Abstract
We theoretically investigate the polaron physics of an impurity immersed in a twodimensional Fermi sea, interacting via a pwave interaction at finite temperature. In the unitary limit with a divergent scattering area, we find a welldefined repulsive Fermi polaron at short interaction range, which shows a remarkable thermal stability with increasing temperature. The appearance of such a stable repulsive Fermi polaron in the resonantly interacting limit can be attributed to the existence of a quasibound dressed molecule state hidden in the twoparticle continuum, although there is no bound state in the twoparticle limit. We show that the repulsive Fermi polaron disappears when the interaction range increases or when the scattering area is tuned to the weakly interacting regime. The large interaction range and small scattering area instead stabilize attractive Fermi polarons.
1 Introduction
Fermi polaron is a wellestablished research area that addresses the fate of an impurity when it moves and interacts with the surrounding environment of a quantum manyfermion system [1]. The ability of a free particlelike motion of the impurity under interactions can be intuitively described by a fundamental concept of manybody physics — quasiparticle [2]. For many decades, the determination of quasiparticle properties, such as energy, residue, and lifetime, provides valuable insights on intriguing quantum manyfermion systems in the different branches of physics [3,4,5,6,7,8,9,10].
Over the past two decades, the investigation of Fermi polarons with ultracold atoms attracts particular interests [6, 7, 11,12,13,14,15,16,17]. Due to the unprecedented controllability on the interatomic interaction (i.e., through Feshbach resonance [18]), purity and dimensionality of ultracold atomic Fermi gases [19, 20], new interesting features of Fermi polarons can be revealed in a quantitative manner [21,22,23]. For example, repulsive Fermi polaron, which is an excited polaron state with welldefined residue and lifetime, has been theoretically predicted [24, 25] and experimentally observed in an interacting imbalanced spin1/2 Fermi gas with swave contact interactions [12, 13, 15]. This repulsive branch appears on the tightly bound side of the Feshbach resonance, with a deep twobody bound state. Towards the strongly interacting unitary limit at resonance where the twobody bound state dissolves, the decay rate of repulsive Fermi polaron increases too rapidly to behave like a welldefined quasiparticle [15, 25].
In this work, we predict the existence of a welldefined repulsive Fermi polaron in a twodimensional Fermi gas, under a resonant pwave interaction between impurity and fermions, in the absence of a twobody bound state. This resonant unitary limit is described by an infinitely large scattering area \(a_{p}=\pm \infty\), together with a nonzero effective interaction range \(R_{p}\) [26, 27]. The repulsive Fermi polaron appears at small interaction range, due to a quasibound dressed molecule state hidden in the twoparticle continuum, although no twobody bound state can exist.
To describe the repulsive polaron branch, we apply a manybody Tmatrix theory [28,29,30,31] that allows us to go beyond the earlier zerotemperature study of pwave Fermi polarons (in three dimensions) [32] and to explore their finitetemperature properties. Remarkably, the pwave repulsive Fermi polaron appears to be robust against thermal fluctuations. This thermal stability of pwave repulsive Fermi polaron could be crucial for its experimental observation, since a pwave interacting Fermi gas often suffers severe atom loss at low temperature [33,34,35].
The rest of the paper is organized as follows. In the next section (Section 2), we outline the model Hamiltonian for the pwave interacting Fermi polaron and present the manybody Tmatrix approach that is capable to describe the oneparticlehole excitation at finite temperature, which is the key ingredient of polaron physics. In Section 3, we first discuss the zeromomentum spectral function of the impurity and show the appearance of the pwave repulsive Fermi polaron in the unitary limit. We then present the spectral function of the molecule to reveal the existence of a quasibound dressed molecule state. Finally, we explore the parameter space for pwave repulsive Fermi polaron, by changing either the scattering area or the effective interaction range. The conclusions follow in Section 4.
