Neutron Polaron in Neutron Matter

• Neutron polaron is a quasiparticle formed by a spin-down neutron dressed by excitations in a spin-up Fermi sea, serving as a key model for impurity behavior in low-density nuclear systems.
• It is studied using methods such as Brueckner-Hartree-Fock, quantum Monte Carlo, and auxiliary-field techniques to extract observables like energy, effective mass, and quasiparticle residue, closely matching cold-atom benchmarks.
• The analysis of neutron polarons informs calibrations of nuclear energy-density functionals and contributes to understanding astrophysical phenomena in neutron-star crusts.

Searching arXiv for recent and foundational papers on neutron polarons. arXiv search query: neutron polaron spin-polarized neutron matter The neutron polaron is a quasiparticle formed by a single spin-down neutron immersed in a zero-temperature Fermi sea of spin-up neutrons, so that the impurity is dressed by excitations of the surrounding medium. In the low-density regime, the dominant interaction is the neutron-neutron $^{1}S_{0}$ channel, whose large scattering length $a_{s}\approx -18.5\,\mathrm{fm}$ places spin-polarized neutron matter close to the unitary limit when $r_{e}<n^{-1/3}<|a_{s}|$. In that regime, the principal observables are the impurity energy, the effective mass, and the quasiparticle residue, and their values are strikingly close to those of the attractive Fermi polaron realized in ultracold atomic gases (Vidana, 2021).

1. Definition and physical regime

In many-body physics, a polaron is a mobile impurity dressed by excitations of a surrounding medium. The neutron polaron is the nuclear analogue in which the impurity is a single spin-down neutron and the medium is a free or weakly structured Fermi gas of spin-up neutrons. In the thermodynamic limit at fixed majority density $\rho_{\uparrow}$, the canonical definition of the polaron energy is

$$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$

so the neutron polaron is equivalently the spin-down chemical potential at infinite polarization (Forbes et al., 2013).

The low-density regime is special because the $^{1}S_{0}$ neutron-neutron interaction dominates and the hierarchy of length scales becomes nearly universal. For Fermi momenta in the interval $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$, neutron matter satisfies $r_{e}<n^{-1/3}<|a_{s}|$, which is the “near unitary” window discussed for spin-polarized neutron matter. In this range, the impurity behaves very similarly to the attractive Fermi polaron at unitarity (Vidana, 2021).

A broader usage appears in the literature on impurities in neutron-rich environments. Reviews of impurity physics in dilute neutron matter place the neutron polaron alongside proton and cluster polarons, emphasizing that the same impurity-medium logic extends to several nuclear systems (Tajima et al., 2023). A related but distinct realization occurs in dilute $\alpha$ matter, where a neutron immersed in an $\alpha$ condensate behaves as a $a_{s}\approx -18.5\,\mathrm{fm}$0-wave Bose polaron rather than a Fermi polaron (Tajima et al., 2024).

2. Microscopic formulations

A central approach for the canonical spin-down-in-spin-up system is the Brueckner-Hartree-Fock (BHF) ladder approximation. There the basic quantity is the in-medium $a_{s}\approx -18.5\,\mathrm{fm}$1-matrix $a_{s}\approx -18.5\,\mathrm{fm}$2, which solves the Bethe-Goldstone equation for an impurity neutron and a majority neutron. The calculations reported for low-density neutron polarons use either the chiral two-body nucleon-nucleon interaction of Entem and Machleidt at $a_{s}\approx -18.5\,\mathrm{fm}$3LO with a $a_{s}\approx -18.5\,\mathrm{fm}$4 cut-off (EM500) or the Argonne V18 phenomenological potential, with both $a_{s}\approx -18.5\,\mathrm{fm}$5 and $a_{s}\approx -18.5\,\mathrm{fm}$6 restricted to the $a_{s}\approx -18.5\,\mathrm{fm}$7 channel. Three-nucleon forces are expected to be irrelevant at $a_{s}\approx -18.5\,\mathrm{fm}$8 and are therefore neglected (Vidana, 2021).

