Wilson Ratio of Fermi Gases in One Dimension

We calculate the Wilson ratio of the one-dimensional Fermi gas with spin imbalance. The Wilson ratio of attractively interacting fermions is solely determined by the density stiffness and sound velocity of pairs and of excess fermions for the two-component Tomonaga-Luttinger liquid phase. The ratio exhibits anomalous enhancement at the two critical points due to the sudden change in the density of states. Despite a breakdown of the quasiparticle description in one dimension, two important features of the Fermi liquid are retained; namely, the specific heat is linearly proportional to temperature, whereas the susceptibility is independent of temperature. In contrast to the phenomenological Tomonaga-Luttinger liquid parameter, the Wilson ratio provides a powerful parameter for testing universal quantum liquids of interacting fermions in one, two, and three dimensions.

Figure 1

Contour plot of the Wilson ratio $R_W$ [Eq. (1)] of the attractive Fermi gas for dimensionless interaction $|\gamma |=10$ as a function of the reduced temperature $t=T/{\epsilon }_b$ and magnetic field. ${\epsilon }_b$ is the binding energy. The result [Eq. (2)] provides a criterion for the two-component TLL phase in the region below the dashed lines, where $R_W$ is temperature independent. The dashed lines indicate the crossover temperature $T^{*}\sim |H-H_c|$ separating the relativistic liquid from the nonrelativistic liquid. $R_W=0$ for both the TLL of pairs (PP) and the TLL of excess fermions (F). In the critical regimes (CR), $R_W$ gives a temperature-dependent scaling. However, near the two critical points, the ratio reveals anomalous enhancement, discussed further in the text. The inset shows the enhancement at the lower critical point.

Figure 2

The dimensionless susceptibility vs magnetic field for $|\gamma |=10$ at different temperatures. The susceptibility is independent of temperature for $T<H-H_{c1}$ and $T<H_{c2}-H$. Round peaks of the susceptibility in the vicinity of the two critical points are observed at low temperatures. The inset shows the susceptibility for $|\gamma |=5$ and $10$ at $T=0$. The pink crosses denote the analytic result [Eq. (4)] which is in excellent agreement with the numerical results obtained from the field-magnetization relation [19] (red circles) and from the dressed energy equations [28] (blue lines).

Figure 3

Dimensionless specific heat vs polarization for $|\gamma |=10$ at different temperatures. The deviation from linear temperature dependence [Eq. (8)] (red crosses) indicates the breakdown of the two-component TLL. The inset shows a round peak evolved near $H_{c1}$ at $T=0.00001{ϵ}_b$.

Figure 4

Wilson Ration vs polarization for $|\gamma |=10$ at different temperatures. The numerical result obtained from the equation of state fully agrees with the Wilson ratio [Eq. (2)] (red crosses) for the two-component TLL phase. The deviations from the result [Eq. (2)] characterize the crossover temperature $T^{*}$. Anomalous behavior is found near $P=0$ and $P=1$ (see the inset for the region near the critical point $H_{c1}$).