08-ilang-propagation / 07-bond-inertia
07bond inertiaverified
Determines how the cost of a bond made by oscillator cells depends on separation and joint motion, with which factor it enters the inertia of the bound pair, and whether the pair falls like its parts.
check_bound_pair.py9 KB · raw
"""Check (ii) of 07-bond-inertia: bound level E_b(P) and its curvature from the exact
diagonalization of the effective joint contract (7.2)-(7.4) in the sector P, against
the formulas (7.10)-(7.14) of item 2.
Ring of L cells; bodies (sigma1, sigma2) = (+,+) with rest costs M1, M2 of their levels,
tau = 1; field c0 = tau/r, m_h = c0/ell. Grips: square (4 F0 F2 = F1^2) and linear (F2 = 0).
F1 is set (proportional to c0) so that the static bond is the same for every r; two
binding strengths: a 'deep' one (a_b < ell) and a 'shallow' one (a_b > ell).
Self-energies of the single bodies are not included (they cancel in D at this order).
"""
import json
import time
from pathlib import Path
import numpy as np
from scipy.linalg import eigh
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
AIRY1 = 1.0187929716 # first zero of Ai'
def setup(L, ell, r, tau, M1, M2, F0, F1, F2, sg1=1, sg2=1):
c0 = tau / r; mh = c0 / ell; kap = c0**2
n = np.arange(L); qs = 2 * np.pi * n / L
om = np.sqrt(mh**2 + 4 * kap * np.sin(qs / 2)**2)
w0 = np.sum(1 / (2 * L * om)); abar = F0 + F2 * w0
Mb1, Mb2 = abar * M1, abar * M2
I1, I2 = sg1 * Mb1 / tau**2, sg2 * Mb2 / tau**2 # signed inertias (7.1)
return dict(L=L, n=n, om=om, c0=c0, mh=mh, kap=kap, tau=tau, r=r, M1=M1, M2=M2,
F0=F0, F1=F1, F2=F2, Mb1=Mb1, Mb2=Mb2, I1=I1, I2=I2, I=I1 + I2,
mred=I1 * I2 / (I1 + I2), mu12=sg1 * sg2 * M1 * M2, sg1=sg1, sg2=sg2,
rest=sg1 * Mb1 + sg2 * Mb2)
def hamiltonian(P, s, static=False):
L, n, om, tau = s["L"], s["n"], s["om"], s["tau"]
ks = 2 * np.pi * n / L
p1 = P * s["I1"] / s["I"] + ks; p2 = P * s["I2"] / s["I"] - ks
e1 = s["sg1"] * np.sqrt(s["Mb1"]**2 + tau**2 * np.sin(p1)**2)
e2 = s["sg2"] * np.sqrt(s["Mb2"]**2 + tau**2 * np.sin(p2)**2)
th1 = np.arctan2(tau * np.sin(p1), s["Mb1"]); th2 = np.arctan2(tau * np.sin(p2), s["Mb2"])
if static:
C = np.ones((L, L)); D1 = np.zeros((L, L)); D2 = np.zeros((L, L))
else:
C = np.cos((th1[:, None] + th1[None, :]) / 2) * np.cos((th2[:, None] + th2[None, :]) / 2)
D1 = e1[None, :] - e1[:, None]; D2 = e2[:, None] - e2[None, :] # rows k', cols k
m = (n[None, :] - n[:, None]) % L # q = k - k'
wq2 = om[m]**2
U = -s["mu12"] * s["F1"]**2 * C / (2 * L) * (1 / (wq2 - D1**2) + 1 / (wq2 - D2**2))
ratio = max(np.max(np.abs(D1) / np.sqrt(wq2)), np.max(np.abs(D2) / np.sqrt(wq2)))
if s["F2"] != 0:
for mi in range(L):
kp = (n - mi) % L
ob = om[(mi - n) % L]; Om = om + ob; w = Om / (om * ob)
d2 = np.concatenate([D1[kp, n], D2[kp, n]])**2
