arXiv:cond-mat/0701635 · Physical Review B

Global phase diagram for the spin-1 antiferromagnet with uniaxial anisotropy on the kagome lattice

Cenke Xu, Joel E. Moore

Published Sep 24, 2007• 15 citations

Abstract

The phase diagram of the $XXZ$ spin-1 quantum magnet on the kagome lattice is studied for all cases where the $z$-component nearest-neighbor spin interaction ${J}_{z}$ is antiferromagnetic. Besides ${J}_{z}$ and the nearest-neighbor in-plane spin interaction ${J}_{xy}$, the system is also parametrized by an on-site anisotropic term $D{({S}^{z})}^{2}$. In the zero magnetic field case, the six previously introduced phases, found using various methods, are the nondegenerate gapped photon phase, which breaks no space symmetry or spin symmetry; the sixfold degenerate phase with plaquette order, which breaks both time-reversal symmetry and translational symmetry; the ``superfluid'' (ferromagnetic) phase with an in-plane global U(1) symmetry broken when ${J}_{xy}<0$; the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order when ${J}_{xy}>0$; the nematic phase when $D<0$ and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the sixfold plaquette phases. The transition between nematic phase and the sixfold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the sixfold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, organic compound $m\text{\ensuremath{-}}\mathrm{MPYNN}∙\mathrm{B}{\mathrm{F}}_{4}$, or optical lattices of atoms with three degenerate states on every site.

DOI: 10.1103/physrevb.76.104427

Related papers

Geometric criticality for transitions between plaquette phases in integer-spin kagome XXZ antiferromagnets

Cenke Xu, Joel E. Moore

arXiv:cond-mat/0501580Physical Review BAug 31, 200511 citations

Chiral spin state and nematic ferromagnet in the spin-1 Kitaev- Γ model

Qiang Luo, Jize Zhao, Jinbin Li +1 more

arXiv:2403.08382Physical review. B./Physical review. BJul 8, 20248 citations

Plaquette-triplon analysis of magnetic disorder and order in a trimerized spin-1 kagome Heisenberg antiferromagnet

Pratyay Ghosh, Akhilesh Kumar Verma, Brijesh Kumar

arXiv:1507.08038Physical review. B./Physical review. BJan 19, 201616 citations

Spin- S kagome quantum antiferromagnets in a field with tensor networks

Thibaut Picot, Marc Ziegler, Román Orús +1 more

arXiv:1508.07189Physical review. B./Physical review. BFeb 19, 2016109 citations

Nematic and supernematic phases in kagome quantum antiferromagnets under the influence of a magnetic field

Thibaut Picot, Didier Poilblanc

arXiv:1406.7205Physical Review BFeb 17, 201541 citations

Spin- 1 2 kagome XXZ model in a field: Competition between lattice nematic and solid orders

Augustine Kshetrimayum, Thibaut Picot, Román Orús +1 more

arXiv:1608.00437Physical review. B./Physical review. BDec 20, 201625 citations

arXiv:cond-mat/0701635 · Physical Review B

Global phase diagram for the spin-1 antiferromagnet with uniaxial anisotropy on the kagome lattice

Cenke Xu, Joel E. Moore

Published Sep 24, 2007• 15 citations

arXiv HTML PDF

Abstract

The phase diagram of the

XXZ

spin-1 quantum magnet on the kagome lattice is studied for all cases where the

z

-component nearest-neighbor spin interaction

{J}_{z}

is antiferromagnetic. Besides

{J}_{z}

and the nearest-neighbor in-plane spin interaction

{J}_{xy}

, the system is also parametrized by an on-site anisotropic term

D{({S}^{z})}^{2}

. In the zero magnetic field case, the six previously introduced phases, found using various methods, are the nondegenerate gapped photon phase, which breaks no space symmetry or spin symmetry; the sixfold degenerate phase with plaquette order, which breaks both time-reversal symmetry and translational symmetry; the ``superfluid'' (ferromagnetic) phase with an in-plane global U(1) symmetry broken when

{J}_{xy}<0

; the

\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}

order when

{J}_{xy}>0

; the nematic phase when

D<0

and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the sixfold plaquette phases. The transition between nematic phase and the sixfold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along

\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}

, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the sixfold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, organic compound

m\text{\ensuremath{-}}\mathrm{MPYNN}∙\mathrm{B}{\mathrm{F}}_{4}

, or optical lattices of atoms with three degenerate states on every site.

