Few vs many-body physics of an impurity immersed in a superfluid of spin 1/2 attractive fermions

TL;DR

This paper explores the behavior of an impurity in a superfluid of strongly correlated spin 1/2 fermions, analyzing stability, three-body interactions, and energy corrections beyond mean-field approximations.

Contribution

It relates the stability of dimer and trimer states to the three-body problem and highlights the divergence of beyond-mean-field energy corrections requiring regularization.

Findings

Stability diagram of dimer and trimer states linked to three-body physics.

Beyond-mean-field energy corrections are divergent and need regularization.

Proper accounting of three-body physics is essential for accurate impurity energy calculations.

Abstract

In this article we investigate the properties of an impurity immersed in a superfluid of strongly correlated spin 1/2 fermions. For resonant interactions, we first relate the stability diagram of dimer and trimer states to the three-body problem for an impurity interacting with a pair of fermions. Then we calculate the beyond-mean-field corrections to the energy of a weakly interacting impurity. We show that these corrections are divergent and have to be regularized by properly accounting for three-body physics in the problem.

Equations

$$\hat{H}={\hat{H}}_{\mathrm{imp}}+{\hat{H}}_{\mathrm{mb}}+{\hat{H}}_{\mathrm{int}},$$

$${\hat{H}}_{\mathrm{int}}=g_0^{\prime }\sum _{\sigma =\uparrow ,\downarrow }\int d^3\mathbf{r}{\hat{\psi }}_{\sigma }^{†}(\mathbf{r}){\hat{\psi }}_{\sigma }(\mathbf{r}){\hat{\varphi }}^{†}(\mathbf{r})\hat{\varphi }(\mathbf{r}),$$

$$\frac{1}{g_0^{\prime }}=\frac{1}{g^{\prime }}-\frac{1}{\Omega }\sum _{k<\Lambda }\frac{1}{{\epsilon }_k^{(\mathrm{r})}},$$

$$g_0^{\prime }=g^{\prime }+\frac{{g^{\prime }}^2}{\Omega }\sum _{k<\Lambda }\frac{1}{{\epsilon }_k^{(\mathrm{r})}}+o({g^{\prime }}^2).$$

$$\Delta E_{\mathrm{pert}}=g^{\prime }n+\frac{{g^{\prime }}^{2}n}{\Omega }\sum _{\mathbf{q}}\left[\frac{1}{{\epsilon }_q^{(\mathrm{r})}}-\chi (\mathbf{q},{\epsilon }_q^{(\mathrm{i})})\right].$$

$$\chi (\mathbf{q},E)=\frac{1}{N}\sum _{\alpha }\frac{{\left|⟨\alpha |{\hat{\rho }}_{-\mathbf{q}}|0⟩\right|}^2}{E_{\alpha }-E_0+E},$$

$$\chi (\mathbf{q},E)\simeq \frac{S(q)}{{\epsilon }_q^{(\mathrm{f})}+E},$$

$$\left(\begin{array}[]{c}\hat{T}_{1}\\ \hat{T}_{2}\\ \hat{T}_{3}\end{array}\right)=\left(\begin{array}[]{c}\hat{t}_{1}\\ \hat{t}_{2}\\ \hat{t}_{3}\end{array}\right)+\left(\begin{array}[]{ccc}0&\hat{t}_{1}&\hat{t}_{1}\\ \hat{t}_{2}&0&\hat{t}_{2}\\ \hat{t}_{3}&\hat{t}_{3}&0\end{array}\right)\hat{G}_{0}\left(\begin{array}[]{c}\hat{T}_{1}\\ \hat{T}_{2}\\ \hat{T}_{3}\end{array}\right)$$

$${\Gamma }_{\mathrm{Born}}\underset{\Lambda \to \infty }{\sim }\frac{m_f^3}{{\hslash }^6}{g^{\prime }}^2\kappa (\eta =m_b/m_f)\ln (\Lambda )$$

