S_xx, S_yy with U1Irrep symmetry

[Yue-Zhengyuan]

We (with @pbrehmer @leburgel) found that the sub-lattice rotated J1-J2 Hamiltonian with U(1) spin symmetry evaluates without error. It turns out that in MPSKitModels S_xx, S_yy both evaluates with U1Irrep. But upon a closer look, the results do not look right. For example, compare the Trivial and U1Irrep version of S_xx:

julia> using Pkg, TensorKit, MPSKitModels

julia> Pkg.status("MPSKitModels")
Status `~/.julia/environments/v1.11/Project.toml`
  [ca635005] MPSKitModels v0.4.3

julia> Pkg.status("TensorKit")
Status `~/.julia/environments/v1.11/Project.toml`
⌃ [07d1fe3e] TensorKit v0.14.9
Info Packages marked with ⌃ have new versions available and may be upgradable.

julia> S_xx(Trivial)
TensorMap((ℂ^2 ⊗ ℂ^2) ← (ℂ^2 ⊗ ℂ^2)):
[:, :, 1, 1] =
 0.0 + 0.0im   0.0 + 0.0im
 0.0 + 0.0im  0.25 + 0.0im

[:, :, 2, 1] =
 0.0 + 0.0im  0.25 + 0.0im
 0.0 + 0.0im   0.0 + 0.0im

[:, :, 1, 2] =
  0.0 + 0.0im  0.0 + 0.0im
 0.25 + 0.0im  0.0 + 0.0im

[:, :, 2, 2] =
 0.25 + 0.0im  0.0 + 0.0im
  0.0 + 0.0im  0.0 + 0.0im


julia> S_xx(U1Irrep)
TensorMap((Rep[U₁](1/2=>1, -1/2=>1) ⊗ Rep[U₁](1/2=>1, -1/2=>1)) ← (Rep[U₁](1/2=>1, -1/2=>1) ⊗ Rep[U₁](1/2=>1, -1/2=>1))):
* Data for sector (Irrep[U₁](-1/2), Irrep[U₁](1/2)) ← (Irrep[U₁](-1/2), Irrep[U₁](1/2)):
[:, :, 1, 1] =
 0.0 + 0.0im
* Data for sector (Irrep[U₁](1/2), Irrep[U₁](-1/2)) ← (Irrep[U₁](-1/2), Irrep[U₁](1/2)):
[:, :, 1, 1] =
 0.25 + 0.0im
* Data for sector (Irrep[U₁](-1/2), Irrep[U₁](1/2)) ← (Irrep[U₁](1/2), Irrep[U₁](-1/2)):
[:, :, 1, 1] =
 0.25 + 0.0im
* Data for sector (Irrep[U₁](1/2), Irrep[U₁](-1/2)) ← (Irrep[U₁](1/2), Irrep[U₁](-1/2)):
[:, :, 1, 1] =
 0.0 + 0.0im
* Data for sector (Irrep[U₁](1/2), Irrep[U₁](1/2)) ← (Irrep[U₁](1/2), Irrep[U₁](1/2)):
[:, :, 1, 1] =
 0.0 + 0.0im
* Data for sector (Irrep[U₁](-1/2), Irrep[U₁](-1/2)) ← (Irrep[U₁](-1/2), Irrep[U₁](-1/2)):
[:, :, 1, 1] =
 0.0 + 0.0im

In the Trivial version, there are 4 nonzero elements, corresponding to

[2, 2, 1, 1]:   |↓↓⟩ ← |↑↑⟩
[1, 2, 2, 1]:   |↑↓⟩ ← |↓↑⟩
[2, 1, 1, 2]:   |↓↑⟩ ← |↑↓⟩
[1, 1, 2, 2]:   |↑↑⟩ ← |↓↓⟩

But the U1Irrep version of S_xx appears to be obtained by projecting out the part that does not preserve z-components of the total spin, removing the elements corresponding to |↓↓⟩ ← |↑↑⟩ and |↑↑⟩ ← |↓↓⟩, with only 2 nonzero elements left. @lkdvos Is this an error when generating 2-spin functions, or an intended design?

[lkdvos]

I think this is an old mistake on my part. Both xx and yy just can't be created while conserving the symmetry, I don't really know or remember why I thought it could be done. In TensorKitTensors I'm pretty sure that we no longer provide this and correctly error out.

[Yue-Zhengyuan]

Then we need to rewrite PEPSKit.j1_j2_model with sub-lattice rotation and U1 spin symmetry. I'll try making a PR there by explicitly specifying the tensor elements.

[lkdvos]

That's probably not necessary, if you can rewrite it using plusmin and min plus it should be fine (it's probably the difference instead of the sum, but best to double check)

[Yue-Zhengyuan]

Well I was too optimistic... The elements of rotated $(S_x, S_y, S_z)_1 \cdot (-S_x, S_y, -S_z)_2$ turn out to be

julia> S = Trivial
Trivial

julia> - S_xx(S) + S_yy(S) - S_zz(S)
TensorMap((ℂ^2 ⊗ ℂ^2) ← (ℂ^2 ⊗ ℂ^2)):
[:, :, 1, 1] =
 -0.25 + 0.0im   0.0 + 0.0im
   0.0 + 0.0im  -0.5 + 0.0im

[:, :, 2, 1] =
  0.0 + 0.0im  0.0 + 0.0im
 0.25 + 0.0im  0.0 + 0.0im

[:, :, 1, 2] =
 0.0 + 0.0im  0.25 + 0.0im
 0.0 + 0.0im   0.0 + 0.0im

[:, :, 2, 2] =
 -0.5 + 0.0im    0.0 + 0.0im
  0.0 + 0.0im  -0.25 + 0.0im

There are elements at [2, 2, 1, 1], [1, 1, 2, 2] that breaks the spin U(1) symmetry. @pbrehmer @leburgel We need to think more carefully on how to deal with U(1) with sub-lattice rotation that can keep physical space the same on each site...

