08-ilang-propagation / 08-tilt
08tiltverified
Determines whether a push on one cell orients rotation-symmetric exchange cells far away, how small deviations match the anchored oscillator field, and which sources have a long-range mutual cost.
ed_push.py6 KB · raw
"""Check (item 1): exact diagonalization of exchange cells with a push.
C = J sum_<xy> S_x.S_y - h_p S^3_h (J = 1, n = e_3), spin S = 1/2 and S = 1.
Clusters: rings of even size L (S=1/2: 8,12,16,20; S=1: 6,8,10,12) and the
16-object periodic cubic cluster Z^3 / Lambda, Lambda spanned by (2,2,0),(2,0,2),(0,2,2).
C commutes with M = sum_x S^3_x. The two lowest eigenvalues are computed in EVERY sector
M = -NS, ..., NS, at h_p = 0 and at the sampled push strengths HPS. In each sector the lowest
eigenvalue is simple (Perron-Frobenius after the Marshall sign change; the push is diagonal),
so the lowest eigenvector of C is unique iff the lowest sector value lies strictly below the
lowest values of all other sectors and below the second value of its own sector.
Output: output/ed_push.txt
"""
from pathlib import Path
from collections import deque
import itertools
import numpy as np
import scipy.sparse as sp
from scipy.sparse.linalg import eigsh
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
HPS = [0.01, 0.1, 0.3, 1.0, 3.0, 10.0]
def ring(L):
links = [(i, (i + 1) % L) for i in range(L)]
eps = np.array([(-1) ** i for i in range(L)])
return links, eps
def cubic16():
A = np.array([[2, 2, 0], [2, 0, 2], [0, 2, 2]]).T # columns are the periods
Ainv = np.linalg.inv(A)
def same(u, v):
c = Ainv @ (np.array(u) - np.array(v))
return np.allclose(c, np.round(c))
reps = []
for p in itertools.product(range(4), repeat=3):
if not any(same(p, q) for q in reps):
reps.append(p)
assert len(reps) == 16
def idx(p):
return [k for k, q in enumerate(reps) if same(p, q)][0]
links = set()
for k, p in enumerate(reps):
nb = set()
for d in np.eye(3, dtype=int):
for s in (1, -1):
j = idx(tuple(np.array(p) + s * d))
nb.add(j)
links.add(tuple(sorted((k, j))))
assert len(nb) == 6
links = sorted(links)
assert len(links) == 48
eps = np.array([(-1) ** (sum(p) % 2) for p in reps])
for a, b in links:
assert eps[a] != eps[b]
return links, eps, reps
def distances(links, n, h=0):
adj = [[] for _ in range(n)]
for a, b in links:
adj[a].append(b)
adj[b].append(a)
d = [-1] * n
d[h] = 0
q = deque([h])
while q:
u = q.popleft()
for v in adj[u]:
if d[v] < 0:
d[v] = d[u] + 1
q.append(v)
return np.array(d)
def sector(n, S2, M2):
"""Basis states (digit arrays, digit = m + S) with 2*sum(m) = M2; S2 = 2S."""
d = S2 + 1
codes = np.arange(d ** n, dtype=np.int64)
digits = np.zeros((codes.size, n), dtype=np.int8)
c = codes.copy()
for i in range(n):
digits[:, i] = c % d
c //= d
tot = 2 * digits.sum(axis=1).astype(np.int64) - n * S2
keep = tot == M2
return codes[keep], digits[keep]
def build(n, S2, M2, links, J=1.0):
S = S2 / 2.0
d = S2 + 1
codes, digits = sector(n, S2, M2)
m = digits.astype(float) - S
diag = np.zeros(codes.size)
rows, cols, vals = [], [], []
for a, b in links:
diag += J * m[:, a] * m[:, b]
for i, j in ((a, b), (b, a)): # S^+_i S^-_j
ok = (digits[:, i] < S2) & (digits[:, j] > 0)
src = np.nonzero(ok)[0]
new = codes[src] + d ** i - d ** j
dst = np.searchsorted(codes, new)
amp = 0.5 * J * np.sqrt(S * (S + 1) - m[src, i] * (m[src, i] + 1)) \
* np.sqrt(S * (S + 1) - m[src, j] * (m[src, j] - 1))
rows.append(dst); cols.append(src); vals.append(amp)
if rows:
off = sp.csr_matrix((np.concatenate(vals), (np.concatenate(rows), np.concatenate(cols))),
shape=(codes.size, codes.size))
else:
off = sp.csr_matrix((codes.size, codes.size))
return off, diag, m
def lowest(H, k=1):
if H.shape[0] <= 1500:
w, v = np.linalg.eigh(H.toarray())
return w[:k], v[:, :k]
w, v = eigsh(H, k=k, which="SA", tol=1e-12)
o = np.argsort(w)
return w[o], v[:, o]
def run(name, n, S2, links, eps, out):
dist = distances(links, n)
far = np.nonzero(dist == dist.max())[0]
out.append(f"\n## {name}: N={n}, S={S2}/2, links={len(links)}, far sites {[int(f) for f in far]}"
f" at distance {dist.max()}; sectors searched: all M = {-n * S2 / 2:g}, ..., {n * S2 / 2:g}")
hps = [0.0] + HPS
res = {hp: [] for hp in hps}
for M2 in range(-n * S2, n * S2 + 1, 2):
off, diag, m = build(n, S2, M2, links)
for hp in hps:
H = off + sp.diags(diag - hp * m[:, 0])
k = 2 if H.shape[0] > 1 else 1
w, v = lowest(H, k=k)
e2 = w[1] if k == 2 else np.inf
p = np.abs(v[:, 0]) ** 2
sz = p @ m
res[hp].append((w[0], e2, M2, float(np.mean(eps[far] * sz[far])), float(sz[0])))
out.append("h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)")
for hp in hps:
best = min(res[hp], key=lambda t: t[0])
nxt = min([best[1]] + [t[0] for t in res[hp] if t[2] != best[2]])
uniq = best[0] < nxt - 1e-9
out.append(f"{hp:6.2f} {best[2] / 2:5.1f} {best[0]:.10f} {nxt:.10f} {nxt - best[0]:.3e} {uniq!s:6s}"
f" {best[3]: .6f} {best[4]: .6f}")
b0 = min(res[0.0], key=lambda t: t[0])
n0 = min([b0[1]] + [t[0] for t in res[0.0] if t[2] != b0[2]])
out.append(f"no push: lowest eigenvector in M=0 and unique (a total-spin singlet): {b0[2] == 0 and b0[0] < n0 - 1e-9}")
def main():
out = ["# ED check, item 1: exchange cells, J=1, push -h_p S^3_h at h=0 (class +); all sectors of M searched"]
for L in (8, 12, 16, 20):
links, eps = ring(L)
run(f"ring L={L}", L, 1, links, eps, out)
links, eps, reps = cubic16()
out.append("\n16-object cubic cluster sites (representatives mod Lambda): " + str(reps))
out.append("links: each site x to x+e_i, i=1,2,3 (mod Lambda); 48 links, 6 distinct neighbours each")
run("cubic cluster 16", 16, 1, links, eps, out)
for L in (6, 8, 10, 12):
links, eps = ring(L)
run(f"ring L={L}", L, 2, links, eps, out)
(OUT / "ed_push.txt").write_text("\n".join(out) + "\n")
print("\n".join(out))
if __name__ == "__main__":
main()
spinwave_mutual.py17 KB · raw
"""Check (item 3, leading order in 1/S): mutual cost of two held sources on the cubic torus.
