Download Fermion Many-Body Systems I

Outline Intro EFT Dilute Summary Refs
Fermion Many-Body Systems I
Dick Furnstahl
Department of Physics
Ohio State University
June, 2004
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Outline
Overview of Fermion Many-Body Systems
Effective Low-Energy Theories
EFT for Dilute Fermi Systems
Summary I: Fermion Many-body Systems
References and Resources
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
Examples of Fermion Many-Body Systems
Collections of “fundamental” fermions (electrons, quarks, . . . )
or composites of odd number of fermions (e.g., proton)
Isolated atoms or molecules
electrons interacting via long-range (screened) Coulomb
find charge distribution, binding energy, bond lengths, . . .
Bulk solid-state materials
metals, insulators, semiconductors, superconductors, . . .
Liquid 3 He (superfluid!)
Cold fermionic atoms in (optical) traps
Atomic nuclei
Neutron stars
color superconducting quark matter
neutron matter
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
Examples of Fermion Many-Body Systems
Collections of “fundamental” fermions (electrons, quarks, . . . )
or composites of odd number of fermions (e.g., proton)
Isolated atoms or molecules
electrons interacting via long-range (screened) Coulomb
find charge distribution, binding energy, bond lengths, . . .
Bulk solid-state materials
metals, insulators, semiconductors, superconductors, . . .
Liquid 3 He (superfluid!)
Cold fermionic atoms in (optical) traps
Atomic nuclei
Neutron stars
color superconducting quark matter
neutron matter
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
126
Limits of nuclear
existence
82
protons
roc
r-p
rp
28
20
50 ss
ce
ro
-p
82
50
8
28
2
20
2 8 =10
A
Ab initio
few-body
calculations
12
A=
neutrons
A~
s
es
ory
he
l T Field
a
n
tio ean
nc
Fu ent M
y
t
t
nsi sis
De lfcon
Se
60
0Ñω Shell
Model
Many-body approaches
for ordinary nuclei
No-Core Shell Model
G-matrix
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
How Are Cold Atoms Like Neutron Stars?
Regal et al., ultracold fermions
Dick Furnstahl
Chandra X-Ray Observatory
image of pulsar in 3C58
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
Nuclear and Cold Atom Many-Body Problems
50
40
30
20
10
0
-10
n-n potential (MeV)
O2-O2 potential (meV)
Lennard-Jones and nucleon-nucleon potentials
500
n-n system
400
O -O system
2
300
2
200
100
0
-100
0
0
1
2
3
n-n distance (fm)
0.4
0.8
O2-O2 distance (nm)
1.2
[figure borrowed from J. Dobaczewski]
Are there universal features of such many-body systems?
How can we deal with “hard cores” in many-body systems?
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
(Nuclear) Many-Body Physics: “Old” Approach
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
(Nuclear) Many-Body Physics: “Old” Approach
One Hamiltonian for all
problems and energy/length
scales
Dick Furnstahl
For nuclear structure, protons
and neutrons with a local
potential fit to NN data
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
(Nuclear) Many-Body Physics: “Old” Approach
One Hamiltonian for all
problems and energy/length
scales
Find the “best” potential
Dick Furnstahl
For nuclear structure, protons
and neutrons with a local
potential fit to NN data
NN potential with χ2 /dof ≈ 1 up
to ∼ 300 MeV energy
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
(Nuclear) Many-Body Physics: “Old” Approach
One Hamiltonian for all
problems and energy/length
scales
Find the “best” potential
Two-body data may be
sufficient; many-body forces as
last resort
Dick Furnstahl
For nuclear structure, protons
and neutrons with a local
potential fit to NN data
NN potential with χ2 /dof ≈ 1 up
to ∼ 300 MeV energy
Let phenomenology dictate
whether three-body forces are
needed (answer: yes!)
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
(Nuclear) Many-Body Physics: “Old” Approach
One Hamiltonian for all
problems and energy/length
scales
Find the “best” potential
Two-body data may be
sufficient; many-body forces as
last resort
Avoid divergences
Dick Furnstahl
For nuclear structure, protons
and neutrons with a local
potential fit to NN data
NN potential with χ2 /dof ≈ 1 up
to ∼ 300 MeV energy
Let phenomenology dictate
whether three-body forces are
needed (answer: yes!)
