Homework Set 2

Problem 2: Magnons in one-dimensional Heisenberg model

Consider the low-energy excitations (magnons) above the ground state of a one-dimensional spin-S ferromagnet described by the isotropic Heisenberg model ($J>0$):

$\hat{H}=-\frac{J}{{\hslash }^2}\sum _{n=1}^N{\widehat{\mathbf{𝐒}}}_n\cdot {\widehat{\mathbf{𝐒}}}_{n+1}=-\frac{J}{2{\hslash }^2}\sum _{n=1}^N({\hat{S}}_n^{+}\cdot {\hat{S}}_{n+1}^{-}+{\hat{S}}_n^{-}\cdot {\hat{S}}_{n+1}^{+}+2{\hat{S}}_n^z\cdot {\hat{S}}_{n+1}^z)$

The periodic boundary conditions, ${\widehat{\mathbf{𝐒}}}_{n+1}={\widehat{\mathbf{𝐒}}}_1$ and ${\widehat{\mathbf{𝐒}}}_N={\widehat{\mathbf{𝐒}}}_0$, are imposed on spin operators.

(a) Apply the Holstein-Primakoff transformation, in the approximation where the density of magnons is small so that

${\hat{S}}_n^{+}\approx \hslash \sqrt{2S}{\hat{a}}_n^{†}$

${\hat{S}}_n^{-}\approx \hslash \sqrt{2S}{\hat{a}}_n$

${\hat{S}}_n^z=\hslash (S-{\hat{a}}_n^{†}{\hat{a}}_n)$

and then expand the Hamiltonian above to the quadratic order in boson operators.

(b) Using the following Fourier transform

${\hat{a}}_n=\frac{1}{\sqrt{N}}\sum _k{e}^{ikna}{\hat{a}}_k$

diagonalize the approximative Hamiltonian you obtained in (a) to find the magnon energy-momentum dispersion $\hslash {\omega }_k$ in terms of $J,S,a$ parameters. Here $k=2\pi n/Na$ with $a$ as the lattice constant and $n=0,\pm 1,\dots ,(N-1)/2,\pm N/2$.

(c) What is the total number of such non-interacting magnons at temperature ? You should simply write the integral expression without fully evaluating it.

NOTE: Useful formula from Fourier analysis: $\frac{1}{N}\sum _{n=1}^N{e}^{i(k^{\prime }-k)na}={\delta }_{k{k}^{\prime }}$.

Problem 3: Roton excitation in a superfluid

A real superfluid is more complicated than a weakly interacting boson gas because the interaction is strong and can extend over a finite range instead of being just a delta function used in the class. Suppose that interaction takes the form of a square function

Failed to parse (syntax error): {\displaystyle V(\mathbf{r} - \mathbf{r}') = U \mathrm{rect} \left(\frac{x-x'}{2r_0)\right) \left(\frac{y-y'}{2r_0) \right) \left(\frac{z-z'}{2r_0} \right) }

where $\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}(t)=1$ when $|t|<1/2$ and $\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}(t)=0$ when $|t|>1/2$. The chosen interaction is highly anisotropic to make the calculations below tractable, but the qualitative features of the result will remain the same if we choose a more realistic isotropic form of the interaction potential energy.

(a) Write the second quantized Hamiltonian in terms of

${\hat{b}}_{}=\frac{1}{\sqrt{V}}\int d^{3}r{e}^{-i\mathbf{𝐤}\cdot \mathbf{𝐫}}\hat{\mathrm{\Psi }}(\mathbf{𝐫})$

operator anihilating bosons in the momentum operator eigenstates and its Hermitian conjugate ${\hat{b}}_{\mathbf{𝐤}}^{†}$.