05-ilang-free-fall / 11-bound-pair
11bound pairverified
Determines the bound level of two bodies of different types with a binding, its curvature against the inertias and the cost, and the clock rates of the parts, with no field.
check_bound_pair.py58 KB · raw
"""Check for 05-ilang-free-fall/11-bound-pair, derivation v3 (role: check).
Parts 1 and 2 are the script of v2, unchanged (only the output folder is that of v3).
Part 3 (at the end) checks what is new in v3:
- I3: a wide and a tight pair at fixed epsilon: the remainders of the leading-order forms
against the isolation of the level (wide) and against |w11|/M (tight);
- I2: equal data and a non-diagonal on-site binding: a function that meets the definition of
a bound level and takes the value of the flat mixed bands at P = 0.
Part 1 is the check of v1, unchanged: numerical diagonalization of the fibre contract H(P)
of (11.1) on the finite range r = -L..L (open ends), for diagonal on-site bindings, and
comparison with the derivation:
- exact relations (11.5)-(11.9) of item 2 (all parameter sets),
- the bound-level equation (11.10) of item 3(a) (all sets),
- the closed forms (11.12), (11.14), (11.16) of items 3 and 4 (sets in the regime R),
- the scaling of the deviations with epsilon (scan).
Two solvers are used for the same matrix: a dense one (all eigenvectors, once per
set and P) and a banded shift-and-invert one (for the finite differences and the
doubled range); their bound levels are compared.
In the outputs of Part 1, "Gamma" is the expectation value of the symmetry Xi of Step 6.
Part 2 checks what is new in v2:
- I3: the reading statistics of the clock of a part, (11.19), with the rate at the mean level,
the rate to first order (11.25) and its closed form in the regime R (11.26);
- I4: the order eps^6 of the contribution of w22 to the bound level;
- I5: the bound eigenvector at j(P) = 0;
- I2: twofold bound levels: (11.23), (11.24) and the identities for every eigenvector;
- I1: the operator C_bod of (11.21) on a small system given in the places.
Outputs: output/check_bound_pair.json, output/check_bound_pair.txt
"""
import json
import time
from pathlib import Path
import numpy as np
from scipy.linalg import solve_banded
from scipy.optimize import brentq, minimize_scalar
OUT = Path(__file__).parent / "output"
OUT.mkdir(parents=True, exist_ok=True)
T_START = time.time()
s0 = np.eye(2)
sx = np.array([[0.0, 1.0], [1.0, 0.0]])
sz = np.diag([1.0, -1.0])
SX1, SX2 = np.kron(sx, s0), np.kron(s0, sx)
SZ1, SZ2 = np.kron(sz, s0), np.kron(s0, sz)
def fibre_ops(P, p, L, eta):
"""Fibre operators on r=-L..L; index 4*(r+L)+2*(s1-1)+(s2-1)."""
n = 2 * L + 1
S = np.eye(n, k=-1) # (S psi)(r) = psi(r-1)
Sd = S.T
e1, e2 = np.exp(1j * eta * P), np.exp(1j * (1 - eta) * P)
A1 = 0.5j * p["tau1"] * np.kron(e1 * Sd - np.conj(e1) * S, SX1)
A2 = 0.5j * p["tau2"] * np.kron(e2 * S - np.conj(e2) * Sd, SX2)
V1 = -0.5 * p["tau1"] * np.kron(e1 * Sd + np.conj(e1) * S, SX1)
V2 = -0.5 * p["tau2"] * np.kron(e2 * S + np.conj(e2) * Sd, SX2)
Z1, Z2 = np.kron(np.eye(n), SZ1), np.kron(np.eye(n), SZ2)
U = np.zeros((4 * n, 4 * n))
U[4 * L:4 * L + 4, 4 * L:4 * L + 4] = np.diag(p["w"]) # w11, w12, w21, w22
return A1, A2, V1, V2, Z1, Z2, U
def bands(P, p, nq=4001):
"""The four two-body bands (11.2) at the total label P: (lower edge, upper edge)."""
q = np.linspace(-np.pi, np.pi, nq)
om1 = np.sqrt(p["M1"] ** 2 + p["tau1"] ** 2 * np.sin(0.5 * P + q) ** 2)
om2 = np.sqrt(p["M2"] ** 2 + p["tau2"] ** 2 * np.sin(0.5 * P - q) ** 2)
return {f"{a:+d}{b:+d}": (float((a * om1 + b * om2).min()), float((a * om1 + b * om2).max()))
for a in (1, -1) for b in (1, -1)}
def edge(P, p):
"""Edge of the (++) band next to the level: lower edge for w11<0, upper edge for w11>0."""
return bands(P, p)["+1+1"][0 if p["w"][0] < 0 else 1]
def solve_full(P, p, L, eta=0.5):
"""Dense diagonalization; bound level = eigenvector of largest weight at (r=0, s=11)."""
A1, A2, V1, V2, Z1, Z2, U = fibre_ops(P, p, L, eta)
H = A1 + A2 + p["M1"] * Z1 + p["M2"] * Z2 + U
E, X = np.linalg.eigh(H)
b = int(np.argmax(np.abs(X[4 * L, :]) ** 2))
psi = X[:, b]
ex = lambda O: float(np.real(np.vdot(psi, O @ psi)))
amp = X.conj().T @ ((eta * V1 + (1 - eta) * V2) @ psi)
mask = np.ones(len(E), bool)
mask[b] = False
resolvent_sum = float(np.sum(np.abs(amp[mask]) ** 2 / (E[mask] - E[b])))
d2H = -(eta ** 2) * ex(A1) - (1 - eta) ** 2 * ex(A2)
r = np.repeat(np.arange(-L, L + 1), 4)
Gam = ((-1.0) ** np.abs(r)) * np.tile(np.diag(SZ1 @ SZ2), 2 * L + 1) # symmetry Xi of Step 6
return dict(E=float(E[b]), s3_1=ex(Z1), s3_2=ex(Z2), A1=ex(A1), A2=ex(A2), U=ex(U),
V1=ex(V1), V2=ex(V2), Epp_formula=d2H - 2 * resolvent_sum,
weight0=float(np.abs(psi[4 * L]) ** 2),
tail=float(np.sum(np.abs(psi[np.abs(r) > L // 2]) ** 2)),
Gamma=float(np.real(np.sum(Gam * np.abs(psi) ** 2))))
def solve(P, p, L, ref, eta=0.5, theta=1.0):
"""Banded solver: the eigenvalue of H(P) nearest to ref, by inverse iteration from the
vector (r=0, s=11), four steps at the shift ref and then Rayleigh-quotient steps."""
n = 4 * (2 * L + 1)
low = np.zeros((8, n), complex) # lower band storage: low[k, j] = H[j+k, j]
low[0, :] = np.tile(np.diag(p["M1"] * SZ1 + p["M2"] * SZ2), 2 * L + 1)
low[0, 4 * L:4 * L + 4] += theta * np.array(p["w"])
e1, e2 = np.exp(1j * eta * P), np.exp(1j * (1 - eta) * P)
B = -0.5j * p["tau1"] * np.conj(e1) * SX1 + 0.5j * p["tau2"] * e2 * SX2 # block (r+1, r): coefficient of S
for s_row in range(4):
for s_col in range(4):
low[4 + s_row - s_col, s_col:n - 4:4] = B[s_row, s_col]
full = np.zeros((15, n), complex) # storage of solve_banded: full[7+i-j, j] = H[i, j]
for k in range(8):
full[7 + k, :n - k] = low[k, :n - k]
if k:
full[7 - k, k:] = np.conj(low[k, :n - k])
def matvec(v):
out = low[0] * v
for k in range(1, 8):
out[k:] += low[k, :n - k] * v[:n - k]
out[:n - k] += np.conj(low[k, :n - k]) * v[k:]
return out
v = np.zeros(n, complex)
v[4 * L] = 1.0
shift, E = ref, ref
for it in range(12):
m = full.copy()
m[7] -= shift
try:
v = solve_banded((7, 7), m, v)
except np.linalg.LinAlgError:
break
v /= np.linalg.norm(v)
Hv = matvec(v)
E = float(np.real(np.vdot(v, Hv)))
if np.linalg.norm(Hv - E * v) < 1e-13:
break
if it >= 3:
shift = E
assert abs(v[4 * L]) ** 2 > 0.05, "banded solver left the bound level"
return E
def d1(f, x, h):
return (-f(x + 2 * h) + 8 * f(x + h) - 8 * f(x - h) + f(x - 2 * h)) / (12 * h)
def d2(f, x, h):
return (-f(x + 2 * h) + 16 * f(x + h) - 30 * f(x) + 16 * f(x - h) - f(x - 2 * h)) / (12 * h * h)
def green(E, P, p, nq=4096, eta=0.5):
"""Coefficients of (11.10): G_{s1 s2}(E,P) and g_x(E,P), as sums over q."""
q = -np.pi + 2 * np.pi * np.arange(nq) / nq
p1, p2 = eta * P + q, (1 - eta) * P - q
om1 = np.sqrt(p["M1"] ** 2 + p["tau1"] ** 2 * np.sin(p1) ** 2)
om2 = np.sqrt(p["M2"] ** 2 + p["tau2"] ** 2 * np.sin(p2) ** 2)
G, gx = {}, 0.0
for sg1 in (1, -1):
for sg2 in (1, -1):
den = 4 * (sg1 * om1 + sg2 * om2 - E)
gx += np.mean(sg1 * sg2 * p["tau1"] * p["tau2"] * np.sin(p1) * np.sin(p2) / (om1 * om2 * den))
for t1 in (1, 2):
for t2 in (1, 2):
c1, c2 = (1 if t1 == 1 else -1), (1 if t2 == 1 else -1)
G[(t1, t2)] = G.get((t1, t2), 0.0) + np.mean(
(1 + c1 * sg1 * p["M1"] / om1) * (1 + c2 * sg2 * p["M2"] / om2) / den)
return G, gx
def green_direct(E, P, p, nq=4096, eta=0.5):
"""<r=0|(H0(P)-E)^(-1)|r=0> as a 4x4 matrix, by inverting h_P(q)-E on the q grid."""
q = -np.pi + 2 * np.pi * np.arange(nq) / nq
p1, p2 = eta * P + q, (1 - eta) * P - q
h = (-p["tau1"] * np.sin(p1))[:, None, None] * SX1 + (-p["tau2"] * np.sin(p2))[:, None, None] * SX2 \
+ p["M1"] * SZ1 + p["M2"] * SZ2 - E * np.eye(4)
return np.linalg.inv(h).mean(axis=0)
def F_plus(E, P, p):
"""The factor F_+ of (11.10) (sector Xi=+1)."""
G, gx = green(E, P, p)
w11, _, _, w22 = p["w"]
return (1 + w11 * G[(1, 1)]) * (1 + w22 * G[(2, 2)]) - w11 * w22 * gx ** 2
def root_F_plus(Eb, P, p, gap):
lo, hi = Eb - 0.3 * gap, Eb + 0.3 * gap
flo, fhi = F_plus(lo, P, p), F_plus(hi, P, p)
if flo * fhi > 0:
return None
for _ in range(100):
mid = 0.5 * (lo + hi)
fm = F_plus(mid, P, p)
if flo * fm <= 0:
hi, fhi = mid, fm
else:
lo, flo = mid, fm
return 0.5 * (lo + hi)
def closed(P, p):
"""Closed forms of item 3 (leading order in R); expectation values at P=0."""
M1, M2, t1, t2 = p["M1"], p["M2"], p["tau1"], p["tau2"]
w = p["w"][0]
sg = np.sign(w)
a1, a2 = t1 ** 2 / (4 * M1), t2 ** 2 / (4 * M2)
a = a1 + a2
j = np.sqrt(a1 ** 2 + a2 ** 2 + 2 * a1 * a2 * np.cos(2 * P))
root0 = np.sqrt(w * w + a * a)
c = 1 + sg * a / root0
return dict(E=M1 + M2 + a + sg * np.sqrt(w * w + j * j),
Ep=-sg * 2 * a1 * a2 * np.sin(2 * P) / np.sqrt(w * w + j * j),
Epp0=-sg * 4 * a1 * a2 / root0, Ebind=a + sg * root0,
s3_1=1 - a1 / M1 * c, s3_2=1 - a2 / M2 * c, A1=2 * a1 * c, A2=2 * a2 * c,
U=sg * w * w / root0, a=a, u=abs(w) / a)
def with_(p, **kw):
q = dict(p)
q.update(kw)
return q
SETS = {
"A_equal_R": dict(M1=1.0, M2=1.0, tau1=0.10, tau2=0.10, w=[-0.0040, 0.0030, -0.0020, 0.0060]),
"B_unequal_R": dict(M1=1.0, M2=1.7, tau1=0.20, tau2=0.12, w=[-0.0100, 0.0050, -0.0040, 0.0080]),
"C_above_band_R": dict(M1=1.0, M2=0.8, tau1=0.10, tau2=0.15, w=[0.0200, -0.0050, 0.0030, -0.0100]),
"D_wide_R": dict(M1=1.3, M2=1.0, tau1=0.15, tau2=0.20, w=[-0.0040, 0.0020, 0.0010, 0.0030]),
"E_order_one": dict(M1=1.0, M2=1.3, tau1=1.00, tau2=0.80, w=[-0.6000, 0.2000, -0.1500, 0.3000]),
}
REGIME_R = {"A_equal_R", "B_unequal_R", "C_above_band_R", "D_wide_R"}
P_VALUES = [-0.30, -0.15, 0.0, 0.15, 0.30]
L = 120
HP, HX = 0.01, 1e-3
results, lines = {}, []
def say(s):
lines.append(s)
print(s, flush=True)
say(f"range of r: -L..L with L={L} (open ends); convergence: comparison with L={2 * L} and weight at |r|>L/2")
say(f"finite differences: 5-point, step {HP} in P, relative step {HX} in M_i, tau_i, theta; convention eta=1/2")
for name, p in SETS.items():
say(f"\n=== set {name}: {p} max tau_i/M_i = {max(p['tau1'] / p['M1'], p['tau2'] / p['M2']):.3f}")
res = dict(params=p, P={})
keys = ("dM", "dtau", "dtheta", "euler", "V1", "V2", "V12", "Epp", "eq310", "green", "conv", "tail", "eta",
"solvers", "closed_E", "closed_Ep")
worst = dict.fromkeys(keys, 0.0)
for P in P_VALUES:
o = solve_full(P, p, L)
bd = bands(P, p)
gap = min(abs(o["E"] - e) for lohi in bd.values() for e in lohi)
inside = any(lo <= o["E"] <= hi for lo, hi in bd.values())
e0 = edge(P, p)
def fE(PP=P, pp=p, **kw):
"""Banded bound level at changed P or data; the reference follows the band edge."""
