{*}$, and the velocity of the sound modes, which is obtained by direct evaluation of the energy of the first excited state above the Fermi energy [$\left(v-k_F/m^{*}\right)/v$] as a function of the number of particles $N$ for ${\gamma }_{LL}=0.5$. The solid black line is the $1/N$ fit, $a-b/N$, that gives $a=0.09$.