| import OAI.Probability.ClassicalON.BondLaw |
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| namespace OAI |
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| universe uE uV |
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| noncomputable section |
| open scoped BigOperators Classical |
| namespace ClassicalON |
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| def signFieldEquiv : Bool ≃ SignField where |
| toFun b := if b then 0 else 1 |
| invFun a := a=0 |
| left_inv b := by cases b <;> decide |
| right_inv a := by |
| have ha : a=0 ∨ a=1 := by fin_cases a <;> [exact Or.inl rfl; exact Or.inr rfl] |
| rcases ha with rfl | rfl <;> simp |
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| @[simp] theorem signFieldEquiv_true : signFieldEquiv true=0 := rfl |
| @[simp] theorem signFieldEquiv_false : signFieldEquiv false=1 := rfl |
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| def fieldSign (a : SignField) : ℝ := if a=0 then 1 else -1 |
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| theorem signField_cases (a : SignField) : a=0 ∨ a=1 := by |
| fin_cases a <;> [exact Or.inl rfl; exact Or.inr rfl] |
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| @[simp] theorem fieldSign_zero : fieldSign 0=1 := by simp [fieldSign] |
| @[simp] theorem fieldSign_one : fieldSign 1= -1 := by simp [fieldSign] |
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| theorem fieldSign_add (a b : SignField) : fieldSign (a+b)=fieldSign a*fieldSign b := by |
| have htwo : (1 : SignField)+1=0 := rfl |
| rcases signField_cases a with rfl | rfl <;> rcases signField_cases b with rfl | rfl <;> simp [htwo] |
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| theorem fieldSign_sq (a : SignField) : fieldSign a*fieldSign a=1 := by |
| rcases signField_cases a with rfl | rfl <;> simp |
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| variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] |
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| def compatibleSigns (left right : E → V) (η : E → Bool) : Set (V → Bool) := |
| {s | ∀ e,η e=true → s (left e)=s (right e)} |
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| def compatibleSignsEquiv (left right : E → V) (η : E → Bool) : |
| compatibleSigns left right η ≃ bondSubspace left right η := |
| (Equiv.piCongrRight (fun _ : V => signFieldEquiv)).subtypeEquiv (by |
| intro s |
| change (∀ e,η e=true → s (left e)=s (right e)) ↔ |
| ∀ e,η e=true → signFieldEquiv (s (left e))=signFieldEquiv (s (right e)) |
| simp only [Equiv.apply_eq_iff_eq]) |
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| omit [Fintype E] in |
| theorem compatibleSigns_card (left right : E → V) (η : E → Bool) : |
| Fintype.card (compatibleSigns left right η)=2^bondRank left right η := by |
| rw [Fintype.card_congr (compatibleSignsEquiv left right η),Module.card_eq_pow_finrank (K := SignField)] |
| simp [bondRank,SignField] |
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| omit [Fintype E] in |
| theorem compatibleSigns_sum (left right : E → V) (η : E → Bool) : |
| (∑ s : V → Bool,if s∈compatibleSigns left right η then (1:ℝ) else 0)=clusterWeight left right η := by |
| rw [Finset.sum_boole,← Fintype.card_subtype] |
| unfold clusterWeight |
| exact_mod_cast compatibleSigns_card left right η |
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| omit [Fintype V] [Fintype E] in |
| def bondConnected (left right : E → V) (η : E → Bool) (x y : V) : Prop := |
| ∀ s∈bondSubspace left right η,s x=s y |
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| omit [Fintype V] [Fintype E] in |
| theorem bondConnected_mono (left right : E → V) (x y : V) : |
| Monotone (fun η => bondConnected left right η x y) := by |
| intro η ξ h hη s hs |
| exact hη s (bondSubspace_antitone left right h hs) |
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| omit [Fintype E] in |
| theorem bond_sign_correlation (left right : E → V) (η : E → Bool) (x y : V) : |
| (∑ s : bondSubspace left right η,fieldSign (s.val x)*fieldSign (s.val y))= |
| if bondConnected left right η x y then clusterWeight left right η else 0 := by |
| split_ifs with h |
| · have he (s : bondSubspace left right η) : fieldSign (s.val x)*fieldSign (s.val y)=1 := by |
| rw [h s.val s.property]; exact fieldSign_sq _ |
| simp_rw [he] |
| simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] |
| unfold clusterWeight bondRank |
| norm_cast |
| rw [Module.card_eq_pow_finrank (K := SignField)] |
| simp [SignField] |
| · obtain ⟨s,hs,hxy⟩ : ∃ s∈bondSubspace left right η,s x≠s y := by |
| unfold bondConnected at h |
| push Not at h |
| exact h |
| let u : bondSubspace left right η := ⟨s,hs⟩ |
| have hc : fieldSign (s x)*fieldSign (s y)= -1 := by |
| have hlocal (a b : SignField) (hab : a≠b) : fieldSign a*fieldSign b= -1 := by |
| rcases signField_cases a with rfl | rfl <;> rcases signField_cases b with rfl | rfl <;> simp_all |
| exact hlocal _ _ hxy |
| have hh := Equiv.sum_comp (Equiv.addRight u) (fun t : bondSubspace left right η => fieldSign (t.val x)*fieldSign (t.val y)) |
| have he (t : bondSubspace left right η) : |
| fieldSign ((t+u).val x)*fieldSign ((t+u).val y)= -(fieldSign (t.val x)*fieldSign (t.val y)) := by |
| change fieldSign (t.val x+s x)*fieldSign (t.val y+s y)= _ |
| rw [fieldSign_add,fieldSign_add,mul_mul_mul_comm,hc,mul_neg_one] |
| change (∑ t : bondSubspace left right η,fieldSign ((t+u).val x)*fieldSign ((t+u).val y))=_ at hh |
| simp_rw [he] at hh |
| rw [Finset.sum_neg_distrib] at hh |
| linarith |
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| end ClassicalON |
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| end |
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| end OAI |
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