# Is there a "critical temperature" for 2D Boson gas?

• Zhang Bei
Zhang Bei
Homework Statement
Use the fact that the density of states is constant in d = 2 dimensions to show that
Bose-Einstein condensation does not occur no matter how low the temperature. (From David Tong's Lectures)
Relevant Equations
$$\langle N \rangle = \int_0^\infty dE\,\frac{g(E)}{z^{-1}\exp(\beta E)-1}$$
The density of states of a 2D gas in a box is
$$g(E)=\frac{Am}{2\pi\hbar^2}\quad.$$

From this we can obtain
$$T=-\frac{2\pi\hbar^2N}{mAk_B\log (1-z)}$$
Inserting $z \to 1$ gives $T_c=0$. We conclude that the 2D boson gas doesn't form BEC.However, on the other hand, according to the Bose-Einstein distribution, the ground state has an expected occupancy number of
$$\langle n_0 \rangle =\frac{1}{z^{-1}-1}$$If we equate this with $N$, we would obtain a "critical temperature" of
$$T=\frac{2\pi\hbar^2N}{mAk_B\log (N+1)}$$
For $1mol$ of "helium" gas in an $1m^2$ area has a critical temperature of $8380K$.What went wrong? Does this mean the BE distribution needs to be modified in this case? Since z can be pushed arbitrarily close to 1, aren't we always able to force $n_0$ to reach $N$?

Last edited:
Zhang Bei said:
From this we can obtain
$$T=-\frac{2\pi\hbar^2N}{mAk_B\log (1-z)}$$
Inserting $z \to 1$ gives $T_c=0$. We conclude that the 2D boson gas doesn't form BEC.However, on the other hand, according to the Bose-Einstein distribution, the ground state has an expected occupancy number of
$$\langle n_0 \rangle =\frac{1}{z^{-1}-1}$$If we equate this with $N$, we would obtain a "critical temperature" of
$$T=\frac{2\pi\hbar^2N}{mAk_B\log (N+1)}$$
How did you get that last equation from the first 2?

## What is a 2D Boson gas?

A 2D Boson gas is a system of bosons (particles that follow Bose-Einstein statistics) confined to move in two dimensions. This restriction to two dimensions can significantly alter the physical properties of the system compared to three-dimensional systems.

## What is meant by "critical temperature" in the context of a 2D Boson gas?

The "critical temperature" refers to the temperature below which a significant fraction of the bosons occupy the lowest quantum state, leading to Bose-Einstein condensation (BEC). In three dimensions, this results in a macroscopic occupation of the ground state, but the situation is more complex in two dimensions.

## Is there a true Bose-Einstein condensation in a 2D Boson gas?

In an ideal, homogeneous 2D Boson gas, true Bose-Einstein condensation does not occur at any finite temperature due to enhanced thermal fluctuations. However, in the presence of interactions or confinement (such as in a harmonic trap), a quasi-condensate or a Berezinskii-Kosterlitz-Thouless (BKT) transition can occur, leading to a phase with some characteristics of BEC.

## What is the Berezinskii-Kosterlitz-Thouless (BKT) transition?

The BKT transition is a phase transition that occurs in two-dimensional systems with continuous symmetry. For a 2D Boson gas, this transition is characterized by the formation of vortex-antivortex pairs at low temperatures, leading to a quasi-long-range order rather than true long-range order. This transition occurs at a finite temperature, often referred to as the BKT temperature.

## How is the critical temperature for the BKT transition in a 2D Boson gas determined?

The critical temperature for the BKT transition in a 2D Boson gas can be estimated using theoretical models and numerical simulations. It depends on factors such as the density of the bosons and the strength of interactions between them. Experimentally, it can be observed through changes in properties like superfluid density and the behavior of vortices.

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