RG and me

Love at first bite

NSF DMR 0103639

The RG: What ?

Z(a;b;::) =Z

dxZ

dy e¡ a(x2+y2)¡ b(x2+y2)2 :::

Z(a0;b0; ::) =Z

dx e¡ a0x2¡ b0x4 ::: where

e¡ a0x2¡ b0x4 ::: =Z

dy e¡ a(x2+y2)¡ b(x2+y2)2 :::

hX (x)ia;b:: = hX (x)ia0;b0::

a a’, b b’ etc is renormalization Ignoring y is the tree level

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

The RG: Why?

• If we make a’=a by rescaling x, then

• if b’>b it is relevant

• If b’<b it is irrelevant

• If b’=b it is marginal

• Useful for Hamiltonian perturbations

The RG: How to at one loop.

= +

v = v0(¤) + v20(¤)

Z ¤

0

dkk4

v0(¤ ¡ d¤) = v0(¤) + v20(¤)

Z ¤

¤¡ d¤

dkk4

Tree One loop

Nonrelativistic fermions

H0 =Z

Ãy(K ;µ)(K 2

2m¡

K 2F

2m)Ã(K ;µ)K dK dµ

'Z ¤

¡ ¤

Z 2¼

0Ãy(k;µ)(vF k)Ã(k;µ)dkdµ

(1)

K F absorbed in ÃK 2 ¡ K 2

F

2m=

K + K F

2m¢(K ¡ K F ) ' vF k

Where is the low energy physics ?What is the role of interactions?

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.

The shell game

H0 =Z ¤

¡ ¤

Z 2¼

0Ãy(k;µ)(vF k)Ã(k;µ)dkdµ (1)

Z =Z

dÃd¹ÃeS0

S0 =Z 1

¡ 1

Z ¤

¡ ¤

Z 2¼

0

¹Ã(! ;k;µ)(i! ¡ vF k)Ã(! ;k;µ)d! dkdµ (2)

Gaussian Fixed Point

S0 =Z 1

¡ 1

Z ¤

¡ ¤

Z 2¼

0¹Ã(! ;k;µ)(i! ¡ vF k)Ã(! ;k;µ)d! dkdµ

If we lower cutoff by s, So goes into itself under the transformation

! 0= s! k0= sk Ã0= s¡ 3=2Ã

Fate of interactions

• Add

S4 =Z

¹Ã(4) ¹Ã(3)Ã(2)Ã(1)u(4;3;2;1)3Y

i=1

dki d! i dµi

Only quartic interaction survives tree-level RG

Also u cannot depend on k or !

So consider u(µ1;µ2;µ3)

The survivors at tree level

1

2

3

4

1=3 and 2=4

1

2

3

4

1+2=0 and 3+4=0

Forward Scattering BCS Channel

u(µ1 ¡ µ2) = F (µ) u(µ1 ¡ µ3) = V(µ) (1)

F (µ) is Landau's F function

One loop

• F remains marginal

• V flows as follows:

dV(µ1 ¡ µ3)dt

= ¡Z

V(µ1 ¡ µ)V(µ¡ µ3)dµ (1)

dVm

dt= ¡ cmV2

m

How to solve fixed point theoryA small parameter 1/N

= +

One loop

+ +

i

i

i

i

i

i

i

i

i

i

jj

j

j jj

k

k

i

j

j

j

j

j

i

j

Large N for Fermi liquid

• N= K_F/Just sum bubbles

i

j

 =Â0

1¡ F0Â0

i

j

New directions for Fermions(with Senthil)

d=2

d=3

Other applications

• Matter at finite density

(Alford, Rajagopal, Wilczek, Hsu)

• Nuclear physics

(Schwenk, Brown, Kuo)

• Quantum dots with random matrices

Murthy, Mathur and Myself

One loop F is marginal

One loop

+ +

i

i

i

i

i

i

j

j

k

k

j

j

j

j

k P-k

k

k+Q

Q

• V flows as follows

Flow of V

k -kk

-k