Spin excitation spectra of two dimensional $S=1/2$ Heisenberg model with checkerboard structures.

Yining Xu, Zijian Xiong, Han-Qing Wu, Dao‐Xin Yao

Published Dec 2, 2018• 0 citations

Abstract

We study the excitation spectra of two-dimensional spin-$1/2$ Heisenberg model with checkerboard structures using stochastic analytic continuation of imaginary-time correlation function obtained from quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions $J_{1}$ and $J_{2}$, and the tuning parameter $g$ is defined as $J_{2}/J_{1}$. The dynamic structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, wave theory is employed to explain some features of numerical results, especially for the low-energy part. When $g$ is close to $1$, the features of the excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them are discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\times 3$ checkerboard model has distinctive excitation features, like a gap between a low-energy gapless branch and a gapped high-energy part exists when $g$ is small. The gapless branch in this case can be regarded as wave in N$\mathrm{\acute{e}}$el order formed by block spin in each $3\times 3$ plaquette with effective exchange interaction originated from renormalization. One finding is that the continuum also appears in this low-energy branch, which may originated from nearly deconfined process in block spin.

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Spin excitation spectra of two dimensional $S=1/2$ Heisenberg model with checkerboard structures.Spin excitation spectra of two dimensional $S=1/2$ Heisenberg model with checkerboard structures.

Yining Xu, Zijian Xiong, Han-Qing Wu, Dao‐Xin Yao

Published Dec 2, 2018• 0 citations

OpenAlex

Abstract

We study the excitation spectra of two-dimensional spin-

1/2

Heisenberg model with checkerboard structures using stochastic analytic continuation of imaginary-time correlation function obtained from quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions

J_{1}

and

J_{2}

, and the tuning parameter

g

is defined as

J_{2}/J_{1}

. The dynamic structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, wave theory is employed to explain some features of numerical results, especially for the low-energy part. When

g

is close to

1

, the features of the excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them are discussed. In contrast to the other checkerboard structures investigated in this paper, the

3\times 3

checkerboard model has distinctive excitation features, like a gap between a low-energy gapless branch and a gapped high-energy part exists when

g

is small. The gapless branch in this case can be regarded as wave in N

\mathrm{\acute{e}}

el order formed by block spin in each

3\times 3

plaquette with effective exchange interaction originated from renormalization. One finding is that the continuum also appears in this low-energy branch, which may originated from nearly deconfined process in block spin.

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Spin excitation spectra of the two-dimensional S=12 Heisenberg model with a checkerboard structure

Yining Xu, Zijian Xiong, Han-Qing Wu +1 more

We study the spin excitation spectra of the two-dimensional spin- $\frac{1}{2}$ Heisenberg model with a checkerboard structure using stochastic analytic continuation of the imaginary-time correlation function obtained from a quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions ${J}_{1}$ and ${J}_{2}$ , and the tuning parameter $g$ is defined as ${J}_{2}/{J}_{1}$ . The dynamic spin structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, spin wave theory is employed to explain some features of numerical results, especially for the low-energy part. When $g$ is close to 1, the features of the spin excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them is discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\ifmmode\times\else\texttimes\fi{}3$ checkerboard model has distinctive excitation features, such as a gap between a low-energy gapless branch and a gapped high-energy part that exists when $g$ is small. The gapless branch in this case can be regarded as a spin wave in N\'eel order formed by a ``block spin'' in each $3\ifmmode\times\else\texttimes\fi{}3$ plaquette with an effective exchange interaction originating from renormalization. One unexpected finding is that the continuum also appears in this low-energy branch.We study the spin excitation spectra of the two-dimensional spin-$\frac{1}{2}$ Heisenberg model with a checkerboard structure using stochastic analytic continuation of the imaginary-time correlation function obtained from a quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions ${J}_{1}$ and ${J}_{2}$, and the tuning parameter $g$ is defined as ${J}_{2}/{J}_{1}$. The dynamic spin structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, spin wave theory is employed to explain some features of numerical results, especially for the low-energy part. When $g$ is close to 1, the features of the spin excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them is discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\ifmmode\times\else\texttimes\fi{}3$ checkerboard model has distinctive excitation features, such as a gap between a low-energy gapless branch and a gapped high-energy part that exists when $g$ is small. The gapless branch in this case can be regarded as a spin wave in N\'eel order formed by a ``block spin'' in each $3\ifmmode\times\else\texttimes\fi{}3$ plaquette with an effective exchange interaction originating from renormalization. One unexpected finding is that the continuum also appears in this low-energy branch.

