| import OAI.Probability.InvariantIsing.Cavity.CavityBaseReplacement |
| import OAI.Probability.InvariantIsing.Cavity.CavityRetainedStack |
|
|
| namespace OAI |
|
|
| /-! The replaced interaction preserves retained eigenvectors, gives the |
| special axes their assigned eigenvalues, and annihilates the cavity. -/ |
|
|
| noncomputable section |
| open scoped Matrix |
|
|
| namespace InvariantIsing |
|
|
| lemma cavityBaseReplacement_retained {r s d : ℕ} {ι : Type*} [Fintype ι] |
| (J : Matrix (Fin r) (Fin r) ℝ) (W : Matrix (Fin r) (Fin s) ℝ) |
| (D : Matrix (Fin s) (Fin s) ℝ) (B : Matrix (Fin s) (Fin d) ℝ) |
| (A₀ : Matrix (Fin d) (Fin d) ℝ) (R : Matrix (Fin r) ι ℝ) (L : Matrix ι ι ℝ) |
| (hJR : J * R = R * L) (hWR : W.transpose * R = 0) : |
| cavityBaseReplacement J W D B A₀ * R = R * L := by |
| simp only [cavityBaseReplacement, Matrix.add_mul, Matrix.sub_mul, Matrix.transpose_mul, |
| Matrix.mul_assoc, hWR, Matrix.mul_zero, sub_zero, add_zero, hJR] |
|
|
| lemma cavityBaseReplacement_special {r s d : ℕ} |
| (J : Matrix (Fin r) (Fin r) ℝ) (W : Matrix (Fin r) (Fin s) ℝ) |
| (D : Matrix (Fin s) (Fin s) ℝ) (B : Matrix (Fin s) (Fin d) ℝ) |
| (A₀ : Matrix (Fin d) (Fin d) ℝ) |
| (hJW : J * W = W * D) (hW : W.transpose * W = 1) (hB : B.transpose * B = 1) : |
| cavityBaseReplacement J W D B A₀ * (W * B) = (W * B) * A₀ := by |
| have hJB : J * (W * B) = W * D * B := by rw [← Matrix.mul_assoc, hJW] |
| have hWB : W.transpose * (W * B) = B := by rw [← Matrix.mul_assoc, hW, Matrix.one_mul] |
| have hBB : (W * B).transpose * (W * B) = 1 := by |
| rw [Matrix.transpose_mul, Matrix.mul_assoc, hWB, hB] |
| rw [cavityBaseReplacement, Matrix.add_mul, Matrix.sub_mul, hJB, |
| Matrix.mul_assoc (W * D) W.transpose (W * B), hWB, sub_self, zero_add, |
| Matrix.mul_assoc (W * B * A₀) (W * B).transpose (W * B), hBB, Matrix.mul_one] |
|
|
| lemma cavityBaseReplacement_cavity {r s d n : ℕ} |
| (J : Matrix (Fin r) (Fin r) ℝ) (W : Matrix (Fin r) (Fin s) ℝ) |
| (D : Matrix (Fin s) (Fin s) ℝ) (B : Matrix (Fin s) (Fin d) ℝ) |
| (E : Matrix (Fin s) (Fin n) ℝ) (A₀ : Matrix (Fin d) (Fin d) ℝ) |
| (hJW : J * W = W * D) (hW : W.transpose * W = 1) (hBE : B.transpose * E = 0) : |
| cavityBaseReplacement J W D B A₀ * (W * E) = 0 := by |
| have hJE : J * (W * E) = W * D * E := by rw [← Matrix.mul_assoc, hJW] |
| have hWE : W.transpose * (W * E) = E := by rw [← Matrix.mul_assoc, hW, Matrix.one_mul] |
| have hBB : (W * B).transpose * (W * E) = 0 := by |
| rw [Matrix.transpose_mul, Matrix.mul_assoc, hWE, hBE] |
| rw [cavityBaseReplacement, Matrix.add_mul, Matrix.sub_mul, hJE, |
| Matrix.mul_assoc (W * D) W.transpose (W * E), hWE, sub_self, zero_add, |
| Matrix.mul_assoc (W * B * A₀) (W * B).transpose (W * E), hBB, Matrix.mul_zero] |
|
|
| lemma cavityBaseReplacement_conjugate {r s d : ℕ} |
| (J U : Matrix (Fin r) (Fin r) ℝ) (W : Matrix (Fin r) (Fin s) ℝ) |
| (D : Matrix (Fin s) (Fin s) ℝ) (B : Matrix (Fin s) (Fin d) ℝ) |
| (A₀ : Matrix (Fin d) (Fin d) ℝ) : |
| cavityBaseReplacement (U * J * U.transpose) (U * W) D B A₀ = |
| U * cavityBaseReplacement J W D B A₀ * U.transpose := by |
| simp only [cavityBaseReplacement, Matrix.transpose_mul, Matrix.mul_add, |
| Matrix.add_mul, Matrix.mul_sub, Matrix.sub_mul, Matrix.mul_assoc] |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
