Thermodynamics, contact, and density profiles of the repulsive Gaudin-Yang model

We address the problem of computing the thermodynamic properties of the repulsive one-dimensional two-component Fermi gas with contact interaction, also known as the Gaudin-Yang model. Using a specific lattice embedding and the quantum transfer matrix we derive an exact system of only two nonlinear integral equations for the thermodynamics of the homogeneous model which is valid for all temperatures and values of the chemical potential, magnetic field, and coupling strength. This system allows for an easy and extremely accurate calculation of thermodynamic properties circumventing the difficulties associated with the truncation of the thermodynamic Bethe ansatz system of equations. We present extensive results for the densities, polarization, magnetic susceptibility, specific heat, interaction energy, Tan contact, and local correlation function of opposite spins. Our results show that at low and intermediate temperatures the experimentally accessible contact is a nonmonotonic function of the coupling strength. As a function of the temperature the contact presents a pronounced local minimum in the Tonks-Girardeau regime which signals an abrupt change of the momentum distribution in a small interval of temperature. The density profiles of the system in the presence of a harmonic trapping potential are computed using the exact solution of the homogeneous model coupled with the local density approximation. We find that at finite temperature the density profile presents a double shell structure (partially polarized center and fully polarized wings) only when the polarization in the center of the trap is above a critical value which is monotonically increasing with temperature.

Figure 1

(Left panel) Temperature dependence of the grand-canonical potential calculated using Eq. (5) for $\mu =10,H=5$ and $c=1$ (dotted black line), $c=10$ (dashed black line), and $c=50$ (black line). In the limit of impenetrable particles we reproduce Takahashi's result (8) (dashed red line). (Right panel) Relative error $|\varphi -{\varphi }_{TBA}|/|\varphi |$ with $\varphi$ the grand-canonical potential given by Eq. (5) and ${\varphi }_{TBA}$ calculated from the TBA equations (4) for $c=1,\mu =1$ and magnetic fields $H=\{0.01,0.1,0.5\}$. Grand-canonical potential per length and temperature in units of ${\varphi }_0$ and $T_0$ [55].

Figure 2

Plots of the polarization (first row), densities of spin-up and spin-down particles (second row), susceptibility per length normalized by $n/T_d$ (third row), and specific heat per length normalized by $n{k}_B$ (bottom row) as functions of the reduced temperature $\tau =T/T_d=T/n^2$ for four values of the dimensionless coupling strength $\gamma =\{0,1,3,\infty \}$ and magnetic field $H=\{1,2.5,10,50\}$. The total density is fixed to $n=1$.

Figure 3

Plots of the zero-temperature energy density $\mathcal{E}$ at fixed polarization (upper panels, black continuous line) in units of $\mathcal{E}\left(\infty \right)=n^3{\pi }^2/3$ and dimensionless contact $s=\mathcal{C}/k_F^4$ (lower panels, black continuous line) with $k_F=\pi n/2$ as functions of the dimensionless coupling constant $\gamma .$ The dashed blue and dot-dashed red lines represent the asymptotic expansions in the strong and weak interaction limit given by Eqs. (13), (14), and (15). The results presented in the top row can be seen as complementing Fig. 1 of [75].

Figure 4

Plots of the dimensionless contact $s$ (upper panels) local correlation function of opposite spins $g_{\uparrow \downarrow }^{\left(2\right)}\left(0\right)$ (middle panels) and interaction energy $\mathcal{I}$ (lower panels) as functions of the dimensionless coupling constant $\gamma$ for $\tau =\{0,1.3,3,20\}$ and fixed polarization $P=0$ (red line), $P=0.25$ (blue dashed line), $P=0.5$ (black dot-dashed line), and $P=0.75$ (green dotted line). The red disks, blue squares, black triangles, and green diamonds represent the values of the dimensionless contact $s(\gamma =\infty )$ computed from the momentum distribution derived from the Fredholm determinants (17) for $P=\{0,0.25,0.5,0.75\}$. The total density is fixed to $n=1/2$.

Figure 5

Reduced temperature dependence of the dimensionless contact $s$ for $\gamma =\{0.2,2,20,200\}$ and fixed polarization $P=0$ (red line), $P=0.25$ (blue dashed line), $P=0.5$ (black dot-dashed line), and $P=0.75$ (green dotted line). The red disks, blue squares, black triangles, and green diamonds represent the values of the dimensionless contact $s(\tau =0)$ computed from the integral equations (12) for $P=\{0,0.25,0.5,0.75\}$.

Figure 6

Plots of the normalized total density $n\left(x\right)/n\left(0\right)$ (blue continuous line), normalized density of up spins $n_{\uparrow }\left(x\right)/n\left(0\right)$ (blue dashed line), normalized density of down spins $n_{\downarrow }\left(x\right)/n\left(0\right)$ (blue dotted line), and their difference $[n_{\uparrow }\left(x\right)-n_{\downarrow }\left(x\right)]/n\left(0\right)$ (red dot-dashed line) for an inhomogeneous Gaudin-Yang model with $\tau \left(0\right)=0.5$ and polarizations $P\left(0\right)=0.15$ (upper panels) and $P\left(0\right)=0.6$ (lower panels). The insets depict the variation of the polarization $P\left(x\right)=[n_{\uparrow }\left(x\right)-n_{\downarrow }\left(x\right)]/[n_{\uparrow }\left(x\right)+n_{\downarrow }\left(x\right)]$ in the trap. For all values of the coupling strength $\gamma \left(0\right)=\{0,0.61,1.83,\infty \}$, the total density in the center of the trap is $n\left(0\right)=\sqrt{24}$.

Figure 7

Relevant contours and distribution of Bethe roots $\left\{v_i^{\left(1\right)}\right\},\left\{v_j^{\left(2\right)}\right\}$ ($•,\blacksquare$) and holes $\left\{v_i^{\prime \left(1\right)}\right\},\left\{v_j^{\prime \left(2\right)}\right\}$ ($\circ ,\square$) for the largest eigenvalue [$\gamma \in (0,\pi /2)$]. The upper and lower edges of $\mathcal{C}$ are parallel with the real axis and intersect the imaginary axis at $\pm i(\gamma -\varepsilon )/2$ with $\varepsilon \to 0$.