S = (1 / (Om[None, :]**2 - d2[:, None])) @ w
U[kp, n] += -s["mu12"] * s["F2"]**2 * C[kp, n] / (2 * L**2) * (S[:L] + S[L:])
ratio = max(ratio, np.sqrt(d2.max()) / Om.min())
if static:
return U, ratio # static mutual kernel only
return U + np.diag(e1 + e2), ratio
def analyse(s):
L, tau, r = s["L"], s["tau"], s["r"]
h = 0.015 * abs(s["I"]) * tau # P <= 3h keeps the pair velocity P/I below 0.045 tau
H0, ratio = hamiltonian(0.0, s)
up = s["mred"] < 0 # bound level above the band (mixed branches)
sel = [L - 2, L - 1] if up else [0, 1]
E, V = eigh(H0, subset_by_index=sel)
if up:
E, V = E[::-1], V[:, ::-1]
E0, psi = E[0], V[:, 0]
Ps = h * np.arange(1, 4); EP = []
for P in Ps:
HP, rt = hamiltonian(P, s); ratio = max(ratio, rt)
j = L - 1 if up else 0
EP.append(eigh(HP, eigvals_only=True, subset_by_index=[j, j])[0])
EP = np.array(EP)
A = np.stack([Ps**2, Ps**4, Ps**6], axis=1)
a, b, c = np.linalg.solve(A, EP - E0)
Ipair = 1 / (2 * a)
# relative label measured from the nearest band minimum (0 or pi: the two valleys of Model D)
ks = 2 * np.pi * s["n"] / L; ks = (ks + np.pi / 2) % np.pi - np.pi / 2
T = np.sum(np.abs(psi)**2 * ks**2) / (2 * s["mred"])
w_pi = np.sum(np.abs(psi[np.abs(2 * np.pi * s["n"] / L - np.pi) < np.pi / 2])**2)
dE = E0 - s["rest"]
Ust, _ = hamiltonian(0.0, s, static=True) # static kernel U~(k-k')/L
Uexp = psi @ Ust @ psi
Uq = L * Ust[0, :] # U~(q) at q = 2 pi j / L
Ur = np.real(np.fft.ifft(Uq)) # U(r), r = 0..L-1
rr = np.where(s["n"] > L // 2, s["n"] - L, s["n"])
rUp = rr * (np.roll(Ur, -1) - np.roll(Ur, 1)) / 2
psir = np.fft.ifft(psi) * np.sqrt(L) # psi(r) = L^-1/2 sum_k psi_k e^{ikr}
rUp_exp = np.sum(np.abs(psir)**2 * rUp)
I1, I2 = s["I1"], s["I2"]
D_num = Ipair - I1 - I2
D_formula = ((2 - r**2) * dE + (1 - r**2) * T) / tau**2 + 4 * I1 * I2 * T
D_general = (3 * T + 2 * Uexp) / tau**2 - (Uexp + rUp_exp) / s["c0"]**2 + 4 * I1 * I2 * T
lam = s["mu12"] * s["F1"]**2 / (2 * s["kap"])
g = -Uq[0]
rad = np.sqrt(np.sum(np.abs(psir)**2 * rr**2))
EPf = E0 + Ps**2 / (2 * (I1 + I2 + D_formula))
return dict(r=r, F1=s["F1"], F2=s["F2"], ell=s["c0"] / s["mh"], L=L,
branches=[s["sg1"], s["sg2"]], m_red=s["mred"],
E_b0=E0, gap_to_next=abs(E[1] - E[0]), weight_pi_valley=w_pi, dE=dE, T=T,
U_exp=Uexp, virial_2T=2 * T, virial_rUp=rUp_exp, rms_size=rad,
dE_deep_asym=Ur[0] + np.sign(s["mred"]) * AIRY1
* (lam**2 / (2 * abs(s["mred"])))**(1 / 3),
dE_shallow_asym=-s["mred"] * g**2 / 2,
I1=I1, I2=I2, I_pair=Ipair, D_num=D_num, D_formula=D_formula,
D_general=D_general, D_deep_asym=(2 - r**2) * dE / tau**2,
D_shallow_asym=dE / tau**2 - 4 * I1 * I2 * dE,
rel_dev_D=abs(D_num - D_formula) / abs(D_formula),
# D at first order in dE/M: I^2 (1/I - 1/I_pair) removes the trivial D^2/I term
D_num_lin=(I1 + I2)**2 * (1 / (I1 + I2) - 1 / Ipair),