DOI: 10.1103/physrevb.76.104427

Related papers

Geometric criticality for transitions between plaquette phases in integer-spin kagome XXZ antiferromagnets

Cenke Xu, Joel E. Moore

The phase diagram of the uniaxially anisotropic $s=1$ antiferromagnet on the kagome lattice includes a critical line exactly described by the classical three-color model. This line is distinct from the standard geometric classical criticality that appears in the classical limit $(s\ensuremath{\rightarrow}\ensuremath{\infty})$ of the two-dimensional XY model; the $s=1$ geometric $T=0$ critical line separates two unconventional plaquette-ordered phases that survive to nonzero temperature. The experimentally important correlations at finite temperature and the nature of the transitions into these ordered phases are obtained using the mapping to the three-color model and a combination of perturbation theory and a variational ansatz for the ordered phases. The ordered phases show sixfold symmetry breaking and are similar to phases proposed for the honeycomb lattice dimer model and $s=1∕2$ XXZ model. The same mapping and phase transition can be realized also for integer spins $s\ensuremath{\geqslant}2$ but then require strong on-site anisotropy in the Hamiltonian.The phase diagram of the uniaxially anisotropic $s=1$ antiferromagnet on the kagome lattice includes a critical line exactly described by the classical three-color model. This line is distinct from the standard geometric classical criticality that appears in the classical limit $(s\ensuremath{\rightarrow}\ensuremath{\infty})$ of the two-dimensional XY model; the $s=1$ geometric $T=0$ critical line separates two unconventional plaquette-ordered phases that survive to nonzero temperature. The experimentally important correlations at finite temperature and the nature of the transitions into these ordered phases are obtained using the mapping to the three-color model and a combination of perturbation theory and a variational ansatz for the ordered phases. The ordered phases show sixfold symmetry breaking and are similar to phases proposed for the honeycomb lattice dimer model and $s=1∕2$ XXZ model. The same mapping and phase transition can be realized also for integer spins $s\ensuremath{\geqslant}2$ but then require strong on-site anisotropy in the Hamiltonian.