$$\begin{array}{rl}\kappa (\eta )=& \frac{\sqrt{{\eta }^3(\eta +2)}}{2{\pi }^3(\eta +1{)}^2}-\frac{\eta }{2{\pi }^3}\arctan \left(\frac{1}{\sqrt{\eta (\eta +2)}}\right)\\ & -\frac{4}{{\pi }^3}\sqrt{\frac{\eta }{\eta +2}}\arctan {\left(\sqrt{\frac{\eta }{\eta +2}}\right)}^2\end{array}$$

$${\Gamma }_{\mathrm{Born}}-{\Gamma }_{\mathrm{Faddeev}}\underset{\Lambda \to \infty }{=}\frac{m_f^3}{{\hslash }^6}{g^{\prime }}^2\kappa (\eta )\ln (\Lambda R_3)+o(1),$$

$${\hat{H}}_{3b}=g_3(\Lambda )\int d^3\mathbf{r}{\hat{\psi }}_1^{†}(\mathbf{r}){\hat{\psi }}_2^{†}(\mathbf{r}){\hat{\psi }}_3^{†}(\mathbf{r}){\hat{\psi }}_3(\mathbf{r}){\hat{\psi }}_2(\mathbf{r}){\hat{\psi }}_1(\mathbf{r})$$

$${\Gamma }_{3\mathrm{b}}=g_3(\Lambda ){\left(\frac{1}{\Omega }\sum _{k<\Lambda }\frac{1}{2{\epsilon }_k^{(\mathrm{f})}}\right)}^2.$$

$$g_3(\Lambda ){\left(\frac{1}{\Omega }\sum _{k<\Lambda }\frac{1}{2{\epsilon }_k^{(\mathrm{f})}}\right)}^2=-\frac{m_f^3}{{\hslash }^6}{g^{\prime }}^2\kappa (\eta )\ln (\Lambda R_3)$$

$$\Delta E_{3b}=g_3(\Lambda )⟨0|{\hat{\psi }}_1^{†}(\mathbf{r}){\hat{\psi }}_2^{†}(\mathbf{r}){\hat{\psi }}_2(\mathbf{r}){\hat{\psi }}_1(\mathbf{r})|0⟩$$

$$\Delta E_{3b}=-{g^{\prime }}^2\kappa (\eta )\frac{m_{\mathrm{f}}{C}_2}{{\hslash }^2\Omega }\ln (\Lambda R_3).$$

$$\begin{split}\Delta E=g^{\prime}n\biggl{[}1+&k_{F}a^{\prime}F\left(\frac{1}{k_{F}a}\right)\\ &-2\pi\frac{m_{\rm f}}{m_{\rm r}}\kappa(\eta)\frac{a^{\prime}C_{2}}{N}\ln(k_{F}R_{3})+...\biggr{]},\end{split}$$

$$\begin{split}F\left(\frac{1}{k_{F}a}\right)\underset{\Lambda\rightarrow\infty}{=}\frac{2\pi}{k_{F}}\biggl{[}\frac{\hbar^{2}}{m_{\rm r}}\int_{q<\Lambda}&\frac{d^{3}\bm{q}}{(2\pi)^{3}}\left(\frac{1}{\varepsilon^{\rm(r)}_{q}}-\chi(q,\varepsilon^{(\rm i)}_{q})\right)\\ &-\frac{m_{\rm f}}{m_{\rm r}}\kappa(\eta)\frac{C_{2}}{N}\ln(\Lambda/k_{F})\biggr{]}\end{split}$$

$$F\left(\frac{1}{k_{F}a}\right)\underset{a\to 0^{+}}{=}8{\pi }^2\kappa (\eta )\frac{m_{\mathrm{f}}}{m_{\mathrm{r}}}\frac{\ln \left(k_{F}a\right)}{k_{F}a}+...$$

$$g_{\mathrm{ad}}=2g^{\prime }\left[1-8{\pi }^2\kappa (\eta )\frac{m_{\mathrm{f}}}{m_{\mathrm{r}}}\frac{a^{\prime }}{a}\left(\ln (R_3/a)+C_{\mathrm{ad}}\right)...\right],$$

$$\chi (q,{\epsilon }_q^{(\mathrm{i})})\underset{q\to \infty }{=}\frac{1}{{\epsilon }_q^{(\mathrm{r})}}\left[1-{\pi }^2\kappa (\eta )\frac{m_{\mathrm{f}}}{m_{\mathrm{r}}}\frac{C_2}{Nq}+...\right]$$

