Geometric phases and the magnetization process in quantum antiferromagnets

The physics underlying the magnetization process of quantum antiferromagnets is revisited from the viewpoint of geometric phases. A continuum variant of the Lieb-Schultz-Mattis-type approach to the problem is put forth, where the commensurability condition of Oshikawa et al. derives from a Berry connection formulation of the system’s crystal momentum. We then go on to formulate an effective field theory which can deal with higher dimensional cases as well. We find that a topological term, whose principle function is to assign Berry phase factors to space-time vortex objects, ultimately controls the magnetic behavior of the system. We further show how our effective action maps into a ${\mathbf{Z}}_2$ gauge theory under certain conditions, which in turn allows for the occurrence of a fractionalized phase with topological order.

Figure 1

(Color online) Schematic view of the slowly varying angular variables $\left\{{\varphi }_j\right\}$, and the magnetization per site $m$.

Figure 2

(Color online) A vortex loop event $C_j$ in $\left(2+1\right)d$. Projected onto the $xy$ plane, this corresponds to a vortex-antivortex pair created at point 1, which pair-annihilates at point 2.

Figure 3

(Color online) A typical spin state found at an $m=1/2$ plateau in spin-1 chains $\left(S-m=1/2\right)$. Lattice translation symmetry is broken spontaneously and as a consequence we have two degenerate ground states (a) and (b). Half (arrows) of each spin-1 degree of freedom is frozen by a strong magnetic field and the remaining fluctuating half forms valence-bond solid states (Ref. 57). The lowest excitation is a kink which carries an $S^z$ quantum number of 1/2.

Figure 4

(Color online) Two-dimensional analog of the spin-1 partially-polarized valence-bond-solid state shown in Fig. 3. Again, the unpolarized “fractions” of the spin-1’s form a valence-bond pattern. Dashed lines denote domain walls separating four different valence-bond patterns. At the core of the ${\mathbb{Z}}_4$-vortex is a fractionalized excitation carrying $S^z=1/2$ (large arrow). The resemblance of the spatial patterns as well as the fractionalized nature of the core excitation with those of Refs. 58, 59 is evident.

Figure 5

Schematic phase diagram of the fractionalized boson coupled to ${\mathbb{Z}}_2$-gauge theory for $\left(1+1\right)d$ (inset) and $\left(2+1\right)d$. The large-$K_{XY}$ region is dominated by an $XY$-ordered [or, superfluid (SF)] phase, which, in $\left(1+1\right)d$, is replaced with an algebraic Tomonaga-Luttinger (TL) liquid phase. In the small-$K_{XY}$ plateaus region, there is always a valence-bond (VB) phase with broken translation symmetry. Both in $\left(1+1\right)d$ and in $\left(2+1\right)d$, the VB phase supports topological excitations [kinks in $\left(1+1\right)d$ and ${\mathbb{Z}}_4$ vortices in $\left(2+1\right)d$]. On top of this, there is a featureless plateau phase with fractionalized (i.e., $\delta S^z=1/2$) bosons in $\left(2+1\right)d$.