by Juraj Hasik, Glen Bigan Mbeng
with contributions by Wei-Lin Tu, Seydou-Samba Diop, Sen Niu, Yi Xi

peps-torch performs optimization of infinite Projected entangled-pair states (iPEPS) by direct energy minimization. The energy is evaluated using environments obtained by the corner-transfer matrix (CTM) algorithm. Afterwards, the gradients are computed by reverse-mode automatic differentiation (AD).

• abelian symmetries, with implementation power by YASTN

  Allows definition of abelian-symmetric iPEPS in terms of block-sparse tensors, computing their symmetric environments, and their optimization with gradient-based methods.

• complex tensors with PyTorch 1.11+

  extended L-BFGS optimizer for handling complex-valued tensors.

• rich set of models and their examples

  various spin models on square and Kagome lattices, using both dense and abelian-symmetric tensors. Wide set of examples which show how to optimize and compute observables for these models.

For the full documentation, continue to jurajhasik.github.io/peps-torch

Ex. 1) Optimize one-site (C4v) symmetric iPEPS with bond dimension D=2 and environment dimension X=32 for J1-J2 model at J2/J1=0.3 run

python examples/j1j2/optim_j1j2_c4v.py --bond_dim 2 --chi 32 --seed 123 --j2 0.3 --out_prefix ex-c4v

Using the resulting state ex-c4v_state.json, compute the observables such as spin-spin and dimer-dimer correlations for distance up to 20 sites

python examples/j1j2/ctmrg_j1j2_c4v.py --instate ex-c4v_state.json --chi 48 --j2 0.3 --corrf_r 20

Ex. 2) Optimize iPEPS with 2x2 unit cell containing four distinct on-site tensors by running

python examples/j1j2/optim_j1j2.py --tiling 4SITE --bond_dim 2 --chi 32 --seed 123 --j2 0.3 \
--CTMARGS_fwd_checkpoint_move --OPTARGS_tolerance_grad 1.0e-8 --out_prefix ex-4site

The memory requirements of AD would increase sharply if all the intermediate variables are stored. Instead, by passing --CTMARGS_fwd_checkpoint_move flag we opt for recomputing them on the fly while saving only the intermediate tensors of the CTM environment.

Compute observables and spin-spin correlation functions in horizontal and vertical direction of the resulting state

python examples/j1j2/ctmrg_j1j2.py --tiling 4SITE --chi 48 --j2 0.3 --instate ex-4site_state.json --corrf_r 20

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]. Run

python examples/j1j2/optim_j1j2_u1_c4v.py --bond_dim 4 --u1_class B --chi 32 --j2 0.2 \
--OPTARGS_line_search backtracking --OPTARGS_line_search_svd_method SYMARP --CTMARGS_fwd_checkpoint_move \
--out_prefix ex-u1b

The optimization is performed together with backtracking linesearch. Moreover, the CTM steps during linesearching are accelerated by using partial eigenvalue decomposition (SCIPY's Arnoldi) instead of full-rank one.

Using the resulting state ex-u1b_state.json, compute the observables such as leading part of transfer matrix spectrum or spin-spin and dimer-dimer correlations for distance up to 20 sites

python examples/j1j2/ctmrg_j1j2_u1_c4v.py --instate ex-u1b_state.json --j2 0.2 --chi 32 --top_n 4 --corrf_r 20

Ex. 4) Optimize complex-valued one-site (C4v) symmetric iPEPS. Simply pass the flag --GLOBALARGS_dtype complex128. For instance, the Ex. 1 becomes

python examples/j1j2/optim_j1j2_c4v.py --GLOBALARGS_dtype complex128 --bond_dim 3 --chi 32 \
--seed 123 --j2 0.5 --out_prefix ex-c4v

The computation of expectations values can be done right away, as the state stored in ex-c4v_state.json carries the datatype (dtype) of its tensors

python examples/j1j2/ctmrg_j1j2_c4v.py --GLOBALARGS_dtype complex128 --instate ex-c4v_state.json \
--chi 48 --j2 0.5 --corrf_r 20

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. Run

python examples/j1j2/abelian/optim_j1j2_u1.py --tiling BIPARTITE \
--instate test-input/abelian/c4v/BFGS100LS_U1B_D3-chi72-j20.0-run0-iRNDseed321_blocks_2site_state.json \
--chi 32 --j2 0.3 --CTMARGS_fwd_checkpoint_move --OPTARGS_tolerance_grad 1.0e-8 --out_prefix ex-2site-u1

The computation of expectations values can be carried out by invoking

python examples/j1j2/abelian/ctmrg_j1j2_u1.py --tiling BIPARTITE --chi 48 --j2 0.3 --instate ex-2site-u1_state.json

• spin systems
• arbitrary rectangular unit cells
• abelian-symmetric tensors
• both real- and complex-valued tensors

• PyTorch 1.11+ (see https://pytorch.org/)
• (optional) YASTN (see https://github.com/yastn/yastn)
• (optional) scipy 1.3.+
• (optional) opt_einsum
• (optional) ArrayFire (see https://github.com/arrayfire/arrayfire)

YASTN is linked to peps-torch as a git submodule. To obtain it, you can use git:

git submodule update --init --recursive

• PyTorch 1.11+
• sphinx
• sphinx_rtd_theme

All the dependencies can be installed through conda (see https://docs.conda.io).

Afterwards, build documentation as follows:

cd docs && make html

The generated documentation is found at docs/build/html/index.html

Inspired by the pioneering work of Hai-Jun Liao, Jin-Guo Liu, Lei Wang, and Tao Xiang, Phys. Rev. X 9, 031041 or arXiv version arXiv:1903.09650

References:

1. Corner Transfer Matrix Renormalization Group Method, T. Nishino and K. Okunishi, Journal of the Physical Society of Japan 65, 891 (1996) or arXiv version arXiv:cond-mat/9507087
2. Faster Methods for Contracting Infinite 2D Tensor Networks,
   M.T. Fishman, L. Vanderstraeten, V. Zauner-Stauber, J. Haegeman, and F. Verstraete, Phys. Rev. B 98, 235148 (2018) or arXiv version arXiv:1711.05881
3. Competing States in the t-J Model: Uniform d-Wave State versus Stripe State (Supplemental Material), P. Corboz, T. M. Rice, and M. Troyer, Phys. Rev. Lett. 113, 046402 (2014) or arXiv version arXiv:1402.2859
4. Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry, M. Mambrini, R. Orus, and D. Poilblanc, Phys. Rev. B 94, 205124 (2016) or arXiv version arXiv:1608.06003
5. Investigation of the Néel phase of the frustrated Heisenberg antiferromagnet by differentiable symmetric tensor networks, J. Hasik, D. Poilblanc, and F. Becca, SciPost Phys. 10, 012 (2021)