Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0502146 · Physical Review B

SU(2)-invariant spin- 12 Hamiltonians with resonating and other valence bond phases

Kumar S. Raman, Roderich Moessner, Shivaji L. Sondhi

Published Aug 5, 2005• 67 citations
arXivHTMLPDF

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) ${Z}_{2}$ resonating valence bond (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.

DOI
10.1103/physrevb.72.064413

Related papers

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

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

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.

•Jan 1, 2005•1 citation

From exotic phases to microscopic Hamiltonians

Roderich Moessner

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.

arXiv:cond-mat/0510498•AIP conference proceedings•Jan 1, 2006•1 citation

Three-dimensional resonating-valence-bond liquids and their excitations

Roderich Moessner, Shivaji L. Sondhi

We show that there are two types of resonating-valence-bond (RVB) liquid phases present in three-dimensional quantum dimer models, corresponding to the deconfining phases of $U(1)$ and ${Z}_{2}$ gauge theories in $d=3+1.$ The former is found on the bipartite cubic lattice and is the generalization of the critical point in the square lattice quantum dimer model found originally by Rokhsar and Kivelson. The latter exists on the nonbipartite face-centered cubic lattice and generalizes the RVB phase found earlier by us on the triangular lattice. We discuss the excitation spectrum and the nature of the ordering in both cases. Both phases exhibit gapped spinons. In the $U(1)$ case we find a collective, linearly dispersing, transverse excitation, which is the photon of the low-energy Maxwell Lagrangian and we identify the ordering as quantum order in Wen's sense. In the ${Z}_{2}$ case all collective excitations are gapped and, as in $d=2,$ the low-energy description of this topologically ordered state is the purely topological BF action. As a byproduct of this analysis, we unearth a further gapless excitation, the pi0n, in the square lattice quantum dimer model at its critical point.

arXiv:cond-mat/0307592•Physical review. B, Condensed matter•Nov 18, 2003•170 citations

Spin Hamiltonians with Resonating-Valence-Bond Ground States

Jennifer Cano, Paul Fendley

Quantum dimer models exhibit quantum critical points and liquid states when the ground state is the resonating-valence bond (RVB) state. We construct SU(2)-invariant spin-1/2 Hamiltonians with the same RVB ground state. The main technical obstacle overcome is the fact that different dimer configurations in the spin model are not orthogonal. We show that the physics depends on how dimers are related to the spins, and find a Hamiltonian that is likely to be quantum critical.

arXiv:0910.5708•Physical Review Letters•Aug 5, 2010•37 citations

Entanglement in gapless resonating-valence-bond states

Jean-Marie Stéphan, Hyejin Ju, Paul Fendley +1 more

We study resonating-valence-bond (RVB) states on the square lattice of spins and of dimers, as well as SU ( N )-invariant states that interpolate between the two. These states are ground states of gapless models, although the SU (2)-invariant spin RVB state is also believed to be a gapped liquid in its spinful sector. We show that the gapless behavior in spin and dimer RVB states is qualitatively similar by studying the Rényi entropy for splitting a torus into two cylinders. We compute this exactly for dimers, showing it behaves similarly to the familiar one-dimensional log term, although not identically. We extend the exact computation to an effective theory believed to interpolate among these states. By numerical calculations for the SU (2) RVB state and its SU ( N )-invariant generalizations, we provide further support for this belief. We also show how the entanglement entropy behaves qualitatively differently for different values of the Rényi index n , with large values of n proving a more sensitive probe here, by virtue of exhibiting a striking even/odd effect.

arXiv:1207.3820•New Journal of Physics•Jan 11, 2013•38 citations

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

Ji-Yao Chen, Didier Poilblanc

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∕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 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 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 show that there are two types of resonating-valence-bond (RVB) liquid phases present in three-dimensional quantum dimer models, corresponding to the deconfining phases of U(1)U(1)U(1) and Z2{Z}_{2}Z2​ gauge theories in d=3+1.d=3+1.d=3+1. The former is found on the bipartite cubic lattice and is the generalization of the critical point in the square lattice quantum dimer model found originally by Rokhsar and Kivelson. The latter exists on the nonbipartite face-centered cubic lattice and generalizes the RVB phase found earlier by us on the triangular lattice. We discuss the excitation spectrum and the nature of the ordering in both cases. Both phases exhibit gapped spinons. In the U(1)U(1)U(1) case we find a collective, linearly dispersing, transverse excitation, which is the photon of the low-energy Maxwell Lagrangian and we identify the ordering as quantum order in Wen's sense. In the Z2{Z}_{2}Z2​ case all collective excitations are gapped and, as in d=2,d=2,d=2, the low-energy description of this topologically ordered state is the purely topological BF action. As a byproduct of this analysis, we unearth a further gapless excitation, the pi0n, in the square lattice quantum dimer model at its critical point.
We study resonating-valence-bond (RVB) states on the square lattice of spins and of dimers, as well as SU ( N )-invariant states that interpolate between the two. These states are ground states of gapless models, although the SU (2)-invariant spin RVB state is also believed to be a gapped liquid in its spinful sector. We show that the gapless behavior in spin and dimer RVB states is qualitatively similar by studying the Rényi entropy for splitting a torus into two cylinders. We compute this exactly for dimers, showing it behaves similarly to the familiar one-dimensional log term, although not identically. We extend the exact computation to an effective theory believed to interpolate among these states. By numerical calculations for the SU (2) RVB state and its SU ( N )-invariant generalizations, we provide further support for this belief. We also show how the entanglement entropy behaves qualitatively differently for different values of the Rényi index n , with large values of n proving a more sensitive probe here, by virtue of exhibiting a striking even/odd effect.
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.