Units J S = 1. C_2 = 1/2 P^T T P + 1/2 Q^T V Q, T_k = z(1+gamma_k), V_k = z(1-gamma_k); q1 = Q, q2 = E P.
(1) Transverse direction sources (linear in q): separation part from ghat; single objects, two-object cells,
and a single object against a cell along the separation (midpoint against end-point separation).
(2) Static quadratic sources dH_i = 1/2 Q^T A_i Q + 1/2 P^T B_i P (along-n sources on objects and cells; changed
links R2 in four orientation pairs): E_12 = -2 int_0^inf dtau <dH_1(tau) dH_2(0)>_c, Wick with G_QQ, G_PP and
G_QP = (i/2) F (tau > 0); far from the push: the two zero modes (k = 0, k = Q) are left out. R2 is compared
with the continuum formula at the separation of the link midpoints.
(3) Held spin R1 with the retarded vertex of Step 12, K(nu) = a(nu) 1 + b(nu) eps (per unit |g'| s_b):
a = nu^2/(nu^2+D^2), b = sgn(sigma) eps_x D nu/(nu^2+D^2), D = |g'|/J;
E_12 = -1/2 int dnu/2pi Tr[K1 G12 K2 G21], lattice G(nu) at (r,0,0) by summing over (k2,k3) for each k1,
tori L = 32, 64, 128, 256; against the continuum crossover function
J S E/(|g'1 g'2| s1 s2) = -kappa_par F(beta)/r^3, beta = 2 r D / sqrt(2z) (A), and against the same with the
off-diagonal part G12 = G0 (1 + gam eps), gam = -(eps1+eps2) nu/T_0, of the continuum covariance (B).
Output: output/spinwave_mutual.txt
"""
from pathlib import Path
import numpy as np
from scipy.integrate import quad
from scipy.special import ive
from scipy import fft as sfft
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
Z = 6
CS = np.sqrt(2 * Z) # speed of the copies, units J S = 1
KAP = np.sqrt(2 * Z) / (32 * np.pi ** 3) # kappa_par
PRE = np.sqrt(3) / (8 * np.pi ** 3) # prefactor of (8.17)
def kgrid(L):
k = 2 * np.pi * np.arange(L) / L
c = np.cos(k)
g = (c[:, None, None] + c[None, :, None] + c[None, None, :]) / 3.0
return g
def ghat(L):
k = 2 * np.pi * np.arange(L) / L
lap = 2 * ((1 - np.cos(k))[:, None, None] + (1 - np.cos(k))[None, :, None] + (1 - np.cos(k))[None, None, :])
inv = np.zeros_like(lap)
inv[lap > 1e-12] = 1.0 / lap[lap > 1e-12]
return np.real(np.fft.ifftn(inv))
def ghat_inf(x):
"""Infinite cubic lattice: (Lap^{-1})(x) = int_0^inf prod_i e^{-2t} I_{x_i}(2t) dt."""
f = lambda t: np.prod([ive(abs(int(c)), 2 * t) for c in x])
return quad(f, 0, np.inf, limit=400, epsabs=1e-13, epsrel=1e-11)[0]
def eps(x):
return (-1) ** (int(sum(x)) % 2)
def check_58(L=6, eta=0.7):
n = L ** 3
co = np.array(np.unravel_index(np.arange(n), (L,) * 3)).T
adj = np.zeros((n, n))
for d in range(3):
sh = co.copy(); sh[:, d] = (sh[:, d] + 1) % L
j = np.ravel_multi_index(sh.T, (L,) * 3)
adj[np.arange(n), j] += 1; adj[j, np.arange(n)] += 1
V = Z * np.eye(n) - adj
V[0, 0] += eta
Ga = np.linalg.inv(V)
g = ghat(L).ravel()
dif = lambda a, b: np.ravel_multi_index(tuple((co[a] - co[b]) % L), (L,) * 3)
err = 0.0
for a, b in [(5, 17), (40, 101), (3, 3), (0, 77)]:
f = 1 / eta + g[dif(a, b)] - g[dif(a, 0)] - g[dif(b, 0)] + g[0]
err = max(err, abs(f - Ga[a, b]))
return err
def pert(kind, x, e=None):
"""Return (A, B) as lists of (site, site, value), per unit strength."""
x = tuple(x)
if kind == "along":
A = [(x, x, -eps(x))]
return A, A
if kind == "along_cell":
x2 = (x[0], x[1] + 1, x[2])
A = [(x, x, -eps(x)), (x2, x2, -eps(x))]
return A, A
if kind == "link":
y = tuple(np.array(x) - np.array(e))
A = [(x, x, 1), (y, y, 1), (x, y, -1), (y, x, -1)]
B = [(x, x, 1), (y, y, 1), (x, y, 1), (y, x, 1)]
return A, B
raise ValueError(kind)
def contraction(M1, M2, L):
w, i1, i2 = [], [], []
for a, b, u in M1:
for c, d, v in M2:
w.append(u * v)
i1.append(np.ravel_multi_index(tuple(np.mod(np.array(a) - np.array(c), L)), (L,) * 3))
i2.append(np.ravel_multi_index(tuple(np.mod(np.array(b) - np.array(d), L)), (L,) * 3))
return np.array(w), np.array(i1), np.array(i2)
def mutual_quadratic(L, configs, nodes=300):
g = kgrid(L)
om = Z * np.sqrt(np.clip(1 - g ** 2, 0, None))
mask = om > 1e-9
Tk = np.where(mask, Z * (1 + g), 0.0)
Vk = np.where(mask, Z * (1 - g), 0.0)
omm = np.where(mask, om, 1.0)
pre = []
for (A1, B1), (A2, B2) in configs:
pre.append((contraction(A1, A2, L), contraction(B1, B2, L),
contraction(A1, B2, L), contraction(B1, A2, L)))
s, ws = np.polynomial.legendre.leggauss(nodes)
lo, hi = np.log(1e-7), np.log(200.0)
sv = 0.5 * (hi - lo) * s + 0.5 * (hi + lo)
wv = 0.5 * (hi - lo) * ws
acc = np.zeros(len(configs))
for sj, wj in zip(sv, wv):
tau = np.exp(sj)
ex = np.where(mask, np.exp(-omm * tau), 0.0)
GQ = sfft.ifftn(Tk / (2 * omm) * ex, workers=-1).real.ravel()
GP = sfft.ifftn(Vk / (2 * omm) * ex, workers=-1).real.ravel()
F = sfft.ifftn(ex, workers=-1).real.ravel()
for n, (qq, pp, qp, pq) in enumerate(pre):
val = 0.5 * np.sum(qq[0] * GQ[qq[1]] * GQ[qq[2]]) + 0.5 * np.sum(pp[0] * GP[pp[1]] * GP[pp[2]]) \
- 0.125 * (np.sum(qp[0] * F[qp[1]] * F[qp[2]]) + np.sum(pq[0] * F[pq[1]] * F[pq[2]]))
acc[n] += wj * tau * val
return -2 * acc
def cont_link(e1, e2, e12, R):
"""Continuum (8.17) integral: J E/(delta1 delta2 S) for two changed links (unit vectors e1, e2 from y to x,
e12 = eps_x1 eps_x2) whose midpoints are separated by the vector R."""
