Exact plaquette singlet phases in an orthogonal-plaquette model

We introduce a quantum spin-$1/2$ model hosting two exact plaquette singlet ground states in extended parameter regimes. There is an exact phase transition between both phases, at which the system has an extensive ground-state degeneracy. Further, the model exhibits an extensive number of conserved quantities, which allow the prediction of additional phases. We exploit this feature and explore the phase diagram in detail for two specific parameter regimes. The model is based solely on Heisenberg interactions and seems sufficiently simple to be realized in quantum materials. A general scheme to determine exact singlet product states is briefly discussed.

Figure 1

(a) Orthogonal-plaquette model Eq. (1) realizing exact plaquette product ground states with singlets on four-spin $J$ plaquettes (solid black lines). Red dots illustrate spins $1/2^{\prime }s$ interacting via Heisenberg interactions shown as lines. The orthogonal $J_h$ bonds (dashed purple) together with the in-plane $J^{\prime }$ bonds (solid blue) yield the SSM. (b) Bird's eye view of the orthogonal-plaquette model Eq. (1). The $J^{\prime }$ couplings present in the SSM form a square lattice. (c) A five-spin pyramid with halved intraplaquette couplings contains a four-spin plaquette $p$. The full model Eq. (1) can be decomposed into a sum of such pyramids Eq. (6).

Figure 2

Phase diagram of the orthogonal-plaquette model Eq. (1) for $J_h=J_v$. The exact plaquette singlet phases $|s_{\text{t}}=0,s_{\text{v}}=1\rangle$ (light-green region) and $|s_{\text{t}}=0,s_{\text{v}}=0\rangle$ (light-red region) are analytically proven to occur in the areas below the dashed black line. The light-yellow and white areas yield the possible extension of an exact dimer singlet state $s_{\text{v}}=0$ ($s_{\text{v}}^p=0\forall p$) in a product with an EPP or AFM on the Shastry-Sutherland lattice, respectively. All dotted lines are only sketched. For the phase transition between $|s_{\text{v}}=0;\text{EPP,}\text{SSM}\rangle$ and $|s_{\text{v}}=0;\text{AFM,}\text{SSM}\rangle$ as well as $|s_{\text{t}}=0,s_{\text{v}}=0\rangle$, numerical errors from iPEPS are also included [38]. For the phase transition between $|s_{\text{v}}=0;\text{EPP,}\text{SSM}\rangle$ and $|s_{\text{t}}=0,s_{\text{v}}=1\rangle$, the uncertainties from the series expansions of the EPP ground-state energy are neglectable on this scale and are therefore not shown. The AFM with $s_{\text{v}}=1$ ($s_{\text{v}}^p=1\forall p$, light-blue region) is included, but its existence is only clear in the limit $J^{\prime }\gg J,J_h,J_v$. There might be other phases at intermediate coupling values. The given quantum numbers for all phases are exact. The phases are depicted such that couplings with vanishing contributions are not shown. Entangled local units are shaded blue for singlets and red for triplets. They are all exactly apart from the plaquettes in $|s_{\text{v}}=0;\text{EPP,}\text{SSM}\rangle$.

Figure 3

Phase diagram of the orthogonal-plaquette model Eq. (1) with $J_h=0$. The exact plaquette singlet phase $|s_{\text{t}}=0,s_{\text{v}}=1\rangle$ (light green) is analytically proven to occur in the area surrounded by the dashed black line. The light-orange and light-cyan areas yield the extension of products from dimer singlets $s_{\text{v}}=0$ with an AFM and FM, respectively. A pure FM is shown in light gray. There might be other phases in these areas. The AFM with $s_{\text{v}}=1$ ($s_{\text{v}}^p=1\forall p$, light-blue) is sketched, and is only clear in the limit $J^{\prime }\gg J,J_v$. Quantum numbers and correlations can be understood as in Fig. 2.