Spontaneous separation of large-spin Fermi gas in the harmonic trap: a density functional study

The component separation of the trapped large-spin Fermi gas is studied within density functional theory. The ground state and ferromagnetic transition in the gas, with and without the spin mixing collision, are calculated. In the absence of spin mixing, two patterns of separation are observed as the interaction between atoms increases, whereas only one of them corresponds to a ferromagnetic transition. The phase diagram suggests that the pattern which the system chooses depends on the interaction strength in the collision channels. With the presence of spin mixing, the distribution of phase region changes because of the interplay between different collision channels. Specifically, the spin exchange benefits the FM transition, while it suppresses the component separation of CS-II pattern.

display novel phase behaviors or new patterns for the formation of domain and texture, which can cast new lights on the understanding about the FM transition in quantum gas.

Model and Theory
In this work, we consider a confined LSFG, which consists of atoms with hyperfine spin = f 3 2 . The spinor character implies that the Fermi gas can be treated as a four-components mixture with pseudo-spin σ = ± ± , . Accordingly, the Hamiltonian of the system can be given by: Here ω = V m r trap 1 2 2 2 is the spin-independent external potential applied by the trap and V (2) the spin-dependent pair potential between atoms. Ψ σ is the atomic field annihilation operator associating the hyperfine spin state σ | = = 〉 f m , f 3 2 . Note that in Eq. (1), the tilde is used to distinguish the qualities from their reduced forms, which will be defined below. The factor 1 2 is added in the second term to avoid overcounting in the summation. In the low energy regime, the s-wave scattering dominates the collision processes and the interaction can be modeled by the contact potential, i.e., (2) . The coupling coefficient U ijkl can be obtained from the two-body interaction model = +ˆÛ g P g P 0 0 2 2 , with the projection operator P F . Moreover, with a F denoting the s-wave scattering length in total F spin channel. Note that only the even values of F are relevant due to the symmetry of the wave functions in the s-wave channel.
In order to specify the contribution from different collision channels, it is convenient to decompose the interaction Hamiltonian into three parts, i.e., H inter , H intra and H mix : with t = inter, intra, mix and A t the corresponding coupling parameters. The possible collision channels in the spin- 3 2 Fermi gas have been shown in Fig. 1. Specifically, H inter and H intra describe respectively the contribution from the atom collision between atoms of the symmetrical and asymmetrical spin orientation, while H mix takes into account the contribution form the spin mixing collision. Projecting the two-body interaction to the total spin space, one obtains the coupling coefficients in terms of g F 40 : = . Note that in this work, we choose ∼ A inter and ∼ A intra as the independent parameters, which give the third one by Interaction between atoms of the same spin orientation is absent because of the Pauli exclusion.
On the theoretical side, density functional theory 41 is a powerful tool which of several theoretical superiorities, including exact mathematical framework and inexpensive numerical cost. With the proper approximation for the exchange-correlation energy, good performance has been shown in its application to the spin-1 2 Fermi gas [42][43][44] . Among the available treatments, local density approximation (LDA) is favored due to its relative simplicity and efficiency in prediction of FM transition. Especially for the two-components trapped Fermi gas, the critical scattering length predicted by LDA is in good agreement with that obtained from experiments 8   surface tension term in LDA may be important, especially near the boundary of the atom cloud. However, it depends largely on the atom number in the cloud. For a gas with large atom number, the inclusion of surface effect leads to nonsignificant difference in its comparison with LDA results 44 . Therefore, it is believed that the application of LDA in LSFG can also provide qualitative predictions for the FM transition, though the correlation in it should be even more complex. Prior to performing calculations for the ground state, we construct firstly the density functional for the LSFG in the following form: Obviously, the kinetic contribution is given in the Thomas-Fermi form, which treats the kinetic energy only as a correction. Actually, the validity of this approximation is restricted by the criterion are respectively the number of atoms, the s-wave scattering length and the quantum mechanical length scale for the oscillator. Thence, the Thomas-Fermi approximation for the kinetic contribution can be effective so long as N is large enough. The detail for the construction of the interaction energy functional is described in the section Method.
For simplicity in the further calculation, it is necessary to transform the related qualities in Eq.
(3) to their reduced forms. Here we introduce the parameters c 1 ~ c 6 , which satisfy: N and λ σ  are respectively the particle number and chemical potential of spin-σ component. These six above-defined parameters can reduce the total energy to the following form: To ensure the conservation of the particle number of each spin component, the Lagrange multiplier λ σ should be introduced, which relates to the chemical potential of the spin-σ component. Therefore, the density of each component in their ground state can be determined by the Euler equations derived from the variational principle: with ∫ = σ σ r r N d n ( ) the reduced particle number of the spin σ-component.

