Energy Gap of Neutral Excitations Implies Vanishing Charge Susceptibility

In quantum many-body systems with a U(1) symmetry, such as particle number conservation and axial spin conservation, there are two distinct types of excitations: charge-neutral excitations and charged excitations. The energy gaps of these excitations may be independent from each other in strongly correlated systems. The static susceptibility of the U(1) charge vanishes when the charged excitations are all gapped, but its relation to the neutral excitations is not obvious. Here we show that a finite excitation gap of the neutral excitations is, in fact, sufficient to prove that the charge susceptibility vanishes (i.e., the system is incompressible). This result gives a partial explanation for why the celebrated quantization condition $n(S-m_z)\in \mathbb{Z}$ at magnetization plateaus works even in spatial dimensions greater than one.

Figure 1

Illustration of the neutral excitation gap ${\mathrm{\Delta }}_0(\mu )$ and the charged excitation gap ${\mathrm{\Delta }}_1(\mu )$.

Figure 2

The setup for computing the static charge susceptibility $\chi (\mu )$ from a correlation function. The field $\mu$ is changed by $\delta \mu$ only in the subregion $\mathrm{\Gamma }$ of the total system $\mathrm{\Lambda }$. Consequently, an extra charge $\delta Q_{\mathrm{\Gamma }}$ flows into $\mathrm{\Gamma }$ from the charge reservoir $\mathrm{\Lambda }\backslash \mathrm{\Gamma }$.

Figure 3

When the quasiparticle picture is valid, one can construct gapless neutral excitations using gapless charged excitations, assuming that both positively and negatively charged gapless excitations exist and can be localized.