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:2301.00829 · Physical Review Research

Dynamical properties of quantum many-body systems with long-range interactions

Menghan Song, Jiarui Zhao, Chengkang Zhou, Zi Yang Meng

Published Jul 25, 2023• 21 citations
arXivHTMLPDF

Abstract

Employing large-scale quantum Monte Carlo simulations, we systematically compute the energy spectra of the two-dimensional (2D) spin-1/2 Heisenberg model with long-range interactions. With the $1/{r}^{\ensuremath{\alpha}}$ ferromagnetic and staggered antiferromagnetic interactions, we find the explicit range in $\ensuremath{\alpha}$ for the short-range Goldstone-type (gapless), anomalous Goldstone-type (gapless), and Higgs-type (gapped) spectra. Accompanied by the spin-wave analysis, our numerical results vividly reveal how the long-range interactions alter the usual linear and quadratic magnon dispersions in 2D quantum magnets and give rise to anomalous dynamical exponents. Moreover, we find the explicit case where the gapped excitation emerges at a noninteger decay exponent $\ensuremath{\alpha}$ for the antiferromagnetic Hamiltonian. This work provides the first set of unbiased dynamical data of long-range quantum many-body systems and suggests that many universally accepted low-energy customs for short-range systems need to be substantially modified for long-range ones, which are of immediate relevance to the ongoing experimental efforts from quantum simulators to 2D quantum moir\'e materials.

DOI
10.1103/physrevresearch.5.033046

Related papers

Study of the excited states of the quantum antiferromagnets

Mohammad G. Merdan

We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. 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, 2013•0 citations

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

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

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.

•Dec 2, 2018•0 citations

Longitudinal excitations in quantum antiferromagnets

Yang Xian

By extending our recently proposed magnon-density waves to low dimensions, we investigate, using a microscopic many-body approach, the longitudinal excitations of the quasi-one-dimensional (quasi-1d) and quasi-2d Heisenberg antiferromagnetic systems on a bipartite lattice with a general spin quantum number. We obtain the full energy spectrum of the longitudinal mode as a function of the coupling constants in the original lattice Hamiltonian and find that it always has a nonzero energy gap if the ground state has a long-range order and becomes gapless for the pure isotropic 1d model. The numerical value of the minimum gap in our approximation agrees with that of a longitudinal mode observed in the quasi-1d antiferromagnetic compound KCuF3 at low temperature. It will be interesting to compare values of the energy spectrum at other momenta if their experimental results are available.

arXiv:1105.1959•Journal of Physics Condensed Matter•Aug 12, 2011•7 citations

Dynamics and interaction of magnons in the two dimensional Heisenberg antiferromagnet

