Bose-Einstein Condensation in lower dimensions

Bose-Einstein condensation occurs at 3 dimensions. However, it is not possible to happen at 1 or 2 dimensions; in fact I am able to prove this myself. What is the explanation for this?

Answer

For simplicity, I will consider non-interacting gases. The idea here is that there is a bound on the density of excited states in 3 and higher dimensions, whereas no such bound exists in 1 and 2 dimensions. This bound enforces a condensation of the gas particles into the ground state for 3 and higher dimensions but not in lower.

The details are a bit cumbersome to write down from scratch, so I will refer to some standard details and sketch out the non-trivial part. Check out eqns (7.31) and (7.32) of Mehran Kardar, Statistical physics of particles, where the average occupation number of a non relativistic gas is derived. Write down the same in $d$ dimensions. If you now convert the integral for the average occupation number into a dimensionless form, you will arrive at the generalization of eqn (7.34) for $d-$ dimensions given by

$$n = \dfrac{g}{\lambda^d} f^1_{d/2}(z)$$

where

$$f^1_{m}(z) = \dfrac{1}{(m-1)!} \int_{0}^{\infty} \dfrac{dx x^{m-1}}{z^{-1}e^x - 1}$$

This integral is finite for all values of $z$ if $d \geq 3$, and hence has a maximum value , whereas the same is not true for lower dimensions. In $d \geq 3$, we therefore have a bound on the density of excited states at $z = 1$, given by

$$n _x = \dfrac{g}{\lambda^d} f^1_{d/2}(z) \leq n* = \dfrac{g}{\lambda^d} \zeta_{d/2}$$

where $\zeta_{d/2}$ is the max value of the number density of excited states at $z = 1$.

Since the integral is not finite and therefore no such maximum value exists for $d = 1,2$, hence there is no BEC in lower dimensions for this system.