Chain

A high-performance DMRG (density matrix renormalization group) solver for 1D anyon chains — quantum many-body systems whose Hilbert spaces are not tensor-factorized over sites, and therefore cannot be handled by standard spin-chain codes out of the box.

Written in C++ on top of ITensor, with a later Julia port for prototyping new fusion categories.

Research output: Numerical Evidence for a Haagerup Conformal Field Theory, Phys. Rev. Lett. 128, 231603 (2022).

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What it does

Given a fusion category (the algebraic data defining the anyon model) and a chain geometry, the solver:

• Constructs the Hamiltonian as a matrix product operator (MPO) directly from the category's F-symbols — no hard-coded constants.
• Runs DMRG to find the ground state and low-lying excited states, targeting excited states via orthogonality penalties.
• Measures translation and topological-symmetry eigenvalues alongside the energy spectrum, so that states can be organized into momentum and symmetry sectors.
• Supports multiple boundary conditions: periodic, open, Dirichlet, and sine-squared deformed (SSD), including the SSD-to-periodic "hot start" protocol.
• Handles quantum-number (QN) conservation where the category admits a grading, which dramatically reduces bond dimension at fixed accuracy.

Built-in models: golden (Fibonacci), Haagerup, Haagerup with QN conservation. The Julia side additionally supports user-defined fusion categories via a generic AnyonModel interface.

Why it's non-trivial

Anyonic Hilbert spaces are labeled by fusion trees rather than local spin configurations. This means:

• Local operators don't decompose as tensor products; every term in the Hamiltonian involves an F-move that depends on the global fusion-tree state.
• The MPO construction has to be generated from categorical data at runtime.
• Symmetry sectors are labeled by non-invertible operators (e.g., the $\rho$ line of the Haagerup category), which the code diagonalizes jointly with energy and translation.

The solver reaches bond dimensions of $\sim 1600$ on chains up to $L \sim 16$ with SVD cutoffs of $10^{-8}$, which was sufficient to extract the central charge of the emergent CFT in the published work.

Tech stack

Layer	Tool
Core solver	C++17, ITensor v3
Linear algebra backend	OpenBLAS / MKL (configurable at build time)
Build system	CMake ≥ 2.8
Data format	ITensor binary (C++), JLD2 (Julia)
Prototyping / generic categories	Julia 1.x
Output analysis	Mathematica-compatible matrix dumps (.m)

Repository layout

src/          C++ DMRG driver, MPO construction, measurement modes
julia/        Julia port with generic AnyonModel interface
extern/       Vendored ITensor headers and libraries (see INSTALL.md)
CMakeLists*   Build configuration (template + Mac variant)

Install

See INSTALL.md for the full build procedure. In short: vendor ITensor under extern/itensor/, point CMakeLists.txt at your BLAS installation, then:

mkdir build && cd build
cmake ..
make

The Julia version has no build step:

julia --project=julia -e 'using Pkg; Pkg.instantiate()'

Quick start (C++)

Simulate a length-6 periodic Haagerup chain with QN conservation under the pure-$\rho$ projector, in the neutral sector, and measure the energy / translation / $\rho$ spectrum after each state:

./Chain -s haagerupq -b p -l 6 -d 1600 -c 1e-8 -p 1e-2 -j 0,-1,0 -u 2 -q 0 -n 1 -m 5

Output files:

Extension	Contents
.an	Diagonalized spectrum: {energy, momentum, ρ}
.ee	Entanglement entropy curve
.en	Raw (undiagonalized) energies
.m	Energy / translation / ρ matrices in Mathematica format

Full CLI reference is in INSTALL.md.

Quick start (Julia)

julia --project=julia julia/run_chain.jl -l 6 -d 100 -s 5 -b p -j 1.0 -u 2.0 -m haagerup -o results.jld2

Or, using the generic interface to run a Fibonacci chain:

using Chain
energy = run_dmrg(L=6, model=Model.fibonacci_model(), U=2.0)

Define your own fusion category by supplying F-symbols and quantum dimensions:

using Chain.Model
m = AnyonModel(2; fsymbols=myF, qdims=[1.0, phi])

Citing

If this code is useful for your work, please cite:

@article{HuangLinOhmoriTachikawaTezuka2022,
  title   = {Numerical Evidence for a Haagerup Conformal Field Theory},
  author  = {Huang, Tzu-Chen and Lin, Ying-Hsuan and Ohmori, Kantaro
             and Tachikawa, Yuji and Tezuka, Masaki},
  journal = {Phys. Rev. Lett.},
  volume  = {128},
  number  = {23},
  pages   = {231603},
  year    = {2022},
  doi     = {10.1103/PhysRevLett.128.231603},
}