return solve(PP, pp, L, o["E"] + edge(PP, pp) - e0, **kw)
solvers = abs(fE() - o["E"])
conv = abs(solve(P, p, 2 * L, o["E"]) - o["E"])
eta_dep = max(abs(fE(eta=e) - o["E"]) for e in (0.0, 1.0, 0.3))
# item 2(a): derivatives of E_b against expectation values
dM1 = d1(lambda x: fE(pp=with_(p, M1=x)), p["M1"], HX * p["M1"])
dM2 = d1(lambda x: fE(pp=with_(p, M2=x)), p["M2"], HX * p["M2"])
dt1 = p["tau1"] * d1(lambda x: fE(pp=with_(p, tau1=x)), p["tau1"], HX * p["tau1"])
dt2 = p["tau2"] * d1(lambda x: fE(pp=with_(p, tau2=x)), p["tau2"], HX * p["tau2"])
dth = d1(lambda x: fE(theta=x), 1.0, HX)
euler = p["M1"] * o["s3_1"] + p["M2"] * o["s3_2"] + o["A1"] + o["A2"] + o["U"] - o["E"]
# items 2(b), 2(c)
Ep = d1(lambda x: fE(PP=x), P, HP)
Epp = d2(lambda x: fE(PP=x), P, HP)
# item 3(a): coefficients against a direct inversion, root against E_b
G, gx = green(o["E"], P, p)
Gd = green_direct(o["E"], P, p)
gerr = max(abs(G[(1, 1)] - Gd[0, 0]), abs(G[(1, 2)] - Gd[1, 1]), abs(G[(2, 1)] - Gd[2, 2]),
abs(G[(2, 2)] - Gd[3, 3]), abs(gx - Gd[0, 3]), abs(gx - Gd[1, 2]),
abs(Gd[0, 1]), abs(Gd[0, 2]), abs(Gd[1, 3]), abs(Gd[2, 3])) / np.abs(Gd).max()
root = root_F_plus(o["E"], P, p, gap)
c = closed(P, p)
res["P"][f"{P:+.2f}"] = dict(
E=o["E"], isolation=gap, inside_band=inside, weight0=o["weight0"], Gamma=o["Gamma"],
tail=o["tail"], conv_2L=conv, eta_dependence=eta_dep, dense_minus_banded=solvers,
dM1=[dM1, o["s3_1"]], dM2=[dM2, o["s3_2"]], dtau1=[dt1, o["A1"]], dtau2=[dt2, o["A2"]],
dtheta=[dth, o["U"]], euler_residual=euler, Ep=[Ep, o["V1"], o["V2"]], Epp=[Epp, o["Epp_formula"]],
green_rel_error=float(gerr), eq_3a_root=root,
eq_3a_root_minus_E=(None if root is None else root - o["E"]),
closed_E=float(c["E"]), closed_Ep=float(c["Ep"]))
for k, v in (("dM", max(abs(dM1 - o["s3_1"]), abs(dM2 - o["s3_2"]))),
("dtau", max(abs(dt1 - o["A1"]), abs(dt2 - o["A2"]))), ("dtheta", abs(dth - o["U"])),
("euler", abs(euler)), ("V1", abs(Ep - o["V1"])), ("V2", abs(Ep - o["V2"])),
("V12", abs(o["V1"] - o["V2"])), ("Epp", abs(Epp - o["Epp_formula"])),
("eq310", np.inf if root is None else abs(root - o["E"])), ("green", float(gerr)),
("conv", conv), ("tail", o["tail"]), ("eta", eta_dep), ("solvers", solvers),
("closed_E", abs(c["E"] - o["E"])), ("closed_Ep", abs(c["Ep"] - Ep))):
worst[k] = max(worst[k], float(v))
say(f"P={P:+.2f} E_b={o['E']:.12f} isolation={gap:.3e} in a band: {inside} weight(r=0,11)={o['weight0']:.4f}"
f" Gamma={o['Gamma']:+.6f} E_b'={Ep:+.6e} <V1>={o['V1']:+.6e} <V2>={o['V2']:+.6e}"
f" E_b''={Epp:+.6e} (11.9)={o['Epp_formula']:+.6e}")
res["max_abs_deviation"] = worst
say("max |deviation| over the P values:")
say(f" 2(a): dE/dM_i - <sigma3_i> {worst['dM']:.2e}; tau_i dE/dtau_i - <A_i> {worst['dtau']:.2e};"
f" dE/dtheta - <U> {worst['dtheta']:.2e}; Euler residual {worst['euler']:.2e}")
say(f" 2(b): E_b' - <V1> {worst['V1']:.2e}; E_b' - <V2> {worst['V2']:.2e}; <V1> - <V2> {worst['V12']:.2e}")
say(f" 2(c): E_b'' (finite difference) - formula (11.9) {worst['Epp']:.2e}")
say(f" 3(a): coefficients of (11.10) against direct inversion (relative) {worst['green']:.2e};"
f" root of (11.10) - E_b {worst['eq310']:.2e}")
say(f" numerics: |E_b(L) - E_b(2L)| {worst['conv']:.2e}; weight at |r|>L/2 {worst['tail']:.2e};"
f" dependence on eta {worst['eta']:.2e}; dense - banded {worst['solvers']:.2e}")
# closed forms at P = 0
o0 = solve_full(0.0, p, L)
c0 = closed(0.0, p)
Epp0 = o0["Epp_formula"]
eps2 = max(p["tau1"] / p["M1"], p["tau2"] / p["M2"]) ** 2
cmp0 = {k: [o0[k], float(c0[k])] for k in ("s3_1", "s3_2", "A1", "A2", "U")}
cmp0["Ebind"] = [o0["E"] - p["M1"] - p["M2"], float(c0["Ebind"])]
cmp0["Epp0"] = [Epp0, float(c0["Epp0"])]
m1, m2 = p["M1"] / p["tau1"] ** 2, p["M2"] / p["tau2"] ** 2
mb = 1.0 / Epp0
item4 = dict(mb_over_m1_plus_m2=[mb / (m1 + m2), float(-np.sign(p["w"][0]) * np.sqrt(1 + c0["u"] ** 2))],
mb_minus_m1_m2=[mb - m1 - m2, -4 * m1 * m2 * (o0["E"] - p["M1"] - p["M2"])])
res["closed_forms_P0"] = dict(u=float(c0["u"]), a=float(c0["a"]), eps2=eps2, compare=cmp0, item4=item4,
max_dE_over_P=worst["closed_E"], max_dEp_over_P=worst["closed_Ep"],
rel_dev_Epp0=float(abs(Epp0 - c0["Epp0"]) / abs(c0["Epp0"])))
tag = "regime R" if name in REGIME_R else "control, outside R"
say(f"closed forms (items 3, 4), {tag}: |w11|/a = {c0['u']:.3f}, (max tau/M)^2 = {eps2:.3e}")
say(f" max_P |E_b - (11.12)| = {worst['closed_E']:.3e} = {worst['closed_E'] / (eps2 ** 2):.3f} * (tau/M)^4;"
f" max_P |E_b' - d(11.12)/dP| = {worst['closed_Ep']:.3e}")
say(f" E_b''(0): numeric {Epp0:+.8e} closed {c0['Epp0']:+.8e} rel.dev {abs(Epp0 - c0['Epp0']) / abs(c0['Epp0']):.3e}"
f" = {abs(Epp0 - c0['Epp0']) / abs(c0['Epp0']) / eps2:.3f} * (tau/M)^2")
for k, (num, cl) in cmp0.items():
say(f" {k:6s} numeric {num:+.10e} closed {cl:+.10e} diff {num - cl:+.2e}")
say(f" m_b/(m_1+m_2): numeric {item4['mb_over_m1_plus_m2'][0]:+.6f} (11.16) {item4['mb_over_m1_plus_m2'][1]:+.6f};"
f" m_b-(m_1+m_2): numeric {item4['mb_minus_m1_m2'][0]:+.6e} -4 m_1 m_2 E_bind {item4['mb_minus_m1_m2'][1]:+.6e}")
results[name] = res
# epsilon scan: tau_i = eps*tauhat_i, w = eps^2*what, for the data of set B (eps = 1)
say("\n=== epsilon scan on set B_unequal_R (tau -> eps*tau, w -> eps^2*w)")
base = SETS["B_unequal_R"]
scan, prev = [], None
for eps in (1.0, 0.5, 0.25, 0.125):
p = with_(base, tau1=eps * base["tau1"], tau2=eps * base["tau2"], w=[eps * eps * x for x in base["w"]])
o0, c0 = solve_full(0.0, p, L), closed(0.0, p)
row = dict(eps=eps, dE_P0=o0["E"] - c0["E"], dE_P03=solve_full(0.3, p, L)["E"] - closed(0.3, p)["E"],
rel_dEpp0=(o0["Epp_formula"] - c0["Epp0"]) / c0["Epp0"], d_s3_1=o0["s3_1"] - c0["s3_1"],
d_s3_2=o0["s3_2"] - c0["s3_2"])
row = {k: float(v) for k, v in row.items()}
if prev:
row["ratio_to_previous"] = {k: prev[k] / row[k] for k in row if k != "eps"}
scan.append(row)
prev = {k: v for k, v in row.items() if k != "ratio_to_previous"}
say(f"eps={eps:6.3f} E_b-(11.12): P=0 {row['dE_P0']:+.4e}, P=0.3 {row['dE_P03']:+.4e}; rel.dev E_b''(0) {row['rel_dEpp0']:+.4e};"
f" <sigma3_i>-(11.14): {row['d_s3_1']:+.4e}, {row['d_s3_2']:+.4e}")
if "ratio_to_previous" in row:
say(" ratios to the previous eps: " + ", ".join(f"{k} {v:.2f}" for k, v in row["ratio_to_previous"].items()))
say("expected for halving eps: 16 for E_b and <sigma3_i> (next order eps^4), 4 for the relative deviation of E_b''(0)")
results["epsilon_scan_set_B"] = scan
results["settings"] = dict(L=L, L_check=2 * L, P_values=P_VALUES, eta=0.5, h_P=HP, h_rel_parameters=HX)
# ---------------------------------------------------------------------------
# Part 2: checks of what is new in derivation v2
# ---------------------------------------------------------------------------
new = {}
def dense(P, p, L, eta=0.5, W0=None, theta=1.0):
"""Dense H(P) with all eigenvectors and its parts. W0: a general on-site binding
(4x4, order 11, 12, 21, 22) used instead of the diagonal p['w']."""
A1, A2, V1, V2, Z1, Z2, U = fibre_ops(P, p, L, eta)
if W0 is not None:
U = np.zeros_like(A1)
U[4 * L:4 * L + 4, 4 * L:4 * L + 4] = W0
H = A1 + A2 + p["M1"] * Z1 + p["M2"] * Z2 + theta * U
E, X = np.linalg.eigh(H)
return E, X, dict(A1=A1, A2=A2, V1=V1, V2=V2, Z1=Z1, Z2=Z2, U=U)
def P_ring(a, vartheta, N):
"""P_N(a; vartheta) of (5.1) of 04-ilang-time/05-moving-clock."""
k = np.arange(N)
return float(abs(np.mean(np.exp(2j * np.pi * k * (a - vartheta) / N))) ** 2)
def closed_d2M(p, part):
"""Second derivative of the closed form (11.12) at P=0 with respect to M_part (used in (11.26))."""
M1, M2, t1, t2 = p["M1"], p["M2"], p["tau1"], p["tau2"]
w = p["w"][0]
sg = np.sign(w)
a1, a2 = t1 ** 2 / (4 * M1), t2 ** 2 / (4 * M2)
a, ai, Mi = a1 + a2, (a1 if part == 1 else a2), (M1 if part == 1 else M2)
u = abs(w) / a
lam = 1 + sg / np.sqrt(1 + u * u)
return float(ai / Mi ** 2 * (2 * lam + sg * ai / a * u * u / (1 + u * u) ** 1.5))
def clock_check(p, part, N, D, ticks):
"""Item 5(a), (11.19) and (11.25): exact reading statistics of the clock label of one part
for a pair in the bound level at P=0 (sharp total label, no records), against
P_N(a; Gamma*lambda/lambda_0) with three choices of Gamma."""
key = f"M{part}"
m = p[key]
lam0 = 2 * np.pi * (N - 1) / (N * D) # tick parameter with the level spread d_{N-1} = D
d = D * np.arange(N) / (N - 1)
dbar = 0.5 * D
lev = [dense(0.0, with_(p, **{key: m + dk}), L)[:2] for dk in d]
E0, X0, ops = dense(0.0, p, L)
b = int(np.argmax(np.abs(X0[4 * L, :]) ** 2))
psi0 = X0[:, b]
Z = ops[f"Z{part}"]
g0 = np.inf # isolation of the bound level over the levels of the clock
for E, X in lev:
bk = int(np.argmax(np.abs(X[4 * L, :]) ** 2))
g0 = min(g0, float(np.min(np.abs(np.delete(E, bk) - E[bk]))))
G0 = float(np.real(np.vdot(psi0, Z @ psi0))) # <sigma3_i>_0 at M_i = m_i (rate of v1)
amp = X0.conj().T @ (Z @ psi0)
mask = np.ones(len(E0), bool)
mask[b] = False
d2E = float(-2 * np.sum(np.abs(amp[mask]) ** 2 / (E0[mask] - E0[b]))) # d^2 E_b / d M_i^2 at m_i
Gmean = solve_full(0.0, with_(p, **{key: m + dbar}), L)[f"s3_{part}"] # <sigma3_i>_0 at the mean level
Gfirst = G0 + dbar * d2E # (11.25)
coef = [X.conj().T @ psi0 for E, X in lev] # start: eigenvector of the level 0 on every level
q = np.zeros((len(ticks), N))
for it, t in enumerate(ticks):
vec = np.array([X @ (np.exp(-1j * E * t * lam0) * c) for (E, X), c in zip(lev, coef)])
for a in range(N):
q[it, a] = np.linalg.norm(np.exp(2j * np.pi * np.arange(N) * a / N) @ vec / N) ** 2
def table(G):
return np.array([[P_ring(a, G * t, N) for a in range(N)] for t in ticks])
def dev(G, tmax=None):
sel = np.ones(len(ticks), bool) if tmax is None else (np.asarray(ticks) <= tmax + 1e-12)
return float(np.max(np.abs(q - table(G))[sel]))
fit = minimize_scalar(lambda G: float(np.sum((q - table(G)) ** 2)), bounds=(Gmean - 0.05, Gmean + 0.05),
method="bounded", options=dict(xatol=1e-13))
cm = closed(0.0, with_(p, **{key: m + dbar}))
c0 = closed(0.0, p)
return dict(part=part, N=N, D=D, lambda0=lam0, g0=g0, D_over_g0=D / g0, D_over_M=D / m, sum_q_minus_1=float(
np.max(np.abs(q.sum(axis=1) - 1))), Gamma_level0=G0, Gamma_mean_level=Gmean, Gamma_first_order=Gfirst,
Gamma_fit=float(fit.x), d2E_dM2=d2E, d2E_dM2_closed=closed_d2M(p, part),
Gamma_closed_11_20_at_mean=float(cm[f"s3_{part}"]),
Gamma_closed_11_26=float(c0[f"s3_{part}"] + dbar * closed_d2M(p, part)),
dev_mean_level=dev(Gmean), dev_level0=dev(G0), dev_first_order=dev(Gfirst),
dev_mean_level_first_revolution=dev(Gmean, N), dev_level0_first_revolution=dev(G0, N))
say("\n=== new in v2, I3: reading statistics of the clock of a part, (11.19), (11.25), (11.26)")
say("start: bound eigenvector of the level 0 on every level, clock label in |0>, total label P=0 sharp, no records;")
say("readings at lambda = t*lambda_0, t = 0.25, 0.5, ..., 2N; deviation = max over a and t of |q - P_N(a; Gamma*t)|")
clock_runs = []
for name, part, N, D in (("E_order_one", 1, 4, 0.06), ("B_unequal_R", 1, 4, 0.10), ("B_unequal_R", 2, 5, 0.10)):
prev = None
for DD in (D, D / 2, D / 4):
ticks = list(np.arange(1, 8 * N + 1) * 0.25)
r = clock_check(SETS[name], part, N, DD, ticks)
r["set"] = name
if prev:
r["ratio_to_previous"] = dict(dev_mean_level=prev["dev_mean_level"] / r["dev_mean_level"],
dev_level0=prev["dev_level0"] / r["dev_level0"],
dev_first_order=prev["dev_first_order"] / r["dev_first_order"])
clock_runs.append(r)
prev = r
say(f"set {name}, clock of part {part}, N={N}, spread D={DD:.4f} (D/g0={r['D_over_g0']:.3f}, D/m={r['D_over_M']:.4f},"
f" lambda_0={r['lambda0']:.2f})")
say(f" Gamma: level 0 (v1) {r['Gamma_level0']:.10f}; mean level (11.19) {r['Gamma_mean_level']:.10f};"
f" first order (11.25) {r['Gamma_first_order']:.10f}; least-squares fit {r['Gamma_fit']:.10f}")
say(f" d2E_b/dM^2 at m: spectral sum {r['d2E_dM2']:+.8e}; closed form of (11.26) {r['d2E_dM2_closed']:+.8e};"
f" closed (11.20) at the mean level {r['Gamma_closed_11_20_at_mean']:.10f}; closed (11.26) {r['Gamma_closed_11_26']:.10f}")
say(f" deviation of the statistics: Gamma at the mean level {r['dev_mean_level']:.3e} (t<=N: {r['dev_mean_level_first_revolution']:.3e});"
f" first order {r['dev_first_order']:.3e}; Gamma of the level 0 {r['dev_level0']:.3e} (t<=N: {r['dev_level0_first_revolution']:.3e})")
if prev and "ratio_to_previous" in r:
say(" ratios to the previous spread: " + ", ".join(f"{k} {v:.2f}" for k, v in r["ratio_to_previous"].items()))
say("expected for halving the spread at fixed lambda/lambda_0: 4 with Gamma at the mean level, 2 with Gamma of the level 0")
new["I3_clock"] = clock_runs
say("\n=== new in v2, I4: order of the contribution of w22 (set B, tau -> eps*tau, w -> eps^2*w)")
rows, prev = [], None
for eps in (2.0, 1.0, 0.5, 0.25):
p = with_(base, tau1=eps * base["tau1"], tau2=eps * base["tau2"], w=[eps * eps * x for x in base["w"]])
pz = with_(p, w=p["w"][:3] + [0.0])
Ea, Xa, _ = dense(0.0, p, L)
Ez, Xz, _ = dense(0.0, pz, L)
ba, bz = int(np.argmax(np.abs(Xa[4 * L, :]) ** 2)), int(np.argmax(np.abs(Xz[4 * L, :]) ** 2))
dE = float(Ea[ba] - Ez[bz])
wt = 0.5 * float(np.abs(Xa[4 * L + 3, ba]) ** 2 + np.abs(Xz[4 * L + 3, bz]) ** 2) # |<22|psi(0)>|^2
row = dict(eps=eps, w22=p["w"][3], dE=dE, w22_times_weight=p["w"][3] * wt, weight_22_at_0=wt)
if prev:
row["ratio_to_previous"] = dict(dE=prev["dE"] / dE, weight_22_at_0=prev["weight_22_at_0"] / wt)
rows.append(row)
prev = row
say(f"eps={eps:6.3f} E_b(w22) - E_b(w22=0) = {dE:+.6e}; w22*|<22|psi(0)>|^2 = {row['w22_times_weight']:+.6e};"
f" |<22|psi(0)>|^2 = {wt:.6e}" + ("" if "ratio_to_previous" not in row else
f"; ratios to the previous eps: dE {row['ratio_to_previous']['dE']:.2f}, weight {row['ratio_to_previous']['weight_22_at_0']:.2f}"))
say("expected for halving eps: 64 for the contribution of w22 to E_b (order eps^6), 16 for the weight (order eps^4)")
new["I4_w22"] = rows
say("\n=== new in v2, I5: equal data (a_1 = a_2) at P = pi/2, where j(P) = 0 (set A, tau -> eps*tau, w -> eps^2*w)")
rows, prev = [], None
baseA = SETS["A_equal_R"]
for eps in (1.0, 0.5, 0.25):
p = with_(baseA, tau1=eps * baseA["tau1"], tau2=eps * baseA["tau2"], w=[eps * eps * x for x in baseA["w"]])
E, X, _ = dense(0.5 * np.pi, p, L)
b = int(np.argmax(np.abs(X[4 * L, :]) ** 2))
a = p["tau1"] ** 2 / (4 * p["M1"]) + p["tau2"] ** 2 / (4 * p["M2"])
row = dict(eps=eps, E=float(E[b]), closed_E=p["M1"] + p["M2"] + a + p["w"][0],
dE=float(E[b] - (p["M1"] + p["M2"] + a + p["w"][0])),
one_minus_weight_r0_11=float(1 - np.abs(X[4 * L, b]) ** 2),
weight_r_not_0_in_11=float(np.sum(np.abs(X[0::4, b]) ** 2) - np.abs(X[4 * L, b]) ** 2))
if prev:
row["ratio_to_previous"] = dict(dE=prev["dE"] / row["dE"],
one_minus_weight=prev["one_minus_weight_r0_11"] / row["one_minus_weight_r0_11"])
rows.append(row)
prev = row
say(f"eps={eps:6.3f} E_b = {row['E']:.12f} M_1+M_2+a+w11 = {row['closed_E']:.12f} difference {row['dE']:+.3e};"
f" 1 - weight(r=0,11) = {row['one_minus_weight_r0_11']:.3e}; weight of 11 at r != 0: {row['weight_r_not_0_in_11']:.3e}"
+ ("" if "ratio_to_previous" not in row else f"; ratios: dE {row['ratio_to_previous']['dE']:.2f},"
f" 1-weight {row['ratio_to_previous']['one_minus_weight']:.2f}"))
say("expected: psi_P = delta_{r,0}|11> up to O(eps) (1 - weight of order eps^2), E_b = M_1+M_2+a+w11 up to O(eps^4)")
new["I5_j_zero"] = rows
def degenerate_check(p, P0, E0):
"""I2: a twofold bound level. On-site binding W0 = -G(E0,P0)^(-1) on span(|11>,|22>) (G of Step 6),
diagonal on |12>,|21>; then 1 + G W0 vanishes on that span and E0 is a twofold level of H(P0)."""