arXiv:cond-mat/0501580Physical Review BAug 31, 200511 citations

Chiral spin state and nematic ferromagnet in the spin-1 Kitaev- Γ model

Qiang Luo, Jize Zhao, Jinbin Li +1 more

The higher-spin Kitaev magnets, in which the Kitaev interaction and off-diagonal exchange couplings are overwhelmingly large, have emerged as a fertile avenue to explore exotic phases and unusual excitations. In this paper, we study the quantum phase diagram of the spin-1 Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model on the honeycomb lattice using density-matrix renormalization group. It harbors six distinct phases and the intriguing findings are three magnetically ordered phases in which both time-reversal symmetry and lattice symmetry, albeit of different sorts, are broken spontaneously. The chiral spin state originates from the order-by-disorder effect and exhibits an almost saturated scalar spin chirality at the quantum level. Depending on the relative strength of the two interactions, it also features a columnarlike or plaquettelike dimer pattern as a consequence of the translational symmetry breaking. In parallel, the nematic ferromagnets are situated at the ferromagnetic Kitaev side and possess small but finite ferromagnetic ordering. The lattice-rotational symmetry breaking enforces nonequivalent bond energy along one of the three bonds. Although the intrinsic difference between the two nematic ferromagnets remains elusive, the discontinuities in the von Neumann entropy, hexagonal plaquette operator, and Wilson loop operator convincingly suggest that they are separated via a first-order phase transition.The higher-spin Kitaev magnets, in which the Kitaev interaction and off-diagonal exchange couplings are overwhelmingly large, have emerged as a fertile avenue to explore exotic phases and unusual excitations. In this paper, we study the quantum phase diagram of the spin-1 Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model on the honeycomb lattice using density-matrix renormalization group. It harbors six distinct phases and the intriguing findings are three magnetically ordered phases in which both time-reversal symmetry and lattice symmetry, albeit of different sorts, are broken spontaneously. The chiral spin state originates from the order-by-disorder effect and exhibits an almost saturated scalar spin chirality at the quantum level. Depending on the relative strength of the two interactions, it also features a columnarlike or plaquettelike dimer pattern as a consequence of the translational symmetry breaking. In parallel, the nematic ferromagnets are situated at the ferromagnetic Kitaev side and possess small but finite ferromagnetic ordering. The lattice-rotational symmetry breaking enforces nonequivalent bond energy along one of the three bonds. Although the intrinsic difference between the two nematic ferromagnets remains elusive, the discontinuities in the von Neumann entropy, hexagonal plaquette operator, and Wilson loop operator convincingly suggest that they are separated via a first-order phase transition.

arXiv:2403.08382Physical review. B./Physical review. BJul 8, 20248 citations

Plaquette-triplon analysis of magnetic disorder and order in a trimerized spin-1 kagome Heisenberg antiferromagnet

