Translation-Invariant Parent Hamiltonians of Valence Bond Crystals

We present a general method to construct translation-invariant and SU(2) symmetric antiferromagnetic parent Hamiltonians of valence bond crystals (VBCs). The method is based on a canonical mapping transforming $S=1/2$ spin operators into a bilinear form of a new set of dimer fermion operators. We construct parent Hamiltonians of the columnar and the staggered VBCs on the square lattice, for which the VBC is an eigenstate in all regimes and the exact ground state in some region of the phase diagram. We study the departure from the exact VBC regime upon tuning the anisotropy by means of the hierarchical mean field theory and exact diagonalization on finite clusters. In both Hamiltonians, the VBC phase extends over the exact regime and transits to a columnar antiferromagnet (CAFM) through a window of intermediate phases, revealing an intriguing competition of correlation lengths at the VBC-CAFM transition. The method can be readily applied to construct other VBC parent Hamiltonians in different lattices and dimensions.

Figure 1

Schematic representation of the SVBC (a) and CVBC (b) states and their respective parent Hamiltonians, together with their phase diagrams, (c) and (d). Green ovals represent dimers, thick solid lines represent the Heisenberg interaction among NN spins along the $x$ (black line) and $y$ (red line) directions, second-NN spins (dotted blue line), and a subset of fourth-NN spins (dashed green line). Pairs of orange wavy and spring-shaped lines represent anisotropic four-spin interactions.

Figure 2

Scaling of the GS phase diagram of the SVBC parent Hamiltonian Eq. (12) as computed with (a) ED and (b) HMFT. Inset (a): Singlet-triplet gap computed with ED at $J_y/J_x=1.4$. (c) Energy per site (units of $J_x$) computed with HMFT for different cluster sizes of rhomboid shape. (d) Magnetization $M$ and dimerization $\mathcal{D}$ as computed with 24-HMFT.

Figure 3

Scaling of the GS phase diagram of the CVBC parent Hamiltonian Eq. (13) as computed with (a) ED and (b) HMFT. Inset (a) Singlet-triplet gap computed with ED at $J_y/J_x=1$. (c) Energy per site (units of $J_x$) computed with HMFT for different cluster sizes of square shape. (d) Magnetization $M$ as computed with $6\times 4$-HMFT.