Michael Powalski

In this thesis the elementary excitations of the spin-1=2 Heisenberg antiferromagnet on the square lattice are studied. This fundamental model is a paradigmatic example for long range ordered quantum phases and the spontaneous breaking of a continuous symmetry in condensed matter. The elementary excitations at long wavelengths are well understood by quantized spin waves, i.e., magnons, which are the gapless Goldstone bosons of the broken SU(2) symmetry. However, recent findings reveal an anomalous energy dip at short wavelengths, which cannot be explained by the conventional spin wave theory. The nature of spin waves at short wavelengths remains unclear to this day. A central aim of this thesis is the derivation of an effective magnon description for the square lattice antiferromagnet which is quantitatively valid for all length scales. For this purpose, the method of continuous unitary transformations (CUT) is extended to its use for gapless excitations. CUT methods provide a powerful tool for the derivation of effective quasi-particle descriptions by means of flow equations. However, typical approaches are truncated in real space restricting their scope of application to gapped quantum phases. In order to circumvent these limitations a CUT approach in momentum space is developed. The scaling dimension of operator terms is established as a suitable truncation criterion for gapless quasi-particles. Moreover, a diagrammatic representation of operator terms is introduced which simplifies the derivation and interpretation of flow equations in momentum space. The developed approach is used to transform the Heisenberg Hamiltonian into an effective Hamiltonian which conserves the number of magnons. In addition, effective observables required for the theoretical description of inelastic neutron scattering are derived. The initial magnon operators before the transformation are defined by the Dyson-Maleev representation which leads to a superficially non-hermitian Hamiltonian. As a result, the transformation constitutes a continuous similarity transformation (CST). This formalism enables the formulation of spin wave interactions by means of quartic boson operators. A crucial step of the transformation is the proper renormalization of the spin wave interaction which is achieved by a selfsimilar CST for scaling dimension d = 2. The resulting effective magnon description is consistent at all length scales. In particular, the magnon dispersion exhibits an energy dip at short wavelengths which is in quantitative agreement with recent numerical and experimental data. Further analysis of the spectral properties indicates a strong attraction between spin waves giving rise to a resonance in the longitudinal two-magnon channel. This resonance can be interpreted as the amplitude or Higgs mode of the continuously broken SU(2) symmetry with finite lifetime. The origin of the anomalous dispersion is ascribed to a scattering between the Higgs resonance and single magnon states. The theoretical results are directly compared with experimental data obtained by inelastic neutron scattering and various numerical results which support this view. The quantitative agreement between the effective spin wave approach and numerous numerical and experimental findings shows that magnons are a valid description of the elementary excitations at all length scales. Therefore, this thesis provides a final settlement of the question about the nature of the elementary excitations in the square lattice quantum antiferromagnet.

•Jan 1, 2017•0 citations

Low-energy properties of anisotropic two-dimensional spin- 12 Heisenberg models in staggered magnetic fields

Bin Xi, Shijie Hu, Jize Zhao +3 more

We present a systematic study of the anisotropic spin-1/2 Heisenberg model in staggered magnetic fields in two dimensions. To mimic real materials, we consider a system of coupled, antiferromagnetic chains, whose interchain interaction can be either ferro- or antiferromagnetic. When the staggered field is commensurate with the magnetic interactions, an energy gap opens immediately and follows a power law as a function of the applied field, similar to the situation in one dimension. When the field competes with the interactions, a quantum phase transition (QPT) occurs from a gapless, magnetically ordered phase at low fields to a gapped, disordered regime. We use a continuous-time Monte Carlo method to compute the staggered moment of the ordered phases and the spin gap of the disordered phases. We deduce the phase diagrams as functions of the anisotropy ratio and the applied field, and calculate the scaling behavior of the models in both quantities. We show that in the competitive case, the staggered field acts to maintain a regime of quasi-1D behavior around the QPT, and we discuss as a consequence the nature of the crossover from one-dimensional (1D) to 2D physics.

arXiv:1106.4605•Physical Review B•Oct 10, 2011•11 citations

Dynamics of correlations in two-dimensional quantum spin models with long-range interactions: a phase-space Monte-Carlo study

Johannes Schachenmayer, Alexander Pikovski, A M Rey

Interacting quantum spin models are remarkably useful for describing different types of physical, chemical, and biological systems.Significant understanding of their equilibrium properties has been achieved to date, especially for the case of spin models with short-range couplings.However, progress toward the development of a comparable understanding in long-range interacting models, in particular out-of-equilibrium, remains limited.In a recent work, we proposed a semiclassical numerical method to study spin models, the discrete truncated Wigner approximation (DTWA), and demonstrated its capability to correctly capture the dynamics of one-and two-point correlations in one-dimensional (1D) systems.Here we go one step forward and use the DTWA method to study the dynamics of correlations in two-dimensional (2D) systems with many spins and different types of long-range couplings, in regimes where other numerical methods are generally unreliable.We compute spatial and time-dependent correlations for spin-couplings that decay with distance as a power-law and determine the velocity at which correlations propagate through the system.Sharp changes in the behavior of those velocities are found as a function of the power-law decay exponent.Our predictions are relevant for a broad range of systems including solid state materials, atom-photon systems and ultracold gases of polar molecules, trapped ions, Rydberg, and magnetic atoms.We validate the DTWA predictions for small 2D systems and 1D systems, but ultimately, in the spirt of quantum simulation, experiments will be needed to confirm our predictions for large 2D systems.