In BHF, the impurity single-particle energy is written as

$a_{s}\approx -18.5\,\mathrm{fm}$9

with the self-consistent mean field

$r_{e}<n^{-1/3}<|a_{s}|$0

This formulation explicitly encodes repeated impurity-medium scattering in the medium (Vidana, 2021).

Quantum Monte Carlo methods provide an ab initio benchmark. Fixed-node GFMC and AFDMC calculations treat the impurity problem directly in finite periodic boxes and extract $r_{e}<n^{-1/3}<|a_{s}|$1 from the energy difference $r_{e}<n^{-1/3}<|a_{s}|$2. In the 2013 benchmark study, the Hamiltonian includes the majority kinetic energy, the impurity kinetic energy, and the neutron-neutron interaction between the impurity and each majority particle, with either the $r_{e}<n^{-1/3}<|a_{s}|$3 Argonne $r_{e}<n^{-1/3}<|a_{s}|$4 channel or a modified Pöschl-Teller potential fit to the same $r_{e}<n^{-1/3}<|a_{s}|$5 and $r_{e}<n^{-1/3}<|a_{s}|$6 (Forbes et al., 2013).

A more recent lattice treatment uses auxiliary-field quantum Monte Carlo (AFQMC) on a simple cubic lattice with periodic boundary conditions. The Hamiltonian is

$r_{e}<n^{-1/3}<|a_{s}|$7

with dispersion

$r_{e}<n^{-1/3}<|a_{s}|$8

The parameters $r_{e}<n^{-1/3}<|a_{s}|$9 and $\rho_{\uparrow}$0 are tuned to the desired two-body continuum parameters $\rho_{\uparrow}$1 using Lüscher’s formula, and a parametric matrix model is then used to emulate two-body AFQMC energies and accelerate the tuning procedure (Curry et al., 6 Oct 2025).

3. Quasiparticle observables

The standard observables are the polaron energy $\rho_{\uparrow}$2, the effective mass $\rho_{\uparrow}$3, and the quasiparticle residue $\rho_{\uparrow}$4. In the BHF formulation,

$\rho_{\uparrow}$5

so the dimensionless ratio $\rho_{\uparrow}$6 plays the role of the universal constant $\rho_{\uparrow}$7 familiar from the unitary Fermi polaron (Vidana, 2021).

The effective mass is defined by the low-momentum expansion

$\rho_{\uparrow}$8

The quasiparticle residue, which measures the discontinuity in the impurity momentum distribution, obeys the Fermi-liquid relation

$\rho_{\uparrow}$9

These definitions are the direct neutron-matter counterparts of the observables used in ultracold-atom polaron physics (Vidana, 2021).

Effective-range effects are non-negligible in neutron matter and enter analytic low-density expansions. For neutrons with $$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$0 and $$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$1, the perturbative expansion quoted in effective-field-theory treatments is

$$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$2

This makes explicit that the neutron polaron is not exactly the zero-range unitary polaron even when its observables are numerically close to the universal values (Forbes et al., 2013).

The lattice AFQMC study uses the same basic energy-difference definition,

$$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$3

and notes that an effective mass would follow from the dispersion

$$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$4

but $$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$5 was not computed in that work; residues $$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$6 were likewise not reported (Curry et al., 6 Oct 2025).

4. Benchmark results and near-unitary behavior

In the low-density window $$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$7, BHF calculations yield nearly density-independent dimensionless quasiparticle parameters. The reported ranges are as follows (Vidana, 2021).

