NTHU-Qubit

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====== 10910PHYS522000 ======
====== 統計力學二 Statistical Mechanics (II) ======

===== Instructor =====
  * Prof. Pochung Chen
  * R521, Physics building
  * pcchen {at} phys {dot} nthu {dot} edu {dot} tw

===== Teaching Assistant =====
  * 林彥彤 aronton0502 {at} gmail {dot} com

===== Time =====
  * R5R6R7

===== Room =====
  * Room 501

===== Grade =====
  * TBA

===== References =====
  * Daniel Arovas, [[https://courses.physics.ucsd.edu/2018/Spring/physics210a/LECTURES/BOOK_STATMECH.pdf|Lecture Notes on Thermodynamics and Statistical Mechanics]], (A Work in Progress), in PDF (2020).
    * [[https://phys.libretexts.org/Under_Construction/Arovas_Texts/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09%3A_Renormalization|Chapter 9: Renormalization]]. 
  * R. J. Baxter, [[http://physics.anu.edu.au/theophys/_files/Exactly.pdf|Exactly Solved Models in Statistical Mechanics]], in PDF (1989)
    * Chapter 2, Chapter 6.
  * R. K. Pathria, Paul D. Beale, [[https://www.amazon.com/Statistical-Mechanics-Third-R-Pathria/dp/0123821886/ |Statistical Mechanics]], 3rd Edition (2011).
    * Chapter 12, Chapter 13, Chapter 14.
==== Critical phenomena, Renormalization ====
  * Kim Christensen and Nicholas R. Moloney, [[https://www.amazon.co.uk/Complexity-Criticality-Imperial-College-Advanced/dp/1860945171 | Complexity and Criticality]], 1st Edition, (2005).
    * Chapter 2.
  * Michael Cross, [[http://www.pmaweb.caltech.edu/~mcc/Ph127/index.html|Statistical Physics]].
    * Lecture 8, 9, 10, 11.
  * Leonid Levitov, [[http://www.mit.edu/~levitov/8.334/|Statistical Physics II]], (MIT, 2003).
    * Lecture 5, 6, 7.
    * [[Ising Model]]
    * [[https://www.dropbox.com/s/68f8ii034vbxlrk/sandvik3.pdf?dl=0|Systematic finite-size scaling methods for analyzing critical points (and ordered phases)]], Anders W Sandvik, Boston University.
===== Topics =====
  * Phase transitions.
     * Abrupt phase transitions.
     * Continuous phase transitions.
  * Mean-field theories.
  * Landau theory.
  * Renormalization group.
  * Tensor network.
  * Exact solutions.
  * Models:
     * Ising model.
     * Potts model.
     * XY model.

