Phase Transitions in Definite Total Spin States of Two-Component Fermi Gases

Second-order phase transitions have no latent heat and are characterized by a change in symmetry. In addition to the conventional symmetric and antisymmetric states under permutations of bosons and fermions, mathematical group-representation theory allows for non-Abelian permutation symmetry. Such symmetry can be hidden in states with defined total spins of spinor gases, which can be formed in optical cavities. The present work shows that the symmetry reveals itself in spin-independent or coordinate-independent properties of these gases, namely as non-Abelian entropy in thermodynamic properties. In weakly interacting Fermi gases, two phases appear associated with fermionic and non-Abelian symmetry under permutations of particle states, respectively. The second-order transitions between the phases are characterized by discontinuities in specific heat. Unlike other phase transitions, the present ones are not caused by interactions and can appear even in ideal gases. Similar effects in Bose gases and strong interactions are discussed.

Figure 1

Cells with average energies ${\overline{ϵ}}_i$ in a single-body energy spectrum. The circles denote the level occupation.

Figure 2

(a) Three allowed Weyl tableaux for ${ϵ}_1={ϵ}_2<{ϵ}_3<{ϵ}_4<{ϵ}_5<{ϵ}_6$ corresponding to the unsaturated phase. The black cells form Young tableaux corresponding to a non-Abelian irrep. (b) A Weyl tableau for ${ϵ}_1={ϵ}_2<{ϵ}_3={ϵ}_4<{ϵ}_5<{ϵ}_6$ corresponding to the saturated phase. The black cells form a one-column Young tableau corresponding to an antisymmetric irrep. (c) The total number of double-occupied levels $q_2$ (blue long dash), the maximal allowed value of $q_2$ (green horizontal short dash), non-Abelian entropy $\ln f_S(q_1)$ (black solid line), and specific heat (per atom) $C_v$ (red dotted-dashed line) at the temperature $T$ for $N=10^2$ two-dimensional particles in a flat potential with the total spin $S=20$. The temperature scale $T_0$ is given by Eq. (4).

Figure 3

Specific heat (per atom) at the temperature $T$ for the state with the defined many-body spin $S$ (black solid lines) and the state with defined individual spins and the total spin projection $S_z=S$ (red dashed lines) of $N=10^2$ two-dimensional particles in a flat potential. The blue line shows the boundary between the saturated (S) and unsaturated (U) phases. The temperature scale $T_0$ is given by Eq. (4).

Figure 4

(a) The ratio of the specific heat on the phase boundary to the one for the gas with defined individual spins in a three-dimensional (3D) harmonic trap for the saturated (solid lines) and unsaturated (dashed lines) phases with $N=10^2$ (black), $N=10^3$ (blue), and $N=10^4$ (red) particles. The temperature scale $T_k(N)$ is defined by Eq. (4). (b) The same ratios for the 3D gas in a flat potential; the black solid and blue long-dashed lines correspond to the saturated and unsaturated phases, respectively. The ratios for a two-dimensional (2D) harmonic trapping are plotted by the red dotted-dashed and magenta short-dashed lines, respectively. All plots are for $N=10^2$. (c) The relative change in the specific heat at the phase boundary for $N=10^2$ particles in flat potentials (black solid and blue long-dashed lines for the 2D and 3D cases, respectively) and in harmonic traps (red dotted-dashed and magenta short-dashed lines, respectively).