dev_over_dE=abs(D_num - D_formula) / (abs(dE) / tau**2),
dev_lin_over_dE=abs((I1 + I2)**2 * (1 / (I1 + I2) - 1 / Ipair) - D_formula)
/ (abs(dE) / tau**2),
expected_scale=abs(dE) / abs(s["rest"]) + (tau / min(s["Mb1"], s["Mb2"]) / rad)**2,
T_over_dE=T / dE,
P=list(Ps), E_b_P_num=list(EP), E_b_P_formula=list(EPf),
max_EP_dev_over_shift=float(np.max(np.abs(EP - EPf) / (EP - E0))),
quartic_coeff=b, max_transfer_over_quantum=ratio,
D_over_I=D_num / (I1 + I2))
import sys
L = int(sys.argv[1]) if len(sys.argv) > 1 else 1024
ell = float(sys.argv[2]) if len(sys.argv) > 2 else 30.0
tau, M1, M2, F0 = 1.0, 0.5, 0.4, 1.0
results = []
t0 = time.time()
runs = [((1, 1), g) for g in ("square", "linear")] + [((1, -1), "linear")]
# (name, ell, target size a_b): deep a_b = ell/3; shallow a_b = 3 ell; very shallow a_b = 11 ell
regimes = [("deep", ell, ell / 3), ("shallow", ell, 3 * ell), ("v-shallow", 8.0, 88.0)]
for (sg1, sg2), grip in runs:
mred0 = abs((M1 * M2 / (sg1 * M1 + sg2 * M2))) / tau**2
for strength, ell_, ab in regimes:
if sg1 != sg2 and strength == "v-shallow":
continue
for r in (0.5, 1.0, 1.4):
c0 = tau / r; mh = c0 / ell_
if strength == "deep":
lam = 1 / (2 * mred0 * ab**3); F1 = c0 * np.sqrt(2 * lam / (M1 * M2))
else:
gg = 1 / (mred0 * ab); F1 = mh * np.sqrt(gg / (M1 * M2))
F2 = F1**2 / (4 * F0) if grip == "square" else 0.0
s = setup(L, ell_, r, tau, M1, M2, F0, F1, F2, sg1, sg2)
res = analyse(s); res.update(strength=strength, grip=grip)
results.append(res)
print(f"{sg1:+d}{sg2:+d} {strength:9s} {grip:6s} r={r:.2f} size={res['rms_size']:.1f} "
f"dE={res['dE']:.3e} T/dE={res['T_over_dE']:.3f} D_num={res['D_num']:.4e} "
f"D_lin={res['D_num_lin']:.4e} D_form={res['D_formula']:.4e} "
f"D_gen={res['D_general']:.4e} dev/dE={res['dev_over_dE']:.3f} "
f"devlin/dE={res['dev_lin_over_dE']:.3f} scale={res['expected_scale']:.3f} "
f"D/I={res['D_over_I']:.3f} EPdev={res['max_EP_dev_over_shift']:.3f} "
f"ratio={res['max_transfer_over_quantum']:.2f} gap={res['gap_to_next']:.2e} "
f"wpi={res['weight_pi_valley']:.2f} t={time.time() - t0:.0f}s", flush=True)
summary = {"parameters": {"L": L, "tau": tau, "M_k1": M1, "M_k2": M2, "F0": F0,
"regimes": regimes, "r": [0.5, 1.0, 1.4], "P": "h, 2h, 3h with h = 0.015 |I| tau"},
"max_dev_lin_over_dE": max(x["dev_lin_over_dE"] for x in results),
"max_dev_over_dE": max(x["dev_over_dE"] for x in results),
"results": results}
(OUT / "bound_pair_check.json").write_text(json.dumps(summary, indent=1, default=float))
print("max dev/dE:", summary["max_dev_over_dE"], " max devlin/dE:", summary["max_dev_lin_over_dE"])
check_mutual_term.py7 KB · raw
"""Check (i) of 07-bond-inertia: the mutual term (7.3)-(7.4) against second-order
perturbation theory in the total contract, restricted to at most two field quanta.
