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SU(2)-invariant spin-1/2 Hamiltonians with RVB and other valence bond phases

Raman, K. S., Roderich Moessner, Sondhi, S. L.

Published Jan 1, 2005• 1 citation
OpenAlex

Abstract

We construct a family of rotationally invariant, local, S=1/2 Klein Hamiltonians on various lattices that exhibit ground state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well understood quantum dimer models on the corresponding lattices. Specifically, we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of the reference lattice. This strategy leads to Hamiltonians that exhibit i) $Z_2$ RVB phases in two dimensions, ii) U(1) RVB phases with a gapless ``photon'' in three dimensions, and iii) a Cantor deconfined region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a $Z_2$ RVB phase and the other a U(1) RVB phase.

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Topological Z2 resonating-valence-bond spin liquid on the square lattice

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A one-parameter family of long-range resonating-valence-bond (RVB) state on the square lattice was previously proposed to describe a critical spin liquid (SL) phase of the spin-1/2 frustrated Heisenberg model. We provide evidence that this RVB state in fact also realizes a topological (long-range entangled) ${\mathbb{Z}}_{2}$ SL, limited by two transitions to critical SL phases. The topological phase is naturally connected to the ${\mathbb{Z}}_{2}$ gauge symmetry of the local tensor. This Rapid Communication shows that, on one hand, spin-1/2 topological SL with ${C}_{4v}$ point-group symmetry and $SU(2)$ spin rotation symmetry exists on the square lattice and, on the other hand, criticality and nonbipartiteness are compatible. We also point out that strong similarities between our phase diagram and the ones of classical interacting dimer models suggest both can be described by similar Kosterlitz-Thouless transitions. This scenario is further supported by the analysis of the one-dimensional boundary state. Forms of parent Hamiltonians hosting the ${\mathbb{Z}}_{2}$ SL are suggested.

arXiv:1709.10019•Physical review. B./Physical review. B•Apr 11, 2018•26 citations
We construct a family of rotationally invariant, local, S=1/2 Klein Hamiltonians on various lattices that exhibit ground state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well understood quantum dimer models on the corresponding lattices. Specifically, we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of the reference lattice. This strategy leads to Hamiltonians that exhibit i) Z2Z_2Z2​ RVB phases in two dimensions, ii) U(1) RVB phases with a gapless ``photon'' in three dimensions, and iii) a Cantor deconfined region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a Z2Z_2Z2​ RVB phase and the other a U(1) RVB phase.
We construct a family of rotationally invariant, local, S=1∕2S=1∕2S=1∕2 Klein Hamiltonians on various lattices that exhibit ground-state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well-understood quantum dimer models on the corresponding lattices. Specifically, we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of the reference lattice. This strategy leads to Hamiltonians that exhibit (i) Z2{Z}_{2}Z2​ resonating valence bond (RVB) phases in two dimensions, (ii) U(1)U(1)U(1) RVB phases with a gapless ``photon'' in three dimensions, and (iii) a Cantor deconfined region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a Z2{Z}_{2}Z2​ RVB phase and the other a U(1)U(1)U(1) RVB phase.
We report recent analytical progress in the quest for spin models realising exotic phases. We focus on the question of ‘reverse‐engineering’ a local, SU(2) invariant S=1/2 Hamiltonian to exhibit phases predicted on the basis of effective models, such as large‐N or quantum dimer models. This aim is to provide a point‐of‐principle demonstration of the possibility of constructing such microscopic lattice Hamiltonians, as well as to complement and guide numerical (and experimental) approaches to the same question. In particular, we demonstrate how to utilise peturbed Klein Hamiltonians to generate effective quantum dimer models. These models use local multi‐spin interactions and, to obtain a controlled theory, a decoration procedure involving the insertion of Majumdar‐Ghosh chainlets on the bonds of the lattice. The phases we thus realise include deconfined resonating valence bond liquids, a devil’s staircase of interleaved phases which exhibits Cantor deconfinement, as well as a three‐dimensional U(1) liquid phase exhibiting photonic excitations.
We construct an exact spin-111 parent Hamiltonian for constrained valence-bond loop manifolds on the checkerboard and pyrochlore lattices. The Hamiltonian is local, SU(2)- and time-reversal-invariant, and built from positive-semidefinite projectors acting on triangular faces. Each projector removes only the maximally polarized state of a triangle, so the model is frustration-free. Its zero-energy states are generated by an AKLT-like construction in which each spin-111 moment is resolved into two virtual spin-12\tfrac1221​ degrees of freedom, singlets are formed inside every crossed plaquette or tetrahedron, and the physical spin-111 Hilbert space is recovered by projection. The resulting ground states are fully packed singlet-loop states on the corner-sharing lattice. Thus, a loop or dimer constraint, usually introduced as part of an effective Rokhsar--Kivelson description, appears here as the exact zero-energy manifold of a microscopic spin Hamiltonian. We analyze spin correlations within this manifold and show that, for a fixed loop covering, they are determined by loop connectivity. We also project symmetry-allowed perturbations into the ground-state manifold and derive the resulting low-energy pseudospin dynamics. The checkerboard and pyrochlore cases differ sharply. On the pyrochlore lattice, tetrahedral symmetry removes simple local bias terms, and the leading nontrivial next-nearest-neighbour Heisenberg perturbation gives an emergent spin-12\tfrac1221​ XY model on the diamond lattice of tetrahedron centers. These results give an exact spin-111 microscopic starting point for constrained valence-bond physics in two and three dimensions, and show how loop, dimer, and gauge-theoretic descriptions can be approached from a small-spin, SU(2)-invariant frustrated magnet.
A class of local SU(2)SU(2)SU(2)-invariant spin-12\frac{1}{2}21​ Hamiltonians is studied that has ground states within the space of nearest-neighbor valence-bond states on the kagome lattice. Cases include ``generalized Klein'' models without obvious non-valence-bond ground states, as well as a ``resonating valence-bond'' Hamiltonian whose unique ground states within the nearest-neighbor valence-bond space are four topologically degenerate ``Sutherland-Rokhsar-Kivelson'' (SRK) -type wave functions, which are expected to describe a gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquid. The proof of this uniqueness is intimately related to the linear independence of the nearest-neighbor valence-bond states on quite general and arbitrarily large kagome lattices, which is also established in this work. It is argued that the SRK ground states are also unique within the entire Hilbert space depending on properties of the generalized Klein models. Applications of the strategies developed in this work to other lattice types are also discussed.
A one-parameter family of long-range resonating-valence-bond (RVB) state on the square lattice was previously proposed to describe a critical spin liquid (SL) phase of the spin-1/2 frustrated Heisenberg model. We provide evidence that this RVB state in fact also realizes a topological (long-range entangled) Z2{\mathbb{Z}}_{2}Z2​ SL, limited by two transitions to critical SL phases. The topological phase is naturally connected to the Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry of the local tensor. This Rapid Communication shows that, on one hand, spin-1/2 topological SL with C4v{C}_{4v}C4v​ point-group symmetry and SU(2)SU(2)SU(2) spin rotation symmetry exists on the square lattice and, on the other hand, criticality and nonbipartiteness are compatible. We also point out that strong similarities between our phase diagram and the ones of classical interacting dimer models suggest both can be described by similar Kosterlitz-Thouless transitions. This scenario is further supported by the analysis of the one-dimensional boundary state. Forms of parent Hamiltonians hosting the Z2{\mathbb{Z}}_{2}Z2​ SL are suggested.