Gd = green_direct(E0, P0, p)
blk = np.ix_([0, 3], [0, 3])
W0 = np.diag([0.0, 0.15, -0.10, 0.0]).astype(complex)
W0[blk] = -np.linalg.inv(Gd[blk])
W0 = 0.5 * (W0 + W0.conj().T)
par0 = dict(P=P0, M1=p["M1"], M2=p["M2"], tau1=p["tau1"], tau2=p["tau2"], theta=1.0)
eta = 0.5
def at(**kw):
v = dict(par0)
v.update(kw)
pp = with_(p, M1=v["M1"], M2=v["M2"], tau1=v["tau1"], tau2=v["tau2"])
E, X, ops = dense(v["P"], pp, L, eta=eta, W0=W0, theta=v["theta"])
return E, X, ops, np.argsort(np.abs(E - E0))
E, X, ops, idx = at()
pair = idx[:2]
Eb = float(E[pair].mean())
Phi = X[:, pair] # orthonormal basis of the level, as returned by the solver
rest = np.ones(len(E), bool)
rest[pair] = False
Veta = eta * ops["V1"] + (1 - eta) * ops["V2"]
dH = dict(M1=ops["Z1"], M2=ops["Z2"], tau1=ops["A1"] / p["tau1"], tau2=ops["A2"] / p["tau2"], theta=ops["U"], P=Veta)
amp = X[:, rest].conj().T @ (Veta @ Phi)
h2P = Phi.conj().T @ ((-(eta ** 2) * ops["A1"] - (1 - eta) ** 2 * ops["A2"]) @ Phi) \
- 2 * amp.conj().T @ (amp / (E[rest] - Eb)[:, None]) # second-order matrix of (11.24)
steps = dict(P=HP, M1=HX * p["M1"], M2=HX * p["M2"], tau1=HX * p["tau1"], tau2=HX * p["tau2"], theta=HX)
r_lab = np.repeat(np.arange(-L, L + 1), 4)
out = dict(P0=P0, E0=E0, W0_block_11_22=np.real(W0[blk]).tolist(), level=Eb, splitting=float(abs(E[pair[0]] - E[pair[1]])),
isolation=float(abs(E[idx[2]] - Eb)), tail=float(np.sum(np.abs(Phi[np.abs(r_lab) > L // 2]) ** 2)), xi={})
bases = {}
for xi, O in dH.items():
h = Phi.conj().T @ (O @ Phi)
e, c = np.linalg.eigh(h)
second_order = bool(xi == "P" and abs(e[1] - e[0]) < 1e-8)
if second_order: # equal first derivatives: branch vectors from the second order
c = c @ np.linalg.eigh(c.conj().T @ h2P @ c)[1]
B = Phi @ c # branch vectors of the parameter xi
bases[xi] = c
vals, quality = {}, 1.0
for s in (-2, -1, 1, 2):
E_, X_, _, idx_ = at(**{xi: par0[xi] + s * steps[xi]})
pr = idx_[:2]
ov = np.abs(B.conj().T @ X_[:, pr]) ** 2 # [branch, eigenvector]
j0 = int(np.argmax(ov[0]))
vals[(0, s)], vals[(1, s)] = float(E_[pr[j0]]), float(E_[pr[1 - j0]])
quality = min(quality, float(ov[0, j0]), float(ov[1, 1 - j0]))
hh = steps[xi]
res = dict(branch_overlap_min=quality, second_order_branches=second_order,
matrix_eigenvalues=[float(np.real(np.vdot(c[:, n], h @ c[:, n]))) for n in (0, 1)],
slopes_finite_difference=[(-vals[(n, 2)] + 8 * vals[(n, 1)] - 8 * vals[(n, -1)] + vals[(n, -2)]) / (12 * hh)
for n in (0, 1)])
res["max_abs_deviation"] = float(max(abs(a - b) for a, b in zip(res["matrix_eigenvalues"], res["slopes_finite_difference"])))
if xi == "P":
res["second_derivative_11_24"] = [float(np.real(np.vdot(c[:, n], h2P @ c[:, n]))) for n in (0, 1)]
res["second_derivative_finite_difference"] = [
(-vals[(n, 2)] + 16 * vals[(n, 1)] - 30 * Eb + 16 * vals[(n, -1)] - vals[(n, -2)]) / (12 * hh * hh) for n in (0, 1)]
res["max_abs_deviation_second"] = float(max(abs(a - b) for a, b in zip(
res["second_derivative_11_24"], res["second_derivative_finite_difference"])))
# the term that (11.9) with Q_P = 1 - |psi><psi| would contain for the other vector of the level
res["term_of_11_9_from_the_level"] = dict(numerator=float(abs(np.vdot(Phi[:, 1], Veta @ Phi[:, 0])) ** 2),
denominator=float(E[pair[1]] - E[pair[0]]))
out["xi"][xi] = res
# identities for every eigenvector of the level
rng = np.random.default_rng(11)
worst = dict(euler=0.0, V12=0.0, mean_rule=0.0)
for _ in range(5):
cc = rng.normal(size=2) + 1j * rng.normal(size=2)
cc /= np.linalg.norm(cc)
psi = Phi @ cc
ex = lambda O: float(np.real(np.vdot(psi, O @ psi)))
worst["euler"] = max(worst["euler"], abs(p["M1"] * ex(ops["Z1"]) + p["M2"] * ex(ops["Z2"]) + ex(ops["A1"])
+ ex(ops["A2"]) + ex(ops["U"]) - Eb))
worst["V12"] = max(worst["V12"], abs(ex(ops["V1"]) - ex(ops["V2"])))
for xi in ("M1", "theta"):
wts = np.abs(bases[xi].conj().T @ cc) ** 2
worst["mean_rule"] = max(worst["mean_rule"], abs(ex(dH[xi]) - float(wts @ np.array(out["xi"][xi]["slopes_finite_difference"]))))
out["every_eigenvector"] = dict(worst, norm_Pi_V1_minus_V2_Pi=float(np.linalg.norm(Phi.conj().T @ ((ops["V1"] - ops["V2"]) @ Phi))))
# the branch vectors of different parameters
out["overlap_of_branch_0_of_M1_with_branch_0_of"] = {xi: float(abs(np.vdot(bases["M1"][:, 0], bases[xi][:, 0])) ** 2)
for xi in dH if xi != "M1"}
hU = Phi.conj().T @ (ops["U"] @ Phi)
out["U_in_the_branch_vectors_of_M1"] = [float(np.real(np.vdot(bases["M1"][:, n], hU @ bases["M1"][:, n]))) for n in (0, 1)]
return out
say("\n=== new in v2, I2: twofold bound levels (on-site binding W0 = -G(E0,P0)^(-1) on span(|11>,|22>))")
pX = dict(M1=1.0, M2=0.7, tau1=0.5, tau2=0.35, w=[0.0, 0.0, 0.0, 0.0])
deg = {}
for label, P0 in (("X_P0_0.3", 0.3), ("X_P0_0", 0.0)):
o = degenerate_check(pX, P0, 1.0)
deg[label] = o
bd = {k: (round(v[0], 4), round(v[1], 4)) for k, v in bands(P0, pX).items()}
say(f"set {label}: M = {pX['M1']}, {pX['M2']}; tau = {pX['tau1']}, {pX['tau2']}; P0 = {P0};"
f" W0 on (11,22) = {np.round(o['W0_block_11_22'], 6).tolist()}; bands {bd}")
say(f" level {o['level']:.12f}, splitting of the two eigenvalues {o['splitting']:.2e}, distance to the next eigenvalue {o['isolation']:.4f},"
f" weight at |r|>L/2 {o['tail']:.1e}")
for xi, r in o["xi"].items():
say(f" {xi:5s}: eigenvalues of Pi dH Pi {r['matrix_eigenvalues'][0]:+.9f}, {r['matrix_eigenvalues'][1]:+.9f};"
f" slopes of the branches (finite differences) {r['slopes_finite_difference'][0]:+.9f}, {r['slopes_finite_difference'][1]:+.9f};"
f" max dev {r['max_abs_deviation']:.1e}; overlap with the branch vectors >= {r['branch_overlap_min']:.6f}"
+ ("; branch vectors from the second order" if r["second_order_branches"] else ""))
if xi == "P":
say(f" E'' of the branches: (11.24) {r['second_derivative_11_24'][0]:+.8f}, {r['second_derivative_11_24'][1]:+.8f};"
f" finite differences {r['second_derivative_finite_difference'][0]:+.8f}, {r['second_derivative_finite_difference'][1]:+.8f};"
f" max dev {r['max_abs_deviation_second']:.1e}")
t = r["term_of_11_9_from_the_level"]
say(f" (11.9) with Q_P = 1 - |psi><psi|, term of the other vector of the level in the basis of the solver:"
f" |<psi'|V_eta|psi>|^2 = {t['numerator']:.3e} over E' - E = {t['denominator']:.3e}")
ev = o["every_eigenvector"]
say(f" every eigenvector (5 random vectors of the level): residual of (11.6) {ev['euler']:.1e}; <V1> - <V2> {ev['V12']:.1e};"
f" ||Pi (V1 - V2) Pi|| {ev['norm_Pi_V1_minus_V2_Pi']:.1e}; <dH> - weighted mean of the branch slopes (M1, theta) {ev['mean_rule']:.1e}")
say(" |<branch 0 of M1 | branch 0 of xi>|^2: " + ", ".join(f"{k} {v:.4f}" for k, v in o["overlap_of_branch_0_of_M1_with_branch_0_of"].items()))
say(f" <U> in the two branch vectors of M1: {o['U_in_the_branch_vectors_of_M1'][0]:+.6f}, {o['U_in_the_branch_vectors_of_M1'][1]:+.6f}"
f" against the slopes of the branches of theta: {o['xi']['theta']['slopes_finite_difference'][0]:+.6f}, {o['xi']['theta']['slopes_finite_difference'][1]:+.6f}")
new["I2_degenerate"] = deg
def listing_check():
"""I1: C_bod = <chi|C|chi> of (11.21) on a small system given in the places (two bodies with a branch
label on an open chain of 4 positions, a medium with 3 places)."""
rng = np.random.default_rng(5)
n, dc = 4, 3
hop = np.eye(n, k=-1)
def body(tau, M):
return 0.5 * tau * (np.kron(hop, -1j * sx) + np.kron(hop.T, 1j * sx)) + M * np.kron(np.eye(n), sz)
I8, Ic = np.eye(2 * n), np.eye(dc)
xh = np.kron(np.diag(np.arange(n, dtype=float)), np.eye(2))
C1, C2 = np.kron(body(0.6, 1.0), I8), np.kron(I8, body(0.45, 0.7))
x12 = np.kron(xh, I8) + np.kron(I8, xh)
U = np.zeros((64, 64))
wv = np.array([[-0.5, 0.2], [-0.15, 0.3]])
for x in range(n):
for s1 in range(2):
for s2 in range(2):
i = (2 * x + s1) * 8 + 2 * x + s2
U[i, i] = wv[s1, s2]
Cpair = C1 + C2 + U
Xr = rng.normal(size=(dc, dc)) + 1j * rng.normal(size=(dc, dc))
Xr = 0.5 * (Xr + Xr.conj().T)
chi = rng.normal(size=dc) + 1j * rng.normal(size=dc)
chi /= np.linalg.norm(chi)
mean = float(np.real(np.vdot(chi, Xr @ chi)))
Xc = Xr - mean * Ic # <chi|Xc|chi> = 0
phi = rng.normal(size=64) + 1j * rng.normal(size=64)
phi /= np.linalg.norm(phi)
alpha = np.kron(0.3 * np.kron(np.eye(n), sz) + 0.2 * np.eye(8), I8) # h*sigma3 of b_1 and a constant
beta = np.kron(I8, 0.25 * 0.5 * 0.45 * (np.kron(hop, -1j * sx) + np.kron(hop.T, 1j * sx)) - 0.1 * np.eye(8))
def bod(C):
return np.einsum("a,iajb,b->ij", chi.conj(), C.reshape(64, dc, 64, dc), chi)
def evolve(G, v, lam):
E, Y = np.linalg.eigh(G)
return Y @ (np.exp(-1j * E * lam) * (Y.conj().T @ v))
out, lam = dict(scan=[]), 3.0
for kappa in (0.04, 0.02, 0.01):
row = dict(kappa=kappa)
for tag, Xm in (("centred", Xc), ("not_centred", Xr)):
C = np.kron(Cpair, Ic) + kappa * np.kron(x12, Xm) # listing of the question
CB = np.kron(Cpair - alpha - beta, Ic) + (kappa * np.kron(x12, Xm) + np.kron(alpha + beta, Ic)) # another listing
Cbod = bod(C)
Psi = evolve(C, np.kron(phi, chi), lam).reshape(64, dc)
view = Psi @ Psi.conj().T
proj = lambda v: np.outer(v, v.conj())
row[tag] = dict(
total_contract_listing_B_minus_A=float(np.linalg.norm(CB - C)),
Cbod_listing_B_minus_A=float(np.linalg.norm(bod(CB) - Cbod)),
Cbod_minus_listed_Cpair_A=float(np.linalg.norm(Cbod - Cpair)),
Cbod_minus_listed_Cpair_A_minus_gradient=float(np.linalg.norm(
Cbod - Cpair - kappa * (mean if tag == "not_centred" else 0.0) * x12)),
Cbod_minus_listed_Cpair_B=float(np.linalg.norm(Cbod - (Cpair - alpha - beta))),
view_minus_evolution_by_Cbod=float(np.linalg.norm(view - proj(evolve(Cbod, phi, lam)))),
view_minus_evolution_by_listed_Cpair_A=float(np.linalg.norm(view - proj(evolve(Cpair, phi, lam)))),
view_minus_evolution_by_listed_Cpair_B=float(np.linalg.norm(view - proj(evolve(Cpair - alpha - beta, phi, lam)))))
out["scan"].append(row)
out["mean_of_X_not_centred"] = mean
return out
say("\n=== new in v2, I1: the operator C_bod = <chi|C|chi> of (11.21), small system in the places (dimension 64 x 3)")
lc = listing_check()
say(f"listing A: the terms of the question; listing B: h*sigma3 of b_1, a constant and a part of the motion term of b_2 listed"
f" in the record terms; <chi|X|chi> = {lc['mean_of_X_not_centred']:+.4f} for the medium that is not centred; lambda = 3")
for row in lc["scan"]:
for tag in ("centred", "not_centred"):
r = row[tag]
say(f"kappa={row['kappa']:.3f} {tag:11s}: ||C_B - C_A|| {r['total_contract_listing_B_minus_A']:.1e};"
f" ||C_bod(B) - C_bod(A)|| {r['Cbod_listing_B_minus_A']:.1e}; ||C_bod - C_pair(A)|| {r['Cbod_minus_listed_Cpair_A']:.3e}"
f" (minus the gradient: {r['Cbod_minus_listed_Cpair_A_minus_gradient']:.1e}); ||C_bod - C_pair(B)|| {r['Cbod_minus_listed_Cpair_B']:.3f};"
f" view against evolution by C_bod {r['view_minus_evolution_by_Cbod']:.3e}, by C_pair(A) {r['view_minus_evolution_by_listed_Cpair_A']:.3e},"
f" by C_pair(B) {r['view_minus_evolution_by_listed_Cpair_B']:.3f}")
say("expected for halving kappa: the view against the evolution by C_bod falls by 4 (second order);"
" by the listed C_pair(A) with a medium that is not centred it falls by 2")
new["I1_listing"] = lc
results["new_in_v2"] = new
# ---------------------------------------------------------------------------
# Part 3: checks of what is new in derivation v3
# ---------------------------------------------------------------------------
v3 = {}
NQ3 = 2 ** 15
def omega(M, tau, x):
return np.sqrt(M * M + tau * tau * np.sin(x) ** 2)
def lead(P, p):
"""Leading order in R, (11.12), relative to M_1+M_2: the level x, its distance iso from the band
and the band edge next to it (lower edge for w11 < 0, upper edge for w11 > 0)."""
a1, a2 = p["tau1"] ** 2 / (4 * p["M1"]), p["tau2"] ** 2 / (4 * p["M2"])
w = p["w"][0]
sg = np.sign(w)
j = np.sqrt(a1 * a1 + a2 * a2 + 2 * a1 * a2 * np.cos(2 * P))
root = np.sqrt(w * w + j * j)
return dict(a1=a1, a2=a2, a=a1 + a2, j=float(j), x=float(a1 + a2 + sg * root), iso=float(root - j),
edge=float(a1 + a2 + sg * j))
def edge_rel(P, p, upper=False):
"""Exact edge of the band B_++(P) minus (M_1+M_2): the minimum (upper: the maximum) over q of
(omega_1 - M_1) + (omega_2 - M_2), each difference formed without cancellation."""
sgn = -1.0 if upper else 1.0
def f(q):
s1, s2 = p["tau1"] * np.sin(0.5 * P + q), p["tau2"] * np.sin(0.5 * P - q)
return sgn * (s1 ** 2 / (np.sqrt(p["M1"] ** 2 + s1 ** 2) + p["M1"])
+ s2 ** 2 / (np.sqrt(p["M2"] ** 2 + s2 ** 2) + p["M2"]))
q = np.linspace(-np.pi, np.pi, 8001)
k = int(np.argmin(f(q)))
r = minimize_scalar(f, bounds=(q[max(k - 1, 0)], q[min(k + 1, 8000)]), method="bounded", options=dict(xatol=1e-13))
return sgn * min(float(r.fun), float(f(q[k])))
def F_plus_rel(x, P, p, nq=NQ3):
"""The factor F_+ of (11.10) at E = M_1+M_2+x on the infinite chain (mean over nq values of q, eta = 1/2).
