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v460_seam_s3_lto.py
verification/v460_seam_s3_lto.py
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"""v460 -- SEAM.S3.LTO.01: the EXPLICIT flat-band commuting-projector (local-topological-order)
realisation of the S3 input, advancing v424 sub-step (i) ("realise the raw seam as a
Z/2-reflection commuting-projector LTO") from an abstract existence statement to an explicit,
gap-protected quasi-local projector net carrying the Z/2 reflection and the mu4 clock.
NPW26 [arXiv:2605.10693] prove the intrinsic Bisognano-Wichmann lemma (LTO-RP) for
COMMUTING-PROJECTOR (local-topological-order) models with a Z/2 reflection. v424 reduced the
keystone BW residual to that theorem modulo two open sub-steps; v426 settled sub-step (ii) (the
invertible beta=1-KMS variant) and v455 gave the uniform-in-N BW tower. This module attacks the
COMMUTING-PROJECTOR side of sub-step (i): it exhibits, on the SAME explicit v367 p+ip collar, the
frustration-free FLAT-BAND parent (the commuting-projector form NPW26 use), shows the projector
net is gap-protected QUASI-LOCAL, and shows the flattened modular Hamiltonian inherits the v426/
v455 Z/2 reflection and mu4 clock EXACTLY. It also makes precise WHY only a QUASI-local (not a
strictly finite-range) commuting projector is possible -- the chiral (c_-!=0) obstruction
(Kapustin-Fidkowski [CMP 373 (2020) 763, arXiv:1810.07756]; for free fermions C!=0 forbids
exponentially-localised Wannier functions, Thouless 1984) -- so the realisation is exactly the
quasi-local LTO net of NPW26. It does NOT supply the continuum scaling-limit theorem (MMST,
v336) and does NOT close SEAM.EQUIV.01.
Flattening dictionary (spectral sign):
Bloch: h(k) gapped -> Q(k)=h(k)/|h(k)| (flat bands +-1), P(k)=(I-Q)/2 the occupied projector;
modular: K (v455 collar) -> K_flat=sign(K)=P_+ - P_-, the commuting-projector parent of the state.
sign is ODD => Theta K Theta=-K => Theta K_flat Theta=-K_flat (BW inherited);
rho commutes with K => [rho, f(K)]=0 for any f => [rho, K_flat]=0 (mu4 inherited).
[E] 1. FLAT-BAND COMMUTING-PROJECTOR PARENT. Q(k)=h(k)/|h(k)| has Q^2=I (flat +-1 spectrum) and
P(k)=(I-Q)/2 is an exact projector P^2=P EQUAL to the occupied ground-state projector of
v367's p+ip h(k) -- the gapped collar has a frustration-free flat-band (commuting-projector)
parent with the same ground state (verified to machine precision over a k-grid).
[E] 2. GAP-PROTECTED QUASI-LOCALITY. the real-space projector kernel P(R) (R=(n,0)) decays
rapidly in the gapped phase (effective xi~1.14 at M=1, exp-fit R^2>0.99, |P(20,0)|<1e-6) and
its xi GROWS as the gap closes (xi: 1.14 at gap 2.0 -> 2.7 at gap 0.02); at the GAPLESS point
(M=2) the kernel decays as a clean power law |R|^{-2.1} (R^2>0.99, tail ~2e-4 at |R|=24) --
the commuting-projector net is quasi-local exactly BECAUSE the bulk is gapped.
[E] 3. Z/2 REFLECTION + mu4 INHERITED. flattening the v455 collar modular Hamiltonian K gives
K_flat^2=I, P_-^2=P_-, and Theta K_flat Theta=-K_flat (BW, u_Theta=J) AND [rho,K_flat]=0
(mu4 a modular symmetry) hold EXACTLY (0 to machine precision) -- the commuting-projector
parent carries the same Z/2 reflection and mu4 clock as v426/v455.
[E] 4. NEG CONTROLS (teeth). a side-EVEN reflection gives ||Theta_e K_flat Theta_e + K_flat||
=2||K_flat||!=0 (BW fails) and an off-mu4 K mixing two OPPOSITE-sign marks gives
[rho,sign(K)]!=0 (mu4 fails) -- not every reflection / not every K works.
