[ITensors] [BUG] Autofermion system incompatible with simple (no QN) Indices (when factorizing)

[b-kloss]

Description of bug

Enabling the Autofermion system (ITensors.enable_auto_fermion) leads to errors when performing DMRG on a system without quantum number conservation enforced.

Minimal code demonstrating the bug or unexpected behavior

This is a copy of examples/dmrg/1d_heisenberg.jl up to ITensors.enable_auto_fermion() at the top of the code block.

Minimal runnable code

```julia using ITensors using Printf using Random Random.seed!(1234) let N = 100 ITensors.enable_auto_fermion() # Create N spin-one degrees of freedom sites = siteinds("S=1", N) # Alternatively can make spin-half sites instead #sites = siteinds("S=1/2", N) # Input operator terms which define a Hamiltonian os = OpSum() for j in 1:(N - 1) os += "Sz", j, "Sz", j + 1 os += 0.5, "S+", j, "S-", j + 1 os += 0.5, "S-", j, "S+", j + 1 end # Convert these terms to an MPO tensor network H = MPO(os, sites) # Create an initial random matrix product state psi0 = randomMPS(sites; linkdims=10) # Plan to do 5 DMRG sweeps: nsweeps = 5 # Set maximum MPS bond dimensions for each sweep maxdim = [10, 20, 100, 100, 200] # Set maximum truncation error allowed when adapting bond dimensions cutoff = [1E-11] # Run the DMRG algorithm, returning energy and optimized MPS energy, psi = dmrg(H, psi0; nsweeps, maxdim, cutoff) @printf("Final energy = %.12f\n", energy) end ```

Expected output or behavior

DMRG should run successfully.

Actual output or behavior An error is raised in the factorization, regarding the expected arrow-directions when the autofermion system is enabled. I expect the solution to be change the default arrow-direction when either interpreting a non-QN Index as a QNIndex with trivial QN or converting one to the other.

Output of minimal runnable code

```julia ERROR: LoadError: With auto_fermion enabled, left inds in eigen must have Out arrows Stacktrace: [1] error(s::String) @ Base ./error.jl:35 [2] eigen(A::ITensor, Linds::Vector{Index{Int64}}, Rinds::Vector{Index{Int64}}; mindim::Int64, maxdim::Int64, cutoff::Float64, use_absolute_cutoff::Nothing, use_relative_cutoff::Nothing, ishermitian::Bool, tags::TagSet, lefttags::Nothing, righttags::Nothing, plev::Nothing, leftplev::Nothing, rightplev::Nothing) @ ITensors ~/.julia/v1.9/packages/ITensors/Gf9aD/src/tensor_operations/matrix_decomposition.jl:358 [3] factorize_eigen(A::ITensor, Linds::Tuple{Index{Int64}, Index{Int64}}; ortho::String, eigen_perturbation::Nothing, mindim::Int64, maxdim::Int64, cutoff::Float64, tags::TagSet, use_absolute_cutoff::Nothing, use_relative_cutoff::Nothing) @ ITensors ~/.julia/v1.9/packages/ITensors/Gf9aD/src/tensor_operations/matrix_decomposition.jl:680 [4] factorize(A::ITensor, Linds::Tuple{Index{Int64}, Index{Int64}}; mindim::Int64, maxdim::Int64, cutoff::Float64, ortho::String, tags::TagSet, plev::Nothing, which_decomp::Nothing, eigen_perturbation::Nothing, svd_alg::Nothing, use_absolute_cutoff::Nothing, use_relative_cutoff::Nothing, min_blockdim::Nothing, singular_values!::Nothing, dir::Nothing) @ ITensors ~/.julia/v1.9/packages/ITensors/Gf9aD/src/tensor_operations/matrix_decomposition.jl:827 [5] replacebond!(M::MPS, b::Int64, phi::ITensor; normalize::Bool, swapsites::Nothing, ortho::String, which_decomp::Nothing, mindim::Int64, maxdim::Int64, cutoff::Float64, eigen_perturbation::Nothing, svd_alg::Nothing, use_absolute_cutoff::Nothing, use_relative_cutoff::Nothing, min_blockdim::Nothing) @ ITensors ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/mps.jl:559 [6] replacebond! @ ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/mps.jl:530 [inlined] [7] #replacebond!#956 @ ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/mps.jl:606 [inlined] [8] macro expansion @ ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/dmrg.jl:277 [inlined] [9] macro expansion @ ~/.julia/v1.9/packages/TimerOutputs/RsWnF/src/TimerOutput.jl:253 [inlined] [10] macro expansion @ ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/dmrg.jl:276 [inlined] [11] macro expansion @ ./timing.jl:393 [inlined] [12] dmrg(PH::ProjMPO, psi0::MPS, sweeps::Sweeps; which_decomp::Nothing, svd_alg::Nothing, observer::NoObserver, outputlevel::Int64, write_when_maxdim_exceeds::Nothing, write_path::String, eigsolve_tol::Float64, eigsolve_krylovdim::Int64, eigsolve_maxiter::Int64, eigsolve_verbosity::Int64, eigsolve_which_eigenvalue::Symbol, ishermitian::Bool) @ ITensors ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/dmrg.jl:204 [13] dmrg @ ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/dmrg.jl:156 [inlined] [14] #dmrg#1053 @ ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/dmrg.jl:27 [inlined] [15] dmrg @ ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/dmrg.jl:20 [inlined] [16] #dmrg#1059 @ ~/.julia/v1.9/packages/ITensors/Gf9aD/src/mps/dmrg.jl:391 [inlined] [17] top-level scope @ ~/.julia/v1.9/dev/ITensors/examples/dmrg/1d_heisenberg_af.jl:36 [18] include(fname::String) @ Base.MainInclude ./client.jl:478 [19] top-level scope @ REPL[3]:1 in expression starting at /mnt/home/bkloss/.julia/v1.9/dev/ITensors/examples/dmrg/1d_heisenberg_af.jl:7 ```

Version information

• Output from versioninfo():

  julia> versioninfo()
  Commit 8e63055292* (2023-05-07 11:25 UTC)
  Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: 32 × Intel(R) Xeon(R) Gold 6234 CPU @ 3.30GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-14.0.6 (ORCJIT, cascadelake)
  Threads: 1 on 32 virtual cores

• Output from using Pkg; Pkg.status("ITensors"):

  julia> using Pkg; Pkg.status("ITensors")
  Status `~/projects/ITensors_issues/af_vs_noqn/Project.toml`
  [9136182c] ITensors v0.3.54

[mtfishman]

@emstoudenmire could you take a look?

[mtfishman]

Closed by #1380.