Finite range and upper branch effects on itinerant ferromagnetism in repulsive Fermi gases: Bethe–Goldstone ladder resummation approach

Highlights

• Nonperturbative interaction energy is obtained within the Bethe–Goldstone ladder resummation approach.

• Positive and negative effective ranges have opposite effects on the critical gas parameter.

• The upper branch Fermi gas exhibits an energy maximum and reentrant ferromagnetic transition.

• The ferromagnetic phase disappears for large and negative effective ranges.

Abstract

We investigate the ferromagnetic transition in repulsive Fermi gases at zero temperature with upper branch and effective range effects. Based on a general effective Lagrangian that reproduces precisely the two-body $s$-wave scattering phase shift, we obtain a nonperturbative expression of the energy density as a function of the polarization by using the Bethe–Goldstone ladder resummation. For hard sphere potential, the predicted critical gas parameter $k_{F}a=0.816$ and the spin susceptibility agree well with the results from fixed-node diffusion Monte Carlo calculations. In general, positive and negative effective ranges have opposite effects on the critical gas parameter $k_{F}a$: While a positive effective range reduces the critical gas parameter, a negative effective range increases it. For attractive potential or Feshbach resonance model, the many-body upper branch exhibits an energy maximum at $k_{F}a=\alpha$ with $\alpha =1.34$ from the Bethe–Goldstone ladder resummation, which is qualitatively consistent with experimental results. The many-body T-matrix has a positive-energy pole for $k_{F}a>\alpha$ and it becomes impossible to distinguish the bound state and the scattering state. These positive-energy bound states become occupied and therefore the upper branch reaches an energy maximum at $k_{F}a=\alpha$. In the zero range limit, there exists a narrow window ($0.86<k_{F}a<1.56$) for the ferromagnetic phase. At sufficiently large negative effective range, the ferromagnetic phase disappears. On the other hand, the appearance of positive-energy bound state resonantly enhances the two-body decay rate around $k_{F}a=\alpha$ and may prevent the study of equilibrium phases and ferromagnetism of the upper branch Fermi gas.

Introduction

Itinerant ferromagnetism in repulsive Fermi systems is a longstanding problem in many-body physics, which can be dated back to the basic model proposed by Stoner  [1]. The physical picture of the ferromagnetism in repulsive Fermi systems can be understood as a result of the competition between the repulsive interaction and the Pauli exclusion principle. The former tends to induce polarization or magnetization and reduce the interaction energy, while the latter prefers balanced spin populations to reduce the kinetic energy. The reduced interaction energy for a polarized state finally overcomes the gain in kinetic energy at a critical repulsion where the ferromagnetic phase transition (FMPT) occurs. It is generally thought that a dilute spin-$\frac{1}{2}$ (two-component) Fermi gas with short-ranged repulsive interaction may serve as a clean system to simulate the Stoner model  [2].

Quantitatively, to study the FMPT in cold repulsive Fermi gases, we need to know the energy density $\mathcal{E}$ of the system as a function of the spin polarization or magnetization $x=(n_{\uparrow }-n_{\downarrow })/(n_{\uparrow }+n_{\downarrow })$ for given interaction strength  [2]. Formally, the energy density can be expressed as

$$\mathcal{E}\left(x\right)=\frac{3}{5}n{E}_{F}f\left(x\right)\text{,}$$

where $E_F=k_F^2/\left(2M\right)$ is the Fermi energy with $M$ being the fermion mass, $k_F$ is the Fermi momentum related to the total density $n=n_{\uparrow }+n_{\downarrow }$ by $n=k_F^3/\left(3{\pi }^2\right)$. The dimensionless function $f\left(x\right)$ represents the energy landscape with respect to the magnetization $x$. If we consider only the $s$-wave contribution, the function $f\left(x\right)$ depends on the $s$-wave gas parameters $k_{F}a$, $k_F{r}_e$, etc. Here $a$ is the $s$-wave scattering length and $r_e$ is the $s$-wave effective range. The naive mean-field or Hartree–Fock approximation predicts a critical gas parameter $k_{F}a=\pi /2$ in the dilute limit  [2].

