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arXiv:2309.14782

Perturbative computation of thermal characteristics of the Stoner phase transition

Oskar Grocholski, Piotr H. Chankowski

Published Jan 1, 2023• 0 citations
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Abstract

We apply the thermal (imaginary time) perturbative expansion to the relevant effective field theory to compute characteristics of the phase transition to the ordered state which can occur at low temperatures in the gas of (nonrelativistic) spin 1/2 fermions interacting through a short-range spin independent repulsive binary interaction potential. We show how to obtain a systematic expansion of the system's free energy depending on the densities $n_+$ and $n_-$ of spin-up and spin-down fermions. In this paper we truncate this expansion at the second order and determine, by numerically minimizing the free energy, the equilibrium proportions of $n_+$ and $n_-$ (that is, the system's polarization) as functions of the temperature, the system's overall density $n = n_+ + n_-$ and the strength of the interaction.

DOI
10.3204/pubdb-2023-05943

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Applying the previously developed systematic thermal (imaginary time) perturbative expansion to the relevant effective field theory we compute the free energy FFF of the diluted gas of (nonrelativistic) spin 1/21/21/2 fermions interacting through a spin-independent repulsive two-body potential as a function of the numbers N+N_+N+​ and N−N_-N−​ of spin up and spin down fermions (i.e. as a function of the system's polarization) and the temperature TTT. We give the complete order (kFa0)3(k_{\rm F}a_0)^3(kF​a0​)3 (kFk_{\rm F}kF​ is the Fermi wave vector and a0a_0a0​ is the sss-wave scattering length characterizing the interaction potential) contribution to FFF. We also extend the computation beyond a fixed order by resumming to all orders in the parameter kFa0k_{\rm F}a_0kF​a0​ the contributions to FFF of two infinite sets of Feynman diagrams: the so-called particle-particle rings and the particle-hole rings. We find that including the second one of these two contributions has a dramatic consequence for the transition of the system from the paramagnetic to the ferromagnetic phase (the so called Stoner phase transition): in this approximation the phase transition simply disappears. This result does not contradict the expectation that a transition to the magnetically ordered state should occur in truly repulsive systems. The ppp-wave and higher scattering lengths, as well as other parameters chacterizing the interaction potential, are in such systems generally of the same order of magnitude as a0a_0a0​ and contributions depending on them should be, therefore, also included in FFF. Our results may, however, have implications for the search of the itinerant ferromagnetism of cold atomic gases in which large a0a_0a0​, much larger than all other parameters, is artificially created by exploiting the physics of the Feshbach resonance.
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Third order corrections to the ground state energy of the polarized diluted gas of spin 1/21/21/2 fermions
We present the results of the computation of the third order corrections to the ground state energy of the diluted polarized gas of nonrelativistic spin 1/21/21/2 fermions interacting through a spin-independent repulsive two-body potential. The corrections are computed within the effective field theory approach which does not require specifying the interaction potential explicitly but only to characterize it by only a few parameters - the scattering lengths a0a_0a0​, a1,…a_1,\dotsa1​,… and effective radii r0,…r_0,\dotsr0​,… - measurable in low energy fermion-fermion elastic scattering. The corrections are computed semi-analytically, that is are expressed in terms of two functions of the system's polarization. The functions are given by the integrals which can be easily evaluated using the Mathematica built-in routines for numerical integration.
The effective field theory approach simplifies the perturbative computation of the ground-state energy of the diluted gas of repulsively interacting fermions allowing in the case of the unpolarized system to easily rederive the classic results up to the (kFa0)2{({k}_{\mathrm{F}}{a}_{0})}^{2}(kF​a0​)2 order (where kF{k}_{\mathrm{F}}kF​ is the system's Fermi momentum and a0{a}_{0}a0​ the sss-wave scattering length) and (with more labor) to extend it up to the order (kFa0)4{({k}_{\mathrm{F}}{a}_{0})}^{4}(kF​a0​)4. The analogous expansion of the ground-state energy of the polarized gas of spin 1/21/21/2 fermions is known only up to the kFa0{k}_{\mathrm{F}}{a}_{0}kF​a0​ order (where kF{k}_{\mathrm{F}}kF​ stands for ${k}_{\mathrm{F}\ensuremath{\uparrow}}$ or ${k}_{\mathrm{F}\ensuremath{\downarrow}}$); the order (kFa0)2{({k}_{\mathrm{F}}{a}_{0})}^{2}(kF​a0​)2 contribution has been computed (analytically) only using a specific (hard-core type) interaction potential. Here we show that the effective field theory method also allows to easily obtain the order (kFa0)2{({k}_{\mathrm{F}}{a}_{0})}^{2}(kF​a0​)2 correction to the ground state of the polarized gas in a way applicable to all repulsive interactions.
Recent advances in rapidly quenched ultracold atomic Fermi gases near a Feshbach resonance have brought about a number of interesting problems in the context of observing the long-sought Stoner ferromagnetic phase transition. The possibility of experimentally obtaining a ``quasirepulsive'' regime in the upper branch of the energy spectrum due to the rapid quench is currently being debated, and the Stoner transition has mainly been investigated theoretically by using perturbation theory or at high polarization due to the limited theoretical approaches in the strongly repulsive regime. In this work, we present a nonperturbative theoretical approach to the quasirepulsive upper branch of a Fermi gas near a broad Feshbach resonance, and we determine the finite-temperature phase diagram for the Stoner instability. Our results agree well with the known quantum Monte Carlo simulations at zero temperature, and we recover the known virial expansion prediction at high temperature for arbitrary interaction strengths. At resonance, we find that the Stoner transition temperature becomes of the order of the Fermi temperature, around which the molecule formation rate becomes vanishingly small. This suggests a feasible way to observe Stoner ferromagnetism in the nondegenerate temperature regime.