arXiv:1501.06593•New Journal of Physics•Jun 10, 2015•69 citations
Employing large-scale quantum Monte Carlo simulations, we systematically compute the energy spectra of the two-dimensional (2D) spin-1/2 Heisenberg model with long-range interactions. With the 1/r\ensuremathα1/{r}^{\ensuremath{\alpha}}1/r\ensuremathα ferromagnetic and staggered antiferromagnetic interactions, we find the explicit range in \ensuremathα\ensuremath{\alpha}\ensuremathα for the short-range Goldstone-type (gapless), anomalous Goldstone-type (gapless), and Higgs-type (gapped) spectra. Accompanied by the spin-wave analysis, our numerical results vividly reveal how the long-range interactions alter the usual linear and quadratic magnon dispersions in 2D quantum magnets and give rise to anomalous dynamical exponents. Moreover, we find the explicit case where the gapped excitation emerges at a noninteger decay exponent \ensuremathα\ensuremath{\alpha}\ensuremathα for the antiferromagnetic Hamiltonian. This work provides the first set of unbiased dynamical data of long-range quantum many-body systems and suggests that many universally accepted low-energy customs for short-range systems need to be substantially modified for long-range ones, which are of immediate relevance to the ongoing experimental efforts from quantum simulators to 2D quantum moir'e materials.
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. 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 Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/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.
Spin excitation spectra of two dimensional S=1/2S=1/2S=1/2 Heisenberg model with checkerboard structures.
We study the excitation spectra of two-dimensional spin-1/21/21/2 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 J1J_{1}J1​ and J2J_{2}J2​, and the tuning parameter ggg is defined as J2/J1J_{2}/J_{1}J2​/J1​. 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 ggg is close to 111, 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 3×33\times 33×3 checkerboard model has distinctive excitation features, like a gap between a low-energy gapless branch and a gapped high-energy part exists when ggg is small. The gapless branch in this case can be regarded as wave in Neˊ\mathrm{\acute{e}}eˊel order formed by block spin in each 3×33\times 33×3 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.
By extending our recently proposed magnon-density waves to low dimensions, we investigate, using a microscopic many-body approach, the longitudinal excitations of the quasi-one-dimensional (quasi-1d) and quasi-2d Heisenberg antiferromagnetic systems on a bipartite lattice with a general spin quantum number. We obtain the full energy spectrum of the longitudinal mode as a function of the coupling constants in the original lattice Hamiltonian and find that it always has a nonzero energy gap if the ground state has a long-range order and becomes gapless for the pure isotropic 1d model. The numerical value of the minimum gap in our approximation agrees with that of a longitudinal mode observed in the quasi-1d antiferromagnetic compound KCuF3 at low temperature. It will be interesting to compare values of the energy spectrum at other momenta if their experimental results are available.
In this thesis the elementary excitations of the spin-1=2 Heisenberg antiferromagnet on the square lattice are studied. This fundamental model is a paradigmatic example for long range ordered quantum phases and the spontaneous breaking of a continuous symmetry in condensed matter. The elementary excitations at long wavelengths are well understood by quantized spin waves, i.e., magnons, which are the gapless Goldstone bosons of the broken SU(2) symmetry. However, recent findings reveal an anomalous energy dip at short wavelengths, which cannot be explained by the conventional spin wave theory. The nature of spin waves at short wavelengths remains unclear to this day. A central aim of this thesis is the derivation of an effective magnon description for the square lattice antiferromagnet which is quantitatively valid for all length scales. For this purpose, the method of continuous unitary transformations (CUT) is extended to its use for gapless excitations. CUT methods provide a powerful tool for the derivation of effective quasi-particle descriptions by means of flow equations. However, typical approaches are truncated in real space restricting their scope of application to gapped quantum phases. In order to circumvent these limitations a CUT approach in momentum space is developed. The scaling dimension of operator terms is established as a suitable truncation criterion for gapless quasi-particles. Moreover, a diagrammatic representation of operator terms is introduced which simplifies the derivation and interpretation