Infinite PEPS states for J1-J2 model on a square lattice

This repository contains two datasets

• single-site_pg-C4v-A1
• single-site_pg-C4v-A1_internal_U1, basis for SciPost Phys. 10, 012 (2021)

representing translationary invariant infinite PEPS states defined by a single rank-5 tensor A^s_uldr. The index s runs over two states of spin-1/2 degree of freedom and the virtual indices u,l,d,r, for up, left, down and right directions on square lattice, run over values from 0 to D-1 with D being the bond dimension.

This tensor transforms as A₁ irreducible representation of C4_v point group (symmetry group of square lattice) under 90° rotations and reflections represented by permutations of virtual indices

A^s_uldr=A^s_ldru=A^s_ruld=A^s_urdl=A^s_dlur.

These single-site translationally invariant PEPS are endowed with antiferromagnetic correlations by applying a unitary rotation -iσ^y to spins¹ on every sublattice-B site of a square lattice under bipartite tiling

B^s_uldr=-iσ^y_skA^k_uldr.

The tensors in the dataset single-site_pg-C4v-A1_internal_U1 also possess U(1) symmetry [see SciPost Phys. 10, 012 (2021)]. The states are stored in plain text format (JSON).

Observables

Each state is accompanied by corresponding *.dat file containing the value of selected observables as evaluated by corner transfer matrix algorithm implemented in peps-torch library. For a set of environment bond dimensions χ, those are

• energy per site of J₁-J₂ model²
• on-site magnetization m=|⟨S⟩|, with S=(S^z,S^x,S^y) the vector of spin-1/2 operators
• leading eigenvalues λ₀, λ₁,... of (width-1) transfer matrix. The spectrum is normalized (λ₀=1) and the leading correlation length can be obtained as ξ = -1/ln(λ₁)

Reading and exporting states

To parse the states use the Python script ipeps_io.py which can export the dense tensor A^s_uldr to either NumPy's *.npz format or MATLAB's *.mat format (requires SciPy)

python ipeps_io.py --instate path/to/json/file --format mat [--out optional/name/for/exported/file]
python ipeps_io.py --instate path/to/json/file --format npz [--out optional/name/for/exported/file]

Or access the (NumPy) tensor directly in the interactive mode

>>> from ipeps_io import load_from_pepstorch_json_dense
>>> A=load_from_pepstorch_json_dense("single-site_pg-C4v-A1/j20.0/state_1s_A1_j20.0_D3_chi_opt108.json")
>>> type(A)
<class 'numpy.ndarray'>
>>> A.shape
(2, 3, 3, 3, 3)
>>> A[0,0,0,:,:]
array([[-0.49333601,  0.1460243 ,  0.03569442],
       [ 0.1460243 , -0.26334648, -0.01241685],
       [ 0.03569442, -0.01241685,  0.40762404]])

It is also possible to use ipeps_io.py script to export U(1)-symmetric states of single-site_pg-C4v-A1_internal_U1 dataset to commonly adopted block-sparse form. See single-site_pg-C4v-A1_internal_U1/README.md.

Footnotes

1. In practice it is more convenient to instead rotate either operators or reduced density matrices. ↩

2. both nearest- and next-nearest neighbour spin exchange terms are evaluated in 2x2 patch embedded in CTM environment ↩