Add “form factor” to suppress
high-energy intermediate states;
don’t consider cutoff
dependence
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Systems Traditional
(Nuclear) Many-Body Physics: “Old” Approach
One Hamiltonian for all
problems and energy/length
scales
Find the “best” potential
Two-body data may be
sufficient; many-body forces as
last resort
Avoid divergences
Choose diagrams by “art”
Dick Furnstahl
For nuclear structure, protons
and neutrons with a local
potential fit to NN data
NN potential with χ2 /dof ≈ 1 up
to ∼ 300 MeV energy
Let phenomenology dictate
whether three-body forces are
needed (answer: yes!)
Add “form factor” to suppress
high-energy intermediate states;
don’t consider cutoff
dependence
Use physical arguments (often
handwaving) to justify the
subset of diagrams used
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Resolution and the Pointillists
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Resolution and the Pointillists
George Seurat painted using closely spaced small dots
(≈ 0.4 mm wide) of pure pigment
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Resolution and the Pointillists
George Seurat painted using closely spaced small dots
(≈ 0.4 mm wide) of pure pigment
Why do the dots blend together?
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Diffraction and Resolution
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Diffraction and Resolution
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Diffraction and Resolution
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Diffraction and Resolution
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Diffraction and Resolution
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Diffraction and Resolution
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Diffraction and Resolution
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Wavelength and Resolution
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Principles of Effective Low-Energy Theories
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Principles of Effective Low-Energy Theories
If system is probed at low energies, fine details not resolved
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Principles of Effective Low-Energy Theories
If system is probed at low energies, fine details not resolved
use low-energy variables for low-energy processes
short-distance structure can be replaced by something simpler
without distorting low-energy observables
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
“Simple” Many-Body Problem: Hard Spheres
Infinite potential at radius R
R
Scattering solutions are
simple:
u0(r) = sin[k(r-R)]
2
Ek = k /M
0
R
r
>>R
What is the energy / particle
of the many-body system at a
given density?
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Quick Review of Scattering
P/2 + k
P/2 + k0
P/2 − k
0
sin(kr+δ)
0
P/2 − k
r
R
r →∞
ikr
Relative motion in frame with P = 0: ψ(r ) −→ eik·r + f (k, θ) er
where k 2 = k 0 2 = MEk and cos θ = k̂ · k̂ 0
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Quick Review of Scattering
P/2 + k
P/2 + k0
P/2 − k
0
sin(kr+δ)
0
P/2 − k
r
R
r →∞
ikr
Relative motion in frame with P = 0: ψ(r ) −→ eik·r + f (k, θ) er
where k 2 = k 0 2 = MEk and cos θ = k̂ · k̂ 0
Differential cross section is dσ/dΩ = |f (k, θ)|2
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Quick Review of Scattering
P/2 + k
P/2 + k0
P/2 − k
0
sin(kr+δ)
0
P/2 − k
r
R
r →∞
ikr
Relative motion in frame with P = 0: ψ(r ) −→ eik·r + f (k, θ) er
where k 2 = k 0 2 = MEk and cos θ = k̂ · k̂ 0
Differential cross section is dσ/dΩ = |f (k, θ)|2
P
Central V =⇒ partial waves: f (k, θ) = l (2l + 1)fl (k)Pl (cos θ)
where fl (k ) =
eiδl (k) sin δl (k)
1
=
k
k cot δl (k) − ik
and the S-wave phase shift is defined by
r →∞
u0 (r ) −→ sin[kr + δ0 (k)]
Dick Furnstahl
=⇒
δ0 (k ) = −kR for hard sphere
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
At Low Energies: Effective Range Expansion
As first shown by Schwinger, k l+1 cot δl (k) has a power series
expansion. For l = 0:
k cot δ0 = −
1
1
+ r0 k 2 − Pr03 k 4 + · · ·
a0
2
which defines the scattering length a0 and the effective range r0
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
At Low Energies: Effective Range Expansion
As first shown by Schwinger, k l+1 cot δl (k) has a power series
expansion. For l = 0:
k cot δ0 = −
1
1
+ r0 k 2 − Pr03 k 4 + · · ·
a0
2
which defines the scattering length a0 and the effective range r0
While r0 ∼ R, the range of the potential, a0 can be anything
if a0 ∼ R, it is called “natural”
|a0 | R (unnatural) is particularly interesting =⇒ tomorrow!