2 Model Hamiltonian and manybody Tmatrix approach
We consider a highly imbalanced spin1/2 Fermi gas of ultracold atoms near a pwave Feshbach resonance in two dimensions. In the extreme limit of vanishing population of minority atoms, we treat minority atoms as individual impurities, interacting with a noninteracting Fermi sea of majority atoms via an interaction potential \(V_{p}\left( \textbf{k},\textbf{k}'\right)\). Our system can then be welldescribed by the singlechannel model Hamiltonian,
where \(c_{\textbf{k}}^{\dagger }\) and \(d_{\textbf{k}}^{\dagger }\) are the creation field operators for the majority of atoms and the single impurity, respectively. The first two terms in the Hamiltonian are the singleparticle terms with dispersion relations \(\xi _{\textbf{k}}=\hbar ^{2}\textbf{k}^{2}/(2m)\mu =\epsilon _{\textbf{k}}\mu\) and \(E_{\textbf{k}}=\hbar ^{2}\textbf{k}^{2}/(2m)\), while the last term describes the interaction term. Here, \(\mu\) is the chemical potential of the majority of atoms and at finite temperature T is given by, \(\mu =k_{B}T\ln \left[ \exp (\varepsilon _{F}/k_{B}T)1\right]\). At low temperature, the chemical potential approaches the Fermi energy \(\varepsilon _{F}=\hbar ^{2}k_{F}^{2}/(2m)=2\pi n\hbar ^{2}/m\), where n is the density of the majority of atoms in two dimensions.
For the pwave interaction potential \(V_{p}\left( \textbf{k},\textbf{k}'\right)\), for concreteness, we choose the chiral \(p_{x}+ip_{y}\) channel, by assuming that experimentally one can carefully tune the magnetic field very close to a pwave Feshbach resonance with azimuthal quantum number \(m=+1\). In practice, it is convenient to take a separable interaction potential in the form [27, 36],
where \(\lambda\) is the interaction strength and
is a dimensionless \(p_{x}+ip_{y}\) form factor function with a cutoff momentum \(k_{0}\), a polar angle \(\varphi _{{\textbf {k}}}=\arctan (k_{y}/k_{x})\) and an exponent n that is introduced for the convenience of numerical calculations. The wavevector k is measured in units of the Fermi wavevector \(k_{F}=(4\pi n)^{1/2}\).
2.1 Twobody Tmatrix and the renormalization of pwave interaction
While it is convenient to use the set of the three parameters (\(\lambda ,k_{0},n\)) to describe the pwave interaction potential, physically we would better use the scattering area \(a_{p}\) (in two dimensions) and the effective range of interaction \(R_{p}\), which are formally defined through the pwave phase shift \(\delta _{p}\left( k\right)\) in the lowenergy limit (i.e., \(k\rightarrow 0\)) [27, 33],
The pwave phase shift \(\delta _{p}\left( k\right)\) can be easily obtained by calculating the lowenergy twobody Tmatrix in vacuum [27], \(T_{2}^{(\text {vac})}({\textbf {k}},{\textbf {k}};E)=\left \Gamma ({\textbf {k}})\right ^{2}\tilde{T}_{2}^{(\text {vac})}(E)\) with
where \(E\equiv \hbar ^{2}k^{2}/m\). Using the wellknown relation \([T_{2}^{(\text {vac})}({\textbf {k}},{\textbf {k}};E)]^{1}=m[\cot \delta _{p}(k)i]/(4\hbar ^{2})\), we find that in the lowenergy limit \(k\rightarrow 0\),
where \(\mathscr {P}\) stands for taking the Cauchy principal value. By carrying out the summation over \(\textbf{p}\), we find that [27]
Throughout this work, we take an exponent \(n=1.0\). For the given scattering area \(a_{p}\) and effective range \(R_{p}\), we then determine the interaction strength \(\lambda\) and the cutoff momentum \(k_{0}\) by using the above two expressions. Physical results of course will not depend on the choice of the exponent n. In Appendix 1, we explicitly show the nindependence of the separable interaction potential \(V_{p}\left( \textbf{k},\textbf{k}'\right)\), by calculating the impurity selfenergy.
