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:1606.02255

Comment on "$Z_{2}$ spin liquid phase on the kagome lattice: a new saddle point", by Tao Li [arXiv:1601.02165 (2016)]

Yasir Iqbal, Didier Poilblanc, Federico Becca

Published Jun 5, 2016• 6 citations
arXivHTMLPDF

Abstract

In a recent paper [arXiv:1601.02165], Tao Li claimed that a gapped $\mathbb{Z}_2$ spin liquid, obtained using projected Gutzwiller fermionic wave functions, can be stabilized in the Heisenberg model on the kagome lattice. We perform very accurate variational calculations and confirm with unprecedented accuracy the fact, that the $\mathbb{Z}_{2}$ spin liquid is a local energy minimum that goes away with system size, thus reaffirming the scenario of a gapless $U(1)$ Dirac spin liquid ground state.

DOI
10.48550/arxiv.1606.02255

Related papers

Projected wave function study of Z 2 spin liquids on the kagome lattice for the spin- 1 2 quantum Heisenberg antiferromagnet

Yasir Iqbal, Federico Becca, Didier Poilblanc

Motivated by recent density-matrix renormalization group (DMRG) calculations [Yan, Huse, and White, Science 332, 1173 (2011)], which claimed that the ground state of the nearest-neighbor spin-$1/2$ Heisenberg antiferromagnet on the kagome lattice geometry is a fully gapped spin liquid with numerical signatures of ${\mathbb{Z}}_{2}$ gauge structure, and a further theoretical work [Lu, Ran, and Lee, Phys. Rev. B 83, 224413 (2011)], which gave a classification of all Schwinger-fermion mean-field fully symmetric ${\mathbb{Z}}_{2}$ spin liquids on the kagome lattice, we have thoroughly studied Gutzwiller-projected fermionic wave functions by using quantum variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. In particular, we investigated the energetics of all ${\mathbb{Z}}_{2}$ candidates (gapped and gapless) that lie in the neighborhood of the energetically competitive U(1) gapless spin liquids. By using a state-of-the-art optimization method, we were able to conclusively show that the U(1) Dirac state is remarkably stable with respect to all ${\mathbb{Z}}_{2}$ spin liquids in its neighborhood, and in particular for opening a gap toward the so-called ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state, which was conjectured to describe the ground state obtained by the DMRG method. Finally, we also considered the addition of a small second nearest-neighbor exchange coupling of both antiferromagnetic and ferromagnetic type, and obtained similar results, namely, a U(1) Dirac spin-liquid ground state.

arXiv:1105.0341•Physical Review B•Jul 20, 2011•172 citations

Variational Monte Carlo study of a gapless spin liquid in the spin- 12 XXZ antiferromagnetic model on the kagome lattice

Wen-Jun Hu, Shou-Shu Gong, Federico Becca +1 more

By using the variational Monte Carlo technique, we study the spin-$\frac{1}{2}$ XXZ antiferromagnetic model (with easy-plane anisotropy) on the kagome lattice. A class of Gutzwiller projected fermionic states with a spin Jastrow factor is considered to describe either spin liquids [with $U(1)$ or ${Z}_{2}$ symmetry] or magnetically ordered phases [with $\mathbf{q}=(0,0)$ or $\mathbf{q}=(4\ensuremath{\pi}/3,0)$]. We find that the magnetic states are not stable in the thermodynamic limit. Moreover, there is no energy gain to break the gauge symmetry from $U(1)$ to ${Z}_{2}$ within the spin-liquid states, as previously found in the Heisenberg model. The best variational wave function is therefore the $U(1)$ Dirac state, supplemented by the spin Jastrow factor. Furthermore, a vanishing $S=2$ spin gap is obtained at the variational level, in the whole regime from the $XY$ to the Heisenberg model.

