⇦Chinese Physics Letters, 2017, Vol. 34, No. 2, Article code 020501 Spin-1/2 Fermion Gas in One-Dimensional Harmonic Trap with Attractive Delta Function Interaction * Ya-Hui Wang(王亚辉)1**, Zhong-Qi Ma(马中骐)2 Affiliations 1School of Physics and Telecommunication Engineering, Shaanxi Sci-Tech University, Hanzhong 723000 2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049 Received 23 November 2016 *Supported by the National Natural Science Foundation of China under Grant No 11405101 and 11647018, and the Natural Science Foundation of Shaanxi Province under Grant No 2013JK0641.
**Corresponding author. Email: wangyahui8312469@163.com Citation Text: Wang Y H and Ma Z Q 2017 Chin. Phys. Lett. 34 020501 Abstract In terms of the Thomas–Fermi method, we solve the ground state energy of the $N$-body 1D harmonically trapped spin-1/2 fermion gas with the attractive $\delta$-function interaction in the limit $N \to \infty$. DOI:10.1088/0256-307X/34/2/020501 PACS:05.30.Fk, 67.85.-d, 03.75.Hh © 2017 Chinese Physics Society Article Text In recent research[1-3] the ground state energies of the 1D harmonically trapped $N$-body systems, both the spinless Boson gas and the spin-1/2 fermion gas, with the repulsive $\delta$-function interaction in the limit $N \rightarrow \infty$ for fixed $\sqrt{N}/g$ were solved. In solving the problem in the trap, the Thomas–Fermi method was used to treat first the local problem in each interval $dx$, and then to find the best values of the local thermodynamic variables at each $x$ to reach minimum total energy for the whole system in the trap. In this Letter, we generalize the calculation to the fermion gas with the attractive $\delta$-function interaction where the pair of fermions with different spinor states is attracted as a molecule. It is evident that the ground state energy of Boson gas with the attractive $\delta$-function interaction is negative infinity. The Hamiltonian of the 1D harmonically trapped fermion gas with the attractive $\delta$-function interaction is $$\begin{alignat}{1} H=\,&H_{1}+\sum_{i=1}^N V(x_{i}),~V(x_{i})=x^{2}_{i}/2.~~ \tag {1} \end{alignat} $$ $$\begin{alignat}{1} H_{1}=\,&\sum_{i=1}^N-\frac{1}{2} \frac {\partial^2}{\partial x^{2}_{i}}+g \sum_{i>j}^N \delta (x_{i}-x_{j}),~{\rm for}~g < 0.~~ \tag {2} \end{alignat} $$ The ground state space wave functions are assumed to belong to the representation $[2^{M}, 1^{N-2M}]$ of the permutation group $S_{N}$, where $M$ is the number of spin-down fermions. The trapless spin-1/2 fermion problem for $\mathcal{N}$ particles in a periodic 1D interval $\mathcal{L}$ was solved in 1967[4] using Bethe's hypothesis, resulting in two coupled Fredholm equations for the repulsive $\delta$-function interaction. The Fredholm equations were generalized to those for the attractive $\delta$-function interaction in 1970.