Intuition behind Linked Cluster Theorem: connected vs. non-connected diagrams

Linked-cluster theorem$^1$ $$\ln Z~=~\frac{i}{\hbar} W_c, \tag{1}$$ where $Z$ is the partition function and $W_c$ is the generating functional of connected diagrams.

Proof. We will use a replica trick, cf. Ref. 1.

1. Recall that if a theory consists of $n$ independent sub-theories with partition functions $Z_1, \ldots, Z_n$ (i.e. interactions are only allowed within each sub-theories), then the partition function for the full theory is the product $Z_1 \cdots Z_n$.

2. Introduce $n$ copies of the original theory under investigation, where $n\in\mathbb{N}$ is a positive integer. The replica partition function becomes just a power $$\sum\left\{\text{all replica diagrams}\right\} ~=~Z^n,\tag{2}$$ because different copies do not interact. Each field $\phi^{\alpha}_{(i)}(x)$ in the replica theory now carries a copy label $i\in\{1, \ldots, n\}$, and doesn't talk to other copies.

3. Given a Feynman diagram $D$ in the original theory, the contributions to the corresponding replica Feynman diagram should be multiplied with a factor $n^{\#(D)}$, where $\#(D)$ denotes the number of connected components of $D$. In other words, $$ \sum\left\{\text{all replica diagrams}\right\}~=~1+ n\sum\left\{\text{connected original diagrams}\right\} +{\cal O}(n^2). \tag{3}$$ In eq. (3) we have implicitly normalized the partition function such that $1$ is the value of the empty diagram, which by definition is not connected.

4. Equivalently, by Taylor expansion, $$ \ln\sum\left\{\text{all replica diagrams}\right\}~\stackrel{(3)}{=}~ n\sum\left\{\text{connected original diagrams}\right\} +{\cal O}(n^2). \tag{4}$$

5. Combining eqs. (2) & (4) yield $$\ln Z - \sum\left\{\text{connected original diagrams}\right\} ~\stackrel{(2)+(4)}{=}~{\cal O}(n^1) .\tag{5}$$ The LHS. of eq. (5) is independent of $n$, i.e. it is a constant wrt. $n$. But since the RHS. of eq. (5) has no ${\cal O}(n^0)$ terms, the constant must be zero. (Alternatively, we may formally treat the integer $n$ as a real number, and take the limit $n\to 0^{+}.$) This yields the linked-cluster theorem (1). $\Box$

See also this and this related Phys.SE posts.

References:

1. X.G. Wen, QFT of many-body systems, (2004); p. 143.

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$^1$NB: Conventionally in QFT, one allows for a multiplicative normalization factor in the partition function $Z$, which hence corresponds to an additive constant in $W_c$, cf. eq. (1).