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
At Low Energies: Effective Range Expansion
As first shown by Schwinger, k l+1 cot δl (k) has a power series
expansion. For l = 0:
k cot δ0 = −
1
1
+ r0 k 2 − Pr03 k 4 + · · ·
a0
2
which defines the scattering length a0 and the effective range r0
While r0 ∼ R, the range of the potential, a0 can be anything
if a0 ∼ R, it is called “natural”
|a0 | R (unnatural) is particularly interesting =⇒ tomorrow!
The effective range expansion for hard sphere scattering is:
k cot(−kR) = −
1
1
+ Rk 2 + · · ·
R
3
=⇒
a0 = R
so the low-energy effective theory is natural
Dick Furnstahl
Fermion Many-Body Systems I
r0 = 2R/3
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
In Search of a Perturbative Expansion
If a0 is natural, then low-energy scattering simplifies further
For scattering at momentum k 1/R, we should recover a
perturbative expansion in kR for scattering amplitude:
f0 (k ) ∝
1
−→ a0 1 − ia0 k − (a02 − a0 r0 /2)k 2 + O(k 3 a03 )
k cot δ(k) − ik
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
In Search of a Perturbative Expansion
If a0 is natural, then low-energy scattering simplifies further
For scattering at momentum k 1/R, we should recover a
perturbative expansion in kR for scattering amplitude:
f0 (k ) ∝
1
−→ a0 1 − ia0 k − (a02 − a0 r0 /2)k 2 + O(k 3 a03 )
k cot δ(k) − ik
Can we reproduce this simple expansion for the hard-sphere?
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
In Search of a Perturbative Expansion
If a0 is natural, then low-energy scattering simplifies further
For scattering at momentum k 1/R, we should recover a
perturbative expansion in kR for scattering amplitude:
f0 (k ) ∝
1
−→ a0 1 − ia0 k − (a02 − a0 r0 /2)k 2 + O(k 3 a03 )
k cot δ(k) − ik
Can we reproduce this simple expansion for the hard-sphere?
Perturbation theory in the hard-sphere potential won’t work:
=⇒
0
hk|V |k0 i ∝
Z
0
dx eik·x V (x) e−ik ·x −→ ∞
R
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
In Search of a Perturbative Expansion (cont.)
Standard solution: Solve the scattering problem
nonperturbatively, then expand in kR
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
In Search of a Perturbative Expansion (cont.)
Standard solution: Solve the scattering problem
nonperturbatively, then expand in kR
For our example, this is easy =⇒ use δ0 (k) = −kR:
f0 (k ) ∝
1
−→ a0 − ia02 k − (a03 − a02 r0 /2)k 2 + O(k 3 a03 )
k cot δ(k ) − ik
−→ 1 − ikR − 2k 2 R 2 /3 + O(k 3 R 3 )
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
In Search of a Perturbative Expansion (cont.)
Standard solution: Solve the scattering problem
nonperturbatively, then expand in kR
For our example, this is easy =⇒ use δ0 (k) = −kR:
f0 (k ) ∝
1
−→ a0 − ia02 k − (a03 − a02 r0 /2)k 2 + O(k 3 a03 )
k cot δ(k ) − ik
−→ 1 − ikR − 2k 2 R 2 /3 + O(k 3 R 3 )
Easy for 2–2 scattering, but not for the many-body problem!
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
In Search of a Perturbative Expansion (cont.)
Standard solution: Solve the scattering problem
nonperturbatively, then expand in kR
For our example, this is easy =⇒ use δ0 (k) = −kR:
f0 (k ) ∝
1
−→ a0 − ia02 k − (a03 − a02 r0 /2)k 2 + O(k 3 a03 )
k cot δ(k ) − ik
−→ 1 − ikR − 2k 2 R 2 /3 + O(k 3 R 3 )
Easy for 2–2 scattering, but not for the many-body problem!
EFT approach: k 1/R means we probe at low resolution
=⇒ replace potential with a simpler but general interaction
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
EFT for “Natural” Short-Range Interaction
A simple, general interaction is a sum of delta functions and
derivatives of delta functions. In momentum space,
hk|Veft |k0 i = C0 +
1
2
C2 (k2 + k0 ) + C20 k · k0 + · · ·
2
Or, Left has most general local (contact) interactions:
Left
→2i
−
h ∂
↔
∇
C0 † 2 C2 = ψ i
+
ψ−
(ψ ψ) +
(ψψ)† (ψ ∇ 2 ψ) + h.c.