R = np.asarray(R, float); Rn = np.linalg.norm(R); Rh = R / Rn
E1 = np.r_[0, e1]; E2 = np.r_[0, e2]; t0 = np.r_[1.0, 0, 0, 0]
M1 = np.outer(E1, E1) - (2 / Z) * np.outer(t0, t0)
M2 = np.outer(E2, E2) - (2 / Z) * np.outer(t0, t0)
def C(t):
X = np.r_[t, Rh]; rho = np.linalg.norm(X); n = X / rho
I = np.eye(4) - 4 * np.outer(n, n)
cross = I[0, 0] * (E1 @ I @ E2) - (I @ E1)[0] * (I @ E2)[0]
return (4 * np.trace(M1 @ I @ M2 @ I) - (16 / Z) * e12 * cross) / rho ** 8
val = quad(C, -np.inf, np.inf, epsabs=1e-14, epsrel=1e-12)[0]
return -np.sqrt(2 * Z) / (16 * np.pi ** 4) * val / Rn ** 7
def F_hat(beta, vs):
"""Continuum crossover function of (8.18) for equal held spins (beta1 = beta2 = beta), vs = sgn(s1 s2) e1 e2."""
f = lambda u: np.exp(-u) * u ** 2 * (u ** 2 - vs * beta ** 2) / (u ** 2 + beta ** 2) ** 2
if beta == 0:
return 1.0
return quad(f, 0, beta, limit=400, epsabs=1e-15, epsrel=1e-12)[0] + \
quad(f, beta, np.inf, limit=400, epsabs=1e-15, epsrel=1e-12)[0]
def F_full(beta, r, e1, e2, s=1):
"""(8.18) including the off-diagonal part of the continuum covariance, G12 = G0 (1 + gam eps),
gam = -(e1+e2) nu/T_0 with nu = c u/(2r); held spins with sgn(sigma1) = sgn(sigma2) = s."""
T0 = 2 * Z
def f(u):
a = u ** 2 / (u ** 2 + beta ** 2)
b1 = s * e1 * beta * u / (u ** 2 + beta ** 2)
b2 = s * e2 * beta * u / (u ** 2 + beta ** 2)
gam = (e1 + e2) * CS * u / (2 * r * T0)
return np.exp(-u) * ((a * a - b1 * b2) * (1 - gam ** 2) + 2 * gam * a * (b1 + b2))
if beta == 0:
return quad(f, 0, np.inf, limit=400, epsabs=1e-15, epsrel=1e-12)[0]
return quad(f, 0, beta, limit=400, epsabs=1e-15, epsrel=1e-12)[0] + \
quad(f, beta, np.inf, limit=400, epsabs=1e-15, epsrel=1e-12)[0]
def r1_lattice(L, rs, deltas, nodes=240):
"""Lattice value of J S E_12/(|g'1 g'2| s1 s2) for held spins with sgn(sigma1) = sgn(sigma2) = +1 at 0 and
(r,0,0). G^QQ, G^PP, G^QP at (r,0,0) for all frequency nodes by summing over (k2,k3) for each k1
(zero modes k = 0, Q left out)."""
s, ws = np.polynomial.legendre.leggauss(nodes)
lo, hi = np.log(1e-5), np.log(300.0)
nus = np.exp(0.5 * (hi - lo) * s + 0.5 * (hi + lo))
wv = 0.5 * (hi - lo) * ws
k = 2 * np.pi * np.arange(L) / L
c = np.cos(k)
C23 = (c[:, None] + c[None, :]).ravel()
nu2 = nus ** 2
SQ = np.zeros((L, nodes)); SP = np.zeros((L, nodes)); SX = np.zeros((L, nodes))
for i1 in range(L):
g = (c[i1] + C23) / 3.0
om2 = Z ** 2 * (1 - g ** 2)
m = om2 > 1e-12
g, om2 = g[m], om2[m]
den = 1.0 / (nu2[:, None] + om2[None, :])
SQ[i1] = den @ (Z * (1 + g))
SP[i1] = den @ (Z * (1 - g))
SX[i1] = -nus * den.sum(axis=1)
cosm = np.cos(np.outer(rs, k)) / L ** 3
GQ, GP, GX = cosm @ SQ, cosm @ SP, cosm @ SX # (len(rs), nodes)
acc = np.zeros((len(deltas), len(rs)))
for ir, r in enumerate(rs):
e1, e2 = 1, (-1) ** r
g0, g1 = GQ[ir], e2 * GX[ir] # G12 = [[GQ, e2 GX], [-e1 GX, e1 e2 GP]]
gp = e1 * e2 * GP[ir]
for idl, D in enumerate(deltas):
a = nu2 / (nu2 + D ** 2)
b = D * nus / (nu2 + D ** 2)
# Tr[K1 G12 K2 G21], K_i = a 1 + e_i b eps, G21 = G12(-nu)^T, written out for 2x2 matrices
G12 = np.array([[g0, g1], [-e1 * GX[ir], gp]])
G21 = np.array([[g0, e1 * GX[ir]], [-g1, gp]])
K1 = np.array([[a, e1 * b], [-e1 * b, a]])
K2 = np.array([[a, e2 * b], [-e2 * b, a]])
M = np.einsum("ijn,jkn,kln,lin->n", K1, G12, K2, G21)
acc[idl, ir] = np.sum(wv * nus * M)
return -acc / (2 * np.pi)
def main():
out = ["# Mutual cost of two held sources, leading order in 1/S, cubic torus (units as stated per block)"]
out.append(f"\n(5.8) with kappa_h = J S, kappa_a = h_p against direct inverse (L=6): max error {check_58():.2e}")
out.append("\n## (1) Transverse sources: separation part J E_r/(g1 g2 (m1perp.m2perp))")
out.append("single objects: -eps1 eps2 ghat(r); cells (x, x+e2), + on the first object: -eps1 eps2 sum ghat")
out.append(f"ghat_inf(0) = {ghat_inf((0, 0, 0)):.7f} (gamma_3 = 0.2527310)")
out.append("r eps1eps2 single: J E_r/(g1g2) 4 pi r ghat_inf(r) cells: J E_r/(g1g2) pi r sum ghat_inf"
" torus check: ghat_L(r)-ghat_L(8) for L=16, 32, 64 vs ghat_inf(r)-ghat_inf(8)")
gl = {L: ghat(L) for L in (16, 32, 64)}
g8 = ghat_inf((8, 0, 0))
gi = {r: ghat_inf((r, 0, 0)) for r in range(2, 10)}
for r in range(2, 9):
e12 = (-1) ** r
gs = gi[r]
gc = 2 * gs + ghat_inf((r, 1, 0)) + ghat_inf((r, -1, 0))
tor = " ".join(f"{gl[L][r, 0, 0] - gl[L][8, 0, 0]:.6f}" for L in (16, 32, 64))
out.append(f"{r:2d} {e12:+d} {-e12 * gs: .6f} {4 * np.pi * r * gs:.4f}"
f" {-e12 * gc: .6f} {np.pi * r * gc:.4f} {tor} {gs - g8:.6f}")
out.append("single object at 0, cell (r e1, (r+1) e1) along the separation: 4 pi R [ghat(r)+ghat(r+1)]/2 with"
