| import OAI.Probability.ClassicalON.SpinBonds |
|
|
| namespace OAI |
|
|
| universe uE uV |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
| namespace ClassicalON |
|
|
| variable {V : Type uV} {E : Type uE} [Fintype V] |
|
|
| def pinRestriction (B : Set V) : (V → SignField) →ₗ[SignField] (B → SignField) where |
| toFun s := fun v => s v |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| def bondPinRank (left right : E → V) (B : Set V) (η : E → Bool) : ℕ := |
| Module.finrank SignField ((bondSubspace left right η).map (pinRestriction B)) |
| |
| def bondPinFactor (left right : E → V) (B : Set V) (η : E → Bool) : ℝ := |
| (2^bondPinRank left right B η)⁻¹ |
|
|
| omit [Fintype V] in |
| theorem bondPinFactor_pos (left right : E → V) (B : Set V) (η : E → Bool) : |
| 0 < bondPinFactor left right B η := by unfold bondPinFactor; positivity |
|
|
| theorem bondPinRank_antitone (left right : E → V) (B : Set V) : |
| Antitone (bondPinRank left right B) := by |
| intro η ξ h |
| apply Submodule.finrank_mono |
| exact Submodule.map_mono (bondSubspace_antitone left right h) |
|
|
| theorem bondPinFactor_monotone (left right : E → V) (B : Set V) : |
| Monotone (bondPinFactor left right B) := by |
| intro η ξ h |
| apply inv_anti₀ (by positivity : 0 < (2:ℝ)^bondPinRank left right B ξ) |
| exact pow_le_pow_right₀ (by norm_num) (bondPinRank_antitone left right B h) |
|
|
| def pinnedBondSubspace (left right : E → V) (B : Set V) (η : E → Bool) : |
| Submodule SignField (V → SignField) := |
| bondSubspace left right η⊓LinearMap.ker (pinRestriction B) |
| |
| theorem pinnedBond_rank_add (left right : E → V) (B : Set V) (η : E → Bool) : |
| Module.finrank SignField (pinnedBondSubspace left right B η)+bondPinRank left right B η = |
| bondRank left right η := by |
| let K := bondSubspace left right η |
| let f := (pinRestriction B).domRestrict K |
| have hk : LinearMap.ker f=(pinnedBondSubspace left right B η).comap K.subtype := by |
| ext s |
| simp [f,pinnedBondSubspace,K] |
| have hh := LinearMap.finrank_range_add_finrank_ker f |
| rw [show LinearMap.range f=K.map (pinRestriction B) from LinearMap.range_domRestrict K _,hk, |
| (Submodule.comapSubtypeEquivOfLe (show pinnedBondSubspace left right B η ≤ K from inf_le_left)).finrank_eq] at hh |
| exact (Nat.add_comm _ _).trans hh |
|
|
| theorem pinnedBond_card (left right : E → V) (B : Set V) (η : E → Bool) : |
| (Fintype.card (pinnedBondSubspace left right B η) : ℝ)= |
| clusterWeight left right η*bondPinFactor left right B η := by |
| rw [Module.card_eq_pow_finrank (K := SignField)] |
| simp only [SignField,ZMod.card,Nat.cast_pow,Nat.cast_ofNat] |
| unfold clusterWeight bondPinFactor |
| rw [← pinnedBond_rank_add left right B η,pow_add,mul_assoc,mul_inv_cancel₀ (by positivity),mul_one] |
| |
| end ClassicalON |
|
|
| end |
|
|
| end OAI |
|
|
