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arXiv:1807.02399 · Physical review. B./Physical review. B

Quantum dynamics of the square-lattice Heisenberg model

Ruben Verresen, Frank Pollmann, Roderich Moessner

Published Oct 1, 2018• 44 citations
arXivHTMLPDF

Abstract

Despite nearly a century of study of the $S=1/2$ Heisenberg model on the square lattice, there is still disagreement on the nature of its high-energy excitations. By tuning toward the Heisenberg model from the exactly soluble Ising limit, we find that the strongly attractive magnon interactions of the latter naturally account for a number of spectral features of the Heisenberg model. This claim is backed up both numerically and analytically. Using the density matrix renormalization group method, we obtain the dynamical structure factor for a cylindrical geometry, allowing us to continuously connect both limits. Remarkably, a semiquantitative description of certain observed features arises already at the lowest nontrivial order in perturbation theory around the Ising limit. Moreover, our analysis uncovers that high-energy magnons are localized on a single sublattice, which is related to the entanglement properties of the ground state.

DOI
10.1103/physrevb.98.155102

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Despite nearly a century of study of the S=1/2S=1/2S=1/2 Heisenberg model on the square lattice, there is still disagreement on the nature of its high-energy excitations. By tuning toward the Heisenberg model from the exactly soluble Ising limit, we find that the strongly attractive magnon interactions of the latter naturally account for a number of spectral features of the Heisenberg model. This claim is backed up both numerically and analytically. Using the density matrix renormalization group method, we obtain the dynamical structure factor for a cylindrical geometry, allowing us to continuously connect both limits. Remarkably, a semiquantitative description of certain observed features arises already at the lowest nontrivial order in perturbation theory around the Ising limit. Moreover, our analysis uncovers that high-energy magnons are localized on a single sublattice, which is related to the entanglement properties of the ground state.
We study the phase diagram and the dynamical spin structure factor of the spin-1/2 J1−−J3{J}_{1}--{J}_{3}J1​−−J3​ 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 XXX and MMM 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 J1−−J3{J}_{1}--{J}_{3}J1​−−J3​ Heisenberg model, the uniform interactions and the spontaneously translational symmetry breaking of the PVBS phase make the difference in the excitation spectra.
Spin waves (magnons) in two dimensions are the potential glue in high-temperature superconductors so that their quantitative understanding is mandatory. Yet even for the fundamental case of the undoped Heisenberg model on the square lattice a consistent picture is still lacking. Significant spectral continua are taken as evidence of the existence of fractional excitations (spinons), but descriptions in terms of spinons fail to show the established absence of an energy gap. Here a fully consistent picture of the dynamics in the square-lattice quantum antiferromagnet is provided which agrees with the experimental findings. The key step is to capture (i) the strong attractive interaction between the spin waves and (ii) the vertex corrections of the observables.
Understanding the nature of the excitation spectrum in quantum spin liquids is of fundamental importance, in particular for the experimental detection of candidate materials.However, current theoretical and numerical techniques have limited capabilities, especially in obtaining the dynamical structure factor, which gives a crucial characterization of the ultimate nature of the quantum state and may be directly assessed by inelastic neutron scattering.In this work, we investigate the low-energy properties of the S ¼ 1=2 Heisenberg model on the triangular lattice, including both nearest-neighbor J 1 and next-nearestneighbor J 2 superexchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models.For J 2 ¼ 0, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations at K points (i.e., at the corners of the Brillouin zone).The strong renormalization of the magnon branch (also including rotonlike minima around the M points, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a nontrivial magnetic π flux threading half of the triangular plaquettes.When increasing the frustrating ratio J 2 =J 1 , we detect a progressive softening of the magnon branch at M, which eventually becomes gapless within the spin-liquid phase.This feature is captured by the band structure of the unprojected wave function (with two Dirac points for each spin component).In addition, we observe an intense signal at low energies around the K points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids.
We employ spin-cluster perturbation theory to investigate the evolution of the spin excitation spectra of the mixed-dimensional antiferromagnetic s=12s=\frac{1}{2}s=21​ Heisenberg models from the one-dimensional chain to the two-dimensional square and honeycomb lattices by varying the spin interactions ${J}_{\ensuremath{\perp}}$ across the chains relative to ${J}_{\ensuremath{\parallel}}$ along the chains. In addition to the well-defined magnon excitations, the spectra of the N\'eel phases for both lattices exhibit anomalous high-energy continua that evolve continuously from the continuum of one-dimensional chain through increasing ${J}_{\ensuremath{\perp}}$. Notably, the key features of the continua at the two-dimensional limit for ${J}_{\ensuremath{\perp}}/{J}_{\ensuremath{\parallel}}=1$ are well consistent with experimental observations in real materials. Our results indicate that these high-energy continua in the N\'eel phases are a result of spinon deconfinement, which is further supported by a random-phase-approximation analysis based on the spinon Hubbard models.
Abstract Motivated by the need for high-precision benchmark results for basic quantum many-body models, several ground state properties of the square lattice S = 1 / 2 Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures T sufficiently low to realize the T → 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L × L lattices with L ∈ [ 6 , 96 ] are tabulated versus L and extrapolations to infinite system size are carried out. The relative statistical errors of finite-size energies are below 10 −7 . The extrapolated ground state energy density is e 0 = − 0.669 441 857 ( 7 ) , where the number within parentheses is the statistical error (one standard deviation) of the preceding digit. This value represents an improvement in precision of three orders of magnitude over the previous best result. The leading and subleading finite-size corrections to e 0 are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is m s = 0.307 447 ( 2 ) , which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in L −1 correction to m s 2 agrees with the value from chiral perturbation theory and the presence of a factor ln γ ⁡ ( L ) in the second-order correction is also confirmed, with the previously unknown value of the exponent being γ = 0.82 ( 4 ) . The finite-size corrections to the staggered susceptibility point to logarithmic corrections in this quantity as well. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.
We compute the zero temperature dynamical structure factor S(q,ω)S({\bf q},\omega)S(q,ω) of the triangular lattice Heisenberg model using a Schwinger Boson approach that includes the Gaussian fluctuations (1/N1/N1/N correction) of the saddle point solution. While the ground state of this model exhibits a well-known 120∘^{\circ}∘ magnetic ordering, the excitation spectrum revealed by S(q,ω)S({\bf q},\omega)S(q,ω) has a strong quantum character that is not captured by low-order 1/S1/S1/S expansions. The low-energy magnons consist of two spinon bound states confined by gauge fluctuations of the auxiliary fields, which can produce internal structure of the magnon peaks. In addition, the high-energy continuum of spinon modes extends up to about three times the single-magnon bandwidth.