Pratyay Ghosh, Akhilesh Kumar Verma, Brijesh Kumar

A spin-1 Heisenberg model on trimerized kagome lattice is studied by doing a low-energy bosonic theory in terms of plaquette triplons defined on its triangular unit cells. The model considered has an intratriangle antiferromagnetic exchange interaction $J$ (set to 1) and two intertriangle couplings ${J}^{\ensuremath{'}}>0$ (nearest neighbor) and ${J}^{\ensuremath{''}}$ (next nearest neighbor; of both signs). The triplon analysis performed on this model investigates the stability of the trimerized singlet ground state (which is exact in the absence of intertriangle couplings) in the ${J}^{\ensuremath{'}}\ensuremath{-}{J}^{\ensuremath{''}}$ plane. It gives a quantum phase diagram that has two gapless antiferromagnetically ordered phases separated by the spin-gapped trimerized singlet phase. The trimerized singlet ground state is found to be stable on ${J}^{\ensuremath{''}}=0$ line (the nearest-neighbor case), and on both sides of it for ${J}^{\ensuremath{''}}\ensuremath{\ne}0$, in an extended region bounded by the critical lines of transition to the gapless antiferromagnetic phases. The gapless phase in the negative ${J}^{\ensuremath{''}}$ region has a coplanar ${120}^{\ensuremath{\circ}}$ antiferromagnetic order with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ structure. In this phase, all the magnetic moments are of equal length, and the angle between any two of them on a triangle is exactly ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this case has a unit cell consisting of three triangles. The other gapless phase, in the positive ${J}^{\ensuremath{''}}$ region, is found to exhibit a different coplanar antiferromagnetic order with ordering wave vector $\mathbf{q}=(0,0)$ . Here, two magnetic moments in a triangle are of the same magnitude, but shorter than the third. While the angle between two short moments is ${120}^{\ensuremath{\circ}}\ensuremath{-}2\ensuremath{\delta}$, it is ${120}^{\ensuremath{\circ}}+\ensuremath{\delta}$ between a short and the long one. Only when ${J}^{\ensuremath{''}}={J}^{\ensuremath{'}}$, their magnitudes become equal and the relative angles ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this $\mathbf{q}=(0,0)$ phase has the translational symmetry of the kagome lattice with triangular unit cells of reduced (isosceles) symmetry. This reduction in the point-group symmetry is found to show up as a difference in the intensities of certain Bragg peaks, whose ratio ${I}_{(1,0)}/{I}_{(0,1)}=4{sin}^{2}(\frac{\ensuremath{\pi}}{6}+\ensuremath{\delta})$ presents an experimental measure of the deviation $\ensuremath{\delta}$ from the ${120}^{\ensuremath{\circ}}$ order.A spin-1 Heisenberg model on trimerized kagome lattice is studied by doing a low-energy bosonic theory in terms of plaquette triplons defined on its triangular unit cells. The model considered has an intratriangle antiferromagnetic exchange interaction $J$ (set to 1) and two intertriangle couplings ${J}^{\ensuremath{'}}>0$ (nearest neighbor) and ${J}^{\ensuremath{''}}$ (next nearest neighbor; of both signs). The triplon analysis performed on this model investigates the stability of the trimerized singlet ground state (which is exact in the absence of intertriangle couplings) in the ${J}^{\ensuremath{'}}\ensuremath{-}{J}^{\ensuremath{''}}$ plane. It gives a quantum phase diagram that has two gapless antiferromagnetically ordered phases separated by the spin-gapped trimerized singlet phase. The trimerized singlet ground state is found to be stable on ${J}^{\ensuremath{''}}=0$ line (the nearest-neighbor case), and on both sides of it for ${J}^{\ensuremath{''}}\ensuremath{\ne}0$, in an extended region bounded by the critical lines of transition to the gapless antiferromagnetic phases. The gapless phase in the negative ${J}^{\ensuremath{''}}$ region has a coplanar ${120}^{\ensuremath{\circ}}$ antiferromagnetic order with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ structure. In this phase, all the magnetic moments are of equal length, and the angle between any two of them on a triangle is exactly ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this case has a unit cell consisting of three triangles. The other gapless phase, in the positive ${J}^{\ensuremath{''}}$ region, is found to exhibit a different coplanar antiferromagnetic order with ordering wave vector $\mathbf{q}=(0,0)$. Here, two magnetic moments in a triangle are of the same magnitude, but shorter than the third. While the angle between two short moments is ${120}^{\ensuremath{\circ}}\ensuremath{-}2\ensuremath{\delta}$, it is ${120}^{\ensuremath{\circ}}+\ensuremath{\delta}$ between a short and the long one. Only when ${J}^{\ensuremath{''}}={J}^{\ensuremath{'}}$, their magnitudes become equal and the relative angles ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this $\mathbf{q}=(0,0)$ phase has the translational symmetry of the kagome lattice with triangular unit cells of reduced (isosceles) symmetry. This reduction in the point-group symmetry is found to show up as a difference in the intensities of certain Bragg peaks, whose ratio ${I}_{(1,0)}/{I}_{(0,1)}=4{sin}^{2}(\frac{\ensuremath{\pi}}{6}+\ensuremath{\delta})$ presents an experimental measure of the deviation $\ensuremath{\delta}$ from the ${120}^{\ensuremath{\circ}}$ order.