arXiv:1505.06276•Physical Review B•Nov 11, 2015•6 citations

Gapless spin-liquid phase in the kagome spin- 12 Heisenberg antiferromagnet

Yasir Iqbal, Federico Becca, Sandro Sorella +1 more

We study the energy and the static spin structure factor of the ground state of the spin-$1/2$ quantum Heisenberg antiferromagnetic model on the kagome lattice. By the iterative application of a few Lanczos steps on accurate projected fermionic wave functions and the Green's function Monte Carlo technique, we find that a gapless (algebraic) $U(1)$ Dirac spin liquid is competitive with previously proposed gapped (topological) ${\mathbb{Z}}_{2}$ spin liquids. By performing a finite-size extrapolation of the ground-state energy, we obtain an energy per site $E/J=\ensuremath{-}0.4365(2)$, which is equal, within three error bars, to the estimates given by the density-matrix renormalization group (DMRG). Our estimate is obtained for a translationally invariant system, and, therefore, does not suffer from boundary effects, like in DMRG. Moreover, on finite toric clusters at the pure variational level, our energies are lower compared to those from DMRG calculations.

arXiv:1209.1858•Physical Review B•Feb 14, 2013•331 citations

Z 2 spin liquids in the S = 1 2 Heisenberg model on the kagome lattice: A projective symmetry-group study of Schwinger fermion mean-field states

Yuan-Ming Lu, Ying Ran, Patrick A. Lee

Due to strong geometric frustration and quantum fluctuation, the $S=1/2$ quantum Heisenberg antiferromagnet on the $\text{kagome}$ lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg $S=1/2,\text{kagome}$ lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of ${\mathbb{Z}}_{2}$ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of ${\mathbb{Z}}_{2}$ SLs on the $\text{kagome}$ lattice. Among them we found only one gapped ${\mathbb{Z}}_{2}$ SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.

arXiv:1104.1432•Physical Review B•Jun 22, 2011•147 citations

Persistence of the gapless spin liquid in the breathing kagome Heisenberg antiferromagnet

Yasir Iqbal, Didier Poilblanc, Ronny Thomale +1 more

The nature of the ground state of the spin $S=1/2$ Heisenberg antiferromagnet on the kagome lattice with breathing anisotropy (i.e., with different superexchange couplings ${J}_{▵}$ and ${J}_{▿}$ within elementary up- and down-pointing triangles) is investigated within the framework of Gutzwiller projected fermionic wave functions and Monte Carlo methods. We analyze the stability of the $\text{U}(1)$ Dirac spin liquid with respect to the presence of fermionic pairing that leads to a gapped ${\mathbb{Z}}_{2}$ spin liquid. For several values of the ratio ${J}_{▿}/{J}_{▵}$, the size scaling of the energy gain due to the pairing fields and the variational parameters are reported. Our results show that the energy gain of the gapped spin liquid with respect to the gapless state either vanishes for large enough system size or scales to zero in the thermodynamic limit. Similarly, the optimized pairing amplitudes (responsible for opening the spin gap) are shown to vanish in the thermodynamic limit. Our outcome is corroborated by the application of one and two Lanczos steps to the gapless and gapped wave functions, for which no energy gain of the gapped state is detected when improving the quality of the variational states. Finally, we discuss the competition with the ``simplex'' ${\mathbb{Z}}_{2}$ resonating-valence-bond spin liquid, valence-bond crystal, and nematic states in the strongly anisotropic regime, i.e., ${J}_{▿}\ensuremath{\ll}{J}_{▵}$.

arXiv:1712.04579•Physical review. B./Physical review. B•Mar 13, 2018•40 citations

Gutzwiller projected states for the J 1 − J 2 Heisenberg model on the Kagome lattice: Achievements and pitfalls

Yasir Iqbal, Francesco Ferrari, Aishwarya Chauhan +3 more

We assess the ground-state phase diagram of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the Kagome lattice by employing Gutzwiller-projected fermionic wave functions. Within this framework, different states can be represented, defined by distinct unprojected fermionic Hamiltonians that include hopping and pairing terms, as well as a coupling to local Zeeman fields to generate magnetic order. For ${J}_{2}=0$, the so-called U(1) Dirac state, in which only hopping is present (such as to generate a $\ensuremath{\pi}$-flux in the hexagons), has been shown to accurately describe the exact ground state [Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B 87, 060405(R) (2013); Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Phys. Rev. X 7, 031020 (2017).]. Here we show that its accuracy improves in the presence of a small antiferromagnetic superexchange ${J}_{2}$, leading to a finite region where the gapless spin liquid is stable; then, for ${J}_{2}/{J}_{1}=0.11(1)$, a first-order transition to a magnetic phase with pitch vector $\mathbf{q}=(0,0)$ is detected by allowing magnetic order within the fermionic Hamiltonian. Instead, for small ferromagnetic values of $|{J}_{2}|/{J}_{1}$, the situation is more contradictory. While the U(1) Dirac state remains stable against several perturbations in the fermionic part (i.e., dimerization patterns or chiral terms), its accuracy clearly deteriorates on small systems, most notably on 36 sites where exact diagonalization is possible. Then, on increasing the ratio $|{J}_{2}|/{J}_{1}$, a magnetically ordered state with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ periodicity eventually overcomes the U(1) Dirac spin liquid. Within the ferromagnetic ${J}_{2}$ regime, evidence is shown in favor of a first-order transition at ${J}_{2}/{J}_{1}=\ensuremath{-}0.065(5)$.

