Statistical Mechanics

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