Infinite PEPS and PESS states for Heisenberg model with Dzyaloshinskii–Moriya interaction on a Kagome lattice

This repository contains a dataset of optimized 2D tensor network states representing ground states of antiferromagnetic Heisenberg model with Dzyaloshinskii–Moriya interaction (DMI) on Kagome lattice, obtained in SciPost Phys. 14, 139 (2023).

This dataset contains two families of states and their observables:

IPEPS - a single rank-5 tensor accounts for 3 spins on each up-pointing triangle of Kagome lattice
IPESS - a set of five rank-3 tensors together accounting for 3 spins on each up-pointing triangle: two tensors without physical degrees of freedom and three bond tensor, one for each spin on the up-pointing triangle

For each dataset the states are stored in the corresponding folder, i.e. IPEPS/J<nearest-neighbour-exchange>_JD<DMI-strength> or IPESS/J<nearest-neighbour-exchange>_JD<DMI-strength>. States, or more specifically the tensors, are stored in plain text format (JSON). Below, we describe the individual datasets in more detail.

iPEPS

The single-site iPEPS ansatz adopted in SciPost Phys. 14, 139 (2023) coarse-grains Kagome lattice to an effective square lattice by grouping three spin-1/2's on each up-pointing triangle into a single degree of freedom with local Hilbert space ℋ = ⊗₃ ℋ(spin-1/2) of dimension 8.

The on-site tensor a^s_uldr adopts the following index convention

   u
   |
l--\                    
    \
    s0--s2--r      = a[s,u,l,d,r]
     | /
     |/   <- up-pointing triangle, where s0, s1, and s2 denote position of spins
    s1
     |               
     d

where physical index s enumerates states of three spin-1/2's in the basis s₀⊗s₁⊗s₂. The indices u,l,d,r of the virtual degrees of freedom, each of bond dimension D, are associated with up, left, down, and right directions on a square lattice.

Finally, the Kagome lattice is made up from the pattern above. Replacing each up-pointing triangle by tensor a and contracting the neighbouring tensors over their virtual indices creates an iPEPS tensor network defined on a square lattice

         |            
       --\          
          \           |
          s0--s2--  --\    
           | /         \            |  |
           |/          s0--s2-- = --a--a--
          s1            | /         |  |
           |            |/        --a--a--
           |           s1           |  |
         --\            |
            \           |
            s0--s2-- --\      
             | /        \
             |/         s0--s2--
            s1          | /
             |          |/
                       s1
                        |

Note that not all edges of Kagome lattice are explicitly shown here. Due to coarse-graining, some nearest-neighbour bonds (edges) of Kagome lattice become next-nearest neighbour bonds on the effective square lattice.

iPESS

iPESS ansatz endows the rank-5 on-site tensor a^s_uldr with more structure, further constraining the variational freedom.

The on-site tensor is obtained by contraction of 5 smaller rank-3 tensors:

two rank-3 trivalent tensors T_u and T_d residing within up- and down-pointing triangles. These tensors have only virtual indices of bond dimension D.
three rank-3 bond tensors B_a, B_b, and B_c residing on corners shared between different triangles of Kagome lattice. Each bond tensor is associated to one of the three spins of an up-pointing triangle and carries a physical index corresponding to its spin-1/2 degree of freedom.

To obtain tensor a^s_uldr, contract the iPESS tensors as follows

                2(l)   1(u)
                   \   /
                    T_u                      u
                     |                       |
                     0(i)                 l--\
                     2(i)                     \
                     |                         m--o--r
                     B_c==0(m)       =         | /        = a[m,n,o,u,l,d,r] = a[s=(m,n,o),u,l,d,r]
                     |                         |/
                     1(j)                      n
                     0(j)                      |
                     |                         d
                    T_d
                   /  \                   where index conventions for iPESS tensors are:
                1(k)  2(x)              
              1(k)     1(x)               T_u[i,u,l]
               /          \               T_d[j,k,x]
       0(n)==B_b          B_a==0(o)       B_a[o,x,r]
             /              \             B_b[n,k,d]
          2(d)             2(r)           B_c[m,j,i]

# Simple snippet demonstrating the contraction and reshaping of the result into an on-site tensor a^s_uldr

A= einsum('iul,mji,jkx,nkd,oxr->mnouldr', T_u, B_c, T_d, B_b, B_a)
A= A.reshape(8,D,D,D,D) # where D is the bond dimension

here, the first index of each bond tensor is physical index. Neighbouring trivalent tensors are connected by contractions with bond tensors creating the iPESS tensor network.

This choice represents the geometry of Kagome lattice more faithfully, treating the up- and down-pointing triangles on equal footing. Moreover, it allows for imposing further point-group symmetries on trivalent and/or bond tensors. It is also common starting point for imposing internal symmetries on iPESS tensor

Observables

Each state is accompanied with a simple plain-text *.dat file containing selected observables. These are evaluated using corner transfer matrix algorithm implemented in peps-torch. For a set of increasing environment bond dimensions χ, they are

energy per site of Heisenberg antiferromagnet with DMI (see Eq.1 of SciPost Phys. 14, 139 (2023))
energies of up- and down-pointing triangles. Their difference indicates inversion symmetry breaking
magnetization m=|⟨S⟩|, with S=(S^z,S^x,S^y) the vector of spin-1/2 operators, for each of the three spins within up-pointing triangle
individual spin components S^z, S⁺, S^- for each spin
Nearest-neighbour spin-spin correlations of spins within down- ⟨S.S⟩_down,01, ⟨S.S⟩_down,02, ⟨S.S⟩_down,12 and up-pointing triangle ⟨S.S⟩_up,01, ⟨S.S⟩_up,02, ⟨S.S⟩_up,12
leading eigenvalues λ_0,x, λ_1,x, λ_2,x of (width-1) horizontal and λ_0,y, λ_1,y, λ_2,y of vertical transfer matrix. The spectra are normalized (λ_0,*=1) and the leading correlation length can be obtained as ξ_x = -1/ln(λ_1,x) and ξ_y = -1/ln(λ_1,y)

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("IPEPS/J1.0_JD0.0/IPEPS_J1.0_JD0.0_D3_chi_opt40.json")
>>> type(A)
<class 'numpy.ndarray'>
>>> A.shape
(8, 3, 3, 3, 3)
>>> A[:,0,0,0,0]
array([-0.00782555,  0.12528113, -0.13526041, -0.01993145, -0.02498749,
    0.11935324, -0.12645546, -0.00031023])

When exporting iPESS states, both usual rank-5 on-site tensor a^s_uldr is exported as well as the individual iPESS tensors, stored under the keys defined in section above.

>>> from ipeps_io import load_peps_from_json_dense, load_pess_from_json_dense
>>> A,ipess_tensors=load_pess_from_json_dense("IPESS/J1.0_JD0.0/IPESS_J1.0_JD0.0_D4_chi_opt20.json")
>>> ipess_tensors.keys()
dict_keys(['T_u', 'T_d', 'B_c', 'B_a', 'B_b'])
>>> ipess_tensors['T_u'][:,:,0]
array([[-0.47092122,  1.39894781,  0.39612967, -0.89794985],
    [ 0.45824881, -0.77460775, -0.83686949, -0.71734594],
    [ 0.24671555, -0.47313889, -3.47609289,  0.39378996],
    [ 1.03717042,  0.58974852,  0.09221363, -0.84499218]])

jurajHasik/j1jD_kagome_ipeps_states

Infinite PEPS and PESS states for Heisenberg model with Dzyaloshinskii–Moriya interaction on a Kagome lattice

iPEPS

iPESS

Observables

Reading and exporting states