∂t
2M
2
16
†
+
↔
C20 ↔ †
D0 † 3
(ψ ∇ ψ) · (ψ ∇ ψ) −
(ψ ψ) + . . .
8
6
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
EFT for “Natural” Short-Range Interaction
A simple, general interaction is a sum of delta functions and
derivatives of delta functions. In momentum space,
hk|Veft |k0 i = C0 +
1
2
C2 (k2 + k0 ) + C20 k · k0 + · · ·
2
Or, Left has most general local (contact) interactions:
Left
→2i
−
h ∂
↔
∇
C0 † 2 C2 = ψ i
+
ψ−
(ψ ψ) +
(ψψ)† (ψ ∇ 2 ψ) + h.c.
∂t
2M
2
16
†
+
↔
C20 ↔ †
D0 † 3
(ψ ∇ ψ) · (ψ ∇ ψ) −
(ψ ψ) + . . .
8
6
Dimensional analysis =⇒ C2i ∼ 4π R 2i+1 ,
M
Dick Furnstahl
D2i ∼ 4π R 2i+4
M
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Feynman Rules for EFT Vertices
− 2i
→
h ∂
↔
∇
C0 † 2 C2 Left = ψ † i
+
ψ−
(ψ ψ) +
(ψψ)† (ψ ∇ 2 ψ) + h.c.
∂t
2M
2
16
+
↔
C20 ↔ †
D0 † 3
(ψ ∇ ψ) · (ψ ∇ ψ) −
(ψ ψ) + . . .
8
6
P/2 + k0
P/2 + k
=
P/2 − k
0
P/2 − k0
−ihk |VEFT |ki
+
− iC0
=
+
−iC2
+
k2 + k 0 2
2
+
−iC20 k · k0
···
−iD0
Dick Furnstahl
Fermion Many-Body Systems I
···
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Renormalization
Reproduce f0 (k ) in perturbation theory (Born series):
f0 (k) ∝ a0 − ia02 k − (a03 − a02 r0 /2)k 2 + O(k 3 a03 )
(0)
Consider the leading potential VEFT (x) = C0 δ(x) or
(0)
hk|Veft |k0 i =⇒
=⇒ C0
Choosing C0 ∝ a0 gets the first term. Now hk|VG0 V |k0 i:
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Renormalization
Reproduce f0 (k ) in perturbation theory (Born series):
f0 (k) ∝ a0 − ia02 k − (a03 − a02 r0 /2)k 2 + O(k 3 a03 )
(0)
Consider the leading potential VEFT (x) = C0 δ(x) or
(0)
hk|Veft |k0 i =⇒
=⇒ C0
Choosing C0 ∝ a0 gets the first term. Now hk|VG0 V |k0 i:
Z
=⇒
d 3q
1
−→ ∞!
3
2
(2π) k − q 2 + i
=⇒ Linear divergence!
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Renormalization
Reproduce f0 (k ) in perturbation theory (Born series):
f0 (k) ∝ a0 − ia02 k − (a03 − a02 r0 /2)k 2 + O(k 3 a03 )
(0)
Consider the leading potential VEFT (x) = C0 δ(x) or
(0)
hk|Veft |k0 i =⇒
=⇒ C0
Choosing C0 ∝ a0 gets the first term. Now hk|VG0 V |k0 i:
Z
=⇒
Λc
d 3q
1
Λc
ik
−→
−
+ O(k 2 /Λc )
(2π)3 k 2 − q 2 + i
2π 2
4π
=⇒ If cutoff at Λc , then can absorb into C0 , but all powers of k 2
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Renormalization
Reproduce f0 (k ) in perturbation theory (Born series):
f0 (k) ∝ a0 − ia02 k − (a03 − a02 r0 /2)k 2 + O(k 3 a03 )
(0)
Consider the leading potential VEFT (x) = C0 δ(x) or
(0)
hk|Veft |k0 i =⇒
=⇒ C0
Choosing C0 ∝ a0 gets the first term. Now hk|VG0 V |k0 i:
Z
=⇒
dDq
1
ik
D→3
−→ −
3
2
2
(2π) k − q + i
4π
Dimensional regularization with minimal subtraction
=⇒ only one power of k !