" R = midpoint separation r+1/2, and with R = end-point separation r")
for r in range(2, 9):
s2 = gi[r] + gi[r + 1]
out.append(f"{r:2d} midpoint: {2 * np.pi * (r + 0.5) * s2:.5f} end point: {2 * np.pi * r * s2:.5f}")
out.append("\n## Continuum coefficients")
out.append(f"along n, single: J S E/(g1 g2) = -eps1 eps2 kappa_par / r^3, kappa_par = sqrt(2z)/(32 pi^3) = {KAP:.7f}")
out.append(f"held spin, r >> l_b: S E/(J sig1 sig2) = +eps1 eps2 lambda / r^5, lambda = 3 sqrt3/(8 pi^3) = "
f"{3 * np.sqrt(3) / (8 * np.pi ** 3):.7f}")
ex, ey, ez = [1, 0, 0], [0, 1, 0], [0, 0, 1]
geo = {"par (e1,e1)": (ex, ex, lambda e: 43 / 6 + 2.5 * e),
"perp same (e2,e2)": (ey, ey, lambda e: 4 / 3 - 5 / 12 * e),
"perp cross (e2,e3)": (ey, ez, lambda e: 1 / 12),
"mixed (e1,e2)": (ex, ey, lambda e: -0.5)}
out.append("changed links, separation r e1: J E/(d1 d2 S) * r^7 = -sqrt3/(8 pi^3) a; numerical integral against a:")
for nm, (u, v, af) in geo.items():
for e in (1, -1):
num = cont_link(u, v, e, [1, 0, 0])
out.append(f" {nm:20s} eps1eps2={e:+d}: integral {num:.8f} closed form a={af(e):.6f}: {-PRE * af(e):.8f}")
out.append("\n## (2) Static quadratic sources, lattice (zero modes left out); ratio = lattice / continuum")
out.append("columns: r, eps1eps2, along single [J S E/(g1g2), ratio], along cell [.., ratio vs 4x], then J E/(d1d2S)"
" and ratio for links: par (e1 at 0, e1 at r e1), perp same (e2, e2), perp cross (e2, e3), mixed (e1, e2);"
" link continuum at the midpoint separation R")
rs = list(range(2, 13))
Rvec = {"par": lambda r: [r, 0, 0], "perp": lambda r: [r, 0, 0],
"cross": lambda r: [r, 0.5, -0.5], "mixed": lambda r: [r + 0.5, -0.5, 0]}
cref = {}
for r in rs:
e = (-1) ** r
cref[r] = [-e * KAP / r ** 3, -4 * e * KAP / r ** 3,
cont_link(ex, ex, e, Rvec["par"](r)), cont_link(ey, ey, e, Rvec["perp"](r)),
cont_link(ey, ez, e, Rvec["cross"](r)), cont_link(ex, ey, e, Rvec["mixed"](r))]
ratios = {}
for L in (32, 64, 128):
configs = []
for r in rs:
x2 = (r, 0, 0)
configs += [(pert("along", (0, 0, 0)), pert("along", x2)),
(pert("along_cell", (0, 0, 0)), pert("along_cell", x2)),
(pert("link", (0, 0, 0), ex), pert("link", x2, ex)),
(pert("link", (0, 0, 0), ey), pert("link", x2, ey)),
(pert("link", (0, 0, 0), ey), pert("link", x2, ez)),
(pert("link", (0, 0, 0), ex), pert("link", x2, ey))]
res = mutual_quadratic(L, configs).reshape(len(rs), 6)
out.append(f"L = {L}")
ratios[L] = np.zeros_like(res)
for i, r in enumerate(rs):
p = np.array(cref[r])
ratios[L][i] = res[i] / p
row = " ".join(f"{res[i, j]: .3e} {res[i, j] / p[j]:6.3f}" for j in range(6))
out.append(f"{r:2d} {(-1) ** r:+d} {row}")
if L == 32:
along32 = res[:, 0].copy()
out.append("ratio extrapolated to L -> infinity: Aitken from L = 32, 64, 128 (geometric finite-size error);"
" the L = 128 value where the change from 64 to 128 is below 1e-3 or not geometric")
for i, r in enumerate(rs):
x1, x2, x3 = ratios[32][i], ratios[64][i], ratios[128][i]
d1, d2 = x2 - x1, x3 - x2
geo_ = (np.abs(d2) > 1e-3) & (d1 * d2 > 0) & (np.abs(d2) < np.abs(d1))
exv = np.where(geo_, x3 + d2 / np.where(geo_, d1 / np.where(geo_, d2, 1.0) - 1, 1.0), x3)
out.append(f"{r:2d} " + " ".join(f"{v:6.3f}" for v in exv))
out.append("\n## (3) Held spin R1, retarded vertex (Step 12); J S E/(|g'1 g'2| s1 s2), sgn sigma1 = sgn sigma2,"
" so vs = eps1 eps2 = (-1)^r")
out.append("continuum crossover function F(beta) (equal held spins), with its small-beta form 1 - pi (3+vs) beta/4"
" and large-beta form -2 vs/beta^2 + 24 (1+2 vs)/beta^4")
for vs in (1, -1):
for beta in (0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, 100.0):
Fv = F_hat(beta, vs)
out.append(f" vs={vs:+d} beta={beta:7.2f}: F = {Fv: .6e} small-beta form {1 - np.pi * (3 + vs) * beta / 4: .6e}"
f" large-beta form {-2 * vs / beta ** 2 + 24 * (1 + 2 * vs) / beta ** 4: .6e}")
lo_, hi_ = 0.1, 10.0
for _ in range(80):
mid = 0.5 * (lo_ + hi_)
if F_hat(mid, 1) > 0:
lo_ = mid
else:
hi_ = mid
out.append(f" vs=+1: F changes sign at beta_0 = {0.5 * (lo_ + hi_):.5f} (r_0 = beta_0 l_b / 2); vs=-1: F > 0 at every beta")
deltas = [0.0, 0.2, 0.5, 2.0]
out.append("lattice against continuum: D = |g'|/J = " + ", ".join(f"{d:g}" for d in deltas)
+ f"; l_b = sqrt(2z)/D = " + ", ".join("inf" if d == 0 else f"{CS / d:.2f}" for d in deltas))
out.append("continuum A = (8.18): -kappa_par F(beta)/r^3; continuum B = (8.18) with the off-diagonal part of the"
" covariance (nonzero for eps1 = eps2 only)")
out.append("columns per D: lattice value, ratio lattice/A, ratio lattice/B")
contA = np.array([[-KAP * F_hat(2 * r * D / CS, (-1) ** r) / r ** 3 for r in rs] for D in deltas])