Ring of L = 16 cells, two Model D bodies (clock level fixed, so S_i -> M_i sigma_3),
grip F(q) = F0 + F1 q + F2 q^2. The unperturbed contract H0 contains the motion terms,
the first-order grip abar*M_i*sigma_3 (abar = F0 + F2 w0) and the field; the
perturbation is V = sum_i sum_x |x><x| M_i sigma_3 (dF(q_x) - F2 w0). The mutual
term is the part of the symmetric (des Cloizeaux) second-order effective contract with
one vertex on each body. Everything is built numerically in the position basis of the
bodies and the occupation basis of the field modes (<= 2 quanta).
"""
import json
from itertools import combinations_with_replacement
from pathlib import Path
import numpy as np
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
def fock(L):
states = [()] + [(j,) for j in range(L)] + list(combinations_with_replacement(range(L), 2))
index = {s: i for i, s in enumerate(states)}
B = np.zeros((L, len(states), len(states)))
for i, s in enumerate(states):
for j in set(s):
t = list(s); t.remove(j)
B[j, index[tuple(sorted(t))], i] = np.sqrt(s.count(j))
return states, B
def body(L, tau, Mbar):
"""Position-basis contract (index 2x+s) and its plane-wave eigenvectors."""
s1 = np.array([[0, 1], [1, 0]], complex); s3 = np.diag([1.0, -1.0]).astype(complex)
h = np.zeros((2 * L, 2 * L), complex)
for x in range(L):
y = (x + 1) % L
h[2*y:2*y+2, 2*x:2*x+2] += 0.5 * tau * (-1j * s1)
h[2*x:2*x+2, 2*y:2*y+2] += 0.5 * tau * (1j * s1)
h[2*x:2*x+2, 2*x:2*x+2] += Mbar * s3
W = np.zeros((2 * L, 2 * L), complex); e = np.zeros(2 * L); lab = []
xs = np.arange(L)
for n in range(L):
p = 2 * np.pi * n / L
pw = np.exp(1j * p * xs) / np.sqrt(L)
blk = np.array([[np.vdot(np.kron(pw, np.eye(2)[a]), h @ np.kron(pw, np.eye(2)[b]))
for b in range(2)] for a in range(2)])
ev, U = np.linalg.eigh(blk)
for col, band in ((1, +1), (0, -1)):
u = U[:, col]
ref = u[0] if band == +1 else u[1] # phase convention of Step 1
u = u * abs(ref) / ref if abs(ref) > 1e-12 else u
v = np.kron(pw, u); k = len(lab)
W[:, k] = v; e[k] = ev[col]; lab.append((n, band))
resid = np.max(np.abs(h @ W - W * e))
return W, e, lab, resid
def run(case):
L, kap, mh, tau = case["L"], case["kappa_h"], case["m_h"], case["tau"]
M1, M2, sg1, sg2 = case["M1"], case["M2"], case["sigma1"], case["sigma2"]
F0, F1, F2 = case["F0"], case["F1"], case["F2"]
qs = 2 * np.pi * np.arange(L) / L
om = np.sqrt(mh**2 + 4 * kap * np.sin(qs / 2)**2)
w0 = np.sum(1 / (2 * L * om))
abar = F0 + F2 * w0
states, B = fock(L)
EF = np.array([sum(om[j] for j in s) for s in states])