Results and Discussion
From the energetic point of view, component separation and spin mixing are two ways for the LSFG to lower its total energy. As in spin- 1 2 Fermi gas, the competition between repulsive interaction and kinetic energy is responsible for the FM transition. The former tends to induce polarization, while the latter prefers to equally populate each component in local regions. In the LSFG, more collision channels between different spin components are opened, which may supply new alternatives to lower the total energy. In the following, the FM transition and spin mixing process are studied through the calculation of the ground density profile of each component. Note that in this work, the particle number of each component is set as 10 6 , which can be reduced to N σ = 0.1 in the calculation.
Coupling with A inter = A intra and A mix = 0. For simplicity, calculations are firstly performed for the LSFG with regular collision channels. Specifically, in the absence of spin mixing collision, the interaction energy E int can be given by: Note that with further simplification of A inter = A intra , the energy functional in Eq. (6) shows SU(4) symmetry, which indicates that there should exist degenerate ground states because the total energy keeps unchanged as the order of the subscripts is arbitrarily exchanged. The results in Fig. 2 suggest that the coupling strength is the critical factor for the occurrence of CS, which does not appear until the coupling is as strong as A inter = A intra = 1.08. This is similar to the spin-1 2 Fermi gas, whose FM transition occurs at the Stoner point. As expected, the CS state is enhanced as the coupling strength is increased. Note that the density profile shown in Fig. 2 is only one of the degenerate states of the LSFG because our further calculation verifies that all of the possible degenerate states have the same ground energy. Therefore, it is concluded that in the case of A inter = A intra , the LSFG behaves like a two-components system, because the energy consumption in the FM transition is the same as that in the CS between spins with asymmetric orientation.

