Renormalization Group Theory

The goal of this section is to introduce several concepts of Renormalization Group Theory and to illustrate such concepts with the 1-dimensional Ising model.

Consider the task of computing the canonical partition function $Z$ of the one-dimensional Ising model in the absence of an external magnetic field. According to Eq. (286),

$$Z(K, N) = \sum_{S_1} \sum_{S_2} ... \sum_{S_N} e^{K(S_1 S_2 +S_2 S_3 + ... + S_N S_1 )},$$ (317)

where coupling parameter$K \equiv \beta J$ and N is the total number of spins. Note that according to Eq. (317),

$$\lim_{K \rightarrow 0} Z(K,N) = \prod_{j=1}^N \sum_{S_j=-1}^1 1 = 2^N.$$ (318)

The renormalization group strategy for the 1-dimensional Ising model can be described as follows.
Step (1). Sum over the even numbered spins in Eq. (317). Note that summing, e.g., over $S_2$ we obtain

$$Z(K, N) = \sum_{S_1,S_3,S_4,S_5,S_6,S_7...} [e^{K (S_1+ S_3)}+e^{-K (S_1+ S_3)} ] e^{K S_3 S_4} e^{K S_4 S_5} e^{K S_5 S_6} e^{K S_6 S_7} ...,$$ (319)

summing over $S_2$ and $S_4$ we obtain,

$$Z(K, N) = \sum_{S_1,S_3,S_5,S_6,S_7...} [e^{K(S_1+ S_3)}+e^{-K(S_1+ S_3)}] [e^{K(S_3+ S_5)} + e^{-K(S_3+ S_5)}] e^{K S_5 S_6} e^{K S_6 S_7} ...,$$ (320)

and summing over all even numbered spins we obtain

$$Z(K, N) = \sum_{S_1,S_3,S_5,S_7...} [e^{K(S_1+ S_3)}+e^{-K(S_1+ S...... [e^{K(S_3+ S_5)} + e^{-K(S_3+ S_5)}] [e^{K(S_5+ S_7)} + e^{-K(S_7+ S_9)}]... .$$ (321)

Step (2). Rewrite the remaining sum (i.e., the sum over odd numbered spins introduced by Eq. (321)) by implementing the Kandanoff transformation

$$e^{K(S+S')}+e^{-K(S+S')} = f(K) e^{K' SS'},$$ (322)

where both $f(K)$ and $K'$ are functions of $K$. Substituting Eq. (322) into Eq. (321) we obtain

$$Z(K, N) = f(K)^{N/2} \sum_{S_1,S_3,S_5,S_7...} e^{K' S_1 S_3} e^{K' S_3 S_5} e^{K' S_5 S_7}... . = f(K)^{N/2} Z(K',N/2).$$ (323)

Note that such transformation allow us to rewrite the partition function $Z(K,N)$ in terms of a renormalized partition function $Z(K',N/2)$ (i.e., a partition function with new parameters that describes an Ising model with half the number of spins and a different coupling parameter $K'$).
 

In order to determine the renormalization group equations (i.e., $K'$ and $f(K)$ as a function of K) and show that $K' < K$, we note that when $S=S'=\pm 1$, Eq. (322) gives

$$e^{2 K}+e^{-2 K} = f(K) e^{K'},$$ (324)

and when$S=-S'=\pm 1$, Eq. (322) gives

$$2 = f(K) e^{-K'}.$$ (325)

Therefore, solving for $f(K)$ in Eq. (325) and substituting into Eq. (324) we obtain

$$K' = \frac{1}{2} ( (2 K)),$$ (326)

and substituting Eq. (326) into Eq. (325) we obtain

$$f(K) = 2 ^{\frac{1}{2}}(2 K).$$ (327)

Eqs. (326) and (327) are called renormalization group equations since they provide the renormalization scheme.
Step (3). Go to (1), replacing $Z(K,N)$ by $Z(K',N/2)$.

Step (3) is repeated each time on the subsequent (renormalized) partition function (i.e., $Z(K'',N/4)$, $Z(K''',N/8)$, $Z(K^{IV},N/8)$, $Z(K^{V},N/8)$, ... etc.) until the renormalized parameters become approximately constant (i.e., until the renormalized parameters reach a fixed point and become invariant under the Kadanoff transformation). Note that, according to Eq. (326), $K > K' > K''> K'''$, etc., so after a few iterations the coupling parameter becomes negligibly small and the partition function can be approximated by using Eq. (318) as follows:

$$\begin{array}{ll}\text{ln} Z(K, N) &\approx \frac{N}{2} \tex......cosh}^{1/2} (2K^{V}) ] + \frac{N}{2^6} \text{ln} 2.\end{array}$$ (328)

The renormalization group strategy thus involve computing the total sum, introduced by Eq. (317), step by step. The success of the approach relies on the fact that after a few iterations the sum converges to an expression that can be easily computed.

Sometimes the partition function is known for a specific value of the coupling parameter (e.g., for $K' \approx 0$ in the 1-dimensional Ising model). The renormalization group theory can then be implemented to compute the partition function of the system for a different value $K$ of the coupling constant. This is accomplished by inverting Eq. (326) as follows:

$$K = \frac{1}{2} ^{-1}[ (2 K')].$$ (329)

and computing $Z(K,N)$ from $Z(K',N/2)$ according to Eq. (323).

One could also define the function $g(K)$ as follows

$$N g(K) \equiv Z(K,N),$$ (330)

and substituting Eq. (329) into Eq. (323) we obtain

$$N g(K) = \frac{N}{2} 2 + \frac{N}{2} ( ^{\frac{1}{2}}(2 K)) + \frac{N}{2} g(K').$$ (331)

Therefore, given the partition function $Z(K',N)$ for a system with coupling constant $K'$, one can compute $g(K')$ and K according to Eqs. (330) and (329), respectively. The partition function $Z(K,N) =$   exp$(N g(K))$ is then obtained by substituting the values of $g(K')$ and K in Eq. (330). Note that according to this procedure, $K > K'$ and the subsequent iterations give larger and larger values of $K$. This indicates that the flow of $K$ has only two fixed points at K$=0$ (e.g., at infinite temperature) and K$=\infty$ (e.g., at 0 K).
 

Systems with phase transitions, however, have nontrivial fixed points at intermediate values of $K$. For instance, following a similar procedure, as the one described in this section, it is possible to show that the 2-dimensional Ising model has an additional fixed point $K_c$ and that the heat capacity $C=\frac{d^2}{dk^2}g(k)$ divergers at $K_c$. Thus, $K_c$ determines the critical temperature where the system undergoes a phase transition and spontaneosly magnetizes.