vac = np.zeros(len(states)); vac[0] = 1
X = []
for x in range(L):
qx = sum((B[j] * np.exp(1j * qs[j] * x) + B[j].T * np.exp(-1j * qs[j] * x))
/ np.sqrt(2 * L * om[j]) for j in range(L))
chi = F1 * (qx @ vac) + F2 * (qx @ (qx @ vac) - w0 * vac)
X.append(chi)
X = np.array(X) # (x, fock)
assert abs(X[:, 0]).max() < 1e-12 # no vacuum component
W1, e1, lab1, r1 = body(L, tau, abar * M1)
W2, e2, lab2, r2 = body(L, tau, abar * M2)
s3 = np.diag([1.0, -1.0])
def Y(psi1, psi2):
B1 = np.zeros((2 * L, L), complex); B2 = np.zeros((2 * L, L), complex)
for x in range(L):
B1[2*x:2*x+2, x] = M1 * s3 @ psi1[2*x:2*x+2]
B2[2*x:2*x+2, x] = M2 * s3 @ psi2[2*x:2*x+2]
V1 = np.einsum('ax,b,xc->abc', B1, psi2, X)
V2 = np.einsum('a,bx,xc->abc', psi1, B2, X)
tr = lambda T: np.einsum('ap,bq,abc->pqc', W1.conj(), W2.conj(), T)
return tr(V1), tr(V2)
En = e1[:, None, None] + e2[None, :, None] + EF[None, None, :]
idx1 = {l: i for i, l in enumerate(lab1)}; idx2 = {l: i for i, l in enumerate(lab2)}
def eps(M, sg, p):
return sg * np.sqrt((abar * M)**2 + tau**2 * np.sin(p)**2)
def th(M, p):
return np.arctan2(tau * np.sin(p), abar * M)
rows = []
for (n1, n2, m) in case["labels"]:
i1, i2 = idx1[(n1 % L, sg1)], idx2[(n2 % L, sg2)]
f1, f2 = idx1[((n1 - m) % L, sg1)], idx2[((n2 + m) % L, sg2)]
Yi = Y(W1[:, i1], W2[:, i2]); Yf = Y(W1[:, f1], W2[:, f2])
Ei, Ef = e1[i1] + e2[i2], e1[f1] + e2[f2]
cross = (Yf[0].conj() * Yi[1] + Yf[1].conj() * Yi[0])[:, :, 1:] # Q: >= 1 quantum
num = 0.5 * np.sum(cross * (1 / (Ei - En[:, :, 1:]) + 1 / (Ef - En[:, :, 1:])))
# closed form (7.3)-(7.4)
p1, p2, q = qs[n1 % L], qs[n2 % L], qs[m % L]
p1f, p2f = p1 - q, p2 + q
D1 = eps(M1, sg1, p1) - eps(M1, sg1, p1f); D2 = eps(M2, sg2, p2f) - eps(M2, sg2, p2)
c1 = np.cos((th(M1, p1) + th(M1, p1f)) / 2); c2 = np.cos((th(M2, p2) + th(M2, p2f)) / 2)
mu12 = sg1 * M1 * sg2 * M2
wq = om[m % L]
U1 = -mu12 * F1**2 * c1 * c2 / (2 * L) * (1 / (wq**2 - D1**2) + 1 / (wq**2 - D2**2))
ob = om[(m - np.arange(L)) % L]; Om = om + ob
U2 = -mu12 * F2**2 * c1 * c2 / (2 * L**2) * np.sum(Om / (om * ob) * (
1 / (Om**2 - D1**2) + 1 / (Om**2 - D2**2)))
form = U1 + U2
rows.append({"labels_n1_n2_m": [n1, n2, m], "numerical": [num.real, num.imag],
"formula": form, "rel_dev": abs(num - form) / abs(form)})
return {"case": {k: v for k, v in case.items() if k != "labels"}, "abar": abar,
"eigvec_residual": max(r1, r2), "rows": rows,
"max_rel_dev": max(r["rel_dev"] for r in rows)}
labels = [(0, 0, 1), (1, -2, 3), (3, -5, 2), (-2, 4, -1), (5, 2, 7), (2, 2, 0), (-3, 1, 0), (6, -6, 8)]
