Figure 1: Ising model on two coupled lines.

Statistical Physics 3 December 8th, 2010

Series 10

We apply a renormalization group decimation technique to the system of 2N spins on two lines with one interaction costant (K ₁ ) between spins in the same line and another (K ₂ ) between the spins of the two lines as in Fig. 1.

Figure 1: Ising model on two coupled lines.

Consider the following Hamiltonian:

−β H =

N

∑

i=1

[K ₁ (s _i s _i+1 + t _i t _i+1 ) + K ₂ (s _i t _i + s _i+1 t _i+1 ) + 2C]

The decimation procedure consists into integrate one spin over two, for example the spins with even index i. The renormalized Hamlitonian −β H ⁰ is defined from:

e ^{−β H}

^({s

^,t

^{},odd i)} = ∑

{s

,t

}even i

e ^{−β H}

1. Show that

−β H ⁰ = ∑

odd i

[K ₂ (s _i t _i + s _i+2 t _i+2 ) + 4C + H e _i ] where

H e _i = log[e ^2K

2 cosh(K ₁ (s _i + t _i + s _i+2 + t _i+2 )) + e ^−2K

2 cosh(K ₁ (s _i − t _i + s _i+2 − t _i+2 ))]

2. the term H e _i is a function of only (s _i + s _i+2 ) and (t _i + t _i+2 ); moreover it is invariant the transfor- mation (s,t) → (−s, −t) as well as the transformation (s,t) → (t, s). We want to rewrite H e _i in a more convenient form; given those symmetry considerations we can guess a form like:

H e _i = D + B(s _i + s _i+2 )(t _i + t _i+2 ) + A(s _i + s _i+2 ) ² + A(t _i + t _i+2 ) ² + E(s _i + s _i+2 ) ² (t _i + t _i+2 ) ² (1) Show that the four constants D, B, E, A are enough for taking into account all the 16 cases for {s _i , s _i+2 ,t _i ,t _i+2 }.

1

Statistical Physics 3 December 8th, 2010

The renormalized Hamiltonian has more coupling constants: in particular, devoloping the equa- tion (1) you see that the diagonal interactions (s _i t _i+2 + s _i+2 t _i ) and the square interactions (s _i t _i+2 s _i+2 t _i ) appeared. This is a typical feature of the renormalization procedures, named proliferation of cou- plings.

Verify that the introduced new coupling constants are enough for defining a complete set of renor- malization equations without further proliferation

3. starting from the Hamiltonian

−β H =

N

∑

i=1

[K ₁ (s _i s _i+1 + t _i t _i+1 ) + K ₂ (s _i t _i + s _i+1 t _i+1 ) + K ₃ (s _i t _i+1 + s _i+1 t _i ) + K ₄ (s _i t _i+1 s _i+1 t _i ) + 2C]

repeat the decimation procedure;

4. find the equation of renormalization;

5. verify that the infinite temperature K ₁ = K ₂ = K ₃ = K ₄ = 0 is a fixed point of the system.

2