Homework Set 5

Problem 1: Heisenberg model for a linear chain of three spins

Consider three spin-1/2 located on three lattice sites of a linear chain with periodic boundary conditions (i.e., spins on site 1 and 3 are assumed to interact via exchange coupling $J$). With both the spin-spin interaction and Zeeman term with a magnetic field $B$ in the z-direction, the Hamiltonian of the system is given by

${\hat {H}}={\frac {J}{\hbar ^{2}}}\sum _{\langle jl\rangle }{\hat {\mathbf {S} }}_{j}\cdot {\hat {\mathbf {S} }}_{l}-{\frac {g\mu _{B}B}{\hbar }}\sum _{j=1}^{3}{\hat {S}}_{j}^{z}$

where $\langle jl\rangle$ indicates the summation over nearest neighbor only and ${\hat {\mathbf {S} }}=({\hat {S}}_{x},{\hat {S}}_{y},{\hat {S}}_{z})={\frac {\hbar }{2}}({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ is the vector operator for spin-1/2 with $({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ being the vector of Pauli matrices.

(a) Show that the dot product of two vector operators representing spin-1/2 on two different sites can be written as ${\hat {\mathbf {S} }}_{j}\cdot {\hat {\mathbf {S} }}_{l}=({\hat {S}}_{j}^{+}{\hat {S}}_{l}^{-}+{\hat {S}}_{j}^{-}{\hat {S}}_{l}^{+})/2+{\hat {S}}_{j}^{z}{\hat {S}}_{l}^{z}$, where the rising and lowering spin-1/2 operators are defined by ${\hat {S}}_{j}^{\pm }={\hat {S}}_{j}^{x}\pm i{\hat {S}}_{j}^{y}$.

(b) Write down the matrix representation of the Hamiltonian in the basis consisting of the following vectors:

$|1\rangle =|\uparrow \uparrow \uparrow \rangle ,\ |2\rangle =|\uparrow \uparrow \downarrow \rangle ,\ |3\rangle =|\uparrow \downarrow \uparrow \rangle ,\ |4\rangle =|\downarrow \uparrow \uparrow \rangle ,\ |5\rangle =|\uparrow \downarrow \downarrow \rangle ,\ |6\rangle =|\downarrow \uparrow \downarrow \rangle ,\ |7\rangle =|\downarrow \downarrow \uparrow \rangle ,|8\rangle =|\downarrow \downarrow \downarrow \rangle .$

(c) Find eigenenergies of this Hamiltonian. HINT: The $8\times 8$ Hamiltonian matrix in (a) will consist of four blocks of size $1\times 1$, $3\times 3$, $3\times 3$, and $1\times 1$ along the main diagonal. So, the first and last block give eigenenergies directly, while the second and third $3\times 3$ blocks have to be diagonalized individually to find the corresponding eigenenergies (which can be done using Mathematica or Maple).

(d) Using result obtained in (c), compute the canonical partition function for this Hamiltonian and its magnetization. What is magnetization in the limit $B\rightarrow 0$.

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). HINT: You can start by fixing the the first spin on the left to be up and then consider the four microstates of two other spins. By symmetry, the same result is obtained if the first spin is down.

(b) For one-dimensional chain of $N$ Ising spins and with free boundary conditions, show that $G(r)=\langle s_{k}s_{k+r}\rangle =(\tanh \beta J)^{r}$. HINT: You can use the following trick (which also helps to find the partition function $Z_{N}$ given below for the Ising model with open boundaries via elementary means):

$\langle s_{k}s_{k+r}\rangle =\langle s_{k}s_{k+1}^{2}...s_{k+r-1}^{2}s_{k+r}\rangle$

which is an identity since square of an Ising spin variable is equal to 1. Then change variables $s_{k}s_{k+1}\rightarrow \sigma _{k}$; $s_{k+1}s_{k+2}\rightarrow \sigma _{k+1}$, and so on. This allows one to rewrite:

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

as the r-th power of the average value $\langle \sigma \rangle$ of a single $\sigma$ variable introduced by the substitution above.

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

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

and show that it diverges exponentially 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.

Problem 4: Mean-field theory of XY ferromagnet

A ferromagnetic XY model consists of unit classical spins, $\mathbf {S} _{i}=(S_{i}^{x},S_{i}^{y})=(\cos \phi ,\sin \phi )$ so that $|\mathbf {S} |=1$ 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 $\mathbf {h}$ 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 $\chi =\partial m/\partial h|_{h=0}$ per spin where $m=\langle S_{i}\rangle$. HINT: If you select the direction of external field as $\mathbf {h} =(h,0)$, then the single spin Hamiltonian is $H=-\mathbf {h} \cdot \mathbf {S}$ and $m=\langle S\rangle =(\langle \cos \phi \rangle ,\langle \sin \phi \rangle )=(\langle \cos \phi \rangle ,0)$.

(b) Use your result in (a) to calculate the transition temperature in the ferromagnetic states in mean field theory for $h=0$ 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., $\chi$).