How Does Periodic Potential Affect the Energy Spectrum of a Bose Gas?

[physicus]

Homework Statement

Suppose an ideal bose gas sees a periodic potential with a period a in both x and y directions. Its eigenstates are altered from the free-particle form. The lowest band has energies
$\epsilon_\vec{k}=2t(2-cos(k_xa)-cos(k_ya))$
where t is an energy scale that depends on the amplitude of the periodic potential, and the wavevectors $\vec{k}$ are restricted to the first Brillouin zone: $\pi/a <k_{x,y}<\pi/a$.

(i) Show that the density of states $g(\epsilon)$ is zero for $\epsilon>8t$ and $\epsilon<0$

(ii) Show that $g(\epsilon) \simeq L^2/4\pi ta^2$ at low energies $0\leq\epsilon\ll 8t$

Hint: Sketch the dispersion relation above consider the approximate form of $\epsilon_{\vec{k}}$ at lom wavelengths ($ka \ll 1$).

Homework Equations

The Attempt at a Solution

Number (i) is quite obvious since the cosine can only take values between -1 and 1. Therefore the expression in brackets can only take value between 0 and 4 which shows that there are no states with energies < 0 or energies > 8t.

(ii) For long wavelengths we can approximate
$\epsilon_\vec{k}=2t(2-cos(k_xa)-cos(k_ya)) = -2t(k_x^2 a^2+k_y^2a^2) = -2ta^2\vec{k}^2$
But how does that help me to get the density of states? I don't know how to begin. Can someone give me an ansatz?


Cheers, physicus

 

[TSny]

(ii) For long wavelengths we can approximate
$\epsilon_\vec{k}=2t(2-cos(k_xa)-cos(k_ya)) = -2t(k_x^2 a^2+k_y^2a^2) = -2ta^2\vec{k}^2$
But how does that help me to get the density of states? I don't know how to begin. Can someone give me an ansatz?

Check your result for your approximation of $\epsilon_\vec{k}$. I think you might be off by a sign and a factor of 2.

To proceed to the density of states, compare your approximate result for $\epsilon_\vec{k}$ with $\epsilon_{\vec{k},free}$ for a free particle of mass m moving in 2 dimensions.