Discovery of Fractionalized Neutral Spin-1/2 Excitation of Topological Order

After the discovery of fraction quantum Hall states in the 1980s,^[1] it became more and more clear that Landau symmetry breaking theory does not describe all possible quantum phases of matter. The new quantum phases of matter were called topologically ordered phases^[2,3] (for gapped cases) or quantum ordered phases^[4] (for gapless cases), which correspond to patterns of many-body entanglement.^[5–7] One may wonder: besides quantum Hall systems, are there other systems that realize the new topological/quantum order?

In the 1980s and 1990s, it was shown theoretically that topological orders can be realized in spin liquids, such as the chiral spin liquids^[8,9] and Z₂-spin liquids.^[10,11] Also, stable quantum ordered phases can be realized in algebraic spin liquids.^[12–15] The topological/quantum ordered states are not easy to detect since they are not characterized by local order parameters. On the other hand, the absence of local order parameters lead to a strange way to discover topological/quantum ordered states: one tries to detect any kind of order parameters and phases transitions as the temperature is lower to zero. If one finds nothing, then one can declare that a certain topological/quantum ordered state is discovered (if the trivial ground state can be ruled out). In fact, such a strategy was used by Y. Lee, which led to a discovery of herbertsmithite as a possible spin liquid candidate on Kagome lattice.^[16] A few years earlier, another spin liquid candidate was discovered in organic Mott insulator of triangular lattice.^[17] The above two are 2-dimensional spin liquids. A 3-dimensional spin liquid candidate was found in hyperkagome antiferromagnet.^[18] Recently, a very promising spin liquid was discovered in honeycomb lattice α-RuCl₃ with strong spin-orbital coupling.^[19–25]

One of the most important properties of a spin liquid is whether the spin liquid is gapped or gapless. If the spin liquid is gapped, then the next important question is whether the spin liquid has fractionalized spin-1/2 quasiparticles or not. The appearance of spin-1/2 excitations implies a non-trivial topological order in the spin liquid. However, one challenge to study herbertsmithite in more detail is to reduce the influence of magnetic impurities. The 5–10% magnetic impurities in herbertsmithite make it difficult to determine if the spin liquid is gapped or gapless.^[26] In a recent work, Ref. [27], published by Chinese Physics Letters, a new kind of Kagome spin liquid was found in a new material Cu₃Zn(OH)₆FBr. The new material allows one to measure Knight shift via ¹⁹F NMR measurements (with I = 1/2 nuclear spin). The intrinsic Cu-spin magnetic susceptibility from Knight shift reveals a small spin gap of 8 K (compared to the spin coupling of 200 K). The small spin gap is consistent with a recent numerical calculation which found a long correlation length in the Heisenberg model on Kagome lattice.^[28] Furthermore, the magnetic field dependence of spin gap indicates that the thermally excited spin excitations carry fractionalized spin-1/2.

Just like the direct discovery of fractional charge via noise measurement,^[29] the discovery of a totally new fractionalized neutral spin-1/2 excitation is a very exciting result. This result suggests that the Kagome spin liquid is the Z₂-spin liquid with a Z₂ topological order.^[10,11] The SO(3) symmetric Z₂ topological order features emergent spin-1/2, emergent fermions, etc.^[10,11] However, at moment, it is not clear whether the observed spin-1/2 excitation is a boson or a fermion. Hopefully, more detailed future experiments can resolve this issue. I also like to remark that the spin liquid in α-RuCl₃ does not have the SO(3) spin rotation symmetry. In this case, it is harder to directly detect the fractionalization of topological order.