| import Mathlib |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators |
| import OAI.MathematicalPhysics.PEPSSubvolume.ScaleTiling |
|
|
| namespace OAI |
|
|
| /-! Strict subvolume exponent from uniform square-scale recurrence. -/ |
|
|
| noncomputable section |
| open scoped BigOperators ComplexOrder |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator |
| open scoped BigOperators |
| open scoped Topology |
| open Filter |
| open scoped MatrixOrder |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open scoped ComplexOrder BigOperators Matrix.Norms.L2Operator |
| open Matrix |
| open Filter Topology |
| open Set Filter Complex Complex.HadamardThreeLines |
| open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder |
| open PolynomialPEPS.PinnedEntropy |
|
|
| namespace PolynomialPEPS.Subvolume.SubvolumeArithmetic |
|
|
| theorem power_bound (B : ℕ) (hB : 1<B) (T A : ℝ) |
| (hBT : (B:ℝ)<T) (hTB : T<(B:ℝ)^2) (hA : 0≤A) |
| (f : ℕ → ℝ) (hf : ∀ k n : ℕ, n≤B^k → f n≤A*T^k) : |
| let e := Real.logb (B:ℝ) T-1 |
| 0<e ∧ e<1 ∧ ∀ n : ℕ, 0<n → f n≤(A*T)*Real.rpow n (1+e) := by |
| have hBr : (1:ℝ)<B := by exact_mod_cast hB |
| have hB0 : (0:ℝ)<B := lt_trans zero_lt_one hBr |
| have hT0 : 0<T := hB0.trans hBT |
| let a := Real.logb (B:ℝ) T |
| have ha1 : 1<a := by |
| apply (Real.lt_logb_iff_rpow_lt hBr hT0).mpr |
| simpa only [Real.rpow_one] using hBT |
| have ha2 : a<2 := by |
| apply (Real.logb_lt_iff_lt_rpow hBr hT0).mpr |
| simpa only [Real.rpow_two] using hTB |
| change 0<a-1 ∧ a-1<1 ∧ _ |
| refine ⟨by linarith only [ha1],by linarith only [ha2],?_⟩ |
| intro n hn |
| let k := Nat.log B n |
| have hlow : B^k≤n := Nat.pow_log_le_self B (ne_of_gt hn) |
| have hhigh : n≤B^(k+1) := le_of_lt (Nat.lt_pow_succ_log_self hB n) |
| have hh := hf (k+1) n hhigh |
| have he : Real.rpow ((B^k:ℕ):ℝ) a=T^k := by |
| rw [Nat.cast_pow,Real.rpow_eq_pow,← Real.rpow_pow_comm (le_of_lt hB0)] |
| dsimp only [a] |
| rw [Real.rpow_logb hB0 (ne_of_gt hBr) hT0] |
| have hpower : T^k≤Real.rpow (n:ℝ) a := by |
| rw [← he] |
| exact Real.rpow_le_rpow (Nat.cast_nonneg _) (by exact_mod_cast hlow) |
| (le_trans zero_le_one (le_of_lt ha1)) |
| have hstep := mul_le_mul_of_nonneg_left hpower (mul_nonneg hA (le_of_lt hT0)) |
| have haa : 1+(a-1)=a := by ring |
| change f n≤(A*T)*Real.rpow n (1+(a-1)) |
| rw [haa] |
| calc |
| f n≤A*T^(k+1) := hh |
| _=(A*T)*T^k := by rw [pow_succ]; ring |
| _≤(A*T)*Real.rpow n a := hstep |
|
|
| end PolynomialPEPS.Subvolume.SubvolumeArithmetic |
|
|
| namespace PolynomialPEPS.Subvolume.RectangleTiling |
| open scoped BigOperators |