Full text

The physics of an impurity immersed in a many-body ensemble is one of the simplest although non-trivial paradigms in many-body physics. One of the first examples of such a system is the polaron problem which was introduced by Landau and Pekar landau1948effective to describe the interaction of an electron with the acoustic excitations of a surrounding crystal. Likewise, in magnetic compounds Kondo’s Effect arises from the interaction of magnetic impurities with the background Fermi sea kondo1964resistance ; anderson1961localized . Similar situations occur in high-energy physics, e.g. in neutron stars to interpret the interaction of a proton with a superfluid of neutrons zuo20041s0 , or in quantum chromodynamics where the so-called Polyakov loop describes the properties of a test color charge immersed in a hot gluonic medium fukushima2017polyakov . Finally, impurity problems can be used as prototypes for more complex many body-systems levinsen2017universality , as illustrated by the dynamical mean-field theory Georges96dynamical .

The recent advent of strongly correlated quantum gases permitted by the control of interactions in these systems have opened a new research avenue for the physics of impurities bloch2008many ; zwerger2012BCSBEC ; Chevy2010Unitary ; massignan2014polarons . Experiments on strongly polarized Fermi gases zwierlein2006fsi ; partridge2006pap ; nascimbene2009pol were interpreted by the introduction of the so-called Fermi polarons, a quasi-particle describing the properties of an impurity immersed in an ensemble of spin-polarized fermions and dressed by a cloud of particle-hole excitations of the surrounding Fermi Sea chevy2006upa ; lobo2006nsp ; prokof'ev08fpb . More recently, the physics of Bose polarons (impurities immersed in a Bose-Einstein condensate) was explored using radio-frequency spectroscopy jorgensen2016observation ; hu2016bose . Contrary to the Fermi polaron, this system is subject to an Efimov effect efimov73energy and three-body interactions play an important role in the strongly correlated regime levinsen2015impurity . Finally recent experiments on dual superfluids have raised the question of the behaviour of an impurity immersed in a superfluid of spin 1/2 fermions Ferrier2014Mixture ; roy2017two ; yao2016observation . In these experiments, the polaron was weakly coupled to the background superfluid and the interaction could be accurately modeled within mean-field approximation. Further theoretical works explored the strongly coupled regime using mean-field theory to describe the fermionic superfluid nishida2015polaronic ; yi2015polarons . They highlighted the role of Efimov physics in the phase diagram of the system and as a consequence some results were plagued by unphysical ultraviolet divergences. In this letter we address this problem without making any assumption on the properties of the superfluid component. We calculate the first beyond-mean-field corrections to the energy of the polaron and we show that the logarithmic divergence arising from three-body physics can be cured within an effective field theory approach introduced previously in the study of beyond mean-field corrections in Bose gases braaten1999quantum ; braaten2002dilute .

Qualitatively speaking, the phase diagram of the impurity can be decomposed in three different regions when the strength of the impurity/fermion interaction is varied (see Fig. 1). For a weak attraction, the impurity can be described as a polaronic quasi-particle. When attraction is increased, the impurity binds to an existing Cooper pair and the polaronic branch connects to the resonant Efimov trimer states. Earlier variational calculations suggest that the transition between the polaron and trimeron states is a smooth crossover nishida2015polaronic ; yi2015polarons . Finally, in the strongly attractive regime, impurity/fermion attraction overcomes Cooper pairing leading to a dimeron state describing an impurity/fermion dimer immersed in a fermionic superfluid medium.

