In the McGuire model — a one-dimensional gas of free fermions interacting with a single mobile impurity — the impurity Green's functions are exactly representable by Fredholm determinants at arbitrary temperature, and the equal-time (static) one-body function, its large-distance asymptotics via effective form factors, and the impurity momentum distribution were computed in Ref. [1], which declared the time-dependent generalization its next step. This problem asks for the time-domain completion of that program: derive the large-time asymptotics of the finite-temperature impurity Green's function at fixed spin rapidity, with uniformity across the winding transition of the effective phase shift; determine the finite-temperature impurity spectral function and its edge structure; and establish the infinite-coupling limit and whether the logarithmic-in-time spreading of the impurity predicted by Zvonarev, Cheianov and Giamarchi is recovered. No derivation of these asymptotics exists in the literature: the problem is open.

Background

A mobile impurity — a single distinguishable particle immersed in a quantum fluid, forming the quasiparticle known as a Fermi polaron — is a paradigmatic probe of one-dimensional (1D) many-body physics. Its central observable is the impurity one-body (Green's) function, ⟨ψimp(x,t)ψimp†(0,0)⟩\langle\psi_{\rm imp}(x,t)\psi^\dagger_{\rm imp}(0,0)\rangle, whose Fourier transform in position and time is the impurity spectral function, the quantity probed by radio-frequency and photoemission spectroscopy of ultracold atoms. The exactly solvable realization is the McGuire model, i.e. the fully polarized sector of the Yang-Gaudin model: NN free spinless fermions (or, equivalently, hard-core bosons) of density ρ0\rho_0 interacting with one impurity through a δ\delta-function potential of strength gg, with dimensionless coupling γ=mg/ρ0\gamma = mg/\rho_0. Its exact solution by the Bethe ansatz (a diagonalization scheme in which every eigenstate is parametrized by rapidities) labels each state by fermionic rapidities {kj}\{k_j\} plus a spin rapidity Λ\Lambda that fixes how the total momentum is shared between the gas and the impurity and determines the impurity-gas phase shift δ(k)=π/2−arctan⁡(Λ−αk)\delta(k)=\pi/2-\arctan(\Lambda-\alpha k) with α=2π/γ\alpha=2\pi/\gamma.

At finite temperature T=1/βT=1/\beta, the thermal impurity one-body function was computed by Gamayun, Panfil and Sant'Ana [Phys. Rev. A 106, 023305 (2022)] as a thermal average over Λ\Lambda of Fredholm determinants — determinants det⁡(1+V^)\det(1+\hat V) of integral operators, the standard exact representation of free-fermion-type correlation functions — with kernels built from the thermal Fermi weight σ(k)\sigma(k). Its large-distance asymptotics were extracted by the effective form factors (EFF) approach — form factors being the matrix elements of the impurity field operator between exact Bethe-ansatz eigenstates — a heuristic that replaces the exact finite-temperature determinant by a simplified 'effective form factor' kernel chosen to reproduce the same thermodynamic limit, whose associated matrix Riemann-Hilbert problem (a boundary-value problem for analytic matrix functions with prescribed jump conditions across a contour) is exactly solvable, thereby circumventing the nonlinear steepest-descent analysis (the standard asymptotic technique for such determinants) of the original kernel. The structure of the asymptotics is governed by the winding number of the effective phase shift: for ∣Λ∣<Λc=αμ|\Lambda|<\Lambda_c=\alpha\sqrt{\mu} (with μ\mu the chemical potential of the gas) it winds once around the origin, while for ∣Λ∣>Λc|\Lambda|>\Lambda_c it does not — a topological dichotomy analogous to the magnetic-field dichotomy of the Ising model and the XY spin chain.

