lattice-system

Roadmap history: P0 through P1j

Historical implementation record normalized from the former roadmap table. Active work is governed by tracking Issues.

P0: Project skeleton, CI, documentation infrastructure

Done

P1a: Finite-volume quantum spin operator algebra (Pauli, onSite, commutativity)

Done

P1b: Finite-chain quantum Ising Hamiltonian, Hermiticity, Gibbs state instantiation (Hermiticity, commutativity with H, β = 0 closed form, expectation realness for Hermitian observables, conservation ⟨[H, A]⟩ = 0, energy expectation as bond + transverse-field decomposition, energy expectation real, ⟨H · O⟩ real for Hermitian O, ⟨H^n⟩ real for any n : ℕ)

Done

P1c (Tasaki §2.1): Spin-1/2 operators Ŝ^(α) and the commutator algebra

Done

P1d (Tasaki §2.1): Basis states |ψ^↑⟩, |ψ^↓⟩, raising/lowering Ŝ^± (S = 1/2)

Done

P1d’ (Tasaki §2.1): S = 1 matrix representations (eq. (2.1.9))

Done

P1d’’ (Tasaki §2.1): Problem 2.1.a for S = 1/2 (Pauli basis of M_2(ℂ))

Done

P1d’’’ (Tasaki §2.1): Problem 2.1.a for S ≥ 1 (polynomial basis of M_{2S+1}(ℂ) via Lagrange interpolation in Ŝ^(3) and Ŝ^± ladder action)

Done for general S ≥ 1 — spinS_adjoin_eq_top (Issue #458 closed in PR #490). Algebra spanned: Algebra.adjoin ℂ {Ŝ^{(1)}, Ŝ^{(2)}, Ŝ^{(3)}} = ⊤.

P1e (Tasaki §2.1): S = 1/2 rotation Û^(α)_θ closed form, Û_0, adjoint, Û_{2π}

Done

P1e’: Rotation group law and unitarity

Done

P1e’’ (Tasaki §2.1): Û^(α)_θ = exp(-iθŜ^(α)) via Matrix.exp_diagonal + Matrix.exp_conj (Problem 2.1.b, all 3 axes)

Done

P1e’’’ (Tasaki §2.1): π-rotations: Û^(α)_π = -2i·Ŝ^(α), anticommutation at distinct axes

Done

P1e’’’’ (Tasaki §2.1): Û^(α)_π · Û^(β)_π = Û^(γ)_π; conjugations (Û^(α)_π)†·Ŝ^(β)·Û^(α)_π = ±Ŝ^(β)

Done

P1e’’’’’ (Tasaki §2.1): General θ transformation (Û^(α)_θ)† Ŝ^(β) Û^(α)_θ = cos θ · Ŝ^(β) - sin θ · ε^{αβγ} Ŝ^(γ) (eq. (2.1.16))

Done

P1e’’’’’’ (Tasaki §2.1): Z₂ × Z₂ representation (eqs. (2.1.27)-(2.1.34)): S = 1/2 projective + S = 1 genuine

Done

P1d-S1 (Tasaki §2.1): S = 1 basis states and Ŝ^(3), Ŝ^± actions (eqs. (2.1.2)–(2.1.6) for S = 1)

Done

P1f (Tasaki §2.2): Abstract lattice Λ, site operators Ŝ_x^(α), distinct-site commutation (eq. (2.2.6), x ≠ y)

Done

P1f-same (Tasaki §2.2): Same-site commutation [Ŝ_x^(α), Ŝ_x^(β)] = i·ε^{αβγ} Ŝ_x^(γ) (eq. (2.2.6), x = y)

Done

P1f’ (Tasaki §2.2): Total spin operator Ŝ_tot^(α) (eq. (2.2.7)) and Hermiticity

Done

P1f’-pm (Tasaki §2.2): Total raising/lowering Ŝ^±_tot = Σ_x Ŝ_x^± (eq. (2.2.8))

Done

P1f-mag (Tasaki §2.2): Total magnetization |σ| := Σ_x spinSign(σ_x) (eq. (2.2.2))

Done

P1f’’ (Tasaki §2.2): Global rotation Û^(α)_θ = exp(-iθ Ŝ_tot^(α)) (eq. (2.2.11))

Done (proved without axioms)

P1f’’’ (Tasaki §2.2): SU(2) / U(1) invariance (eqs. (2.2.12)-(2.2.13))

Done (commutativity totalSpinHalfRot{α}_commute_of_commute, unitarity totalSpinHalfRot{α}_conjTranspose_mul_self, and finite-form invariance totalSpinHalfRot{α}_conj_eq_self_of_commute all proved without axioms)

P1f’’’’ (Tasaki §2.2): Two-site inner product Ŝ_x · Ŝ_y raising/lowering decomposition (eq. (2.2.16))

Done

P1f’’’’’ (Tasaki §2.2): SU(2) invariance of Ŝ_x · Ŝ_y and eigenvalues (eqs. (2.2.17)–(2.2.19))

