# 物理代写|统计力学代写Statistical mechanics代考|PHYS3006

## 物理代写|统计力学代写Statistical mechanics代考|THE ONE-DIMENSIONAL ISING MODEL

The Ising model, conceived in 1924 as a model of magnetism, has come to occupy a special place in theoretical physics with an enormous literature. ${ }^{33}$ Consider a one-dimensional crystalline latticea set of uniformly spaced points (lattice sites) separated by a distance $a$. Referring to Fig. $6.19$, at each lattice site assign the value of a variable that can take one of two values, conventionally denoted $\sigma_i=\pm 1, i=1, \cdots, N$, where $N$ is the number of lattice sites. The variables $\sigma_i$ can be visualized as vertical arrows, up or down (as in Fig. 6.19), and for that reason are known as Ising spins. Real spin- $\frac{1}{2}$ particles have two values of the projection of their spin vectors $S$ onto a preselected $z$-axis, $S_z=\pm \frac{1}{2} \hbar$, and thus $S_z$ can be written $S_z=\frac{1}{2} \hbar \sigma$, but that is the extent to which Ising spins have any relation to quantum spins. Ising spins are two-valued classical variables. In this section we consider one-dimensional systems of Ising spins, which can be solved exactly. Ising spins on two-dimensional lattices can also be solved exactly, but the mathematics is more difficult. We touch on the two-dimensional Ising model in Chapter 7; what we learn here will help.

Paramagnetism was treated in Chapter 5 in which independent magnetic moments interact with an externally applied magnetic field. Many other types of magnetic phenomena occur as a result of interactions between moments located on lattice sites of crystals. In ferromagnets, moments at widely separated sites become aligned, spontaneously producing (at sufficiently low temperatures) a magnetized sample in the absence of an applied field. In antiferromagnets, moments become antialigned at different sites, in which there is a spontaneous ordering of the individual moments, even though there is no net magnetization. The Heisenberg spin Hamiltonian models the coupling of spins $S_i, S_j$ located at lattice sites $(i, j)$ in the form $H=-\sum_{i j} J_{i j} S_i \cdot S_j$, where the coefficients $J_{i j}$ are the exchange coupling constants. ${ }^{34}$ Positive (negative) exchange coefficients promote ferromagnetic (antiferromagnetic) ordering. The microscopic underpinnings of the interaction $J_{i j}$ is a complicated business we won’t venture into. ${ }^{35}$ In statistical mechanics, the coupling coefficients are taken as given parameters. The symbol $S_i$ strictly speaking refers to a quantum-mechanical operator, but in many cases is approximated as a classical vector. The Ising model replaces $S_i$ with its $z$-component, normalized to unit magnitude.

We take as the Hamiltonian for a one-dimensional system of Ising spins having nearest-neighbor interactions, ${ }^{36}$
$$H\left(\sigma_1, \cdots, \sigma_N\right)=-J \sum_{i=1}^{N-1} \sigma_i \sigma_{i+1}-b \sum_{i=1}^N \sigma_i$$

## 物理代写|统计力学代写Statistical mechanics代考|The transfer matrix method

The method of analysis leading to the recursion relation in Eq. (6.69) does not generalize to finite magnetic fields (try it!). We now present a more general technique for calculating the partition function of spin models, the transfer matrix method, which applies to systems satisfying periodic boundary conditions. Figure $6.22$ shows a system of $N$ Ising spins that wraps around on itself ${ }^{40}$ with $\sigma_{N+1} \equiv \sigma_1$. The Hamiltonian
$$H=-J \sum_{i=1}^N \sigma_i \sigma_{i+1}-b \sum_{i=1}^N \sigma_i . \quad\left(\sigma_{N+1} \equiv \sigma_1\right)$$
The only difference between Eqs. (6.75) and (6.68) is the spin interaction $\sigma_N \sigma_1$.