2.2 Manybody Tmatrix theory of Fermi polarons
We now turn to solve the Fermi polaron problem by applying the manybody Tmatrix theory with the ladder approximation [28,29,30]. The theory has been outlined in greater detail in the previous work for Fermi polarons with an swave interaction potential [31], so here we only briefly sketch the key ingredients and emphasize the changes due to the pwave interaction potential.
Following the same derivation as in the previous work [31], it is easy to show that the manybody Tmatrix is given by
where
By taking \(\textbf{q}=0\), \(\mu =0\) and the FermiDirac distribution function \(f(x)=0\) (i.e., by considering the twobody scattering in vacuum), \(\tilde{T}_{2}(\textbf{q};\omega )\) reduces to the twobody Tmatrix Eq. (5). In other words, the manybody Tmatrix can be understood as the effective interaction in the medium. According to the interaction Hamiltonian in Eq. (1), diagrammatically this effective interaction comes with two incoming legs with momenta \(\textbf{q}/2\pm \textbf{k}'\) and with two outgoing legs with momenta \(\textbf{q}/2\pm \textbf{k}\). Therefore, by winding back the outgoing leg of majority atoms (i.e., with the momentum \(\textbf{q}/2+\textbf{k}\)) and connecting it to the incoming leg with the momentum \(\textbf{q}/2+\textbf{k}'\) (so \(\textbf{k}=\textbf{k}'\)) [31], we obtain the approximate impurity selfenergy in the manybody Tmatrix theory,
where \(\mathscr {G}\) is the noninteracting Green function of the Fermi sea and we use \(\mathcal {Q}=(\textbf{q},i\nu _{n})\) and \(\mathcal {K}'=(\textbf{k}',i\omega _{m'})\) to denote the fourdimensional momenta with the bosonic Matsubara frequency \(\nu _{n}=2\pi nk_{B}T\) and the fermionic Matsubara frequency \(\omega _{m'}=\pi (2m'+1)k_{B}T\) at finite temperature T. We have also used the abbreviation \(\sum _{\mathcal {Q}}\equiv \sum _{\textbf{q}}k_{B}T\sum _{i\nu _{n}}\). For the impurity selfenergy, it is convenient to change to \(\mathcal {K}=\mathcal {Q}/2\mathcal {K}'=(\textbf{k},i\omega _{m}\)), so \(\Sigma (\mathcal {K})=\sum _{\mathcal {Q}}G(\mathcal {Q}\mathcal {K})\tilde{T}_{2}(\mathcal {Q})\left \Gamma (\textbf{q}/2{\textbf {k}})\right ^{2}\). By summing over the bosonic Matsubara frequency \(\nu _{n}\) and taking the analytic continuation \(i\omega _{m}\rightarrow \omega +i0^{+}\) [31], we find the impurity selfenergy,
In comparison with the case of an swave interaction potential [31], the manybody Tmatrix Eq. (10) and the impurity selfenergy Eq. (12) take essentially the same forms, apart from the additional interaction form factors \(\Gamma ({\textbf {k}})\) and \(\Gamma ^{*}({\textbf {k'}})\) that are necessary to characterize the pwave interaction [27, 36]. These interaction form factors do not introduce too many numerical workloads, and the numerical difficulty still lies on the handling of the pole structure that might appear in the summation over the momentum \(\textbf{p}\) in \(\tilde{T}_{2}^{1}(\textbf{q},\omega )\). A detailed discussion of \(\tilde{T}_{2}^{1}(\textbf{q},\omega )\) is included in Appendix 2. We note that the summation over the momentum \(\textbf{q}\) in the selfenergy \(\Sigma (\textbf{k},\omega )\) may also suffer from the existence of a welldefined molecule state, which manifests itself as a pole or a deltapeak of \(\tilde{T}_{2}(\textbf{q},\omega )\).
3 Results and discussions
Once the impurity selfenergy is obtained, we directly calculate the impurity Green function [14, 31],
Fermi polarons can be wellcharacterized by the impurity spectral function
where each quasiparticle is visualized by the appearance of a sharp spectral peak, with its width reflecting the lifetime or decay rate of quasiparticle [25, 28]. Mathematically, we may determine the quasiparticle energy \(\mathcal {E}_{P}(\textbf{k})\), either the attractive polaron energy or the repulsive polaron energy, from the pole position of the impurity Green function (\(\omega \rightarrow \mathcal {E}_{P}(\mathbf {k)}\)), i.e.,