The denominator of the (+,+) term is formed without cancellation:
omega_1+omega_2-E = (omega_1-M_1)+(omega_2-M_2)-x."""
q = -np.pi + 2 * np.pi * np.arange(nq) / nq
s1, s2 = p["tau1"] * np.sin(0.5 * P + q), p["tau2"] * np.sin(0.5 * P - q)
om1, om2 = np.sqrt(p["M1"] ** 2 + s1 ** 2), np.sqrt(p["M2"] ** 2 + s2 ** 2)
G11 = G22 = gx = 0.0
for sg1 in (1, -1):
for sg2 in (1, -1):
if sg1 == 1 and sg2 == 1:
den = 4 * (s1 ** 2 / (om1 + p["M1"]) + s2 ** 2 / (om2 + p["M2"]) - x)
else:
den = 4 * (sg1 * om1 + sg2 * om2 - p["M1"] - p["M2"] - x)
G11 += np.mean((1 + sg1 * p["M1"] / om1) * (1 + sg2 * p["M2"] / om2) / den)
G22 += np.mean((1 - sg1 * p["M1"] / om1) * (1 - sg2 * p["M2"] / om2) / den)
gx += np.mean(sg1 * sg2 * s1 * s2 / (om1 * om2 * den))
w11, _, _, w22 = p["w"]
return (1 + w11 * G11) * (1 + w22 * G22) - w11 * w22 * gx ** 2
def level_rel(P, p, nq=NQ3):
"""Bound level next to B_++(P) on the infinite chain, relative to M_1+M_2: the root x of F_+ below the
lower edge (w11 < 0) or above the upper edge (w11 > 0). Returns (x, exact edge, leading-order values)."""
up = p["w"][0] > 0
e, ld = edge_rel(P, p, upper=up), lead(P, p)
s = 1.0 if up else -1.0
lo, hi = sorted((e + s * 0.02 * ld["iso"], e + s * 30 * ld["iso"]))
f = lambda x: F_plus_rel(x, P, p, nq)
if f(lo) * f(hi) > 0:
return None, e, ld
return float(brentq(f, lo, hi, xtol=1e-25, rtol=1e-15, maxiter=300)), e, ld
def scaled(base, epsp, w11_over_a):
"""Data of a set with tau_i -> epsp*tau_i, w11 = w11_over_a * a and the other three w -> epsp^2 * w."""
p = with_(base, tau1=epsp * base["tau1"], tau2=epsp * base["tau2"])
a = p["tau1"] ** 2 / (4 * p["M1"]) + p["tau2"] ** 2 / (4 * p["M2"])
p["w"] = [w11_over_a * a] + [epsp ** 2 * x for x in base["w"][1:]]
return p, a
def wide_scan(base, epsp, sign, us, h=0.01):
"""I3, wide pair at fixed epsilon: the level on the infinite chain from the root of F_+ of (11.10)."""
rows = []
for u in us:
p, a = scaled(base, epsp, sign * u)
w = p["w"][0]
x0, e0, ld0 = level_rel(0.0, p)
row = dict(eps_factor=epsp, sign=sign, u=u, w11=w, a=a, found=x0 is not None)
if x0 is None:
rows.append(row)
continue
iso = abs(x0 - e0)
row.update(E_bind=x0, E_bind_leading=ld0["x"], isolation=iso, isolation_leading=ld0["iso"],
rel_dev_isolation=(iso - ld0["iso"]) / ld0["iso"], remainder=x0 - ld0["x"],
remainder_over_isolation=abs(x0 - ld0["x"]) / iso, rel_dev_E_bind=(x0 - ld0["x"]) / ld0["x"])
if sign < 0:
xs = {k: level_rel(k * h, p)[0] for k in (-2, -1, 1, 2)}
Epp = (-xs[2] + 16 * xs[1] - 30 * x0 + 16 * xs[-1] - xs[-2]) / (12 * h * h)
Epp_lead = 4 * ld0["a1"] * ld0["a2"] / np.sqrt(w * w + a * a)
m1, m2 = p["M1"] / p["tau1"] ** 2, p["M2"] / p["tau2"] ** 2
dm, fn = 1.0 / Epp - m1 - m2, (m1 + m2) * (np.sqrt(1 + u * u) - 1)
x1, e1, ld1 = level_rel(0.5 * np.pi, p)
row.update(wide_form=-w * w / (2 * a), rel_dev_wide_form=(x0 + w * w / (2 * a)) / x0,
Epp0=Epp, rel_dev_Epp0=float((Epp - Epp_lead) / Epp_lead),
mb_minus_m1_m2=dm, minus_4_m1_m2_E_bind=-4 * m1 * m2 * x0,
rel_dev_11_16=(dm + 4 * m1 * m2 * x0) / dm, leading_function=float(fn),
rel_dev_leading_function=float((dm - fn) / fn),
P_half_pi=dict(isolation=e1 - x1, isolation_leading=ld1["iso"], remainder=x1 - ld1["x"],
remainder_over_isolation=abs(x1 - ld1["x"]) / (e1 - x1),
closed_form_minus_exact_edge=ld1["x"] - e1))
else:
row.update(E_bind_minus_2a=x0 - 2 * a, w2_over_2a=w * w / (2 * a))
rows.append(row)
return rows
def tight_scan(base, epsp, us):
"""I3, tight pair at fixed epsilon: dense diagonalization at P = 0 against the leading-order forms."""
rows = []
for u in us:
p, a = scaled(base, epsp, -u)
w = p["w"][0]
o, c = solve_full(0.0, p, L), closed(0.0, p)
r = abs(w) / min(p["M1"], p["M2"])
X, Xl, Xt = o["E"] - p["M1"] - p["M2"] - w, float(c["Ebind"]) - w, a + a * a / (2 * w)
rows.append(dict(eps_factor=epsp, u=u, w11=w, w11_over_M=r,
in_a_band=any(lo <= o["E"] <= hi for lo, hi in bands(0.0, p).values()),
weight0=o["weight0"], tail=o["tail"], E_bind_minus_w11=[X, Xl, Xt],
rel_dev_leading=(X - Xl) / Xl, rel_dev_tight_form=(X - Xt) / X,
rel_dev_Epp0=float((o["Epp_formula"] - c["Epp0"]) / c["Epp0"]),
rel_dev_1_minus_s3=[float(((1 - o[k]) - (1 - c[k])) / (1 - c[k])) for k in ("s3_1", "s3_2")]))
return rows
say("\n=== new in v3, I3: wide pair at fixed epsilon (data of set B_unequal_R, w11 = -u*a or +u*a, the other three w fixed)")
say(f"level on the infinite chain: root of F_+ of (11.10), mean over {NQ3} values of q; x = E_b - (M_1+M_2);"
" E_b''(0) from 5-point differences of the roots, step 0.01")
say("cross-check of the root against the dense diagonalization (L = 120), eps' = 1:")
cross = []
for sign, u in ((-1, 1.0), (-1, 0.5), (1, 1.0), (1, 0.5)):
p, a = scaled(SETS["B_unequal_R"], 1.0, sign * u)
d = level_rel(0.0, p)[0] - (solve_full(0.0, p, L)["E"] - p["M1"] - p["M2"])
cross.append(dict(sign=sign, u=u, root_minus_dense=d))
say(f" sign {sign:+d}, u = {u}: root - dense = {d:+.2e}")
v3["I3_root_against_dense"] = cross
US_WIDE = (2.0, 1.0, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125)
wide = []
for epsp in (1.0, 0.5):
eps_ = epsp * max(SETS["B_unequal_R"]["tau1"] / SETS["B_unequal_R"]["M1"], SETS["B_unequal_R"]["tau2"] / SETS["B_unequal_R"]["M2"])
say(f"eps' = {epsp} (max tau_i/M_i = {eps_:.2f}), w11 < 0, level below the band:")
rows = wide_scan(SETS["B_unequal_R"], epsp, -1, US_WIDE)
for r in rows:
if not r["found"]:
say(f" u={r['u']:9.6f}: no root found")
continue
q = r["P_half_pi"]
say(f" u={r['u']:9.6f} eps^2/u^2={eps_ ** 2 / r['u'] ** 2:9.3e} P=0: isolation {r['isolation']:.4e}, remainder x-(11.12) {r['remainder']:+.3e}"
f" (over isolation {r['remainder_over_isolation']:.2e}), rel.dev E_bind {r['rel_dev_E_bind']:+.2e}, wide form {r['rel_dev_wide_form']:+.2e},"
f" E_b''(0) {r['rel_dev_Epp0']:+.2e}; m_b-(m_1+m_2) = {r['mb_minus_m1_m2']:.6e} against -4 m_1 m_2 E_bind (rel.dev {r['rel_dev_11_16']:+.2e})"
f" and the leading function (rel.dev {r['rel_dev_leading_function']:+.2e}); P=pi/2: isolation {q['isolation']:.4e},"
f" remainder {q['remainder']:+.3e} (over isolation {q['remainder_over_isolation']:.2e}), (11.12) minus the exact edge {q['closed_form_minus_exact_edge']:+.3e}")
wide += rows
say(f"eps' = {epsp}, w11 > 0, level above the band, P = 0:")
rows = wide_scan(SETS["B_unequal_R"], epsp, 1, US_WIDE)
for r in rows:
if not r["found"]:
say(f" u={r['u']:9.6f}: no root found")
continue
say(f" u={r['u']:9.6f} eps^2/u^2={eps_ ** 2 / r['u'] ** 2:9.3e} isolation {r['isolation']:.4e} (leading {r['isolation_leading']:.4e},"
f" rel.dev {r['rel_dev_isolation']:+.2e}), remainder x-(11.12) {r['remainder']:+.3e} (over isolation {r['remainder_over_isolation']:.2e}),"
f" E_bind-2a = {r['E_bind_minus_2a']:+.4e} against w11^2/(2a) = {r['w2_over_2a']:.4e}")
wide += rows
say("reading: (11.30) is sufficient, not necessary. Below the band at P = 0 the remainder stays far below the isolation for"
" every u (the lower edge at P = 0 is exactly M_1+M_2); at P = pi/2 and above the band it tends to a constant of order"
" eps^4 and exceeds the isolation for u of the order of eps")
v3["I3_wide"] = wide
say("\n=== new in v3, I3: tight pair at fixed epsilon (set B_unequal_R, w11 = -u*a), dense diagonalization, L = 120, P = 0")
tight = []
for epsp in (1.0, 0.5):
rows = tight_scan(SETS["B_unequal_R"], epsp, (2, 4, 8, 16, 32, 64, 128))
for r in rows:
say(f" eps'={epsp} u={r['u']:4d} w11={r['w11']:+.4f} |w11|/min M = {r['w11_over_M']:.4f} in a band: {r['in_a_band']}"
f" weight(r=0,11)={r['weight0']:.4f} E_bind-w11 = {r['E_bind_minus_w11'][0]:.6e}: rel.dev from the leading form {r['rel_dev_leading']:+.3e}"
f" (= {r['rel_dev_leading'] / r['w11_over_M']:+.3f} |w11|/M), from the tight form a+a^2/(2w) {r['rel_dev_tight_form']:+.3e};"
f" rel.dev E_b''(0) {r['rel_dev_Epp0']:+.3e} (= {r['rel_dev_Epp0'] / r['w11_over_M']:+.3f} |w11|/M);"
f" rel.dev 1-<sigma3_i> {r['rel_dev_1_minus_s3'][0]:+.3e}, {r['rel_dev_1_minus_s3'][1]:+.3e}")
tight += rows
say("reading: at fixed epsilon the relative remainders of the leading forms grow in proportion to |w11|/M; the forms fail for |w11| of order M")
v3["I3_tight"] = tight
def flat_member(M=1.0, tau=1.0, w=1.0, Lf=240):
"""I2: equal data and the on-site binding W0 = w |v><v| with v = (|11> - |22>)/sqrt(2) (not diagonal).
The bound levels outside the bands solve 1 + w <v|G(E,P)|v> = 0 (Step 6 with this W0)."""
p = dict(M1=M, M2=M, tau1=tau, tau2=tau, w=[0.0, 0.0, 0.0, 0.0])
v = np.array([1.0, 0.0, 0.0, -1.0]) / np.sqrt(2)
W0 = w * np.outer(v, v)
def Delta(P):
"""Upper end of the mixed bands [-Delta, Delta]: max over q of omega(P/2+q) - omega(P/2-q)."""
