quantum number conservation in `make_time_mpo`

[ZongYongyue]

Hi Lukas,

Is it possible to apply quantum number conservation in make_time_mpo?

Here, I would like to evolve the system fixed at the sector of N = 4, but the result of tr(rho) is actually on the full Hilbert space rather than N Hilbert space.

using TensorKit
using MPSKit
using MPSKitModels:FiniteChain, hubbard_model


H_u1_su2 = hubbard_model(ComplexF64, U1Irrep, SU2Irrep, FiniteChain(4); U=8.0, t=1.0);
charges = fill(FermionParity(1) ⊠ U1Irrep(1) ⊠ SU2Irrep(0), 4);
H_u1_su2 = MPSKit.add_physical_charge(H_u1_su2, charges);

rho = make_time_mpo(H_u1_su2, -0.01*im, TaylorCluster(; N=2))
tr(MPSKit.DenseMPO(rho))
#236.91308189073806 + 0.0im

#ED result of rho in full hilbert space is 236.91352429093988
#ED result of rho in 4-particle hilbert space is 65.43362134529468

[lkdvos]

In principle the rho you construct here does conserve quantum numbers, and will still respect the symmetry. However, I don't know if it is as straightforward to use the same trick we use for states.
In the case of a state, we are implicitly making use of the fact that our states have to transform trivially, so by default they have charge zero and we can shift that around a bit.
For an operator however, this condition means that the total charge of the operator is zero, in other words it doesn't change the charge of whatever it is acting on. Since the charges are appearing on both sides (domain and codomain) of the operator, we can't easily single out a given sector, since a shift in the domain would simply be cancelled by a shift in the codomain (this is exactly why we are allowed to shift the charges when considering states in the first place!)

In some sense, I think one would like to use, instead of a full trace which can be viewed as contracting with the identity, a projected trace on the given subspace. However, I don't think you can write down that projector as a local operator, in the sense that my naive considerations would mean that the bond dimension would simply scale with system size.

Long story short, I'm not sure if this is possible, but this might also be because of my lack of knowledge.

[ZongYongyue]

Thank you for your detailed reply!

I agree with that a projected trace is needed on the given subspace. But I'm not familiar with the implement of make_time_mpo , my naive idea is, if $e^{-\beta H}$ is evolved from identity operator $I$, time evolution from a projected operator $P_N$ might restrict the system on the given subspace? And the trace $Tr(P_N e^{-\beta H})$ is what we want?

[lkdvos]

Ah I see what you mean. The issue is that this isn't how that function works, it more or less just builds the Taylor expansion in a smart way, so there is no real evolution going on.

One thing you might try is to multiply the resulting MPO with a projector that is just the tensor product of local projectors on the subspace of a given charge, but note that this probably isn't actually the correct subspace. For example, the 2 particle sector for 2 sites should contain (2,0), (0,2) but also (1,1) particle per site, and this latter state would not be included

[ZongYongyue]

I see, thank you for your kind reply again ❤️

[Jutho]

Just to add, the projector onto a given particle number subspace is indeed not something that can be efficiently represented as an MPO with a fixed bond dimension; the bond dimension needs to scale linearly in the number of particles N. For a finite system, that might still be reasonable, but it is not something you can do in the thermodynamic limit.

If you want N particles in a system of L sites, then the bond between site L/2 and L/2 + 1 represents the information of the number of particles on the left half of the system, which can really take all values from 0 to N (assuming bosons, or fermions and L/2 >= N), i.e. all the configurations going from all the particles are on the right half (zero on the left) to all the particles are on the left side, and everything in between.

[lkdvos]

Actually I just remembered that I experimented with these kinds of states at some point, but I did not make the connection with the projector as that is an operator.
If I'm understanding this correctly, you could trace the operator by taking the expectation value of this operator with the Dicke states, since that should be the correct projector, at a bond dimension that scales linearly in the number of particles (but not system size).
Credits go to https://github.com/ryanlevy for that, and you can find the code here:

https://github.com/lkdvos/DickeStates

[ZongYongyue]

That's so kind of you, this is sure to be very helpful to me!

[ZongYongyue]

Just to add, the projector onto a given particle number subspace is indeed not something that can be efficiently represented as an MPO with a fixed bond dimension; the bond dimension needs to scale linearly in the number of particles N. For a finite system, that might still be reasonable, but it is not something you can do in the thermodynamic limit.

If you want N particles in a system of L sites, then the bond between site L/2 and L/2 + 1 represents the information of the number of particles on the left half of the system, which can really take all values from 0 to N (assuming bosons, or fermions and L/2 >= N), i.e. all the configurations going from all the particles are on the right half (zero on the left) to all the particles are on the left side, and everything in between.

Thank you for your further reply!😄