[5,6] Recently we[7] have presented an accurate approximate expression for the ground state energy $\mathcal{E}_{1}$ of these Fredholm equations with the attractive $\delta$-function interaction, $$\begin{align} \mathcal{E}_{1}=\,&\mathcal{N} g^{2} \zeta (\beta, \xi),~~ \tag {3} \end{align} $$ $$\begin{align} \zeta (\beta, \xi)=\,&2\beta^{2}(1-3\xi+3\xi^{2})\Big\{\pi^{2}\\ &-8\xi^{3} \Big[\frac{3}{2|\beta| (1-3\xi+3\xi^{2})}\\ &+\frac{3(1-3\xi)}{16000 \xi(1-3\xi+3\xi^{2})^{2} \beta^{2}}+ \frac{F(\xi)}{|\beta|^{3}}\\ &+\frac{2-12 \xi+24 \xi^{2}-15 \xi^{3}}{25\pi \xi \beta^{4}} +|\beta|^{-5}\Big]\Big\}\\ &\times\Big\{12+16\xi^{2}(1-3\xi+3\xi^{2})\\ &\times\Big[\frac{9(1-2\xi)(1+\xi)}{2\pi^{2}\xi (1-3\xi+3\xi^{2})^{2}|\beta|}\\ &+\frac{4(2-12 \xi+24 \xi^{2}-15 \xi^{3})}{25\pi \xi\beta^{2}}+\frac{4}{|\beta|^{3}}\Big]\Big\}^{-1}, \\ F(\xi)=\,&\{27(1-2\xi)(1+\xi)-8\pi^{4}(2-12 \xi\\ &+24 \xi^{2}-15 \xi^{3})(1-3\xi+3\xi^{2})^{2}\}\\ &\times \{24\pi^{2}\xi(1-3\xi+3\xi^{2})^{2}\}^{-1},~~ \tag {4} \end{align} $$ where $\mathcal{N}_{1}=\mathcal{N}-\mathcal{M}$ and $\mathcal{N}_{2}=\mathcal{M}$ are the numbers of the spin-up and spin-down fermions in the interval $\mathcal{L}$, respectively, $$\begin{alignat}{1} \xi=\mathcal{M}/\mathcal{N},~~\rho=\mathcal{N}/ \mathcal{L}, ~~\beta=\mathcal{N}/(\mathcal{L}g)=\rho/g,~~ \tag {5} \end{alignat} $$ and the symmetry for the system in the interval is denoted by $[2^{\mathcal{M}},1^{\mathcal{N}-\mathcal{M}}]$. In the Thomas–Fermi method, $\mathcal{E}_{1}/\mathcal{L}$ is the energy density of the trapless problem with the attractive $\delta$-function interaction in the interval $dx$. Equilibrium demands $\delta E=0$ under constraints $$ \int g \beta dx=N,~~~~\int g \beta \xi dx=M,~~ \tag {6} $$ where the total energy $E$ of the ground state of the system is $$\begin{align} E=\,&\int dx\frac{\mathcal{N}_{1}+\mathcal{N}_{2}}{\mathcal{L}} g^{2} \zeta \Big(\frac{\mathcal{N}_{1}+\mathcal{N}_{2}}{\mathcal{L}g}, \frac{\mathcal{N}_{2}}{\mathcal{N}_{1}+\mathcal{N}_{2}}\Big) \\ &+\int dx\frac{\mathcal{N}_{1}+\mathcal{N}_{2}}{\mathcal{L}} V(x).~~ \tag {7} \end{align} $$ Following the calculation in Ref. [3], we also obtain two variation relations $$\begin{align} \frac{\partial \zeta} {\partial \xi}=-A,~~ \tag {8} \end{align} $$ $$\begin{align} \frac{x^2}{2g^{2}}+\zeta+\beta\frac{\partial \zeta}{\partial \beta}+A \xi =K^{2},~~ \tag {9} \end{align} $$ where $K^{2}$ and $A$ are Lagrange's multipliers which are independent of $x$. The value of $\xi$ for the whole system in the trap is $$ \xi_{\rm S}=M/N=[\int g \beta \xi dx]/[\int g \beta dx ].~~ \tag {10} $$ Note that in the Thomas–Fermi method, the variation $d \mathcal{E}_1$ of the energy of the trapless fermions in length $dx$ is $$ d \mathcal{E}_1=-pd \mathcal{L}+\mu_1 d \mathcal{N}_1+\mu_2 d \mathcal{N}_2, $$ where $p$ is the pressure, $\mu_{1}$ and $\mu_{2}$ are two chemical potentials with respect to the numbers $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$, respectively.
cpl-34-2-020501-fig1.png
Fig. 1. The curve of $\omega(\beta,A)$ versus $|\beta|$ for two different $A$.
