# FQHE in iTensor?

+1 vote

Is it possible to investigate fractional quantum Hall systems in iTensor?

I have in mind mostly the cyllindrical geometry. Then, in Landau gauge the system can be expressed as 1D fermionic chain, with every Landau gauge function being a spinless fermionic site which can be either occupied or empty. So this can be doable using the HubbardSite class. However, every such site can be labeled with a momentum quantum number which is different for every site. And this is something that iTensor site class cannot do. So the question is, is there some trick which can make this work? Is there any way to use the momentum conservation in iTensor in such a system?

Hi, good question. So this should certainly be possible to do with ITensor. The way to handle the momentum quantum number should be that label the allowed y-periodic momenta by integers, then create a "slot" in the QNs which you set up to have a ZN addition rule, where N is the maximum allowed momentum (dependent on the transverse system size Ly of course).

This paper which you've probably seen has some helpful details:
https://arxiv.org/abs/1512.03318

To set up the QNs, you can do something like this:

First site IQIndex along a column of your cylinder (qn values are spin, charge, momentum):

IQIndex("S1",Index("e1",1),QN({0,1},{0,-1},{0,Nmax}),
Index("up1",1),QN({1,1},{1,-1},{0,Nmax}),
Index("dn1",1),QN({-1,1},{1,-1},{0,Nmax}),
Index("ud1",1),QN({0,1},{2,-1},{0,Nmax}))


Second site IQIndex

IQIndex("S1",Index("e1",1),QN({0,1},{0,-1},{1,Nmax}),
Index("up1",1),QN({1,1},{1,-1},{1,Nmax}),
Index("dn1",1),QN({-1,1},{1,-1},{1,Nmax}),
Index("ud1",1),QN({0,1},{2,-1},{1,Nmax}))


To understand the notation inside the QN better, see the QN documentation page:
http://itensor.org/docs.cgi?page=classes/qn