Since the size of the ground-state Efimov trimer is typically much smaller than the interparticle spacing, its binding energy is much larger than the Fermi energy of the fermionic superfluid. As a consequence, except when the Efimov trimer becomes resonant with the atomic continuum, the internal structure of the trimer is only weakly affected by the many-body environment. A first insight on the phase diagram of the system can thus be obtained from the study of the three-body problem to determine the stability domain of the Efimov trimers with respect to the free-atom and atom-dimer continuum. In this pursuit, we use a two-channel model similar to the one presented in e.g. gogolin2008analytical in the case of the bosonic Efimov problem SuppMat . For the sake of simplicity, we assume here that the masses of the fermions and the impurity are the same, and that the impurity interacts the same way with both spin states of the fermionic ensemble (these assumptions are well satisified in the experiments reported in Ferrier2014Mixture ). The properties of the system are therefore characterized by three different length scales: the fermion-fermion scattering length ($a$), the fermion-impurity scattering length ($a^{\prime}$) and the effective range of the interaction potential ($R_{e}$) 111Note that we define $R_{e}$ as in gogolin2008analytical that is positive for a two-channel model and whose sign is opposite to the traditional definition of the effective range.. The corresponding phase diagram is displayed in Fig. 2. When the fermion/fermion interaction strength is varied, the superfluid explores the BEC-BCS crossover zwerger2012BCSBEC that connects the weakly attractive regime ($a\rightarrow 0^{-}$) where the fermions form loosely bound Cooper pairs described by BCS (Bardeen-Cooper-Schrieffer) theory, to the strongly attractive limit ($a\rightarrow 0^{+}$) where they form a Bose-Einstein Condensate (BEC) of deeply bound dimers. As a consequence, the polaronic state smoothly evolves from a Fermi polaron (an impurity immersed in a non-interacting Fermi sea) to a Bose polaron (an impurity immersed in a BEC of dimers). The trimeron stability region is obtained by a numerical resolution of the three-body problem SuppMat . The polaron-dimeron frontier simply corresponds to a competition between fermion/fermion and fermion/impurity pairings.

Since the trimeron and dimeron regimes are dominated by few-body physics, we now focus on the energy of the polaronic branch that is more strongly affected by the presence of the superfluid. We furthermore assume that the impurity-fermion interaction can be treated perturbatively.

Consider thus an impurity of mass $m_{\rm i}$ immersed in a bath of spin 1/2 fermions of mass $m_{\rm f}$. We write the Hamiltonian of the system as

[TABLE]

where $\hat{H}_{\rm imp}$ (resp. $\hat{H}_{\rm mb}$) is the Hamiltonian of the impurity (resp. many-body background) alone, and $\hat{H}_{\rm int}$ describes the interaction between the two subsystems. We label the eigenstates of the impurity by its momentum $\bm{q}$ and its eigenvalues are $\varepsilon^{(\rm i)}_{q}=\hbar^{2}q^{2}/2m_{\rm i}$. The eigenstates and eigenvalues of $\hat{H}_{\rm mb}$ are denoted $|\alpha\rangle$ and $E_{\alpha}$, where by definition $\alpha=0$ corresponds to the ground state of the fermionic superfluid.

Assuming for simplicity an identical contact interaction between the impurity and each spin component of the many-body ensemble, we write

[TABLE]

where $\hat{\psi}_{\sigma}$ and $\hat{\phi}$ are the field operators for spin $\sigma$ particles of the many-body ensemble and of the impurity respectively. In this expression, the bare and physical coupling constants $g^{\prime}_{0}$ and $g^{\prime}$ are related through

[TABLE]

where $\Omega$ is the quantization volume, $\Lambda$ is some ultraviolet cutoff and $\epsilon^{({\rm r})}_{k}=\hbar^{2}k^{2}/2m_{\rm r}$, with $m_{\rm r}$ the impurity-fermion reduced mass. Assuming that the contact interaction can be treated perturbatively, we have up to second order

[TABLE]

Calculating the energy $\Delta E$ of the polaron to that same order, we have

[TABLE]

where $n$ is the particle density in the many-body medium and

[TABLE]

with $\hat{\rho}_{\bm{q}}=\sum_{\sigma}\int d^{3}\bm{r}\hat{\psi}_{\sigma}^{\dagger}(\bm{r})\hat{\psi}_{\sigma}(\bm{r})e^{i\bm{q}\cdot\bm{r}}$.

