Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1210.8179 · Europhysics Letters (EPL)

Polaron residue and spatial structure in a Fermi gas

C. Trefzger, Yvan Castin

Published Feb 1, 2013• 15 citations
arXivHTMLPDF

Abstract

We study the problem of a mobile impurity of mass $M$ interacting {\sl via} a s-wave broad or narrow Feshbach resonance with a Fermi sea of particles of mass $m$. Truncating the Hilbert space to at most one pair of particle-hole excitations of the Fermi sea, we determine ground state properties of the polaronic branch other than its energy, namely the polaron quasiparticle residue $Z$, and the impurity-to-fermion pair correlation function $G(x)$. We show that $G(x)$ deviates from unity at large distances as $-(A\_4+B\_4 \cos 2 k\_F x)/(k\_F x)^4$, where $k\_F$ is the Fermi momentum; since $A\_4>0$ and $B\_4>0$, the polaron has a diverging rms radius and exhibits Friedel-like oscillations. In the weakly attractive limit, we obtain analytical results, that in particular detect the failure of the Hilbert space truncation for a diverging mass impurity, as expected from Anderson orthogonality catastrophe; at distances between $\sim 1/k\_F$ and the asymptotic distance where the $1/x^4$ law applies, they reveal that $G(x)$ exhibits an intriguing multiscale structure.

DOI
10.1209/0295-5075/101/30006

Related papers

Impurity in a Fermi sea on a narrow Feshbach resonance: A variational study of the polaronic and dimeronic branches

C. Trefzger, Yvan Castin

We study the problem of a single impurity of mass $M$ immersed in a Fermi sea of particles of mass $m$. The impurity and the fermions interact through an $s$-wave narrow Feshbach resonance, so that the Feshbach length ${R}_{*}$ naturally appears in the system. We use simple variational ans\"atze, limited to at most one pair of particle-hole excitations of the Fermi sea, and we determine for the polaronic and dimeronic branches the phase diagram between absolute ground state, local minimum, thermodynamically unstable regions (with negative effective mass), and regions of complex energies (with negative imaginary part). We also determine the closed-channel population which is experimentally accessible. Finally, we identify a nontrivial weakly attractive limit where analytical results can be obtained, in particular for the crossing point between the polaronic and dimeronic energy branches.

arXiv:1112.4364•Physical Review A•May 10, 2012•51 citations

Energy, decay rate, and effective masses for a moving polaron in a Fermi sea: Explicit results in the weakly attractive limit

C. Trefzger, Yvan Castin

We study the properties of an impurity of mass $M$ moving through a spatially homogeneous three-dimensional fully polarized Fermi gas of particles of mass $m$. In the weakly attractive limit, where the effective coupling constant $g\to0^-$ and perturbation theory can be used, both for a broad and a narrow Feshbach resonance, we obtain an explicit analytical expression for the complex energy $ΔE(\KK)$ of the moving impurity up to order two included in $g$. This also gives access to its longitudinal and transverse effective masses $m_\parallel^*(\KK)$, $m_\perp^*(\KK)$, as functions of the impurity wave vector $\KK$. Depending on the modulus of $\KK$ and on the impurity-to-fermion mass ratio $M/m$ we identify four regions separated by singularities in derivatives with respect to $\KK$ of the second-order term of $ΔE(\KK)$, and we discuss the physical origin of these regions. Remarkably, the second-order term of $m_\parallel^*(\KK)$ presents points of non-differentiability, replaced by a logarithmic divergence for $M=m$, when $\KK$ is on the Fermi surface of the fermions. We also discuss the third-order contribution and relevance for cold atom experiments.

arXiv:1307.6326•Europhysics Letters (EPL)•Dec 1, 2013•7 citations

Quasi-particle residue and charge of the one-dimensional Fermi polaron

Giuliano Orso, L. Barisic, Ekaterina Gradova +2 more

We consider a mobile impurity coupled to an ideal Fermi gas in one spatial dimension through an attractive contact interaction. We calculate the quasi-particle residue $Z$ exactly, based on Bethe Ansatz and diagrammatic Monte Carlo methods, and with varational Ansatz up to one particle-hole excitation of the Fermi sea. We find that the exact quasi-particle residue vanishes in the thermodynamic limit as a power law in the number of particles, consistent with the Luttinger-liquid paradigm and the breakdown of Fermi-liquid theory. The variational Ansatz, however, predicts a finite value of $Z$, even in the thermodynamic limit. We also study how the presence of the impurity affects the density of the spin-up sea by calculating the pair correlation function. Subtracting the homogeneous background and integrating over all distances gives the charge $Q$. This charge turns out to grow continuously from 0 at zero coupling to 1 in the strong-coupling limit. The varational Ansatz predicts $Q=0$ at all couplings. So, although the variational Ansatz has been shown to be remarkably accurate for the energy and the effective mass, it fails even qualitatively when predicting $Z$ and the pair correlation function in the thermodynamic limit.