arXiv:1507.08038Physical review. B./Physical review. BJan 19, 201616 citations

Spin- S kagome quantum antiferromagnets in a field with tensor networks

Thibaut Picot, Marc Ziegler, Román Orús +1 more

Spin- $S$ Heisenberg quantum antiferromagnets on the kagome lattice offer, when placed in a magnetic field, a fantastic playground to observe exotic phases of matter with (magnetic analogs of) superfluid, charge, bond, or nematic orders, or a coexistence of several of the latter. In this context, we have obtained the (zero-temperature) phase diagrams up to $S=2$ directly in the thermodynamic limit owing to infinite projected entangled pair states, a tensor network numerical tool. We find incompressible phases characterized by a magnetization plateau versus field and stabilized by spontaneous breaking of point group or lattice translation symmetry(ies). The nature of such phases may be semiclassical, as the plateaus at the $\frac{1}{3}\mathrm{th},\left(1\ensuremath{-}\frac{2}{9S}\right)\mathrm{th}$, and $\left(1\ensuremath{-}\frac{1}{9S}\right)\mathrm{th}$ of the saturated magnetization (the latter followed by a macroscopic magnetization jump), or fully quantum as the spin- $\frac{1}{2}$ $\frac{1}{9}$ plateau exhibiting a coexistence of charge and bond orders. Upon restoration of the spin rotation $U(1)$ symmetry, a finite compressibility appears, although lattice symmetry breaking persists. For integer spin values we also identify spin gapped phases at low enough fields, such as the $S=2$ (topologically trivial) spin liquid with no symmetry breaking, neither spin nor lattice.Spin-$S$ Heisenberg quantum antiferromagnets on the kagome lattice offer, when placed in a magnetic field, a fantastic playground to observe exotic phases of matter with (magnetic analogs of) superfluid, charge, bond, or nematic orders, or a coexistence of several of the latter. In this context, we have obtained the (zero-temperature) phase diagrams up to $S=2$ directly in the thermodynamic limit owing to infinite projected entangled pair states, a tensor network numerical tool. We find incompressible phases characterized by a magnetization plateau versus field and stabilized by spontaneous breaking of point group or lattice translation symmetry(ies). The nature of such phases may be semiclassical, as the plateaus at the $\frac{1}{3}\mathrm{th},\left(1\ensuremath{-}\frac{2}{9S}\right)\mathrm{th}$, and $\left(1\ensuremath{-}\frac{1}{9S}\right)\mathrm{th}$ of the saturated magnetization (the latter followed by a macroscopic magnetization jump), or fully quantum as the spin-$\frac{1}{2}$ $\frac{1}{9}$ plateau exhibiting a coexistence of charge and bond orders. Upon restoration of the spin rotation $U(1)$ symmetry, a finite compressibility appears, although lattice symmetry breaking persists. For integer spin values we also identify spin gapped phases at low enough fields, such as the $S=2$ (topologically trivial) spin liquid with no symmetry breaking, neither spin nor lattice.

arXiv:1508.07189Physical review. B./Physical review. BFeb 19, 2016109 citations

Nematic and supernematic phases in kagome quantum antiferromagnets under the influence of a magnetic field

Thibaut Picot, Didier Poilblanc

Optimizing translationally invariant infinite-projected entangled pair states, we investigate the spin-2 Affleck-Kennedy-Lieb-Tasaki and spin-1 Heisenberg models on the kagome lattice as a function of magnetic field. We found that the magnetization curves offer a wide variety of compressible and incompressible phases. Incompressible nematic phases breaking the lattice ${C}_{3}$ rotation, for which we propose simple qualitative pictures, give rise to magnetization plateaus at reduced magnetization ${m}_{z}=\frac{5}{6}$ and $\frac{1}{3}$ for spin-2 and -1, respectively, in addition to the ${m}_{z}=0$ plateaus characteristic of zero-field gapped spin liquids. Moving away from the plateaus, we observe a rich variety of compressible superfluid nematic, named ``supernematic,'' phases breaking spontaneously both point group and spin-U(1) symmetries, as well as a superfluid phase preserving lattice symmetries. We also identify the nature, continuous or first order, of the various phase transitions. Possible connections to experimental spin-1 systems are discussed.Optimizing translationally invariant infinite-projected entangled pair states, we investigate the spin-2 Affleck-Kennedy-Lieb-Tasaki and spin-1 Heisenberg models on the kagome lattice as a function of magnetic field. We found that the magnetization curves offer a wide variety of compressible and incompressible phases. Incompressible nematic phases breaking the lattice ${C}_{3}$ rotation, for which we propose simple qualitative pictures, give rise to magnetization plateaus at reduced magnetization ${m}_{z}=\frac{5}{6}$ and $\frac{1}{3}$ for spin-2 and -1, respectively, in addition to the ${m}_{z}=0$ plateaus characteristic of zero-field gapped spin liquids. Moving away from the plateaus, we observe a rich variety of compressible superfluid nematic, named ``supernematic,'' phases breaking spontaneously both point group and spin-U(1) symmetries, as well as a superfluid phase preserving lattice symmetries. We also identify the nature, continuous or first order, of the various phase transitions. Possible connections to experimental spin-1 systems are discussed.