By expanding the selfenergy near the zero momentum \(\textbf{k}=0\) and the polaron energy \(\mathcal {E}_{P}\equiv \mathcal {E}_{P}\left( \textbf{0}\right)\), we directly determines various quasiparticle properties, including the polaron residue,
and the polaron decay rate,
3.1 Repulsive Fermi polaron in the unitary limit
Let us first focus on the unitary limit, where the scattering area diverges, i.e., \(a_{p}=\pm \infty\). In Fig. 1a, we report the temperature evolution of the zeromomentum spectral function \(A(\textbf{k}=0,\omega\)) at a small interaction range \(k_{F}R_{p}=0.1\), in the form of a twodimensional contour plot in the linear scale. The onedimensional plots of the spectral functions at the three typical temperatures \(T=0.1T_{F}\), \(0.2T_{F}\) and \(0.5T_{F}\) are shown in Fig. 1b, by using black solid, red dashed, and blue dotdashed lines, respectively.
At very low temperature (i.e., \(T\sim 0\)), we find two dominant peaks in the spectral function at the positions \(\omega \simeq 0.31\varepsilon _{F}\) and \(\omega \simeq +0.37\varepsilon _{F}\), which correspond to the attractive Fermi polaron and the repulsive Fermi polaron [21,22,23, 25], respectively. By increasing temperature, both polaron states show a red shift in energy, similar to the swave case [31]. The initially sharp attractive polaron peak quickly dissolves into a broad distribution at \(T\sim 0.3T_{F}\) and eventually disappears at \(T\sim 0.6T_{F}\). In sharp contrast, the repulsive polaron appears to be very robust against thermal fluctuations. In particular, once the temperature is larger than \(0.6T_{F}\), the position and the width of the repulsive polaron peak essentially do not change with temperature.
To confirm the existence of a repulsive Fermi polaron, in Fig. 2, we present the real part (a) and the imaginary part (b) of the selfenergy at the same three typical temperatures as in Fig. 1b. At zero momentum (i.e., \(E_{\textbf{k}}=0\)), a pole of the impurity Green function occurs when \(\omega =\text {Re}\Sigma (k=\textbf{0},\omega )\). Therefore, in Fig. 2a, the intercept between the green dotted line (i.e., \(y=\omega\)) and the curves \(\text {Re}\Sigma (k=\textbf{0},\omega )\) determines the polaron energy. On the negative frequency side (\(\omega <0\)), we find that the green dotted line fails to intercept with the curve \(\text {Re}\Sigma (k=\textbf{0},\omega )\) at the temperate \(T=0.5_{F}\), consistent with our earlier observation that the attractive polaron develops into a broad structure once \(T>0.3T_{F}\). On the positive frequency side (\(\omega >0\)), the green dotted line always crosses with the curves \(\text {Re}\Sigma (k=\textbf{0},\omega )\) at different temperatures, indicating the persistence of the repulsive polarons with increasing temperature. The position of the crossing points or the repulsive polaron energies do not change too much as temperature increases. Remarkably, at those repulsive polaron energies, the imaginary part of the selfenergy \(\text {Im}\Sigma (k=\textbf{0},\omega )\) turns out to be reasonably small, and more importantly to be temperature insensitive.
The thermally stable repulsive polaron is not expected to appear in the unitary limit, because a welldefined twobody bound state does not exist [14, 25]. This situation might be compared with an swave Fermi polaron in three dimensions [31]. In that case, in the same unitary limit, where the swave scattering length \(a_{s}\) diverges, the impurity spectral function only shows an extremely broad background at positive energy, without any signal for a repulsive polaron. The analysis of the real part of the selfenergy confirms the absence of a repulsive polaron, since no solution exists for the condition \(\omega =\text {Re}\Sigma (k=\textbf{0},\omega )\) when \(\omega >0\) [31]. Under the swave interaction between the impurity and the Fermi sea, repulsive polaron only appears in the tightly bound limit, where a welldefined twobody bound state exists. Towards the swave unitary limit, the decay rate of the swave repulsive polaron will quickly increase with both the scattering length \(a_{s}\) and temperature. As a result, there is no meaningful swave repulsive polaron near the strongly interacting regime of the unitary limit.