[C]/[O] 5. CHIRAL OBSTRUCTION + VERDICT. a chiral invertible phase (c_-=8!=0; FHS Chern |C|=1
here, 0 trivial control) admits NO STRICTLY finite-range commuting-projector Hamiltonian
(Kapustin-Fidkowski; C!=0 forbids exponentially-localised Wannier), so the realisation is the
QUASI-LOCAL projector net -- precisely NPW26's LTO setting. This advances v424 sub-step (i)
from "abstract LTO realisation" to "explicit gap-protected quasi-local commuting-projector net
carrying the Z/2 reflection and mu4 clock"; strict finite-range locality is the cited residual.
SEAM.EQUIV.01 stays [O] (the continuum scaling-limit theorem, MMST v336, is the backbone).
Python-only (numpy; Fukui-Hatsugai-Suzuki Chern + spectral flattening; the det/level
discriminators are Wolfram-mirrored via v89/v281/v422/v424/v454).
"""
import numpy as np
from tfpt_constants import check, summary, reset, g_car, N_fam
SX = np.array([[0, 1], [1, 0]], complex)
SY = np.array([[0, -1j], [1j, 0]], complex)
SZ = np.array([[1, 0], [0, -1]], complex)
def _d(kx, ky, M):
return np.array([np.sin(kx), np.sin(ky), M - np.cos(kx) - np.cos(ky)])
def _hbloch(kx, ky, M):
dx, dy, dz = _d(kx, ky, M)
return dx * SX + dy * SY + dz * SZ
def _Q(kx, ky, M):
"""Flattened Bloch Hamiltonian Q=h/|h| (the commuting-projector / flat-band parent)."""
n = _d(kx, ky, M)
n = n / np.linalg.norm(n)
return n[0] * SX + n[1] * SY + n[2] * SZ
def _occ_proj(kx, ky, M):
"""Occupied (lower-band) projector of h(k), via eigh (well defined also at gap closing)."""
w, v = np.linalg.eigh(_hbloch(kx, ky, M))
return np.outer(v[:, 0], v[:, 0].conj())
def _chern(M, N=24):
ks = np.linspace(0, 2 * np.pi, N, endpoint=False)
u = [[None] * N for _ in range(N)]
for i, kx in enumerate(ks):
for j, ky in enumerate(ks):
w, v = np.linalg.eigh(_hbloch(kx, ky, M))
u[i][j] = v[:, 0]
F = 0.0
for i in range(N):
for j in range(N):
ip, jp = (i + 1) % N, (j + 1) % N
def link(a, b):
z = np.vdot(a, b)
return z / abs(z)
Ux = link(u[i][j], u[ip][j])
Uy = link(u[ip][j], u[ip][jp])
Ux2 = link(u[i][jp], u[ip][jp])
Uy2 = link(u[i][j], u[i][jp])
F += np.angle(Ux * Uy / (Ux2 * Uy2))
return F / (2 * np.pi)
def _proj_decay(M, N=160, nmax=26):
"""|P(n,0)| of the occupied-band real-space projector kernel."""
ks = np.linspace(0, 2 * np.pi, N, endpoint=False)
P = np.zeros((N, N, 2, 2), complex)
for i, kx in enumerate(ks):
for j, ky in enumerate(ks):
P[i, j] = _occ_proj(kx, ky, M)
out = []
for n in range(1, nmax + 1):
Pn = np.einsum('ijab,i->ab', P, np.exp(1j * ks * n)) / (N * N)
out.append(np.linalg.norm(Pn))
return np.array(out)
def _fit_exp(mags, nmin=2, nmax=12):
ns = np.arange(1, len(mags) + 1)
sel = (ns >= nmin) & (ns <= nmax) & (mags > 1e-12)
n, lm = ns[sel], np.log(mags[sel])
A = np.vstack([n, np.ones_like(n)]).T
(s, c), *_ = np.linalg.lstsq(A, lm, rcond=None)
pred = A @ [s, c]
r2 = 1 - np.sum((lm - pred) ** 2) / np.sum((lm - lm.mean()) ** 2)
return -1.0 / s, r2
def _fit_pow(mags, nmin=4, nmax=22):
ns = np.arange(1, len(mags) + 1)
sel = (ns >= nmin) & (ns <= nmax) & (mags > 1e-12)
n, lm = np.log(ns[sel]), np.log(mags[sel])
A = np.vstack([n, np.ones_like(n)]).T
(s, c), *_ = np.linalg.lstsq(A, lm, rcond=None)
pred = A @ [s, c]
r2 = 1 - np.sum((lm - pred) ** 2) / np.sum((lm - lm.mean()) ** 2)
return s, r2
def _matsign(A):
w, V = np.linalg.eigh(A)
return (V * np.sign(w)) @ V.conj().T
def _collar_K(m, k):
"""The v455 collar one-particle modular Hamiltonian K (reflection-odd, mu4-even, gapped),
its reflection Theta (mode swap) and the mu4 clock rho."""
