Chapter 9: Renormalization.

Chapter 2, Chapter 6.

Chapter 12, Chapter 13, Chapter 14.

Chapter 2.

Lecture 8, 9, 10, 11.

Lecture 5, 6, 7.

Ising Model

Systematic finite-size scaling methods for analyzing critical points (and ordered phases), Anders W Sandvik, Boston University.

Abrupt phase transitions.

Continuous phase transitions.

Ising model.

Potts model.

XY model.

The van der Waals system.

Fluids, Magnets, and the Ising Model.

Arovas, Chapter 7.2, 7.3.

Mean Field Theory.

Arovas, Chapter 7.4.

No class.

Variational Density Matrix Method.

Arovas, Chapter 7.5.

Variational Density Matrix Method.

Arovas, Chapter 7.6.

Variational Density Matrix Method.

Landau Theory of Phase Transition.

Arovas, Chapter 7.5, 7.6.

Landau Theory of Phase Transition.

Ising model.

Arovas, Chapter 7.6, 6.2.

Ising model.

Transfer matrix, Peierls’ argument.

Arovas, Chapter 6.2.

Pathria, Chapter 13.2.

Baxter, Chapter 2.

2D Ising model.

High temperature expansion, low temperature expansion.

Duality transformation.

Pathria, Chapter 13.4.

Baxter, Chapter 6.

Renormalization group.

Renormalization group.

Scaling.

Renormalization group.

Renormalization group.

Renormalization group.

Finite size scaling.

Renormalization group.

Finite size scaling.

Finite size scaling.

Tensor renormalization group.

Pathria (3rd ed), Pb. 12.2, 12.3.

Derive Eq. 7.66 of Arovas' note.

Derive Eq. 7.117 of Arovas' note.

An arbitrary function $G(s)$ of the Ising spin variable $s=\pm 1$ can always be expanded as $G_b(s)=a+s b$, with $a$ the even part of $G$ in $s$ and $b$ the odd part. Show that $$Tr_{s^\prime}\left[e^{Kss^\prime}G_b(s^\prime)\right]=2(a\cosh(K)+b s \sinh(K))=2\cosh(K)G_{b^\prime}(s),$$ where $b^\prime=b\tanh(K)$, by using $Tr_{s^\prime}[1]=2$ and $Tr_{s^\prime}[s^\prime]=0$.

Consider a two-dimensional Ising model of $m$ rows of $n$ spins each, with the Hamiltonian $\beta H = K\sum_{\langle i,j\rangle} S_i S_j$. Denoting the partition function of the system as $Z_{n,m}$, one has $Z_{n,m}=\text{Tr}[T^m_n]$, where the transfer matrix $T_n$ is a $2^n \times 2^n$ matrix. When $m \rightarrow \infty$, (i.e. an infinite strip of width $n$), The correlation length of the system is $$ \frac{1}{\xi(n)}=\ln \frac{\lambda_0}{\lambda_1},$$ where $\lambda_0$ is the largest eigenvalue and $\lambda_1$ is the second largest eigenvalue of $T_n$. According to the exact solution, one has $$\lambda_0=(2\sinh(2K))^{\frac{n}{2}}\exp\left(\frac{1}{2}(\gamma_1+\gamma_3 + \cdots + \gamma_{2n-1})\right),$$ $$\lambda_1=(2\sinh(2K))^{\frac{n}{2}}\exp\left(\frac{1}{2}(\gamma_0+\gamma_2 + \cdots + \gamma_{2n-2})\right),$$ where $$\gamma_0=2(K-K^*)=2K+\ln(\tanh(K))$$ and $$\cosh(\gamma_l)=\cosh(2K)\cosh(2K^*)-\sinh(2K)\sinh(2K^*)\cos(l\pi/n)$$ or $$\cosh(\gamma_l)=\cosh(2K)\coth(2K)-\cos(l\pi/n).$$ for $l\neq 0$.

Write down the transfer matrix for the case of $n=2$ and $n=4$. Diagonalize the transfer matrix numerically and compare your results to the exact expression. Plot $\lambda_0$ and $\lambda_1$ as a function of temperature.

Calculate and plot $\xi(n, T)/n$ for the case of $n=2$ and $n=4$ and determine their crossing point. The crossing point should be around $T^*(n=2)=1/0.4266$.

Denote the derivate of $\xi(n, T)/n$ as $s(n, T)$, i.e., $$s(n, T)=\frac{d}{dT} \frac{\xi(n,T)}{n}.$$ then one can estimate $1/\nu$ by $$\frac{1}{\nu^*(n)} = \frac{1}{\ln(2)}\ln\left(\frac{s(n=4,T^*(n=2))}{s(n=2, T^*(n=2))}\right).$$ Evaluate $\nu^*(n=2)$ using results above.

Challenge: Repeat the same calculation with larger $n$. For example $n=4$ and $n=8$, or $n=8$ and $n=16$, etc. (Keep the ratio to 2). You should observe that as $n$ increases, your results will approach exact $T_c=2/\ln(1+\sqrt{2})$ and $\nu=1$.