Quantity	EM500	Argonne V18
$$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$8	$$E_{\rm pol}=E_{N_{\uparrow}+1}-E_{N_{\uparrow}} \;\xrightarrow{N_{\uparrow}\to\infty}\; \mu_{\downarrow}(\rho_{\uparrow},0),$$9 to $^{1}S_{0}$0	$^{1}S_{0}$1 to $^{1}S_{0}$2
$^{1}S_{0}$3	$^{1}S_{0}$4 to $^{1}S_{0}$5	$^{1}S_{0}$6 to $^{1}S_{0}$7
$^{1}S_{0}$8	$^{1}S_{0}$9 to $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$0	$0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$1 to $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$2

The significance of these numbers lies in their proximity to the attractive Fermi polaron at unitarity. The cold-atom benchmarks quoted in the same context are $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$3, $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$4, and $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$5; experiments give $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$6 and $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$7, while diagrammatic QMC yields $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$8. The neutron-polaron values therefore reproduce the universal unitary numbers within the quoted ranges, despite finite $0.25\lesssim k_{F}\lesssim 0.45\,\mathrm{fm}^{-1}$9 and finite $r_{e}<n^{-1/3}<|a_{s}|$0 (Vidana, 2021).

Earlier QMC and EFT benchmarks show the same trend. For $r_{e}<n^{-1/3}<|a_{s}|$1, fixed-node QMC found $r_{e}<n^{-1/3}<|a_{s}|$2, and the dimensionless energy was reported as $r_{e}<n^{-1/3}<|a_{s}|$3 at $r_{e}<n^{-1/3}<|a_{s}|$4 and $r_{e}<n^{-1/3}<|a_{s}|$5 at $r_{e}<n^{-1/3}<|a_{s}|$6. At $r_{e}<n^{-1/3}<|a_{s}|$7, the QMC points lie within a few percent of both dEFT and perturbative or resummed chiral-EFT bands; beyond $r_{e}<n^{-1/3}<|a_{s}|$8, the QMC energies become slightly more attractive than one-particle-one-hole dEFT and resummed-ladder EFT (Forbes et al., 2013).

The lattice AFQMC study extends the density range and reports, after extrapolating to the continuum, $r_{e}<n^{-1/3}<|a_{s}|$9 at $\alpha$0, $\alpha$1 at $\alpha$2, and $\alpha$3 at $\alpha$4. At low density, AFQMC agrees with previous diffusion-Monte-Carlo and EFT results at the $\alpha$5 level. At higher density, the energy deepens more rapidly than phenomenological BHF; the authors state that this may signal missing correlations in BHF or deficiencies in chiral EFT at higher $\alpha$6 (Curry et al., 6 Oct 2025).

A common misconception is that the neutron polaron is simply the unitary Fermi polaron under another name. The numerical agreement is strong only in the near-unitary low-density window where $\alpha$7. Outside that window, effective-range effects and density dependence become increasingly important, and the neutron-polaron energy departs from a density-independent universal constant (Vidana, 2021).

5. Beyond the canonical Fermi-sea problem

The term “neutron polaron” also appears in a distinct setting: a neutron immersed in dilute $\alpha$8 matter. In that problem the medium is a Bose-condensed gas of $\alpha$9 particles and the dominant coupling is resonant $\alpha$0-wave neutron-$\alpha$1 scattering in the $\alpha$2 channel, described by a two-channel Hamiltonian involving neutrons $\alpha$3, $\alpha$4’s, and the closed-channel $\alpha$5 resonance $\alpha$6. The empirical parameters used to fix the interaction are $\alpha$7, $\alpha$8, and $\alpha$9 (Tajima et al., 2024).

In this $a_{s}\approx -18.5\,\mathrm{fm}$00-wave Bose-polaron problem, the self-energy analysis yields $a_{s}\approx -18.5\,\mathrm{fm}$01, hence $a_{s}\approx -18.5\,\mathrm{fm}$02, and the effective mass becomes

$a_{s}\approx -18.5\,\mathrm{fm}$03

For small $a_{s}\approx -18.5\,\mathrm{fm}$04,

$a_{s}\approx -18.5\,\mathrm{fm}$05

The denominator vanishes at

$a_{s}\approx -18.5\,\mathrm{fm}$06

signaling a breakdown of the homogeneous-polaron picture. The quoted density scale is $a_{s}\approx -18.5\,\mathrm{fm}$07, while relevant dilute-$a_{s}\approx -18.5\,\mathrm{fm}$08 densities are typically $a_{s}\approx -18.5\,\mathrm{fm}$09–$a_{s}\approx -18.5\,\mathrm{fm}$10 (Tajima et al., 2024).

