Casey Chu
• Statistical Mechanics
• I took these notes in winter 2017 for Physics 212, taught by Professor Stephen Shenker.
• Ising model
• The Ising model is a lattice $\Lambda$ of spins $s_i = \pm 1$ with Hamiltonian $H(\{ s_i \}) = -J\sum_{\langle i\, j \rangle} s_i s_j - \mu B\sum_i s_i,$ where $\langle i\, j \rangle$ indexes over neighboring sites $s_i$ and $s_j$. Thus the partition function is $Z = \sum_{\{ s_i \}} e^{-\beta H} = \sum_{\{ s_i \}} \exp\Big( k\sum_{\langle i\, j \rangle} s_is_j + h\sum_i s_i \Big),$ where we’ve nondimensionalized by setting $k = \beta J$ and $h = \beta \mu B$. In the following, let $M$ be the number of bonds, and $N$ be the number of sites.
• In the high-$k$/low-temperature regime, the system is ordered, dominated by states in which most spins are pointing in the same direction. In the low-$k$/high-temperature regime, the system is disordered, and many states are equally important.
• Useful diagostics of order are magnetization $\langle s_0 \rangle$ in the symmetry-breaking limit $h \to 0^+$ and the spin-spin correlation function $\langle s_0s_r \rangle$.
• High-temperature expansion ($k \ll k_c$)
• In the following, set $h = 0^+$. Note that for $s = \pm 1$, the following identity holds: $e^{ks} = (\cosh k)(1 + s \tanh k).$ We can use this to rewrite the partition function as \begin{aligned} Z &= \sum_{\{ s_i \}} \exp\Big( k\sum_{\langle i\, j\rangle} s_i s_j \Big)\\ &= \sum_{\{ s_i \}}\prod_{\langle i\, j\rangle} (\cosh k)(1 + s_is_j \tanh k) \\ &= (\cosh k)^{M} \sum_{\{ s_i \}}\prod_{\langle i\, j\rangle} (1 + s_is_j \tanh k). \end{aligned} The product expands into a power series in $\tanh k$. Noticing that $\sum_{s = \pm 1} s = 0, \qquad \sum_{s = \pm 1} s^2 = 2,$ we see that terms corresponding to open paths on the lattice vanish: $(\tanh k) \sum_{\{ s_i \}} s_1s_2 = 0,$ but terms corresponding to closed paths on the lattice survive: $(\tanh k)^4 \sum_{\{ s_i \}} (s_1s_2)(s_2s_3)(s_3s_4)(s_4s_1) = (\tanh k)^4\, 2^N.$ More abstractly, $Z = (\cosh k)^M 2^N \sum_{\Gamma \in \Lambda} (\tanh k)^{|\Gamma|},$ where the sum indices over closed (but perhaps disconnected) paths $\Gamma$ in the lattice.
• For the 2D lattice, for example, we obtain $Z = (\cosh k)^M2^N(1 + N(\tanh k)^4 + 2N(\tanh k)^6 + \cdots).$
• The magnetization is clearly $0$, and the spin-spin correlation function can be seen as $\frac{1}{Z}$ times the above expression for the partition function that counts closed paths, except that the sites $s_0$ and $s_r$ are “pre-covered.” Thus this counts paths from $s_0$ to $s_r$ (as well as disconnected closed paths, which tend to cancel). The first contribution is the path of length $r$ directly from $s_0$ to $s_r$, so $\langle s_0 s_r \rangle \simeq (\tanh k)^r$.
• Low-temperature expansion ($k \gg k_c$)
• In the low-temperature regime, we can explicitly list the most important Boltzmann factors. For the 2D lattice, the all-up/all-down states dominate, followed by the $2N$ states in which only spin differs from the rest, followed by the $4N$ states in which two adjacent spins differ from the rest, and so on, so that $Z = 2(e^{kM} + N e^{kM - 4(2k)} + 2N e^{kM - 6(2k)} + \cdots).$ The $e^{kM}$ is the Boltzmann weight for the ordered all up/all down state, and the $e^{-n2k}$ is the cost of breaking $n$ bonds. For example, in the state in which only one spin differs from the rest, $4$ incident bonds are broken.
• In fact, the broken bonds surrounding an island of flipped spins can be thought of as, again, closed paths, but this time in the dual lattice. Abstractly, we write $Z = 2e^{kM} \sum_{\Gamma \in \Lambda^*} (e^{-2k})^{|\Gamma|}.$ This is Kramers-Wannier duality. We can define an involution with the relation $e^{-2k} = \tanh k^*$ that identifies a lattice at $k$ with its dual lattice at $k^*$. More symmetrically, this relation is $\sinh 2k\sinh 2k^* = 1$. Since the 2D lattice is its own dual, if we assume that there is one critical point, we can solve for $k_c$ by setting $k = k^*$.