The existence of a thermally robust repulsive polaron, without a twobody bound state, is therefore an unique feature of pwave Fermi polarons. To understand its formation mechanics, we consider the dressed molecule state in the presence of the manybody environment of a Fermi sea, which is an analog of a Cooper pair in the limit of an extreme population imbalance. In the manybody Tmatrix theory, the dressed molecule state is simply described by its effective Green function, i.e, the manybody Tmatrix \(T_{2}(\textbf{q},\omega )\) [28, 29, 31]. Thus, we introduce a molecule spectral function,
In Fig. 3, we report the twodimensional contour plots of the molecule spectral function in the logarithmic scale across the unitary limit at the temperature \(0.2T_{F}\), with the inverse scattering areas \(1/(k_{F}^{2}a_{p})=0.5\) (a), 0.0 (b), and \(+0.5\) (c). To be consistent with the results in Fig. 1, we have taken the same effective interaction range \(k_{F}R_{p}=0.1\).
On the molecule side of the Feshbach resonance in Fig. 3c, it is ready to see a sharp peak starting at energy \(\omega \sim 1.2\varepsilon _{F}\), which is wellseparated from the twoparticle continuum. Moreover, at the momentum \(q
In the unitary limit on resonance (see Fig. 3b), although a separate undamped peak does not exist, we do observe that a sharp peak emerges at energy \(\omega \sim 0.6\varepsilon _{F}\) and becomes broader when the momentum q is larger than \(0.5k_{F}\). We would like to attribute the existence of a welldefined repulsive polaron in the unitary limit to this quasibound dressed molecule state, which is hidden slightly above the bottom of the twoparticle scattering continuum. Although it is a quasibound molecule state, it effectively depletes fermions surrounding the dressed molecule and eventually leads to the excited repulsive polaron [25]. This idea is also supported by the weak temperature dependence of the quasibound dressed molecule state (not shown in the figure), which may explain the observed thermal stability of the repulsive polaron.
We finally consider a negative scattering area, as shown in Fig. 3a. The dressed molecule state now becomes less welldefined, with a much broader peak and with its energy blueshifted to \(\omega \sim 0\) at zero momentum. In this situation, the repulsive polaron may fail to exist, as we shall see.
To complete our analysis of the repulsive polaron in the unitary limit, we show in Fig. 4 the temperature dependence of the energy (in the main figure) and the decay rate (in the inset) of both attractive Fermi polaron and repulsive Fermi polaron. As illustrated by the red stars in the inset, the decay rate of the attractive Fermi polaron rapidly increases with increasing temperature. It becomes larger than the Fermi energy when \(T>0.3T_{F}\), in agreement with the observation in Fig. 1a that the attractive polaron ceases to exist at this temperature. On the other hand, as indicated by black dots in the inset, the decay rate of the repulsive Fermi polaron increases from \(0.1\varepsilon _{F}\) at \(T\sim 0\) to about \(0.5\varepsilon _{F}\) at \(T\sim 0.7T_{F}\). By further increasing temperature, the decay rate only slightly decreases, with a polaron energy almost fixed at \(0.27\varepsilon _{F}\) (see the main figure). This provides a quantitative measure of the thermal robustness of the repulsive Fermi polaron in the unitary limit.
3.2 Parameter space for repulsive Fermi polarons
We now turn to explore the parameter window of repulsive Fermi polarons, for a pwave interaction strength that does not support a twobody bound state. We focus on the cases of a fixed temperature \(T=0.2T_{F}\), but with varying effective interaction range \(k_{F}R_{p}\) and with varying scattering area \(1/(k_{F}^{2}a_{p})\).
