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Physical review. B./Physical review. B

Dynamical structure factor of the SU(4) algebraic spin liquid on the honeycomb lattice

Dániel Vörös, Karlo Penc

Published Dec 7, 2023• 3 citations
Publisher

Abstract

We compute the momentum-resolved dynamical spin structure factor $S(k,\ensuremath{\omega})$ of the SU(4) Heisenberg model on the honeycomb lattice assuming the $\ensuremath{\pi}$-flux Dirac spin liquid ground state by using two methods: (i) variationally using Gutzwiller projected particle-hole excitations of the $\ensuremath{\pi}$-flux Fermi sea, and (ii) in the noninteracting parton mean-field picture. The two approaches produce qualitatively similar results. Based on this analogy, we argue that the energy spectrum of the projected excitations is a gapless continuum of fractional excitations. Quantitatively, the Gutzwiller projection shifts the weight from higher to lower energies, thus emphasizing the lower edge of the continuum. In the mean-field approach, we obtained the $1/{\text{distance}}^{4}$ decay of the spin correlation function, and the local correlations show ${S}_{\text{MF}}^{33}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{3}$ behavior.

DOI
10.1103/physrevb.108.214407

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We compute the momentum-resolved dynamical spin structure factor S(k,\ensuremathω)S(k,\ensuremath{\omega})S(k,\ensuremathω) of the SU(4) Heisenberg model on the honeycomb lattice assuming the \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux Dirac spin liquid ground state by using two methods: (i) variationally using Gutzwiller projected particle-hole excitations of the \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux Fermi sea, and (ii) in the noninteracting parton mean-field picture. The two approaches produce qualitatively similar results. Based on this analogy, we argue that the energy spectrum of the projected excitations is a gapless continuum of fractional excitations. Quantitatively, the Gutzwiller projection shifts the weight from higher to lower energies, thus emphasizing the lower edge of the continuum. In the mean-field approach, we obtained the 1/distance41/{\text{distance}}^{4}1/distance4 decay of the spin correlation function, and the local correlations show SMF33(\ensuremathω)\ensuremath∝\ensuremathω3{S}_{\text{MF}}^{33}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{3}SMF33​(\ensuremathω)\ensuremath∝\ensuremathω3 behavior.
We compute the momentum resolved dynamical spin structure factor S(k,ω)S(k,ω)S(k,ω) of the SU(4) Heisenberg model on the honeycomb lattice assuming the πππ-flux Dirac spin liquid ground state by two methods: (i) variationally using Gutzwiller projected particle-hole excitations of the πππ-flux Fermi sea and (ii) in the non-interacting parton mean-field picture. The two approaches produce qualitatively similar results. Based on this analogy, we argue that the energy spectrum of the projected excitations is a gapless continuum of fractional excitations. Quantitatively, the Gutzwiller projection shifts the weight from higher to lower energies, thus emphasizing the lower edge of the continuum. In the mean-field approach, we obtained the 1/distance41/\text{distance}^41/distance4 decay of the spin correlation function, and the local correlations show SMF33(ω)∝ω3S^{33}_{\text{MF}}(ω)\propto ω^3SMF33​(ω)∝ω3 behavior.
By combining the density matrix renormalization group (DMRG) method with Gutzwiller projected wave functions, we study the SU(4) symmetric spin-orbital model on the honeycomb lattice. We find that the ground states can be well described by a Gutzwiller projected $\ensuremath{\pi}$-flux state with Dirac-type gapless excitations at one quarter filling. Although these Dirac points are gapped by emergent gauge fluxes on finite cylinders, they govern the critical behavior in the thermodynamic limit. By inserting a $\ensuremath{\theta}=\ensuremath{\pi}$ spin flux to twist the boundary condition, we can shift the gapless sector to the ground state, which provides compelling evidence for the presence of a gapless Dirac spin-orbital liquid.
We perform large-scale density matrix renormalization group simulations of the SU(4) Heisenberg model on the honeycomb lattice to address the long-standing question of its ground state in an unbiased and quantitatively controlled manner. We find reliable numerical evidence that the ground state is a gapped spin-orbital liquid, presumably with a Z_{4} topological order, characterized by a finite topological entanglement entropy close to ln(4), the absence of both SU(4) and lattice symmetry breaking, and a variationally optimized ground-state energy well below the previously proposed π-flux variational state. By exploiting full SU(4) symmetry and keeping up to 12 800 SU(4) multiplets, corresponding to more than 1×10^{6} U(1) states, we achieve unprecedented accuracy for two-dimensional SU(4) quantum magnets. Finite-size scaling of energies and entanglement entropies supports a robust gapped phase in the two-dimensional limit, while a gapless critical state on narrow cylinders is identified as a proximate remnant of a Dirac spin-orbital liquid. Our results find the SU(4) honeycomb Heisenberg model a realization of a gapped topological spin-orbital liquid and provide convincing numerical evidence for topological order in a highly symmetric two-dimensional quantum magnet.
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 compute the dynamical spin structure factor $S(k,\ensuremath{\omega})$ of the SU(3) Heisenberg chain variationally using a truncated Hilbert space spanned by the Gutzwiller projected particle-hole excitations of the Fermi sea, introduced by Dalla Piazza et al. [Nat. Phys. 11, 62 (2015)], with a modified importance sampling. We check the reliability of the method by comparing the $S(k,\ensuremath{\omega})$ to exact diagonalization results for 18 sites and to the two-soliton continuum of the Bethe ansatz for 72 sites. We get an excellent agreement in both cases. Detailed analysis of the finite-size effects shows that the method captures the critical Wess-Zumino-Witten SU(3)1{}_{1}1​ behavior and reproduces the correct exponent, with the exception of the size dependence of the weight of the bottom of the conformal tower. We also calculate the single-mode approximation for the SU(N)\mathrm{SU}(N)SU(N) Heisenberg model and determine the velocity of excitations. Finally, we apply the method to the SU(3) Haldane-Shastry model and find that the variational method gives the exact wave function for the lowest excitation at $k=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}/3$.
We present an augmented parton mean-field theory which (i) reproduces the exact ground state, spectrum, and dynamics of the quantum spin-liquid phase of Kitaev's honeycomb model, and (ii) is amenable to the inclusion of integrability breaking terms, allowing a perturbation theory from a controlled starting point. Thus, we exemplarily study dynamical spin correlations of the honeycomb Kitaev quantum spin liquid within the $K\ensuremath{-}J\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model, which includes Heisenberg and symmetric-anisotropic (pseudodipolar) interactions. This allows us to trace changes of the correlations in the regime of slowly moving fluxes, where the theory captures the dominant deviations when integrability is lost. These include an asymmetric shift together with a broadening of the dominant peak in the response as a function of frequency, the generation of further-neighbor correlations and their structure in real and spin space, and a resulting loss of an approximate rotational symmetry of the structure factor in reciprocal space. We discuss the limitations of this approach and also view the neutron-scattering experiments on the putative proximate quantum spin-liquid material $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ in the light of the results from this extended parton theory.