Maps and Symmetrizers

Not every workflow stays inside one basis. Sometimes you want to move amplitudes between a full tensor-product basis and a symmetry-reduced basis, or between two different symmetry sectors that share the same local Hilbert space.

That is what DoubleBasis is for.

DoubleBasis

DoubleBasis(B1, B2) creates a basis-like object representing maps from coordinates in B2 to coordinates in B1.

Typical uses:

• project a full-space vector into a symmetry sector,
• lift sector amplitudes back into a less reduced basis,
• compare two sector descriptions through their common embedding.

The two bases must have the same chain length and local on-site dimension.

Symmetrizers

The most direct basis-only map is symmetrizer(B), where B is a DoubleBasis.

It returns the overlap map between the embeddings of the two bases into the full tensor-product basis.

In the common case where B2 is less symmetric than B1, it behaves like a projection or symmetrization matrix into the target sector.

Example: Full Space To A Symmetry Sector

using EDKit

L = 8
Bfull = TensorBasis(L = L, base = 2)
Bsector = basis(L = L, N = 4, p = 1)
T = DoubleBasis(Bsector, Bfull)

P = symmetrizer(T)

P now maps coordinates from the full basis into the chosen symmetry sector.

Applying A DoubleBasis

DoubleBasis itself can also act on a vector:

v_sector = T(v_full)

This is convenient when you want the basis map as an operation, not just as an explicit matrix.

When To Reach For This Layer

Use DoubleBasis and symmetrizer when:

• you want to compare states across basis choices,
• you need an explicit projection map,
• you want to verify that reduced-basis calculations agree with full-space calculations,
• or you want to build inter-basis operators.

If you only need a Hamiltonian inside one symmetry sector, you usually do not need this layer at all.