"""v472 -- ALPHA.QUILLEN.DETLINE.01: occupied-state Berry line over the U(1)-twist moduli of the finite collar model. Its FHS Chern integer matches the Bloch Chern number. The continuum Quillen comparison stays open. The SAME collar Hamiltonian h(k) = sin kx SX + sin ky SY + (M - cos kx - cos ky) SZ (v367/v470) is put in real space on an L x L torus with twisted boundary conditions. Over the twist torus, P = 1_{(-infinity,0)}(H) defines the occupied bundle; L_occ = det Ran P is its determinant (many-body ground-state) LINE. The FHS links use local occupied frames. A gap defines a smooth projector and a ground-state ray, not a global nonvanishing Slater-vector section. A nonzero first Chern class excludes a global nonvanishing section of L_occ. Keep the sign convention explicit: the FHS continuum convention is C_Berry = integral F_occ/(2*pi*i), A_occ = trace(U^dagger dU). With nabla = d + A_occ and c1 = i F_occ/(2*pi), c1[B] = -C_Berry. Identifying a finite-mesh C_FHS with C_Berry also requires mesh control. [E] 1. BLOCH LEVEL RE-VERIFIED: FHS Chern over the BZ gives C(M=1) = 1, C(M=3) = 0, C(M=-1) = -1 (the v470/v367 integers). [E] 2. OCCUPIED BERRY LINE COMPUTED: twist-torus C_FHS = 1 at M = 1, to 1e-9, for L = 4 AND L = 6. [E] 3. CONTROLS: M = 3 gives 0 and M = -1 gives -1. [E] 4. TWO TORI, ONE INTEGER: twist and Bloch Chern numbers agree for all three M (the finite Niu-Thouless-Wu response check). [E] 5. SAMPLED GAP: the gap stays open at every sampled twist for M=1. The global bound also follows analytically from |d(k)|^2 = 1 + 2(1-cos kx)(1-cos ky) >= 1, hence gap >= 2. This neither constructs a Dai-Freed vector nor trivializes L_occ. [C] 6. INTEGRATED RESPONSE: |C_FHS| = k0 = 1 in this collar model. An integrated Chern number does not determine individual holonomies or identify the exact zeta-determinant variation. [O] 7. CONTINUUM COMPARISON: specify a chiral Fredholm family D^+, its Quillen line (including its zero locus), and a comparison with L_occ at the level of metrics and connections. A Dai-Freed vector requires filling data and lives in the pulled-back inverse boundary line. Norm-resolvent convergence and a gap alone do not control the ultraviolet contribution to a zeta determinant. ALPHA.QUILLEN.EXACT.01 stays [O]; the alpha-value claim is unchanged. References: FHS JPSJ 74 (2005), Niu-Thouless-Wu PRB 31 (1985); Quillen (1985), Bismut-Freed CMP 106 (1986), Dai-Freed JMP 35 (1994) for the distinct continuum determinant-line structures. Numerical routines: dense eigh and FHS on the twist grid. """ import numpy as np from tfpt_constants import check, summary, reset SX = np.array([[0, 1], [1, 0]], complex) SY = np.array([[0, -1j], [1j, 0]], complex) SZ = np.array([[1, 0], [0, -1]], complex) # real-space hoppings reproducing h(k) = sin kx SX + sin ky SY + (M - cos kx - cos ky) SZ TX = (-SZ - 1j * SX) / 2 # c^dag_{r+x} TX c_r + h.c. TY = (-SZ - 1j * SY) / 2 N_GRID = 8 # FHS grid on the twist torus TOL = 1e-9 def _hamiltonian(L, M, thx, thy): """Collar model on an L x L torus with U(1) twists (thx, thy).""" n = L * L H = np.zeros((2 * n, 2 * n), complex) def blk(x, y): return 2 * ((x % L) + L * (y % L)) for x in range(L): for y in range(L): i = blk(x, y) H[i:i + 2, i:i + 2] += M * SZ for (dx, dy, T, th) in ((1, 0, TX, thx), (0, 1, TY, thy)): j = blk(x + dx, y + dy) phase = np.exp(1j * th) if (x + dx == L or y + dy == L) else 1.0 H[j:j + 2, i:i + 2] += phase * T H[i:i + 2, j:j + 2] += np.conj(phase) * T.conj().T return H def _occupied_frame(L, M, thx, thy): w, v = np.linalg.eigh(_hamiltonian(L, M, thx, thy)) n_occ = int(np.sum(w < 0)) return v[:, :n_occ], n_occ, w[n_occ] - w[n_occ - 1] def detline_chern(L, M, n_grid=N_GRID): """FHS Chern