arXiv:1406.7205Physical Review BFeb 17, 201541 citations

Spin- 1 2 kagome XXZ model in a field: Competition between lattice nematic and solid orders

Augustine Kshetrimayum, Thibaut Picot, Román Orús +1 more

We study numerically the spin-1/2 XXZ model in a field on an infinite kagome lattice. We use different algorithms based on infinite projected entangled pair states (iPEPSs) for this, namely, (i) an approach with simplex tensors and a 9-site unit cell, and (ii) an approach based on coarse-graining three spins in the kagome lattice and mapping it to a square-lattice model with local and nearest-neighbor interactions, with the usual PEPS tensors, 6- and 12-site unit cells. Similarly to our previous calculation at the SU(2)-symmetric point (Heisenberg Hamiltonian), for any anisotropy from the Ising limit to the XY limit, we also observe the emergence of magnetization plateaus as a function of the magnetic field, at ${m}_{z}=\frac{1}{3}$ using 6-, 9-, and 12-site PEPS unit cells, and at ${m}_{z}=\frac{1}{9},\phantom{\rule{0.28em}{0ex}}\frac{5}{9}$ , and $\frac{7}{9}$ using a 9-site PEPS unit cell, the latter setup being able to accommodate $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ solid order. We also find that, at ${m}_{z}=\frac{1}{3}$ , (lattice) nematic and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC-order states are degenerate within the accuracy of the nine-site simplex method, for all anisotropy. The 6- and 12-site coarse-grained PEPS methods produce almost-degenerate nematic and $1\ifmmode\times\else\texttimes\fi{}2$ VBC-solid orders. We also find that, within our accuracy, the six-site coarse-grained PEPS method gives slightly lower energies, which can be explained by the larger amount of entanglement this approach can handle, even in cases where the PEPS unit cell is not commensurate with the expected ground-state unit cell. Furthermore, we do not observe chiral spin liquid behaviors at and close to the XY point, as has been recently proposed. Our results are the first tensor network investigations of the XXZ model in a field and reveal the subtle competition between nearby magnetic orders in numerical simulations of frustrated quantum antiferromagnets, as well as the delicate interplay between energy optimization and symmetry in tensor network numerical simulations.We study numerically the spin-1/2 XXZ model in a field on an infinite kagome lattice. We use different algorithms based on infinite projected entangled pair states (iPEPSs) for this, namely, (i) an approach with simplex tensors and a 9-site unit cell, and (ii) an approach based on coarse-graining three spins in the kagome lattice and mapping it to a square-lattice model with local and nearest-neighbor interactions, with the usual PEPS tensors, 6- and 12-site unit cells. Similarly to our previous calculation at the SU(2)-symmetric point (Heisenberg Hamiltonian), for any anisotropy from the Ising limit to the XY limit, we also observe the emergence of magnetization plateaus as a function of the magnetic field, at ${m}_{z}=\frac{1}{3}$ using 6-, 9-, and 12-site PEPS unit cells, and at ${m}_{z}=\frac{1}{9},\phantom{\rule{0.28em}{0ex}}\frac{5}{9}$, and $\frac{7}{9}$ using a 9-site PEPS unit cell, the latter setup being able to accommodate $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ solid order. We also find that, at ${m}_{z}=\frac{1}{3}$, (lattice) nematic and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC-order states are degenerate within the accuracy of the nine-site simplex method, for all anisotropy. The 6- and 12-site coarse-grained PEPS methods produce almost-degenerate nematic and $1\ifmmode\times\else\texttimes\fi{}2$ VBC-solid orders. We also find that, within our accuracy, the six-site coarse-grained PEPS method gives slightly lower energies, which can be explained by the larger amount of entanglement this approach can handle, even in cases where the PEPS unit cell is not commensurate with the expected ground-state unit cell. Furthermore, we do not observe chiral spin liquid behaviors at and close to the XY point, as has been recently proposed. Our results are the first tensor network investigations of the XXZ model in a field and reveal the subtle competition between nearby magnetic orders in numerical simulations of frustrated quantum antiferromagnets, as well as the delicate interplay between energy optimization and symmetry in tensor network numerical simulations.

