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arXiv:2608.30539

Mass-gap functional determinant approach for mobile Fermi polarons

Emilio Ramos Rodríguez, Eugen Dizer, Xin Chen, Richard Schmidt

cond-mat.quant-gascond-mat.str-elmath-phquant-phSubmitted Aug 31, 2026• Announced Sep 1, 2026• 0 citations
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Abstract

We extend the functional determinant approach (FDA), previously restricted to static impurities, to the case of finite-mass impurities by using the mass-gap description of Fermi polarons [Phys. Rev. Lett. 135, 193401 (2025)]. The quadratic structure of the mass-gap model enables the exact evaluation of many-body spectra and Ramsey dynamics for mobile impurities. We show that this mass-gap FDA smoothly interpolates between the Fermi-edge singularity for infinitely heavy impurities and the emergence of quasiparticle weight for finite impurity mass. Our results demonstrate that the mass-gap FDA provides a transparent and computationally efficient framework to describe the quantum dynamics of mobile impurities.

Comments
13 pages, 6 figures

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We extend the functional determinant approach (FDA), previously restricted to static impurities, to the case of finite-mass impurities by using the mass-gap description of Fermi polarons [Phys. Rev. Lett. 135, 193401 (2025)]. The quadratic structure of the mass-gap model enables the exact evaluation of many-body spectra and Ramsey dynamics for mobile impurities. We show that this mass-gap FDA smoothly interpolates between the Fermi-edge singularity for infinitely heavy impurities and the emergence of quasiparticle weight for finite impurity mass. Our results demonstrate that the mass-gap FDA provides a transparent and computationally efficient framework to describe the quantum dynamics of mobile impurities.
Recently, a mass-gap description of mobile impurities in a Fermi gas was introduced, which connects Anderson's orthogonality catastrophe for static impurities to the quasiparticle picture of Fermi polarons through a recoil-induced energy gap in the fermionic dispersion. That description, however, was restricted to zero temperature and did not address dynamics. Here we generalize the mass-gap model to finite temperature by combining the Lee--Low--Pines transformation with a self-consistent Hartree--Fock decoupling of the recoil-induced interaction, and we study the quench dynamics within this framework using the functional-determinant approach. At finite temperature the effective mass gap obeys the self-consistency equation Δ(T)=2UFtanh⁡[Δ(T)/4kBT]Δ(T)=2U_F\tanh[Δ(T)/4k_B T]Δ(T)=2UF​tanh[Δ(T)/4kB​T], with UF=kF2/2MU_F=k_F^2/2MUF​=kF2​/2M and MMM the impurity mass. This equation admits a nonzero solution below the characteristic temperature T∗=UF/(2kB)T^*=U_F/(2k_B)T∗=UF​/(2kB​) and closes as (T∗−T)1/2(T^*-T)^{1/2}(T∗−T)1/2. We identify this closing as the mean-field signature of the thermal melting of the polaron and molecule quasiparticles. Computing the Ramsey response S(t)S(t)S(t) after a sudden quench of the impurity--fermion interaction, we find that its long-time oscillations---quantum beats between the bound and in-gap states---disappear precisely above T∗T^*T∗. Our work ties the thermodynamic and dynamical fingerprints of polaron formation to a single temperature-dependent mean-field parameter.
We present a unified theory that connects the quasiparticle picture of Fermi polarons for mobile impurities to the Anderson orthogonality catastrophe for static impurities. By operator reordering of the underlying many-body Hamiltonian, we obtain a modified fermionic dispersion relation that features a recoil-induced energy gap, which we call the "mass gap." We show that the resulting mean-field Hamiltonian exhibits an in-gap state for finite impurity mass, which takes a key role in Fermi polaron and molecule formation. We identify the mass gap as the microscopic origin of the quasiparticle weight of Fermi polarons and derive a power-law scaling of the weight with the impurity-to-fermion mass ratio. The associated in-gap state is shown to give rise to the emergence of the polaron-to-molecule transition away from the limiting case of the Anderson orthogonality catastrophe in which the transition is absent.
