Homework Set 5

Problem 1: Ferromagnetic quantum spin chain: Many-body eigenstates and eigenenergies for three interacting spins with periodic boundary conditions

Consider three spin-1/2 located on three lattice sites of a quantum Heisenberg ferromagnetic chain with periodic boundary conditions, i.e., spins on site 1 and 3 are assumed to interact via exchange coupling $J>0$. 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

${\hat {H}}_{\mathrm {FM} }=-{\frac {J}{\hbar ^{2}}}\left({\hat {\mathbf {S} }}_{1}\cdot {\hat {\mathbf {S} }}_{2}+{\hat {\mathbf {S} }}_{2}\cdot {\hat {\mathbf {S} }}_{3}+{\hat {\mathbf {S} }}_{3}\cdot {\hat {\mathbf {S} }}_{1}\right)-{\frac {g\mu _{B}B}{\hbar }}\sum _{j=1}^{3}{\hat {S}}_{j}^{z}$

where ${\hat {\mathbf {S} }}_{j}=({\hat {S}}_{j}^{x},{\hat {S}}_{j}^{y},{\hat {S}}_{j}^{z})={\frac {\hbar }{2}}({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ is the vector operator for spin-1/2 with $({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ being the vector of the 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: Antiferromagnetic quantum spin chain: Néel state vs. valence-bond variational ground-state

In contemporary research on antiferromagnetic spintronics it is common to consider antiferromagnets as materials where magnetic moments assemble into the so-called Néel state where they point in opposite directions on neighboring atoms, as illustrated below:

Although Néel state is the ground-state of classical Heisenberg Hamiltonian, the corresponding ket

$|\mathrm {Neel} \rangle =|\uparrow _{1}\rangle \otimes |\downarrow _{2}\rangle \otimes |\uparrow _{3}\rangle \otimes |\downarrow _{4}\rangle \otimes \cdots \otimes |\uparrow _{N-1}\rangle \otimes |\downarrow _{N}\rangle$

is not the true ground-eigenstate (i.e., eigenstate corresponding to the lowest energy eigenvalue) of the quantum Heisenberg Hamiltonian -> one often talks about quantum spin fluctuation (or zero point motion) lowering the energy of $|\mathrm {Neel} \rangle$ toward the true ground-state.

Consider a quantum spin chain of $N$ spins $S=1/2$ in the Figure above, where $N$ is even integer and periodic boundary conditions are applied (so, site $N+1$ is the same as site 1), with antiferromagnetic nearest-neighbor exchange interaction, $J<0$. The chain is described by the following quantum Heisenberg Hamiltonian:

${\hat {H}}_{\mathrm {AFM} }=-J\sum _{n=1}^{N-1}{\hat {\boldsymbol {\sigma }}}_{n}\cdot {\hat {\boldsymbol {\sigma }}}_{n+1}$

where ${\hat {\boldsymbol {\sigma }}}=({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ is the vector of the Pauli matrices.

(a) Find expectation value of this Hamiltonian in the Néel state, $\langle \mathrm {Neel} |{\hat {H}}_{\mathrm {AFM} }|\mathrm {Neel} \rangle$.

(b) A variational ground-state could be chosen as the so-called "valence-bond" state composed of tensor products of dimer states

$|-\rangle =\prod _{n=1}^{N/2}\otimes \,{\frac {|\uparrow _{2n}\rangle \otimes |\downarrow _{2n-1}\rangle -|\downarrow _{2n}\rangle \otimes |\uparrow _{2n-1}\rangle }{\sqrt {2}}}$

$|+\rangle =\prod _{n=1}^{N/2}\otimes \,{\frac {|\downarrow _{2n}\rangle \otimes |\uparrow _{2n+1}\rangle -|\uparrow _{2n}\rangle \otimes |\downarrow _{2n+1}\rangle }{\sqrt {2}}}$

Find expectation values $\langle \pm |{\hat {H}}_{\mathrm {AFM} }|\pm \rangle$ and show that both satisfy $\langle \pm |{\hat {H}}_{\mathrm {AFM} }|\pm \rangle <\langle \mathrm {Neel} |{\hat {H}}_{\mathrm {AFM} }|\mathrm {Neel} \rangle$. This means that valence-bond state provides tighter bound on the ground-state energy. While it is not in general true that variational states with lower energy are more similar to true ground-state, numerical simulations do suggest that ground-state is indeed similar to the simple valence-bond state $|\pm \rangle$. Since you should find that $|+\rangle$ is degenerate with $|-\rangle \rangle$, this shows that valence-bond state is still not a true ground-state since it violates Marshall theorem according to which ground-state of antiferromagnetic chain must be nondegenerate.

Problem 3: Spin-spin correlation function for classical Ising model in one dimension

We can gain further insight into the properties of the one-dimensional classical 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 4: Mean-field theory of classical 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., $\chi$).

NOTE: Although XY model does not seem to describe any real ferromagnet, it is the simplest model to study critical phenomena in the universality class characterized by two-component order parameters. For example, this model on cubic lattice has been used to study critical behavior of the superfluid to normal phase transition in liquid $^{4}\mathrm {He}$ and displays global $U(1)$ symmetry, which amounts to a change $\phi _{i}\rightarrow \phi _{i}+\alpha$ on every site, with $\alpha$ being a real number. The two-dimensional XY model has no spontaneous magnetization (i.e., long-range order) at finite temperatures, and therefore no ordinary phase transitions. Nevertheless, it exhibits a special type of Berezinsky–Kosterlitz–Thouless transition where the correlation function decays exponentially in the "paramagnetic" phase and slowly, as power law, in the low-temperature critical phase.

Problem 5: Magnetic phases of the ground state in the mean-field solution of the Hubbard model of interacting electrons

Reproduce Figure 6 from:

• Y. Claveau, B. Arnaud, and S. Di Matteo, Mean-field solution of the Hubbard model: the magnetic phase diagram, Eur. J. Phys. 35, 035023 (2014). [PDF]

This Figure represents ground state phase diagram where lines delineate boundary between different types of magnetic ordering (or no ordering in the case of paramagnetic phase) in the celebrate Hubbard model of interacting lattice electrons solved by mean-field theory. Since such solution requires self-consistent loop in Figure 3 of the above references, you will need to use some simple coding of numerics via Mathematica, Matlab or Python. The transition between different phases at zero temperature would be an example of quantum phase transition.