contB = np.array([[-KAP * F_full(2 * r * D / CS, r, 1, (-1) ** r) / r ** 3 for r in rs] for D in deltas])
r1 = {}
for L in (32, 64, 128, 256):
r1[L] = r1_lattice(L, rs, deltas)
out.append(f"L = {L}")
for ir, r in enumerate(rs):
row = [f"{r1[L][idl, ir]: .4e} {r1[L][idl, ir] / contA[idl, ir]:6.3f} {r1[L][idl, ir] / contB[idl, ir]:6.3f}"
for idl in range(len(deltas))]
out.append(f"{r:2d} {(-1) ** r:+d} " + " ".join(row))
if L == 32:
dev = np.max(np.abs(r1[L][0] - np.array([(-1) ** r for r in rs]) * along32) / np.abs(along32))
out.append(f"cross-check D = 0 (frequency domain) against the along-n single column (time domain), L = 32:"
f" max relative difference {dev:.2e}")
out.append("ratio lattice/B extrapolated to L -> infinity: Aitken from L = 64, 128, 256 (as in block (2))")
for ir, r in enumerate(rs):
x1, x2, x3 = (r1[L][:, ir] / contB[:, ir] for L in (64, 128, 256))
d1, d2 = x2 - x1, x3 - x2
geo_ = (np.abs(d2) > 1e-3) & (d1 * d2 > 0) & (np.abs(d2) < np.abs(d1))
exv = np.where(geo_, x3 + d2 / np.where(geo_, d1 / np.where(geo_, d2, 1.0) - 1, 1.0), x3)
out.append(f"{r:2d} {(-1) ** r:+d} " + " ".join(f"{v:7.3f}" for v in exv)
+ " (continuum B: " + " ".join(f"{contB[idl, ir]: .3e}" for idl in range(len(deltas))) + ")")
(OUT / "spinwave_mutual.txt").write_text("\n".join(out) + "\n")
print("\n".join(out))
if __name__ == "__main__":
main()
spinwave_tilt.py3 KB · raw
"""Check (item 1(b), leading order in 1/S): first correction to m(x) far from the push.
Quadratic form (units J S = 1): C_2 = 1/2 P^T T P + 1/2 Q^T V Q with
V = Lap + eta P_h, T = E V E = Lap^+ + eta P_h, eta = h_p/(J S), E = diag(eps).
Ground-state covariances: <QQ^T> = 1/2 T^(1/2) (T^(1/2) V T^(1/2))^(-1/2) T^(1/2), same for P with T<->V.
First correction: m(x) = eps_h (S - dS(x)), dS(x) = (<Q_x^2> + <P_x^2> - 1)/2.
Output: output/spinwave_tilt.txt
"""
from pathlib import Path
import numpy as np
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
def lattice(L, D):
n = L ** D
coords = np.array(np.unravel_index(np.arange(n), (L,) * D)).T
adj = np.zeros((n, n))
for d in range(D):
sh = coords.copy()
sh[:, d] = (sh[:, d] + 1) % L
j = np.ravel_multi_index(sh.T, (L,) * D)
adj[np.arange(n), j] += 1
adj[j, np.arange(n)] += 1
eps = (-1.0) ** coords.sum(axis=1)
return coords, adj, eps
def sqrtm_sym(A, power):
w, U = np.linalg.eigh(A)
return (U * w ** power) @ U.T
def dS_push(L, D, eta):
coords, adj, eps = lattice(L, D)
z = 2 * D
n = L ** D
V = z * np.eye(n) - adj
V[0, 0] += eta
T = (eps[:, None] * V) * eps[None, :]
Vh = sqrtm_sym(V, 0.5)
Th = (eps[:, None] * Vh) * eps[None, :]
covQ = 0.5 * Th @ sqrtm_sym(Th @ V @ Th, -0.5) @ Th
covP = 0.5 * Vh @ sqrtm_sym(Vh @ T @ Vh, -0.5) @ Vh
far = np.ravel_multi_index(tuple(np.full(D, L // 2)), (L,) * D)
dS = 0.5 * (np.diag(covQ) + np.diag(covP) - 1)
return dS[far], dS[0], np.max(np.abs(np.diag(covQ) - np.diag(covP)))
def dS3_bulk(L):
k = 2 * np.pi * (np.arange(L) + 0.5) / L
c = np.cos(k)
tot = 0.0
for c1 in c:
g = (c1 + c[:, None] + c[None, :]) / 3.0
tot += np.sum(1.0 / np.sqrt(1.0 - g ** 2))
W = tot / L ** 3
return 0.5 * (W - 1.0)
def main():
out = ["# Leading order in 1/S: first correction dS, m = eps_h (S - dS) + O(1/S)"]
out.append("\n## Cubic torus, L -> infinity, no push (midpoint sums): dS_3 = (W - 1)/2")
vals = {}
for L in (64, 128, 256):
vals[L] = dS3_bulk(L)
out.append(f"L={L:4d}: dS_3 = {vals[L]:.6f}")
rich = (4 * vals[256] - vals[128]) / 3
out.append(f"Richardson (error ~ L^-2): dS_3 = {rich:.5f}")
out.append("\n## Cubic torus with push (exact Gaussian), far site (L/2,L/2,L/2)")
out.append("L eta=h_p/(JS) dS(far) dS(h) zero-mode estimate sqrt(2z/(N eta))/2 max|<Q^2>-<P^2>|")
for L in (8, 12, 16):
for eta in (0.1, 1.0, 10.0):
f, h, sym = dS_push(L, 3, eta)
est = 0.5 * np.sqrt(12.0 / (L ** 3 * eta))
out.append(f"{L:3d} {eta:6.2f} {f:.5f} {h:.5f} {est:.5f} {sym:.1e}")
out.append("\n## Ring with push (exact Gaussian), eta = 1, far site L/2")
out.append("L dS(L/2) dS(L/2) - (1/pi) ln(L/2)")
ring = []
for L in (64, 128, 256, 512, 1024):
f, h, _ = dS_push(L, 1, 1.0)
ring.append((L, f))
out.append(f"{L:5d} {f:.5f} {f - np.log(L / 2) / np.pi:.5f}")
sl = np.polyfit(np.log([r[0] for r in ring[-3:]]), [r[1] for r in ring[-3:]], 1)[0]
out.append(f"slope d dS / d ln L (last three sizes) = {sl:.5f}; 1/pi = {1 / np.pi:.5f} (Dirichlet image at the push)")
(OUT / "spinwave_tilt.txt").write_text("\n".join(out) + "\n")
print("\n".join(out))
if __name__ == "__main__":
main()
output/ed_push.txt9 KB · raw