arXiv:1608.00437Physical review. B./Physical review. BDec 20, 201625 citations

The phase diagram of the

XXZ

spin-1 quantum magnet on the kagome lattice is studied for all cases where the

z

-component nearest-neighbor spin interaction

{J}_{z}

is antiferromagnetic. Besides

{J}_{z}

and the nearest-neighbor in-plane spin interaction

{J}_{xy}

, the system is also parametrized by an on-site anisotropic term

D{({S}^{z})}^{2}

{J}_{xy}<0

; the

\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}

order when

{J}_{xy}>0

; the nematic phase when

D<0

\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}

m\text{\ensuremath{-}}\mathrm{MPYNN}∙\mathrm{B}{\mathrm{F}}_{4}

, or optical lattices of atoms with three degenerate states on every site.

The phase diagram of the uniaxially anisotropic

s=1

antiferromagnet on the kagome lattice includes a critical line exactly described by the classical three-color model. This line is distinct from the standard geometric classical criticality that appears in the classical limit $(s\ensuremath{\rightarrow}\ensuremath{\infty})$ of the two-dimensional XY model; the

s=1

geometric

T=0

critical line separates two unconventional plaquette-ordered phases that survive to nonzero temperature. The experimentally important correlations at finite temperature and the nature of the transitions into these ordered phases are obtained using the mapping to the three-color model and a combination of perturbation theory and a variational ansatz for the ordered phases. The ordered phases show sixfold symmetry breaking and are similar to phases proposed for the honeycomb lattice dimer model and

s=1∕2

XXZ model. The same mapping and phase transition can be realized also for integer spins $s\ensuremath{\geqslant}2$ but then require strong on-site anisotropy in the Hamiltonian.

J

(set to 1) and two intertriangle couplings ${J}^{\ensuremath{'}}>0$ (nearest neighbor) and ${J}^{\ensuremath{''}}$ (next nearest neighbor; of both signs). The triplon analysis performed on this model investigates the stability of the trimerized singlet ground state (which is exact in the absence of intertriangle couplings) in the ${J}^{\ensuremath{'}}\ensuremath{-}{J}^{\ensuremath{''}}$ plane. It gives a quantum phase diagram that has two gapless antiferromagnetically ordered phases separated by the spin-gapped trimerized singlet phase. The trimerized singlet ground state is found to be stable on ${J}^{\ensuremath{''}}=0$ line (the nearest-neighbor case), and on both sides of it for ${J}^{\ensuremath{''}}\ensuremath{\ne}0$, in an extended region bounded by the critical lines of transition to the gapless antiferromagnetic phases. The gapless phase in the negative ${J}^{\ensuremath{''}}$ region has a coplanar ${120}^{\ensuremath{\circ}}$ antiferromagnetic order with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ structure. In this phase, all the magnetic moments are of equal length, and the angle between any two of them on a triangle is exactly ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this case has a unit cell consisting of three triangles. The other gapless phase, in the positive ${J}^{\ensuremath{''}}$ region, is found to exhibit a different coplanar antiferromagnetic order with ordering wave vector

\mathbf{q}=(0,0)

. Here, two magnetic moments in a triangle are of the same magnitude, but shorter than the third. While the angle between two short moments is ${120}^{\ensuremath{\circ}}\ensuremath{-}2\ensuremath{\delta}$, it is ${120}^{\ensuremath{\circ}}+\ensuremath{\delta}$ between a short and the long one. Only when ${J}^{\ensuremath{''}}={J}^{\ensuremath{'}}$, their magnitudes become equal and the relative angles ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this

