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:1711.10717 · Physical review. B./Physical review. B

Entanglement studies of resonating valence bonds on the frustrated square lattice

Julia Wildeboer, Alexander Seidel

Published Jun 3, 2020• 1 citation
arXivHTMLPDF

Abstract

We study a short-range resonating valence bond (RVB) wave function with diagonal links on the square lattice that permits sign-problem free wave function Monte Carlo studies. Special attention is given to entanglement properties, in particular, the study of minimum entropy states (MES) according to the method of Zhang et al. [Phys. Rev. B 85, 235151 (2012)]. We provide evidence that the MES associated with the RVB wave functions can be lifted from an associated quantum dimer picture of these wave functions, where MES states are certain linear combinations of eigenstates of a 't Hooft ``magnetic loop''-type operator. From this identification, we calculate a value consistent with $ln(2)$ for the topological entanglement entropy directly for the RVB states via wave function Monte Carlo. This corroborates the ${\mathbb{Z}}_{2}$ nature of the RVB states. We furthermore define and elaborate on the concept of a ``pre-Kasteleyn'' orientation that may be useful for the study of lattices with nonplanar topology in general.

DOI
10.1103/physrevb.101.224406

Related papers

Topological Z2 resonating-valence-bond quantum spin liquid on the ruby lattice

Saeed S. Jahromi, Román Orús

We construct a short-range resonating-valence-bond (RVB) state on the ruby lattice, using projected entangled-pair states with bond dimension $D=3$. By introducing nonlocal moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining a nonzero topological contribution, $\ensuremath{\gamma}=\ensuremath{-}\mathrm{ln}\phantom{\rule{4pt}{0ex}}2$, consistent with that of a ${\mathbb{Z}}_{2}$ topological quantum spin liquid. We also calculate the ground-state energy of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function.

arXiv:1912.06215•Physical review. B./Physical review. B•Mar 9, 2020•13 citations

Topological and entanglement properties of resonating valence bond wave functions

Didier Poilblanc, Norbert Schuch, David Pérez-Garcı́a +1 more

We examine in details the connections between topological and entanglement properties of short-range resonating valence bond (RVB) wave functions using projected entangled pair states (PEPS) on kagome and square lattices on (quasi)infinite cylinders with generalized boundary conditions (and perimeters with up to 20 lattice spacings). By making use of disconnected topological sectors in the space of dimer lattice coverings, we explicitly derive (orthogonal) ``minimally entangled'' PEPS RVB states. For the kagome lattice, using the quantum Heisenberg antiferromagnet as a reference model, we obtain the finite-size scaling with increasing cylinder perimeter of the vanishing energy separations between these states. In particular, we extract two separate (vanishing) energy scales corresponding (i) to insert a vison line between the two ends of the cylinder and (ii) to pull out and freeze a spin at either end. We also investigate the relations between bulk and boundary properties and show that, for a bipartition of the cylinder, the boundary Hamiltonian defined on the edge can be written as a product of a highly nonlocal projector, which fundamentally depends upon boundary conditions, with an emergent (local) SU$(2)$-invariant one-dimensional (superfluid) $t$-$J$ Hamiltonian, which arises due to the symmetry properties of the auxiliary spins at the edge. This multiplicative structure, a consequence of the disconnected topological sectors in the space of dimer lattice coverings, is characteristic of the topological nature of the states. For minimally entangled RVB states, it is shown that the entanglement spectrum, which reflects the properties of the (gapless or gapped) edge modes, is a subset of the spectrum of the local Hamiltonian, e.g., half of it for the kagome RVB state, providing a simple argument on the origin of the topological entanglement entropy ${S}_{0}=\ensuremath{-}\mathrm{ln}2$ of the ${\mathbb{Z}}_{2}$ spin liquid. We propose to use these features to probe topological phases in microscopic Hamiltonians, and some results are compared to existing density matrix renormalization group data.

arXiv:1202.0947•Physical Review B•Jul 6, 2012•140 citations

Frustrated RVB states in 2D: classifications and short-range correlations

Fan Yang, Hong Yao

Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (non-bipartite or $Z_2$) bosonic RVB states we studied, indicating that they are gapped $Z_2$ quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.

arXiv:1204.6381•Apr 28, 2012•1 citation

Frustrated Resonating Valence Bond States in Two Dimensions: Classification and Short-Range Correlations

Fan Yang, Hong Yao

Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (nonbipartite or Z2) bosonic RVB states we studied, indicating that they are gapped Z2 quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.