arXiv:2504.17558•Apr 24, 2025•1 citation

Highly polarized one-dimensional Fermi gases near a narrow p -wave resonance

Yinfeng Ma, Xiaoling Cui

Based on the recently developed interaction renormalization for the one-dimensional $p$-wave interaction, we study the problem of a single impurity immersed in a highly polarized Fermi sea. They interact through a narrow $p$-wave Feshbach resonance, so the effective range ${r}_{0}$ naturally appears in the system. We use the variational approach limited to single-particle--hole excitations on top of the unperturbed Fermi sea. The polaron exhibits two branches of solutions, namely, the attractive and repulsive branches, with varying scattering length across the resonance. We calculate the energy spectrum, residue, and effective mass for each of the branches. We compare the polaronic energy with the energy of the dressed molecule and find that the molecular state is energetically favored when increasing the interaction strength. The critical interaction strength for the polaron-to-molecule transition will shift to the BCS side of the $p$-wave resonance as the effective range increases.

arXiv:2007.06186•Physical Review A•Dec 26, 2019•6 citations

Self-energy of an impurity in an ideal Fermi gas to second order in the interaction strength

C. Trefzger, Yvan Castin

We study in three dimensions the problem of a spatially homogeneous zero-temperature ideal Fermi gas of spin-polarized particles of mass $m$ perturbed by the presence of a single distinguishable impurity of mass $M$. The interaction between the impurity and the fermions involves only the partial $s$ wave through the scattering length $a$ and has negligible range $b$ compared to the inverse Fermi wave number $1/{k}_{\mathrm{F}}$ of the gas. Through the interactions with the Fermi gas the impurity gives birth to a quasiparticle, which will be here a Fermi polaron (or more precisely a monomeron). We consider the general case of an impurity moving with wave vector $\mathbf{K}\ensuremath{\ne}\mathbf{0}$: Then the quasiparticle acquires a finite lifetime in its initial momentum channel because it can radiate particle-hole pairs in the Fermi sea. A description of the system using a variational approach, based on a finite number of particle-hole excitations of the Fermi sea, then becomes inappropriate around $\mathbf{K}=\mathbf{0}$. We rely thus upon perturbation theory, where the small and negative parameter ${k}_{\mathrm{F}}a\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ excludes any branches other than the monomeronic one in the ground state (as, e.g., the dimeronic one), and allows us a systematic study of the system. We calculate the impurity self-energy ${\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$ up to second order included in $a$. Remarkably, we obtain an analytical explicit expression for ${\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$, allowing us to study its derivatives in the plane $(K,\ensuremath{\omega})$. These present interesting singularities, which in general appear in the third-order derivatives ${\ensuremath{\partial}}^{3}{\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$. In the special case of equal masses, $M=m$, singularities appear already in the physically more accessible second-order derivatives ${\ensuremath{\partial}}^{2}{\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$; using a self-consistent heuristic approach based on ${\ensuremath{\Sigma}}^{(2)}$ we then regularize the divergence of the second-order derivative ${\ensuremath{\partial}}_{K}^{2}\ensuremath{\Delta}E(\mathbf{K})$ of the complex energy of the quasiparticle found in Trefzger and Castin [Europhys. Lett. 104, 50005 (2013)] at $K={k}_{\mathrm{F}}$, and we predict an interesting scaling law in the neighborhood of $K={k}_{\mathrm{F}}$. As a by product of our theory we have access to all moments of the momentum of the particle-hole pair emitted by the impurity while damping its motion in the Fermi sea at the level of Fermi's golden rule.