For the nature of the FMPT, Belitz, Kirkpatrick, and Vojta (BKV)  [3] have argued that the phase transition in clean itinerant ferromagnets is generically of first order at low temperatures, if it occurs at weak coupling. This is because of the coupling of the order parameter to the gapless modes that leads to a nonanalytic term in the free energy. The general form of the Ginzburg–Landau free energy for clean itinerant ferromagnets takes the form $f_{\mathrm{GL}}\left(x\right)=\alpha x^2+\upsilon x^{4}ln\left|x\right|+\beta x^4+O\left(x^6\right)$, where we can keep $\beta >0$. If the coefficient $\upsilon$ is positive, the phase transition is always of first order. On the other hand, for negative $\upsilon$, one always finds a second-order phase transition. For many solid-state systems where the FMPT occurs at weak coupling, the assumption of $\upsilon >0$ is true according to the perturbative calculation  [3]. However, for dilute Fermi gases where the critical gas parameter $k_{F}a$ is expected to be of order $O\left(1\right)$, the assumption of a positive $\upsilon$ is not reliable. In Ref.  [4], the authors found that the FMPT is of second order when the ladder diagrams are resummed to all orders in the gas parameter $k_{F}a$. Similar conclusion was also obtained in  [5] by using the lowest order constrained variational (LOCV) approach [6].

There have been some quantum Monte Carlo calculations  [7], [8], [9], [10] and numerous theoretical studies  [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41] of itinerant ferromagnetism in atomic Fermi gases. To study the ferromagnetism experimentally, one needs to realize a two-component “repulsive” gas of fermionic atoms by rapidly quenching the atoms to the upper branch (scattering state) at the BEC side of a Feshbach resonance  [42]. However, the term “upper branch” only has clear definition for two-body systems. Even for three-body systems, exact solution of the energy levels in a harmonic trap shows that there are many avoided crossings between the lowest two branches as one approaches the resonance ($a\to \infty$), making it difficult to identify a repulsive Fermi system  [20]. For many-body system, the upper branch can have clear meaning for $a>0$ in the high temperature limit where the virial expansion to the second order in the fugacity is sufficient to describe the system  [43]. Therefore, it is a theoretical challenge to understand the many-body upper branch at low temperature and its influence on the FMPT.

The upper branch Fermi gas has been experimentally studied with different densities, temperatures, and trap depths  [44], [45]. The interaction energy has also been measured  [45]. In addition to the strong atom loss near the resonance, the interaction energy was found to increase and then decrease as one approaches the resonance from the repulsive side, showing a maximum before reaching resonance  [45]. Hence there exists a region where the energy derivative $\partial \mathcal{E}/\partial (-1/a)<0$ and Tan’s adiabatic relation is violated  [46]. These features of the upper branch Fermi gas at high temperatures have been theoretically explained by Shenoy and Ho  [47] by using an extended Nozi$\acute{e}$res–Schmitt-Rink (NSR) approach where the bound state contribution is subtracted. At low temperature, a recent measurement for the narrow resonance of ⁶Li  [48] found that the energy maximum becomes even sharper than the high temperature case. However, a unified theoretical approach to study the upper branch, energy maximum, and ferromagnetic transition at low temperature is still lacking.

On the other hand, it was experimentally found that the rapid decay into bound pairs prevent the study of equilibrium phases of the upper-branch Fermi gas  [49]. One possibility to suppress the pair formation is to use a narrow Feshbach resonance where the pairs have dominately closed channel character and therefore a much smaller overlap matrix element with the atoms  [49]. However, the narrow Feshbach resonances are characterized by a large negative effective range  [50], [51], [52]. Therefore, we need to study the effective range effects on the ferromagnetic transition. Another possible way to overcome this difficulty is not to use the upper branch of a Feshbach resonance but use the background repulsive interaction without Feshbach resonance effect in some atoms. The background scattering length is usually small. For example, the gas parameter $k_{F}a$ in the current experiments of ¹³⁷Yb is about $0.13$   [41], [53]. To reach the critical gas parameter of the FMPT, we therefore need much higher density. Then the effective range effect becomes important. In both cases, it is quite necessary to study the effective range effects on the FMPT.

In this paper, we study the upper branch and finite range effects on the FMPT by using a general effective Lagrangian which reproduce precisely the $s$-wave scattering phase shift for a given interaction potential. The energy density of the many-body system as a function of the polarization is obtained by using the Bethe–Goldstone ladder resummation  [54], [55], [56], [57], [58], [59] which allows us to study the FMPT and the upper branch of attractive potentials nonperturbatively. The paper is organized as follows. In Section  2, we construct a general effective Lagrangian for two-body scattering. In Section  3, the nonperturbative energy density of the many-body system is derived in the Bethe–Goldstone ladder approximation. In Section  4, we discuss the properties of the upper branch Fermi gas for attractive potentials. The results for some model potentials and for the Fashbach resonance model are presented in Sections  5 Results for some potential models, 6 Feshbach resonance model, respectively. We summarize in Section  7. We use the units $ħ=1$ throughout the paper.