We solve $\xi$ from Eq. (8) as $\xi=\omega(\beta,A)$, where $\zeta$ is given in Eq. (4). Figure 1 shows the curve of $\omega$ versus $|\beta|$ for a given $A$, where $\omega$ decreases monotonically from infinity to zero as $\beta$ decreases from infinity to a positive number $\beta_{0}$. Thus $\beta_{0}$ is a function of $A$ satisfying $$\begin{align} \frac{\partial \zeta(\beta_{0},\xi)} {\partial \xi}\Big|_{\xi=0}=-A,~~ \tag {11} \end{align} $$ and $\omega(\beta,A)$ can be rewritten as ${\it \Omega}(\beta,\beta_{0})=\omega(\beta,A)$. As $\beta$ decreases from $\beta_{0}$ again, $\xi=\omega(\beta,A)$ becomes negative which is not allowed. That is, Eq. (8) no longer holds for $\beta < \beta_{0}$ because $\partial \xi=0$. Thus we obtain $$ \xi(\beta,\beta_{0}) =\begin{cases} \!\! {\it \Omega}(\beta,\beta_{0}), &\beta \geq \beta_0,\\\!\! 0, &0\leq \beta < \beta_{0}. \end{cases}~~ \tag {12} $$ Making use of Eq. (12), Eq. (9) becomes $$\begin{align} K^2-\frac{x^2}{2g^2}=\,&T(\beta,\beta_{0})\\ =\,&\begin{cases} \!\! \zeta+\beta \frac{\partial \zeta}{\partial \beta} +A {\it \Omega},& \beta\geq \beta_{0},\\\!\! \frac{\pi^{2}\beta^{2}}{2},&\beta < \beta_{0}. \end{cases}~~ \tag {13} \end{align} $$ Figure 2 shows that $T(\beta,\beta_{0})$ is monotonic in $\beta$ at fixed $\beta_{0}$. Thus from Eq. (13), $T(\beta,\beta_{0})$ assumes its maximum value $K^{2}$ when $x=0$, $$ K^{2}=T(\beta_{\max},\beta_{0}).~~ \tag {14} $$
cpl-34-2-020501-fig2.png
Fig. 2. The curve of $T(\beta,\beta_{0})$ versus $|\beta|$ for two different $\beta_{0}$.
cpl-34-2-020501-fig3.png
Fig. 3. The curve of $\beta_{0}$ versus $\beta_{\max}$ for different $M/N$.
Now, two Lagrange multipliers $A$ and $K^{2}$ can be replaced equivalently by two other parameters $\beta_{0}$ and $\beta_{\max}$. As $x$ increases from $x=0$ to $x=x_{\max}=\sqrt{2} gK$, $\beta$ decreases from $\beta_{\max}$ to 0. By integration by parts we rewrite the constraints (6) as $$\begin{alignat}{1} \frac {N}{g^2} =\,&2\sqrt{2}\int_{0}^{\beta_{\max}} \sqrt{K^{2}-T}d\beta,~~ \tag {15} \end{alignat} $$ $$\begin{alignat}{1} \frac {M}{g^2} =\,&2\sqrt{2}\int_{\beta_{0}}^{\beta_{\max}} \sqrt{K^{2}-T}\Big[\xi+\beta \frac{d\xi}{d\beta}\Big]d\beta.~~ \tag {16} \end{alignat} $$ Making numerical calculations with Eqs. (15) and (16) we obtain Fig. 3 showing the similar property to the repulsive interaction, the curves of equal $\xi_{\rm S}=M/N$ in $(\beta_{\max}, \beta_0)$ space are monotonic and along each curve, $N/g^2$ and $M/g^2$ both increase monotonically with increasing $\beta_{\max}$. All physically allowed values of $(N/g^2, M/g^2)$ are mapped one-to-one to the space in $(\beta_{\max}$, $\beta_0)$ bounded by $\beta_0=0$ with $\xi_{\rm S}=1/2$, and $\beta_{\max}=\beta_0$ with $\xi_{\rm S}=0$.
cpl-34-2-020501-fig4.png
Fig. 4. (a) The curves of $\rho/\sqrt{2N}$ versus $x/\sqrt{2N}$ for $\xi_{\rm S}=0$. (b) The curves of $\rho/\sqrt{2N}$ versus $x/\sqrt{2N}$ for $\xi_{\rm S}=1/2$. (c) The curves of $\rho/\sqrt{2N}$ versus $x/\sqrt{2N}$ for $\xi_{\rm S}=1/10$.
Letting $\gamma=\sqrt{N}/g$, we can rewrite Eq. (13) as $$ \frac{K^2}{\gamma^2}- \Big(\frac {x}{\sqrt{2N}}\Big)^2=\frac{1}{\gamma^2} T \Big(\sqrt{2} \gamma \frac{\rho}{\sqrt{2N}}, \beta_{0}\Big),~~ \tag {17} $$ giving a relationship between $\rho/\sqrt{2N}$ and $x/\sqrt{2N}$ for fermions. For $\xi_{\rm S}=0$ (i.e., $\beta_{\max}=\beta_0$), this relationship is a half ellipse and is the same for all values of $g/\sqrt{N}$ (Fig. 4(a)). The curves for $\xi_{\rm S}=1/2$ (i.e., $\beta_{0}=0$) and for the remaining $\xi_{\rm S}$, that is, $\xi_{\rm S}=0.1$, are given in Figs. 4(b) and 4(c), respectively, for several values of $g/\sqrt{N}$, where we do not observe a kink evidently. Since $\xi$ is determined from Eq. (12) when $\beta=\rho g$ and $\beta_0$ are given, the curve of $\xi$ versus $x/\sqrt{2N}$ (see Fig. 5 for $\xi_{\rm S}=0.1$) can be drawn from Eq. (17), which gives the dependence of $\rho/\sqrt{2N}$ on $x/\sqrt{2N}$ with given $\xi_{\rm S}$. Although a kink exists in the density $\delta$-function interaction and for the attractive one, the kink does not occur evidently in the $\rho$–$x$ curves for the attractive $\delta$-function interaction.
