Semiconductor Z2 Invariant Calculator

Compute explicit inversion-symmetric 2D Fu–Kane parity formula over caller occupied Kramers pairs at four TRIM; not generic Wilson loop, 3D indices or discovery of band topology.

Description

The two-dimensional Fu–Kane parity construction uses occupied Kramers-pair parities at four time-reversal-invariant momenta.

The two-dimensional Fu–Kane parity construction uses occupied Kramers-pair parities at four time-reversal-invariant momenta.^{[1, 2]}

It applies only after inversion and time-reversal symmetry and the occupied pairs have been established.

When to use Semiconductor Z2 Invariant Calculator

Use this tool when you need to reproduce the stated calculation from measured records or independently selected model inputs. Prepare measured device or specimen records under independently matched branch, temperature, frame, and unit conventions.

How to use Semiconductor Z2 Invariant Calculator and understand its results

Prepare measured device or specimen records under independently matched branch, temperature, frame, and unit conventions.
Enter callerTrimOccupiedKramersPairParities, callerInversionAndTimeReversalSymmetryVerified, following the field table and rules below, then run the tool.
Read trimParityProducts, fuKaneTwoDimensionalZ2. Use the output meanings below to interpret units, signs, flags, and any nested result fields.
Correct reported input issues and run again. For a successful result, check the stated evidence coverage and assumptions before using it in your review.

Input field	Meaning	Value type	Requirement
callerTrimOccupiedKramersPairParities	Caller trim occupied kramers pair parities	numeric matrix	Required
callerInversionAndTimeReversalSymmetryVerified	Caller inversion and time reversal symmetry verified	boolean	Required

Enter a rectangular matrix with exactly four nonempty rows of ±1 parities, one column per occupied Kramers pair, and true for independently verified symmetries. Multiply each row and then all four products; a negative overall product gives Z₂=1, otherwise zero. It does not compute three-dimensional indices or discover symmetry.

δ_i=ΠoccupiedPairs parity_i,pair; (−1)^ν=ΠfourTRIM δ_i

Symbols follow the input and output meanings and the calculation rules above.

Output field	Meaning	Value type
trimParityProducts	Trim parity products	list of number
fuKaneTwoDimensionalZ2	Fu kane two dimensional z2	number

json

{
  "input": {
    "callerTrimOccupiedKramersPairParities": [
      [
        1,
        -1
      ],
      [
        1,
        1
      ],
      [
        1,
        1
      ],
      [
        1,
        1
      ]
    ],
    "callerInversionAndTimeReversalSymmetryVerified": true
  },
  "output": {
    "trimParityProducts": [
      -1,
      1,
      1,
      1
    ],
    "fuKaneTwoDimensionalZ2": 1
  }
}

Limitations and assumptions

Matched measurement branch, material state, calibration, and conventions must be independently established. Results do not certify device performance or operating conditions.

Missing required fields, unknown fields, wrong types, and violated calculation rules are input issues. Numeric values must be finite; nonfinite results are rejected. Evidence arrays must be dense, with no missing cells. Numeric matrices are limited to 128 rows and 128 entries per row, with rectangularity required where stated.

Alternative or Complementary methods

For a related review, use Semiconductor Weyl Node Separation Calculator: Weyl-node separation is a displacement between independently identified nodes in one reciprocal-space frame. A directed vector preserves orientation alongside its Euclidean magnitude. For a complementary step, use Semiconductor Valley Hall Conductivity Calculator: A valley-restricted conductivity estimate integrates sampled Berry curvature with k-space cell area and occupation. Its sign and scale depend on the independently declared valley and conductivity convention.

References

Semiconductor Z2 Invariant Calculator — tool.io
Topological Insulators with Inversion Symmetry — Fu and Kane

Similar or alternative tools

Semiconductor Weyl Node Separation Calculator
Weyl-node separation is a displacement between independently identified nodes in one reciprocal-space frame. A directed vector preserves orientation alongside its Euclidean magnitude.
Semiconductor Valley Hall Conductivity Calculator
A valley-restricted conductivity estimate integrates sampled Berry curvature with k-space cell area and occupation. Its sign and scale depend on the independently declared valley and conductivity convention.