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arXiv:2503.14643 · Physical review. B./Physical review. B

Confined and deconfined spinon excitations in the rectangular-lattice quantum antiferromagnet

N. E. Shaik, Ellen Fogh, B. Dalla Piazza, Bruce Normand, Dmitri A. Ivanov, Henrik Moodysson Rønnow

Published Jul 16, 2025• 1 citation
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Abstract

Fractionalization remains one of the most fascinating manifestations of strong interactions in quantum many-body systems. In quantum magnetism, the existence of spinons—collective magnetic excitations that behave as quasiparticles with fractional quantum numbers—is proven in spin chains, but the criteria for their appearance in higher dimensions remain disputed. Motivated by experiments reporting the observation of spinons at high energies in the square-lattice Heisenberg antiferromagnet, we adopt the approach of extrapolating from where spinons are well defined. We study the dynamical properties of a Gutzwiller-projected wave function, the staggered-flux state, on a rectangular spin-1/2 Heisenberg lattice as a function of the spatial coupling ratio, γ = J y / J x . By studying the spectrum and the spinon separation distribution we show how, as the system evolves from one-dimensional (1D) towards 2D, the spinons become progressively more confined over most of reciprocal space, but remain deconfined at specific wave vectors.

DOI
10.1103/zntl-4622

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Fractionalization remains one of the most fascinating manifestations of strong interactions in quantum many-body systems. In quantum magnetism, the existence of spinons—collective magnetic excitations that behave as quasiparticles with fractional quantum numbers—is proven in spin chains, but the criteria for their appearance in higher dimensions remain disputed. Motivated by experiments reporting the observation of spinons at high energies in the square-lattice Heisenberg antiferromagnet, we adopt the approach of extrapolating from where spinons are well defined. We study the dynamical properties of a Gutzwiller-projected wave function, the staggered-flux state, on a rectangular spin-1/2 Heisenberg lattice as a function of the spatial coupling ratio, γ = J y / J x . By studying the spectrum and the spinon separation distribution we show how, as the system evolves from one-dimensional (1D) towards 2D, the spinons become progressively more confined over most of reciprocal space, but remain deconfined at specific wave vectors.
This thesis presents a theoretical study of the Heisenberg model on a two dimensional lattice with asymmetric couplings along x and y directions. By gradually varying the coupling along y-direction we traverse the 1D, quasi 1D and 2D domains of this model. Primary focus of the research revolves around exploring the merits and de-merits of the staggered-flux state, a prototypical disordered resonating valance bond state, as a candidate wavefunction within this domain. Novel features like the fractionalization of spin-1 `magnon' excitations into spin-1/2 `spinon' excitations, which qualitatively explain the experimentally observed (π,0)(\pi,0)(π,0) quantum anomaly, demand further investigation into this novel wave function. Theoretical framework involving construction of a Gutzwiller projected staggered flux state, a wavefunction depending on variational parameters, is developed. A numerical method based on MonteCarlo simulation is used to optimize the variational wave function and calculate the relevant physical observables. Exploring the gradual evolution of these observables with the coupling ratio γ=Jy/Jx\gamma=J_y/J_xγ=Jy​/Jx​ provides us with a clear picture of the evolution of spin excitations on this RVB state, as the dimensionality of the model is changed from 1D to 2D.
The familiar spin-12\frac{1}{2}21​ quantum Heisenberg antiferromanget in a two-dimensional (2D) square lattice is shown, within the nonlinear sigma model approximations, to be another state of matter that has excitations with fractional quantum numbers above a finite energy gap. The 1-skyrmion with an energy $\ensuremath{\approx}2\ensuremath{\pi}J$ is shown to be made of two ``deconfined spinons'' or ``O(3) vortices.'' The many skyrmion operator and the wave functions that we have found are strikingly similar to quantum Hall quasiparticle operators and wave functions. We also predict the presence of finite energy ``spin-SSS spinon'' for a general spin S Heisenberg antiferromagnets in 2D. Some consequences are briefly discussed.
We investigate the spin-12\frac{1}{2}21​ Heisenberg model on a rectangular lattice, using the Gutzwiller projected variational wave function known as the staggered flux state. Using Monte Carlo techniques, the variational parameters and instantaneous spin-spin correlation function for different coupling anisotropies $\ensuremath{\gamma}={J}_{y}/{J}_{x}$ are calculated. We observe a gradual evolution of the ground state energy towards a value which is very close to the one-dimensional (1D) estimate provided by the Bethe ansatz and a good agreement between the finite-size scaling of the energies. The spin-spin correlation functions exhibit a power-law decay with varying exponents for different anisotropies. Though the lack of N\'eel order makes the staggered flux state energetically unfavorable in the symmetric case $\ensuremath{\gamma}=1$, it appears to capture the essence of the system close to 1D. Hence we believe that the staggered flux state provides an interesting starting point to explore the crossover from quantum disordered chains to the N\'eel ordered two-dimensional square lattices.
Using large-scale determinant quantum Monte Carlo simulations in combination with the stochastic analytical continuation, we study two-particle dynamical correlation functions in the anisotropic square lattice of weakly coupled one-dimensional (1D) Hubbard chains at half filling and in the presence of weak frustration. The evolution of the static spin structure factor upon increasing the interchain coupling is suggestive of the transition from the power-law decay of spin-spin correlations in the 1D limit to long-range antiferromagnetic order in the quasi-1D regime and at T=0T=0T=0. In the numerically accessible regime of interchain couplings, the charge sector remains gapped. The low-energy momentum dependence of the spin excitations is well described by the linear spin-wave theory with the largest intensity located around the antiferromagnetic wave vector. This magnon mode corresponds to a bound state of two spinons. At higher energies the spinons deconfine and we observe signatures of the two-spinon continuum which progressively fade away as a function of interchain hopping.
Abstract Using quantum Monte Carlo, exact diagonalization, and perturbation theory, we study the spectrum of the S = 1/2 antiferromagnetic Heisenberg trimer chain by varying the ratio g = J2/J1 of the intertrimer and intratrimer coupling strengths. The doublet ground states of trimers form effective interacting S = 1/2 degrees of freedom described by a Heisenberg chain. Therefore, the conventional two-spinon continuum of width ∝ J1 when g = 1 evolves into to a similar continuum of width ∝ J2 when g → 0. The intermediate-energy and high-energy modes are termed doublons and quartons which fractionalize with increasing g to form the conventional spinon continuum. In particular, at g ≈ 0.716, the gap between the low-energy spinon branch and the high-energy band with mixed doublons, quartons, and spinons closes. These features should be observable in inelastic neutron scattering experiments if a quasi-one-dimensional quantum magnet with the linear trimer structure and J2 < J1 can be identified. Our results may open a window for exploring the high-energy fractional excitations.
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.