Emergence and Disruption of Spin-Charge Separation in One-Dimensional Repulsive Fermions

At low temperature, collective excitations of one-dimensional (1D) interacting fermions exhibit spin-charge separation, a unique feature predicted by the Tomonaga-Luttinger liquid (TLL) theory, but a rigorous understanding remains challenging. Using the thermodynamic Bethe ansatz (TBA) formalism, we analytically derive universal properties of a 1D repulsive spin-$1/2$ Fermi gas with arbitrary interaction strength. We show how spin-charge separation emerges from the exact TBA formalism, and how it is disrupted by the interplay between the two degrees of freedom that brings us beyond the TLL paradigm. Based on the exact low-lying excitation spectra, we further evaluate the spin and charge dynamical structure factors (DSFs). The peaks of the DSFs exhibit distinguishable propagating velocities of spin and charge as functions of interaction strength, which can be observed by Bragg spectroscopy with ultracold atoms.

Figure 1

(a) Contour plot of the Wilson ratio (WR) in the $\tilde{\mu }-\tilde{H}$ plane for the repulsive Fermi gas at $\tilde{T}=0.005$. Here the dimensionless quantities are $\tilde{T}=(T/|c{|}^2)$, $\tilde{\mu }=(\mu /|c{|}^2)$, $\tilde{H}=(H/|c{|}^2)$. The values of the WR given by Eq. (4) elegantly mark three quantum phases: mixed phase (MP), full-polarized phase (FP), and vacuum at zero temperature. At low temperatures, the phase boundaries are indicated by sudden enhancements of the WR, which match well with the zero temperature phase boundaries (black dashed lines). The inset shows the WR vs magnetic field $\tilde{H}$ at $\tilde{\mu }=0.3$ and $\tilde{T}=0.005$, where a sudden enhancement of the WR is observed.

Figure 2

Phase diagram in the $\tilde{T}\text{−}\tilde{H}$ plane: contour plot of specific heat. We set the dimensionless chemical potentials $\tilde{\mu }=2.5$, ${\tilde{H}}_c=2.9145$. The black dashed lines denote the peak positions of specific heat, and the dot-dashed line shows the boundary of the linear $T$ dependence of specific heat. The crossover regions between QC and the TLL are labeled as COR1 and COR2.

Figure 3

Exact low energy excitation spectra in charge (yellow green) and spin (dark green) at $\gamma =c/n=5.03(a_s=700a_0)$ with the Fermi surface $k_F=n\pi$, density $n=N/L=3\times {10}^6(1/\mathrm{m})$, $\mathrm{\Delta }E=\hslash \omega$. The yellow green shows the particle-hole continuum excitation. The black solid lines indicate the thresholds of particle-hole excitation which remarkably manifest the free fermion-like dispersion (9) with an effective mass $m^{*}\approx 1.27m$ at low energy. The black dashed line in the charge excitation stands for the charge velocity $v_c$. The dark green shows the two-spinon excitation, where the black dashed lines stand for the spin velocities $v_s$ near $\mathrm{\Delta }K=0$ and $\hslash k_F$, respectively. The two red dashed lines indicates the positions of excitation momenta in charge and spin sectors for Fig. 4.

Figure 4

Normalized charge and spin DSFs of a homogeneous Fermi gas with parameters corresponding to these of Ref. [24]: length $L=20\mu \mathrm{m}$, particle numbers $N=60$, temperature $T=120\mathrm{nk}$, and various interaction strengths $a_s=400a_0$, $500a_0$, $600a_0$, $700a_0$. Here $a_s$ is the 3D scattering length, which is related to the 1D interaction strength by $c=-2{\hslash }^2/m{a}_{1\mathrm{D}}$ with $a_{1\mathrm{D}}=(-a_{\perp }^2/2a_s)[1-C(a_s/a_{\perp })]$ [52]. In converting to dimensional quantities, we have assumed the atoms are ${}^6\mathrm{Li}$ with transverse harmonic confinement ${\omega }_{\perp }=(2\pi )198\mathrm{kHz}$. (a) Normalized charge DSF [Eq. (10)] vs Bragg frequency $\omega /2\pi$ at $q=1.47\mu {\mathrm{m}}^{-1}$. (b) The empty circles denote the peak frequency of each spectrum vs $\gamma$. The corresponding peak charge velocity $\omega /q$ is given by the right axis. The dashed line is the charge sound velocity obtained from TBA. (c) Normalized spin DSF [Eq. (11)] vs Bragg frequency $\omega /2\pi$ at $\delta k=1.47\mu {\mathrm{m}}^{-1}$. (d) The empty circles denote the peak frequency of each spectrum vs $\gamma$. The corresponding peak spin velocity $\omega /\delta k$ is given by the right axis. Stars are spin sound velocity obtain from the TBA.