arXiv:1205.3838 · Physical Review B

Gapped spin-liquid phase in the J1−J2 Heisenberg model by a bosonic resonating valence-bond ansatz

Tao Li, Federico Becca, Wen-Jun Hu, Sandro Sorella

Published Aug 7, 2012• 62 citations

Abstract

We study the ground-state phase diagram of the spin-$1/2$${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice with an accurate bosonic resonating-valence-bond (RVB) wave function. In contrast to the RVB ansatz based on Schwinger fermions, the representation based on Schwinger bosons, supplemented by a variational Monte Carlo technique enforcing the exact projection onto the physical subspace, is able to describe a fully gapped spin liquid in the strongly frustrated regime. In particular, a fully symmetric ${Z}_{2}$ spin liquid is stable between two antiferromagnetic phases; a continuous transition at ${J}_{2}=0.4{J}_{1}$, when the Marshall sign rule begins to be essentially violated, and a first-order transition around ${J}_{2}=0.6{J}_{1}$ are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around ${J}_{2}=0.51{J}_{1}$, when the lowest spinon excitation moves from the $\ensuremath{\Gamma}$ to the $M$ point, i.e., $k=(\ensuremath{\pi},0)$.

DOI: 10.1103/physrevb.86.075111

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arXiv:1205.3838 · Physical Review B

Gapped spin-liquid phase in the J1−J2 Heisenberg model by a bosonic resonating valence-bond ansatz

Tao Li, Federico Becca, Wen-Jun Hu, Sandro Sorella

Published Aug 7, 2012• 62 citations

arXiv HTML PDF

Abstract

We study the ground-state phase diagram of the spin-

1/2$${J}_{1}\ensuremath{-}{J}_{2}

Heisenberg model on the square lattice with an accurate bosonic resonating-valence-bond (RVB) wave function. In contrast to the RVB ansatz based on Schwinger fermions, the representation based on Schwinger bosons, supplemented by a variational Monte Carlo technique enforcing the exact projection onto the physical subspace, is able to describe a fully gapped spin liquid in the strongly frustrated regime. In particular, a fully symmetric

{Z}_{2}

spin liquid is stable between two antiferromagnetic phases; a continuous transition at

{J}_{2}=0.4{J}_{1}

, when the Marshall sign rule begins to be essentially violated, and a first-order transition around

{J}_{2}=0.6{J}_{1}

are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around

{J}_{2}=0.51{J}_{1}

, when the lowest spinon excitation moves from the

\ensuremath{\Gamma}

to the

M

point, i.e.,

k=(\ensuremath{\pi},0)

DOI: 10.1103/physrevb.86.075111

Related papers

Quantum Spin Liquid in Spin 1/2J1–J2Heisenberg Model on Square Lattice: Many-Variable Variational Monte Carlo Study Combined with Quantum-Number Projections

Satoshi Morita, Ryui Kaneko, Masatoshi Imada

The nature of quantum spin liquids is studied for the spin- $1/2$ antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, $J_1$ , and those between next-nearest-neighbor sites, $J_2$ . We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region $0.4< J_2/J_1\le 0.6$ sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of $0.4 < J_2/J_1 \le 0.5$ , while the gapped valence-bond-crystal (VBC) phase is stabilized for $0.5 < J_2/J_1 \le 0.6$ . The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the $C_{4v}$ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation $\langle S(0)S(r)\rangle \propto 1/r^{z+η}$ at a long distance $r$ appears to increase from $z+η\sim 1$ at $J_2/J_1\sim0.4$ toward the continuous transition to the VBC phase at $J_1/J_1\sim0.5$ . Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.The nature of quantum spin liquids is studied for the spin-$1/2$ antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, $J_1$, and those between next-nearest-neighbor sites, $J_2$. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region $0.4< J_2/J_1\le 0.6$ sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of $0.4 < J_2/J_1 \le 0.5$, while the gapped valence-bond-crystal (VBC) phase is stabilized for $0.5 < J_2/J_1 \le 0.6$. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the $C_{4v}$ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation $\langle S(0)S(r)\rangle \propto 1/r^{z+η}$ at a long distance $r$ appears to increase from $z+η\sim 1$ at $J_2/J_1\sim0.4$ toward the continuous transition to the VBC phase at $J_1/J_1\sim0.5$. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.

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A gapped $Z_{2}$ spin liquid phase with a $U(1)$ mean field ansatz: a Bosonic resonating valence-bond descriptionA gapped $Z_{2}$ spin liquid phase with a $U(1)$ mean field ansatz: a Bosonic resonating valence-bond description

Tao Li

Gapped $Z_{2}$ spin liquid as the simplest spin liquid has been proved to be the most difficult to realize in realistic models. Here we show that the frustration from a third-neighbor exchange $J_{3}$ on the spin- $\frac{1}{2}$ $J_{1}-J_{2}$ model on the square lattice may serve to stabilize such a long-sought state. We argue that a Bosonic RVB description is more appropriate than a Fermonic RVB description for such a gapped spin liquid phase. We show that while the mean field ansatz of the proposed state has a $U(1)$ gauge symmetry, the gauge fluctuation spectrum on it is actually gapped and the state should be understood rather as a $Z_{2}$ spin liquid state with topological order. The state is thus locally stable with respect to gauge fluctuation and can emerge continuously from a collinear Neel ordered phase.Gapped $Z_{2}$ spin liquid as the simplest spin liquid has been proved to be the most difficult to realize in realistic models. Here we show that the frustration from a third-neighbor exchange $J_{3}$ on the spin-$\frac{1}{2}$ $J_{1}-J_{2}$ model on the square lattice may serve to stabilize such a long-sought state. We argue that a Bosonic RVB description is more appropriate than a Fermonic RVB description for such a gapped spin liquid phase. We show that while the mean field ansatz of the proposed state has a $U(1)$ gauge symmetry, the gauge fluctuation spectrum on it is actually gapped and the state should be understood rather as a $Z_{2}$ spin liquid state with topological order. The state is thus locally stable with respect to gauge fluctuation and can emerge continuously from a collinear Neel ordered phase.