f = lambda q: -(omega(M, tau, 0.5 * P + q) - omega(M, tau, 0.5 * P - q))
q = np.linspace(-np.pi, np.pi, 8001)
k = int(np.argmin(f(q)))
r = minimize_scalar(f, bounds=(q[max(k - 1, 0)], q[min(k + 1, 8000)]), method="bounded", options=dict(xatol=1e-13))
return -min(float(r.fun), float(f(q[k])))
def gv(E, P, nq):
G, gx = green(E, P, p, nq=nq)
return 0.5 * (G[(1, 1)] + G[(2, 2)]) - gx
def root_above(P, nq):
D = Delta(P)
f = lambda E: 1 + w * gv(E, P, nq)
offs = D * np.logspace(-10, 0, 81)
vals = np.array([f(D + o) for o in offs])
k = np.where((vals[:-1] < 0) & (vals[1:] > 0))[0]
below = np.array([f(-D - o) for o in offs])
nb = int(np.sum(below[:-1] * below[1:] < 0))
if len(k) == 0:
return None, D, 0, nb
return float(brentq(f, D + offs[k[0]], D + offs[k[0] + 1], xtol=1e-16, rtol=1e-15)), D, int(len(k)), nb
out = dict(M=M, tau=tau, w=w, c_star=float(np.sqrt(M * M + tau * tau) - M), L=Lf, P=[])
wt = 0.0 # weight of v on the kernel of h_0(q) (the flat bands at P = 0)
for q in np.linspace(-np.pi, np.pi, 721):
e, x = np.linalg.eigh(-tau * np.sin(q) * SX1 + tau * np.sin(q) * SX2 + M * SZ1 + M * SZ2)
wt = max(wt, float(np.sum(np.abs(x[:, np.abs(e) < 1e-9].conj().T @ v) ** 2)))
out["max_weight_of_v_on_the_flat_bands_at_P0"] = wt
for P in (0.8, 0.4, 0.2, 0.1, 0.05, -0.2):
r1, D, n1, nb = root_above(P, 2 ** 15)
r2 = root_above(P, 2 ** 17)[0]
row = dict(P=P, Delta=D, sign_changes_above=n1, sign_changes_below=nb, root=r1, root_finer_q=r2)
if r1 is not None:
row.update(root_minus_Delta=r1 - D, root_minus_Delta_over_P3=(r1 - D) / abs(P) ** 3, root_over_P=r1 / abs(P))
out["P"].append(row)
f0 = lambda E: 1 + w * gv(E, 0.0, 2 ** 15)
out["P0_discrete_root"] = float(brentq(f0, -2 * M + 1e-6, -1e-6))
out["P0_equation_at_plus_minus_0.3"] = [float(f0(0.3)), float(f0(-0.3))]
rr = np.repeat(np.arange(-Lf, Lf + 1), 4)
vec0 = np.concatenate([np.zeros(4 * Lf), v, np.zeros(4 * Lf)])
out["dense"] = []
for P in (0.8, 0.4):
E, X, _ = dense(P, p, Lf, W0=W0)
root, D = root_above(P, 2 ** 15)[:2]
i = int(np.argmin(np.abs(E - root)))
out["dense"].append(dict(P=P, E=float(E[i]), E_minus_root=float(E[i] - root),
tail=float(np.sum(np.abs(X[np.abs(rr) > Lf // 2, i]) ** 2)),
weight_on_v_at_r0=float(abs(np.vdot(vec0, X[:, i])) ** 2),
next_below=float(E[i - 1]), next_above=float(E[i + 1]),
eigenvalues_between_Delta_and_1=int(np.sum((E > D + 1e-12) & (E < 1.0))),
eigenvalues_between_minus_1_and_minus_Delta=int(np.sum((E < -D - 1e-12) & (E > -1.0)))))
E, X, _ = dense(0.0, p, Lf, W0=W0)
zero = np.abs(E) < 1e-9
rest = E[~zero]
out["dense_P0"] = dict(dimension=int(len(E)), eigenvalues_at_zero=int(np.sum(zero)), two_times_sites=int(2 * (2 * Lf + 1)),
max_abs_of_them=float(np.max(np.abs(E[zero]))), nearest_other_eigenvalue=float(rest[np.argmin(np.abs(rest))]))
return out
say("\n=== new in v3, I2: equal data, on-site binding W0 = w |v><v|, v = (|11> - |22>)/sqrt(2): the flat point of the mixed bands")
fm = flat_member()
say(f"M_1 = M_2 = {fm['M']}, tau_1 = tau_2 = {fm['tau']}, w = {fm['w']}; c* = Omega - M = {fm['c_star']:.6f};"
f" largest weight of v on the flat bands at P = 0 over q: {fm['max_weight_of_v_on_the_flat_bands_at_P0']:.1e}")
for r in fm["P"]:
if r["root"] is None:
say(f" P={r['P']:+.2f}: mixed bands [-{r['Delta']:.10f}, {r['Delta']:.10f}]; no root above")
continue
say(f" P={r['P']:+.2f}: mixed bands [-Delta, Delta], Delta = {r['Delta']:.10f}; root of 1 + w g_v above them: E = {r['root']:.12f}"
f" (finer q grid: {r['root_finer_q']:.12f}); E - Delta = {r['root_minus_Delta']:.4e} = {r['root_minus_Delta_over_P3']:.4e} |P|^3;"
f" E/|P| = {r['root_over_P']:.6f}; sign changes above: {r['sign_changes_above']}, below -Delta: {r['sign_changes_below']}")
for r in fm["dense"]:
say(f" dense, L = {fm['L']}, P={r['P']:+.2f}: eigenvalue {r['E']:.12f}, minus the root {r['E_minus_root']:+.1e}; weight at |r|>L/2 {r['tail']:.1e};"
f" weight on (r=0, v) {r['weight_on_v_at_r0']:.4f}; neighbours {r['next_below']:.8f}, {r['next_above']:.8f};"
f" eigenvalues in (Delta, 1): {r['eigenvalues_between_Delta_and_1']}, in (-1, -Delta): {r['eigenvalues_between_minus_1_and_minus_Delta']}")
d0 = fm["dense_P0"]
say(f" dense, L = {fm['L']}, P=0: dimension {d0['dimension']}; eigenvalues with |E| < 1e-9: {d0['eigenvalues_at_zero']}"
f" (2 x number of r: {d0['two_times_sites']}), largest of them {d0['max_abs_of_them']:.1e}; nearest other eigenvalue"
f" {d0['nearest_other_eigenvalue']:+.8f}; root of 1 + w g_v(E,0): {fm['P0_discrete_root']:+.8f};"
f" 1 + w g_v(E,0) at E = +0.3, -0.3: {fm['P0_equation_at_plus_minus_0.3'][0]:.4f}, {fm['P0_equation_at_plus_minus_0.3'][1]:.4f}")
say("reading: for P != 0 one simple eigenvalue lies just above the mixed bands and tends to 0 as c*|P|; at P = 0 the value 0 is the"
" flat point (infinite multiplicity on the infinite chain), isolated from the rest of the spectrum")
v3["I2_flat_member"] = fm
results["new_in_v3"] = v3
say(f"\nrun time: {time.time() - T_START:.0f} s")
(OUT / "check_bound_pair.json").write_text(json.dumps(results, indent=1), encoding="utf-8")
(OUT / "check_bound_pair.txt").write_text("\n".join(lines) + "\n", encoding="utf-8")
output/check_bound_pair.json104 KB · raw
{
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"P0_discrete_root": -1.816160202538259,
"P0_equation_at_plus_minus_0.3": [
1.053944184776615,
0.9460558152233852
],
"dense": [
{
"P": 0.8,
"E": 0.3003429150020165,
"E_minus_root": -6.106226635438361e-16,
"tail": 2.5895460575642306e-08,
"weight_on_v_at_r0": 0.006282874936772755,
"next_below": 0.29713618685757637,
"next_above": 2.12475875363549,
"eigenvalues_between_Delta_and_1": 1,
"eigenvalues_between_minus_1_and_minus_Delta": 0
},
{
"P": 0.4,
"E": 0.16171255341103397,
"E_minus_root": -5.822511917052964e-11,
"tail": 0.00019208097907288073,
"weight_on_v_at_r0": 0.0008160833844589366,
"next_below": 0.16130089343034437,
"next_above": 2.037558991872855,
"eigenvalues_between_Delta_and_1": 1,
"eigenvalues_between_minus_1_and_minus_Delta": 0
}
],
"dense_P0": {
"dimension": 1924,
"eigenvalues_at_zero": 962,
"two_times_sites": 962,
"max_abs_of_them": 7.048522331707946e-15,
"nearest_other_eigenvalue": -1.8161602025381942
}
}
}
}output/check_bound_pair.txt48 KB · raw
range of r: -L..L with L=120 (open ends); convergence: comparison with L=240 and weight at |r|>L/2
finite differences: 5-point, step 0.01 in P, relative step 0.001 in M_i, tau_i, theta; convention eta=1/2
=== set A_equal_R: {'M1': 1.0, 'M2': 1.0, 'tau1': 0.1, 'tau2': 0.1, 'w': [-0.004, 0.003, -0.002, 0.006]} max tau_i/M_i = 0.100
P=-0.30 E_b=1.998771748883 isolation=1.452e-03 in a band: False weight(r=0,11)=0.6408 Gamma=+1.000000 E_b'=-1.133090e-03 <V1>=-1.133090e-03 <V2>=-1.133090e-03 E_b''=+3.517649e-03 (11.9)=+3.517649e-03
P=-0.15 E_b=1.998642641792 isolation=1.414e-03 in a band: False weight(r=0,11)=0.6278 Gamma=+1.000000 E_b'=-5.810445e-04 <V1>=-5.810445e-04 <V2>=-5.810445e-04 E_b''=+3.809572e-03 (11.9)=+3.809572e-03
P=+0.00 E_b=1.998598883527 isolation=1.401e-03 in a band: False weight(r=0,11)=0.6235 Gamma=+1.000000 E_b'=+1.850372e-15 <V1>=+7.893878e-21 <V2>=-2.053590e-20 E_b''=+3.905596e-03 (11.9)=+3.905596e-03
P=+0.15 E_b=1.998642641792 isolation=1.414e-03 in a band: False weight(r=0,11)=0.6278 Gamma=+1.000000 E_b'=+5.810445e-04 <V1>=+5.810445e-04 <V2>=+5.810445e-04 E_b''=+3.809572e-03 (11.9)=+3.809572e-03
P=+0.30 E_b=1.998771748883 isolation=1.452e-03 in a band: False weight(r=0,11)=0.6408 Gamma=+1.000000 E_b'=+1.133090e-03 <V1>=+1.133090e-03 <V2>=+1.133090e-03 E_b''=+3.517649e-03 (11.9)=+3.517649e-03
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 1.25e-12; tau_i dE/dtau_i - <A_i> 1.37e-12; dE/dtheta - <U> 8.96e-13; Euler residual 4.22e-15
2(b): E_b' - <V1> 1.59e-12; E_b' - <V2> 1.59e-12; <V1> - <V2> 6.51e-19
2(c): E_b'' (finite difference) - formula (11.9) 1.59e-11
3(a): coefficients of (11.10) against direct inversion (relative) 1.19e-14; root of (11.10) - E_b 1.55e-15
numerics: |E_b(L) - E_b(2L)| 2.00e-15; weight at |r|>L/2 3.17e-20; dependence on eta 2.44e-15; dense - banded 2.44e-15
closed forms (items 3, 4), regime R: |w11|/a = 0.800, (max tau/M)^2 = 1.000e-02
max_P |E_b - (11.12)| = 2.053e-06 = 0.021 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 2.354e-07
E_b''(0): numeric +3.90559613e-03 closed +3.90434405e-03 rel.dev 3.207e-04 = 0.032 * (tau/M)^2
s3_1 numeric +9.9945154586e-01 closed +9.9945217202e-01 diff -6.26e-07
s3_2 numeric +9.9945154586e-01 closed +9.9945217202e-01 diff -6.26e-07
A1 numeric +1.0948993397e-03 closed +1.0956559528e-03 diff -7.57e-07
A2 numeric +1.0948993397e-03 closed +1.0956559528e-03 diff -7.57e-07
U numeric -2.4940068754e-03 closed -2.4987801902e-03 diff +4.77e-06
Ebind numeric -1.4011164735e-03 closed -1.4031242374e-03 diff +2.01e-06
Epp0 numeric +3.9055961250e-03 closed +3.9043440472e-03 diff +1.25e-06
m_b/(m_1+m_2): numeric +1.280214 (11.16) +1.280625; m_b-(m_1+m_2): numeric +5.604286e+01 -4 m_1 m_2 E_bind +5.604466e+01
=== set B_unequal_R: {'M1': 1.0, 'M2': 1.7, 'tau1': 0.2, 'tau2': 0.12, 'w': [-0.01, 0.005, -0.004, 0.008]} max tau_i/M_i = 0.200
P=-0.30 E_b=2.696652071452 isolation=3.657e-03 in a band: False weight(r=0,11)=0.6441 Gamma=+1.000000 E_b'=-1.546134e-03 <V1>=-1.546134e-03 <V2>=-1.546134e-03 E_b''=+4.673251e-03 (11.9)=+4.673251e-03
P=-0.15 E_b=2.696475088077 isolation=3.603e-03 in a band: False weight(r=0,11)=0.6368 Gamma=+1.000000 E_b'=-8.001290e-04 <V1>=-8.001290e-04 <V2>=-8.001290e-04 E_b''=+5.213785e-03 (11.9)=+5.213785e-03
P=+0.00 E_b=2.696414739677 isolation=3.585e-03 in a band: False weight(r=0,11)=0.6344 Gamma=+1.000000 E_b'=+1.110223e-14 <V1>=-1.752348e-19 <V2>=+3.687192e-20 E_b''=+5.394417e-03 (11.9)=+5.394417e-03
P=+0.15 E_b=2.696475088077 isolation=3.603e-03 in a band: False weight(r=0,11)=0.6368 Gamma=+1.000000 E_b'=+8.001290e-04 <V1>=+8.001290e-04 <V2>=+8.001290e-04 E_b''=+5.213785e-03 (11.9)=+5.213785e-03
P=+0.30 E_b=2.696652071452 isolation=3.657e-03 in a band: False weight(r=0,11)=0.6441 Gamma=+1.000000 E_b'=+1.546134e-03 <V1>=+1.546134e-03 <V2>=+1.546134e-03 E_b''=+4.673251e-03 (11.9)=+4.673251e-03
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 1.49e-12; tau_i dE/dtau_i - <A_i> 1.56e-12; dE/dtheta - <U> 6.98e-13; Euler residual 4.44e-15
2(b): E_b' - <V1> 8.50e-13; E_b' - <V2> 8.50e-13; <V1> - <V2> 7.35e-17
2(c): E_b'' (finite difference) - formula (11.9) 1.83e-11
3(a): coefficients of (11.10) against direct inversion (relative) 4.28e-14; root of (11.10) - E_b 4.88e-15
numerics: |E_b(L) - E_b(2L)| 3.11e-15; weight at |r|>L/2 9.86e-21; dependence on eta 6.22e-15; dense - banded 3.55e-15
closed forms (items 3, 4), regime R: |w11|/a = 0.825, (max tau/M)^2 = 4.000e-02
max_P |E_b - (11.12)| = 8.267e-06 = 0.005 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 6.697e-07
E_b''(0): numeric +5.39441742e-03 closed +5.39148145e-03 rel.dev 5.446e-04 = 0.014 * (tau/M)^2
s3_1 numeric +9.9771238948e-01 closed +9.9771281374e-01 diff -4.24e-07
s3_2 numeric +9.9971332360e-01 closed +9.9971509098e-01 diff -1.77e-06
A1 numeric +4.5625732894e-03 closed +4.5743725191e-03 diff -1.18e-05
A2 numeric +9.7104322032e-04 closed +9.6869065111e-04 diff +2.35e-06
U numeric -6.3439164227e-03 closed -6.3649433780e-03 diff +2.10e-05
Ebind numeric -3.5852603230e-03 closed -3.5934117929e-03 diff +8.15e-06
Epp0 numeric +5.3944174247e-03 closed +5.3914814496e-03 diff +2.94e-06
m_b/(m_1+m_2): numeric +1.295838 (11.16) +1.296544; m_b-(m_1+m_2): numeric +4.232127e+01 -4 m_1 m_2 E_bind +4.232599e+01
=== set C_above_band_R: {'M1': 1.0, 'M2': 0.8, 'tau1': 0.1, 'tau2': 0.15, 'w': [0.02, -0.005, 0.003, -0.01]} max tau_i/M_i = 0.187
P=-0.30 E_b=1.831346531561 isolation=1.274e-02 in a band: False weight(r=0,11)=0.9037 Gamma=+1.000000 E_b'=+8.867732e-04 <V1>=+8.867732e-04 <V2>=+8.867732e-04 E_b''=-2.629234e-03 (11.9)=-2.629234e-03
P=-0.15 E_b=1.831448371991 isolation=1.260e-02 in a band: False weight(r=0,11)=0.8994 Gamma=+1.000000 E_b'=+4.619112e-04 <V1>=+4.619112e-04 <V2>=+4.619112e-04 E_b''=-2.996418e-03 (11.9)=-2.996418e-03
P=+0.00 E_b=1.831483249143 isolation=1.255e-02 in a band: False weight(r=0,11)=0.8980 Gamma=+1.000000 E_b'=+1.850372e-15 <V1>=-1.604183e-19 <V2>=+5.735736e-19 E_b''=-3.121011e-03 (11.9)=-3.121011e-03
P=+0.15 E_b=1.831448371991 isolation=1.260e-02 in a band: False weight(r=0,11)=0.8994 Gamma=+1.000000 E_b'=-4.619112e-04 <V1>=-4.619112e-04 <V2>=-4.619112e-04 E_b''=-2.996418e-03 (11.9)=-2.996418e-03
P=+0.30 E_b=1.831346531561 isolation=1.274e-02 in a band: False weight(r=0,11)=0.9037 Gamma=+1.000000 E_b'=-8.867732e-04 <V1>=-8.867732e-04 <V2>=-8.867732e-04 E_b''=-2.629234e-03 (11.9)=-2.629234e-03
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 1.04e-12; tau_i dE/dtau_i - <A_i> 1.30e-12; dE/dtheta - <U> 1.11e-12; Euler residual 1.78e-15
2(b): E_b' - <V1> 2.74e-12; E_b' - <V2> 2.74e-12; <V1> - <V2> 8.55e-17
2(c): E_b'' (finite difference) - formula (11.9) 1.26e-11
3(a): coefficients of (11.10) against direct inversion (relative) 4.97e-15; root of (11.10) - E_b 8.88e-16
numerics: |E_b(L) - E_b(2L)| 2.22e-16; weight at |r|>L/2 5.56e-28; dependence on eta 1.33e-15; dense - banded 4.44e-16
closed forms (items 3, 4), regime R: |w11|/a = 2.098, (max tau/M)^2 = 3.516e-02
max_P |E_b - (11.12)| = 2.030e-04 = 0.164 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 1.488e-05
E_b''(0): numeric -3.12101105e-03 closed -3.17366056e-03 rel.dev 1.659e-02 = 0.472 * (tau/M)^2
s3_1 numeric +9.9650760383e-01 closed +9.9642448170e-01 diff +8.31e-05
