# 物理代写|统计力学代写Statistical mechanics代考|THE RENORMALIZATION GROUP

## 物理代写|统计力学代写Statistical mechanics代考|THE RENORMALIZATION GROUP

We now look at renormalization from a more general perspective. We start by enlarging the class of Hamiltonians to include all the types of spin interactions $S_i(\sigma)$ generated by renormalization transformations, ${ }^{32}$

Renormalization transformations have two parts: 1) stretch the lattice constant $a \rightarrow L a$ and thin the number of spins $N \rightarrow N / L^d$; and 2) find the effective couplings $\boldsymbol{K}^{\prime}$ among the remaining degrees of freedom (such that Eqs. (8.19) and (8.20) are satisfied). Starting with a model characterized by couplings $\boldsymbol{K}$, find the renormalized couplings $K^{\prime}$, an operation that we symbolize
$$\boldsymbol{K}^{\prime}=\mathcal{R}_L(\boldsymbol{K})$$
That is, $\mathcal{R}_L$ is an operator that maps the coupling vector $\boldsymbol{K}$ to that of the transformed system $\boldsymbol{K}^{\prime}$ associated with scale change $L$. Representing renormalizations with an operator is standard, ${ }^{33}$ but it’s abstract. An equivalent but more concrete way of writing Eq. (8.71) is
$$K_i^{\prime}=f_i^{(L)}(\boldsymbol{K})$$
where $i$ runs over the range of the index in Eq. (8.70). For each renormalized coupling $K_i^{\prime}$ associated with scale change $L$, there is a function $f_i^{(L)}$ of all couplings $\boldsymbol{K}$ (see Eqs. (8.38), (8.40), or (8.65)). $\mathcal{R}_L$ is thus a collection of functions that act on the components of $\boldsymbol{K}, \mathcal{R}_L \leftrightarrow\left{f_i^{(L)}\right}$. We’ll use both ways of writing the transformation.

$$-\beta H=\mathcal{H}=\sum_i K_i S_i(\sigma)$$
We could have the sum of spins $S_1 \equiv \sum_{k=1}^N \sigma_k$, near-neighbor two-spin interactions $S_{2, n n} \equiv$ $\sum_{\langle i j\rangle} \sigma_i \sigma_j$, next-nearest neighbor two-spin interactions $S_{2, n n n} \equiv \sum_{i j \in n n n} \sigma_i \sigma_j$, three, and fourspin interactions, etc. See Exercise 8.29. All generated interactions must be included in the model to have a consistent theory (see Section 8.3.2). We can collect these interactions in a set $\left{S_i\right}$ labeled by an index $i$. As we’ve learned (Section 8.3), we should include a constant in the Hamiltonian; call it $S_0 \equiv 1$. The couplings $K_i$ in Eq. (8.70) are dimensionless parameters associated with spin interactions $S_i$. In what follows, we “package” the couplings $\left{K_i\right}$ into a vector, $\boldsymbol{K} \equiv\left(K_0, K_1, K_2, \cdots\right)$. A system characterized by couplings $\left(K_0, K_1, \cdots\right)$ is represented by a vector $\boldsymbol{K}$ in a space of all possible couplings, coupling space (or parameter space).

## 物理代写|统计力学代写Statistical mechanics代考|REAL-SPACE RENORMALIZATION

Consider a set ${\sigma}$ of $N$ interacting Ising spins on a lattice of lattice constant $a$, and suppose we wish to find a new set ${\mu}$ of $N^{\prime}$ Ising spins and their interactions on a scaled lattice with lattice constant $L a$, where $N^{\prime}=N / L^d$. For that purpose, we define the real space mapping function $T[\mu \mid \sigma]$, such that
$$\mathrm{e}^{\mathcal{H}^{\prime}(\mu)}=\sum_{{\sigma}} T[\mu \mid \sigma] \mathrm{e}^{\mathcal{H}(\sigma)}$$

A requirement on the function $T[\mu \mid \sigma]$ is that
$$\sum_{{\mu}} T[\mu \mid \sigma] \equiv \prod_{i=1}^{N^{\prime}} \sum_{\mu_i=-1}^1 T[\mu \mid \sigma]=1$$
Equation (8.84) implies, from Eq. (8.83), that
$$Z_{N^{\prime}}\left(\boldsymbol{K}^{\prime}\right)=Z_N(\boldsymbol{K})$$
Equation (8.85) appears to differ from Eq. (8.39), but there is no discrepancy: In (8.39) we separated the constant term $N^{\prime} K_0$ from the renormalized Hamiltonian $\mathcal{H}^{\prime}$; in (8.85) the generated constant is part of $\mathcal{H}^{\prime}$. By combining Eq. (8.85) with Eq. (8.83), we have for the block-spin probability distribution $P(\mu)$,
$$P(\mu)=\langle T[\mu \mid \sigma]\rangle$$
The quantity $T[\mu \mid \sigma]$ therefore plays the role of a conditional probability (Section 3.2) that the $\mu-$ spins have their values, given that the configuration of $\sigma$-spins is known.