cpl-34-2-020501-fig5.png
Fig. 5. The curves of $\xi$ versus $x/\sqrt{2N}$.
cpl-34-2-020501-fig6.png
Fig. 6. The curves of $E/N^{2}$ versus $|g/\sqrt{N}|$.
The energy $E$ of the ground state of the whole system as $N\rightarrow \infty$ at fixed $\sqrt{N}/g$ can be calculated from Eq. (7), $$\begin{alignat}{1} \frac{E}{g^{4}} =\,&g^{-2}\int_{-x_{\max}}^{x_{\max}}\rho[\zeta+ x^{2}/(2g^{2})]dx\\ =\,&2\sqrt{2}\int_{0}^{\beta_{\max}}\sqrt{K^{2}-T}\\ &\times \Big\{K^{2}-\frac{d}{d \beta}\Big(\beta^{2}\frac{\partial \zeta}{\partial \beta}\Big) +A\frac{d{\it \Omega}}{d \beta}\Big\}d\beta,~~ \tag {18} \end{alignat} $$ where $K^{2}$, $\beta_{0}$ and $A$, are constants. It is clear from Eqs. (15) and (18) that the ratio $E/N^{2}$ is dependent only on parameters $\beta_{\max}$ and $\beta_{0}$. That is, the conclusion that $E/N^{2}$ is dependent only on $\sqrt{N}/g$ and $M/N= \xi_{\rm S}$ as $N\rightarrow \infty$ proves the conjecture made in a recent study.[8] Figure 6 shows the curves of $E/N^2$ versus $g/\sqrt{N}=1/ \gamma $ for several values of $\xi_{\rm S}$ for spin-1/2 fermions. The curves connect with those for the repulsive interaction at $g/\sqrt{N}=0$, and now we calculate the value of $E/N^2$ for all values of $g/\sqrt{N}$ and $\xi_{\rm S}$ for the trapped spin-1/2 fermion gas, with both the repulsive $\delta$-function interaction and the attractive one. We believe that the Thomas–Fermi method used in the present work, as well as in Refs. [1–3], gives the exact limit when $N \rightarrow \infty$ for fixed $g/\sqrt{N}$. Although at present we do not have rigorous proof of this statement, the statement in a ferromagnetic state, $M=0$, can be proved by directly checking because in this state the ground state energy of the $N$-body 1D harmonically trapped spin-1/2 fermion gas with the $\delta$-function interaction, for both repulsive and attractive interactions can be exactly solved. The curves in this study contain errors due to inaccuracies in fitting Eq. (4) and in subsequent integrations. However, for any required final accuracy, all these inaccuracies can be eliminated by giving sufficient computer time. In this sense, we have the exact limit as $N \rightarrow \infty$ at fixed $g/ \sqrt{N}$, for both repulsive and attractive interactions. From the variation relations of the Thomas–Fermi method, we find that kinks do not occur evidently in the density distribution $\rho(x)$ for fermion gas (Fig. 4(c)) with the attractive $\delta$-function interaction. We thank Professor Chen N. Y. for his valuable advice. References Spinless Bosons in a 1D Harmonic Trap with Repulsive Delta Function Interparticle Interaction I: General TheorySpinless Bosons in a 1D Harmonic Trap with Repulsive Delta Function Interparticle Interaction II: Numerical SolutionsSpin ½ Fermions in 1D Harmonic Trap with Repulsive Delta Function Interparticle InteractionSome Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function InteractionMany-Body Problem of Attractive Fermions with Arbitrary Spin in One DimensionGround State Energy of 1D Attractive δ-Function Interacting Fermi GasGround State Energy for Fermions in a 1D Harmonic Trap with Delta Function Interaction
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