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arXiv:2206.08018 · Physical review. B./Physical review. B

Confinement and oscillating deconfinement crossover of two-magnon excitations in quantum spin chains quantified by spin entanglement entropy

Zhao-Yang Dong, Jian‐Xin Li

Published Jun 15, 2022• 2 citations
arXivHTMLPDF

Abstract

We introduce the spin entanglement entropy (EE) to characterize spin excitations. The scalings of EEs are elaborated as $lnN, lnN+{D}_{k}$, and $2lnN+D$ for the single-magnon states, two-magnon bound states, and two-magnon continuum, respectively, based on the Bethe ansatz solutions of the ferromagnetic spin chain with $N$ sites. The $lnN$ divergence for the bound states reveals that the two magnons emerge as a new quasiparticle. More importantly, the nonzero intercepts (${D}_{k},D$) embody the many-body effects of two-magnon states. In particular, an exact relation between the intercept of EEs and an observable quantity, the two-magnon distance, is established for the bound states. In such a case, we quantify the two-magnon confinement in a spin system by the increasing entanglement with the distance as a particle physics analog. Moreover, the EEs can also be used to study the evolution of the excitations. When the bound states are immersed in the continuum in the alternating chain, they undergo an oscillating confinement-deconfinement crossover, shown by the oscillations of the EEs and two-magnon distance with the chain length.

DOI
10.1103/physrevb.105.l220402

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We introduce the spin entanglement entropy (EE) to characterize spin excitations. The scalings of EEs are elaborated as lnN,lnN+DklnN, lnN+{D}_{k}lnN,lnN+Dk​, and 2lnN+D2lnN+D2lnN+D for the single-magnon states, two-magnon bound states, and two-magnon continuum, respectively, based on the Bethe ansatz solutions of the ferromagnetic spin chain with NNN sites. The lnNlnNlnN divergence for the bound states reveals that the two magnons emerge as a new quasiparticle. More importantly, the nonzero intercepts (Dk,D{D}_{k},DDk​,D) embody the many-body effects of two-magnon states. In particular, an exact relation between the intercept of EEs and an observable quantity, the two-magnon distance, is established for the bound states. In such a case, we quantify the two-magnon confinement in a spin system by the increasing entanglement with the distance as a particle physics analog. Moreover, the EEs can also be used to study the evolution of the excitations. When the bound states are immersed in the continuum in the alternating chain, they undergo an oscillating confinement-deconfinement crossover, shown by the oscillations of the EEs and two-magnon distance with the chain length.
Confinement is a ubiquitous phenomenon in which constituent particles are bound together into a new quasiparticle. In condensed matter physics, magnon bound states can be regarded as a confinement of magnons, and an oscillating deconfinement crossover of two-magnon bound states scattered by a two-magnon continuum was revealed in the one-dimensional (1D) ferromagnetic chain by the spin entanglement entropy (SEE) analysis. Now, we extend the study to the two-dimensional (2D) and three-dimensional (3D) cases. Although the interactions between magnons in high dimensions are not as effective as in one dimension, two-magnon bound states can still survive around the corners of the Brillouin zone (BZ). In 2D or 3D square lattices, the bound states reduce their dimensionality to be 1D bound states at the BZ edges, which are bound only along the direction parallel to the BZ edge. This is verified by examining the relation between the magnon separation ddd and the intercept DDD of the SEE, which is found to satisfy the 1D relation D=lnd+1D=lnd+1D=lnd+1, but in 2D and 3D lattices. Meanwhile, the 2D bound states satisfy D=2lnd+1.3D=2lnd+1.3D=2lnd+1.3 in both 2D and 3D lattices. Unlike those in the 1D chain, some bound states will enter the continuum directly rather than cling to its bottom. We find that these immersed bound states are neither preserved nor deconfined completely when just meeting the continuum in finite-size systems, which is similar to what occurs in the 1D chain. Their SEE and magnon separation oscillate with respect to the system size. Therefore, these bound states undergo the oscillating deconfinement crossover in the continuum.
We introduce an entanglement entropy analysis to quantitatively identify the confinement and deconfinement of the spinons in the spin excitations of quantum magnets. Our proposal is implemented by the parton construction of a honeycomb-lattice antiferromagnet exhibiting high-energy anomalous spectra. To obtain the quasiparticles of spin excitations for entanglement entropy calculations, we develop an effective Hamiltonian using the random phase approximation. We elaborate quantitatively the deconfinement-to-confinement transition of spinons in the anomalous spectra with the increase of the Hubbard interaction, indicating the avoided fractionalization of magnons in the strong interaction regime. Meanwhile, the Higgs mode at the ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ point is fractionalized into four degenerate spinons, although it appears as a sharp well-defined peak in the spectra. Our work extends our understanding of the deconfinement of the spinon and its coexistence with the magnon in quantum magnets.
Entanglement Entropy of the N = 4 N=4 \mathcal{N}=4 N=4 SYM spin chain
We present a detailed study of the Entanglement Entropy (EE) of excited states in all closed rank one subsectors of N=4 \mathcal{N}=4 N=4 SYM, namely SU(2), SU(1|1) and SL(2). Exploiting the techniques of the Coordinate and the Algebraic Bethe Ansatz we obtain the EE for spin chains with up to seven magnons, at leading order in the coupling expansion but exact in the length of the spin chain and of the part of it that we cut. Focusing on the superconformal primary operator with two magnons in the BMN limit, we derive analytic and exact, in the coupling λ′, expressions for the Renyi and the EE. The interpolating functions for the Renyi and the EE monotonically increase as the coupling increases from the weak coupling λ′ → 0 regime to the strong coupling λ′ → ∞ regime. This results to a violation of a certain bound for the EE that is present at weak coupling and confirms the physical intuition that entanglement increases when the coupling increases.
We investigate the entanglement properties of the excited states of the spin- Heisenberg ( XXX ) chain with isotropic antiferromagnetic interactions, by exploiting the Bethe ansatz solution of the model. We consider eigenstates obtained from both real and complex solutions ('strings') of the Bethe equations. Physically, the former are states of interacting magnons, whereas the latter contain bound states of groups of particles. We first focus on the situation with few particles in the chain. Using exact results and semiclassical arguments, we derive an upper bound S MAX for the entanglement entropy. This exhibits an intermediate behaviour between logarithmic and extensive, and it is saturated for highly-entangled states. As a function of the eigenstate energy, the entanglement entropy is organized in bands. Their number depends on the number of blocks of contiguous Bethe–Takahashi quantum numbers. In the presence of bound states a significant reduction in the entanglement entropy occurs, reflecting that a group of bound particles behaves effectively as a single particle. Interestingly, the associated entanglement spectrum shows edge-related levels. At a finite particle density, the semiclassical bound S MAX becomes inaccurate. For highly-entangled states S A ∝ L c , with L c the chord length, signalling the crossover to extensive entanglement. Finally, we consider eigenstates containing a single pair of bound particles. No significant entanglement reduction occurs, in contrast with the few-particle case.
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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 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.