s3_2 numeric +9.8795706438e-01 closed +9.8742981847e-01 diff +5.27e-04
A1 numeric +7.0534612009e-03 closed +7.1510366031e-03 diff -9.76e-05
A2 numeric +1.9596946814e-02 closed +2.0112290446e-02 diff -5.15e-04
U numeric +1.7959585791e-02 closed +1.8054602308e-02 diff -9.50e-05
Ebind numeric +3.1483249143e-02 closed +3.1686265833e-02 diff -2.03e-04
Epp0 numeric -3.1210110472e-03 closed -3.1736605620e-03 diff +5.26e-05
m_b/(m_1+m_2): numeric -2.363673 (11.16) -2.324461; m_b-(m_1+m_2): numeric -4.559645e+02 -4 m_1 m_2 E_bind -4.477618e+02
=== set D_wide_R: {'M1': 1.3, 'M2': 1.0, 'tau1': 0.15, 'tau2': 0.2, 'w': [-0.004, 0.002, 0.001, 0.003]} max tau_i/M_i = 0.200
P=-0.30 E_b=2.299969840671 isolation=5.677e-04 in a band: False weight(r=0,11)=0.2780 Gamma=+1.000000 E_b'=-3.402251e-03 <V1>=-3.402251e-03 <V2>=-3.402251e-03 E_b''=+1.074662e-02 (11.9)=+1.074662e-02
P=-0.15 E_b=2.299583345287 isolation=5.522e-04 in a band: False weight(r=0,11)=0.2707 Gamma=+1.000000 E_b'=-1.734310e-03 <V1>=-1.734310e-03 <V2>=-1.734310e-03 E_b''=+1.141561e-02 (11.9)=+1.141561e-02
P=+0.00 E_b=2.299452860723 isolation=5.471e-04 in a band: False weight(r=0,11)=0.2684 Gamma=+1.000000 E_b'=+1.110223e-14 <V1>=-2.235218e-19 <V2>=+5.295572e-19 E_b''=+1.163513e-02 (11.9)=+1.163513e-02
P=+0.15 E_b=2.299583345287 isolation=5.522e-04 in a band: False weight(r=0,11)=0.2707 Gamma=+1.000000 E_b'=+1.734310e-03 <V1>=+1.734310e-03 <V2>=+1.734310e-03 E_b''=+1.141561e-02 (11.9)=+1.141561e-02
P=+0.30 E_b=2.299969840671 isolation=5.677e-04 in a band: False weight(r=0,11)=0.2780 Gamma=+1.000000 E_b'=+3.402251e-03 <V1>=+3.402251e-03 <V2>=+3.402251e-03 E_b''=+1.074662e-02 (11.9)=+1.074662e-02
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 2.05e-12; tau_i dE/dtau_i - <A_i> 1.96e-12; dE/dtheta - <U> 2.07e-12; Euler residual 5.33e-15
2(b): E_b' - <V1> 5.04e-12; E_b' - <V2> 5.04e-12; <V1> - <V2> 8.46e-17
2(c): E_b'' (finite difference) - formula (11.9) 9.92e-11
3(a): coefficients of (11.10) against direct inversion (relative) 2.34e-13; root of (11.10) - E_b 1.78e-15
numerics: |E_b(L) - E_b(2L)| 4.44e-15; weight at |r|>L/2 4.85e-08; dependence on eta 4.44e-15; dense - banded 2.22e-15
closed forms (items 3, 4), regime R: |w11|/a = 0.279, (max tau/M)^2 = 4.000e-02
max_P |E_b - (11.12)| = 7.729e-07 = 0.000 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 1.014e-06
E_b''(0): numeric +1.16351317e-02 closed +1.16355522e-02 rel.dev 3.615e-05 = 0.001 * (tau/M)^2
s3_1 numeric +9.9987712967e-01 closed +9.9987739882e-01 diff -2.69e-07
s3_2 numeric +9.9963191441e-01 closed +9.9963165158e-01 diff +2.63e-07
A1 numeric +3.1883906936e-04 closed +3.1876305792e-04 diff +7.60e-08
A2 numeric +7.3524924751e-04 closed +7.3669684496e-04 diff -1.45e-06
U numeric -1.0734105787e-03 closed -1.0756421628e-03 diff +2.23e-06
Ebind numeric -5.4713927724e-04 closed -5.4791221131e-04 diff +7.73e-07
Epp0 numeric +1.1635131656e-02 closed +1.1635552241e-02 diff -4.21e-07
m_b/(m_1+m_2): numeric +1.038281 (11.16) +1.038244; m_b-(m_1+m_2): numeric +3.168822e+00 -4 m_1 m_2 E_bind +3.161249e+00
=== set E_order_one: {'M1': 1.0, 'M2': 1.3, 'tau1': 1.0, 'tau2': 0.8, 'w': [-0.6, 0.2, -0.15, 0.3]} max tau_i/M_i = 1.000
P=-0.30 E_b=2.020243065365 isolation=2.944e-01 in a band: False weight(r=0,11)=0.7017 Gamma=+1.000000 E_b'=-5.330716e-02 <V1>=-5.330716e-02 <V2>=-5.330716e-02 E_b''=+1.580211e-01 (11.9)=+1.580211e-01
P=-0.15 E_b=2.014120790386 isolation=2.896e-01 in a band: False weight(r=0,11)=0.6956 Gamma=+1.000000 E_b'=-2.776997e-02 <V1>=-2.776997e-02 <V2>=-2.776997e-02 E_b''=+1.801270e-01 (11.9)=+1.801270e-01
P=+0.00 E_b=2.012023935088 isolation=2.880e-01 in a band: False weight(r=0,11)=0.6936 Gamma=+1.000000 E_b'=-7.401487e-15 <V1>=-3.140157e-17 <V2>=+1.914190e-17 E_b''=+1.876436e-01 (11.9)=+1.876436e-01
P=+0.15 E_b=2.014120790386 isolation=2.896e-01 in a band: False weight(r=0,11)=0.6956 Gamma=+1.000000 E_b'=+2.776997e-02 <V1>=+2.776997e-02 <V2>=+2.776997e-02 E_b''=+1.801270e-01 (11.9)=+1.801270e-01
P=+0.30 E_b=2.020243065365 isolation=2.944e-01 in a band: False weight(r=0,11)=0.7017 Gamma=+1.000000 E_b'=+5.330716e-02 <V1>=+5.330716e-02 <V2>=+5.330716e-02 E_b''=+1.580211e-01 (11.9)=+1.580211e-01
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 7.62e-13; tau_i dE/dtau_i - <A_i> 1.52e-12; dE/dtheta - <U> 1.15e-12; Euler residual 1.33e-15
2(b): E_b' - <V1> 1.74e-10; E_b' - <V2> 1.74e-10; <V1> - <V2> 3.54e-16
2(c): E_b'' (finite difference) - formula (11.9) 2.40e-10
3(a): coefficients of (11.10) against direct inversion (relative) 3.59e-15; root of (11.10) - E_b 1.33e-15
numerics: |E_b(L) - E_b(2L)| 8.88e-16; weight at |r|>L/2 2.26e-30; dependence on eta 8.88e-16; dense - banded 2.66e-15
closed forms (items 3, 4), control, outside R: |w11|/a = 1.608, (max tau/M)^2 = 1.000e+00
max_P |E_b - (11.12)| = 4.605e-02 = 0.046 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 3.589e-03
E_b''(0): numeric +1.87643618e-01 closed +1.74198841e-01 rel.dev 7.718e-02 = 0.077 * (tau/M)^2
s3_1 numeric +8.5646967127e-01 closed +8.8201005947e-01 diff -2.55e-02
s3_2 numeric +9.3859089119e-01 closed +9.5531741897e-01 diff -1.67e-02
A1 numeric +2.2543811170e-01 closed +2.3597988106e-01 diff -1.05e-02
A2 numeric +1.2495039252e-01 closed +1.1617471068e-01 diff +8.78e-03
U numeric -4.1500239894e-01 closed -5.0953161098e-01 diff +9.45e-02
Ebind numeric -2.8797606491e-01 closed -3.3345431511e-01 diff +4.55e-02
Epp0 numeric +1.8764361764e-01 closed +1.7419884136e-01 diff +1.34e-02
m_b/(m_1+m_2): numeric +1.758104 (11.16) +1.893795; m_b-(m_1+m_2): numeric +2.298001e+00 -4 m_1 m_2 E_bind +2.339806e+00
=== epsilon scan on set B_unequal_R (tau -> eps*tau, w -> eps^2*w)
eps= 1.000 E_b-(11.12): P=0 +8.1515e-06, P=0.3 +8.2675e-06; rel.dev E_b''(0) +5.4456e-04; <sigma3_i>-(11.14): -4.2426e-07, -1.7674e-06
eps= 0.500 E_b-(11.12): P=0 +5.0943e-07, P=0.3 +5.1674e-07; rel.dev E_b''(0) +1.3708e-04; <sigma3_i>-(11.14): -2.7203e-08, -1.1020e-07
ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 3.97, d_s3_1 15.60, d_s3_2 16.04
eps= 0.250 E_b-(11.12): P=0 +3.1839e-08, P=0.3 +3.2297e-08; rel.dev E_b''(0) +3.4328e-05; <sigma3_i>-(11.14): -1.7109e-09, -6.8832e-09
ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 3.99, d_s3_1 15.90, d_s3_2 16.01
eps= 0.125 E_b-(11.12): P=0 +1.9899e-09, P=0.3 +2.0186e-09; rel.dev E_b''(0) +8.5858e-06; <sigma3_i>-(11.14): -1.0710e-10, -4.3014e-10
ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 4.00, d_s3_1 15.97, d_s3_2 16.00
expected for halving eps: 16 for E_b and <sigma3_i> (next order eps^4), 4 for the relative deviation of E_b''(0)
=== new in v2, I3: reading statistics of the clock of a part, (11.19), (11.25), (11.26)
start: bound eigenvector of the level 0 on every level, clock label in |0>, total label P=0 sharp, no records;
readings at lambda = t*lambda_0, t = 0.25, 0.5, ..., 2N; deviation = max over a and t of |q - P_N(a; Gamma*t)|
set E_order_one, clock of part 1, N=4, spread D=0.0600 (D/g0=0.208, D/m=0.0600, lambda_0=78.54)
Gamma: level 0 (v1) 0.8564696713; mean level (11.19) 0.8621903823; first order (11.25) 0.8623337320; least-squares fit 0.8621456724
d2E_b/dM^2 at m: spectral sum +1.95468691e-01; closed form of (11.26) +1.72184621e-01; closed (11.20) at the mean level 0.8870173806; closed (11.26) 0.8871755981
deviation of the statistics: Gamma at the mean level 9.468e-04 (t<=N: 3.598e-04); first order 2.645e-03; Gamma of the level 0 7.098e-02 (t<=N: 3.790e-02)
set E_order_one, clock of part 1, N=4, spread D=0.0300 (D/g0=0.104, D/m=0.0300, lambda_0=157.08)
Gamma: level 0 (v1) 0.8564696713; mean level (11.19) 0.8593655981; first order (11.25) 0.8594017016; least-squares fit 0.8593541025
d2E_b/dM^2 at m: spectral sum +1.95468691e-01; closed form of (11.26) +1.72184621e-01; closed (11.20) at the mean level 0.8845527554; closed (11.26) 0.8845928288
deviation of the statistics: Gamma at the mean level 2.476e-04 (t<=N: 9.235e-05); first order 6.867e-04; Gamma of the level 0 3.606e-02 (t<=N: 1.923e-02)
ratios to the previous spread: dev_mean_level 3.82, dev_level0 1.97, dev_first_order 3.85
set E_order_one, clock of part 1, N=4, spread D=0.0150 (D/g0=0.052, D/m=0.0150, lambda_0=314.16)
Gamma: level 0 (v1) 0.8564696713; mean level (11.19) 0.8579266274; first order (11.25) 0.8579356864; least-squares fit 0.8579237667
d2E_b/dM^2 at m: spectral sum +1.95468691e-01; closed form of (11.26) +1.72184621e-01; closed (11.20) at the mean level 0.8832913598; closed (11.26) 0.8833014441
deviation of the statistics: Gamma at the mean level 6.077e-05 (t<=N: 2.322e-05); first order 1.707e-04; Gamma of the level 0 1.816e-02 (t<=N: 9.686e-03)
ratios to the previous spread: dev_mean_level 4.07, dev_level0 1.99, dev_first_order 4.02
set B_unequal_R, clock of part 1, N=4, spread D=0.1000 (D/g0=27.860, D/m=0.1000, lambda_0=47.12)
Gamma: level 0 (v1) 0.9977123895; mean level (11.19) 0.9978093968; first order (11.25) 0.9978119033; least-squares fit 0.9978328312
d2E_b/dM^2 at m: spectral sum +1.99027585e-03; closed form of (11.26) +1.99577457e-03; closed (11.20) at the mean level 0.9978100720; closed (11.26) 0.9978126025
deviation of the statistics: Gamma at the mean level 4.686e-04 (t<=N: 4.128e-04); first order 4.707e-04; Gamma of the level 0 1.426e-03 (t<=N: 9.115e-04)
set B_unequal_R, clock of part 1, N=4, spread D=0.0500 (D/g0=13.930, D/m=0.0500, lambda_0=94.25)
Gamma: level 0 (v1) 0.9977123895; mean level (11.19) 0.9977615160; first order (11.25) 0.9977621464; least-squares fit 0.9977620037
d2E_b/dM^2 at m: spectral sum +1.99027585e-03; closed form of (11.26) +1.99577457e-03; closed (11.20) at the mean level 0.9977620716; closed (11.26) 0.9977627081
deviation of the statistics: Gamma at the mean level 1.307e-04 (t<=N: 1.251e-04); first order 1.309e-04; Gamma of the level 0 5.874e-04 (t<=N: 3.517e-04)
ratios to the previous spread: dev_mean_level 3.59, dev_level0 2.43, dev_first_order 3.60
set B_unequal_R, clock of part 1, N=4, spread D=0.0250 (D/g0=6.965, D/m=0.0250, lambda_0=188.50)
Gamma: level 0 (v1) 0.9977123895; mean level (11.19) 0.9977371099; first order (11.25) 0.9977372679; least-squares fit 0.9977368502
d2E_b/dM^2 at m: spectral sum +1.99027585e-03; closed form of (11.26) +1.99577457e-03; closed (11.20) at the mean level 0.9977376013; closed (11.26) 0.9977377609
deviation of the statistics: Gamma at the mean level 3.259e-05 (t<=N: 3.259e-05); first order 3.261e-05; Gamma of the level 0 3.021e-04 (t<=N: 1.408e-04)
ratios to the previous spread: dev_mean_level 4.01, dev_level0 1.94, dev_first_order 4.01
set B_unequal_R, clock of part 2, N=5, spread D=0.1000 (D/g0=27.860, D/m=0.0588, lambda_0=50.27)
Gamma: level 0 (v1) 0.9997133236; mean level (11.19) 0.9997276451; first order (11.25) 0.9997282035; least-squares fit 0.9997277090
d2E_b/dM^2 at m: spectral sum +2.97598087e-04; closed form of (11.26) +2.95174829e-04; closed (11.20) at the mean level 0.9997292967; closed (11.26) 0.9997298497
deviation of the statistics: Gamma at the mean level 7.255e-06 (t<=N: 5.690e-06); first order 1.305e-05; Gamma of the level 0 2.199e-04 (t<=N: 1.083e-04)
set B_unequal_R, clock of part 2, N=5, spread D=0.0500 (D/g0=13.930, D/m=0.0294, lambda_0=100.53)
Gamma: level 0 (v1) 0.9997133236; mean level (11.19) 0.9997206216; first order (11.25) 0.9997207635; least-squares fit 0.9997205698
d2E_b/dM^2 at m: spectral sum +2.97598087e-04; closed form of (11.26) +2.95174829e-04; closed (11.20) at the mean level 0.9997223298; closed (11.26) 0.9997224704
deviation of the statistics: Gamma at the mean level 1.889e-06 (t<=N: 1.889e-06); first order 3.433e-06; Gamma of the level 0 1.114e-04 (t<=N: 5.357e-05)
ratios to the previous spread: dev_mean_level 3.84, dev_level0 1.97, dev_first_order 3.80
set B_unequal_R, clock of part 2, N=5, spread D=0.0250 (D/g0=6.965, D/m=0.0147, lambda_0=201.06)
Gamma: level 0 (v1) 0.9997133236; mean level (11.19) 0.9997170078; first order (11.25) 0.9997170436; least-squares fit 0.9997169943
d2E_b/dM^2 at m: spectral sum +2.97598087e-04; closed form of (11.26) +2.95174829e-04; closed (11.20) at the mean level 0.9997187452; closed (11.26) 0.9997187807
deviation of the statistics: Gamma at the mean level 4.854e-07 (t<=N: 4.854e-07); first order 8.020e-07; Gamma of the level 0 5.664e-05 (t<=N: 2.673e-05)
ratios to the previous spread: dev_mean_level 3.89, dev_level0 1.97, dev_first_order 4.28
expected for halving the spread at fixed lambda/lambda_0: 4 with Gamma at the mean level, 2 with Gamma of the level 0
=== new in v2, I4: order of the contribution of w22 (set B, tau -> eps*tau, w -> eps^2*w)
eps= 2.000 E_b(w22) - E_b(w22=0) = +2.867296e-07; w22*|<22|psi(0)>|^2 = +2.867347e-07; |<22|psi(0)>|^2 = 8.960461e-06
eps= 1.000 E_b(w22) - E_b(w22=0) = +4.444423e-09; w22*|<22|psi(0)>|^2 = +4.444423e-09; |<22|psi(0)>|^2 = 5.555528e-07; ratios to the previous eps: dE 64.51, weight 16.13
eps= 0.500 E_b(w22) - E_b(w22=0) = +6.930323e-11; w22*|<22|psi(0)>|^2 = +6.930409e-11; |<22|psi(0)>|^2 = 3.465205e-08; ratios to the previous eps: dE 64.13, weight 16.03
eps= 0.250 E_b(w22) - E_b(w22=0) = +1.080025e-12; w22*|<22|psi(0)>|^2 = +1.082329e-12; |<22|psi(0)>|^2 = 2.164659e-09; ratios to the previous eps: dE 64.17, weight 16.01
expected for halving eps: 64 for the contribution of w22 to E_b (order eps^6), 16 for the weight (order eps^4)
=== new in v2, I5: equal data (a_1 = a_2) at P = pi/2, where j(P) = 0 (set A, tau -> eps*tau, w -> eps^2*w)