# ED check, item 1: exchange cells, J=1, push -h_p S^3_h at h=0 (class +); all sectors of M searched
## ring L=8: N=8, S=1/2, links=8, far sites [4] at distance 4; sectors searched: all M = -4, ..., 4
h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)
0.00 0.0 -3.6510934089 -3.1284190638 5.227e-01 True 0.000000 -0.000000
0.01 0.0 -3.6511215930 -3.1296915912 5.214e-01 True 0.002915 0.005637
0.10 0.0 -3.6539014094 -3.1431227850 5.108e-01 True 0.028913 0.055951
0.30 0.0 -3.6756584654 -3.1846251065 4.910e-01 True 0.081547 0.158772
1.00 0.0 -3.8706747638 -3.4156351786 4.550e-01 True 0.178327 0.361393
3.00 0.0 -4.7406497893 -4.3144845120 4.262e-01 True 0.217820 0.469512
10.00 0.0 -8.1656749710 -7.7600032466 4.057e-01 True 0.223623 0.496231
no push: lowest eigenvector in M=0 and unique (a total-spin singlet): True
## ring L=12: N=12, S=1/2, links=12, far sites [6] at distance 6; sectors searched: all M = -6, ..., 6
h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)
0.00 0.0 -5.3873909174 -5.0315434037 3.558e-01 True 0.000000 -0.000000
0.01 0.0 -5.3874232606 -5.0324032116 3.550e-01 True 0.003078 0.006468
0.10 0.0 -5.3906079037 -5.0424751866 3.481e-01 True 0.030399 0.063994
0.30 0.0 -5.4152047132 -5.0786464182 3.366e-01 True 0.083233 0.177593
1.00 0.0 -5.6229268437 -5.3035888107 3.193e-01 True 0.162829 0.374293
3.00 0.0 -6.5029771989 -6.1995713695 3.034e-01 True 0.187370 0.471246
10.00 0.0 -9.9314563940 -9.6416540673 2.898e-01 True 0.190367 0.496371
no push: lowest eigenvector in M=0 and unique (a total-spin singlet): True
## ring L=16: N=16, S=1/2, links=16, far sites [8] at distance 8; sectors searched: all M = -8, ..., 8
h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)
0.00 0.0 -7.1422963606 -6.8721066784 2.702e-01 True -0.000000 -0.000000
0.01 0.0 -7.1423316575 -6.8727610785 2.696e-01 True 0.003176 0.007059
0.10 0.0 -7.1458016524 -6.8812455850 2.646e-01 True 0.031226 0.069621
0.30 0.0 -7.1722882312 -6.9153754572 2.569e-01 True 0.083171 0.189514
1.00 0.0 -7.3866985161 -7.1395893046 2.471e-01 True 0.150071 0.379989
3.00 0.0 -8.2707749474 -8.0347711943 2.360e-01 True 0.167171 0.471891
10.00 0.0 -11.7005556368 -11.4746815979 2.259e-01 True 0.169000 0.496425
no push: lowest eigenvector in M=0 and unique (a total-spin singlet): True
## ring L=20: N=20, S=1/2, links=20, far sites [10] at distance 10; sectors searched: all M = -10, ..., 10
h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)
0.00 0.0 -8.9043865299 -8.6864409862 2.179e-01 True -0.000000 -0.000000
0.01 0.0 -8.9044241355 -8.6869727301 2.175e-01 True 0.003245 0.007520
0.10 0.0 -8.9081154951 -8.6945612781 2.136e-01 True 0.031764 0.073953
0.30 0.0 -8.9359905145 -8.7278252110 2.082e-01 True 0.082444 0.197877
1.00 0.0 -9.1544799698 -8.9525608798 2.019e-01 True 0.139769 0.383032
3.00 0.0 -10.0405891867 -9.8472945657 1.933e-01 True 0.152531 0.472206
10.00 0.0 -13.4710121778 -13.2857257643 1.853e-01 True 0.153753 0.496452
no push: lowest eigenvector in M=0 and unique (a total-spin singlet): True
16-object cubic cluster sites (representatives mod Lambda): [(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 1, 0), (0, 1, 1), (0, 1, 2), (0, 1, 3), (1, 0, 0), (1, 0, 1), (1, 0, 2), (1, 0, 3), (1, 1, 0), (1, 1, 1), (1, 1, 2), (1, 1, 3)]
links: each site x to x+e_i, i=1,2,3 (mod Lambda); 48 links, 6 distinct neighbours each
## cubic cluster 16: N=16, S=1/2, links=48, far sites [13, 15] at distance 3; sectors searched: all M = -8, ..., 8
h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)
0.00 0.0 -15.5783288133 -14.7965997285 7.817e-01 True -0.000000 -0.000000
0.01 0.0 -15.5783455887 -14.7972341444 7.811e-01 True 0.002435 0.003355
0.10 0.0 -15.5800039442 -14.8037863318 7.762e-01 True 0.024280 0.033454
0.30 0.0 -15.5932340835 -14.8236634632 7.696e-01 True 0.071115 0.098118
1.00 0.0 -15.7272156212 -14.9455491085 7.817e-01 True 0.192088 0.268207
3.00 0.0 -16.4641224768 -15.5885629797 8.756e-01 True 0.294607 0.424892
10.00 0.0 -19.7476521880 -18.7648142770 9.828e-01 True 0.330312 0.487059
no push: lowest eigenvector in M=0 and unique (a total-spin singlet): True
## ring L=6: N=6, S=2/2, links=6, far sites [3] at distance 3; sectors searched: all M = -6, ..., 6
h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)
0.00 0.0 -8.6174231818 -7.8967958192 7.206e-01 True -0.000000 -0.000000
0.01 0.0 -8.6174901294 -7.8984940829 7.190e-01 True 0.010352 0.013389
0.10 0.0 -8.6240752905 -7.9165725682 7.075e-01 True 0.102132 0.132193
0.30 0.0 -8.6745349327 -7.9736946079 7.008e-01 True 0.278218 0.362050
1.00 0.0 -9.0869706859 -8.3170486312 7.699e-01 True 0.546592 0.732523
3.00 0.0 -10.7994879963 -9.8711217775 9.284e-01 True 0.660086 0.919464