• On the 2D lattice, we can calculate the magnetization directly from the expression for the partition function: $\langle s_0 \rangle = \frac{1}{\frac{1}{2}Z}\Big( e^{kM} + [(N-1)e^{kM-4(2k)} - e^{kM-4(2k)} ] + \cdots\Big) \simeq 1 - 2e^{-8k}.$ where the factor of $\frac{1}{2}$ is for symmetry-breaking. For the spin-spin correlation function, the first two factors that don’t cancel correspond to the totally ordered state and the one where $r$ spins are flipped, giving $\langle s_0 s_r \rangle \simeq 1 - e^{-(2r + 4)(2k)}$.
• Mean field theory/effective field theory
• Now coarse-grain by coalescing the individual microstates $\{ s_i \}$ into macrostates determined by a magnetization field $m(x)$, roughly the average spin in a block of spins at a location $x$. An effective Hamiltonian $H[m]$ is given by $e^{-\beta H[m]} = \sum_{\{ s_i \} \cong m} e^{-\beta H(\{ s_i \})},$ and the partition function can now be written as the functional integral $Z = \int Dm\, e^{-\beta H[m]}.$
• The Landau-Ginzburg Hamiltonian is defined by $\beta H[m] = \int dx\,\Big[ \tfrac{1}{2}tm^2 + um^4 + \tfrac{1}{2}k(\nabla m)^2 + \cdots -hm\Big],$ where:
• the integral over space arises because we’d like to express $H$ as a sum of local effects,
• the expansion in powers of $m$ is justified because non-analyticity is washed out by the coarse-graining process (think central limit theorem),
• the gradient terms arise to capture local interactions,
• odd powers of $m$ vanish via the symmetry $H[m] = H[-m]$ when $h = 0$,
• mixed derivative terms vanish via rotational symmetry,
• the highest power of $m$ has positive coefficent (in this case, $u > 0$) to ensure zero probability for large $m$.
• The parameters $t$, $u$, $k$, etc. are phenomenological and depend on the specific model and properties like temperature and pressure.
• We make the mean-field approximation and examine only the most probable configuration, that is, the one that minimizes $\beta H[m]$. When $\beta H$ is minimized, the field is constant ($\nabla m = 0$), and the most probable configuration $m$ can be viewed as the minimum of the function $\psi(m) = \tfrac{1}{2}tm^2 + um^4 - hm.$ Assume $h = 0^+$ for now. For $t > 0$, the minimum is only at $m = 0$, but for $t < 0$, the minimum goes as $m \sim |t|^{1/2}$. If $t$ is an increasing differentiable function of the temperature $T$, we conclude as well that $m = 0$ for $T > T_c$ and $m \sim |T - T_c|^{1/2}$ for $T \le T_c$, at least locally around $T=T_c$.
• Thus we see that a phase transition occurs at $t = 0$. Further (higher-order) phase transitions occur when, for example, both $t = 0$ and $u = 0$, and can be investigated by including more terms in the power series. The mean-field approximation is extremely accurate for the Ising model in dimensions $d \ge 4$.
• For $t > 0$, we can consider small fluctuations around the minimum of $m = 0$. That is, we write $m(x) \approx \phi(x)$, so that \begin{aligned} Z &\approx \int D\phi\, \exp\Big( {-\int dx \, \Big[ \tfrac{1}{2}k (\nabla\phi)^2 + t\phi^2 - h\phi \Big]}\Big) \\ &= \int D\phi\, \exp\Big( {-\int dx \, \Big[ \tfrac{1}{2}\phi (-k\Delta + t) \phi - h \phi \Big]}\Big) \\ &= \int D\phi\, e^{-\frac{1}{2}\langle \phi, K\phi \rangle + \langle h, \phi\rangle } \\ &\propto (\det K)^{-1/2} e^{\frac{1}{2}\langle h, K^{-1}h \rangle}, \end{aligned} where $K = -k\Delta + t$, and the inner product is the $L^2$ inner product. This is an example of a Gaussian integral. We’re ignoring the divergent factor of $(2\pi)^{N/2}$. On a lattice, the Laplacian becomes the discrete Laplacian.
• Viewed as a function of $h(0)$, the free energy $\log Z$ is the moment-generating function of $\phi(0)$. In particular, the correlation function is $\langle \phi(0) \phi(x) \rangle = \frac{\partial}{\partial h(0)} \frac{\partial}{\partial h(x)} \log Z = \langle\delta_x, K^{-1}\delta_0 \rangle = \Phi(x) \sim \frac{1}{|x|}e^{-\sqrt{t}|x|}$ in $d = 3$, where $\Phi$ is the solution to the PDE $K\Phi = \delta_0$, found using, for example, the Fourier transform. (In momentum space with momenta $q$, $K$ is diagonal with eigenvalues $kq^2 + t$, allowing us to easily invert $K$. On a square lattice, the eigenvalues of $-\Delta$ become $2-2\cos q$ instead of $q^2$, erasing rotational symmetry at high momenta/small scales.) At the critical point, the correlation scales as a power law.
• Renormalization group