This setup has a second consequence: two such heavy polarons can form a bound dineutron through the residual $a_{s}\approx -18.5\,\mathrm{fm}$11 neutron-neutron attraction. For $a_{s}\approx -18.5\,\mathrm{fm}$12, the critical effective mass for binding is reported as $a_{s}\approx -18.5\,\mathrm{fm}$13, and because $a_{s}\approx -18.5\,\mathrm{fm}$14 grows linearly with $a_{s}\approx -18.5\,\mathrm{fm}$15, dineutron binding appears above a threshold density of order $a_{s}\approx -18.5\,\mathrm{fm}$16–$a_{s}\approx -18.5\,\mathrm{fm}$17, even though the dineutron is unbound in vacuum (Tajima et al., 2024).

The broader impurity framework in dilute neutron matter includes proton and cluster polarons as well. In that language, the quasiparticle energy of a single proton in neutron matter is associated with the symmetry energy through

$a_{s}\approx -18.5\,\mathrm{fm}$18

and the same logic has been used to argue that nuclei unbound in vacuum can acquire binding in a neutron background, modifying the ordinary chart of nuclides (Tajima et al., 2023). This does not redefine the neutron polaron itself, but it places it within a wider impurity-and-cluster phenomenology.

6. Constraints on functionals and astrophysical relevance

The neutron polaron has become a stringent constraint on the spin-dependent sector of nuclear energy-density functionals. In Skyrme-type functionals, the energy density in neutron matter contains both time-even and time-odd couplings, and the polaron energy is obtained from

$a_{s}\approx -18.5\,\mathrm{fm}$19

Benchmark QMC and EFT results show that existing Skyrme and Gogny functionals, including SIII, SGII, SkM*, SLy4/5, SkO, BSk9, SAMi, and D1N, underestimate $a_{s}\approx -18.5\,\mathrm{fm}$20 by $a_{s}\approx -18.5\,\mathrm{fm}$21 or more and/or have the wrong density dependence (Forbes et al., 2013).

To address this, the time-odd terms of UNEDF1 were refit to the neutron-polaron benchmark while keeping the time-even sector unchanged, producing UNEDF1-pol. The quoted fitted combinations are

$a_{s}\approx -18.5\,\mathrm{fm}$22

$a_{s}\approx -18.5\,\mathrm{fm}$23

$a_{s}\approx -18.5\,\mathrm{fm}$24

with the additional constraint $a_{s}\approx -18.5\,\mathrm{fm}$25 to avoid spin instabilities (Forbes et al., 2013). In this sense, the neutron polaron is not merely a specialized impurity problem; it is a calibration point for the highly polarized limit of nuclear matter.

The astrophysical relevance follows from the density scales at which low-density neutron matter occurs. In neutron-star crusts and outer layers, the density of unbound neutrons spans $a_{s}\approx -18.5\,\mathrm{fm}$26–$a_{s}\approx -18.5\,\mathrm{fm}$27, corresponding to $a_{s}\approx -18.5\,\mathrm{fm}$28. A small admixture of opposite-spin neutrons will then behave as polarons, affecting transport coefficients such as thermal conductivity and viscosity, as well as neutrino opacity and cooling; the same physics is also relevant to the onset of spin-polarized phases in strong magnetic fields, including magnetars (Vidana, 2021).

A plausible implication is that the neutron polaron acts as one of the clearest points of contact between ultracold-atom many-body physics and neutron-rich matter. The near-unitary behavior seen for $a_{s}\approx -18.5\,\mathrm{fm}$29 suggests a limited but nontrivial universality: cold-atom benchmarks can inform nuclear many-body descriptions, while neutron matter supplies a controlled finite-range deformation of the unitary Fermi-polaron problem (Vidana, 2021).