I am a bit confused on the different names people use, sometimes seemingly interchangeably, when talking about Resonating Valence Bond (RVB) wavefunctions. I'm looking for some clarification.

I have heard about fermionic RVB, bosonic RVB, uniform RVB, projected BCS, and spin liquid states. Are these all examples of RVB wavefunctions? Are some special cases of the others? How do these categories overlap?

I believe these are all related in the following way, but am unsure:

Projected BCS wavefunctions are of the form $$ \left| pBCS\right\rangle = P_G P_N \left| BCS\right\rangle = P_G P_N \prod_k (u_k + v_k c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow})\left|0\right\rangle$$ where $P_N$ is a particle number projection operator and $P_G$ is the Gutzwiller projection operator, which projects out the double-occupancy of electrons in the same orbital. This is an example of a fermionic RVB state.

Assume that the electrons are on a lattice, such as in the Hubbard model. When $N$ is the number of lattice sites and $P_G$ forbids double-occupancy, then each site only has a single electron with spin up or down. These electrons are parts of spin singlets that extend between two sites, making this specific $\left| pBCS\right\rangle$ state a spin liquid. But, in general, spin liquids are not exclusively projected BCS wavefunctions.

What I am least clear on are bosonic RVB states and uniform RVB states (which I have heard can be written down as both fermionic and bosonic RVB states).

Am I incorrect in my understanding or missing something important?


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