Neutral and spin/charge gaplessness within the Lieb-Schultz-Mattis construction
by
Physics/124
The Lieb–Schultz–Mattis (LSM) theorem is known as a powerful result that constrains the low-energy spectrum of a quantum many-body system solely from its symmetries. In systems with a U(1) symmetry, however, there are two distinct types of energy gaps. The conventional LSM theorem concerns only the neutral gap, associated with excitations which do not change the total magnetization, whereas the spin gap, associated with excitations that increase or decrease the magnetization, lies outside its direct scope. We previously showed that a vanishing spin gap implies a vanishing neutral gap. The converse statement, however, is highly nontrivial and has not been systematically addressed.
In this work, we investigate the relation between these two gaps in one-dimensional systems at fractional filling with U(1) and translation symmetries. In particular, by combining a local LSM operator with quantum dynamics, we show that if the system is neutral-gapless in the sense of LSM-type excited states, then it must also be spin-gapless. Together with the converse implication, this establishes an equivalence between the existence of a spin gap and that of a neutral gap within the LSM construction. Our result therefore rules out phases where only one of the neutral and spin gaps is finite while the other vanishes, in quantum systems to which our argument applies.