of flow equations in momentum space. The developed approach is used to transform the Heisenberg Hamiltonian into an effective Hamiltonian which conserves the number of magnons. In addition, effective observables required for the theoretical description of inelastic neutron scattering are derived. The initial magnon operators before the transformation are defined by the Dyson-Maleev representation which leads to a superficially non-hermitian Hamiltonian. As a result, the transformation constitutes a continuous similarity transformation (CST). This formalism enables the formulation of spin wave interactions by means of quartic boson operators. A crucial step of the transformation is the proper renormalization of the spin wave interaction which is achieved by a selfsimilar CST for scaling dimension d = 2. The resulting effective magnon description is consistent at all length scales. In particular, the magnon dispersion exhibits an energy dip at short wavelengths which is in quantitative agreement with recent numerical and experimental data. Further analysis of the spectral properties indicates a strong attraction between spin waves giving rise to a resonance in the longitudinal two-magnon channel. This resonance can be interpreted as the amplitude or Higgs mode of the continuously broken SU(2) symmetry with finite lifetime. The origin of the anomalous dispersion is ascribed to a scattering between the Higgs resonance and single magnon states. The theoretical results are directly compared with experimental data obtained by inelastic neutron scattering and various numerical results which support this view. The quantitative agreement between the effective spin wave approach and numerous numerical and experimental findings shows that magnons are a valid description of the elementary excitations at all length scales. Therefore, this thesis provides a final settlement of the question about the nature of the elementary excitations in the square lattice quantum antiferromagnet.
We present a systematic study of the anisotropic spin-1/2 Heisenberg model in staggered magnetic fields in two dimensions. To mimic real materials, we consider a system of coupled, antiferromagnetic chains, whose interchain interaction can be either ferro- or antiferromagnetic. When the staggered field is commensurate with the magnetic interactions, an energy gap opens immediately and follows a power law as a function of the applied field, similar to the situation in one dimension. When the field competes with the interactions, a quantum phase transition (QPT) occurs from a gapless, magnetically ordered phase at low fields to a gapped, disordered regime. We use a continuous-time Monte Carlo method to compute the staggered moment of the ordered phases and the spin gap of the disordered phases. We deduce the phase diagrams as functions of the anisotropy ratio and the applied field, and calculate the scaling behavior of the models in both quantities. We show that in the competitive case, the staggered field acts to maintain a regime of quasi-1D behavior around the QPT, and we discuss as a consequence the nature of the crossover from one-dimensional (1D) to 2D physics.
Interacting quantum spin models are remarkably useful for describing different types of physical, chemical, and biological systems.Significant understanding of their equilibrium properties has been achieved to date, especially for the case of spin models with short-range couplings.However, progress toward the development of a comparable understanding in long-range interacting models, in particular out-of-equilibrium, remains limited.In a recent work, we proposed a semiclassical numerical method to study spin models, the discrete truncated Wigner approximation (DTWA), and demonstrated its capability to correctly capture the dynamics of one-and two-point correlations in one-dimensional (1D) systems.Here we go one step forward and use the DTWA method to study the dynamics of correlations in two-dimensional (2D) systems with many spins and different types of long-range couplings, in regimes where other numerical methods are generally unreliable.We compute spatial and time-dependent correlations for spin-couplings that decay with distance as a power-law and determine the velocity at which correlations propagate through the system.Sharp changes in the behavior of those velocities are found as a function of the power-law decay exponent.Our predictions are relevant for a broad range of systems including solid state materials, atom-photon systems and ultracold gases of polar molecules, trapped ions, Rydberg, and magnetic atoms.We validate the DTWA predictions for small 2D systems and 1D systems, but ultimately, in the spirt of quantum simulation, experiments will be needed to confirm our predictions for large 2D systems.