Spontaneously magnetized Tomonaga-Luttinger liquid in frustrated quantum antiferromagnets

We develop a theory of spontaneously magnetized Tomonaga-Luttinger liquid (SMTLL) in geometrically frustrated quasi-one-dimensional quantum magnets by taking an $S=1/2$ ferrimagnet on a union-jack lattice as an example. We show that a strong frustration leads to a spontaneous magnetization because of the ferrimagnetic nature of lattice. Due to the ferrimagnetic order, the local magnetization has an incommensurate oscillation with the position. We show that the spontaneously magnetized TLL is smoothly connected to the existence of a Nambu-Goldstone boson in the canted ferrimagnetic phase of a two-dimensional frustrated antiferromagnet.