arXiv:1405.6155•Physical Review A•Sep 18, 2014•9 citations

Polarons and dressed molecules near narrow Feshbach resonances

Pietro Massignan

The properties of impurities immersed in a large Fermi sea are naturally described in terms of dressed quasiparticles: attractive and repulsive polarons, and dressed molecules. Motivated by recent experiments on narrow Feshbach resonances, we analyze here how the quasiparticle properties are affected by a non-zero resonance range. We find two interesting analytic results. For large range, the ground-state energy close to resonance is shown to become perturbative in the inverse range. In the limit of broad resonance instead, we provide a new Tan's relation linking the impurity ground-state energy E ↓ to the number of atoms in its dressing cloud Δ N . As a corollary, at unitarity one finds Δ N =− E ↓ / F , with F the Fermi energy of the bath.

arXiv:1112.1029•Europhysics Letters (EPL)•Apr 1, 2012•41 citations
We study the problem of a mobile impurity of mass MMM interacting {\sl via} a s-wave broad or narrow Feshbach resonance with a Fermi sea of particles of mass mmm. Truncating the Hilbert space to at most one pair of particle-hole excitations of the Fermi sea, we determine ground state properties of the polaronic branch other than its energy, namely the polaron quasiparticle residue ZZZ, and the impurity-to-fermion pair correlation function G(x)G(x)G(x). We show that G(x)G(x)G(x) deviates from unity at large distances as −(A_4+B_4cos⁡2k_Fx)/(k_Fx)4-(A\_4+B\_4 \cos 2 k\_F x)/(k\_F x)^4−(A_4+B_4cos2k_Fx)/(k_Fx)4, where k_Fk\_Fk_F is the Fermi momentum; since A_4>0A\_4>0A_4>0 and B_4>0B\_4>0B_4>0, the polaron has a diverging rms radius and exhibits Friedel-like oscillations. In the weakly attractive limit, we obtain analytical results, that in particular detect the failure of the Hilbert space truncation for a diverging mass impurity, as expected from Anderson orthogonality catastrophe; at distances between ∼1/k_F\sim 1/k\_F∼1/k_F and the asymptotic distance where the 1/x41/x^41/x4 law applies, they reveal that G(x)G(x)G(x) exhibits an intriguing multiscale structure.
We study the problem of a single impurity of mass MMM immersed in a Fermi sea of particles of mass mmm. The impurity and the fermions interact through an sss-wave narrow Feshbach resonance, so that the Feshbach length R∗{R}_{*}R∗​ naturally appears in the system. We use simple variational ans\"atze, limited to at most one pair of particle-hole excitations of the Fermi sea, and we determine for the polaronic and dimeronic branches the phase diagram between absolute ground state, local minimum, thermodynamically unstable regions (with negative effective mass), and regions of complex energies (with negative imaginary part). We also determine the closed-channel population which is experimentally accessible. Finally, we identify a nontrivial weakly attractive limit where analytical results can be obtained, in particular for the crossing point between the polaronic and dimeronic energy branches.
We study the properties of an impurity of mass MMM moving through a spatially homogeneous three-dimensional fully polarized Fermi gas of particles of mass mmm. In the weakly attractive limit, where the effective coupling constant g→0−g\to0^-g→0− and perturbation theory can be used, both for a broad and a narrow Feshbach resonance, we obtain an explicit analytical expression for the complex energy $ΔE(\KK)$ of the moving impurity up to order two included in ggg. This also gives access to its longitudinal and transverse effective masses $m_\parallel^*(\KK)$, $m_\perp^*(\KK)$, as functions of the impurity wave vector $\KK$. Depending on the modulus of $\KK$ and on the impurity-to-fermion mass ratio M/mM/mM/m we identify four regions separated by singularities in derivatives with respect to $\KK$ of the second-order term of $ΔE(\KK)$, and we discuss the physical origin of these regions. Remarkably, the second-order term of $m_\parallel^*(\KK)$ presents points of non-differentiability, replaced by a logarithmic divergence for M=mM=mM=m, when $\KK$ is on the Fermi surface of the fermions. We also discuss the third-order contribution and relevance for cold atom experiments.
We consider a mobile impurity coupled to an ideal Fermi gas in one spatial dimension through an attractive contact interaction. We calculate the quasi-particle residue ZZZ exactly, based on Bethe Ansatz and diagrammatic Monte Carlo methods, and with varational Ansatz up to one particle-hole excitation of the Fermi sea. We find that the exact quasi-particle residue vanishes in the thermodynamic limit as a power law in the number of particles, consistent with the Luttinger-liquid paradigm and the breakdown of Fermi-liquid theory. The variational Ansatz, however, predicts a finite value of ZZZ, even in the thermodynamic limit. We also study how the presence of the impurity affects the density of the spin-up sea by calculating the pair correlation function. Subtracting the homogeneous background and integrating over all distances gives the charge QQQ. This charge turns out to grow continuously from 0 at zero coupling to 1 in the strong-coupling limit. The varational Ansatz predicts Q=0Q=0Q=0 at all couplings. So, although the variational Ansatz has been shown to be remarkably accurate for the energy and the effective mass, it fails even qualitatively when predicting ZZZ and the pair correlation function in the thermodynamic limit.