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We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$, the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.

Nov 10, 20130 citations

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.

arXiv:1410.7557Journal of the Physical Society of JapanJan 29, 2015122 citations

Spin excitation spectra of two dimensional

S=1/2

Heisenberg model with checkerboard structures.

We study the excitation spectra of two-dimensional spin-

1/2

J_{1}

and

J_{2}

, and the tuning parameter

g

is defined as

J_{2}/J_{1}

g

is close to

1

3\times 3

checkerboard model has distinctive excitation features, like a gap between a low-energy gapless branch and a gapped high-energy part exists when

g

is small. The gapless branch in this case can be regarded as wave in N

\mathrm{\acute{e}}

el order formed by block spin in each

3\times 3

We study the spin excitation spectra of the two-dimensional spin-

\frac{1}{2}

Heisenberg model with a checkerboard structure using stochastic analytic continuation of the imaginary-time correlation function obtained from a quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions

{J}_{1}

and

{J}_{2}

, and the tuning parameter

g

is defined as

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

. The dynamic spin structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, spin wave theory is employed to explain some features of numerical results, especially for the low-energy part. When

g

is close to 1, the features of the spin excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them is discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\ifmmode\times\else\texttimes\fi{}3$ checkerboard model has distinctive excitation features, such as a gap between a low-energy gapless branch and a gapped high-energy part that exists when

g

is small. The gapless branch in this case can be regarded as a spin wave in N\'eel order formed by a ``block spin'' in each $3\ifmmode\times\else\texttimes\fi{}3$ plaquette with an effective exchange interaction originating from renormalization. One unexpected finding is that the continuum also appears in this low-energy branch.

We investigate the

S=1/2

antiferromagnetic Heisenberg model with a $3\ifmmode\times\else\texttimes\fi{}3$ checkerboard lattice structure in a longitudinal magnetic field. By using the stochastic series expansion quantum Monte Carlo method, we obtain the properties of the nonplateau XY phase; 1/9, 3/9, 5/9, and 7/9 magnetization plateau phases; and fully polarized phase. Then, we determine the precise phase transition critical points belonging to the three-dimensional XY universality class through finite-size scaling. Moreover, we study the longitudinal and transverse dynamic spin structure factors of this model in different phases. For the nonplateau XY phase, the energy spectra present a gap between the low-energy gapless branch and the high-energy part due to the competition of magnetic field and interaction. The gapless branch can be described by the spin wave theory of the effective ``block spin'' model in the canted antiferromagnetic phase. In the magnetization plateau phase, we identify that the low-energy excitations arise from the localized disturbances within the sublattice and are capable of spreading in momentum space. The high-energy quasiparticles could be the multiplet excitations similar to the ``triplon'' in the dimerized system and ``doublon'' and ``quarton'' in the trimerized system. This paper offers theoretical insights and interpretations for the characteristics of the ground state and the inelastic neutron scattering spectra in two-dimensional quantum magnetic materials. Specifically, it focuses on materials with odd-spin configurations in the checkerboard unit cells under the influence of a longitudinal magnetic field.

We study the phase diagram and the dynamical spin structure factor of the spin-1/2

{J}_{1}--{J}_{3}

Heisenberg model on the square lattice using density matrix renormalization group, exact diagonalization (ED), and cluster perturbation theory (CPT). By extrapolating the order parameters and studying the level crossings of the low-lying energy and entanglement spectra, we obtain the phase diagram of this model and identify a narrow region of quantum spin liquid (QSL) phase followed by a plaquette valence-bond solid (PVBS) state in the intermediate region, whose nature has been controversial for many years. More importantly, we use CPT and ED to study the dynamical spin structure factor in the QSL and the PVBS phase. In the QSL phase, the high-energy magnon mode completely turns into some dispersive weak excitations around the

X

and

M

points. For the PVBS phase, the low-energy spectrum is characterized by a gapped triplet excitation, and at the high energy, we find another branch of dispersive excitation with broad continua, which is unlike the plaquette phase in the $2\ifmmode\times\else\texttimes\fi{}2$ checkerboard model. In the latter case, the second branch of excitation is nearly flat due to the weak effective interactions between the local excitations of the plaquettes. And in the

{J}_{1}--{J}_{3}

Heisenberg model, the uniform interactions and the spontaneously translational symmetry breaking of the PVBS phase make the difference in the excitation spectra.

\Gamma(\mathbf q=0)

and at the magnetic ordering wavevectors

\pm[\mathbf Q= (4\pi/3,0)]

, the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.

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.