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:

• TRG (Levin and Nave's original formulation of a TNR scheme)
• BTRG (bond-weighted TRG)
• ATRG (anisotropic TRG)
• ATRG_3D (3D ATRG)
• HOTRG (higher order TRG)
• HOTRG_3D (3D HOTRG)
• LoopTNR (entanglement filtering + loop optimization)
• SLoopTNR (C4 & inversion symmetric LoopTNR)
• ctm_HOTRG (Corner Transfer Matrix environment + HOTRG)
• ctm_TRG (Corner Transfer Matrix environment + TRG)
• c4vCTM (c4v symmetric CTM)
• rCTM (reflection symmetric CTM)

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, truncdim(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

TNRKit includes several common models out of the box.

• 2D Ising model: classical_ising(β; h=0) and classical_ising_symmetric(β), which has a $ℤ_2$ grading on each leg.
• 3D Ising model: classical_ising_3D(β; J=1) and classical_ising_symmetric_3D(β), which has a $ℤ_2$ grading on each leg.
• Potts model: classical_potts(q, β) and classical_potts_symetric(q, β), which has a $ℤ_q$ grading on each leg.
• Six Vertex model: sixvertex(scalartype, spacetype; a=1.0, b=1.0, c=1.0)
• Clock model: classical_clock

Leg-convention

If you want to implement your own model you must respect the leg-convention assumed by all TNRKit schemes. The 2D 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.