| open PolynomialPEPS.Subvolume.HarmonicWeights PolynomialPEPS.Subvolume.GeometricLower |
|
|
| theorem uniform_subvolume_square (q : ℕ) (hq : 2≤q) (J Δ : ℝ) |
| (hJ : 0<J) (hΔ : 0<Δ) : |
| ∃ C e : ℝ, 0<C ∧ 0<e ∧ e<1 ∧ |
| ∀ (L : ℕ) (hv : Vertex L → Operator L q) (he : Edge L → Operator L q) |
| (Ω : State L q) (E₀ : ℝ), |
| ‖Ω‖=1 → IsGridHamiltonian J hv he → |
| asMap (Hamiltonian hv he) Ω=(E₀:ℂ) • Ω → |
| FullSystemGap (Hamiltonian hv he) Ω E₀ Δ → |
| ∀ (lo₁ lo₂ : ℤ) (n : ℕ), 0<n → |
| vonNeumannEntropy Ω (box lo₁ lo₂ 1 n)≤C*Real.rpow n (1+e) := by |
| let M : ℕ := 4*(2^(blocks q J Δ)+1) |
| let B : ℕ := M^2 |
| let T : ℝ := (M:ℝ)^2*((M:ℝ)^2-2) |
| let A : ℝ := 3*Real.log q+4*bufferConstant q J Δ |
| let e : ℝ := Real.logb (B:ℝ) T-1 |
| have hM : (4:ℝ)≤M := by |
| dsimp only [M] |
| exact_mod_cast (Nat.mul_le_mul_left 4 (Nat.succ_le_succ (Nat.zero_le (2^(blocks q J Δ))))) |
| have hM2 : (16:ℝ)≤(M:ℝ)^2 := by nlinarith only [hM] |
| have hBr : (1:ℝ)<B := by dsimp only [B]; push_cast; linarith only [hM2] |
| have hBn : 1<B := by exact_mod_cast hBr |
| have hB0 : (0:ℝ)<B := zero_lt_one.trans hBr |
| have hBT : (B:ℝ)<T := by |
| dsimp only [B,T] |
| push_cast |
| nlinarith only [hM2] |
| have hTB : T<(B:ℝ)^2 := by |
| dsimp only [B,T] |
| push_cast |
| nlinarith only [hM2] |
| have hT : 0<T := hB0.trans hBT |
| have hQ : 0<Real.log q := Real.log_pos (by exact_mod_cast (show 1<q by omega)) |
| have hbuf : 0≤bufferConstant q J Δ := by dsimp only [bufferConstant]; positivity |
| have hA : 0<A := by dsimp only [A]; positivity |
| have he0 : 0<e := by |
| have hh := (Real.lt_logb_iff_rpow_lt hBr hT).mpr |
| (show (B:ℝ)^(1:ℝ)<T by simpa only [Real.rpow_one] using hBT) |
| dsimp only [e] |
| linarith only [hh] |
| have he1 : e<1 := by |
| have hh := (Real.logb_lt_iff_lt_rpow hBr hT).mpr |
| (show T<(B:ℝ)^(2:ℝ) by simpa only [Real.rpow_two] using hTB) |
| dsimp only [e] |
| linarith only [hh] |
| refine ⟨A*T,e,mul_pos hA hT,he0,he1,?_⟩ |
| intro L hv he Ω E₀ hΩ hH hg hgap lo₁ lo₂ n hn |
| have hf : ∀ k n : ℕ, n≤B^k → |
| vonNeumannEntropy Ω (box lo₁ lo₂ 1 n)≤A*T^k := by |
| intro k m hm |
| have hm' : m≤M^(2*k) := by |
| simpa only [B,pow_mul] using hm |
| exact square_entropy_below_scale (by omega) J Δ E₀ (le_of_lt hJ) hΔ |
| hv he Ω hΩ hH hg hgap k m lo₁ lo₂ hm' |
| exact (SubvolumeArithmetic.power_bound B hBn T A hBT hTB (le_of_lt hA) |
| (fun m => vonNeumannEntropy Ω (box lo₁ lo₂ 1 m)) hf).2.2 n hn |
|
|
| end PolynomialPEPS.Subvolume.RectangleTiling |
|
|
| end |
|
|
| end OAI |
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|
