Edge spin fractionalization in open one-dimensional spin- S quantum antiferromagnets

Tai Kai Ng, SJ Qin, Zb Su

Pradip Kattel, Yicheng Tang, J. H. Pixley +1 more

We show that a gapped spin-$S$ chain with antiferromagnetic (AFM) order exhibits, in the thermodynamic limit, exponentially localized fractional $\ifmmode\pm\else\textpm\fi{}\frac{S}{2}$ edge modes when the system possesses U(1) symmetry. We show this for integrable and nonintegrable spin chains, both analytically and numerically. Through exact analytical solutions, we show that an AFM spin-$\frac{1}{2}$ chain with explicitly broken ${\mathbb{Z}}_{2}$ symmetry and an integrable AFM spin-1 chain with spontaneously broken ${\mathbb{Z}}_{2}$ symmetry have $\ifmmode\pm\else\textpm\fi{}\frac{1}{4}$ and $\ifmmode\pm\else\textpm\fi{}\frac{1}{2}$ fractionalized edge modes, respectively. Furthermore, employing the density matrix renormalization group technique, we extend this analysis to generic $XXZ\text{\ensuremath{-}}S$ chains with $S\ensuremath{\le}3$ and demonstrate that these fractional spins are robust quantum observables, substantiated by the observation of variance of the associated fractional spin operators that is consistent with a vanishing functional form in the thermodynamic limit. Moreover, we find that the edge modes are robust to a disorder that couples to the N\'eel order parameter.

Hiroki Yamazaki, K. Katsumata

The magnetic properties of $\mathrm{Cs}{\mathrm{Cr}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{Cl}}_{3} (x=0.073)$ are studied by electron-spin-resonance (ESR), magnetic-susceptibility, and heat-capacity measurements to elucidate the edge state in the finite chains of the $S=2$ quasi-one-dimensional Heisenberg antiferromagnet CsCr${\mathrm{Cl}}_{3}$. The results of the ESR measurements in the temperature range 14-40 K are well explained by an effective $S=1$ fractional spin. A crossover between different Curie constants observed in the susceptibility measurements is also explained by assuming $S=1$ free spins at the ends of the finite chains below 40 K. The appearance of the $S=1$ fractional spin can be understood by the valence-bond-solid model.

R. B. Laughlin, Domenico Giuliano, Raffaele Caracciolo +1 more

Wen-Jia Rao, Guang-Ming Zhang, Kun Yang

Applying a symmetric bulk bipartition to the one-dimensional Affleck-Kennedy-Lieb-Tasaki valence-bond solid (VBS) states for the integer spin-$S$ Haldane gapped phase, we can create an array of fractionalized spin-$S$/2 edge states with the super unit cell $l$ in the reduced bulk system, and the topological properties encoded in the VBS wave functions can be revealed. The entanglement Hamiltonian (EH) with even $l$ corresponds to the quantum antiferromagnetic Heisenberg spin-$S$/2 model. For the even integer spins, the EH still describes the Haldane gapped phase. For the odd integer spins, however, the EH just corresponds to the quantum antiferromagnetic Heisenberg half-odd integer-spin model with spinon excitations, characterizing the critical point separating the topological Haldane phase from the trivial gapped phase. Our results thus demonstrate that the topological bulk property not only determines its fractionalized edge states but also the quantum criticality associated with the topological phase, where the elementary excitations are precisely those fractionalized edge degrees of freedom confined in the bulk of the topological phase.

Jos 'e A. Hoyos, Shailesh Chandrasekharan A, Harold U. Baranger