dvec = np.array([k if i < m // 2 else -k for i in range(m)])
K = np.kron(np.eye(4), np.diag(dvec)).astype(complex)
swap = np.zeros((m, m))
for i in range(m):
swap[i, m - 1 - i] = 1.0
Theta = np.kron(np.eye(4), swap).astype(complex)
rho = np.kron(np.diag([1j ** s for s in range(4)]), np.eye(m))
return K, Theta, rho
def run():
reset()
print("v460 SEAM.S3.LTO.01: flat-band commuting-projector (LTO) realisation of S3 -- "
"advancing v424 sub-step (i)")
# ---- 1. flat-band commuting-projector parent (exact over a k-grid) ----
ks = np.linspace(0.01, 2 * np.pi - 0.01, 30)
eQ2 = eP2 = egs = eflat = 0.0
for kx in ks:
for ky in ks:
Q = _Q(kx, ky, 1.0)
P = (np.eye(2) - Q) / 2
eQ2 = max(eQ2, np.linalg.norm(Q @ Q - np.eye(2)))
eP2 = max(eP2, np.linalg.norm(P @ P - P))
egs = max(egs, np.linalg.norm(P - _occ_proj(kx, ky, 1.0)))
ev = np.sort(np.linalg.eigvalsh(Q).real)
eflat = max(eflat, abs(ev[0] + 1) + abs(ev[1] - 1))
check("FLAT-BAND COMMUTING-PROJECTOR PARENT [E]: the flattened Bloch Q=h/|h| has Q^2=I "
"(||Q^2-I||<%.0e, flat +-1 bands) and P=(I-Q)/2 is an exact projector (||P^2-P||<%.0e) "
"EQUAL to v367's occupied ground-state projector (||P-P_gs||<%.0e) -- the gapped collar "
"has a frustration-free flat-band (commuting-projector) parent with the same ground state"
% (1e-10, 1e-10, 1e-10),
eQ2 < 1e-10 and eP2 < 1e-10 and egs < 1e-10 and eflat < 1e-10)
# ---- 2. gap-protected quasi-locality ----
mags1 = _proj_decay(1.0) # gapped chiral collar (M=1)
xi1, r2_1 = _fit_exp(mags1)
mags_sg = _proj_decay(1.99) # near-gapless (gap 0.02)
xi_sg, _ = _fit_exp(mags_sg)
glm = _proj_decay(2.0) # gapless (M=2)
pslope, r2p = _fit_pow(glm)
gapped_local = (r2_1 > 0.99 and mags1[19] < 1e-6 and 0.8 < xi1 < 1.6)
xi_grows = xi_sg > 1.6 * xi1 # 2.7 vs 1.14
gapless_powerlaw = (r2p > 0.99 and -2.4 < pslope < -1.8 and glm[23] > 1e-5)
check("GAP-PROTECTED QUASI-LOCALITY [E]: the gapped (M=1) projector kernel |P(n,0)| decays "
"rapidly (xi_eff=%.2f, exp R^2=%.3f, |P(20,0)|=%.1e<1e-6) and xi GROWS as the gap closes "
"(xi=%.2f at gap 0.02); the GAPLESS (M=2) kernel decays as a clean power law |R|^{%.2f} "
"(R^2=%.3f, tail |P(24,0)|=%.1e) -- locality is lost without the gap"
% (xi1, r2_1, mags1[19], xi_sg, pslope, r2p, glm[23]),
gapped_local and xi_grows and gapless_powerlaw)
# ---- 3. Z/2 reflection + mu4 inherited by the flattened modular Hamiltonian ----
k = 3.0 * np.log(1.5)
m = 8
K, Theta, rho = _collar_K(m, k)
Kf = _matsign(K)
Pminus = (np.eye(4 * m) - Kf) / 2
e_flat_id = np.linalg.norm(Kf @ Kf - np.eye(4 * m))
e_proj = np.linalg.norm(Pminus @ Pminus - Pminus)
e_bw = np.linalg.norm(Theta @ Kf @ Theta.conj().T + Kf)
e_mu4 = np.linalg.norm(rho @ Kf - Kf @ rho)
check("Z/2 REFLECTION + mu4 INHERITED [E]: flattening the v455 collar modular Hamiltonian gives "
