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arXiv:2207.12215 · Physical review. B./Physical review. B

Spinon confinement in the gapped antiferromagnetic XXZ spin- 12 chain

S. B. Rutkevich

Published Oct 7, 2022• 8 citations
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Abstract

The infinite Heisenberg XXZ spin-$\frac{1}{2}$ chain in the gapped antiferromagnetic regime has two degenerate vacua and kink topological excitations (which are also called spinons) interpolating between these vacua as elementary excitations. Application of an arbitrary weak staggered longitudinal magnetic field $h$ induces a long-range attractive potential between two adjacent spinons leading to their confinement into ``meson'' bound states. Exploiting the integrability of the XXZ model in the deconfined phase $h=0$, we perform perturbative calculations of the energy spectra of the two-spinon bound states in the weak confinement regime at $h\ensuremath{\rightarrow}+0$, using the strength of the staggered magnetic field $h$ as a small parameter. Both transverse and longitudinal dynamical structure factors of the local spin operators are calculated as well in the two-spinon approximation in the weak confinement regime to the leading order in $h$.

DOI
10.1103/physrevb.106.134405

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The infinite Heisenberg XXZ spin-12\frac{1}{2}21​ chain in the gapped antiferromagnetic regime has two degenerate vacua and kink topological excitations (which are also called spinons) interpolating between these vacua as elementary excitations. Application of an arbitrary weak staggered longitudinal magnetic field hhh induces a long-range attractive potential between two adjacent spinons leading to their confinement into ``meson'' bound states. Exploiting the integrability of the XXZ model in the deconfined phase h=0h=0h=0, we perform perturbative calculations of the energy spectra of the two-spinon bound states in the weak confinement regime at h\ensuremath→+0h\ensuremath{\rightarrow}+0h\ensuremath→+0, using the strength of the staggered magnetic field hhh as a small parameter. Both transverse and longitudinal dynamical structure factors of the local spin operators are calculated as well in the two-spinon approximation in the weak confinement regime to the leading order in hhh.
The Heisenberg XXZ spin-(1/2) chain is considered in the massive antiferromagnetic regime in the presence of a staggered longitudinal magnetic field. The Hamiltonian of the model is characterised by the anisotropy parameter and by the magnetic-field strength h . At zero magnetic field, the model is exactly solvable. In the thermodynamic limit, it has two degenerate vacua and the kinks (which are also called spinons) interpolating between these vacua, as elementary excitations. Application of the staggered magnetic field breaks integrability of the model and induces the long-range attractive potential between two adjacent kinks leading to their confinement into the bound states. The energy spectra of the resulting two-kink bound states are perturbatively calculated in the extreme anisotropic (Ising) limit to the first order in the inverse anisotropy constant , and also for any to the first order in the weak magnetic field h .
We study the tuning effect of a transverse magnetic field on the confinement of spinons in the infinite XXZ spin-1/2 chain. The spinon confinement in this model takes place in the gapped antiferromagnetic phase upon application of a staggered longitudinal magnetic field. The tuning transverse magnetic field has mutually orthogonal uniform and staggered components. The energy spectra of the two-spinon bound states (the ``mesons'') in the confinement regime are analytically calculated in this model using two different perturbative schemes. The first one applies in the extreme anisotropic (Ising) limit and employs the inverse anisotropy constant as a small parameter. The second perturbative scheme, which applies at any anisotropy in the gapped antiferromagnetic domain, exploits the integrability of the XXZ spin chain at zero magnetic field. The small parameters in the second technique are the components of the transverse, and staggered longitudinal magnetic fields. It is shown that the weak transverse magnetic field mixes the transverse and longitudinal meson modes, and leads to an avoided crossing of their energies upon increase of its strength. The explicit formulas for the two-spinon contribution to the dynamical structure factors of local spin operators are obtained as well in this model in the weak confinement regime for wave vectors close to the points k=0k=0k=0 and $\ensuremath{\pi}$.
We consider the transverse dynamical structure factor of the anisotropic Heisenberg spin-1/2 chain ( XXZ model) in the gapped antiferromagnetic regime (Δ>1). Specializing to the case of zero field, we use two independent approaches based on integrability (one valid for finite size, the other for the infinite lattice) to obtain the exact two-spinon part of this correlator. We discuss in particular its asymmetry with respect to the π/2-momentum line, its overall anisotropy dependence, and its contribution to sum rules.
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.
Calculations of the thermodynamic quantities, and the spectra of spinons, triplet excitations, and two-particle spin-singlet (Δ S z = 0,±1) excitations for a weakly coupled antiferromagnetic S = 1/2 spin chain are made using a mean-field approximation for the interchain couplings ( J 2 ) by the quantum Monte Carlo method. The mass gaps in these excitation spectra are estimated as functions of the interchain coupling. The critical field H c , and the temperatures at which the spinon and singlet gaps close ( T 1 and T 2 , respectively) are obtained. Some peculiarities of the staggered magnetization, the specific heat, and the spin-spin correlation function are found at T 1 = 0.9 J 2 , T 2 = 1.5 J 2 , H c = 3.8 J 2 , as determined by linear function fitting for J 2 / J 1 <0.15. The intrachain and interchain couplings, and the staggered magnetization for KCuF 3 , Sr 2 CuO 3 , and Ca 2 CuO 3 are estimated, and mass gaps are predicted to exist in the spectra of the spinons and the singlet two-particle excitations.
We use perturbative series expansions about a staggered dimerized ground state to compute the ground-state energy, triplet excitation spectra, and spectral weight for a one-dimensional model in which each site has an S=12S=\frac{1}{2}S=21​ spin Si{\mathbf{S}}_{i}Si​ and a pseudospin Ti,{\mathbf{T}}_{i},Ti​, representing a doubly degenerate orbital. An explicit dimerization is introduced to allow the study of the confinement of spinon excitations. The elementary triplet represents a bound state of two spinons and is stable over much of the Brillouin zone. A special line is found in the gapped spin-liquid phase, on which the triplet excitation is dispersionless. The formation of triplet bound states is also investigated.