I study quantum Hall systems and I haven't studied Fermi liquid theory yet. But I understand the concept of having gap or being gapless. But why do we use the term incompressibility to correspond the presence of gap in the bulk state? Is there any relation to physical compressibility here?
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The electronic compressibility $\kappa$ is defined as $$\kappa =\frac{\partial \rho}{\partial \mu} $$ where $\rho$ is the electron density, and $\mu$ the chemical potential. The region where the $\rho(\mu)$ is constant indeed indicates that there is an energy gap. This can simply be understood by the fact that in the energy gap there is a region where there are no electronic states that can contribute to the electronic density $\rho$, and thus it remains constant. The region of $\mu$ values over which $\rho$ remains constant determines the energy gap.
The criterion often used for insulating behavior is that there is a certain region where $\kappa$ vanishes. However, keep in mind that this criterion for insulating behavior applies to standard insulators, as well as electron-driven insulators (e.g. Mott insulator). Nonetheless, there are insulators where $\kappa$ does not vanishes, these are disorder-induced Anderson insulator. So keep in mind that this is not a universal characterization for insulating behavior!
Simon
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