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Dim. reg. + minimal subtraction =⇒ simple power counting:
P/2 + k
P/2 + k0
P/2 − k
0
=
P/2 − k
iT (k, cos θ)
+
− iC0
−
+
+
+i
M
4π
!2
(C0 )3 k 2
+
− iC2 k 2
Matching: C0 = 4π a0 = 4π R ,
M
M
Dick Furnstahl
M
(C0 )2 k
4π
+
O(k 3 )
− iC20 k 2 cos θ
3
a2 r
C2 = 4π 0 0 = 4π R ,
M 3
M 2
Fermion Many-Body Systems I
···
Outline Intro EFT Dilute Summary Refs
Resolution Low-Energy Scattering
Dim. reg. + minimal subtraction =⇒ simple power counting:
P/2 + k
P/2 + k0
P/2 − k
0
=
P/2 − k
iT (k, cos θ)
+
− iC0
−
+
+
+i
M
4π
!2
(C0 )3 k 2
M
(C0 )2 k
4π
+
− iC2 k 2
+
O(k 3 )
− iC20 k 2 cos θ
3
a2 r
Matching: C0 = 4π a0 = 4π R , C2 = 4π 0 0 = 4π R , · · ·
M 3
M
M
M 2
Recovers expansion order-by-order with perturbative diagrams
one power of k per diagram, natural coefficients
estimate truncation error from dimensional analysis
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
“Simple” Many-Body Problem: Hard Spheres
R
Infinite potential at radius R
sin(kr+δ)
0
r
R
~ 1/ k
Scattering length a0 = R
Dilute nR 3 1 =⇒ kF a0 1
What is the energy / particle?
Dick Furnstahl
Fermion Many-Body Systems I
F
Outline Intro EFT Dilute Summary Refs
Spheres
Noninteracting Fermi Sea at T = 0
Put system in a large box (V = L3 ) with periodic bc’s
spin-isospin degeneracy ν (e.g., for nuclei, ν = 4)
fill momentum states up to Fermi momentum kF
N=ν
kF
X
k
Dick Furnstahl
1,
E =ν
kF
X
~2 k 2
k
2M
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Noninteracting Fermi Sea at T = 0
Put system in a large box (V = L3 ) with periodic bc’s
spin-isospin degeneracy ν (e.g., for nuclei, ν = 4)
fill momentum states up to Fermi momentum kF
N=ν
kF
X
1,
E =ν
k
Note:
R
F (k) dk ≈
P
i
k
F (ki )∆ki =
Dick Furnstahl
kF
X
~2 k 2
P
i
2M
F (ki ) 2π
L ∆ni =
Fermion Many-Body Systems I
2π
L
P
i
F (ki )
Outline Intro EFT Dilute Summary Refs
Spheres
Noninteracting Fermi Sea at T = 0
Put system in a large box (V = L3 ) with periodic bc’s
spin-isospin degeneracy ν (e.g., for nuclei, ν = 4)
fill momentum states up to Fermi momentum kF
N=ν
kF
X
k
1,
E =ν
kF
X
~2 k 2
k
2M
R
P
P
Note: F (k) dk ≈ i F (ki )∆ki = i F (ki ) 2π
L ∆ni =
In 1-D:
L
N=ν
2π
Z
+kF
dk =
−kF
N
νkF
νkF
L =⇒ n =
=
;
π
L
π
2π
L
P
i
F (ki )
E
1 ~2 kF2
=
n
L
3 2M
In 3-D:
V
N=ν
(2π)3
Z
kF
d 3k =
νkF3
N
νkF3
V
=⇒
n
=
=
;
6π 2
V
6π 2
Dick Furnstahl
Fermion Many-Body Systems I
E
3 ~2 kF2
=
n
V
5 2M
Outline Intro EFT Dilute Summary Refs
Spheres
Noninteracting Fermi Sea at T = 0
Put system in a large box (V = L3 ) with periodic bc’s
spin-isospin degeneracy ν (e.g., for nuclei, ν = 4)
fill momentum states up to Fermi momentum kF
N=ν
kF
X
k
1,
E =ν
kF
X
~2 k 2
k
2M
R
P
P
Note: F (k) dk ≈ i F (ki )∆ki = i F (ki ) 2π
L ∆ni =
In 1-D:
L
N=ν
2π
Z
+kF
dk =
−kF
N
νkF
νkF
L =⇒ n =
=
;
π
L
π
2π
L
P
i
F (ki )
E
1 ~2 kF2
=
n
L
3 2M
In 3-D:
V
N=ν
(2π)3
Z
kF
d 3k =
νkF3
N
νkF3
V
=⇒
n
=
=
;
6π 2
V
6π 2
E
3 ~2 kF2
=
n
V
5 2M
Volume/particle V /N = 1/n ∼ 1/kF3 , so spacing ∼ 1/kF
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Sum Over Fermions in the Fermi Sea
(0)
Leading order VEFT (x) = C0 δ(x)
ELO
=⇒
!2
kF
X
C0
=
ν(ν − 1)
1
∝ a0 kF6
2
k
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Sum Over Fermions in the Fermi Sea
(0)
Leading order VEFT (x) = C0 δ(x)
ELO
=⇒
!2
kF
X
C0
=
ν(ν − 1)
1
∝ a0 kF6
2
k
At the next order, we get a linear divergence again:
Z
=⇒
∞
ENLO ∝
kF
Dick Furnstahl
C02
d 3q
(2π)3 k 2 − q 2
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Sum Over Fermions in the Fermi Sea
(0)
Leading order VEFT (x) = C0 δ(x)
ELO
=⇒
!2
kF
X
C0
=
ν(ν − 1)
1
∝ a0 kF6
2
k
At the next order, we get a linear divergence again:
Z
∞
ENLO ∝
=⇒
kF
C02
d 3q
(2π)3 k 2 − q 2
Same renormalization fixes it! Particles −→ holes
Z
∞
kF
1
=
k 2 − q2
Z
0
∞
1
−
k 2 − q2
Z
Dick Furnstahl
0
kF
1
D→3
−→ −
k 2 − q2
Z
0
kF
1
∝ a02 kF7
k 2 − q2
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Feynman Rules for Hugenholtz Diagrams
Ground-state energy density E is sum of Hugenholtz diagrams
same vertices as free space (same renormalization!)
Feynman rules:
1. Each line is assigned conserved ke ≡ (k0 , ~k) and
θ(k − kF )
θ(kF − k)
e
iG0 (k )αβ = iδαβ
+
.
k0 − ω~k + i k0 − ω~k − i
β
δ
α
γ
−→ (δαγ δβδ + δαδ δβγ ) (spin-independent)
2.
3. After spinRsummations, δαα → −g in every closed fermion loop.
+
4. Integrate d 4 k/(2π)4 with eik0 0 for tadpoles
Qlmax
5. Symmetry factor i/(S l=2 (l!)k ) counts vertex permutations
and equivalent l–tuples of lines
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Power Counting
Power counting rules
1. for every propagator: M/k2F
2. for every loop integration: k5F /M
2i+3n−5
3. for every n–body vertex with 2i derivatives: k2i
F /MΛ
Diagram with V2in n–body vertices of each type scales as (kF )β :
β =5+
∞ X
∞
X
(3n + 2i − 5)V2in .
n=2 i=0
e.g.,
=⇒ V02 = 2
=⇒ β = 5 + (3 · 2 + 2 · 0 − 5) · 2 = 7 =⇒ O(k7F )
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
T = 0 Energy Density from Hugenholtz Diagrams
E
kF2 3
= n
V
2M 5
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
T = 0 Energy Density from Hugenholtz Diagrams
O kF6 :
E
kF2 3
2
= n
+ (ν − 1) (kF a0 )
V
2M 5
3π
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
T = 0 Energy Density from Hugenholtz Diagrams
E
kF2 3
2
= n
+ (ν − 1) (kF a0 )
V
2M 5
3π
O kF6 :
O kF7 :
+
+ (ν − 1)
4
(11 − 2 ln 2)(kF a0 )2
35π 2
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
T = 0 Energy Density from Hugenholtz Diagrams
E
kF2 3
2
= n
+ (ν − 1) (kF a0 )
V
2M 5
3π
O kF6 :
O kF7 :
O kF8 :
+
+ (ν − 1)
4
(11 − 2 ln 2)(kF a0 )2
35π 2
+
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
T = 0 Energy Density from Hugenholtz Diagrams
E
kF2 3
2
= n
+ (ν − 1) (kF a0 )
V
2M 5
3π
O kF6 :
+
O kF7 :
+ (ν − 1)
+ (ν − 1) 0.076 + 0.057(ν − 3) (kF a0 )3
+
O kF8 :
+
4
(11 − 2 ln 2)(kF a0 )2
35π 2
+
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
T = 0 Energy Density from Hugenholtz Diagrams
E
kF2 3
2
= n
+ (ν − 1) (kF a0 )
V
2M 5
3π
O kF6 :
+
O kF7 :
+ (ν − 1)
+ (ν − 1) 0.076 + 0.057(ν − 3) (kF a0 )3
+
O kF8 :
4
(11 − 2 ln 2)(kF a0 )2
35π 2
1
(kF r0 )(kF a0 )2
10π
1
3
+ (ν + 1) (kF ap ) + · · ·
5π
+ (ν − 1)
+
+
+
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Looks Like a Power Series in kF ! Is it?