The partition function requires us to evaluate the sum
$$Z_N(K, B)=\sum_{{\sigma}} \exp \left(K \sum_{i=1}^N \sigma_i \sigma_{i+1}+B \sum_{i=1}^N \sigma_i\right),$$
where $B=\beta b$. The exponential in Eq. (6.76) can be factored, ${ }^{41}$ allowing us to write
$$Z_N(K, B)=\sum_{{\sigma}} V\left(\sigma_1, \sigma_2\right) V\left(\sigma_2, \sigma_3\right) \cdots V\left(\sigma_{N-1}, \sigma_N\right) V\left(\sigma_N, \sigma_1\right)$$
where
$$V\left(\sigma_i, \sigma_{i+1}\right) \equiv \exp \left(K \sigma_i \sigma_{i+1}+\frac{1}{2} B\left(\sigma_i+\sigma_{i+1}\right)\right)$$
is symmetric in its arguments, ${ }^{42} V\left(\sigma_i, \sigma_{i+1}\right)=V\left(\sigma_{i+1}, \sigma_i\right)$.
Equation (6.77) is in the form of a product of matrices. We can regard $V\left(\sigma, \sigma^{\prime}\right)$ as the elements of a $2 \times 2$ matrix, $V$, the transfer matrix, ${ }^{43}$ which, in the “up-down” basis $\sigma_j=\pm 1$, has the form
$$\boldsymbol{V}=(+)\left(\begin{array}{cc} \mathrm{e}^{K+B} & (-) \ \mathrm{e}^{-K} & \mathrm{e}^{-K-B} \end{array}\right)$$
Holding $\sigma_1$ fixed in Eq. (6.77) and summing over $\sigma_2, \cdots, \sigma_N, Z$ is related to the trace of an $N$-fold matrix product,
$$Z_N(K, B)=\sum_{\sigma_1} \boldsymbol{V}^N\left(\sigma_1, \sigma_1\right)=\operatorname{Tr} \boldsymbol{V}^N$$

## 物理代写|统计力学代写Statistical mechanics代考|THE ONE-DIMENSIONAL ISING MODEL

Ising 模型于 1924 年被构想为磁性模型，在理论物理学中占有特殊的地位，拥有大量文献。 ${ }^{33}$ 考虑一个一 维晶格集，由间隔一定距离的均匀间隔点 (晶格点) 组成 $a$. 参考图。6.19，在每个格点分配一个变量的 值，该变量可以取两个值之一，通常表示 $\sigma_i=\pm 1, i=1, \cdots, N$ ，在哪里 $N$ 是格点的数量。变量 $\sigma_i$ 可以 可视化为垂直箭头，向上或向下 (如图 $6.19$ 所示)，因此被称为伊辛旋转。真正的旋转- $\frac{1}{2}$ 粒子有两个自 旋向量的投影值 $S$ 到预选的 $z$-轴， $S_z=\pm \frac{1}{2} \hbar$ ，因此 $S_z$ 可以写 $S_z=\frac{1}{2} \hbar \sigma$ ，但这就是伊辛自旋与量子自旋 有任何关系的程度。伊辛自旋是二值经典变量。在本节中，我们考虑可以精确求解的伊辛自旋的一维系 统。二维晶格上的伊辛自旋也可以精确求解，但数学更难。我们将在第 7 章讨论二维伊辛模型；我们在这 里学到的东西会有所帮助。

$$H\left(\sigma_1, \cdots, \sigma_N\right)=-J \sum_{i=1}^{N-1} \sigma_i \sigma_{i+1}-b \sum_{i=1}^N \sigma_i$$

## 物理代写|统计力学代写Statistical mechanics代考|The transfer matrix method

$$H=-J \sum_{i=1}^N \sigma_i \sigma_{i+1}-b \sum_{i=1}^N \sigma_i . \quad\left(\sigma_{N+1} \equiv \sigma_1\right)$$

$$Z_N(K, B)=\sum_\sigma \exp \left(K \sum_{i=1}^N \sigma_i \sigma_{i+1}+B \sum_{i=1}^N \sigma_i\right)$$

$$Z_N(K, B)=\sum_\sigma V\left(\sigma_1, \sigma_2\right) V\left(\sigma_2, \sigma_3\right) \cdots V\left(\sigma_{N-1}, \sigma_N\right) V\left(\sigma_N, \sigma_1\right)$$

$$V\left(\sigma_i, \sigma_{i+1}\right) \equiv \exp \left(K \sigma_i \sigma_{i+1}+\frac{1}{2} B\left(\sigma_i+\sigma_{i+1}\right)\right)$$

$$\boldsymbol{V}=(+)\left(\begin{array}{ll} \mathrm{e}^{K+B} & (-) \mathrm{e}^{-K} \quad \mathrm{e}^{-K-B} \end{array}\right)$$

$$Z_N(K, B)=\sum \boldsymbol{V}^N\left(\sigma_1, \sigma_1\right)=\operatorname{Tr} \boldsymbol{V}^N$$

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