low- and high-energy excitations in the unitary fermi gasson and wingate, annal. of physics 2006...
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![Page 1: Low- and High-Energy Excitations in the Unitary Fermi GasSon and Wingate, Annal. of Physics 2006 Static Density Response Density Functional Rupak and Schafer Nucl.Phys.A816:52-64,2009](https://reader037.vdocuments.site/reader037/viewer/2022090508/601d8b4b6f604e4c17586f68/html5/thumbnails/1.jpg)
Low- and High-Energy Excitations in the Unitary Fermi Gas
Introduction / Motivation Homogeneous Gas Momentum Distribution Quasi-Particle Spectrum Low Energy Excitations and Static Structure Function
Inhomogeneous Gas and Density Functional
Small Trapped Systems High Energy Excitations and Response
Spin Response Density Response
Conclusions / Future
Kevin Schmidt Shiwei Zhang Stefano Gandolfi J. Carlson
C. Regal et al. PRL 2004
Fermi Condensates
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Homogeneous Unitary Fermi Gas
V0 can be tuned across BCS (| V0 | ~ 0) to BEC (- V0 >> EF) Concentrate on unitarity : zero energy bound state infinite scattering length
Values of ξ, δ, t are independent of ρ
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Endres, Kaplan, Lee and Nicholson, arXiv:1203.3169 (2012) PRA (2013)
Equations of State for uniform gas: ξ: Experiments, Analytic, and Computational
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Each step: multiply by exp [ - T dt / 2 ] momentum space Auxiliary field for exp [ -V dt ] coordinate space multiply by exp [ - T dt /2 ] momentum space Use importance sampling with BCS wave function
Lattice actions
T=0 Algorithm: Branching random walk (diffusion) using AFMC
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Homogeneous Gas: ξ and effective range
Carlson, Gandolfi, Schmidt, and Zhang, PRA 2011 Carlson, Gandolfi, and Gezerlis, PTEP 2012
MIT expt
arXiv:1110.3309 (Hu, et al)
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0 1 2 3 4k / kF
0
0.2
0.4
0.6
0.8
1n
(k)
BCSDMC
Momentum Distribution
0 1 2 3k / kF
0
0.1
0.2
0.3
n (k
)
BCSDMC
K4 n(k)
Important for spin excitations and high-energy behavior
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Nuclear Momentum Distributions (see Wiringa’s talk)
Wiringa, et al, 2013 (PRC)
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Spin Excitations: requires high energy
Zweirlein
Spin up, down densities in a trap
ρ
ρ
radius
JC and Reddy, PRL 2005
JC and Reddy, PRL 2007
analyzing MIT data
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S(q) for Unitary Fermi Gas
Simple BCS wave function does not describe low-q behavior added long-range Jastrow to better describe phonons
phonons
contact
Carlson and Gandolfi, PRA, 2014
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S(q) for Unitary Fermi Gas
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Inhomogeneous Matter: Neutron Drops
Harmonic Potential Requires gradient terms, pairing, L.S splitting, …
Gandolfi, et al, PRL 2011
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Low Energy Excitations in the UFG
Low-lying excitations (phonons) Transitions from 3 dimensions to 2 dimensions How do finite systems behave - bulk vs. finite
Density functional Necessary to understand `exotic’ (LOFF,...) phases
Add a spin-independent background potential Examine the system for: weak periodic potentials Harmonic Oscillator potentials (1d, 2d, 3d,...)
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Phonon Dispersion relation
Lowest order determined by next order involves c1 and c2
ξ
EFT treatment Son and Wingate, Annal. of Physics
2006
Static Density Response
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Density Functional
Rupak and Schafer Nucl.Phys.A816:52-64,2009
arXiv:0804.2678 Epsilon expansion at unitarity
compare to free fermions
Note increase in coefficient of gradient term at unitarity compared to free Fermi gas
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Change notation:
Free fermions (BCS limit) Free bosons (BEC limit M = 2m)
see also M. Forbes arXiv:1211.3779 for treatment with Superfluid Local Density Approximation We use only bosonic degrees of freedom no single-particle orbital summation for the density.
The gradient term is exactly like the kinetic term in the Gross–Pitaevskii equation (BEC).
