# Questions on the elementary excitations in the resonating-valence-bond(RVB) states?

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It is known that the RVB states can support spin-charge separations and its elementary excitations are spinons and holons. But it seems that there are some different possibilities for the nature of the statistics of the spinons and holons. Specifically, (1) spinons are fermions while holons are bosons(Kivelson et al);(2)spinons are bosons while holons are fermions(Read et al);(3)spinons and holons are both semions obeying fractional statistics(Laughlin et al).

Do we now have a commonly accepted view for one of the above three possibilities? Or do the quasiparticle statistics depend on the details(e.g. which kind) of the RVB wave functions?

Furthermore, assume RVB states as the ground-states of the antiferromagnetic Heisenberg model, and when we use $t-J$ model to study the lightly doped Mott insulator, for the possibility (1), it's reasonable to adopt slave-boson formalism, while for the possibility (2), it's reasonable to adopt slave-fermion formalism, right? But for the possibility (3), what kind of slave-particle formalism we should adopt? This part may be related to my previous question.

This post imported from StackExchange Physics at 2014-03-09 08:39 (UCT), posted by SE-user K-boy

edited Apr 6, 2015
@ Xiao-Gang Wen Thank you very much.

This post imported from StackExchange Physics at 2014-03-09 08:39 (UCT), posted by SE-user K-boy

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http://www.physicsoverflow.org/12926/what-is-a-resonating-valence-bond-rvb-state?show=12933#a12933

we see the RVB state does not refer to one state, it can refer to many different states (with different topological orders). The RVB states with different topological orders can have different properties. There is a RVB state (with a $Z_2$ topological order) where the spinons are fermions while the holons are bosons. There is another RVB state (with a different $Z_2$ topological order)  where spinons are bosons while holons are fermions. There is a third RVB state (chiral spin liquid) where spinons and holons are both semions obeying fractional statistics.

answered Apr 4, 2014 by (3,475 points)

@Xiao-Gang Wen Thank you Prof.Wen. What about a chiral spin liquid with a $Z_2$ topological order ?

Chiral spin liquid can also be described by the slave-fermion formalism, if you put the fermions into a mean-field ansatz with nontrivial topological band structure (i.e. Chern number).

@Meng Yes, I agree with you. Do you know some existing studies on the $Z_2$ chiral spin liquid (a $Z_2$ spin-liquid with broken time-reversal symmetry) ? Thanks a lot!

There is a large body of literature on chiral spin liquid. A recent VMC study is http://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.041124, and you can find in the references of the paper many recent (exact) numerical studies of chiral spin liquid in Heisenberg models on the kagome lattice.

As to my first comment, I think I can answer it now: The TR symmetry is NOT essential to the topological order for a $Z_2$ SL, since the topological degeneracy of a $Z_2$ SL on a torus is always 4 while the total ground-state degeneracy (TGSD) is TGSD$=2\times 4$ for the spontaneous TR breaking $Z_2$ SL ($Z_2$ CSL) and TGSD$=4$ for the TR symmetric $Z_2$ SL.

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