Magnetization of the antiferromagnetic cactus graph model with an exact dimer ground state

We introduce and explore the magnetization behavior in a quantum spin system on a cactus graph, deemed the cactus graph model, featuring an exact dimer singlet ground state. We analyze the singlet-triplet gap, interactions among excited triplets, and correlated hopping under an external magnetic field using a strong coupling expansion. Employing an effective hard-core boson representation, we unveil density wave magnetization plateaus and supersolid phases, revealing the intricate interplay between interactions and hopping dynamics. Furthermore, we conjecture that correlated hopping gives rise to multitriplet bound states, and potentially engendering to the stabilization of additional low-lying magnetization plateaus.

Figure 1

The cactus graph model (CGM): a graph structure constructed from a Bethe lattice of degree $z=4$. The construction involves generating a graph of quadrangles with shared corners. The depicted diagonals represent the dimer bonds of the model. The singlets residing on the dimer bonds represent an eigenstate of (1).

Figure 2

(a) A speculative quantum phase diagram of (1). The schematics illustrate three states: the exact dimer singlet state, the plaquette singlet state, and the Néel phase. The purple ellipses represent exact dimer singlets, and the orange quadrangles represent plaquette singlets. Like the other exact dimer models [13, 14, 15], the transition out of the dimer singlet phase is likely to be first order. The phase transition from the plaquette singlet phase to the Néel phase may occur at a deconfined quantum critical point [23] or they could be separated by an intermediate spin liquid phase [24]. (b) For the cactus graph model with classical spins, i.e., $S\to \infty$, there exists an incommensurate spiral state for $J^{\prime }<J$, which is different from the Néel state. This state is highly degenerate and incommensurate except for some particular ratios of $J^{\prime }/J$ where $arccos(J^{\prime }/J)$ is a rational fraction of $\pi$. One such configuration for $J^{\prime }=J/2$ is shown.

Figure 3

(a) The singlet's parity and the matrix elements of the $J^{\prime }$ terms [37] restrict the hopping of the triplets only to two of its four adjacent dimers. The possible hopping directions are indicated using arrows on the edges. (b) The interactions between pairs of triplets (reference triplet in black) in the cactus graph model. The interactions are determined within third-order perturbation theory. (c) Correlated hopping in cactus graph model, which is the movement of one triplet (filled in purple or orange) assisted by the other (filled in black).

Figure 4

Magnetization phase diagram of the spin Hamiltonian achieved via Matsubara-Matsuda transformation of the effective hard-core boson Hamiltonian (15) (a) without and (b) with correlated hopping for a 1457-site Cayley tree. The diagrams highlight magnetization plateaus at specific fractional values. Correlated hopping introduces additional supersolid phases that influence the stability of density wave magnetization plateaus. The wavy patterned regions are not accessible within our current calculations. (c) The magnetization as a function of $B$ for fixed $J^{\prime }=0.6J$. The supersolid phase shows continuous growth of magnetization with increasing magnetic field until a new plateau state is reached. (d) The density waves and a representative supersolid state (central 485 sites are shown). The color represents the local densities of the hard-core bosons. The density wave phase exhibits a regular spatial pattern, indicating a stable arrangement of particles. The supersolid phase demonstrates the coexistence of density wave behavior and superfluidity.

Figure 5

(a) Correlated hopping facilitates the motion of two excited triplets (filled in red), leading to two-triplet bound states indicated in light yellow. The vertical bars next to the two-triplet states represent their corresponding contribution to the two-triplet bound-state wave function (for $J^{\prime }=0.6J$), red for positive and blue for negative. (b) The other two-triplet state (marked in light yellow) has an energy lower than $2\mathrm{\Delta }$. The contributions of the corresponding basis states for $J^{\prime }=0.6J$ are also shown in the same convention as in (a). (c) The dot-dashed line represents $\mathrm{\Delta }$, while the dashed line represents the energy associated with two noninteracting dimer triplets ($2\mathrm{\Delta }$), and the solid line corresponds to the energy of a bound state of two triplets, $\mathrm{\Delta }{E}_{2P}$, (in the unit of $J$) as a function of $J^{\prime }/J$. Example of (d) an extended multitriplet bound state (e) a crystal of two-triplet bound states.