Conformal Field Theory Data

TNRKit provides extensive tools for calculating conformal field theory data. Details about the implementation can be found in the TNRKit paper (arxiv/2604.06922).

The core idea behind calculating the central charge, scaling dimensions, and conformal spins is to calculate spectra of transfer matrices constructed from fixed point tensors on a tube geometry. There are different ways to put fixed point tensors on a tube, and the geometry of this tube is characterised by 3 parameters: $[h, L, x]$ where $h$ is the height of the tube, $L$ is the circumference, and $x$ is the horizontal shift. The higher the ratio $\frac{L}{h}$, the higher the resolution but also the more expensive the calculation.

To calculate CFT data we provide the CFTData struct, which can be used in the following ways:

CFTData(scheme; shape=[h, L, x])
CFTData(T::TensorMap; kwargs...) # 1 fixed point tensor
CFTData(TA::TensorMap, TB::TensorMap; kwargs...) # 2x2 checkerboard unitcell

The shapes we provide are: $[1, 1, 0]$, $[\sqrt{2}, 2\sqrt{2}, 0]$, $[1, 4, 1]$, $[1, 8, 1]$, $[\frac{4}{\sqrt{10}}, 2 \sqrt{10}, \frac{2}{\sqrt{10}}]$

The last two shapes require intermediate truncation steps, whose parameters can be tuned by:

CFTData(scheme; shape=[1, 8, 1], trunc = trunc1, truncentanglement=trunc2)

CFTData Struct

The CFTData struct has three fields:

• central_charge: a complex number which is the central charge of the CFT.
• modular_parameter: a complex number which is the modular parameter of one tensor before building the tube transfer matrix.
• scaling_dimensions: a StructuredVector storing the scaling dimensions and conformal spins of the primary fields and their descendants. It can be indexed like an AbstractVector (i.e. with scalars, slices, ...), or with sectors (e.g. Z2Irrep(0)), which will provide the scaling dimensions associated with that sector/charge. To check which sectors you can index the scaling_dimensions with you can use keys(scaling_dimensions).

How Fermionic CFT Data Is Extracted

For a bosonic tensor network, CFTData builds a transfer matrix on the requested tube and diagonalizes it in each ordinary charge sector. For a fermionic tensor network, the same construction has one extra piece of global data: the spin structure around the tube. TNRKit extracts it by evaluating the same transfer matrix with two choices of boundary condition:

• :R: periodic fermions around the tube, obtained with pbc = true. The macro @tensor automatically inserts a fermionic twist to take the supertrace across the tube.
• :NS: antiperiodic fermions around the tube. Internally this is obtained by setting pbc = false, which explicitly inserts an additional fermionic twist to cancel the automatic supertrace twist, leaving the ordinary trace across the tube.

The result is a StructuredVector whose keys are tuples (spin_structure, charge). For fermionic systems without additional symmetries besides the fermion parity, there will be four sectors:

(:NS, FermionParity(0))
(:NS, FermionParity(1))
(:R,  FermionParity(0))
(:R,  FermionParity(1))

The largest eigenvalue (corresponding to the identity field) should be in the NS even sector.