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Static response from weak external potentials
At low q E(V0) determined by compressibility (ξ ) Next order in q determined by cg
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Periodic Potential with momentum k and strength V0
Static Response for BCS and BEC limits
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Why does this work so well far beyond weak coupling?
Assume that density functional is local (not true for Fermions) and scale invariant places strong constraints on ε(ρ).
➯ gradient term alone
works if
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Calculation of cg from weak external potential N=66, k/kF = 0.5, V0 = 0.25 fit gives cg = 0.30 (0.05)
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Beyond the lowest order gradient term
Assume density functional is local Must respect scale invariance Negele-Vautherin-like form (density matrix expansion)
Requires gradient term (cg) be treated non-perturbatively; c4 in perturbation theory
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q = kF
q = kF /2
Eventually larger q, V0 require higher gradient terms
Static response at larger q = 0.5, 1 kF
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What about small systems of trapped Fermions ?
Need ~ 10x greater accuracy than for bulk
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Improved DMC results for trapped fermions
More sophisticated trial wavefunction which includes additional single-particle orbitals & terms which go to SLDA-like pairing. Approaches FN bulk limit of 0.39 No obvious shell closures
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AFMC compared to DMC
Approaching bulk limit of 0.37, no shell closures
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Bands correspond to errors from inhomogeneous bulk
Trapped fermions: LDA + gradient + q4
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Conclusions (Low-Energy Excitations) Low-Energy degrees of freedom are phonons in UFG Scale invariance ties linear response to complete functional cg = 0.30(5) compared to 0.111 for BCS (free fermions) 0.50 for BEC (free bosons of mass 2m) Quadratic corrections important for trapped fermions No evidence for shell structure (large pairing gap) in the unitary Fermi Gas, even for small systems Can treat large systems and dynamics with GP-like equation
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High-Energy Excitations and Quasi-Elastic Response
Spin Response:
Density Response:
S� (q,!) =
Zd!
X
f
h0|⇢†�(q)⇢�(q)|0i�(! � (Ef � E0))
S⇢ (q,!) =
Zd!
X
f
h0|⇢†⇢(q)⇢⇢(q)|0i�(! � (Ef � E0))
⇢�
=
X
i
exp[iq · r] �x
(i)
⇢⇢ =
X
i
exp[iq · r]
Flips spin little FSI
Important FSI
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Analogies to Electron / Neutrino Scattering
12C
A=3,4
See Lovato talk
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0 1 2 3 4k / kF
0
0.2
0.4
0.6
0.8
1
n (k
)
BCSDMC
Momentum Distribution
0 1 2 3k / kF
0
0.1
0.2
0.3
n (k
)
BCSDMC
Can calculate PWIA or Spectral Function Models
Spectral Function From quasiparticle spectrum
Traditional Ingredients to the Response
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0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
Fermi GasDMCBCSExperiment
Spin Response : PWIA
General picture correct, low energy tail overestimated
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0 1 2 30
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
BCS - spectral fnDMC = spectral fnBCS-originalBCS-fit
Spin Response : Spectral Function Approach
Good description of data with spectral function and (adjusted) BCS
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0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
Expt (density)Expt (spin)
Spin versus Density response (Experiment)
Spin,density responses are identical in PWIA, SF
Density
Spin
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0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
Expt (density)BCS
BCS calculations give 2nd peak (pair propagation) But underestimates width
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0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
Expt (density)Expt (spin)BCSEuclidean (GIFT)
Inversion of Euclidean Response w/ GIFT algorithm
Qualitatively correct, but more physics input needed for exact reproduction
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Future
• Dynamics with fermionic `GP’ equations • Better treatment of density response • Multiple species • ... • Improvements to nuclear DFT • Applications to neutrino (nuclear) physics • …
Conclusions
• Low Energy excitations of UFG understood • Can be used to predict even very small clusters • Can be used to calculate dynamics with GP-like equations • High-energy response understood from QMC+BCS,
FSI can be very important … • Many analogies to nuclear physics
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Backup Slide
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Inhomogeneous Matter: Cold atoms and Neutron Drops
Constrain Isovector Gradient Terms in the density functional
Cold atoms versus neutron drops
Comparison of different Hamiltonians