Few-body route to one-dimensional quantum liquids

Gapless many-body quantum systems in one spatial dimension are universally described by the Luttinger liquid effective theory at low energies. Essentially, only two parameters enter the effective low-energy description, namely, the speed of sound and the Luttinger parameter. These are highly system dependent and their calculation requires accurate nonperturbative solutions of the many-body problem. Here we present a simple theoretical method that only uses collisional information to extract the low-energy properties of spinless one-dimensional systems. Our results are in remarkable agreement with available results for integrable models and from large-scale Monte Carlo simulations of one-dimensional helium and hydrogen isotopes. Moreover, we estimate theoretically the critical point for spinodal decomposition in one-dimensional ${}^4\mathrm{He}$ and show that the exponent governing the divergence of the Luttinger parameter near the critical point is exactly 1/2, in excellent agreement with Monte Carlo simulations.

Figure 1

Speed of sound vs Lieb-Liniger constant $\gamma$. We plot the theoretical results from Eqs. (4) and (3) (blue solid line), fourth-order perturbation theory in $1/\gamma$ (red dotted line), and Bogoliubov theory (black dashed line).

Figure 2

Luttinger parameter vs density for ${}^3\mathrm{He}$ (blue solid line), ${}^3\mathrm{H}$ (black double-dash–dotted line), ${}^2\mathrm{H}$ (green dash-dotted line), and ${}^1\mathrm{H}$ (red dashed line). Blue dots $\left({}^3\mathrm{He}\right)$, black triangles $\left({}^3\mathrm{H}\right)$, green diamonds $\left({}^2\mathrm{H}\right)$, and red squares $\left({}^1\mathrm{H}\right)$ are the Monte Carlo data from Refs. [11, 12].

Figure 3

Luttinger parameter vs density for ${}^4\mathrm{He}$ (solid blue line). Red dots are Monte Carlo data of Ref. [10]. The inset shows the Luttinger parameter from Eq. (9) (blue solid line) and the renormalized Luttinger parameter (see the text) with ${\rho }_{*}=0.0296\AA {}^{-1}$ (black dashed line) and with ${\rho }_{*}=0.026\AA {}^{-1}$ (red dash-dotted line). Red dots are the Monte Carlo data of Ref. [10].

Figure 4

Low-density equation of state from Eq. (8) (blue solid line). Red dots are the Monte Carlo data of Ref. [10].