In Fig. 5, we report the evolution of the zeromomentum spectral function \(A(\textbf{k}=0,\omega )\) as the interaction range \(k_{F}R_{p}\) increases in the unitary limit. From the twodimensional contour plot in Fig. 5a, we find that the lower attractive polaron branch becomes increasingly dominant with respect to the increase in the interaction range. However, the interesting new feature of repulsive polaron branch quickly disappears when \(k_{F}R_{p}\) becomes larger than 0.3. At the interaction range \(k_{F}R_{p}=0.5\), as shown in Fig. 5b by the blue dotdashed line, one can observe an extremely broad bump at about \(\omega \sim 0.7\varepsilon _{F}\), as a reminiscent of the repulsive polaron. Therefore, we conclude that a large interaction range does not favor the formation of a repulsive Fermi polaron.
In Fig. 6, we present the zeromomentum spectral function at three inverse scattering areas \(1/(k_{F}^{2}a_{p})=0.05\) (black solid line), \(0.1\) (red dashed line) and \(0.2\) (blue dotdashed line), and at a fixed interaction range \(k_{F}R_{p}=0.1\). By decreasing the inverse scattering area, the attractive Fermi polarons show a blue shift in energy. More importantly, the attractive polaron peak becomes sharper. The repulsive Fermi polarons also show a blue shift in energy. However, their width becomes much wider with increasing inverse scattering area. At \(1/(k_{F}^{2}a_{p})=0.2\), we may hardly identify the peak as a welldefined repulsive polaron. This finding is consistent with the earlier observation from Fig. 3a that the quasibound dressed molecule state becomes less welldefined as the inverse scattering area decreases and moves away from the Feshbach resonance.
4 Conclusions and outlooks
In conclusion, we have investigated pwave Fermi polarons in two dimensions at finite temperature, which potentially can be experimentally realized in a population imbalanced spin1/2 Fermi gas, where minority atoms in one hyperfine state act as impurities and interact with majority atoms in another hyperfine state near a pwave Feshbach resonance. In contrast to the conventional swave case that the existence of a repulsive Fermi polaron requires a twobody bound state [15, 25, 31], a pwave repulsive Fermi polaron can arise in the absence of twobody bound state near the Feshbach resonance, due to a quasibound dressed (manybody) molecule state that is hidden inside the twoparticle scattering continuum. The pwave repulsive polaron shows a remarkable stability against temperature. This extraordinary thermal robustness would be very useful for its experimental observation, since a pwave Fermi gas is not blessed by the Pauli exclusion principle and often has severe atom loss below the Fermi temperature for degeneracy [34, 35].
Instead of using a spin1/2 Fermi gas with two hyperfine states, one may also consider a massimbalanced FermiFermi mixture such as \(^{6}\)Li\(^{40}\)K atomic mixture near Feshbach resonances, with a strong atomdimer attraction occurring between \(^{40}\)K atoms and weaklybound \(^{6}\)Li\(^{40}\)K molecules in odd partialwave channels [37]. This higher partialwave attraction is mainly pwave and is recombinationfree (and therefore stable), as experimentally observed [37]. We may tune down the number of weakly bound \(^{6}\)Li\(^{40}\)K molecules to treat them as impurities. In this case, the mass of impurity is slightly larger than the mass of the fermions in the Fermi sea. Our results of repulsive Fermi polarons, based on equal mass, should be qualitatively applicable. Quantitative predictions however require the extension of our work to account for an arbitrary impurity mass. Another crucial issue of the atomdimer pwave attraction is that weakly bound dimers may spontaneously dissociate on a time scale of about tens millseconds [37]. We would like to leave a careful investigation of these two issues (i.e., the unequal mass effect and the short lifetime of impurity) to a future study.
Availability of data and materials
The data generated during the current study are available from the contributing author upon reasonable request.
References