Abstract In this brief review, we report some new development in the functional determinant approach (FDA), an exact numerical method, in the studies of a heavy quantum impurity immersed in Fermi gases and manipulated with radio-frequency pulses. FDA has been successfully applied to investigate the universal dynamical responses of a heavy impurity in an ultracold ideal Fermi gas in both the time and frequency domain, which allows the exploration of the renowned Anderson’s orthogonality catastrophe (OC). In such a system, OC is induced by the multiple particle-hole excitations of the Fermi sea, which is beyond a simple perturbation picture and manifests itself as the absence of quasiparticles named polarons. More recently, two new directions for studying heavy impurity with FDA have been developed. One is to extend FDA to a strongly correlated background superfluid background, a Bardeen–Cooper–Schrieffer (BCS) superfluid. In this system, Anderson’s orthogonality catastrophe is prohibited due to the suppression of multiple particle-hole excitations by the superfluid gap, which leads to the existence of genuine polaron. The other direction is to generalize the FDA to the case of multiple RF pulses scheme, which extends the well-established 1D Ramsey spectroscopy in ultracold atoms into multidimensional, in the same spirit as the well-known multidimensional nuclear magnetic resonance and optical multidimensional coherent spectroscopy. Multidimensional Ramsey spectroscopy allows us to investigate correlations between spectral peaks of an impurity-medium system that is not accessible in the conventional one-dimensional spectrum.
The functional determinant approach (FDA) is a simple method to compute exactly certain observables for ideal quantum systems and has been successfully applied to the Fermi polaron problem to obtain the dynamical overlap and spectral function. Unfortunately, its application to bosonic ultracold gases is prohibited by the failure of the grand canonical ensemble for these systems. In this paper we show how to circumvent this problem and develop a bosonic FDA. This yields exact injection and ejection spectra for ideal Bose polarons at arbitrary temperatures. While coherent features visible at absolute zero quickly smear out with rising temperature as expected, the linewidth of the main peak is, counterintuitively, found to decrease near unitarity. Furthermore, we provide explicit formulas for the overlap operator, which allow us to compute the necessary determinants for both Bose and Fermi polarons more efficiently than previously possible.
We investigate the system of a heavy impurity embedded in a paired two-component Fermi gas at the crossover from a Bose-Einstein condensate (BEC) to a Bardeen-Cooper-Schrieffer (BCS) superfluid via an extension of the functional determinant approach (FDA). FDA is an exact numerical approach applied to study manifestations of Anderson's orthogonality catastrophe (OC) in the system of a static impurity immersed in an ideal Fermi gas. Here, we extend the FDA to a strongly correlated superfluid background described by a BCS mean-field wave function. In contrast to the ideal Fermi gas case, the pairing gap in the BCS superfluid prevents the OC and leads to genuine polaron signals in the spectrum. Thus our exactly solvable model can provide a deeper understanding of polaron physics. In addition, we find that the polaron spectrum can be used to measure the superfluid pairing gap, and in the case of a magnetic impurity, the energy of the subgap Yu-Shiba-Rusinov (YSR) bound state. Our theoretical predictions can be examined with state-of-art cold-atom experiments.
Motivated by experiments with cold atoms, we investigate a mobile impurity immersed in a Fermi sea in three dimensions at zero temperature by means of the functional renormalization group. We first perform the derivative expansion of the effective action to calculate the ground-state energy and Tan's contact across the polaron-molecule transition for several mass imbalances. Next we study quasiparticle properties of the impurity by using a real-time method recently developed in nuclear physics, which allows one to go beyond the derivative expansion. We obtain the spectral function of the polaron and the effective mass and quasiparticle weight of attractive and repulsive polarons, and clarify how they are affected by mass imbalances.