Coupling with A inter ≠ A intra and A mix = 0.
To get more information about the influence of coupling strength on the component separation, further calculations are performed in the case of A inter ≠ A intra . It is also hoped that the results in this section can provide information which helps us to understand the effect of spin mixing on the FM transition. Obviously, the hidden SU(4) symmetry has been broken due to the difference between A inter and A intra . The results in Figs 3 and 4 show respectively the influence of A intra (A inter ) on the CS for the given coupling parameter A inter (A intra ) = 1.10.
As specifically shown in Fig. 3, when A intra is relatively small, the CS occurs between atoms with asymmetrical spin orientation, i.e., ± 1 2 and ± 3 2 species. This pattern of CS is denoted by CS-I, in which the two components with symmetrical spin orientation always have the same local population. Moreover, with the enhancement of A intra , this type of CS is suppressed, as shown by Fig. 3(b). However, when A intra is large enough, the separation occurs between atoms with symmetrical spin orientation, i.e., + ), which is denoted by CS-II. Unlike the case of CS-I, the further increase of A intra enhances the CS-II pattern, as compared by Fig. 3(c,d). The difference between these two patterns of CS can be understood from Eq. (6), which indicates that the coupling strength A inter relates to the collision between spins with asymmetrical orientation, while A intra to that between spins with symmetrical orientation.
Note that though two patterns of CS state have been observed during the variation of the coupling parameters, only the CS-II pattern corresponds to the FM state because the non-zero local spin magnetic moment is formed only in this case. As a comparison, the ground states for the given A intra and different A inter are also calculated in Fig. 4. The separation with patterns of CS-I and CS-II are also observed during the adjustment of A inter .
Phase diagram of the LSFG with A mix = 0. To obtain more comprehensive understanding about the occurrence of CS, we have calculated the phase diagram in Fig. 5, which depicts the critical coupling strength that   induces the CS state in the LSFG. In the calculation, we choose the two dependent parameters, A inter and A intra , as the variables in the parameter space. Note that in the calculation, only the s-wave contact interaction on the repulsive side is taken into account, that is, both A inter and A intra are set to be positive. Our results suggest that there are three phase regions whose boundaries show nearly in linear pattern. Moreover, a triple point is found at A inter = A intra = 1.08 in the diagram, where three phases coexist. Further, our results show that the CS-II pattern takes place only if A intra is large enough, and that for larger A inter , a larger critical value of A intra is needed to trigger the FM transition. This implies that the coupling between spin ± 3 2 and ± 1 2 components suppresses the occurrence of FM phase.
The distribution of the phase regions can be easily understood according to the energetic analyze about Eq. (6). When the coupling strength A inter (A intra ) is increased, its corresponding contribution to the total ground energy is also enhanced, which prefers to trigger the CS-I(II) pattern because their advantage in the competition with the kinetic energy. To clarify the separation in the phase diagram, we choose six points (A)-(F) in different phase regions. Density profiles correspond to these points are plotted in Fig. 6, which shows the structure evolution from one pattern to another, when one coupling parameter varies while the other keeps unchanged.
Coupling with A mix ≠ 0. Compared to the above investigation with A mix = 0, more information is expected when the spin mixing collision is taken into account. From the energetic point of view, the spin mixing opens another channel for the spin components to lower the total energy. Therefore, the effect of the spin mixing on the CS in the LSFG should be investigated. Before performing calculation, a quality, say δ ≡ − ± ± N N 2 1 2 3 2 , is firstly defined to describe the amount of the atoms that change their spin quantum number from ± 3 2 to ± 1 2 . Accordingly, the ground state with spin mixing can be determined by comparing the total ground energy of the system with different values of δ.
Note that in the calculation, two schemes have been employed to obtain the ground state densities. One is to assume that the separation of the CS-I pattern, while the other assume it of the CS-II pattern. The ground state is determined by comparing the total energy of the two patterns. Following this routine, the calculation is performed for the phase diagram, which shows clearly the effect of spin mixing collision on the phase separation. The phase diagram for A mix ≠ 0 is presented in Fig. 7. Comparing with the results for the case of A mix = 0 in Fig. 5, it is obvious that the boundary lines of the phase region have been rotated around the triple-point S(1.08, 1.08), which is stationary due to the relationship between A mix and the other two coupling parameters: The hidden physics in the diagram can be understood as follow. Firstly, the CS-II pattern separation occurs only in the region above the diagonal dash line, which is in accordance with the case of A mix = 0 shown in Fig. 5. This is because in this region, the energy contribution from intra-component collision dominates over that from inter-component collision. Secondly, for a given A inter < 1.08, the critical value of A intra has been declined because of the introduction the spin mixing collision, which helps the intra-component repulsion in its competition with the kinetic energy. This effect is even more significant especially for a smaller value of A inter , because it leads to a larger A mix . Therefore, the spin mixing collision is benefit to the occurrence of FM transition. Thirdly, for the given A inter > 1.08, the spin mixing collision leads also to significant effect on the critical value of A intra that triggers the CS-I pattern separation. That is, the CS-I pattern separation does not take place for all values of A inter < 1.08. Moreover, with the increase of A inter , a smaller value of A intra is required to triggers the CS-I pattern separation. This is because in the lower-right region of the parameter space, the fact A inter > A intra indicates that A mix should be Summary. In this work, the component separation in trapped LSFG is studied within the framework of density functional theory. The ground state density profile of each spin component is calculated. Our calculation suggests that when the spin mixing collision is absent, two patterns of CS take place, among which only the pattern-II separation corresponds to the itinerant FM state because of the formation of the local polarization in this case. Phase diagram shows that the coupling parameters A inter and A intra relate respectively to CS of pattern-I and pattern-II. On the other hand, when the spin mixing is taken into account, the phase distribution in the parameter space changes due to the newly opened collision channel. The phase diagram shows the interplay between the CS and spin exchange. That is, the spin exchange benefits the occurrence of the CS-II pattern separation, while it suppresses the CS-II pattern separation. Therefore, the spin mixing collision in LSFG plays a positive role in the detection of FM phase. It is hoped that our results can provide useful insight for the investigation of the FM transition in LSFG.
As an end for this section, we comment on the experimental feasibility of the observation of the CS in LSFG. Actually, the experimental setup for spin dynamics in LSFG [33][34][35][36] can be shared to examine the results in this work. The system can be initially prepared with a balanced spin mixture with σ = ± 1 2 , which is confined in a harmonic  . To access the off-diagonal element of the density matrix, short radio-frequency pulses should be applied, which rotates the spin with ϑ in spin space, and then results in a coupling of all possible spin components, whose magnetic quantum numbers ranges from m = − F, ··· , F. Finally, the Stern-Gerlach method can be used to determine the diagonal element 45 , which contains information of the off-diagonal element because they relate with each other through the rotation matrix.

Methods
In this section, we give some details about the construction of the energy functional. As shown in Eq. (2), the interaction Hamiltonian has been decomposed into three parts, i.e., H inter , H intra and H mix . Firstly, we calculate the expectation of H inter :  where the second term on the right-hand side describes the correlation between atoms belong to different spin components. In fact, in most of the studies on the two-components Fermi gas, the second term in Eq. (10) is usually ignored, and the qualitatively correct result can be achieved in prediction for FM transition 6,23,44 . Therefore, it is hoped that the extension of such a treatment to the study of LSFG can also give qualitative results, though the total energy has been overestimated by doing this. Introducing the local density operator ρ = Ψ Ψ σ σ σ˜ † r r r ( ) ( ) ( ), and assuming each spin component is highly occupied, the expectation of H inter is obtained as: