RG and me

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RG and me

• Love at first bite

NSF DMR 0103639
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The RG What ?
a? a, b ? b etc is renormalization
Ignoring y is the tree level
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Who are x and y?

• x is long wavelength, y is short wavelength
• x is low energy modes, y is high energy modes
• x and y stand for many variables, so y is y1,y2
  etc. and we can eliminate y over and over and
  watch the flow

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The RG Why?

• If we make aa by rescaling x, then
• if bgtb it is relevant
• If bltb it is irrelevant
• If bb it is marginal
• Useful for Hamiltonian perturbations

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The RG How to at one loop.
One loop
Tree
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Nonrelativistic fermions
Where is the low energy physics ? What is the
role of interactions?
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Strategy

• 0. Focus on annulus near K_F
• 1. Write an action for H_0
• 2. Find and RG to make it fixed point.
• 3.Add all possible interactions and see
• how they flow under RG.

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The shell game
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Gaussian Fixed Point
If we lower cutoff by s, So goes into itself
under the transformation
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Fate of interactions

• Add

Only quartic interaction survives tree-level RG
Also u cannot depend on k or !
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The survivors at tree level
1
3
1
3
2
2
4
4
13 and 24
120 and 340
Forward Scattering
BCS Channel
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One loop

• F remains marginal
• V flows as follows

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How to solve fixed point theoryA small parameter
1/N
i
j
i
j
i
k
j


k
j
i
i
j
i
j
One loop
i
j
i
j
i
j


i
j
i
j
j
i
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Large N for Fermi liquid

• N K_F/L Just sum bubbles

i
i
j
j
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New directions for Fermions(with Senthil)
d2
d3
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Other applications

• Matter at finite density
• (Alford, Rajagopal, Wilczek, Hsu)
• Nuclear physics
• (Schwenk, Brown, Kuo)
• Quantum dots with random matrices
• Murthy, Mathur and Myself

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One loop F is marginal
i
k
j
Q
k
i
j
One loop
i
j
kQ
j
i
k
P-k


j
i
k
j
i
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Flow of V

• V flows as follows

-k
k
-k
k