eps= 1.000 E_b = 2.001000623470 M_1+M_2+a+w11 = 2.001000000000 difference +6.235e-07; 1 - weight(r=0,11) = 2.495e-03; weight of 11 at r != 0: 3.044e-07
eps= 0.500 E_b = 2.000250039039 M_1+M_2+a+w11 = 2.000250000000 difference +3.904e-08; 1 - weight(r=0,11) = 6.247e-04; weight of 11 at r != 0: 1.906e-08; ratios: dE 15.97, 1-weight 3.99
eps= 0.250 E_b = 2.000062502441 M_1+M_2+a+w11 = 2.000062500000 difference +2.441e-09; 1 - weight(r=0,11) = 1.562e-04; weight of 11 at r != 0: 1.192e-09; ratios: dE 15.99, 1-weight 4.00
expected: psi_P = delta_{r,0}|11> up to O(eps) (1 - weight of order eps^2), E_b = M_1+M_2+a+w11 up to O(eps^4)
=== new in v2, I2: twofold bound levels (on-site binding W0 = -G(E0,P0)^(-1) on span(|11>,|22>))
set X_P0_0.3: M = 1.0, 0.7; tau = 0.5, 0.35; P0 = 0.3; W0 on (11,22) = [[-0.870998, 0.169999], [0.169999, 2.488876]]; bands {'+1+1': (1.7046, 1.8965), '+1-1': (0.2832, 0.3515), '-1+1': (-0.3515, -0.2832), '-1-1': (-1.8965, -1.7046)}
level 1.000000000000, splitting of the two eigenvalues 2.11e-15, distance to the next eigenvalue 0.5368, weight at |r|>L/2 1.3e-30
M1 : eigenvalues of Pi dH Pi -0.628449130, +0.893076371; slopes of the branches (finite differences) -0.628449130, +0.893076371; max dev 1.8e-12; overlap with the branch vectors >= 0.999997
M2 : eigenvalues of Pi dH Pi -0.919383668, +0.837955656; slopes of the branches (finite differences) -0.919383668, +0.837955656; max dev 8.3e-13; overlap with the branch vectors >= 0.999999
tau1 : eigenvalues of Pi dH Pi +0.137013648, +0.601738268; slopes of the branches (finite differences) +0.137013648, +0.601738268; max dev 8.2e-12; overlap with the branch vectors >= 0.999999
tau2 : eigenvalues of Pi dH Pi -0.156239994, +0.533638710; slopes of the branches (finite differences) -0.156239994, +0.533638710; max dev 1.1e-12; overlap with the branch vectors >= 1.000000
theta: eigenvalues of Pi dH Pi -0.726064561, +2.016971420; slopes of the branches (finite differences) -0.726064561, +2.016971420; max dev 4.6e-12; overlap with the branch vectors >= 0.999996
P : eigenvalues of Pi dH Pi -0.031846375, +0.056264981; slopes of the branches (finite differences) -0.031846375, +0.056264981; max dev 2.9e-10; overlap with the branch vectors >= 0.999955
E'' of the branches: (11.24) -0.10636246, +0.17322408; finite differences -0.10636246, +0.17322408; max dev 1.3e-10
(11.9) with Q_P = 1 - |psi><psi|, term of the other vector of the level in the basis of the solver: |<psi'|V_eta|psi>|^2 = 6.549e-04 over E' - E = -2.109e-15
every eigenvector (5 random vectors of the level): residual of (11.6) 1.6e-15; <V1> - <V2> 1.1e-16; ||Pi (V1 - V2) Pi|| 1.6e-16; <dH> - weighted mean of the branch slopes (M1, theta) 4.1e-12
|<branch 0 of M1 | branch 0 of xi>|^2: M2 0.9912, tau1 0.2769, tau2 0.7272, theta 0.0013, P 0.5106
<U> in the two branch vectors of M1: +2.013542, -0.722635 against the slopes of the branches of theta: -0.726065, +2.016971
set X_P0_0: M = 1.0, 0.7; tau = 0.5, 0.35; P0 = 0.0; W0 on (11,22) = [[-0.869086, 0.177794], [0.177794, 2.488618]]; bands {'+1+1': (1.7, 1.9007), '+1-1': (0.3, 0.3354), '-1+1': (-0.3354, -0.3), '-1-1': (-1.9007, -1.7)}
level 1.000000000000, splitting of the two eigenvalues 3.55e-15, distance to the next eigenvalue 0.5397, weight at |r|>L/2 3.2e-30
M1 : eigenvalues of Pi dH Pi -0.631196110, +0.895473254; slopes of the branches (finite differences) -0.631196110, +0.895473254; max dev 1.8e-12; overlap with the branch vectors >= 0.999997
M2 : eigenvalues of Pi dH Pi -0.918901736, +0.845071613; slopes of the branches (finite differences) -0.918901736, +0.845071613; max dev 1.2e-12; overlap with the branch vectors >= 0.999999
tau1 : eigenvalues of Pi dH Pi +0.120869467, +0.605304949; slopes of the branches (finite differences) +0.120869467, +0.605304949; max dev 2.1e-12; overlap with the branch vectors >= 0.999999
tau2 : eigenvalues of Pi dH Pi -0.179004420, +0.538130054; slopes of the branches (finite differences) -0.179004420, +0.538130054; max dev 7.5e-13; overlap with the branch vectors >= 1.000000
theta: eigenvalues of Pi dH Pi -0.725682673, +2.024305435; slopes of the branches (finite differences) -0.725682673, +2.024305435; max dev 3.7e-12; overlap with the branch vectors >= 0.999996
P : eigenvalues of Pi dH Pi -0.000000000, +0.000000000; slopes of the branches (finite differences) +0.000000000, -0.000000000; max dev 1.8e-14; overlap with the branch vectors >= 0.999952; branch vectors from the second order
E'' of the branches: (11.24) -0.10630610, +0.19294341; finite differences -0.10630610, +0.19294341; max dev 4.1e-10
(11.9) with Q_P = 1 - |psi><psi|, term of the other vector of the level in the basis of the solver: |<psi'|V_eta|psi>|^2 = 9.382e-35 over E' - E = -3.553e-15
every eigenvector (5 random vectors of the level): residual of (11.6) 1.6e-15; <V1> - <V2> 3.5e-17; ||Pi (V1 - V2) Pi|| 6.4e-17; <dH> - weighted mean of the branch slopes (M1, theta) 2.9e-12
|<branch 0 of M1 | branch 0 of xi>|^2: M2 0.9904, tau1 0.2843, tau2 0.7171, theta 0.0014, P 0.5112
<U> in the two branch vectors of M1: +2.020503, -0.721880 against the slopes of the branches of theta: -0.725683, +2.024305
=== new in v2, I1: the operator C_bod = <chi|C|chi> of (11.21), small system in the places (dimension 64 x 3)
listing A: the terms of the question; listing B: h*sigma3 of b_1, a constant and a part of the motion term of b_2 listed in the record terms; <chi|X|chi> = -0.6021 for the medium that is not centred; lambda = 3
kappa=0.040 centred : ||C_B - C_A|| 1.3e-15; ||C_bod(B) - C_bod(A)|| 5.2e-16; ||C_bod - C_pair(A)|| 1.339e-15 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.589; view against evolution by C_bod 4.141e-02, by C_pair(A) 4.141e-02, by C_pair(B) 1.025
kappa=0.040 not_centred: ||C_B - C_A|| 1.5e-15; ||C_bod(B) - C_bod(A)|| 1.3e-15; ||C_bod - C_pair(A)|| 6.534e-01 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.491; view against evolution by C_bod 4.147e-02, by C_pair(A) 1.526e-01, by C_pair(B) 1.045
kappa=0.020 centred : ||C_B - C_A|| 1.3e-15; ||C_bod(B) - C_bod(A)|| 1.0e-15; ||C_bod - C_pair(A)|| 1.298e-15 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.589; view against evolution by C_bod 1.052e-02, by C_pair(A) 1.052e-02, by C_pair(B) 1.034
kappa=0.020 not_centred: ||C_B - C_A|| 1.4e-15; ||C_bod(B) - C_bod(A)|| 9.8e-16; ||C_bod - C_pair(A)|| 3.267e-01 (minus the gradient: 1.5e-15); ||C_bod - C_pair(B)|| 2.520; view against evolution by C_bod 1.053e-02, by C_pair(A) 7.442e-02, by C_pair(B) 1.044
kappa=0.010 centred : ||C_B - C_A|| 1.4e-15; ||C_bod(B) - C_bod(A)|| 8.3e-16; ||C_bod - C_pair(A)|| 1.295e-15 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.589; view against evolution by C_bod 2.642e-03, by C_pair(A) 2.642e-03, by C_pair(B) 1.037
kappa=0.010 not_centred: ||C_B - C_A|| 1.3e-15; ||C_bod(B) - C_bod(A)|| 1.2e-15; ||C_bod - C_pair(A)|| 1.633e-01 (minus the gradient: 1.8e-15); ||C_bod - C_pair(B)|| 2.549; view against evolution by C_bod 2.643e-03, by C_pair(A) 3.695e-02, by C_pair(B) 1.041
expected for halving kappa: the view against the evolution by C_bod falls by 4 (second order); by the listed C_pair(A) with a medium that is not centred it falls by 2
=== new in v3, I3: wide pair at fixed epsilon (data of set B_unequal_R, w11 = -u*a or +u*a, the other three w fixed)
level on the infinite chain: root of F_+ of (11.10), mean over 32768 values of q; x = E_b - (M_1+M_2); E_b''(0) from 5-point differences of the roots, step 0.01
cross-check of the root against the dense diagonalization (L = 120), eps' = 1:
sign -1, u = 1.0: root - dense = -3.00e-15
sign -1, u = 0.5: root - dense = +9.69e-16
sign +1, u = 1.0: root - dense = -8.28e-15
sign +1, u = 0.5: root - dense = -2.88e-15
eps' = 1.0 (max tau_i/M_i = 0.20), w11 < 0, level below the band:
u= 2.000000 eps^2/u^2=1.000e-02 P=0: isolation 1.4920e-02, remainder x-(11.12) +5.823e-05 (over isolation 3.90e-03), rel.dev E_bind -3.89e-03, wide form -6.24e-01, E_b''(0) +4.56e-03; m_b-(m_1+m_2) = 1.753737e+02 against -4 m_1 m_2 E_bind (rel.dev -4.36e-03) and the leading function (rel.dev -8.22e-03); P=pi/2: isolation 1.7532e-02, remainder +6.498e-05 (over isolation 3.71e-03), (11.12) minus the exact edge -1.760e-02
u= 1.000000 eps^2/u^2=4.000e-02 P=0: isolation 5.0061e-03, remainder x-(11.12) +1.319e-05 (over isolation 2.64e-03), rel.dev E_bind -2.63e-03, wide form -2.10e-01, E_b''(0) +9.48e-04; m_b-(m_1+m_2) = 5.906384e+01 against -4 m_1 m_2 E_bind (rel.dev -6.08e-04) and the leading function (rel.dev -3.24e-03); P=pi/2: isolation 6.5542e-03, remainder +1.397e-05 (over isolation 2.13e-03), (11.12) minus the exact edge -6.568e-03
u= 0.500000 eps^2/u^2=1.600e-01 P=0: isolation 1.4282e-03, remainder x-(11.12) +2.076e-06 (over isolation 1.45e-03), rel.dev E_bind -1.45e-03, wide form -6.06e-02, E_b''(0) +9.59e-05; m_b-(m_1+m_2) = 1.687008e+01 against -4 m_1 m_2 E_bind (rel.dev +5.44e-04) and the leading function (rel.dev -9.08e-04); P=pi/2: isolation 2.0551e-03, remainder -8.013e-07 (over isolation 3.90e-04), (11.12) minus the exact edge -2.054e-03
u= 0.250000 eps^2/u^2=6.400e-01 P=0: isolation 3.7267e-04, remainder x-(11.12) +2.709e-07 (over isolation 7.27e-04), rel.dev E_bind -7.27e-04, wide form -1.61e-02, E_b''(0) +1.33e-06; m_b-(m_1+m_2) = 4.402540e+00 against -4 m_1 m_2 E_bind (rel.dev +6.82e-04) and the leading function (rel.dev -4.46e-05); P=pi/2: isolation 5.6088e-04, remainder -4.044e-06 (over isolation 7.21e-03), (11.12) minus the exact edge -5.568e-04
u= 0.125000 eps^2/u^2=2.560e+00 P=0: isolation 9.4268e-05, remainder x-(11.12) +3.369e-08 (over isolation 3.57e-04), rel.dev E_bind -3.57e-04, wide form -4.25e-03, E_b''(0) -1.72e-06; m_b-(m_1+m_2) = 1.113538e+00 against -4 m_1 m_2 E_bind (rel.dev +5.80e-04) and the leading function (rel.dev +2.23e-04); P=pi/2: isolation 1.4386e-04, remainder -4.902e-06 (over isolation 3.41e-02), (11.12) minus the exact edge -1.390e-04
u= 0.062500 eps^2/u^2=1.024e+01 P=0: isolation 2.3640e-05, remainder x-(11.12) +4.171e-09 (over isolation 1.76e-04), rel.dev E_bind -1.76e-04, wide form -1.15e-03, E_b''(0) -5.85e-07; m_b-(m_1+m_2) = 2.792169e-01 against -4 m_1 m_2 E_bind (rel.dev +4.77e-04) and the leading function (rel.dev +3.00e-04); P=pi/2: isolation 3.6200e-05, remainder -5.162e-06 (over isolation 1.43e-01), (11.12) minus the exact edge -3.104e-05
u= 0.031250 eps^2/u^2=4.096e+01 P=0: isolation 5.9149e-06, remainder x-(11.12) +5.181e-10 (over isolation 8.76e-05), rel.dev E_bind -8.76e-05, wide form -3.32e-04, E_b''(0) -1.57e-07; m_b-(m_1+m_2) = 6.985683e-02 against -4 m_1 m_2 E_bind (rel.dev +4.10e-04) and the leading function (rel.dev +3.23e-04); P=pi/2: isolation 9.0639e-06, remainder -5.236e-06 (over isolation 5.78e-01), (11.12) minus the exact edge -3.828e-06
u= 0.015625 eps^2/u^2=1.638e+02 P=0: isolation 1.4791e-06, remainder x-(11.12) +6.452e-11 (over isolation 4.36e-05), rel.dev E_bind -4.36e-05, wide form -1.05e-04, E_b''(0) -4.00e-08; m_b-(m_1+m_2) = 1.746749e-02 against -4 m_1 m_2 E_bind (rel.dev +3.71e-04) and the leading function (rel.dev +3.27e-04); P=pi/2: isolation 2.2667e-06, remainder -5.256e-06 (over isolation 2.32e+00), (11.12) minus the exact edge +2.989e-06
u= 0.007812 eps^2/u^2=6.554e+02 P=0: isolation 3.6979e-07, remainder x-(11.12) +8.050e-12 (over isolation 2.18e-05), rel.dev E_bind -2.18e-05, wide form -3.70e-05, E_b''(0) -9.77e-09; m_b-(m_1+m_2) = 4.367040e-03 against -4 m_1 m_2 E_bind (rel.dev +3.42e-04) and the leading function (rel.dev +3.20e-04); P=pi/2: isolation 5.6672e-07, remainder -5.261e-06 (over isolation 9.28e+00), (11.12) minus the exact edge +4.694e-06
eps' = 1.0, w11 > 0, level above the band, P = 0:
u= 2.000000 eps^2/u^2=1.000e-02 isolation 1.4909e-02 (leading 1.4978e-02, rel.dev -4.62e-03), remainder x-(11.12) -2.706e-04 (over isolation 1.81e-02), E_bind-2a = +1.4708e-02 against w11^2/(2a) = 2.4235e-02
u= 1.000000 eps^2/u^2=4.000e-02 isolation 4.9971e-03 (leading 5.0193e-03, rel.dev -4.41e-03), remainder x-(11.12) -2.235e-04 (over isolation 4.47e-02), E_bind-2a = +4.7958e-03 against w11^2/(2a) = 6.0588e-03
u= 0.500000 eps^2/u^2=1.600e-01 isolation 1.4238e-03 (leading 1.4303e-03, rel.dev -4.52e-03), remainder x-(11.12) -2.078e-04 (over isolation 1.46e-01), E_bind-2a = +1.2225e-03 against w11^2/(2a) = 1.5147e-03
u= 0.250000 eps^2/u^2=6.400e-01 isolation 3.7115e-04 (leading 3.7294e-04, rel.dev -4.78e-03), remainder x-(11.12) -2.031e-04 (over isolation 5.47e-01), E_bind-2a = +1.6979e-04 against w11^2/(2a) = 3.7868e-04
u= 0.125000 eps^2/u^2=2.560e+00 isolation 9.3834e-05 (leading 9.4302e-05, rel.dev -4.96e-03), remainder x-(11.12) -2.018e-04 (over isolation 2.15e+00), E_bind-2a = -1.0753e-04 against w11^2/(2a) = 9.4669e-05
u= 0.062500 eps^2/u^2=1.024e+01 isolation 2.3526e-05 (leading 2.3644e-05, rel.dev -4.98e-03), remainder x-(11.12) -2.015e-04 (over isolation 8.56e+00), E_bind-2a = -1.7783e-04 against w11^2/(2a) = 2.3667e-05
u= 0.031250 eps^2/u^2=4.096e+01 isolation 5.8869e-06 (leading 5.9154e-06, rel.dev -4.81e-03), remainder x-(11.12) -2.014e-04 (over isolation 3.42e+01), E_bind-2a = -1.9547e-04 against w11^2/(2a) = 5.9168e-06
u= 0.015625 eps^2/u^2=1.638e+02 isolation 1.4727e-06 (leading 1.4791e-06, rel.dev -4.33e-03), remainder x-(11.12) -2.014e-04 (over isolation 1.37e+02), E_bind-2a = -1.9989e-04 against w11^2/(2a) = 1.4792e-06
u= 0.007812 eps^2/u^2=6.554e+02 isolation 3.6856e-07 (leading 3.6980e-07, rel.dev -3.33e-03), remainder x-(11.12) -2.014e-04 (over isolation 5.46e+02), E_bind-2a = -2.0099e-04 against w11^2/(2a) = 3.6980e-07
