neuralqx.operators.computational.Euclidean3d.jax.volume module

class VolumeOperatorJax(H, vertex)

Bases: ComputationalJaxOperator

JAX-compatible (diagonal) 2+1 “volume” (= area) operator at a single vertex v for U(1)^3.

property max_conn_size: int

The maximum number of non zero ⟨x|O|x’⟩ for every x.

property is_hermitian: bool

Returns true if this operator is hermitian.

property dtype

The dtype of the operator’s matrix elements ⟨σ|Ô|σ’⟩.

tree_flatten()
classmethod tree_unflatten(struct, leaves)
class VolumeOperatorJaxSquared(H, vertex)

Bases: VolumeOperatorJax

JAX-compatible (diagonal) squared 2+1 “volume” operator at a single vertex v for U(1)^3.

This operator represents the functional calculus \(\hat{V}_v^2 = (\hat{V}_v)^2\) applied to the diagonal 2+1D “volume” operator \(\hat{V}_v\) implemented by VolumeOperatorJax.

On a charge basis state \(|\vec m\rangle\), where the underlying operator has the eigenvalue

\[V_v(\vec m) \;=\; \left\| \sum_{(e_1,e_2)\ni v} \epsilon_v(e_1,e_2)\; \vec m_{e_1}\times \vec m_{e_2} \right\| \;\ge 0,\]

this class acts diagonally as

\[\hat{V}_v^2 \,|\vec m\rangle \;=\; \big(V_v(\vec m)\big)^2 \,|\vec m\rangle.\]

Notes

  • The operator is diagonal in the charge basis, hence Hermitian.

  • This is implemented by squaring the diagonal matrix elements produced by the parent operator.

  • In contrast to computing \(V_v(\vec m)\) and then squaring it, one may also implement \(\hat{V}_v^2\) directly as \(\|\vec v\|^2\) to avoid an intermediate square root. This class keeps the implementation minimal and reuses the parent kernel.

class VolumeOperatorJaxSqrt(H, vertex)

Bases: VolumeOperatorJax

JAX-compatible (diagonal) square-root 2+1 “volume” operator at a single vertex v for U(1)^3.

This operator represents the functional calculus \(\sqrt{\hat{V}_v}\) applied to the diagonal 2+1D “volume” operator \(\hat{V}_v\) implemented by VolumeOperatorJax.

On a charge basis state \(|\vec m\rangle\), where the underlying operator has the eigenvalue

\[V_v(\vec m) \;=\; \left\| \sum_{(e_1,e_2)\ni v} \epsilon_v(e_1,e_2)\; \vec m_{e_1}\times \vec m_{e_2} \right\| \;\ge 0,\]

this class acts diagonally as

\[\sqrt{\hat{V}_v}\,|\vec m\rangle \;=\; \sqrt{V_v(\vec m)}\,|\vec m\rangle.\]

Notes

  • In the present 2+1D U(1)^3 setting, \(V_v(\vec m)\) is a Euclidean norm and is therefore non-negative, so the square root is unambiguous.

  • The operator is diagonal in the charge basis, hence Hermitian.

  • This is implemented by applying jnp.sqrt to the diagonal matrix elements produced by the parent operator.