Feature requests

[VictorVanthilt]

Tell us about the features you would like to see implemented!

[Yue-Zhengyuan]

It seems that currently focus of the TNR schemes is to calculate the free energy. But the evolution of scaling dimensions from each RG step is a better way to look into the performance of the schemes and the underlying CFT of the model. So I suggest the abilities to

• Obtain CFT scaling dimension and central charge (not just the normalization factor) from each step of RG, so we can produce a figure like Fig. 3 in the loop-TNR paper 1512.04938.
• Extract CFT data from transfer matrices of a larger size (I guess 2 x 2, 2 x 4 will be enough for most cases), which can be more accurate and stable under RG than the 1 x 1 one.

[VictorVanthilt]

Both of these functionalities already exist here.

But I'm thinking of refactoring this to make a finalize_cft! function to make it more easy to get the cft data at every rg step.

[Yue-Zhengyuan]

In addition, it would be really great if we can add unicode illustrations of the TNR algorithms.

[VictorVanthilt]

I was planning on adding nice figures in the docs but an ascii representation that the user can view in the terminal would also be a nice addition!

[Yue-Zhengyuan]

May I ask did you take fermions into account when writing the code? In principle, we can directly express the partition function of a (1+1)D fermion model as a fermionic tensor network using Trotter decomposition of exp(-βH) (one can do Jordan-Wigner to study the corresponding bosonic model, but doesn't have to). You can find more details in https://uwspace.uwaterloo.ca/items/2707e55c-296d-4200-ae4d-2996e9dd3baa (There the Grassmann tensor formalism is used, but it's not hard to translate to the super vector space formalism used by TensorKit; note that the convention for arrow direction in that thesis is the opposite to TensorKit).

[Yue-Zhengyuan]

The thesis studies the 1D spinless fermion chain with nearest neighbor interaction. It converts exp(-βH) to a square lattice network. It's a bit tricky to tune the imaginary time slice for each row of this network. Then scaling dimension can be extracted from the transfer matrices (twists can be used to change the boundary condition for the transfer matrices, which I don't fully understand yet). You may try this as a test for the CFT part of TNRKit.

[Adwait-Naravane]

The TNR schemes we use to calculate the free energy seem to give us the right value for the free energy for the Gross-Neveu model as compared with the results from https://arxiv.org/pdf/2311.17691. I also tried it for 1+1D Hubbard model https://arxiv.org/abs/2105.00372 but more work needs to be done here. Ultimately, I also want Loop TNR to work for fermionic models so that we can extract something useful like the cft data. In that regard, the additional twists in the cost function you mentioned would be worth looking into.

[Yue-Zhengyuan]

Basically the twist appears in the following case: suppose a fermionic state $|\psi\rangle$ is given by

            ┌---------┐
    |ψ⟩  =  └--┬---┬--┘
               ↓   ↑

Note that the right leg is a dual space. Its norm is given by

            ┌---------┐
            └--┬---┬--┘
               ↓   ↑
    ⟨ψ|ψ⟩ =    |   o (twist)
               ↓   ↑   
            ┌--┴---┴--┐
            └---------┘

A twist should be applied (or in the terminology of 2404.14611, a parity tensor P should be inserted) when contracting the right leg when using @tensor. Currently TNRKit doesn't handle this.

[Adwait-Naravane]

I had a related problem with computing the 2x2 norm. When I compute the 2x2 norm for the tensors of the gross-neveu model, my final answer is not the 4th power of my single tensor norm.

[Yue-Zhengyuan]

@Adwait-Naravane Are you referring to

norm(@tensor scheme.T[7 1; 5 4] * scheme.T[4 2; 6 7] * scheme.T[3 6; 2 8] * scheme.T[8 5; 1 3])

I don't think it should be the same as $|T|^4$ even for bosonic tensors...

[Chenqitrg]

By considering a wider transfer matrix, higher conformal dimensions of CFT can be produced. Maybe by diagonalizing the transfer matrix with a very wide MPS (or MERA), very large conformal dimensions can be calculated.

[Adwait-Naravane]

I would like a function to give me the Gu-Wen ratio defined in https://arxiv.org/abs/0903.1069. I might add it later.

[Adwait-Naravane]

I would like to add a plethora of 2D and 3D spin and field theory models such that real and complex scalar fields, O(N) models, NLSM models and Gauge theories for abelian and non-abelian groups.

[Yue-Zhengyuan]

I'm curious on how (the partition function of) a continuum field theory is going to be expressed in terms of discrete tensors?

[Chenqitrg]

I'm curious on how (the partition function of) a continuum field theory is going to be expressed in terms of discrete tensors?

It is possible to express the tensor whenever the set of symmetry charges is discrete. The descrete charge can happen if the target manifold of the field (like a compact Lie group) is a compact manifold, the Fourier basis of it is infinite but discrete. I indeed tried the 2D XY model before, where charges are labeled by integers. If the model is good enough, keeping finitely many of them will not affect the convergence.