A.S. Alexandrov, J.T. Devreese, Advances in Polaron Physics, vol. 159 (Springer, New York, 2010)

L.D. Landau, Electron Motion in Crystal Lattices. Phys. Z. Sowjetunion 3, 664 (1933)

P. Nozières, C.T. De Dominicis, Singularities in the XRay Absorption and Emission of Metals. III. One Body Theory Exact Solution. Phys. Rev. 178, 1097 (1969)

A.G. Basile, V. Elser, Stability of the ferromagnetic state with respect to a single spin flip: Variational calculations for the \(U=\infty\) Hubbard model on the square lattice. Phys. Rev. B. 41, 4842(R) (1990)

G.M. Zhang, H. Hu, L. Yu, Marginal Fermi Liquid Resonance Induced by a Quantum Magnetic Impurity in dWave Superconductors. Phys. Rev. Lett. 86, 704 (2001)

F. Chevy, Universal phase diagram of a strongly interacting Fermi gas with unbalanced spin populations. Phys. Rev. A. 74, 063628 (2006)

A. Schirotzek, C.H. Wu, A. Sommer, M.W. Zwierlein, Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms. Phys. Rev. Lett. 102, 230402 (2009)

M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. Kroner, E. Demler, A. Imamoglu, Fermi polaronpolaritons in chargetunable atomically thin semiconductors. Nat. Phys. 13, 255 (2017)

Y. Cao, J. Zhou, Fermi polarons in a drivendissipative background medium. Sci. China Phys. Mech. Astron. 65, 110312 (2022)

H. Hu, J. Wang, R. Lalor, X.J. Liu, Twodimensional coherent spectroscopy of trionpolaritons and excitonpolaritons in atomically thin transition metal dichalcogenides. AAPPS Bull. 33, 12 (2023)

Y. Zhang, W. Ong, I. Arakelyan, J.E. Thomas, PolarontoPolaron Transitions in the RadioFrequency Spectrum of a QuasiTwoDimensional Fermi Gas. Phys. Rev. Lett. 108, 235302 (2012)

C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder, P. Massignan, G.M. Bruun, F. Schreck, R. Grimm, Metastability and coherence of repulsive polarons in a strongly interacting Fermi mixture. Nature (London) 485, 615 (2012)

M. Koschorreck, D. Pertot, E. Vogt, B. Fröhlich, M. Feld, M. Köhl, Attractive and repulsive Fermi polarons in two dimensions. Nature (London) 485, 619 (2012)

P. Massignan, M. Zaccanti, G.M. Bruun, Polarons, dressed molecules and itinerant ferromagnetism in ultracold Fermi gases. Rep. Prog. Phys. 77, 034401 (2014)

F. Scazza, G. Valtolina, P. Massignan, A. Recati, A. Amico, A. Burchianti, C. Fort, M. Inguscio, M. Zaccanti, G. Roati, Repulsive Fermi Polarons in a Resonant Mixture of Ultracold \(^{6}\)Li Atoms. Phys. Rev. Lett. 118, 083602 (2017)

R. Schmidt, M. Knap, D.A. Ivanov, J.S. You, M. Cetina, E. Demler, Universal manybody response of heavy impurities coupled to a Fermi sea: a review of recent progress. Rep. Prog. Phys. 81, 024401 (2018)

J. Wang, Functional determinant approach investigations of heavy impurity physics. AAPPS Bull. 33, 20 (2023)

C. Chin, R. Grimm, P. Julienne, E. Tiesinga, Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225 (2010)

I. Bloch, J. Dalibard, W. Zwerger, Manybody physics with ultracold gases. Rev. Mod. Phys. 80, 885 (2008)

H. Hu, X.C. Yao, X.J. Liu, Second sound with ultracold atoms: a brief review. AAPPS Bull. 32, 26 (2022)

O. Goulko, A.S. Mishchenko, N. Prokof’ev, B. Svistunov, Dark continuum in the spectral function of the resonant Fermi polaron. Phys. Rev. A 94, 051605(R) (2016)

J. Wang, X.J. Liu, H. Hu, Exact Quasiparticle Properties of a Heavy Polaron in BCS Fermi Superfluids. Phys. Rev. Lett. 128, 175301 (2022)

J. Wang, X.J. Liu, H. Hu, Heavy polarons in ultracold atomic Fermi superfluids at the BECBCS crossover: Formalism and applications. Phys. Rev. A 105, 043320 (2022)

X. Cui, H. Zhai, Stability of a fully magnetized ferromagnetic state in repulsively interacting ultracold Fermi gases. Phys. Rev. A. 81, 041602(R) (2010)

P. Massignan, G.M. Bruun, Repulsive polarons and itinerant ferromagnetism in strongly polarized Fermi gases. Eur. Phys. J. D 65, 83 (2011)

H. Hu, B.C. Mulkerin, L. He, J. Wang, X.J. Liu, Quantum fluctuations of a resonantly interacting pwave Fermi superfluid in two dimensions. Phys. Rev. A 98, 063605 (2018)

