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arXiv:1508.03451 · Physical review. B./Physical review. B

Emergent spin-1 trimerized valence bond crystal in the spin- 12 Heisenberg model on the star lattice

Shi-Ju Ran, Wei Li, Shou-Shu Gong, Andreas Weichselbaum, Jan von Delft, Gang Su

Published Feb 26, 2018• 23 citations
arXivHTMLPDF

Abstract

We explore the frustrated spin-$\frac{1}{2}$ Heisenberg model on the star lattice with antiferromagnetic (AF) couplings inside each triangle and ferromagnetic (FM) intertriangle couplings (${J}_{e}<0$), and calculate its magnetic and thermodynamic properties. We show that the FM couplings do not sabotage the magnetic disordering of the ground state due to the frustration from the AF interactions inside each triangle, but trigger a fully gapped inversion-symmetry-breaking trimerized valence bond crystal (TVBC) with emergent spin-1 degrees of freedom. We discover that with strengthening ${J}_{e}$, the system exhibits a universal scaling behavior either with or without a magnetic field $h$: the order parameter, the five critical fields that separate the ${J}_{e}\text{\ensuremath{-}}h$ ground-state phase diagram into six phases, and the excitation gap obtained by low-temperature specific heat, all depend exponentially on ${J}_{e}$. Our work implies that the spin-1 VBCs can be stabilized by introducing small FM couplings in the geometrically frustrated spin-$\frac{1}{2}$ systems.

DOI
10.1103/physrevb.97.075146

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We investigate the ground state phase diagram of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the star lattice using infinite projected entangled pair states (iPEPS) and high-order series expansions. The model includes two distinct couplings: Jd{J}_{d}Jd​ on the dimer bonds and Jt{J}_{t}Jt​ on the trimer bonds. While it is established that the system hosts a valence bond solid (VBS) phase for ${J}_{d}\ensuremath{\ge}{J}_{t}$, the ground state phase diagram for Jd<Jt{J}_{d}<{J}_{t}Jd​<Jt​ has remained unsettled. Our iPEPS simulations uncover a first-order phase transition at ${J}_{d}/{J}_{t}\ensuremath{\approx}0.18$, significantly lower than previously reported estimates. Beyond this transition, we identify a close competition between two valence bond crystal (VBC) states; a columnar VBC and a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC, with the latter consistently exhibiting lower energy across all finite bond dimensions. The high-order series expansion supports this by finding that the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC state indeed becomes energetically favorable, but only at sixth order in perturbation theory, revealing the subtle nature of the competition between candidate states.
The spin-12\frac{1}{2}21​ Ising-Heisenberg model on two geometrically related triangles-in-triangles lattices is exactly solved through the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the effective spin-12\frac{1}{2}21​ Ising model on a triangular lattice. Basic thermodynamic quantities were exactly calculated within this rigorous mapping method along with the ground-state and finite-temperature phase diagrams. Apart from the classical ferromagnetic phase, both investigated models exhibit several unconventional quantum ordered and disordered ground states. A mutual competition between two ferromagnetic interactions of basically different character generically leads to the emergence of a quantum ferromagnetic phase, in which a symmetric quantum superposition of three up-up-down states of the Heisenberg trimers accompanies a perfect alignment of all Ising spins. In the highly frustrated regime, we have either found the disordered quantum paramagnetic phase with a substantial residual entropy or a similar but spontaneously ordered phase with a nontrivial criticality at finite temperatures. The latter quantum ground state exhibits a striking coexistence of imperfect spontaneous order with partial disorder, which is evidenced by a quantum reduction of the spontaneous magnetization of Heisenberg spins that indirectly causes a small reduction of the spontaneous magnetization of otherwise classical Ising spins.
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The interplay of frustrated interactions and lattice geometry can lead to a variety of exotic quantum phases. Here we unearth a particularly rich phase diagram of the Kitaev-Heisenberg model on the star lattice, a triangle decorated honeycomb lattice breaking sublattice symmetry. In the antiferromagnetic regime, the interplay of Heisenberg coupling and geometric frustration leads to the formation of valence bond solid (VBS) phases—a singlet VBS and a bond selective triplet VBS stabilized by the Kitaev exchange. We show that the ratio of the Kitaev versus Heisenberg exchange tunes between these VBS phases and chiral quantum spin-liquid regimes. Remarkably, the VBS phases host a whole variety of chiral triplon excitations with high Chern numbers in the presence of a weak magnetic field. We discuss our results in light of a recently synthesized star lattice material and other decorated lattice systems. Published by the American Physical Society 2024
The nature of the 1/9-magnetization plateau of the spin-12\frac{1}{2}21​ kagome Heisenberg antiferromagnet remains controversial due to the exotic physical properties and high complexity induced by the geometrical frustration. Instead of a Z3{\mathbb{Z}}_{3}Z3​ quantum spin liquid revealed on a cylinder, we show on an infinite-size lattice that the 1/9 plateau can be described by a valence bond crystal (VBC) that breaks spatial translational invariance. Consistent results are achieved by two accurate tensor network methods, namely the full-update infinite projected-entangled pair states and the projected-entangled simplex states. The VBC exhibits an hourglass pattern with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ spatial symmetry, demonstrated by magnetizations, bond energies, and three-body correlators. The spatial inversion symmetry in the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC is instantly broken with the presence of the difference between the coupling strengths in the up and down triangles, suggesting the existence of gapless excitations. The gapless nature of the 1/9 plateau is further indicated by the scaling behaviors of the entanglement entropy and the correlation length, which indicate a c=1c=1c=1 conformal field theory.