===== Weekly schedule =====
  * Week 1: 9/17/2020
    * The van der Waals system.
    * Fluids, Magnets, and the Ising Model.
    * Arovas, Chapter 7.2, 7.3. 
  * Week 2: 9/24/2020
    * Mean Field Theory.
    * Arovas, Chapter 7.4. 
  * Week 3: 10/1/2020 中秋節
    * No class.
  * Week 4: 10/8/2020
    * Variational Density Matrix Method.
    * Arovas, Chapter 7.5. 
  * Week 5: 10/15/2020
    * Variational Density Matrix Method.
    * Arovas, Chapter 7.6.
  * Week 6: 10/22/2020
    * Variational Density Matrix Method.
    * Landau Theory of Phase Transition.
    * Arovas, Chapter 7.5, 7.6.
  * Week 7: 10/29/2020
    * Landau Theory of Phase Transition.
    * Ising model.
    * Arovas, Chapter 7.6, 6.2.
  * Week 8: 11/5/2020
    * Ising model.
    * Transfer matrix, Peierls’ argument.
    * Arovas, Chapter 6.2. 
    * Pathria, Chapter 13.2.
    * Baxter, Chapter 2.
  * Week 9: 11/12/2020
    * 2D Ising model.
    * High temperature expansion, low temperature expansion.
    * Duality transformation.
    * Pathria, Chapter 13.4.
    * Baxter, Chapter 6.
  * Week 10: 11/19/2020
    * Renormalization group.
  * Week 11: 11/26/2020
    * Renormalization group.
  * Week 12: 12/3/2020
    * Scaling.
    * Renormalization group.
  * Week 13: 12/10/2020
    * Renormalization group. 
  * Week 14: 12/17/2020
    * Renormalization group.
    * Finite size scaling.
  * Week 15: 12/24/2020
    * Renormalization group.
    * Finite size scaling.
  * Week 16: 12/31/2020
    * Finite size scaling.
    * Tensor renormalization group.
  * Week 17: 1/7/2021
  * Week 18: 1/14/2021
===== Homework =====
  * HW1, due on 10/29/2020
    * Pathria (3rd ed), Pb. 12.2, 12.3.
    * A //ferrimagnet// is a magnetic structure in which there are different types of spins present. Consider a sodium chloride structure in which the $A$ sublattice spins have magnitude $S_A$ and the $B$ sublattice spins have magnitude $S_B$ with $S_B$ < $S_A$ (e.g. $S = 1$ for the $A$ sublattice but $S = 1/2$ for the $B$ sublattice). The Hamiltonian is $$\hat{H}=J\sum_{\langle ij\rangle} \mathbf{S}_i \cdot \mathbf{S}_j + g_A \mu_0 H_{ext} \sum_{i\in A} S^z_i + g_B \mu_0 H_{ext} \sum_{i\in B} S^z_j$$ where $J>0$, so the interactions are antiferromagnetic.
      * Work out the mean field theory for this model. Assume that the spins on the $A$ and $B$ sublattices fluctuate about the mean values $$\langle \mathbf{S}_A \rangle = m_A \hat{z}, \langle \mathbf{S}_B \rangle = m_B \hat{z}$$ and derive a set of coupled mean field equations of the form $$m_A=F_A(\beta g_A \mu_0 H_{ext}+\beta Jzm_B),  m_B=F_B(\beta g_B \mu_0 H_{ext}+\beta Jzm_A)$$ where $z$ is the lattice coordination number and $F_A(x)$ and $F_B(x)$ are related to Brillouin functions.
      * Show graphically that a solution exists, and find the criterion for broken symmetry solutions to exist when $H_{ext} = 0$, i.e. find $T_c$. Then linearize, expanding for small $m_A$, $m_B$, and $H_{ext}$, and solve for $m_A(T)$ and $m_B(T)$ and the susceptibility $$\chi(T)=\frac{1}{2}\frac{\partial}{\partial H}(g_A \mu_0 m_A + g_B \mu_0 m_B)$$ in the region $T > T_c$. Does your $T_c$ depend on the sign of $J$? Why or why not?
  * HW2, due on 11/12/2020
     * Derive Eq.7.30 of Arovas' note. You should go through all the details and you should evaluate all the partial derivatives.
       * (optional) Solve coexistence curve numerically and compare to the analytic form.
     * Derive Eq. 7.66 of Arovas' note.
     * Derive Eq. 7.117 of Arovas' note.
  * HW3, due on 11/26/2020
     * 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 three-state Potts model in one dimension, $$\beta H = -K \sum_i\delta(S_i, S_{i+1})$$ where $S_i=0,1,2$.
       *Evaluate the partition function with a free boundary condition.
       *Evaluate the partition function with a periodic boundary condition.
       *Confirm that the free energy per spin does not depend on the boundary condition in the thermodynamic limit $N\rightarrow \infty$.
     * Evaluate the partition function of the two-leg ladder Ising model,$$\beta H= -K_1 \sum_{i=1}^N S_{1,i}S_{1,i+1} -K_1 \sum_{i=1}^N S_{2,i}S_{2,i+1}-K_2 \sum_{i=1}^N S_{1,i}S_{2,i}$$ with periodic boundary condition.
       * Write down the correlation function $\langle S_{1,i} S_{1,i+r}\rangle$ in terms of the transfer matrix. 
   * HW4, due on 1/14/2021
     * Migdal-Kadanoff Transformation: Consider the Ising model on 2D square lattice and construct a renormalization transformation according to the following two-step process:
       * Step 1. Shift one-half of the horizontal bonds by one lattice spacing to obtain the modified interaction as shown in the fig 1. It is now easy to perform the trace over the spins on the sites denoted by open circles to produce the anisotropic renormalized Hamiltonian shown in fig 2.
       * Step 2. Shift one-half of the vertical bonds by one lattice spacing to obtain fig 3. Once again carry out the trace over the variables at the open circles to obtain the new renormalized Hamiltonian in fig 4. Symmetrize the Hamiltonian by defining $$K^\prime=\frac{1}{2}(2\gamma^\prime+2\gamma)=\gamma^\prime(K)+\gamma(K).$$
       * The rescaling parameter is $b=2$ in this case. Solve for the fixed point of the transformation and obtain the leading thermal exponent $y_t$. Compare with the exact critical temperature and specific heat exponent.
       * [[https://www.dropbox.com/s/ftecz7lsai1iq9w/Migdal-Kadanoff_RG.pdf?dl=0|Figure]]
   * HW5, due on 1/14/2021
     * 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 $\nu=1$.
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