# 物理代写|统计力学代写Statistical mechanics代考|PHYC30017

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

Correlation functions (such as $\left\langle\sigma_i \sigma_j\right\rangle$ ) play an important role in statistical mechanics and will increasingly occupy our attention in this book; they provide spatial, structural information that cannot be obtained from partition functions. ${ }^{47,48}$ We can calculate correlation functions of Ising spins using the transfer matrix method, as we now show. Translational invariance (built into periodic boundary conditions, but attained in any event in the thermodynamic limit) implies that $\left\langle\sigma_i \sigma_j\right\rangle$ is a function of the separation between sites $(i, j),\left\langle\sigma_i \sigma_j\right\rangle=f(|i-j|)$. The quantity $\left\langle\sigma_i \sigma_j\right\rangle$ is in some sense a conditional probability: Given that the spin at site $i$ has value $\sigma_i$, what is the probability that the spin at site $j$ has value $\sigma_j$ ? That is, to what extent is the value of $\sigma_j$ correlated ${ }^{49}$ with the value of $\sigma_i$ ? We expect the closer spins are spatially, the more they are correlated. Correlation functions establish a length, the correlation length, $\xi$, a measure of the range over which correlations persist. We expect for separations far in excess of the correlation length, $|i-j| \gg \xi$, that $\left\langle\sigma_i \sigma_j\right\rangle \rightarrow\langle\sigma\rangle^2$.

Because $V\left(\sigma_1, \sigma_2\right) \cdots V\left(\sigma_N, \sigma_1\right) / Z_N$ is the probability the system is in state $\left(\sigma_1, \cdots, \sigma_N\right)$ (for periodic boundary conditions), the average we wish to calculate is:
\begin{aligned} \left\langle\sigma_i \sigma_j\right\rangle=\frac{1}{Z_N} \sum_{{\sigma}} V\left(\sigma_1, \sigma_2\right) \cdots V\left(\sigma_{i-1}, \sigma_i\right) \sigma_i V\left(\sigma_i, \sigma_{i+1}\right) \cdots \ & \cdots V\left(\sigma_{j-1}, \sigma_j\right) \sigma_j V\left(\sigma_j, \sigma_{j+1}\right) \cdots V\left(\sigma_N, \sigma_1\right) \cdot(6.87) \end{aligned}
We’ve written Eq. (6.87) using transfer matrix symbols, but it’s not in the form of a matrix product (compare with Eq. (6.77)). We need a matrix representation of Ising spins. A matrix represents the action of a linear operator in a given basis of the vector space on which the operator acts. And of course bases are not unique – any set of linearly independent vectors that span the space will do. In Eq. (6.79) we used a basis of up and down spin states, which span a two dimensional space, $|+\rangle \equiv\left(\begin{array}{l}1 \ 0\end{array}\right)$ and $|-\rangle \equiv\left(\begin{array}{c}0 \ 1\end{array}\right)$. With a nod to quantum mechanics, ${ }^{50}$ measuring $\sigma$ at a given site is associated with an operator $S$ that, in the “up-down” basis, is represented by a diagonal matrix with elements $^{51}$
$$S\left(\sigma, \sigma^{\prime}\right) \equiv\left(\begin{array}{cc} 1 & 0 \ 0 & -1 \end{array}\right),$$
which is one of the Pauli spin matrices. ${ }^{52}$

## 物理代写|统计力学代写Statistical mechanics代考|Next-nearest-neighbor model

The transfer matrix method allows us to treat models having interactions that extend beyond nearestneighbors. ${ }^{56}$ We show how to set up the transfer matrix for a model with nearest and next-nearest neighbor interactions ${ }^{57}$ with Hamiltonian
$$H(\sigma)=-J_1 \sum_{i=1}^N \sigma_i \sigma_{i+1}-J_2 \sum_{i=1}^N \sigma_i \sigma_{i+2},$$
where we adopt periodic boundary conditions, $\sigma_{N+1} \equiv \sigma_1$ and $\sigma_{N+2} \equiv \sigma_2$. Figure $6.23$ shows the
Figure 6.23: Nearest-neighbor ( $J_1$, solid lines) and next-nearest neighbor $\left(J_2\right.$, dashed) interactions.
two types of interactions and their connectivity in one dimension.
To set up the transfer matrix, we group spins into cells of two spins apiece, as shown in Fig. 6.24. We label the spins in the $k^{\text {th }}$ cell $\left(\sigma_{k, 1}, \sigma_{k, 2}\right), 1 \leq k \leq N / 2$. A key step is to associate

the degrees of freedom of each cell with a new variable $s_k$ representing the four configurations $(+,+),(+,-),(-,+),(-,-)$. This is a mapping from the $2^N$ degrees of freedom of Ising spins $\left{\sigma_i\right}, 1 \leq i \leq N$, to an equivalent number $4^{N / 2}$ degrees of freedom associated with the cell variables $\left{s_k\right}, 1 \leq k \leq N / 2$.

Besides grouping spins into cells, we also classify interactions as those associated with intra-cell couplings (see Fig. 6.25)
$$V_0\left(s_k\right) \equiv-J_1 \sigma_{k, 1} \sigma_{k, 2},$$
and inter-cell couplings,
$$V_1\left(s_k, s_{k+1}\right) \equiv-J_1 \sigma_{k, 2} \sigma_{k+1,1}-J_2\left(\sigma_{k, 1} \sigma_{k+1,1}+\sigma_{k, 2} \sigma_{k+1,2}\right) .$$
With these definitions, the Hamiltonian can be written as the sum of two terms, one containing all intra-cell interactions and the other containing all inter-cell interactions,
$$H(s)=\sum_{k=1}^{N / 2}\left(V_0\left(s_k\right)+V_1\left(s_k, s_{k+1}\right)\right) \equiv H_0(s)+H_1(s)$$ where s(N/2)+1 ≡ s1. Comparing Eqs.

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

$$\left\langle\sigma_i \sigma_j\right\rangle=\frac{1}{Z_N} \sum_\sigma V\left(\sigma_1, \sigma_2\right) \cdots V\left(\sigma_{i-1}, \sigma_i\right) \sigma_i V\left(\sigma_i, \sigma_{i+1}\right) \cdots \cdots V\left(\sigma_{j-1}, \sigma_j\right) \sigma_j V\left(\sigma_j, \sigma_{j+1}\right) \cdots V\left(\sigma_N, \sigma_1\right)$$

## 物理代写|统计力学代写Statistical mechanics代考|Next-nearest-neighbor model

$$H(\sigma)=-J_1 \sum_{i=1}^N \sigma_i \sigma_{i+1}-J_2 \sum_{i=1}^N \sigma_i \sigma_{i+2}$$
㧴们采用周期性边界条件， $\sigma_{N+1} \equiv \sigma_1$ 和 $\sigma_{N+2} \equiv \sigma_2$. 数字 $6.23$ 显示了

$$V_0\left(s_k\right) \equiv-J_1 \sigma_{k, 1} \sigma_{k, 2},$$

$$V_1\left(s_k, s_{k+1}\right) \equiv-J_1 \sigma_{k, 2} \sigma_{k+1,1}-J_2\left(\sigma_{k, 1} \sigma_{k+1,1}+\sigma_{k, 2} \sigma_{k+1,2}\right)$$

$$H(s)=\sum_{k=1}^{N / 2}\left(V_0\left(s_k\right)+V_1\left(s_k, s_{k+1}\right)\right) \equiv H_0(s)+H_1(s)$$

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