In principle there is a renormalization transformation for every function $T[\mu \mid \sigma]$ satisfying Eq. (8.84); in practice there are a limited number of forms for $T[\mu \mid \sigma]$. Defining transformations as in Eq. (8.83) (with a mapping function) formalizes and generalizes the decimation method, to which it reduces if $T[\mu \mid \sigma]$ is a product of delta functions. The transformation studied in Section 8.3 .1 is effected by Eq. (8.83) if $T[\mu \mid \sigma]=\prod_{i=1}^{N / 2} \delta_{\mu_i, \sigma_{2 i}}$. We now illustrate the application of Eq. (8.83) to the triangular-lattice Ising model.

# 统计力学代考

## 物理代写|统计力学代写Statistical mechanics代考|THE RENORMALIZATION GROUP

$$\boldsymbol{K}^{\prime}=\mathcal{R}L(\boldsymbol{K})$$ 那是， $\mathcal{R}_L$ 是映射耦合向量的算子 $\boldsymbol{K}$ 转换后的系统 $\boldsymbol{K}^{\prime}$ 与规模变化相关 $L$. 用运算符表示 重规范化是标准的， ${ }^{33}$ 但它是抽象的。一种等效但更具体的编写方程式的方法。 (8.71) 是 $$K_i^{\prime}=f_i^{(L)}(\boldsymbol{K})$$ 在哪里 $i$ 运行在 Eq. 中的索引范围内。(8.70)。对于每个重整化耦合 $K_i^{\prime}$ 与规模变化相关 $L$, 有一个函数 $f_i^{(L)}$ 所有联轴器 $\boldsymbol{K}$ (参见等式 (8.38)、 (8.40) 或 (8.65))） $\mathcal{R}_L$ 因此是 们将使用两种方式来编写转换。 $$-\beta H=\mathcal{H}=\sum_i K_i S_i(\sigma)$$ 我们可以得到自旋的总和 $S_1 \equiv \sum{k=1}^N \sigma_k$ ，近邻双自旋相互作用 $S_{2, n n} \equiv \sum_{\langle i j\rangle} \sigma_i \sigma_j$,下 一个最近邻双自旋相互作用 $S_{2, n n n} \equiv \sum_{i j \in n n n} \sigma_i \sigma_j$ 、三和四自旋相互作用等。参见练 $习$ 8.29。所有生成的相互作用都必须包含在模型中以具有一致的理论 (参见第 8.3.2 节)。我们可以将这些交互收集在一个集合中 左{S_i右 $}$ 由索引标记 $i$. 正如我们所了解 的 (第 8.3 节)，我们应该在哈密顿量中包含一个常数；叫它 $S_0 \equiv 1$. 联轴器 $K_i$ 在等式 中 (8.70) 是与自旋相互作用相关的无量纲参数 $S_i$. 下面，我们对联轴器进行”包装” 左{K_i右} 变成一个向量， $\boldsymbol{K} \equiv\left(K_0, K_1, K_2, \cdots\right)$. 以联轴器为特征的系统 $\left(K_0, K_1, \cdots\right)$ 由向量表示 $\boldsymbol{K}$ 在所有可能耦合的空间中，耦合空间（或参数空间）。

## 物理代写|统计力学代写Statistical mechanics代考|REAL-SPACE RENORMALIZATION

$$\mathrm{e}^{\mathcal{H}^{\prime}(\mu)}=\sum_\sigma T[\mu \mid \sigma] \mathrm{e}^{\mathcal{H}(\sigma)}$$

$$\sum_\mu T[\mu \mid \sigma] \equiv \prod_{i=1}^{N^{\prime}} \sum_{\mu_i=-1}^1 T[\mu \mid \sigma]=1$$

$$Z_{N^{\prime}}\left(\boldsymbol{K}^{\prime}\right)=Z_N(\boldsymbol{K})$$

(8.85) 与方程式。(8.83)，我们有块自旋概率分布 $P(\mu)$ ，
$$P(\mu)=\langle T[\mu \mid \sigma]\rangle$$

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