TNRKit

Your one-stop-shop for Tensor Network Renormalization.

Package summary

TNRKit.jl aims to provide as many Tensor Network Renormalization methods as possible. Several models like the classical Ising, Potts and Six Vertex models are provided.

You can use TNRKit for calculating:

1. Partition functions (classical & quantum)
2. CFT data
3. Central charges

Many common TNR schemes have already been implemented:

2D square tensor networks

• TRG (Levin and Nave's Tensor Renormalization Group)
• BTRG (bond-weighted TRG)
• LoopTNR (entanglement filtering + loop optimization)
• SLoopTNR (c4 & inversion symmetric LoopTNR)
• HOTRG (higher order TRG)
• ATRG (anisotropic TRG)

2D square CTM methods

• CTM (Corner Transfer Matrix)
• c4vCTM (c4v symmetric CTM)
• rCTM (reflection symmetric CTM)
• ctm_TRG (Corner Transfer Matrix environment + TRG)
• ctm_HOTRG (Corner Transfer Matrix environment + HOTRG)

2D triangular CTM methods

• c6vCTM_triangular (c6v symmetric CTM on the triangular lattice)
• CTM_triangular (CTM on the triangular lattice)

2D honeycomb CTM methods

• c3vCTM_honeycomb (c3v symmetric CTM on the honeycomb lattice)

Impurity Methods

• ImpurityTRG
• ImpurityHOTRG

3D cubic tensor networks

• ATRG_3D (anisotropic TRG)
• HOTRG_3D (higher order TRG)

Quick Start Guide

1. Choose a (TensorKit!) tensor that respects the leg-convention (see below)
2. Choose a TNR scheme
3. Choose a truncation scheme
4. Choose a stopping criterion

For example:

using TNRKit, TensorKit

T = classical_ising(ising_βc) # partition function of classical Ising model at the critical point
scheme = BTRG(T) # Bond-weighted TRG (excellent choice)
data = run!(scheme, truncrank(16), maxiter(25)) # max bond-dimension of 16, for 25 iterations

data now contains 26 norms of the tensor, 1 for every time the tensor was normalized. (By default there is a normalization step before the first coarse-graining step wich can be turned off by changing the kwarg run!(...; finalize_beginning=false))

Using these norms you could, for example, calculate the free energy of the critical classical Ising model:

f = free_energy(data, ising_βc) # -2.1096504926141826902647832

You could even compare to the exact value, as calculated by the Onsager solution:

julia> abs((f - f_onsager) / f_onsager)
3.1e-07

Pretty impressive for a calculation that takes about 0.3s on a laptop.

Verbosity

There are 3 levels of verbosity implemented in TNRKit:

• Level 0: no TNRKit messages whatsoever.
• Level 1: Info at beginning and end of the simulations (including information on why the simulation stopped, how long it took and how many iterations were performed).
• Level 2: Level 1 + info at every iteration about the last generated finalize output and the iteration number.

to choose the verbosity level, simply use run!(...; verbosity=n). The default is verbosity=1.

Included Models on the square lattice

TNRKit includes several common models out of the box.

• Ising model in 2D: classical_ising(S, β; h=0) where S can be Trivial or Z2Irrep to specify the symmetry.
• Ising model in 2D with impurities: classical_ising_impurity(β; h=0).
• Ising model in 3D: classical_ising_3D(S, β; h=0) where S can be Trivial or Z2Irrep to specify the symmetry.
• Potts model in 2D: classical_potts(S, q, β), where S can be Trivial or ZNIrrep{q} to specify the symmetry.
• Potts model in 2D with impurities: classical_potts_impurity(q, β).
• Six Vertex model: sixvertex(S, elt; a=1.0, b=1.0, c=1.0) where S can be Trivial, U1Irrep or CU1Irrep to specify the symmetry and elt can be any number type (default is Float64).
• Clock model: classical_clock(S, q, β) where S can be Trivial or ZNIrrep{q} to specify the symmetry.
• XY model in 2D: classical_XY(S, β, charge_trunc) where S can be U1Irrep or CU1Irrep to specify the symmetry.
• Real $\phi^4$ model: phi4_real(S, K, μ0, λ, h) where S can be Trivial or Z2Irrep to specify the symmetry.
• Real $\phi^4$ model with impurities: phi4_real_imp1(S, K, μ0, λ, h) and phi4_real_imp2(S, K, μ0, λ, h) where S can be Trivial.
• Complex $\phi^4$ model: phi4_complex(S, K, μ0, λ) where S can be Trivial, Z2Irrep ⊠ Z2Irrep or U1Irrep to specify the symmetry.
• Gross-Neveu model: gross_neveu_start(S, μ, m, g) where S can be FermionParity to specify the symmetry.

Included Models on the triangular lattice

TNRKit includes several common models out of the box.

• Ising model: classical_ising_triangular(S, β; h=0) where S can be Trivial or Z2Irrep to specify the symmetry.

Included Models on the honeycomb lattice

TNRKit includes several common models out of the box.

• Ising model: classical_ising_honeycomb(S, β; h=0) where S can be Trivial or Z2Irrep to specify the symmetry.

If you want to implement your own model you must respect the leg-convention assumed by all TNRKit schemes.

Leg-convention

All the schemes assume that the input tensor lives in the space $V_1 \otimes V_2 \leftarrow V_3 \otimes V_4$ and that the legs are ordered in the following way:

     3
     |
     v
     |
1-<--┼--<-4
     |
     v
     |
     2

The 3D scheme(s) assume that the input tensor lives in the space $V_{\text{D}} \otimes V_{\text{U}} \prime \leftarrow V_{\text{N}} \otimes V_{\text{E}} \otimes V_{\text{S}} \prime \otimes V_{\text{W}} \prime$.

Where D, U, N, E, S, W stand for Down, Up, North, East, South and West.