In the sum, the presence of the two terms allows for a UV cancellation of their $1/q^{2}$ asymptotic behaviours. Indeed, for large $q$ the eigenstates of the many-body Hamiltonian excited by the translation operator $\hat{\rho}_{\bm{q}}$ correspond to free-particle excitations of momentum $\bm{q}$ and energy $\varepsilon^{({\rm f})}_{q}=\hbar^{2}q^{2}/2m_{\rm f}$. We therefore have

[TABLE]

where $S(q)=\sum_{\alpha}|\langle\alpha|\hat{\rho}_{\bm{q}}|0\rangle|^{2}/N$ is the static structure factor of the many-body system. At large momenta, we have $S(q)=1+C_{2}/4Nq+...$, where $C_{2}$ is Tan’s contact parameter of the fermionic system and characterizes its short-range two-body correlations hu2010static ; tan2008large . From this scaling we see that the UV-divergent $1/q^{2}$ contributions in Eq. (5) cancel out. However, the next-to-leading order term in $S(q)$ suggests that this cancellation is not sufficient to regularize the sum that is still log-divergent. This logarithmic behaviour is supported by a directed calculation of $\chi$ using BCS mean-field theory SuppMat and is characteristic of a singularity in the three-body problem for particles with contact interactions that was pointed out first by Wu for bosons wu1959ground and was more recently investigated in the context of cold atoms (see for instance braaten1999quantum ; tan2008three ; hammer2015three ; yi2015polarons ).

To get a better insight on the origin of this singularity, we analyze first the scattering of an impurity with a pair of free fermions. Within Faddeev’s formalism faddeev2013quantum , the corresponding three-body $T$-matrix is written as a sum of three contributions, $\hat{T}_{i=1,2,3}$ solutions of the set of coupled equations

[TABLE]

where $\hat{G}_{0}=1/(z-\hat{H}_{0})$ is the free resolvant operator and $\hat{t}_{i}$ is the two-body $T$-matrix leaving particle $i$ unaffected. The solutions of this equation can be expressed as a series of diagrams where a given two-body $t$-matrix never acts twice in a row and $\hat{T}_{i}$ corresponds to the sum of all diagrams finishing by $\hat{t}_{i}$. Assuming that the impurity is labeled by the index $i=3$ and that its interaction with the other two atoms ($i=1,2$) is weak, we can expand the solutions of Faddeev’s equation with $\hat{t}_{1}$ and $\hat{t}_{2}$. To be consistent with the polaron-energy calculation outlined in previous section, we proceed up to second order in $\hat{t}_{1,2}$.

When the full 2-body $T$-matrices $t_{1,2}=g^{\prime}/\Omega/(1+ia^{\prime}k+a^{\prime}R_{e}k^{2})$ are used, all terms of the expansion are finite. However, when treating them within second order Born’s approximation (i.e. taking simply $t_{1,2}=g^{\prime}(1-ika^{\prime})/\Omega$ and stopping the expansion of the $T$-matrix at second order in $a$’), some diagrams are logarithmically divergent. The singular diagrams are listed in Fig. 3: they all start and end with $t_{3}$ and their contribution can be written as $(t_{3})_{\rm out}\Gamma(t_{3})_{\rm in}$. In Born’s approximation the sums over inner momenta are divergent and the integrals are therefore dominated by the large-k behaviour of $t_{3}$ and $G_{0}$. After a straightforward calculation, we obtain that

[TABLE]

with

[TABLE]

Since $\Gamma$ is finite when the full two-body physics is taken into account we introduce a three-body characteristic length $R_{3}$ such that

[TABLE]

where $\Gamma_{\rm Faddeev}$ corresponds to the value of $\Gamma$ obtained by using the full two-body T-matrices $t_{1,2}$ to calculate the first three diagrams of Fig. (3). In this perturbative approach, $R_{3}/a^{\prime}$ depends on $\eta$ and $R_{e}/a^{\prime}$ and can be computed numerically SuppMat . Since we work in a regime where the polaron is the ground state and Efimov trimers are absent, we do not have to use non-perturbative approaches compatible with Efimov physics and leading to a log-periodic dependence of $R_{3}$ bedaque1999renormalization .