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Looks Like a Power Series in kF ! Is it?
New logarithmic divergences in 3–3 scattering
+
Dick Furnstahl
∝ (C0 )4 ln(k /Λc )
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Looks Like a Power Series in kF ! Is it?
New logarithmic divergences in 3–3 scattering
∝ (C0 )4 ln(k /Λc )
+
Changes in Λc must be absorbed by 3-body coupling D0 (Λc )
=⇒ D0 (Λc ) ∝ (C0 )4 ln(a0 Λc ) + const. [Braaten & Nieto]
d
dΛc
= 0 =⇒ same coefficient!
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Spheres
Looks Like a Power Series in kF ! Is it?
New logarithmic divergences in 3–3 scattering
∝ (C0 )4 ln(k /Λc )
+
Changes in Λc must be absorbed by 3-body coupling D0 (Λc )
=⇒ D0 (Λc ) ∝ (C0 )4 ln(a0 Λc ) + const. [Braaten & Nieto]
d
dΛc
= 0 =⇒ same coefficient!
What does this imply for the energy density?
O kF9 ln(kF ) :
+
+
Dick Furnstahl
···
∝ (ν − 2)(ν − 1) (kF a0 )4 ln(kF a0 )
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Effective Field Theory Ingredients
See “Crossing the Border” [nucl-th/0008064]
1. Use the most general L with low-energy dof’s consistent with
global and local symmetries of underlying theory
∂
Left = ψ † i ∂t
+
∇2
2M
ψ−
C0
†
2
2 (ψ ψ)
Dick Furnstahl
−
D0
†
3
6 (ψ ψ)
+ ...
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Effective Field Theory Ingredients
See “Crossing the Border” [nucl-th/0008064]
1. Use the most general L with low-energy dof’s consistent with
global and local symmetries of underlying theory
∂
Left = ψ † i ∂t
+
∇2
2M
ψ−
C0
†
2
2 (ψ ψ)
−
D0
†
3
6 (ψ ψ)
+ ...
2. Declaration of regularization and renormalization scheme
natural a0 =⇒ dimensional regularization and minimal subtraction
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Effective Field Theory Ingredients
See “Crossing the Border” [nucl-th/0008064]
1. Use the most general L with low-energy dof’s consistent with
global and local symmetries of underlying theory
∂
Left = ψ † i ∂t
+
∇2
2M
ψ−
C0
†
2
2 (ψ ψ)
−
D0
†
3
6 (ψ ψ)
+ ...
2. Declaration of regularization and renormalization scheme
natural a0 =⇒ dimensional regularization and minimal subtraction
3. Well-defined power counting =⇒ small expansion parameters
use the separation of scales =⇒ kF with Λ ∼ 1/R =⇒ kF a0 , etc.
Λ
O kF6 :
E =ρ
O kF7 :
+
kF2 3
2
4
2
+
(kF a0 ) +
(11
−
2
ln
2)(k
a
)
+
·
·
·
F 0
2M 5 3π
35π 2
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
(Nuclear) Many-Body Physics: “Old” vs. “New”
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
(Nuclear) Many-Body Physics: “Old” vs. “New”
One Hamiltonian for all
problems and energy/length
scales
Dick Furnstahl
Infinite # of low-energy
potentials; different resolutions
=⇒ different dof’s and
Hamiltonians
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
(Nuclear) Many-Body Physics: “Old” vs. “New”
One Hamiltonian for all
problems and energy/length
scales
Find the “best” potential
Dick Furnstahl
Infinite # of low-energy
potentials; different resolutions
=⇒ different dof’s and
Hamiltonians
There is no best potential
=⇒ use a convenient one!
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
(Nuclear) Many-Body Physics: “Old” vs. “New”
One Hamiltonian for all
problems and energy/length
scales
Find the “best” potential
Two-body data may be
sufficient; many-body forces as
last resort
Dick Furnstahl
Infinite # of low-energy
potentials; different resolutions
=⇒ different dof’s and
Hamiltonians
There is no best potential
=⇒ use a convenient one!