The time-dependent problem is partly developed but not solved. A time- and temperature-dependent Fredholm-determinant representation of the impurity correlator has existed since 2015-2016 (Gamayun, Pronko, Zvonarev), and the thermal dynamic injection and ejection Green's functions — the linear responses to inserting and removing the impurity, which are related in thermal equilibrium by the Kubo-Martin-Schwinger (KMS) detailed-balance condition — have been written as Λ\Lambda-resolved determinants and shown to obey a refined KMS relation (Gamayun, Panfil, Sant'Ana, 2023). What does not exist is the asymptotic analysis of these time-dependent determinants. At large times and fixed Λ\Lambda, an additional power-law prefactor on top of the exponential decay is expected, as is found exactly in the infinite-coupling case of impenetrable anyons (one-dimensional particles whose exchange phase lies between that of bosons and fermions and which cannot pass through each other); and at infinite coupling the impurity is predicted to spread logarithmically in time (Zvonarev, Cheianov, Giamarchi, 2007, 2009) — an ultraslow diffusion law with no intrinsic time scale. The mathematical control of the EFF method itself has only recently been established, and only for equal-time determinants (Gamayun, Zhuravlev, 2025), where it reproduces the classical Szego, Hartwig-Fisher and Borodin-Okounkov asymptotic formulas for Toeplitz determinants (determinants whose entries depend only on the difference of their indices), together with subleading corrections.

Problem Statement

Starting from the exact Fredholm-determinant representation of the finite-temperature impurity Green's functions of the McGuire model (equivalently, the spin-rapidity-resolved injection and ejection Green's functions together with their thermal average over Λ\Lambda), derive the large-time behavior of the impurity's time-dependent one-body correlation function at arbitrary coupling and temperature, and extract the impurity spectral function at finite temperature. Specifically:

  1. Large-time asymptotics at fixed spin rapidity: obtain the leading asymptotic form of the Λ\Lambda-resolved Green's function, in both the space-like and time-like regimes (i.e. ∣x∣/t|x|/t respectively larger and smaller than the characteristic velocity of the determinant kernel), with control of the uniformity across the winding transition at ∣Λ∣=Λc|\Lambda|=\Lambda_c and of the subsequent thermal average over Λ\Lambda.

  2. Spectral function: from these asymptotics, or by an equivalent exact method, determine the finite-temperature impurity spectral function — its line shape and its edge structure (the impurity analogue of the X-ray-edge problem, where the sudden creation of a potential in a Fermi sea produces a power-law threshold singularity) at arbitrary coupling and temperature.

  3. Infinite-coupling limit: establish the limit of these results as the impurity-gas coupling diverges, and determine whether the logarithmic-in-time spreading of the impurity predicted by Zvonarev, Cheianov and Giamarchi is recovered.

The three deliverables above are required; extensions to a static impurity in a three-dimensional gas and to non-thermal (generalized Gibbs or quench-action) ensembles are out of scope. The task requires the asymptotic results themselves; establishing the mathematical validity of the effective-form-factor (or Riemann-Hilbert) machinery used to obtain them is not required.

References

  1. Oleksandr Gamayun, Milosz Panfil, Felipe Taha Sant'Ana, Mobile impurity in a one-dimensional gas at finite temperatures, Physical Review A 106, 023305 (2022), doi:10.1103/PhysRevA.106.023305, arXiv:2202.07657, https://arxiv.org/abs/2202.07657
  2. Oleksandr Gamayun, Milosz Panfil, Felipe Taha Sant'Ana, Kubo-Martin-Schwinger relation for an interacting mobile impurity, Physical Review Research 5, 043265 (2023), doi:10.1103/PhysRevResearch.5.043265, arXiv:2308.06482, https://arxiv.org/abs/2308.06482
  3. Oleksandr Gamayun, Yuri Zhuravlev, On finite-temperature Fredholm determinants, Physica D: Nonlinear Phenomena 479, 134716 (2025), doi:10.1016/j.physd.2025.134716, arXiv:2411.16401, https://arxiv.org/abs/2411.16401
  4. Oleksandr Gamayun, Oleg Lychkovskiy, One-dimensional Fermi polaron after a kick: two-sided singularity of the momentum distribution, Bragg reflection and other exact results, SciPost Physics 17, 063 (2024), doi:10.21468/SciPostPhys.17.2.063, arXiv:2404.02099, https://arxiv.org/abs/2404.02099
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  11. Yuri Zhuravlev, Eduard Naichuk, Nikolai Iorgov, Oleksandr Gamayun, Large-time and long-distance asymptotics of the thermal correlators of the impenetrable anyonic lattice gas, Physical Review B 105, 085145 (2022), doi:10.1103/PhysRevB.105.085145, arXiv:2110.06860, https://arxiv.org/abs/2110.06860
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