TFPT scientific source excerpt
Original repository path: verification/v1027_signed_det_car_wall.py
Original file SHA-256: 6679297c3c53a86f4ef7d40c169caab64708885e891e68db365acfadfe9f0f4d
Inclusive original lines: 82-153
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000082 | def determinant_canonicalization() -> dict[str, float]:
000083 |     # The matrix regression uses n=4.  The proof is conjugation by the same
000084 |     # explicit even unitary for every even n, including inherited DET32.
000085 |     n = 4
000086 |     canonical = annihilators(n)
000087 |     dimension = 2**n
000088 |     identity = np.eye(dimension)
000089 |     vacuum = identity[:, 0]
000090 |     full = identity[:, -1]
000091 |     p0 = np.outer(vacuum, vacuum)
000092 |     pf = np.outer(full, full)
000093 |     f = np.outer(full, vacuum)
000094 |     omega = (vacuum + full) / math.sqrt(2)
000095 |     p = np.outer(omega, omega)
000096 |     q = identity - p
000097 |     unitary = identity + (1/math.sqrt(2)-1)*(p0+pf) + (f-dagger(f))/math.sqrt(2)
000098 |     check("onsite determinant rotation is unitary",
000099 |           np.linalg.norm(dagger(unitary) @ unitary - identity) < TOL)
000100 |     check("onsite rotation maps empty vacuum to DET ground state",
000101 |           np.linalg.norm(unitary @ vacuum - omega) < TOL)
000102 |     parity = np.diag([(-1)**state.bit_count() for state in range(dimension)])
000103 |     check("even-n rotation preserves fermion parity",
000104 |           np.linalg.norm(unitary @ parity - parity @ unitary) < TOL)
000105 |     dressed = [unitary @ operator @ dagger(unitary) for operator in canonical]
000106 |     car_defect = max(
000107 |         np.linalg.norm(a @ dagger(b) + dagger(b) @ a - (identity if i == j else 0))
000108 |         for i, a in enumerate(dressed) for j, b in enumerate(dressed)
000109 |     )
000110 |     nil_defect = max(np.linalg.norm(a @ b + b @ a) for a in dressed for b in dressed)
000111 |     check("dressed operators satisfy complete CAR",
000112 |           max(car_defect, nil_defect) < TOL, car_defect)
000113 |     check("dressed annihilators annihilate DET vacuum",
000114 |           max(np.linalg.norm(a @ omega) for a in dressed) < TOL)
000115 |     old_excitations = [math.sqrt(2)*q @ dagger(c) @ omega for c in canonical]
000116 |     check("dressed creators reproduce old one-excitation states",
000117 |           max(np.linalg.norm(dagger(a) @ omega - e)
000118 |               for a, e in zip(dressed, old_excitations, strict=True)) < TOL)
000119 |     hubbard = [np.outer(omega, np.conj(e)) for e in old_excitations]
000120 |     hubbard_defect = np.linalg.norm(
000121 |         hubbard[0] @ dagger(hubbard[0]) + dagger(hubbard[0]) @ hubbard[0] - identity, 2
000122 |     )
000123 |     check("old Hubbard-operator CAR mutant rejected",
000124 |           abs(hubbard_defect - 1) < TOL, hubbard_defect)
000125 |     check("Hubbard creators cannot build a two-particle state",
000126 |           np.linalg.norm(dagger(hubbard[0]) @ dagger(hubbard[1]) @ omega) < TOL)
000127 |     check("canonical creators build a normalized two-particle state",
000128 |           abs(np.linalg.norm(dagger(dressed[0]) @ dagger(dressed[1]) @ omega)-1) < TOL)
000129 |     number = sum(dagger(a) @ a for a in canonical)
000130 |     number_a = sum(dagger(a) @ a for a in dressed)
000131 |     formula = number + n/2*(p0-pf-f-dagger(f))
000132 |     check("explicit additive DET number-parent identity",
000133 |           np.linalg.norm(number_a-formula) < TOL)
000134 |     check("Q <= N_a <= nQ",
000135 |           min(np.linalg.eigvalsh(number_a-q).min(),
000136 |               np.linalg.eigvalsh(n*q-number_a).min()) > -TOL)
000137 |     spectrum = np.linalg.eigvalsh(number_a)
000138 |     check("additive parent has the same unique vacuum and gap one",
000139 |           np.sum(abs(spectrum) < TOL) == 1 and abs(spectrum[1]-1) < TOL)
000140 |     check("flat Delta*Q is not silently identified with additive Delta*N_a",
000141 |           np.linalg.norm(number_a-q, 2) > 2.9)
000142 |     charges = [1, -1, 2, -2]
000143 |     charge = sum(value*dagger(a) @ a for value, a in zip(charges, canonical, strict=True))
000144 |     check("neutral-determinant rotation commutes with a traceless gauge generator",
000145 |           np.linalg.norm(unitary @ charge - charge @ unitary) < TOL)
000146 |     check("dressed operators retain their original charges",
000147 |           max(np.linalg.norm(charge @ a - a @ charge + value*a)
000148 |               for a, value in zip(dressed, charges, strict=True)) < TOL)
000149 |     charged_mutant = charge + dagger(canonical[0]) @ canonical[0]
000150 |     check("charged-determinant mutant rejected",
000151 |           np.linalg.norm(unitary @ charged_mutant - charged_mutant @ unitary) > 0.9)
000152 |     return {"canonical_CAR_defect": float(car_defect),
000153 |             "Hubbard_CAR_operator_norm_defect": float(hubbard_defect)}