number of the occupied-state Berry line over the twist torus. Returns (C, min_gap_on_sampled_twists); no global section test.""" ths = np.linspace(0, 2 * np.pi, n_grid, endpoint=False) frames, min_gap, n_ref = {}, np.inf, None for i, tx in enumerate(ths): for j, ty in enumerate(ths): fr, n_occ, gap = _occupied_frame(L, M, tx, ty) frames[(i, j)] = fr min_gap = min(min_gap, gap) n_ref = n_occ if n_ref is None else n_ref assert n_occ == n_ref, "occupation jumped -- gap closed on the torus" def link(a, b): u = np.linalg.det(frames[a].conj().T @ frames[b]) return u / abs(u) F = 0.0 for i in range(n_grid): for j in range(n_grid): ip, jp = (i + 1) % n_grid, (j + 1) % n_grid F += np.angle(link((i, j), (ip, j)) * link((ip, j), (ip, jp)) * np.conj(link((i, jp), (ip, jp))) * np.conj(link((i, j), (i, jp)))) return F / (2 * np.pi), float(min_gap) def bloch_chern(M, N=24): """v470's Bloch-BZ FHS Chern (verbatim convention).""" def occ(kx, ky): d = np.array([np.sin(kx), np.sin(ky), M - np.cos(kx) - np.cos(ky)]) _, v = np.linalg.eigh(d[0] * SX + d[1] * SY + d[2] * SZ) return v[:, 0] ks = np.linspace(0, 2 * np.pi, N, endpoint=False) u = [[occ(kx, ky) for ky in ks] for kx in ks] def ln(a, b): x = np.vdot(a, b) return x / abs(x) F = 0.0 for i in range(N): for j in range(N): ip, jp = (i + 1) % N, (j + 1) % N F += np.angle(ln(u[i][j], u[ip][j]) * ln(u[ip][j], u[ip][jp]) * np.conj(ln(u[i][jp], u[ip][jp])) * np.conj(ln(u[i][j], u[i][jp]))) return F / (2 * np.pi) def run(): reset() print("v472 ALPHA.QUILLEN.DETLINE.01: occupied Berry line over U(1) twists; " "integrated Chern response (continuum comparison open) [E]/[C]/[O]") # 1. Bloch level re-verified c_bz = {M: bloch_chern(M) for M in (1.0, 3.0, -1.0)} check("BLOCH LEVEL RE-VERIFIED [E]: BZ Chern C(M=1) = 1, C(M=3) = 0, " "C(M=-1) = -1 (the v470/v367 integers)", abs(c_bz[1.0] - 1) < TOL and abs(c_bz[3.0]) < TOL and abs(c_bz[-1.0] + 1) < TOL) # 2. the occupied-state Berry line over the twist moduli res = {(M, L): detline_chern(L, M) for M in (1.0, 3.0, -1.0) for L in (4, 6)} for (M, L), (c, gap) in sorted(res.items()): print(" M = %+d, L = %d: C_detline = %+.9f (min Fermi gap %.2f)" % (int(M), L, c, gap)) check("OCCUPIED BERRY LINE [E]: twist C_FHS = 1 within tolerance " "at M = 1 for L = 4 AND L = 6", all(abs(res[(1.0, L)][0] - 1) < TOL for L in (4, 6))) # 3. controls check("CONTROLS [E]: trivial collar M = 3 gives 0; orientation flip " "M = -1 gives -1 (both L)", all(abs(res[(3.0, L)][0]) < TOL for L in (4, 6)) and all(abs(res[(-1.0, L)][0] + 1) < TOL for L in (4, 6))) # 4. two tori, one level check("TWO TORI, ONE LEVEL [E]: twist-moduli integer == Bloch-BZ integer " "for all three M (Niu-Thouless-Wu, exhibited on the S3 collar model)", all(round(res[(M, 6)][0]) == round(c_bz[M]) for M in (1.0, 3.0, -1.0))) # 5. sampled gap; no inference of a global nonvanishing section check("SAMPLED FERMI GAP [E]: gap > 1.9 at M = 1 on all sampled twists " "for L = 4,6; this is not a global-section or Dai-Freed test", all(res[(1.0, L)][1] > 1.9 for L in (4, 6))) # 6. integrated response; individual holonomies are not computed here check("INTEGRATED RESPONSE [C]: occupied C_FHS = k0 = 1 " "in the collar model; Quillen connections, individual holonomies " "and exact zeta-determinant variation are not identified here", round(res[(1.0, 6)][0]) == 1) # 7. honest [O] check("NOT CLOSED [O]: the abstract-seam zeta-determinant identification " "(continuum leg, = the SEAM.EQUIV.01 face) stays open; " "ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 stays [E]", True) return summary("v472 occupied Berry line: twist Chern = Bloch Chern " "(continuum Quillen comparison open) [E]/[C]/[O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)