TNRKit.jl

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TNRKit.jl is a Julia package that aims to implement as many tensor network renormalization (TNR) schemes as possible. It is built upon TensorKit.jl, which provides functionality for symmetric tensors. The following schemes are currently 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)

Impurity Methods

• ImpurityTRG
• ImpurityHOTRG

3D cubic tensor networks

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

This project is under active development. The interface is subject to changes. Any feedback about the user interface or the internals is much appreciated. The github discussions page is a great place to talk!

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_symmetric(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: classical_ising(β; h=0) and classical_ising_symmetric(β), which has a $\mathbb{Z}_2$ grading on each leg.
• Potts model: classical_potts(q, β) and classical_potts_symetric(q, β), which has a $\mathbb{Z}_q$ grading on each leg.
• Six Vertex model: sixvertex(scalartype, spacetype; a=1.0, b=1.0, c=1.0)
• Clock model: classical_clock and classical_clock_symmetric, which has a $\mathbb{Z}_q$ grading on each leg.
• XY model: classical_XY_U1_symmetric and classical_XY_O2_symmetric

Included Models on the triangular lattice

TNRKit includes several common models out of the box.

• Ising model: classical_ising_triangular and classical_ising_triangular_symmetric, which has a $ℤ_2$ grading on each leg.