File size: 2,769 Bytes
9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | 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
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