\mathbf{q}=(0,0)

phase has the translational symmetry of the kagome lattice with triangular unit cells of reduced (isosceles) symmetry. This reduction in the point-group symmetry is found to show up as a difference in the intensities of certain Bragg peaks, whose ratio ${I}_{(1,0)}/{I}_{(0,1)}=4{sin}^{2}(\frac{\ensuremath{\pi}}{6}+\ensuremath{\delta})$ presents an experimental measure of the deviation $\ensuremath{\delta}$ from the ${120}^{\ensuremath{\circ}}$ order.

Spin-

S

Heisenberg quantum antiferromagnets on the kagome lattice offer, when placed in a magnetic field, a fantastic playground to observe exotic phases of matter with (magnetic analogs of) superfluid, charge, bond, or nematic orders, or a coexistence of several of the latter. In this context, we have obtained the (zero-temperature) phase diagrams up to

S=2

directly in the thermodynamic limit owing to infinite projected entangled pair states, a tensor network numerical tool. We find incompressible phases characterized by a magnetization plateau versus field and stabilized by spontaneous breaking of point group or lattice translation symmetry(ies). The nature of such phases may be semiclassical, as the plateaus at the $\frac{1}{3}\mathrm{th},\left(1\ensuremath{-}\frac{2}{9S}\right)\mathrm{th}$, and $\left(1\ensuremath{-}\frac{1}{9S}\right)\mathrm{th}$ of the saturated magnetization (the latter followed by a macroscopic magnetization jump), or fully quantum as the spin-

\frac{1}{2}

\frac{1}{9}

plateau exhibiting a coexistence of charge and bond orders. Upon restoration of the spin rotation

U(1)

symmetry, a finite compressibility appears, although lattice symmetry breaking persists. For integer spin values we also identify spin gapped phases at low enough fields, such as the

S=2

(topologically trivial) spin liquid with no symmetry breaking, neither spin nor lattice.

{C}_{3}

rotation, for which we propose simple qualitative pictures, give rise to magnetization plateaus at reduced magnetization

{m}_{z}=\frac{5}{6}

and

\frac{1}{3}

for spin-2 and -1, respectively, in addition to the

{m}_{z}=0

plateaus characteristic of zero-field gapped spin liquids. Moving away from the plateaus, we observe a rich variety of compressible superfluid nematic, named ``supernematic,'' phases breaking spontaneously both point group and spin-U(1) symmetries, as well as a superfluid phase preserving lattice symmetries. We also identify the nature, continuous or first order, of the various phase transitions. Possible connections to experimental spin-1 systems are discussed.

{m}_{z}=\frac{1}{3}

using 6-, 9-, and 12-site PEPS unit cells, and at

{m}_{z}=\frac{1}{9},\phantom{\rule{0.28em}{0ex}}\frac{5}{9}

, and

\frac{7}{9}

using a 9-site PEPS unit cell, the latter setup being able to accommodate $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ solid order. We also find that, at

{m}_{z}=\frac{1}{3}

, (lattice) nematic and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC-order states are degenerate within the accuracy of the nine-site simplex method, for all anisotropy. The 6- and 12-site coarse-grained PEPS methods produce almost-degenerate nematic and $1\ifmmode\times\else\texttimes\fi{}2$ VBC-solid orders. We also find that, within our accuracy, the six-site coarse-grained PEPS method gives slightly lower energies, which can be explained by the larger amount of entanglement this approach can handle, even in cases where the PEPS unit cell is not commensurate with the expected ground-state unit cell. Furthermore, we do not observe chiral spin liquid behaviors at and close to the XY point, as has been recently proposed. Our results are the first tensor network investigations of the XXZ model in a field and reveal the subtle competition between nearby magnetic orders in numerical simulations of frustrated quantum antiferromagnets, as well as the delicate interplay between energy optimization and symmetry in tensor network numerical simulations.