| import Mathlib |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators |
| import OAI.MathematicalPhysics.PEPSSubvolume.SpectralPowers |
|
|
| namespace OAI |
|
|
| /-! Left and right unitary first variations. -/ |
|
|
| noncomputable section |
| open scoped BigOperators ComplexOrder |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator |
| open scoped BigOperators |
| open scoped Topology |
| open Filter |
| open PolynomialPEPS.PinnedEntropy |
|
|
| namespace PolynomialPEPS.Subvolume.UnitaryStationarity |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator |
| open PolynomialPEPS.Subvolume.SpectralCurve |
| open Filter |
| variable {ι E : Type*} [Fintype ι] [DecidableEq ι] |
| [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
|
|
| theorem real_inner_le_of_norm_le (v w : E) (h : ‖w‖ ≤ ‖v‖) : |
| (inner ℂ v w).re ≤ (inner ℂ v v).re := by |
| calc |
| (inner ℂ v w).re ≤ ‖inner ℂ v w‖ := Complex.re_le_norm _ |
| _ ≤ ‖v‖*‖w‖ := norm_inner_le_norm _ _ |
| _ ≤ ‖v‖*‖v‖ := mul_le_mul_of_nonneg_left h (norm_nonneg _) |
| _ = (inner ℂ v v).re := by |
| change ‖v‖*‖v‖ = RCLike.re (inner ℂ v v) |
| rw [← pow_two, norm_sq_eq_re_inner (𝕜 := ℂ)] |
|
|
| theorem left_moment_real (T : Matrix ι ι ℂ →ₗ[ℂ] E) (F : Matrix ι ι ℂ) |
| (hb : ∀ V : unitary (Matrix ι ι ℂ), ‖T ((V : Matrix ι ι ℂ)*F)‖ ≤ ‖T F‖) |
| (H : Matrix ι ι ℂ) (hH : H.IsHermitian) : |
| (inner ℂ (T F) (T (H*F))).im = 0 := by |
| let U := hH.eigenvectorUnitary |
| let r := hH.eigenvalues |
| let l : Matrix ι ι ℂ →L[ℂ] ℂ := (innerSL ℂ (T F)).comp T.toContinuousLinearMap |
| have hrep : spectralHom U (fun i => (r i : ℂ)) = H := hH.spectral_theorem.symm |
| have hD : HasDerivAt (fun z : ℂ => curve U r (z*Complex.I)*F) |
| (Complex.I • (H*F)) 0 := by |
| have hz : HasDerivAt (fun z : ℂ => z*Complex.I) Complex.I 0 := by |
| simpa using (hasDerivAt_id (0 : ℂ)).mul_const Complex.I |
| have hC := (hasDerivAt_curve_zero U r).scomp_of_eq 0 hz (by simp) |
| simpa [hrep,smul_mul_assoc] using hC.mul_const F |
| have hDl : HasDerivAt (fun z : ℂ => l (curve U r (z*Complex.I)*F)) |
| (Complex.I * l (H*F)) 0 := by |
| simpa +instances [Function.comp_def] using! l.hasFDerivAt.comp_hasDerivAt 0 hD |
| have hm : IsLocalMax (fun t : ℝ => (l (curve U r ((t : ℂ)*Complex.I)*F)).re) 0 := by |
| apply Filter.Eventually.of_forall |
| intro t |
| change (inner ℂ (T F) (T (curve U r ((t : ℂ)*Complex.I)*F))).re ≤ _ |
| have h0 : l (curve U r (((0 : ℝ) : ℂ)*Complex.I)*F) = inner ℂ (T F) (T F) := by |
| simp [l] |
| change _ ≤ (l _).re |
| rw [h0] |
| exact real_inner_le_of_norm_le _ _ (hb |
| ⟨curve U r ((t : ℂ)*Complex.I), curve_mem_unitary U r _ (by simp)⟩) |
| have hz := hm.hasDerivAt_eq_zero hDl.real_of_complex |
| change (Complex.I * inner ℂ (T F) (T (H*F))).re = 0 at hz |
| simpa using hz |
|
|
| theorem right_moment_real (T : Matrix ι ι ℂ →ₗ[ℂ] E) (F : Matrix ι ι ℂ) |
| (hb : ∀ V : unitary (Matrix ι ι ℂ), ‖T (F*(V : Matrix ι ι ℂ))‖ ≤ ‖T F‖) |
| (H : Matrix ι ι ℂ) (hH : H.IsHermitian) : |
| (inner ℂ (T F) (T (F*H))).im = 0 := by |
| let U := hH.eigenvectorUnitary |
| let r := hH.eigenvalues |
| let l : Matrix ι ι ℂ →L[ℂ] ℂ := (innerSL ℂ (T F)).comp T.toContinuousLinearMap |
| have hrep : spectralHom U (fun i => (r i : ℂ)) = H := hH.spectral_theorem.symm |
| have hD : HasDerivAt (fun z : ℂ => F*curve U r (z*Complex.I)) |
| (Complex.I • (F*H)) 0 := by |
| have hz : HasDerivAt (fun z : ℂ => z*Complex.I) Complex.I 0 := by |
| simpa using (hasDerivAt_id (0 : ℂ)).mul_const Complex.I |
| have hC := (hasDerivAt_curve_zero U r).scomp_of_eq 0 hz (by simp) |
| simpa [hrep,mul_smul_comm] using hC.const_mul F |
| have hDl : HasDerivAt (fun z : ℂ => l (F*curve U r (z*Complex.I))) |
| (Complex.I * l (F*H)) 0 := by |
| simpa +instances [Function.comp_def] using! l.hasFDerivAt.comp_hasDerivAt 0 hD |
| have hm : IsLocalMax (fun t : ℝ => (l (F*curve U r ((t : ℂ)*Complex.I))).re) 0 := by |
| apply Filter.Eventually.of_forall |
| intro t |
| change (inner ℂ (T F) (T (F*curve U r ((t : ℂ)*Complex.I)))).re ≤ _ |
| have h0 : l (F*curve U r (((0 : ℝ) : ℂ)*Complex.I)) = inner ℂ (T F) (T F) := by |
| simp [l] |
| change _ ≤ (l _).re |
| rw [h0] |
| exact real_inner_le_of_norm_le _ _ (hb |
| ⟨curve U r ((t : ℂ)*Complex.I), curve_mem_unitary U r _ (by simp)⟩) |
| have hz := hm.hasDerivAt_eq_zero hDl.real_of_complex |
| change (Complex.I * inner ℂ (T F) (T (F*H))).re = 0 at hz |
| simpa using hz |
|
|
| end PolynomialPEPS.Subvolume.UnitaryStationarity |
|
|
| end |
|
|
| end OAI |
|
|