arXiv:2108.02847•Physical review. B./Physical review. B•Oct 6, 2021•34 citations
Comment on "Z2Z_{2}Z2​ spin liquid phase on the kagome lattice: a new saddle point", by Tao Li [arXiv:1601.02165 (2016)]
In a recent paper [arXiv:1601.02165], Tao Li claimed that a gapped Z2\mathbb{Z}_2Z2​ spin liquid, obtained using projected Gutzwiller fermionic wave functions, can be stabilized in the Heisenberg model on the kagome lattice. We perform very accurate variational calculations and confirm with unprecedented accuracy the fact, that the Z2\mathbb{Z}_{2}Z2​ spin liquid is a local energy minimum that goes away with system size, thus reaffirming the scenario of a gapless U(1)U(1)U(1) Dirac spin liquid ground state.
Motivated by recent density-matrix renormalization group (DMRG) calculations [Yan, Huse, and White, Science 332, 1173 (2011)], which claimed that the ground state of the nearest-neighbor spin-1/21/21/2 Heisenberg antiferromagnet on the kagome lattice geometry is a fully gapped spin liquid with numerical signatures of Z2{\mathbb{Z}}_{2}Z2​ gauge structure, and a further theoretical work [Lu, Ran, and Lee, Phys. Rev. B 83, 224413 (2011)], which gave a classification of all Schwinger-fermion mean-field fully symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids on the kagome lattice, we have thoroughly studied Gutzwiller-projected fermionic wave functions by using quantum variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. In particular, we investigated the energetics of all Z2{\mathbb{Z}}_{2}Z2​ candidates (gapped and gapless) that lie in the neighborhood of the energetically competitive U(1) gapless spin liquids. By using a state-of-the-art optimization method, we were able to conclusively show that the U(1) Dirac state is remarkably stable with respect to all Z2{\mathbb{Z}}_{2}Z2​ spin liquids in its neighborhood, and in particular for opening a gap toward the so-called ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state, which was conjectured to describe the ground state obtained by the DMRG method. Finally, we also considered the addition of a small second nearest-neighbor exchange coupling of both antiferromagnetic and ferromagnetic type, and obtained similar results, namely, a U(1) Dirac spin-liquid ground state.
By using the variational Monte Carlo technique, we study the spin-12\frac{1}{2}21​ XXZ antiferromagnetic model (with easy-plane anisotropy) on the kagome lattice. A class of Gutzwiller projected fermionic states with a spin Jastrow factor is considered to describe either spin liquids [with U(1)U(1)U(1) or Z2{Z}_{2}Z2​ symmetry] or magnetically ordered phases [with q=(0,0)\mathbf{q}=(0,0)q=(0,0) or $\mathbf{q}=(4\ensuremath{\pi}/3,0)$]. We find that the magnetic states are not stable in the thermodynamic limit. Moreover, there is no energy gain to break the gauge symmetry from U(1)U(1)U(1) to Z2{Z}_{2}Z2​ within the spin-liquid states, as previously found in the Heisenberg model. The best variational wave function is therefore the U(1)U(1)U(1) Dirac state, supplemented by the spin Jastrow factor. Furthermore, a vanishing S=2S=2S=2 spin gap is obtained at the variational level, in the whole regime from the XYXYXY to the Heisenberg model.
We study the energy and the static spin structure factor of the ground state of the spin-1/21/21/2 quantum Heisenberg antiferromagnetic model on the kagome lattice. By the iterative application of a few Lanczos steps on accurate projected fermionic wave functions and the Green's function Monte Carlo technique, we find that a gapless (algebraic) U(1)U(1)U(1) Dirac spin liquid is competitive with previously proposed gapped (topological) Z2{\mathbb{Z}}_{2}Z2​ spin liquids. By performing a finite-size extrapolation of the ground-state energy, we obtain an energy per site $E/J=\ensuremath{-}0.4365(2)$, which is equal, within three error bars, to the estimates given by the density-matrix renormalization group (DMRG). Our estimate is obtained for a translationally invariant system, and, therefore, does not suffer from boundary effects, like in DMRG. Moreover, on finite toric clusters at the pure variational level, our energies are lower compared to those from DMRG calculations.
