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arXiv:2501.13059 · Physical review. B./Physical review. B

Bound states and deconfined spinons in the dynamical structure factor of the J 1 − J 2 spin-1 chain

Aman Sharma, Mithilesh Nayak, Henrik Moodysson Rønnow, Frédéric Mila

Published Feb 3, 2025• 6 citations
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Abstract

Using a time-dependent density matrix renormalization group approach, we study the dynamical structure factor of the ${J}_{1}\ensuremath{-}{J}_{2}$ spin-1 chain. As ${J}_{2}$ increases, the magnon mode develops incommensurability. The system undergoes a first-order transition at ${J}_{2}=0.76\phantom{\rule{0.16em}{0ex}}{J}_{1}$, and at that point, domain walls lead to a continuum of fractional quasiparticles or spinons. By studying small variations in ${J}_{2}$ around the transition point, we observe the confinement of spinons into bound states in the spectral function and find a smooth evolution of the spectrum into magnon modes away from the phase transition. We employ the single-mode approximation to accurately account for the dispersion of the magnon mode away from the phase transition and describe the associated continua and bound states. We extend the single-mode approximation to describe the dispersion of a spinon at the phase transition point and obtain its dispersion throughout the Brillouin zone. This allows us to relate the incommensurability at and around the transition point to the competition between a negative nearest-neighbor hopping amplitude and a positive next-nearest-neighbor one for the domain wall.

DOI
10.1103/physrevb.111.064404

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Using a time-dependent density matrix renormalization group approach, we study the dynamical structure factor of the J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ spin-1 chain. As J2{J}_{2}J2​ increases, the magnon mode develops incommensurability. The system undergoes a first-order transition at J2=0.76J1{J}_{2}=0.76\phantom{\rule{0.16em}{0ex}}{J}_{1}J2​=0.76J1​, and at that point, domain walls lead to a continuum of fractional quasiparticles or spinons. By studying small variations in J2{J}_{2}J2​ around the transition point, we observe the confinement of spinons into bound states in the spectral function and find a smooth evolution of the spectrum into magnon modes away from the phase transition. We employ the single-mode approximation to accurately account for the dispersion of the magnon mode away from the phase transition and describe the associated continua and bound states. We extend the single-mode approximation to describe the dispersion of a spinon at the phase transition point and obtain its dispersion throughout the Brillouin zone. This allows us to relate the incommensurability at and around the transition point to the competition between a negative nearest-neighbor hopping amplitude and a positive next-nearest-neighbor one for the domain wall.
We investigate the nature of low-energy excitations in a spin chain with antiferrmomagnetic nearest-neighbor J1J_1J1​, next-nearest-neighbor J2J_2J2​, and three-site J3J_3J3​ interactions using the time-dependent density matrix renormalization group and the single mode approximation techniques. In the absence of the J2J_2J2​ interaction, we identify clear distinctions in the spectral functions in the fully dimerized phase across the exactly dimerized line for different magnitudes of the spins. In contrast to the spin-1/21/21/2 chain, where the spinon continuum dominates the spectral functions, the magnon modes are prominent in the spectral functions of the spin-111 and spin-3/23/23/2 chains. Through single mode approximation and valence bond solid approaches, we disentangle magnon and spinon contributions to the spectral functions. After including the J2J_2J2​ interactions, for the spin-111 chain we trace the evolution of the dynamical structure factor along the phase transition line between the Haldane phase and the fully dimerized phase. We find that the excitation spectrum is a continuum along this line and the spectral gap closes as the order of the transition changes from first order to second order. Along the line of first-order transitions, the spinon-like domain walls are deconfined, and the model exhibits their confinement into discrete bound states away from the transition line. A similar phenomenon occurs in the spin-3/23/23/2 chain across the phase transition between partially dimerized to fully dimerized phases, revealing a universal spinon confinement phenomenon across first-order phase transitions. This study presents the dynamical structure factor corresponding to the ground state phase diagram and establishes a unified quasiparticle framework for understanding the fundamental nature of excitations across distinct quantum phases in frustrated J1J_1J1​-J2J_2J2​-J3J_3J3​ Heisenberg spin chains.
We study the dynamical structure factor of the frustrated spin-$3/2\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chains, with particular focus on the partially dimerized phase that emerges between two Kosterlitz-Thouless transitions. Using a valence bond solid Ansatz corroborated by density-matrix renormalization-group simulations, we investigate the nature of magnon and spinon excitations through the single-mode approximation. We show that the magnon develops an incommensurate dispersion at ${J}_{2}\ensuremath{\approx}0.32{J}_{1}$, while the spinons, viewed as domain walls between degenerate valence bond solid states, become incommensurate at ${J}_{2}\ensuremath{\approx}0.4{J}_{1}$ beyond the Lifshitz point $({J}_{2}\ensuremath{\approx}0.388{J}_{1})$. The dynamical structure factor exhibits rich spectral features shaped by the interplay between these excitations, with magnons appearing as resonances embedded in the spinon continuum. The spinon gap shows a nonmonotonic behavior, reaching a peak near the center of the partially dimerized phase and closing at the boundaries, suggesting the appearance of a floating phase as a result of the condensation of incommensurate spinons. Comparative analysis with the spin-5/25/25/2 case confirms the universality of these phenomena across half-integer higher-spin systems. Our results provide detailed insight into how fractionalization and incommensurate condensation govern the spectral properties of frustrated spin chains, offering a unified picture across different spin magnitudes.
Motivated by the rich phase diagrams recently unveiled in frustrated spin-1 chains with competing next-nearest-neighbor and four-spin interactions, we investigate the nature of the elementary excitations of spin-1 chains in the vicinity of phase transitions using the matrix-product state (MPS) representation of elementary excitations. First, we show that both spinons and magnons naturally arise in SU(2)\mathrm{SU}(2)SU(2) invariant spin chains when describing ground states and elementary excitations using MPSs. Then we investigate the nature of the elementary excitations across the first-order transition between the Haldane phase and the topologically trivial next-nearest-neighbor Haldane phase. We show explicitly that spinons deconfine at the transition, and we calculate the dispersion of these deconfined spinons with MPSs. We also show that, immediately away from the transition line, spinons confine, forming dispersive spinon/antispinon bound states on both sides of the transition. Finally, we show that deconfined spinons also appear at the transition between the Haldane phase and the spontaneously dimerized phase when this transition is first order.
We employ a variational Monte Carlo approach to efficiently obtain the dynamical structure factor for the spin-$1/2\phantom{\rule{0.16em}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. Upon increasing the frustrating ratio J2/J1{J}_{2}/{J}_{1}J2​/J1​, the ground state undergoes a continuous transition from a N\'eel antiferromagnet to a Z2{\mathbb{Z}}_{2}Z2​ gapless spin liquid. We identify the characteristic spectral features in both phases and highlight the existence of a broad continuum of excitations in the proximity of the spin-liquid phase. The magnon branch, which dominates the spectrum of the unfrustrated Heisenberg model, gradually loses its spectral weight, thus releasing nearly deconfined spinons, whose signatures are visible even in the magnetically ordered state. Our results provide an important example on how magnons fractionalize into deconfined spinons across a quantum critical point.
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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.