10.00 0.0 -17.5670872881 -16.5260310626 1.041e+00 True 0.694926 0.986112
no push: lowest eigenvector in M=0 and unique (a total-spin singlet): True
## ring L=8: N=8, S=2/2, links=8, far sites [4] at distance 4; sectors searched: all M = -8, ..., 8
h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)
0.00 0.0 -11.3369560779 -10.7434008235 5.936e-01 True 0.000000 0.000000
0.01 0.0 -11.3370291952 -10.7446882357 5.923e-01 True 0.010221 0.014622
0.10 0.0 -11.3442097178 -10.7595852021 5.846e-01 True 0.100476 0.143920
0.30 0.0 -11.3985873076 -10.8127173263 5.859e-01 True 0.267565 0.386678
1.00 0.0 -11.8253272598 -11.1625840354 6.627e-01 True 0.494463 0.745935
3.00 0.0 -13.5497976377 -12.7541078915 7.957e-01 True 0.582260 0.922190
10.00 0.0 -20.3241940732 -19.4463816695 8.778e-01 True 0.609254 0.986461
no push: lowest eigenvector in M=0 and unique (a total-spin singlet): True
## ring L=10: N=10, S=2/2, links=10, far sites [5] at distance 5; sectors searched: all M = -10, ..., 10
h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)
0.00 0.0 -14.0941299549 -13.5693220045 5.248e-01 True 0.000000 0.000000
0.01 0.0 -14.0942062020 -13.5703639701 5.238e-01 True 0.009913 0.015248
0.10 0.0 -14.1016869890 -13.5834553445 5.182e-01 True 0.097223 0.149801
0.30 0.0 -14.1579598525 -13.6349424056 5.230e-01 True 0.255412 0.398181
1.00 0.0 -14.5907289613 -13.9930452621 5.977e-01 True 0.456752 0.751132
3.00 0.0 -16.3197109015 -15.6120227946 7.077e-01 True 0.529673 0.923214
10.00 0.0 -23.0967344366 -22.3278084780 7.689e-01 True 0.550821 0.986603
no push: lowest eigenvector in M=0 and unique (a total-spin singlet): True
## ring L=12: N=12, S=2/2, links=12, far sites [6] at distance 6; sectors searched: all M = -12, ..., 12
h_p/J M_low E_low E_next (all sectors) gap unique m(far)=eps*<S3_far> m(h)
0.00 0.0 -16.8695561395 -16.3853596696 4.842e-01 True -0.000000 -0.000000
0.01 0.0 -16.8696337457 -16.3862386819 4.834e-01 True 0.009055 0.015520
0.10 0.0 -16.8772443685 -16.3981919225 4.791e-01 True 0.088705 0.152333
0.30 0.0 -16.9343124777 -16.4490478042 4.853e-01 True 0.231425 0.402886
1.00 0.0 -17.3693927041 -16.8150715758 5.543e-01 True 0.407686 0.753039
3.00 0.0 -19.1000782201 -18.4546315463 6.454e-01 True 0.470341 0.923626
10.00 0.0 -25.8782395163 -25.1861031627 6.921e-01 True 0.488416 0.986670
no push: lowest eigenvector in M=0 and unique (a total-spin singlet): True
output/spinwave_mutual.txt17 KB · raw
# Mutual cost of two held sources, leading order in 1/S, cubic torus (units as stated per block)
(5.8) with kappa_h = J S, kappa_a = h_p against direct inverse (L=6): max error 4.88e-14
## (1) Transverse sources: separation part J E_r/(g1 g2 (m1perp.m2perp))
single objects: -eps1 eps2 ghat(r); cells (x, x+e2), + on the first object: -eps1 eps2 sum ghat
ghat_inf(0) = 0.2527310 (gamma_3 = 0.2527310)
r eps1eps2 single: J E_r/(g1g2) 4 pi r ghat_inf(r) cells: J E_r/(g1g2) pi r sum ghat_inf torus check: ghat_L(r)-ghat_L(8) for L=16, 32, 64 vs ghat_inf(r)-ghat_inf(8)
2 +1 -0.042889 1.0779 -0.157642 0.9905 0.029359 0.032566 0.032862 0.032901
3 -1 0.027545 1.0384 0.106137 1.0003 0.014233 0.017247 0.017521 0.017557
4 +1 -0.020289 1.0198 -0.079621 1.0005 0.007303 0.010028 0.010269 0.010301
5 -1 0.016101 1.0117 0.063679 1.0003 0.003573 0.005889 0.006087 0.006113
6 +1 -0.013365 1.0077 -0.053058 1.0001 0.001456 0.003213 0.003358 0.003377
7 -1 0.011430 1.0055 0.045475 1.0001 0.000347 0.001353 0.001432 0.001442
8 +1 -0.009988 1.0041 -0.039790 1.0000 0.000000 0.000000 0.000000 0.000000
single object at 0, cell (r e1, (r+1) e1) along the separation: 4 pi R [ghat(r)+ghat(r+1)]/2 with R = midpoint separation r+1/2, and with R = end-point separation r
2 midpoint: 1.10638 end point: 0.88511
3 midpoint: 1.05192 end point: 0.90165
4 midpoint: 1.02890 end point: 0.91458
5 midpoint: 1.01826 end point: 0.92569
6 midpoint: 1.01265 end point: 0.93475
7 midpoint: 1.00932 end point: 0.94203
8 midpoint: 1.00717 end point: 0.94792
## Continuum coefficients
along n, single: J S E/(g1 g2) = -eps1 eps2 kappa_par / r^3, kappa_par = sqrt(2z)/(32 pi^3) = 0.0034913
held spin, r >> l_b: S E/(J sig1 sig2) = +eps1 eps2 lambda / r^5, lambda = 3 sqrt3/(8 pi^3) = 0.0209480
changed links, separation r e1: J E/(d1 d2 S) * r^7 = -sqrt3/(8 pi^3) a; numerical integral against a:
par (e1,e1) eps1eps2=+1: integral -0.06749907 closed form a=9.666667: -0.06749907
par (e1,e1) eps1eps2=-1: integral -0.03258576 closed form a=4.666667: -0.03258576
perp same (e2,e2) eps1eps2=+1: integral -0.00640077 closed form a=0.916667: -0.00640077