eps' = 0.5 (max tau_i/M_i = 0.10), w11 < 0, level below the band:
u= 2.000000 eps^2/u^2=2.500e-03 P=0: isolation 3.7409e-03, remainder x-(11.12) +3.631e-06 (over isolation 9.71e-04), rel.dev E_bind -9.70e-04, wide form -6.20e-01, E_b''(0) +1.15e-03; m_b-(m_1+m_2) = 7.058394e+02 against -4 m_1 m_2 E_bind (rel.dev -1.11e-03) and the leading function (rel.dev -2.07e-03); P=pi/2: isolation 4.3963e-03, remainder +4.056e-06 (over isolation 9.23e-04), (11.12) minus the exact edge -4.400e-03
u= 1.000000 eps^2/u^2=1.000e-02 P=0: isolation 1.2540e-03, remainder x-(11.12) +8.245e-07 (over isolation 6.58e-04), rel.dev E_bind -6.57e-04, wide form -2.08e-01, E_b''(0) +2.39e-04; m_b-(m_1+m_2) = 2.368291e+02 against -4 m_1 m_2 E_bind (rel.dev -1.58e-04) and the leading function (rel.dev -8.15e-04); P=pi/2: isolation 1.6421e-03, remainder +8.743e-07 (over isolation 5.32e-04), (11.12) minus the exact edge -1.643e-03
u= 0.500000 eps^2/u^2=4.000e-02 P=0: isolation 3.5744e-04, remainder x-(11.12) +1.297e-07 (over isolation 3.63e-04), rel.dev E_bind -3.63e-04, wide form -5.94e-02, E_b''(0) +2.42e-05; m_b-(m_1+m_2) = 6.752618e+01 against -4 m_1 m_2 E_bind (rel.dev +1.33e-04) and the leading function (rel.dev -2.29e-04); P=pi/2: isolation 5.1460e-04, remainder -5.014e-08 (over isolation 9.74e-05), (11.12) minus the exact edge -5.146e-04
u= 0.250000 eps^2/u^2=1.600e-01 P=0: isolation 9.3217e-05, remainder x-(11.12) +1.692e-08 (over isolation 1.82e-04), rel.dev E_bind -1.82e-04, wide form -1.56e-02, E_b''(0) +3.82e-07; m_b-(m_1+m_2) = 1.761072e+01 against -4 m_1 m_2 E_bind (rel.dev +1.69e-04) and the leading function (rel.dev -1.28e-05); P=pi/2: isolation 1.4045e-04, remainder -2.532e-07 (over isolation 1.80e-03), (11.12) minus the exact edge -1.402e-04
u= 0.125000 eps^2/u^2=6.400e-01 P=0: isolation 2.3573e-05, remainder x-(11.12) +2.104e-09 (over isolation 8.92e-05), rel.dev E_bind -8.92e-05, wide form -3.98e-03, E_b''(0) -4.18e-07; m_b-(m_1+m_2) = 4.453399e+00 against -4 m_1 m_2 E_bind (rel.dev +1.43e-04) and the leading function (rel.dev +5.41e-05); P=pi/2: isolation 3.6032e-05, remainder -3.069e-07 (over isolation 8.52e-03), (11.12) minus the exact edge -3.572e-05
u= 0.062500 eps^2/u^2=2.560e+00 P=0: isolation 5.9108e-06, remainder x-(11.12) +2.604e-10 (over isolation 4.41e-05), rel.dev E_bind -4.41e-05, wide form -1.02e-03, E_b''(0) -1.43e-07; m_b-(m_1+m_2) = 1.116614e+00 against -4 m_1 m_2 E_bind (rel.dev +1.17e-04) and the leading function (rel.dev +7.34e-05); P=pi/2: isolation 9.0688e-06, remainder -3.232e-07 (over isolation 3.56e-02), (11.12) minus the exact edge -8.746e-06
u= 0.031250 eps^2/u^2=1.024e+01 P=0: isolation 1.4788e-06, remainder x-(11.12) +3.234e-11 (over isolation 2.19e-05), rel.dev E_bind -2.19e-05, wide form -2.66e-04, E_b''(0) -3.83e-08; m_b-(m_1+m_2) = 2.793591e-01 against -4 m_1 m_2 E_bind (rel.dev +1.00e-04) and the leading function (rel.dev +7.85e-05); P=pi/2: isolation 2.2710e-06, remainder -3.278e-07 (over isolation 1.44e-01), (11.12) minus the exact edge -1.943e-06
u= 0.015625 eps^2/u^2=4.096e+01 P=0: isolation 3.6977e-07, remainder x-(11.12) +4.027e-12 (over isolation 1.09e-05), rel.dev E_bind -1.09e-05, wide form -7.19e-05, E_b''(0) -9.50e-09; m_b-(m_1+m_2) = 6.985252e-02 against -4 m_1 m_2 E_bind (rel.dev +8.87e-05) and the leading function (rel.dev +7.78e-05); P=pi/2: isolation 5.6798e-07, remainder -3.291e-07 (over isolation 5.79e-01), (11.12) minus the exact edge -2.389e-07
u= 0.007812 eps^2/u^2=1.638e+02 P=0: isolation 9.2448e-08, remainder x-(11.12) +5.024e-13 (over isolation 5.43e-06), rel.dev E_bind -5.43e-06, wide form -2.07e-05, E_b''(0) -2.09e-09; m_b-(m_1+m_2) = 1.746377e-02 against -4 m_1 m_2 E_bind (rel.dev +7.40e-05) and the leading function (rel.dev +6.86e-05); P=pi/2: isolation 1.4201e-07, remainder -3.294e-07 (over isolation 2.32e+00), (11.12) minus the exact edge +1.874e-07
eps' = 0.5, w11 > 0, level above the band, P = 0:
u= 2.000000 eps^2/u^2=2.500e-03 isolation 3.7401e-03 (leading 3.7446e-03, rel.dev -1.19e-03), remainder x-(11.12) -1.723e-05 (over isolation 4.61e-03), E_bind-2a = +3.7273e-03 against w11^2/(2a) = 6.0588e-03
u= 1.000000 eps^2/u^2=1.000e-02 isolation 1.2534e-03 (leading 1.2548e-03, rel.dev -1.14e-03), remainder x-(11.12) -1.419e-05 (over isolation 1.13e-02), E_bind-2a = +1.2406e-03 against w11^2/(2a) = 1.5147e-03
u= 0.500000 eps^2/u^2=4.000e-02 isolation 3.5716e-04 (leading 3.5757e-04, rel.dev -1.16e-03), remainder x-(11.12) -1.318e-05 (over isolation 3.69e-02), E_bind-2a = +3.4439e-04 against w11^2/(2a) = 3.7868e-04
u= 0.250000 eps^2/u^2=1.600e-01 isolation 9.3119e-05 (leading 9.3234e-05, rel.dev -1.24e-03), remainder x-(11.12) -1.288e-05 (over isolation 1.38e-01), E_bind-2a = +8.0352e-05 against w11^2/(2a) = 9.4669e-05
u= 0.125000 eps^2/u^2=6.400e-01 isolation 2.3545e-05 (leading 2.3576e-05, rel.dev -1.29e-03), remainder x-(11.12) -1.280e-05 (over isolation 5.44e-01), E_bind-2a = +1.0778e-05 against w11^2/(2a) = 2.3667e-05
u= 0.062500 eps^2/u^2=2.560e+00 isolation 5.9032e-06 (leading 5.9111e-06, rel.dev -1.32e-03), remainder x-(11.12) -1.278e-05 (over isolation 2.16e+00), E_bind-2a = -6.8642e-06 against w11^2/(2a) = 5.9168e-06
u= 0.031250 eps^2/u^2=1.024e+01 isolation 1.4769e-06 (leading 1.4788e-06, rel.dev -1.32e-03), remainder x-(11.12) -1.277e-05 (over isolation 8.65e+00), E_bind-2a = -1.1291e-05 against w11^2/(2a) = 1.4792e-06
u= 0.015625 eps^2/u^2=4.096e+01 isolation 3.6930e-07 (leading 3.6978e-07, rel.dev -1.30e-03), remainder x-(11.12) -1.277e-05 (over isolation 3.46e+01), E_bind-2a = -1.2398e-05 against w11^2/(2a) = 3.6980e-07
u= 0.007812 eps^2/u^2=1.638e+02 isolation 9.2334e-08 (leading 9.2449e-08, rel.dev -1.24e-03), remainder x-(11.12) -1.277e-05 (over isolation 1.38e+02), E_bind-2a = -1.2675e-05 against w11^2/(2a) = 9.2450e-08
reading: (11.30) is sufficient, not necessary. Below the band at P = 0 the remainder stays far below the isolation for every u (the lower edge at P = 0 is exactly M_1+M_2); at P = pi/2 and above the band it tends to a constant of order eps^4 and exceeds the isolation for u of the order of eps
=== new in v3, I3: tight pair at fixed epsilon (set B_unequal_R, w11 = -u*a), dense diagonalization, L = 120, P = 0
eps'=1.0 u= 2 w11=-0.0242 |w11|/min M = 0.0242 in a band: False weight(r=0,11)=0.8900 E_bind-w11 = 9.315293e-03: rel.dev from the leading form +6.291e-03 (= +0.260 |w11|/M), from the tight form a+a^2/(2w) +2.437e-02; rel.dev E_b''(0) +4.562e-03 (= +0.188 |w11|/M); rel.dev 1-<sigma3_i> +9.284e-03, +1.328e-02
eps'=1.0 u= 4 w11=-0.0485 |w11|/min M = 0.0485 in a band: False weight(r=0,11)=0.9647 E_bind-w11 = 1.080688e-02: rel.dev from the leading form +1.703e-02 (= +0.351 |w11|/M), from the tight form a+a^2/(2w) +1.887e-02; rel.dev E_b''(0) +1.371e-02 (= +0.283 |w11|/M); rel.dev 1-<sigma3_i> +3.179e-02, +2.730e-02
eps'=1.0 u= 8 w11=-0.0969 |w11|/min M = 0.0969 in a band: False weight(r=0,11)=0.9863 E_bind-w11 = 1.182203e-02: rel.dev from the leading form +4.038e-02 (= +0.416 |w11|/M), from the tight form a+a^2/(2w) +3.906e-02; rel.dev E_b''(0) +3.300e-02 (= +0.340 |w11|/M); rel.dev 1-<sigma3_i> +8.288e-02, +5.724e-02
eps'=1.0 u= 16 w11=-0.1939 |w11|/min M = 0.1939 in a band: False weight(r=0,11)=0.9914 E_bind-w11 = 1.280926e-02: rel.dev from the leading form +9.114e-02 (= +0.470 |w11|/M), from the tight form a+a^2/(2w) +8.356e-02; rel.dev E_b''(0) +7.258e-02 (= +0.374 |w11|/M); rel.dev 1-<sigma3_i> +1.992e-01, +1.216e-01
eps'=1.0 u= 32 w11=-0.3878 |w11|/min M = 0.3878 in a band: False weight(r=0,11)=0.9913 E_bind-w11 = 1.443301e-02: rel.dev from the leading form +2.100e-01 (= +0.542 |w11|/M), from the tight form a+a^2/(2w) +1.735e-01; rel.dev E_b''(0) +1.534e-01 (= +0.396 |w11|/M); rel.dev 1-<sigma3_i> +4.967e-01, +2.691e-01
eps'=1.0 u= 64 w11=-0.7755 |w11|/min M = 0.7755 in a band: False weight(r=0,11)=0.9862 E_bind-w11 = 1.864784e-02: rel.dev from the leading form +5.510e-01 (= +0.711 |w11|/M), from the tight form a+a^2/(2w) +3.553e-01; rel.dev E_b''(0) +2.896e-01 (= +0.373 |w11|/M); rel.dev 1-<sigma3_i> +1.547e+00, +6.647e-01
eps'=1.0 u= 128 w11=-1.5511 |w11|/min M = 1.5511 in a band: False weight(r=0,11)=0.9228 E_bind-w11 = 4.408762e-02: rel.dev from the leading form +2.653e+00 (= +1.710 |w11|/M), from the tight form a+a^2/(2w) +7.262e-01; rel.dev E_b''(0) -2.101e+00 (= -1.355 |w11|/M); rel.dev 1-<sigma3_i> +1.410e+01, +2.168e+00
eps'=0.5 u= 2 w11=-0.0061 |w11|/min M = 0.0061 in a band: False weight(r=0,11)=0.8933 E_bind-w11 = 2.317895e-03: rel.dev from the leading form +1.569e-03 (= +0.259 |w11|/M), from the tight form a+a^2/(2w) +1.978e-02; rel.dev E_b''(0) +1.147e-03 (= +0.189 |w11|/M); rel.dev 1-<sigma3_i> +2.322e-03, +3.302e-03
eps'=0.5 u= 4 w11=-0.0121 |w11|/min M = 0.0121 in a band: False weight(r=0,11)=0.9688 E_bind-w11 = 2.667675e-03: rel.dev from the leading form +4.216e-03 (= +0.348 |w11|/M), from the tight form a+a^2/(2w) +6.350e-03; rel.dev E_b''(0) +3.438e-03 (= +0.284 |w11|/M); rel.dev 1-<sigma3_i> +7.853e-03, +6.741e-03
eps'=0.5 u= 8 w11=-0.0242 |w11|/min M = 0.0242 in a band: False weight(r=0,11)=0.9909 E_bind-w11 = 2.868720e-03: rel.dev from the leading form +9.826e-03 (= +0.405 |w11|/M), from the tight form a+a^2/(2w) +9.986e-03; rel.dev E_b''(0) +8.233e-03 (= +0.340 |w11|/M); rel.dev 1-<sigma3_i> +1.993e-02, +1.391e-02
eps'=0.5 u= 16 w11=-0.0485 |w11|/min M = 0.0485 in a band: False weight(r=0,11)=0.9966 E_bind-w11 = 2.997571e-03: rel.dev from the leading form +2.138e-02 (= +0.441 |w11|/M), from the tight form a+a^2/(2w) +2.096e-02; rel.dev E_b''(0) +1.794e-02 (= +0.370 |w11|/M); rel.dev 1-<sigma3_i> +4.517e-02, +2.858e-02
eps'=0.5 u= 32 w11=-0.0969 |w11|/min M = 0.0969 in a band: False weight(r=0,11)=0.9980 E_bind-w11 = 3.117421e-03: rel.dev from the leading form +4.538e-02 (= +0.468 |w11|/M), from the tight form a+a^2/(2w) +4.342e-02; rel.dev E_b''(0) +3.759e-02 (= +0.388 |w11|/M); rel.dev 1-<sigma3_i> +9.876e-02, +5.893e-02
eps'=0.5 u= 64 w11=-0.1939 |w11|/min M = 0.1939 in a band: False weight(r=0,11)=0.9982 E_bind-w11 = 3.297572e-03: rel.dev from the leading form +9.709e-02 (= +0.501 |w11|/M), from the tight form a+a^2/(2w) +8.850e-02; rel.dev E_b''(0) +7.771e-02 (= +0.401 |w11|/M); rel.dev 1-<sigma3_i> +2.192e-01, +1.238e-01
eps'=0.5 u= 128 w11=-0.3878 |w11|/min M = 0.3878 in a band: False weight(r=0,11)=0.9979 E_bind-w11 = 3.675900e-03: rel.dev from the leading form +2.182e-01 (= +0.563 |w11|/M), from the tight form a+a^2/(2w) +1.791e-01; rel.dev E_b''(0) +1.596e-01 (= +0.412 |w11|/M); rel.dev 1-<sigma3_i> +5.280e-01, +2.728e-01
reading: at fixed epsilon the relative remainders of the leading forms grow in proportion to |w11|/M; the forms fail for |w11| of order M
=== new in v3, I2: equal data, on-site binding W0 = w |v><v|, v = (|11> - |22>)/sqrt(2): the flat point of the mixed bands
M_1 = M_2 = 1.0, tau_1 = tau_2 = 1.0, w = 1.0; c* = Omega - M = 0.414214; largest weight of v on the flat bands at P = 0 over q: 7.7e-32
P=+0.80: mixed bands [-Delta, Delta], Delta = 0.2971386219; root of 1 + w g_v above them: E = 0.300342915002 (finer q grid: 0.300342915002); E - Delta = 3.2043e-03 = 6.2584e-03 |P|^3; E/|P| = 0.375429; sign changes above: 1, below -Delta: 0
P=+0.40: mixed bands [-Delta, Delta], Delta = 0.1613023588; root of 1 + w g_v above them: E = 0.161712553469 (finer q grid: 0.161712553469); E - Delta = 4.1019e-04 = 6.4093e-03 |P|^3; E/|P| = 0.404281; sign changes above: 1, below -Delta: 0
P=+0.20: mixed bands [-Delta, Delta], Delta = 0.0822915312; root of 1 + w g_v above them: E = 0.082343177123 (finer q grid: 0.082343177123); E - Delta = 5.1646e-05 = 6.4557e-03 |P|^3; E/|P| = 0.411716; sign changes above: 1, below -Delta: 0
P=+0.10: mixed bands [-Delta, Delta], Delta = 0.0413523552; root of 1 + w g_v above them: E = 0.041358823110 (finer q grid: 0.041358823110); E - Delta = 6.4680e-06 = 6.4680e-03 |P|^3; E/|P| = 0.413588; sign changes above: 1, below -Delta: 0
P=+0.05: mixed bands [-Delta, Delta], Delta = 0.0207020497; root of 1 + w g_v above them: E = 0.020702858630 (finer q grid: 0.020702858630); E - Delta = 8.0888e-07 = 6.4711e-03 |P|^3; E/|P| = 0.414057; sign changes above: 1, below -Delta: 0
P=-0.20: mixed bands [-Delta, Delta], Delta = 0.0822915312; root of 1 + w g_v above them: E = 0.082343177123 (finer q grid: 0.082343177123); E - Delta = 5.1646e-05 = 6.4557e-03 |P|^3; E/|P| = 0.411716; sign changes above: 1, below -Delta: 0
dense, L = 240, P=+0.80: eigenvalue 0.300342915002, minus the root -6.1e-16; weight at |r|>L/2 2.6e-08; weight on (r=0, v) 0.0063; neighbours 0.29713619, 2.12475875; eigenvalues in (Delta, 1): 1, in (-1, -Delta): 0
dense, L = 240, P=+0.40: eigenvalue 0.161712553411, minus the root -5.8e-11; weight at |r|>L/2 1.9e-04; weight on (r=0, v) 0.0008; neighbours 0.16130089, 2.03755899; eigenvalues in (Delta, 1): 1, in (-1, -Delta): 0
dense, L = 240, P=0: dimension 1924; eigenvalues with |E| < 1e-9: 962 (2 x number of r: 962), largest of them 7.0e-15; nearest other eigenvalue -1.81616020; root of 1 + w g_v(E,0): -1.81616020; 1 + w g_v(E,0) at E = +0.3, -0.3: 1.0539, 0.9461
reading: for P != 0 one simple eigenvalue lies just above the mixed bands and tends to 0 as c*|P|; at P = 0 the value 0 is the flat point (infinite multiplicity on the infinite chain), isolated from the rest of the spectrum
run time: 404 s