fermi polaron

7/24/2019 Fermi Polaron

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1. What is a Fermi Polaron

It is a single impurity interacting with its environment.

It determines the low-temperature behavior of many condensed matter systems.

For Example an electron moving in a crystal lattice, displacing nearby ions and thus

creating a localized polarization.

The electron, together with its surrounding cloud of lattice distortions, forms the lattice

polaron. It is a uasiparticle with an energy and mass that di!er from that of the bare

electron.

in our case the environment is a Fermi gas

rather than phonons of the lattice, i.e. we analyze the problem

of a single spin up-Fermion interacting

with Fermi sea of spin down particles

"e have # $ %pin-down Fermion

& # 'contact interaction(

rho = 4 pi hbar2 a /m a: scattering legth.

& ) * # +o interaction a

& * molecular binding c

-/ & / * 0olaron b


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$/kF # interparticle distance

1ho / * - attractive interaction 2 a / *

2. Polaron wavefunction, energy and quasiparticle residue

V 2r# vander-"aals potential

V 2k# Fourier transform

the actual inter-atomic potential is van der Waals,

which can be modeled by a simple pseudopotential

V(x) = rho delta(x) for low energy scattering

(i.e. measure scattering length a of vdW potential

and plug into pseudopotential to model low

energy scattering). If the scattering length is postitive,

the pseudopotential is repulsive, but one has

to keep in mind that the vdW potential supports an additional

two-particle bound state at low energies in this case

(for a


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k, k,k4,q

#spin state 5,6 7k ) h8k8/8m

9# volume c:k,ck#creation and annihilation operators

k: momentum

Trial wavefunction for the Fermi polaron with zero momentum

0article without interaction ; particle interacting.

The energy is then minimized under variation of the parameters * and kq, with the

constraint of constant norm.

To minimize#

E6# hevy, 0hys. 1ev. ? @A, *BCB8D 28**B.

1esults#


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#uasiparticle residue of a single spin down impurity.

21eal

You didnt introduce the function Sigma here, so Id remove this equation

%pin down uasiparticle on top of a spin down Fermi sea, in the limit of vanishing Fermi

momentum kF6

20robability for not scattering

"ith

"ith

2?nalytic expression for the integral exists but does not provide additional insight

2Fermi energy#

Easily solved numerically


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E6 is negative due to the attractive interactions with the medium.

For $/kFa G we immediately obtain#

Hean eld result

f2E,q # scattering amplitude with energy and momentum E and q

. !adio Frequency "!F# spectrum from the variational $nsat%.

i.e.: transfer impurity from spin up state into a third state which doesnt interact with spin down

Fermi sea

> way to measure polaron binding energy

Jlue# state Kup # spin Lp 2MNO

1ed# state Kdown# %pin down 2/NO


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c), d) : here scattering length is negative a polaron, no molecule)

Fermis golden 1ule #

allows us to directly predict the shape of the impurity.

allows us to compute the probability to transfer the impurity into the third state as a function of RF

frequency (energy)

>hevyPs wave function o!ers a simple way of calculating the 1F spectrum of a single

impurity.

1F operator

promotes the impurity into the free nal state Kf

2second lowest hyperne state of B


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0ossible nal state#

$.

) zero momentum particle in the nal state plus a perfect Fermi sea of up spins

EK* ) * relative to the Fermi energy EF

8.

with q < kF and k > kF

0article with momentum q G k in the nal state and a Fermi sea with a hole at q and an

excited environment particle above the Fermi sea at k

relative to the Fermi energy EF

The matrix elements are

Two components in the 1F spectrum#

The rst part is a delta-peaS shifted by the uasiparticle energy. ?s E6 < *, it is shifted to

higher freuencies# The 1F photon has to supply additional energy to transfer the

impurity out of its attractive environment

?llows determination ofZ by simply integrating the area under the prominent peaS

0eaS is (coherent would not dephase over time.


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fermi polaron

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