Based on the recently developed interaction renormalization for the one-dimensional ppp-wave interaction, we study the problem of a single impurity immersed in a highly polarized Fermi sea. They interact through a narrow ppp-wave Feshbach resonance, so the effective range r0{r}_{0}r0​ naturally appears in the system. We use the variational approach limited to single-particle--hole excitations on top of the unperturbed Fermi sea. The polaron exhibits two branches of solutions, namely, the attractive and repulsive branches, with varying scattering length across the resonance. We calculate the energy spectrum, residue, and effective mass for each of the branches. We compare the polaronic energy with the energy of the dressed molecule and find that the molecular state is energetically favored when increasing the interaction strength. The critical interaction strength for the polaron-to-molecule transition will shift to the BCS side of the ppp-wave resonance as the effective range increases.
We study in three dimensions the problem of a spatially homogeneous zero-temperature ideal Fermi gas of spin-polarized particles of mass mmm perturbed by the presence of a single distinguishable impurity of mass MMM. The interaction between the impurity and the fermions involves only the partial sss wave through the scattering length aaa and has negligible range bbb compared to the inverse Fermi wave number 1/kF1/{k}_{\mathrm{F}}1/kF​ of the gas. Through the interactions with the Fermi gas the impurity gives birth to a quasiparticle, which will be here a Fermi polaron (or more precisely a monomeron). We consider the general case of an impurity moving with wave vector $\mathbf{K}\ensuremath{\ne}\mathbf{0}$: Then the quasiparticle acquires a finite lifetime in its initial momentum channel because it can radiate particle-hole pairs in the Fermi sea. A description of the system using a variational approach, based on a finite number of particle-hole excitations of the Fermi sea, then becomes inappropriate around K=0\mathbf{K}=\mathbf{0}K=0. We rely thus upon perturbation theory, where the small and negative parameter ${k}_{\mathrm{F}}a\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ excludes any branches other than the monomeronic one in the ground state (as, e.g., the dimeronic one), and allows us a systematic study of the system. We calculate the impurity self-energy ${\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$ up to second order included in aaa. Remarkably, we obtain an analytical explicit expression for ${\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$, allowing us to study its derivatives in the plane $(K,\ensuremath{\omega})$. These present interesting singularities, which in general appear in the third-order derivatives ${\ensuremath{\partial}}^{3}{\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$. In the special case of equal masses, M=mM=mM=m, singularities appear already in the physically more accessible second-order derivatives ${\ensuremath{\partial}}^{2}{\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$; using a self-consistent heuristic approach based on ${\ensuremath{\Sigma}}^{(2)}$ we then regularize the divergence of the second-order derivative ${\ensuremath{\partial}}_{K}^{2}\ensuremath{\Delta}E(\mathbf{K})$ of the complex energy of the quasiparticle found in Trefzger and Castin [Europhys. Lett. 104, 50005 (2013)] at K=kFK={k}_{\mathrm{F}}K=kF​, and we predict an interesting scaling law in the neighborhood of K=kFK={k}_{\mathrm{F}}K=kF​. As a by product of our theory we have access to all moments of the momentum of the particle-hole pair emitted by the impurity while damping its motion in the Fermi sea at the level of Fermi's golden rule.
The properties of impurities immersed in a large Fermi sea are naturally described in terms of dressed quasiparticles: attractive and repulsive polarons, and dressed molecules. Motivated by recent experiments on narrow Feshbach resonances, we analyze here how the quasiparticle properties are affected by a non-zero resonance range. We find two interesting analytic results. For large range, the ground-state energy close to resonance is shown to become perturbative in the inverse range. In the limit of broad resonance instead, we provide a new Tan's relation linking the impurity ground-state energy E ↓ to the number of atoms in its dressing cloud Δ N . As a corollary, at unitarity one finds Δ N =− E ↓ / F , with F the Fermi energy of the bath.