Due to strong geometric frustration and quantum fluctuation, the S=1/2S=1/2S=1/2 quantum Heisenberg antiferromagnet on the kagome\text{kagome}kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S=1/2,kagomeS=1/2,\text{kagome}S=1/2,kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2{\mathbb{Z}}_{2}Z2​ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2{\mathbb{Z}}_{2}Z2​ SLs on the kagome\text{kagome}kagome lattice. Among them we found only one gapped Z2{\mathbb{Z}}_{2}Z2​ SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.
The nature of the ground state of the spin S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on the kagome lattice with breathing anisotropy (i.e., with different superexchange couplings J▵{J}_{▵}J▵​ and J▿{J}_{▿}J▿​ within elementary up- and down-pointing triangles) is investigated within the framework of Gutzwiller projected fermionic wave functions and Monte Carlo methods. We analyze the stability of the U(1)\text{U}(1)U(1) Dirac spin liquid with respect to the presence of fermionic pairing that leads to a gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquid. For several values of the ratio J▿/J▵{J}_{▿}/{J}_{▵}J▿​/J▵​, the size scaling of the energy gain due to the pairing fields and the variational parameters are reported. Our results show that the energy gain of the gapped spin liquid with respect to the gapless state either vanishes for large enough system size or scales to zero in the thermodynamic limit. Similarly, the optimized pairing amplitudes (responsible for opening the spin gap) are shown to vanish in the thermodynamic limit. Our outcome is corroborated by the application of one and two Lanczos steps to the gapless and gapped wave functions, for which no energy gain of the gapped state is detected when improving the quality of the variational states. Finally, we discuss the competition with the ``simplex'' Z2{\mathbb{Z}}_{2}Z2​ resonating-valence-bond spin liquid, valence-bond crystal, and nematic states in the strongly anisotropic regime, i.e., ${J}_{▿}\ensuremath{\ll}{J}_{▵}$.
We assess the ground-state phase diagram of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the Kagome lattice by employing Gutzwiller-projected fermionic wave functions. Within this framework, different states can be represented, defined by distinct unprojected fermionic Hamiltonians that include hopping and pairing terms, as well as a coupling to local Zeeman fields to generate magnetic order. For J2=0{J}_{2}=0J2​=0, the so-called U(1) Dirac state, in which only hopping is present (such as to generate a $\ensuremath{\pi}$-flux in the hexagons), has been shown to accurately describe the exact ground state [Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B 87, 060405(R) (2013); Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Phys. Rev. X 7, 031020 (2017).]. Here we show that its accuracy improves in the presence of a small antiferromagnetic superexchange J2{J}_{2}J2​, leading to a finite region where the gapless spin liquid is stable; then, for J2/J1=0.11(1){J}_{2}/{J}_{1}=0.11(1)J2​/J1​=0.11(1), a first-order transition to a magnetic phase with pitch vector q=(0,0)\mathbf{q}=(0,0)q=(0,0) is detected by allowing magnetic order within the fermionic Hamiltonian. Instead, for small ferromagnetic values of ∣J2∣/J1|{J}_{2}|/{J}_{1}∣J2​∣/J1​, the situation is more contradictory. While the U(1) Dirac state remains stable against several perturbations in the fermionic part (i.e., dimerization patterns or chiral terms), its accuracy clearly deteriorates on small systems, most notably on 36 sites where exact diagonalization is possible. Then, on increasing the ratio ∣J2∣/J1|{J}_{2}|/{J}_{1}∣J2​∣/J1​, a magnetically ordered state with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ periodicity eventually overcomes the U(1) Dirac spin liquid. Within the ferromagnetic J2{J}_{2}J2​ regime, evidence is shown in favor of a first-order transition at ${J}_{2}/{J}_{1}=\ensuremath{-}0.065(5)$.