Physical Review Letters•Oct 5, 2012•47 citations

SU(4) topological resonating valence bond spin liquid on the square lattice

Olivier Gauthé, Sylvain Capponi, Didier Poilblanc

We generalize the construction of the spin-1/2 SU(2) resonating valence bond (RVB) state to the case of the self-conjugate $\mathbf{6}$ representation of SU(4). As for the case of SU(2) [J.-Y. Chen and D. Poilblanc, Phys. Rev. B 97, 161107(R) (2018)], we use the projected entangled pair state (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to ${\mathbb{Z}}_{2}$, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is discussed.

arXiv:1901.05905•Physical review. B./Physical review. B•Jun 26, 2019•8 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 study a short-range resonating valence bond (RVB) wave function with diagonal links on the square lattice that permits sign-problem free wave function Monte Carlo studies. Special attention is given to entanglement properties, in particular, the study of minimum entropy states (MES) according to the method of Zhang et al. [Phys. Rev. B 85, 235151 (2012)]. We provide evidence that the MES associated with the RVB wave functions can be lifted from an associated quantum dimer picture of these wave functions, where MES states are certain linear combinations of eigenstates of a 't Hooft magnetic loop''-type operator. From this identification, we calculate a value consistent with $ln(2)$ for the topological entanglement entropy directly for the RVB states via wave function Monte Carlo. This corroborates the ${\mathbb{Z}}_{2}$ nature of the RVB states. We furthermore define and elaborate on the concept of a pre-Kasteleyn'' orientation that may be useful for the study of lattices with nonplanar topology in general.
We construct a short-range resonating-valence-bond (RVB) state on the ruby lattice, using projected entangled-pair states with bond dimension D=3D=3D=3. By introducing nonlocal moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining a nonzero topological contribution, $\ensuremath{\gamma}=\ensuremath{-}\mathrm{ln}\phantom{\rule{4pt}{0ex}}2$, consistent with that of a Z2{\mathbb{Z}}_{2}Z2​ topological quantum spin liquid. We also calculate the ground-state energy of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function.
We examine in details the connections between topological and entanglement properties of short-range resonating valence bond (RVB) wave functions using projected entangled pair states (PEPS) on kagome and square lattices on (quasi)infinite cylinders with generalized boundary conditions (and perimeters with up to 20 lattice spacings). By making use of disconnected topological sectors in the space of dimer lattice coverings, we explicitly derive (orthogonal) ``minimally entangled'' PEPS RVB states. For the kagome lattice, using the quantum Heisenberg antiferromagnet as a reference model, we obtain the finite-size scaling with increasing cylinder perimeter of the vanishing energy separations between these states. In particular, we extract two separate (vanishing) energy scales corresponding (i) to insert a vison line between the two ends of the cylinder and (ii) to pull out and freeze a spin at either end. We also investigate the relations between bulk and boundary properties and show that, for a bipartition of the cylinder, the boundary Hamiltonian defined on the edge can be written as a product of a highly nonlocal projector, which fundamentally depends upon boundary conditions, with an emergent (local) SU(2)(2)(2)-invariant one-dimensional (superfluid) ttt-JJJ Hamiltonian, which arises due to the symmetry properties of the auxiliary spins at the edge. This multiplicative structure, a consequence of the disconnected topological sectors in the space of dimer lattice coverings, is characteristic of the topological nature of the states. For minimally entangled RVB states, it is shown that the entanglement spectrum, which reflects the properties of the (gapless or gapped) edge modes, is a subset of the spectrum of the local Hamiltonian, e.g., half of it for the kagome RVB state, providing a simple argument on the origin of the topological entanglement entropy ${S}_{0}=\ensuremath{-}\mathrm{ln}2$ of the Z2{\mathbb{Z}}_{2}Z2​ spin liquid. We propose to use these features to probe topological phases in microscopic Hamiltonians, and some results are compared to existing density matrix renormalization group data.
Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (non-bipartite or Z2Z_2Z2​) bosonic RVB states we studied, indicating that they are gapped Z2Z_2Z2​ quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.
Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (nonbipartite or Z2) bosonic RVB states we studied, indicating that they are gapped Z2 quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.
We generalize the construction of the spin-1/2 SU(2) resonating valence bond (RVB) state to the case of the self-conjugate 6\mathbf{6}6 representation of SU(4). As for the case of SU(2) [J.-Y. Chen and D. Poilblanc, Phys. Rev. B 97, 161107(R) (2018)], we use the projected entangled pair state (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to Z2{\mathbb{Z}}_{2}Z2​, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is 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.