Strongly Correlated One-Dimensional Bose Fluids

The repulsive one-dimensional contact Bose gas is controlled by $\gamma=mg_{1\mathrm D}/n$: it is a weakly interacting quasicondensate for $\gamma\ll1$, a Luttinger liquid at low energy for every finite repulsion, and a Tonks–Girardeau gas with fermionized spatial correlations as $\gamma\to\infty$. Fermionization changes observables, not the particles’ exchange statistics.

Required background. Use the phase–density EFT and the matched coupling from dimensional crossover and confinement resonances. Helpful background. Luttinger liquids supply the full operator dictionary.

Lieb–Liniger model and scales

For $N$ bosons on a ring of length $L$,

$$H=-\frac{1}{2m}\sum_{j=1}^N\partial_{x_j}^2 +g_{1\mathrm D}\sum_{i<j}\delta(x_i-x_j), \qquad n=\frac NL, \qquad \gamma=\frac{mg_{1\mathrm D}}{n}>0.$$

The Bethe-ansatz solution is exact in the continuum model Lieb and Liniger 1963, pp. 1605–1616. Its ground-state energy density has controlled limits

$$\frac EV\equiv\frac EL \simeq \begin{cases} \dfrac{g_{1\mathrm D}n^2}{2}\left(1-\dfrac{4\sqrt\gamma}{3\pi}+\cdots\right),&\gamma\ll1,\\[6pt] \dfrac{\pi^2n^3}{6m}\left(1-\dfrac4\gamma+\cdots\right),&\gamma\gg1. \end{cases}$$

The strong-coupling leading term is the energy of spinless free fermions. Girardeau’s mapping multiplies a fermion Slater determinant by a sign function, preserving $|\Psi|^2$ and hence density observables while restoring bosonic symmetry Girardeau 1960, pp. 516–523.

Low-energy correlations

At energies below the microscopic curvature scale,

$$H_{\mathrm{LL}}=\frac{v}{2\pi}\int\mathrm dx \left[K(\partial_x\theta)^2+K^{-1}(\partial_x\phi)^2\right].$$

Galilean invariance gives $vK=\pi n/m$. For repulsive contact bosons, $K\simeq\pi/\sqrt\gamma$ at weak coupling and $K\to1$ in the Tonks limit. Consequently the one-body correlator decays as $g_1(x)\sim|x|^{-1/(2K)}$, while the leading $2\pi n$ density oscillation decays as $|x|^{-2K}$. Compressibility fixes the other invariant combination, $K=\pi n/(m v)$.

At every $T>0$, correlations cross from algebraic to exponential beyond the thermal length $v/T$. A trap produces a spatially varying $\gamma(x)$; local-density averaging is controlled only when the density varies slowly compared with the correlation length. Integrability-breaking interactions and losses limit exact Bethe-ansatz predictions.

Exercises

Use $vK=\pi n/m$ to check the Tonks limit.

Solution

Free spinless fermions have $v=v_F=\pi n/m$. Therefore $K=1$, consistent with the strong-coupling limit and with the exponents above. This check concerns long-distance correlations, not exchange statistics.

References

• Marcos Girardeau, “Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension,” Journal of Mathematical Physics 1 (1960) 516–523, doi:10.1063/1.1703687.
• Elliott H. Lieb and Werner Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State,” Physical Review 130 (1963) 1605–1616, doi:10.1103/PhysRev.130.1605.