Done

P1f-2c (Tasaki §2.2 Problem 2.2.c): SU(2)-averaged two-site state = singlet projector (eq. (2.2.15)); integration over Euler angles φ ∈ [0,2π], θ ∈ [0,π]

Done

P1i (Tasaki §2.4): Heisenberg Hamiltonian on the fully-polarised state: H |s…s⟩ = (∑_{x,y} J(x,y)·(if x=y then 3/4 else 1/4)) · |s…s⟩ (eq. (2.4.5), S = 1/2); plus the ladder step Ŝ_tot^± · |s…s⟩ preserves the same H-eigenvalue (eqs. (2.4.7)/(2.4.9), S = 1/2) and its iterated form (Ŝ_tot^±)^k · |s…s⟩ for every k : ℕ; plus [H, Û^(α)_θ] = 0 for the global rotation (eq. (2.4.7) operator-level), the single-axis rotated constant-spin state Û^(α)_θ · |s…s⟩ shares the H-eigenvalue, and the two-axis spin-coherent state Û^(3)_ϕ Û^(2)_θ · |s…s⟩ = |Ξ_θ,ϕ⟩ (eq. (2.4.6) for s = 0); plus the magnetic-quantum-number labelling Ŝtot^(3) · (Ŝtot^-)^k · |↑..↑⟩ = (Smax - k) · (Ŝtot^-)^k · |↑..↑⟩ (eq. (2.4.9), unnormalised, lowering from highest weight) and its dual Ŝtot^(3) · (Ŝtot^+)^k · |↓..↓⟩ = (-Smax + k) · (Ŝtot^+)^k · |↓..↓⟩ (eq. (2.4.9), unnormalised, raising from lowest weight); plus the Casimir invariance Ŝtot² · (Ŝtot^∓)^k · |s..s⟩ = Smax(Smax+1) · (Ŝtot^∓)^k · |s..s⟩ for any constant s. For the matched highest/lowest-weight ladders, the unnormalised iterates (Ŝtot^-)^k · |↑..↑⟩ and (Ŝtot^+)^k · |↓..↓⟩ carry (H, Ŝtot², Ŝtot^(3)) simultaneous eigenvalues (c_J, Smax(Smax+1), Smax∓k); plus the boundary annihilations Ŝtot^- · |↓..↓⟩ = 0 and Ŝtot^+ · |↑..↑⟩ = 0 ensuring the ladder terminates after spanning all 2Smax + 1 = |Λ| + 1 magnetisation sectors — building toward the full |Φ_M⟩ / |Ξ_θ,ϕ⟩ ferromagnetic ground-state space

Done

P1g: Gibbs state ρ = e^{-βH}/Z, Tr(ρ) = 1, ⟨1⟩ = 1, Z(0) = dim, Z(0) ≠ 0, linearity ⟨O₁+O₂⟩ = ⟨O₁⟩+⟨O₂⟩, ⟨c·O⟩ = c·⟨O⟩, ⟨-O⟩ = -⟨O⟩, ⟨A−B⟩ = ⟨A⟩−⟨B⟩, ⟨Σ f⟩ = Σ ⟨f⟩, [ρ, H] = 0, reality of ⟨O⟩ for Hermitian O, conservation ⟨[H,A]⟩ = 0, anticommutator real / commutator imaginary, (⟨H·O⟩).im = 0, β = 0 closed form ρ_0 = I/dim and ⟨A⟩_0 = Tr A / dim, one-parameter group property e^{-(β₁+β₂)H} = e^{-β₁H} · e^{-β₂H} and invertibility, exact discrete semigroup identity e^{-(nβ)H} = (e^{-βH})^n (extended to n : ℤ via gibbsExp_inv)

Done

P1h: Periodic boundary conditions, Heisenberg chain (open and periodic BC), Gibbs state instantiation for both BCs (Hermiticity, commutativity with H, β = 0 closed form, expectation realness for Hermitian observables, conservation ⟨[H, A]⟩ = 0, energy expectation as a bond-sum decomposition, energy expectation real, ⟨H · O⟩ real for Hermitian O, ⟨H^n⟩ real for any n : ℕ)

Done

P1j (Tasaki §2.3): Single-spin and multi-spin time-reversal map Θ̂ := û_2 · K̂ for S = 1/2: explicit formula Θ̂((a, b)ᵀ) = (-b*, a*)ᵀ (Tasaki eq. (2.3.6)), action on |ψ^↑⟩ / |ψ^↓⟩, additivity, antilinearity, single-spin Kramers degeneracy Θ̂² = -1̂ (Tasaki eq. (2.3.8) at half-odd-integer spin), spin sign flip Θ̂(Ŝ^(α) v) = -Ŝ^(α)(Θ̂ v) (Tasaki eq. (2.3.14)), and multi-spin Kramers Θ̂_tot² = (-1)^|Λ| · 1̂ for finite Λ (Tasaki §2.3 lattice extension at S = 1/2)

Done