Generalize infsiteinds to fractional QN densities

[LHerviou]

Hej Mat,

A minor update to fix the issue #68
I am using it as an excuse to ask how to proceed with the following.

On my own fork, I have finished the implementations of a few useful features (paper coming out soon using it). This includes, from trivial to less trivial:

• SU(3) and SU(4) irrep using color conservation (nothing non-Abelian)
• some small fixes in general
• implementation of a few convenient functions for data analysis (QN-resolved analysis of ent spectrum, transfer matrix spectrum + some correlation functions)
• a working iDMRG for InfiniteMPOMatrix
• a working InfiniteMPO constructed from an InfiniteMPOMatrix
• a working iDMRG for InfiniteMPO
• compression of iMPOs using InfiniteMPOMatrix and the approach by Parker, Cao et Zaletel in Phys.Rev.B 102 . It is not yet fully benchmarked so that should clearly be experimental , especially as it is the simplest, non-fail proof version.
• I also plan to do this week some modifications on the subspace expansion as I cannot afford the default version in my new project (the one that required the compression of iMPOs)

It might be a lot for a single merge request. So how would you like to proceed? It will take me one or two weeks anyway to make things a bit cleaner.

[LHerviou]

The failure does not seem to come from my code but from the new @require.

[mtfishman]

Also, I think the issue was from a bug in the JuliaFormatter package which is now fixed, hopefully running the formatter again will fix the CI issue.

[nbaldelli]

Hi @mtfishman, yes this looks like a cleaner solution of what I was trying in my fork, I will try it and close the issue. Thanks @LHerviou !

[mingruyang]

function initstate(n)
    if n%2==0
        if (n/2)%2==1 
            return "Dn"
        else 
            return "Up"
        end
    else
        if (n+1)/2<=19
            if ((n+1)/2)%2==1
                return "Up"
            else
                return "Dn"
            end
        else 
            return "Emp"
        end
    end
end

N = 62
s = infsiteinds(n->isodd(n) ? "tJ" : "S=1/2", N; conserve_qns, initstate)
ψ = InfMPS(s, initstate)

I got
AssertionError: Flux not invariant under one unit cell translation, not implemented

[mtfishman]

@LHerviou @JanReimers I've moved discussions about other topics (InfiniteMPO generation and compression, iDMRG, subspace expansion, etc.) to #73. Let's keep this PR specific to the improvements to infsiteinds.

[nbaldelli]

Hi @LHerviou, any update on this? Otherwise I could try to merge the same fix from my fork.

[LHerviou]

Sorry, somehow everything went into my spams.

I will submit my fix today or tomorrow.
Sorry for the delay

[mingruyang]

For example, as you have mentioned, the error is from that the default two-site subspace expansion does not work in this case, so probably the error message could be "Impossible for the default two-site scheme to do a subspace expansion fails due to the form of the QN conserved Hamiltonian" or pointing our at which line and inside which function the subspace expansion fails. The current error message is difficult for people who are not developer of this repository to trace back where the error happens.

[mtfishman]

@mingruyang could you start a new issue for problems you come across with subspace expansion (with a minimal runnable code example)?

[mtfishman]

Looks good, thanks @LHerviou! Is there a simple test we could add to test out this improvement? Maybe we could modify one of the current tests to have fractional QN densities (the tests already take a while to run so preferably we wouldn't add a new set of VUMPS tests).

[LHerviou]

The simplest test is probably to slightly modify our starting state for Ising or XXZ and check that it knows how to build the correct starting MPS. For example, starting with 3 fermions and 1 hole in XXZ with nsites = 4.

As a side comment, the implementation for Z_n conserved quantities could be improved easily, if you think it is relevant, by instead finding y such that y * nsites = flux [n], if such solution exists.

[mtfishman]

That would be a nice modification to the tests.

That sounds like a good idea to improve the handling of Zn symmetries!

[LHerviou]

I will try to do this this weekend then.

[LHerviou]

Normally this works perfectly, and was running on my side.

It relies on the Chinese remainder theorem for Zp symmetries, i.e. noting N the unit cell, we tried to find y such that
y = 0 mod N and y = flux mod p. Then we just need to find x such that y = N x.

This system admits a solution iff flux = 0 mod( gcd(p, N)), which is not necessarily true. We can find a global multiplier b such that it is true.
The action of the multiplier is flux -> b flux, p -> b p. b = N is a trivial solution. That was more or less what the previous code was doing.
Now I am a bit more subtle: I try to find the smallest b such that:
b flux = 0 mod gcd(N, p b).
It changes nothing but is more satisfactory and natural.

Case when things changed: N = 3, initstate = +--, flux = 1 mod 2.
Previous version: 1 is not divisible by 3, so we mutiply everything by 3. Gives QN ( 4 mod 6, 1 mod 6)
Current version: gcd(2, 3) = 1, so a solution actually exists: 3 x 1 = 1 mod 2 => work with QN (0 mod 2, 1 mod 2)

[mtfishman]

Thanks, the change is pretty self-contained and seems like it will be very important to make sure the code works for more general models so I'm in favor of adding it. Could you add your description above as a comment in the code so we can keep track of things going forward?

[LHerviou]

Well, the construction of \psi already guarantees that the flux is 0. But I added the @test in any case.

[mtfishman]

Thanks @LHerviou! Is this good to merge from your end once tests pass?

[LHerviou]

Yes !