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arXiv:2309.13630Physical review. B./Physical review. BApr 3, 202425 citations

Gapless spin liquid and valence-bond solid in the J1 - J2 Heisenberg model on the square lattice: Insights from singlet and triplet excitations

Francesco Ferrari, Federico Becca

The spin- $1/2$ ${J}_{1}$ - ${J}_{2}$ Heisenberg model on the square lattice represents one of the simplest examples in which the effects of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)], reconcile the contradicting results obtained within different approaches over the years.The spin-$1/2$ ${J}_{1}$-${J}_{2}$ Heisenberg model on the square lattice represents one of the simplest examples in which the effects of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)], reconcile the contradicting results obtained within different approaches over the years.

arXiv:2005.12941Physical review. B./Physical review. BJul 10, 202079 citations

We study the ground-state phase diagram of the spin-

1/2$${J}_{1}\ensuremath{-}{J}_{2}

{Z}_{2}

spin liquid is stable between two antiferromagnetic phases; a continuous transition at

{J}_{2}=0.4{J}_{1}

, when the Marshall sign rule begins to be essentially violated, and a first-order transition around

{J}_{2}=0.6{J}_{1}

are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around

{J}_{2}=0.51{J}_{1}

, when the lowest spinon excitation moves from the

\ensuremath{\Gamma}

to the

M

point, i.e.,

k=(\ensuremath{\pi},0)

The nature of quantum spin liquids is studied for the spin-

1/2

antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites,

J_1

, and those between next-nearest-neighbor sites,

J_2

. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region

0.4< J_2/J_1\le 0.6

sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of

0.4 < J_2/J_1 \le 0.5

, while the gapped valence-bond-crystal (VBC) phase is stabilized for

0.5 < J_2/J_1 \le 0.6

. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the

C_{4v}

symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation

\langle S(0)S(r)\rangle \propto 1/r^{z+η}

at a long distance

r

appears to increase from

z+η\sim 1

J_2/J_1\sim0.4

toward the continuous transition to the VBC phase at

J_1/J_1\sim0.5

. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.

{J}_{2}/{J}_{1}=0.45

and the first order around

{J}_{2}/{J}_{1}=0.6

, can be extracted more clearly than the conventional bosonic RVB state. At the maximally frustrated point

({J}_{2}/{J}_{1}=0.5)

, the AF-fRVB state shows the variational ground-state energy in the thermodynamic limit very close to the one estimated by the projected entangled pair state at the largest bond dimension available. On the other hand, in the frustrated regime $0.4\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$, AF-fRVB states with

\text{ext}\phantom{\rule{0.16em}{0ex}}{s}_{2}

(using the terminology in the field of iron-based superconductors) and

{d}_{xy}

pairing symmetries are degenerate in the thermodynamic limit, implying the existence of gapless Dirac excitations in the spinon spectrum.

A gapped

Z_{2}

spin liquid phase with a

U(1)

mean field ansatz: a Bosonic resonating valence-bond description

Gapped

Z_{2}

spin liquid as the simplest spin liquid has been proved to be the most difficult to realize in realistic models. Here we show that the frustration from a third-neighbor exchange

J_{3}

on the spin-

\frac{1}{2}

J_{1}-J_{2}

model on the square lattice may serve to stabilize such a long-sought state. We argue that a Bosonic RVB description is more appropriate than a Fermonic RVB description for such a gapped spin liquid phase. We show that while the mean field ansatz of the proposed state has a

U(1)

gauge symmetry, the gauge fluctuation spectrum on it is actually gapped and the state should be understood rather as a

Z_{2}

spin liquid state with topological order. The state is thus locally stable with respect to gauge fluctuation and can emerge continuously from a collinear Neel ordered phase.

Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor

{J}_{1}

and second-neighbor

{J}_{2}

antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. N\'eel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio

{J}_{2}/{J}_{1}

, we find strong evidence for a continuous transition to a nonmagnetic phase at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.23$. Close to the transition point, the Gutzwiller-projected uniform resonating valence-bond state gives an excellent approximation to the exact ground-state energy. For $0.23\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.36$, a gapless

{Z}_{2}

spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the N\'eel order melts, this ordered state becomes clearly favored only for ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.3$. Finally, for $0.36\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$, a valence-bond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the N\'eel antiferromagnet from the plaquette valence-bond solid.

{J}_{2}/{J}_{1}=0.535(3)

, suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at

{J}_{2}/{J}_{1}=0.535(3)

in the model. From the crossing of the first derivative of the energies with

{J}_{2}

for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at

{J}_{2}/{J}_{1}=0.610(5)

The spin-

1/2

{J}_{1}

{J}_{2}

Heisenberg model on the square lattice represents one of the simplest examples in which the effects of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)], reconcile the contradicting results obtained within different approaches over the years.