Many-body data needed and
many-body forces inevitable
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
(Nuclear) Many-Body Physics: “Old” vs. “New”
One Hamiltonian for all
problems and energy/length
scales
Find the “best” potential
Two-body data may be
sufficient; many-body forces as
last resort
Avoid divergences
Dick Furnstahl
Infinite # of low-energy
potentials; different resolutions
=⇒ different dof’s and
Hamiltonians
There is no best potential
=⇒ use a convenient one!
Many-body data needed and
many-body forces inevitable
Exploit divergences
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
(Nuclear) Many-Body Physics: “Old” vs. “New”
One Hamiltonian for all
problems and energy/length
scales
Find the “best” potential
Two-body data may be
sufficient; many-body forces as
last resort
Avoid divergences
Choose diagrams by “art”
Dick Furnstahl
Infinite # of low-energy
potentials; different resolutions
=⇒ different dof’s and
Hamiltonians
There is no best potential
=⇒ use a convenient one!
Many-body data needed and
many-body forces inevitable
Exploit divergences
Power counting determines
diagrams and truncation error
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Web Resources
These lectures, including additional notes on each slide, are
available at
http://www.physics.ohio-state.edu/˜ntg/NPSS/
Class notes for a two-quarter course on “Nuclear Many-Body
Physics” given by Dick Furnstahl and Achim Schwenk are
available at
http://www.physics.ohio-state.edu/˜ntg/880/
(username: physics, password: 880.05)
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
References for Many-Body Physics
A.L. Fetter and J.D. Walecka, “Quantum Theory of Many-Particle Systems.” Classic text, but pre-path integrals. Now
available in an inexpensive (about $20) Dover reprint. Get it!
J.W. Negele and H. Orland, “Quantum Many-Particle Systems.” Detailed and careful use of path integrals. Full of good
physics but most of the examples are in the problems, so it can be difficult to learn from.
N. Nagaosa, “Quantum Field Theory in Condensed Matter Physics.” Recent text, covers path integral methods and
symmetry breaking.
A.M. Tsvelik, “Quantum Field Theory in Condensed Matter Physics.” Good on one-dimensional systems.
M. Stone, “The Physics of Quantum Fields.” A combined introduction to quantum field theory as applied to particle
physics problems and to nonrelativistic many-body problems. Some very nice explanations.
R.D. Mattuck, “A Guide to Feynman Diagrams in the Many-Body Problems.” This is a nice, intuitive guide to the
meaning and use of Feynman diagrams.
N. Goldenfeld, “Lectures on Phase Transitions and the Renormalization Group.” The discussion of scaling, dimensional
analysis, and phase transitions is wonderful.
G.D. Mahan, “Many-Particle Physics.” Standard, encyclopedic reference for condensed matter applications.
P. Ring and P. Schuck, “The Nuclear Many-Body Problem.” Somewhat out of date, but still a good, encyclopedic guide
to the nuclear many-body problem. Doesn’t discuss Green’s function methods much and no path integrals.
K. Huang, “Statistical Mechanics.” Excellent choice for general treatment of statistical mechanics, with good sections
on many-body physics.
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
References for Many-Body Effective Field Theory
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Fermi to Dyson in 1953 [recalled in Nature 427 (2004) 297]
Concerning a proposed pseudoscalar meson theory:
“There are two ways of doing calculations in theoretical
physics”, he said. “One way, and this is the way I prefer, is to
have a clear physical picture of the process that you are
calculating. The other way is to have a precise and
self-consistent mathematical formalism. You have neither.”
Dick Furnstahl
Fermion Many-Body Systems I
Outline Intro EFT Dilute Summary Refs
Fermi to Dyson in 1953 [recalled in Nature 427 (2004) 297]
I was slightly stunned, but ventured to ask him why he did
not consider the pseudoscalar meson theory to be a
self-consistent mathematical formalism. He replied,
“Quantum electrodynamics is a good theory because the
forces are weak, and when the formalism is ambiguous we
have a clear physical picture to guide us. With the
pseudoscalar meson theory there is no physical picture, and
the forces are so strong that nothing converges. To reach
your calculated results, you had to introduce arbitrary cut-off
procedures that are not based either on solid physics or on
solid mathematics.”
Dick Furnstahl
Fermion Many-Body Systems I