[lkdvos]

I thought this through a bit more, and actually it's just a matter of doing a rotation around the correct axis, our convention rotates around $S_y$, but we can just as well rotate with $S_z$, which then yields $(S_x, S_y, S_z) \rightarrow (-S_x, -S_y, S_z)$, such that we get for the twosite hamiltonian: $S \cdot S \rightarrow S_zS_z - (S_+S_- + S_-S_+) / 2 $

[Yue-Zhengyuan]

This doesn't quite match the intuition of transforming AFM to FM order. At least when using U1 symmetry, the AFM staggered magnetization is along the z-direction. So you must rotate around an axis in the xy-plane to transform it to FM order. Besides, I don't know if FM order in the xy-plane can be imposed by adding a uniform U(1) charge in the physical legs of each PEPS tensor.

[pbrehmer]

I just gave this a try and at least for dense variational optimization and $U(1)$-symmetric SU evolution the $S_z$-sublattice rotated Hamiltonian does seem to give the correct energies (same as the unrotated $2\times2$ Hamiltonian). Measuring the $S_z$ magnetization also seems to yield correct results, comparing to Juraj's data. However, I'm also not super sure about the physical picture behind this $S_z$ sublattice rotation.

[lkdvos]

I'm also very confused, but at least at first glance everything should be invariant under SU(2), so I should be able to do some re-labeling without changing the result. The question is really if there is an issue with how we rotate and what directions we allow the spontaneous symmetry breaking to happen in (which relates to the choice of "which U(1)" we pick). In any case, I don't think it is possible to create a different hamiltonian with U1 symmetry, as that basis really is spanned by II, SzSz, S+S-, S-S+, and imposing hermitian operators then only leaves 3 parameters, II is an energy shift and there is an overall scaling, so by exhaustion it kind of has to be S+S- + S-S+ + delta * SzSz...

[Yue-Zhengyuan]

@pbrehmer Did you observe FM order in the xy-plane after Sz rotation for both Trivial and U1Irrep?

[pbrehmer]

I only measured the on-site $S_z$ magnetization and that matched. How exactly would you measure $xy$ FM order in a $U(1)$-symmetric way? Just by measuring $S_+ S_- + S_- S_+$?

However, this sublattice rotation also doesn't help the original problem with SU since the virtual spaces are still non-uniform. In fact, if I enforce uniform virtual spaces using truncspace(Vspace), the SU evolution becomes trivial (converges instantly) and I get $E=0.25$ and $M=0.5$ which seems like a product state.

The same thing happens if I add a half-integer charge as @leburgel suggested:

H = add_physical_charge(H, fill(U1Irrep(1 // 2), (2, 2))

Both variational optimization and SU evolution will converge to the $E=0.25$ state.

[lkdvos]

Thinking about it even more, I think I might have convinced myself now that you cant have a single site unit cell and U(1) symmetry, but don't quote me on this 😝

[Yue-Zhengyuan]

Well then how do other people do it with C4v and U1? 😂

@pbrehmer With U1 symmetry I agree that there would be no magnetization in XY plane (S+S- is measuring spin correlation instead). But for the Trivial case, did you find the magnetization in the XY plane, or in some arbitrary direction?

[pbrehmer]

Thinking about it even more, I think I might have convinced myself now that you cant have a single site unit cell and U(1) symmetry, but don't quote me on this 😝

I also have that gut feeling. But this makes me even more confused about how the $U(1)$-symmetric data was actually generated in Juraj's paper. Both on the repo and in the paper it sounds like they used $C_{4v}$ symmetry and a single-site unit cell :D

[pbrehmer]

But for the Trivial case, did you find the magnetization in the XY plane, or in some arbitrary direction?

So in the Trivial case when I do variational optimization (haven't tried SU), if I measure $(S_x, S_y, S_z)$ I do find finite magnetization in $x$ and $y$ direction. That perhaps matches the physical intuition here?

[Yue-Zhengyuan]

In the Trivial case the magnetization in principle could be in an arbitrary direction, so nonzero Sx, Sy are expected. Is it ferromagnetic (same direction for all sites) with sub-lattice rotation?

Maybe we should take a look at the README of https://github.com/jurajHasik/peps-torch, particularly Ex. 3 and 5:

Ex. 3) Optimize one-site iPEPS and impose both C4v symmetry and U(1) symmetry. We choose a particular U(1) class, defined in u1sym/D4_U1_B.txt, given by a set of 25 elementary tensors. The on-site tensor is then constructed as their linear combination. This approach to symmetric iPEPS has been developed in Ref. [4].

Ex. 5) Optimize U(1)-symmetric iPEPS with 2 sites in the unit cell. The U(1)-symmetric tensors are implemented in block-sparse fashion using YAST (see Dependencies). To specify the ansatz, one has to select charge sectors and their sizes for each tensor making up the ansatz. Here, we adopt U(1)-structure determined in Ref. [5]. which is provided in one of the test states, available in test-input/abelian/c4v

Ref. [5] is the paper linked by @pbrehmer above. Or just directly ask him...