- #1
albronco1
- 11
- 5
I need an information about parton (slave-particle) constructions of Z₂ spin liquids and I’m trying to get a couple of things straight.
The usual story: you write a physical operator as a parton bilinear with a gauge redundancy say a U(1) from rephasing the partons and then a charge-2 parton pairing condensate Higgses that U(1) down to Z₂. You end up with Z₂ topological order, partons as the Z₂ charges and visons as the fluxes. That part seems standard enough.
Where I get stuck is the projective symmetry group on top of it. Most of what I researched (Wen and the follow-ups) classifies the symmetry-enriched states through how the space group acts projectively on the partons. Two things I’m unsure about:
First, what if there’s no fixed lattice you’re in the continuum, or one direction isn’t translation-invariant (think a half-line with a position-dependent potential)? Is there a clean way to do PSG without a crystallographic space group, or does the machinery really lean on the lattice? I have a vague memory of PSG-type arguments for Dirac spin liquids in the continuum is that the right thing to look at?
Second, and maybe this is the bigger one: PSG seems to be phrased for a Hamiltonian you classify how symmetry acts on the parton mean-field ansatz at the Hamiltonian level. But what if the natural formulation is Euclidean, i.e. the model is really a path-integral / statistical one, and the Hamiltonian only shows up via the transfer matrix (reflection positivity / Osterwalder–Schrader reconstruction) rather than being fundamental? Does PSG still go through in that case, or does it assume from the start a fundamental, local Hamiltonian? Is there a Euclidean / path-integral way to state PSG, or does it have to live on the reconstructed Hamiltonian?
And a smaller one: if PSG does go through, does it actually pick out which Z₂ state is realized, or just enumerate the symmetry-allowed ones with energetics deciding?
I keep getting confused about how much of PSG depends on having an actual lattice, and on starting from a fundamental Hamiltonian rather than a Euclidean / transfer-matrix one. Any pointers appreciated.
Greetz
KF
The usual story: you write a physical operator as a parton bilinear with a gauge redundancy say a U(1) from rephasing the partons and then a charge-2 parton pairing condensate Higgses that U(1) down to Z₂. You end up with Z₂ topological order, partons as the Z₂ charges and visons as the fluxes. That part seems standard enough.
Where I get stuck is the projective symmetry group on top of it. Most of what I researched (Wen and the follow-ups) classifies the symmetry-enriched states through how the space group acts projectively on the partons. Two things I’m unsure about:
First, what if there’s no fixed lattice you’re in the continuum, or one direction isn’t translation-invariant (think a half-line with a position-dependent potential)? Is there a clean way to do PSG without a crystallographic space group, or does the machinery really lean on the lattice? I have a vague memory of PSG-type arguments for Dirac spin liquids in the continuum is that the right thing to look at?
Second, and maybe this is the bigger one: PSG seems to be phrased for a Hamiltonian you classify how symmetry acts on the parton mean-field ansatz at the Hamiltonian level. But what if the natural formulation is Euclidean, i.e. the model is really a path-integral / statistical one, and the Hamiltonian only shows up via the transfer matrix (reflection positivity / Osterwalder–Schrader reconstruction) rather than being fundamental? Does PSG still go through in that case, or does it assume from the start a fundamental, local Hamiltonian? Is there a Euclidean / path-integral way to state PSG, or does it have to live on the reconstructed Hamiltonian?
And a smaller one: if PSG does go through, does it actually pick out which Z₂ state is realized, or just enumerate the symmetry-allowed ones with energetics deciding?
I keep getting confused about how much of PSG depends on having an actual lattice, and on starting from a fundamental Hamiltonian rather than a Euclidean / transfer-matrix one. Any pointers appreciated.
Greetz
KF