"K_flat^2=I (%.0e), P_-^2=P_- (%.0e) and BOTH Theta K_flat Theta=-K_flat (BW/u_Theta=J, "
"%.0e) and [rho,K_flat]=0 (mu4 modular symmetry, %.0e) EXACTLY -- the commuting-projector "
"parent carries the same Z/2 reflection and mu4 clock as v426/v455 (sign is odd; rho "
"commutes with any f(K))" % (e_flat_id, e_proj, e_bw, e_mu4),
e_flat_id < 1e-10 and e_proj < 1e-10 and e_bw < 1e-10 and e_mu4 < 1e-10)
# ---- 4. neg controls (teeth) ----
swap_even = np.kron(np.eye(4), np.eye(m)) # side-EVEN reflection
bw_neg = np.linalg.norm(swap_even @ Kf @ swap_even.T + Kf) # = 2||K_flat||
Koff = K.copy()
j2 = m + m // 2 # mark1 mode with OPPOSITE K-sign
Koff[0, j2] = Koff[j2, 0] = 0.5
mu4_neg = np.linalg.norm(rho @ _matsign(Koff) - _matsign(Koff) @ rho)
check("NEG CONTROLS [E]: a side-EVEN reflection gives ||Theta_e K_flat Theta_e + K_flat||="
"2||K_flat||=%.2f!=0 (BW fails) and an off-mu4 K mixing two OPPOSITE-sign marks gives "
"||[rho,sign(K)]||=%.2f!=0 (mu4 fails) -- the conditions have teeth"
% (bw_neg, mu4_neg),
bw_neg > 1.0 and mu4_neg > 1e-3)
# ---- 5. chiral obstruction + verdict ----
C_topo = abs(round(abs(_chern(1.0))))
C_triv = abs(round(_chern(3.0)))
c_minus = 2 ** (g_car - 1) * C_topo // 2 # 16/2 = 8
chiral = (C_topo == 1 and C_triv == 0 and c_minus == g_car + N_fam)
check("CHIRAL OBSTRUCTION + VERDICT [C]/[O]: the realised phase is chiral (FHS Chern |C|=%d, "
"trivial control %d; c_-=16/2=%d=g_car+N_fam=rank E8), and a chiral invertible phase "
"(c_-!=0) admits NO strictly finite-range commuting-projector Hamiltonian "
"(Kapustin-Fidkowski; C!=0 forbids exponentially-localised Wannier) -- so the realisation "
"is the QUASI-LOCAL projector net, exactly NPW26's LTO setting. Advances v424 sub-step (i) "
"from 'abstract LTO realisation' to 'explicit gap-protected quasi-local commuting-projector "
"net carrying the Z/2 reflection and mu4 clock'; strict locality is the cited residual. "
"SEAM.EQUIV.01 stays [O] (the continuum theorem MMST v336 is the backbone)"
% (C_topo, C_triv, c_minus),
chiral and gapped_local and e_bw < 1e-10 and eP2 < 1e-10)
return summary("v460 SEAM.S3.LTO.01: the v367 gapped collar has an explicit flat-band "
"commuting-projector parent (Q^2=I, P^2=P=P_gs) that is gap-protected quasi-local "
"(xi~1.14 at M=1, |P(20,0)|<1e-6; power-law |R|^{-2.1} at the gapless point) and "
"whose flattened modular Hamiltonian inherits the Z/2 reflection (Theta K_flat "
"Theta=-K_flat) and mu4 clock ([rho,K_flat]=0) EXACTLY; the chiral c_-=8 obstructs "
"STRICT commuting-projector locality (Kapustin-Fidkowski), so the realisation is "
"the quasi-local LTO net of NPW26 -- advancing v424 sub-step (i); "
"SEAM.EQUIV.01 stays [O]")
if __name__ == "__main__":
raise SystemExit(1 if run() else 0)
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