Following the effective field theory approach discussed in hammer2015three , divergences plaguing Born’s expansion can be cured by introducing an explicit three-body interaction described by a Hamiltonian

[TABLE]

The contribution of this three-body interaction to $\Gamma$ corresponds to the fourth diagram of Fig. 3 and yields the following expression

[TABLE]

Using this three-body interaction to cure Born’s approximation, we must have $\Gamma_{\rm Born}+\Gamma_{\rm 3b}=\Gamma_{\rm Faddeev}$, hence the following expression for the three-body coupling constant

[TABLE]

The introduction of the three-body Hamiltonian implies a new contribution to the second-order energy shift (5). This new term amounts to

[TABLE]

Using Eq. (14) as well as the properties of Tan’s contact parameter, we obtain after a straightforward calculation

[TABLE]

Adding this contribution to Eq. (5), we obtain for the polaron energy $\Delta E=\Delta E_{\rm pert}+\Delta E_{3b}$:

[TABLE]

with

[TABLE]

Eq. (17) and (18) are the main results of this paper. They show that the second order correction of the polaron energy is the sum of two terms: a regular term characterized by the function $F$ defined by Eq. (17), as well as a second term, characterized by a logarithmic singularity and proportional to the fermionic contact parameter.

The function $F$ is in general hard to compute exactly but we can obtain its exact asymptotic expression in the BEC and BCS limits. When the fermions of the background ensemble are weakly interacting, we must recover the Fermi-polaron problem (see Fig. 2). For the mass-balanced case $\eta=1$, we obtain $F(-\infty)=3/2\pi$. In the strongly attractive limit, the fermionic ensemble behaves as a weakly interacting Bose-Einstein condensate of dimers and the polaron energy takes a general mean-field form $g_{\rm ad}n/2$, where $g_{\rm ad}$ is the impurity-dimer s-wave coupling constant and $n/2$ is the dimer density. Since in the BEC limit, $C_{2}/N=4\pi/a$, identifying Eq. (17) with the mean-field impurity-dimer interaction implies that

[TABLE]

and

[TABLE]

where the constant $C_{\rm ad}$ can be obtained from the direct analysis of the atom-dimer scattering problem SuppMat .

Eq. (17) can be used to benchmark previous works on this problem. Ref. yi2015polarons ; nishida2015polaronic were based on a mean-field description of the superfluid component of the system. The mean-field calculation obeys BCS and BEC asymptotic behaviours similar to those predicted by Eq. (17) except for the value of $\kappa$ that does not coincide with the present result since the last term in Eq. (10) is missing within a BCS approach SuppMat . This discrepancy is easily understandable. Indeed, this term corresponds to the third diagram of Fig. (3) where the two fermions interacts between their interaction with the impurity, which contradicts the BCS assumption of non-interacting Bogoliubov excitations. For $\eta=1$, $\kappa/\kappa_{\rm MF}\simeq 15$, showing that BCS approximation underestimates strongly beyond mean-field contributions. Eq. (20) can also be compared to the numerical calculation of the atom-dimer scattering length reported in zhang2014calibration . The comparison between numerics and our analytical result for experimentally relevant mass ratios is shown in Fig. (4) which demonstrates a very good agreement between the two approaches. Note also that Eq. (20) clarifies the range of validity of the perturbative expansion. In addition to the diluteness assumption $k_{F}|a^{\prime}|\ll 1$, the validity of Born’s expansion requires the additional condition $|a^{\prime}|/a\ll 1$ when $a>0$.

Finally, the convergence of Eq. (18) entails that $\chi$ must obey the large momentum asymptotic behavior

[TABLE]

For $m_{\rm i}\rightarrow\infty$, $\varepsilon^{(\rm i)}_{q}=0$, we have $\kappa(\infty)=-1/4\pi$ and we recover the asymptotic result derived in hofmann2017deep using operator product expansion. Note that mean-field theory predicts $\kappa_{\rm MF}(\infty)=0$, and therefore disagrees with this independent result.

Using the mean-field estimate for $F$ at unitarity, we see that the second-order correction to the polaron energy (Eq. (17)) is dominated by the logarithmic contribution. In the case of the polaron oscillation experiments reported in Ferrier2014Mixture , the predicted correction amounts to a 5% shift of the oscillation frequency. Although small, this correction is within the reach of current experimental capabilities and shows that the results presented in this work are necessary to achieve the percent-level agreement between experiment and theory targeted by state of the art precision quantum many-body physics.