Spontaneously magnetized Tomonaga-Luttinger liquid in frustrated quantum antiferromagnets

We develop a theory of spontaneously magnetized Tomonaga-Luttinger (TLL) liquid 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.

Figure 1

The three-leg union-jack ladder (1).

Figure 2

The shaded area depicts the unit cell of the O(3) nonlinear $\sigma$ model.

Figure 3

Dynamical structure factors (a) $S^{\parallel }(k,\omega )$ and (b) $S^{\perp }(k,\omega )$ in the low-energy region. Outside of the shaded area, the dynamical structure factor has zero intensity. The red lines represent the linear dispersion of the SMTLL and the blue dashed curves represent the quadratic dispersions (24) of the nonrelativistic Goldstone mode [$E_{\perp }\left(k\right)$] and the massive mode [$E_{\parallel }\left(k\right)$].

Figure 4

(a) A configuration of trimers (solid rectangles) at $J_2/J_1\gg 1$. Trimers are surrounded by spins (circles). In the low-energy limit, one can regard the trimer as the $S=1/2$ pseudospin. The trimer-trimer interaction is ferromagnetic and the trimer-spin interaction is antiferromagnetic. (b) The effective model that describes the commensurate phase. The solid and blank circles represent trimers ($S=1/2$ pseudospins) and $S=1/2$ spins, respectively. The thick line represents the ferromagnetic coupling (46) of trimers and the thin lines are the antiferromagnetic coupling of trimers and spins.

Figure 5

A schematic magnetization curve of the UJ ladder (1). $J_2^{c1}$ and $J_2^{c2}$ represent quantum critical points. The SMTLL phase exists in the region $J_2^{c1}<J_2<J_2^{c2}$.