Homework Set 5

Problem 1: Three Ising spins

Assume three spins $s_{1},s_{2},s_{3}$ are arranged in an equilateral triangle with each spin interacting with its two neighbors. The energy expression for the Ising model in an external magnetic field $h$ is given by:

$H=-J(s_{1}s_{2}+s_{2}s_{3}+s_{3}s_{1})-h(s_{1}+s_{2}+s_{3})$

(a) Find canonical partition function for this model.

(b) Find average spin $\langle s\rangle$.

(c) Find internal energy $E$.

Problem 2: Spin-spin correlation function in the Ising model

We can gain further insight into the properties of the one-dimensional Ising model of ferromagnetism

$H=-J\sum _{i=1}^{N}s_{i}s_{i+1}$

by calculating the spin-spin correlation function $G(r)$

$G(r)=\langle s_{k}s_{k+r}\rangle -\langle s_{k}\rangle \langle s_{k+r}\rangle$

where $r$ is the separation between the two spins in units of the lattice constant. The statistical average $\langle \ldots \rangle$ is over all microstates. Because all lattice sites are equivalent, $G(r)$ is independent of the choice of specific site $k$ and depends only on the separation r (for a given temperature T and external field h). Since the average value of spin $\langle s_{k}\rangle$ at site $k$ is independent of the choice of that specific site (for periodic boundary conditions) and equals $m=M/N$ ($M$ is magnetization), the correlation function can also be written as:

$G(r)=\langle s_{k}s_{k+r}\rangle -m^{2}$.

The spin-spin correlation function $G(r)$ is a measure of the degree to which a spin at one site is correlated with a spin at another site. If the spins are not correlated, then trivially $G(r)=0$. At high temperatures the interaction between spins is unimportant, and hence the spins are randomly oriented in the absence of an external magnetic field. Thus in the limit $k_{B}T\gg J$, we expect that $G(r)\rightarrow 0$ for any r. For fixed $T$ and $h$, we expect that, if spin $k$ is up, then the two adjacent spins will have a greater probability of being up than down. For spins further away from spin $k$, we expect that the probability that spin at site $k+r$ is up or correlated will decrease. Hence, we expect that $G(r)\rightarrow 0$ as $r\rightarrow \infty$. Note that the physical meaning of the correlation is that it can be used to express magnetic susceptibility $\chi \propto G(r=0)=\langle m^{2}\rangle -\langle m\rangle ^{2}$ as one of the response functions.

(a) Consider an Ising chain of $N=3$ spins with free boundary conditions which is in equilibrium with a heat bath at temperature $T$ and in zero magnetic field $H=0$. Enumerate all $2^{3}$ microstates and calculate $G(r=1)$ and $G(r=2)$ for k = 1 (labeling the first spin on the left).

(b) For one-dimensional chain of $N$ Ising spins and with free boundary conditions, show that $G(r)=(\tanh \beta J)^{r}$. HINT: Start from:

$\langle s_{k}s_{k+r}\rangle ={\frac {1}{Z_{N}}}\sum _{s_{1}=\pm 1}\sum _{s_{N}=\pm 1}s_{k}s_{k+r}\exp[\sum _{i=1}^{N-1}\beta Js_{i}s_{i}+1]$

where the partition function for N spins is:

$Z_{N}=2(2\cosh \beta J)^{N-1}$.

and try to obtain $G(r=1)$, $G(r=2)$, ...

(c) By writing $G(r)=e^{-r/\xi }$ for $r\gg 1$, extract the correlation length from your result in (b):

$\xi =-{\frac {1}{\ln(\tanh \beta J)}}$

in the low temperature limit $\beta J\gg 1$.

Problem 3: Two-dimensional Ising model

(a) Calculate the partition function for the Ising model on a square lattice for $N=4$ spins in the presence of an external magnetic field $H$. Assume that the system is in equilibrium with a heath bath at temperature T and employ periodic boundary conditions in both directions.

(b) The $4^{2}=16$ microstates of the two-dimensional Ising model for $N=4$ can be grouped into four "ordered" states with energies $\pm J$ and 12 "disordered" states with zero energy. Test the hypothesis that the phase transition occurs when the partition function of the disordered states equals that of the ordered states. What is the resulting value of $T_{c}$? This simple reasoning does not work as well for the Ising model in three dimensions.

Mean-field theory of XY ferromagnet

A ferromagnetic XY model consists of unit classical spins, $\mathbf {S} _{i}=(S_{i}^{x},S_{i}^{y})$ and $|\mathbf {S} |=1$ on a three-dimensional cubic lattice with i labeling the site. The spins can point in any direction in the xy plane. The Hamiltonian with nearest neighbor interactions is given by:

$H=-{\frac {1}{2}}J\sum _{i,\delta }\mathbf {S} _{i}\cdot \mathbf {S} _{i+\delta }-\mathbf {h} \cdot \sum _{i}\mathbf {S} _{i}$

with $h$ a field in the xy plane, and with i running over all the sites and $\delta$ running over the six nearest neighbors.

(a) For the noninteracting case $J=0$ calculate the susceptibility $\chi =\partial m/\partial h|_{h=0}$ where $m=|\langle S_{i}\rangle |.(b)Useyourresultin(a)tocalculatethetransitiontemperatureintheferromagneticstatesinmeanfieldtheoryfor<math>h=0$ and $J\neq 0$. HINT: While it is hard to evaluate the integral expression for $m(h)$, it should be easy to evaluate its derivative at $h=0$. The transition to the ferromagnetic state is formulated in terms of $m(h)$, but the transition temperature only depends on the slope at $h=0$ (i.e., $\chi$).