# incompressibility of fully filled Landau level

Suppose Landau level degeneracy is $10^9$, if we force to put ($10^9+1$) particles on the level, what extra energy will we gain? (ignore particles interactions) Like electron degeneracy pressure, degeneracy energy in the white dwarf starts.

Incompressibility says the extra energy would be infinite if we force to do that, right?

Now suppose we decrease B field infinitesimally, so the calculated degeneracy is ($10^9-0.000000001$). Because there is not enough room for $10^9$ particles, there must be one particle jump to the next level and gain a finite energy. This is the statement of incompressibility.

It seems unfair for the particle to jump to next level, just due to 0.000000001 lack of room. Shouldn't we compare the Landau energy gap with certain "degeneracy energy" ?

• I guess, the volume of the system may stretch a little bit, to make 0.000000001 room for the particle. We can compare the stretch energy with Landau gap.
– Jian
Sep 3, 2015 at 23:15

## 1 Answer

1. The extra particle will be pushed into the next higher Landau level. So the extra energy is $\hbar\omega_c$, which is finite. This is unlike the free Fermi gas, where adding an extra electron only increases the energy by infinitesimal amount in the thermodynamic limit.

2. Incompressibility only requires this extra energy to be finite. It does not need to be infinity. See the discussions here: Incompressible quantum liquid.

3. Your statement of decreasing the LL degeneracy by $0.000000001$ is of course impossible since the number of states in one LL should be integer. In other words, you can only increase it or decrease it by one.