User Guide

The package provides three algorithms: trg, ctmrg and variationalipeps-optimisation.

Tensor Renormalization Group

The trg function can currently be used to get the partition function of a classical hamiltonian model on a square lattice. trg uses the principle of coarse graining - low-level detail is discarded in favor of dynamics that dominate the big picture. For an excellent guide to implement trg, check out this tutorial for iTensor.

The input to trg is a rank-4 tensor a that is the tensor-network representation of a model, a cutoff-dimension χ and a number of iterations niter. trg(a, χ, niter) returns the partition function per site.

The result is per site because otherwise the partition function grows exponentially (with the system size) in the number of iterations, leading to numerical problems quickly.

The trg algorithm is fully differentiable with Zygote, which enables us to directly find the first derivative of the partition function with:

julia> using Zygote, TensorNetworkAD

julia> Zygote.gradient(0.5) do β
          trg(model_tensor(Ising(), β), 5, 5)
        end
(1.7502426939979507,)

where we take the derivative of the ising-partition function w.r.t the inverse temperature β using the model_tensor function provided by TensorNetworkAD.

Corner Tensor Renormalization Group

The ctmrg function can be used to find a representation of the environment of an ipeps. The environment can then be used to calculate local quantities for a system of infinite size such as the magnetisation or (short) correlation-lengths. For an introduction, I'd recommend an overview paper by Roman Orus.

To use the function, first whatever tensor represents the bulk of the ipeps needs to be wrapped in a CTMRGRuntime structure which takes care of initializing the environment - either randomly or from the bulk tensor. Currently, there's only one CTMRGRuntime implemented which assumes a bulk-tensor which is invariant under any permutation of its virtual indices.

The runtime-object can then be provided to ctmrg together with a limit to the number of iterations maxit and a tolerance tol. The latter is used to decide convergence - if the sum of absolute differences in consecutive singular values of the corner is less than tol, the algorithm is converged.

A complete example to get the environment of the Ising model is

julia> a = model_tensor(Ising(),0.4);
julia> rt = SquareCTMRGRuntime(a, Val(:random), 10);
julia> rt = ctmrg(rt; tol=1e-6, maxit=100);
julia> corner, edge = rt.corner, rt.edge;

where Val(:random) is used to have the environment initialized with random values.

Variational Ipeps Optimisation

Variational Ipeps Optimisation works by combining ctmrg, automatic differentiation by Zygote and optimisation by Optim. The central function is rather simple and can be found in variationalipeps.jl.

We provide the function optimiseipeps with an IPEPS-object - a thin wrapper around a rank-5 tensor - and minimize the energy function with Optim using the gradient calculated by Zygote. Energy calculation is built on ctmrg so we need to supply its arguments: χ, tol and maxit but we might also modify the optimization algorithm using optimmethod or optimargs. optimargs can be used to e.g. print out the current energy at each step with optimargs = (show_trace = true,).

The convergence is judged by Optim and can be modified with optimargs. A complete example looks like:

julia> using Optim
julia> h = hamiltonian(TFIsing(1.0))
julia> ipeps = SquareIPEPS(randn(2,2,2,2,2))
julia> ipeps = TensorNetworkAD.indexperm_symmetrize(ipeps)
julia> res = optimiseipeps(ipeps, h; χ=5, tol=0, maxit=100,
        optimargs = (Optim.Options(f_tol=1e-6, show_trace=true),))
julia> e = minimum(res)

where we get the hamiltonian h of the transverse field ising model with magnetic field hx = 1, then we create a random initial ipeps with the necessary symmetry and then minimize its energy with with optimiseipeps where we consider it converged if the energy changes by less than 1e-6 between two iterations and we print the energy at each timestep. The ground-state energy is saved in e in the last line.