Thermodynamics of one-dimensional SU(4) and SU(6) fermions with attractive interactions

Motivated by advances in the manipulation and detection of ultracold atoms with multiple internal degrees of freedom, we present a finite-temperature lattice Monte Carlo calculation of the density and pressure equations of state, as well as Tan's contact, of attractively interacting SU(4)- and SU(6)-symmetric fermion systems in one spatial dimension. We also furnish a nonperturbative proof of a universal relation whereby quantities computable in the SU(2) case completely determine the virial coefficients of the SU($N_f$) case. These one-dimensional systems are appealing because they can be experimentally realized in highly constrained traps and because of the dominant role played by correlations. The latter are typically nonperturbative and are crucial for understanding ground states and quantum phase transitions. While quantum fluctuations are typically overpowered by thermal ones in one and two dimensions at any finite temperature, we find that quantum effects do leave their imprint in thermodynamic quantities. Our calculations show that the additional degrees of freedom, relative to the SU(2) case, provide a dramatic enhancement of the density and pressure (in units of their noninteracting counterparts) in a wide region around vanishing $\beta \mu$, where $\beta$ is the inverse temperature and $\mu$ the chemical potential. As shown recently in experiments, the thermodynamics we explore here can be measured in a controlled and precise fashion in highly constrained traps and optical lattices. Our results are a prediction for such experiments in one dimension with atoms of high nuclear spin.

Figure 1

Density $n$ for $N_f^{}=4$ (top) and $N_f^{}=6$ (bottom), in units of the density of the noninteracting system $n_0^{}$ (inset), as a function of the dimensionless parameters $\beta \mu =lnz$ and ${\lambda }^2=\beta g^2$. From bottom to top, the coupling is $\lambda =1.75$, 2.0, 2.25, 2.5, 2.75, 3.0. The data points come from the quantum Monte Carlo calculations and the solid lines are from the fits [see Eq. (32)].

Figure 2

Pressure for $N_f=4$ (top) and $N_f=6$ (bottom) in units of its noninteracting counterpart, as a function of the dimensionless parameters $\beta \mu =lnz$ and ${\lambda }^2=\beta g^2$, obtained by $\beta \mu$ integration of the density [see Eq. (7)]. The values of $\lambda$ shown in this plot are the same as in Fig. 1.

Figure 3

Isothermal compressibility for $N_f=4$ (top) and $N_f=6$ (bottom) in units of its noninteracting counterpart, as a function of the dimensionless parameters $\beta \mu =lnz$ and ${\lambda }^2=\beta g^2$. The values of $\lambda$ range from 0 to 3.0 in steps of 0.125.

Figure 4

Tan's contact $\mathcal{C}$ for $N_f=4$ (top) and $N_f=6$ (bottom), scaled by $\beta {\lambda }_T^{}/\left(2Q_1^{}{\lambda }^2\right)=\pi {\beta }^2/\left(2B_{N_f}L{\lambda }^2\right)$, as in Ref. [21], as a function of $\beta \mu$, for $\lambda =1.75$, 2.0, 2.25, 2.5, 2.75, 3.0, which appear from bottom to top. The value $B_{N_f}$ is the binomial coefficient $N_f$ choose 2; this scale factor was chosen to facilitate comparison between flavors.

Figure 5

Fit parameter $A$ (top), $\xi$ (middle), and $b$ (bottom) of Eq. (32) as a functions of the interaction strength $\lambda$ for $N_f=2,4,6$. The data points are the results obtained by fitting the Monte Carlo data; the solid lines are the fits to these data.

Figure 6

Density $n$ for $N_f=4$, in units of the noninteracting density $n_0^{}$, as a function of $\beta \mu$ at weak coupling ($\lambda =1.0$, top) and at the strongest coupling in this study ($\lambda =3.0$, bottom), for several values of $\beta$. Finite-$\beta$ effects are clearly visible, especially around the maximum. Note the ranges in the $x$ and $y$ axes are different from those of Fig. 1.

Figure 7

Density $n$ for $N_f=6$, in units of the noninteracting density $n_0^{}$, as a function of $\beta \mu$ at weak coupling ($\lambda =1.0$, top) and at the strongest coupling in this study ($\lambda =3.0$, bottom), for several values of $\beta$. Finite-$\beta$ effects are clearly visible, especially around the maximum. Note the ranges in the $x$ and $y$ axes are different from those of Fig. 1.