base = dict(L=16, kappa_h=1.0, m_h=0.3, tau=0.8, M1=0.7, M2=0.5, F0=1.0, F1=0.3, F2=0.2,
sigma1=1, sigma2=1, labels=labels)
cases = [base,
dict(base, F2=base["F1"]**2 / (4 * base["F0"])), # square grip
dict(base, sigma2=-1), # mixed branches
dict(base, tau=1.3)] # r > 1
res = [run(c) for c in cases]
summary = {"max_rel_dev_all": max(r["max_rel_dev"] for r in res), "cases": res}
(OUT / "mutual_term_check.json").write_text(json.dumps(summary, indent=1))
for r in res:
print(r["case"]["tau"], r["case"]["F2"], r["case"]["sigma2"], "max rel dev", r["max_rel_dev"],
"resid", r["eigvec_residual"])
print("overall", summary["max_rel_dev_all"])
output/bound_pair_check.json39 KB · raw
{
"parameters": {
"L": 1024,
"tau": 1.0,
"M_k1": 0.5,
"M_k2": 0.4,
"F0": 1.0,
"regimes": [
[
"deep",
30.0,
10.0
],
[
"shallow",
30.0,
90.0
],
[
"v-shallow",
8.0,
88.0
]
],
"r": [
0.5,
1.0,
1.4
],
"P": "h, 2h, 3h with h = 0.015 |I| tau"
},
"max_dev_lin_over_dE": 0.15616620972696707,
"max_dev_over_dE": 0.1705434907971377,
"results": [
{
"r": 0.5,
"F1": 0.3,
"F2": 0.0225,
"ell": 30.0,
"L": 1024,
"branches": [
1,
1
],
"m_red": 0.22440277537589084,
"E_b0": 0.8618975453057469,
"gap_to_next": 9.59675365588275e-05,
"weight_pi_valley": 0.49999999986346394,
"dE": -0.04693369496661104,
"T": 0.005528448192450895,
"U_exp": -0.05297891874622328,
"virial_2T": 0.01105689638490179,
"virial_rUp": 0.011840173045768837,
"rms_size": 10.262832842569932,
"dE_deep_asym": -0.0447356151085497,
"dE_shallow_asym": -1.842032138095726,
"I1": 0.5049062445957544,
"I2": 0.40392499567660356,
"I_pair": 0.8432940210181585,
"D_num": -0.06553721925419942,
"D_formula": -0.07347764910719537,
"D_general": -0.07457782554994448,
"D_deep_asym": -0.08213396619156932,
"D_shallow_asym": -0.008646265587440027,
"rel_dev_D": 0.10806592139892478,
"D_num_lin": -0.07063049277508511,
"dev_over_dE": 0.1691839915575543,
"dev_lin_over_dE": 0.06066337487674359,
"expected_scale": 0.10983391538639795,
"T_over_dE": -0.1177927328411682,
"P": [
0.013632468604085369,
0.027264937208170737,
0.04089740581225611
],
"E_b_P_num": [
0.8620077257264205,
0.8623381587525126,
0.8628885200932579
],
"E_b_P_formula": [
0.8620087821533566,
0.8623424926961859,
0.8628986769342346
],
"max_EP_dev_over_shift": 0.010249343479491474,
"quartic_coeff": -0.2613189881694042,
"max_transfer_over_quantum": 0.33171097239229597,
"D_over_I": -0.07211153880951458,
"strength": "deep",
"grip": "square"
},
{
"r": 1.0,
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nDot Physics Research