Spontaneous Parity Breaking in Quantum Antiferromagnets on the Triangular Lattice

Abstract

Frustration on the triangular lattice has long been a source of intriguing and often debated phases in many-body systems. Although symmetry analysis has been employed, the role of the seemingly trivial parity symmetry has received little attention. In this work, we show that phases induced by frustration are systematically shaped by an implicit rule of thumb associated with spontaneous parity breaking. This principle enables us to anticipate and rationalize the regimes and conditions under which nontrivial phases emerge. For the spin-$S$ antiferromagnetic XXZ model, we demonstrate that a controversial parity-broken phase appears only at intermediate values of $S$. In bilayer systems, enhanced frustration leads to additional phases, such as supersolids, whose properties can be classified by their characteristic parity features. Benefiting from our improved tensor network contraction techniques, we confirm these results through large-scale tensor-network calculations. This study offers an alternative viewpoint and a systematic approach for examining the interplay between spin, symmetry, and frustration in many-body systems.

Figures (3)

• Figure 1: CTMRG procedure for the triangular lattice. (a–c) Three blocks arising from the triangular-lattice periodicity, each composed of local tensors $M_i$ ($i$ = $A$, $B$, $C$) together with their associated edge tensors $L_i^\mu$ ($\mu =$$l$, $d$, $r$, $u$) and corner tensors $L_i^{\mu\nu}$ ($\mu\nu =$$lu$, $ld$, $rd$, $ru$). (d) Isometries $U_n^\mu$ ($n$ = 1, 2, 3) obtained from singular-value decompositions at the dashed cuts in (a-c) are shown. Solid triangles indicate the position of $U_n^\mu$ for the next step. (e–g) Update of the environment tensors through contractions with $U_n^\mu$ and $U_n^{\mu\dagger}$, yielding the renormalized $L_i^{\mu *}$ and $L_i^{\mu\nu *}$.
• Figure 2: Ground state phase diagrams of the monolayer TLXXZ model for $S =$ 1/2, 1, and 5/2, shown in panels (a-c), respectively. Circles with error bars are the calculated boundaries, while solid curves are guides to the eye. Thick black lines denote parity-breaking phase transitions. Arrows represent spin configurations (FM, Umbrella, V, Y, and UUD) on the $A$, $B$, and $C$ sublattices.
• Figure 3: Magnetization $M_z$ of the bilayer TLXXZ model with varied $h_z$. Inset (a) illustrates the lattice geometry and couplings, and insets (b–i) show representative spin configurations of each phase. The table lists the numbers of singlets (S) and triplets (T), as well as whether parity a and parity b are preserved.