H. Hu, X.J. Liu, Resonantly interacting pwave Fermi superfluid in two dimensions: Tan’s contact and the breathing mode. Phys. Rev. A 100, 023611 (2019)

R. Combescot, A. Recati, C. Lobo, F. Chevy, Normal State of Highly Polarized Fermi Gases: Simple ManyBody Approaches. Phys. Rev. Lett. 98, 180402 (2007)

H. Hu, B.C. Mulkerin, J. Wang, X.J. Liu, Attractive Fermi polarons at nonzero temperatures with a finite impurity concentration. Phys. Rev. A 98, 013626 (2018)

H. Tajima, S. Uchino, Thermal crossover, transition, and coexistence in Fermi polaronic spectroscopies. Phys. Rev. A. 99, 063606 (2019)

H. Hu, X.J. Liu, Fermi polarons at finite temperature: Spectral function and rf spectroscopy. Phys. Rev. A 105, 043303 (2022)

J. Levinsen, P. Massignan, F. Chevy, C. Lobo, pWave Polaron. Phys. Rev. Lett. 109, 075302 (2012)

J. Levinsen, N.R. Cooper, V. Gurarie, Stability of fermionic gases close to a pwave Feshbach resonance. Phys. Rev. A 78, 063616 (2008)

C. Luciuk, S. Trotzky, S. Smale, Z. Yu, S. Zhang, J.H. Thywissen, Evidence for universal relations describing a gas with pwave interactions. Nat. Phys. 12, 599 (2016)

J. Yoshida, T. Saito, M. Waseem, K. Hattori, T. Mukaiyama, Scaling Law for ThreeBody Collisions of Identical Fermions with pWave Interactions. Phys. Rev. Lett. 120, 133401 (2018)

S.S. Botelho, C.A.R. Sa de Melo, Quantum Phase Transition in the BCStoBEC Evolution of pwave Fermi Gases. J. Low Temp. Phys. 140, 409 (2005)

M. Jag, M. Zaccanti, M. Cetina, R.S. Lous, F. Schreck, R. Grimm, D.S. Petrov, J. Levinsen, Observation of a Strong AtomDimer Attraction in a MassImbalanced FermiFermi Mixture. Phys. Rev. Lett. 112, 075302 (2014)
Acknowledgements
The present work is dedicated to the memory of Professor Lee Chang, whose contributions to physical science and education were longstanding and farreaching. He enthusiastically carried out the research on ultracold atomic physics in Tsinghua University twentyfive years ago and guided the authors (HH and XJL) into this fantastic field.
Funding
This research was supported by the Australian Research Council’s (ARC) Discovery Program, Grants No. FT230100229 (J.W.).
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Appendices
Appendix 1: the ndependence of the separable interaction form
Throughout the work, we have taken \(n=1\) in our separable interaction form \(V_{kk'}=\) \(\lambda \Gamma (k)\Gamma (k')\), with a pwave form factor \(\Gamma (k)=(k/k_{F})[1+(k/k_{0})^{2n}]^{3/2}\). After the renormalization, physical results (at low energy) should only depend on the scattering area \(1/(k_{F}^{2}a_{p})\) and interaction range \(k_{F}R_{p}\), independent on the choice of the value of n. In Fig. 7, as an example, we explicitly examine this independence for the impurity selfenergy in the unitary limit \(1/(k_{F}^{2}a_{p})=0\). We find negligible differences when we change the value of n from 1 to 3, as anticipated.
Appendix 2: the numerical calculation of \(T_{2}^{1}(\textbf{q},\omega )\)
In numerical calculations, it is convenient to take the natural units, where \(m=\hbar =k_{B}=1\). In other words, we set the units of energy and momentum as \(\varepsilon _{F}\) and \(k_{F}\), respectively. The inverse of the manybody Tmatrix then takes the form,
where we have defined an angleintegrated function,
The integral Eq. (19) is well defined if \(y\equiv (\omega +\mu )/2q^{2}/4<0.\) In this case, we find that,
Otherwise (\(y\ge 0\)), we may use the identity
to recast the real and imaginary parts of \(T_{2}^{1}(q,\omega )\) into the forms,
Here, by taking the Cauchy principle value, we have defined two integrals,