perp same (e2,e2) eps1eps2=-1: integral -0.01221966 closed form a=1.750000: -0.01221966
perp cross (e2,e3) eps1eps2=+1: integral -0.00058189 closed form a=0.083333: -0.00058189
perp cross (e2,e3) eps1eps2=-1: integral -0.00058189 closed form a=0.083333: -0.00058189
mixed (e1,e2) eps1eps2=+1: integral 0.00349133 closed form a=-0.500000: 0.00349133
mixed (e1,e2) eps1eps2=-1: integral 0.00349133 closed form a=-0.500000: 0.00349133
## (2) Static quadratic sources, lattice (zero modes left out); ratio = lattice / continuum
columns: r, eps1eps2, along single [J S E/(g1g2), ratio], along cell [.., ratio vs 4x], then J E/(d1d2S) and ratio for links: par (e1 at 0, e1 at r e1), perp same (e2, e2), perp cross (e2, e3), mixed (e1, e2); link continuum at the midpoint separation R
L = 32
2 +1 -4.624e-04 1.060 -1.491e-03 0.854 -1.274e-03 2.417 -1.174e-04 2.347 5.574e-06 1.626 1.632e-06 0.342
3 -1 1.198e-04 0.927 4.142e-04 0.801 -2.430e-05 1.631 -1.119e-05 2.002 -7.837e-07 2.114 -1.343e-06 6.120
4 +1 -3.922e-05 0.719 -1.480e-04 0.678 -6.880e-06 1.670 -6.874e-07 1.759 1.380e-08 -1.438 1.286e-07 1.139
5 -1 1.640e-05 0.587 6.200e-05 0.555 -5.446e-07 1.306 -2.119e-07 1.355 -1.049e-08 1.233 -3.651e-09 -0.752
6 +1 -7.170e-06 0.444 -2.790e-05 0.432 -3.006e-07 1.247 -2.761e-08 1.207 -1.060e-09 0.795 9.699e-09 1.102
7 -1 3.351e-06 0.329 1.293e-05 0.318 -4.524e-08 1.143 -1.714e-08 1.155 -8.316e-10 1.098 8.711e-10 0.714
8 +1 -1.510e-06 0.221 -5.918e-06 0.217 -3.768e-08 1.171 -3.017e-09 0.988 -2.295e-10 1.045 1.383e-09 1.042
9 -1 6.659e-07 0.139 2.578e-06 0.135 -7.892e-09 1.158 -2.875e-09 1.125 -1.522e-10 1.199 2.089e-10 0.695
10 +1 -2.585e-07 0.074 -1.014e-06 0.073 -8.418e-09 1.247 -5.211e-10 0.814 -6.725e-11 1.338 2.734e-10 0.921
11 -1 8.809e-08 0.034 3.369e-07 0.032 -2.283e-09 1.366 -7.612e-10 1.214 -4.936e-11 1.606 3.147e-11 0.351
12 +1 -2.467e-08 0.012 -9.778e-08 0.012 -2.906e-09 1.543 -1.184e-10 0.663 -3.074e-11 2.092 5.073e-11 0.589
L = 64
2 +1 -4.970e-04 1.139 -1.616e-03 0.926 -1.274e-03 2.416 -1.174e-04 2.348 5.569e-06 1.624 1.623e-06 0.340
3 -1 1.393e-04 1.077 4.876e-04 0.943 -2.430e-05 1.631 -1.119e-05 2.002 -7.827e-07 2.111 -1.341e-06 6.110
4 +1 -5.146e-05 0.943 -1.957e-04 0.897 -6.868e-06 1.667 -6.922e-07 1.772 1.365e-08 -1.422 1.280e-07 1.134
5 -1 2.483e-05 0.889 9.493e-05 0.850 -5.434e-07 1.303 -2.117e-07 1.353 -1.040e-08 1.222 -3.297e-09 -0.679
6 +1 -1.312e-05 0.812 -5.144e-05 0.796 -2.969e-07 1.231 -2.873e-08 1.256 -1.047e-09 0.785 9.641e-09 1.095
7 -1 7.695e-06 0.756 3.007e-05 0.738 -4.446e-08 1.124 -1.699e-08 1.145 -7.977e-10 1.053 1.003e-09 0.821
8 +1 -4.679e-06 0.686 -1.850e-05 0.678 -3.586e-08 1.114 -3.397e-09 1.113 -2.103e-10 0.958 1.401e-09 1.055
9 -1 2.998e-06 0.626 1.181e-05 0.616 -7.274e-09 1.068 -2.759e-09 1.080 -1.299e-10 1.023 2.856e-10 0.950
10 +1 -1.950e-06 0.559 -7.737e-06 0.554 -7.248e-09 1.074 -6.713e-10 1.049 -4.991e-11 0.993 3.073e-10 1.035
11 -1 1.306e-06 0.498 5.163e-06 0.492 -1.750e-09 1.047 -6.603e-10 1.053 -3.136e-11 1.020 8.712e-11 0.972
12 +1 -8.771e-07 0.434 -3.487e-06 0.431 -1.997e-09 1.060 -1.788e-10 1.001 -1.486e-11 1.011 8.804e-11 1.022
L = 128
2 +1 -5.071e-04 1.162 -1.653e-03 0.947 -1.274e-03 2.416 -1.174e-04 2.348 5.569e-06 1.624 1.623e-06 0.340
3 -1 1.453e-04 1.124 5.105e-04 0.987 -2.430e-05 1.631 -1.119e-05 2.002 -7.827e-07 2.111 -1.341e-06 6.109
4 +1 -5.554e-05 1.018 -2.116e-04 0.970 -6.867e-06 1.667 -6.925e-07 1.773 1.364e-08 -1.421 1.279e-07 1.134
5 -1 2.784e-05 0.997 1.068e-04 0.956 -5.434e-07 1.303 -2.117e-07 1.353 -1.039e-08 1.222 -3.278e-09 -0.675
6 +1 -1.545e-05 0.956 -6.063e-05 0.938 -2.967e-07 1.230 -2.881e-08 1.260 -1.047e-09 0.785 9.634e-09 1.094
7 -1 9.546e-06 0.938 3.740e-05 0.919 -4.445e-08 1.123 -1.699e-08 1.145 -7.968e-10 1.052 1.008e-09 0.826
8 +1 -6.180e-06 0.906 -2.447e-05 0.897 -3.578e-08 1.112 -3.429e-09 1.123 -2.102e-10 0.958 1.399e-09 1.054
9 -1 4.239e-06 0.885 1.674e-05 0.874 -7.265e-09 1.066 -2.757e-09 1.079 -1.294e-10 1.020 2.881e-10 0.958
10 +1 -2.985e-06 0.855 -1.186e-05 0.849 -7.205e-09 1.067 -6.865e-10 1.072 -4.968e-11 0.988 3.070e-10 1.034
11 -1 2.178e-06 0.830 8.638e-06 0.823 -1.743e-09 1.042 -6.589e-10 1.051 -3.104e-11 1.010 8.851e-11 0.987
12 +1 -1.616e-06 0.800 -6.433e-06 0.796 -1.970e-09 1.046 -1.869e-10 1.046 -1.464e-11 0.996 8.812e-11 1.023
ratio extrapolated to L -> infinity: Aitken from L = 32, 64, 128 (geometric finite-size error); the L = 128 value where the change from 64 to 128 is below 1e-3 or not geometric
2 1.172 0.956 2.416 2.348 1.624 0.340
3 1.144 1.007 1.631 2.002 2.111 6.109
4 1.055 1.007 1.667 1.773 -1.421 1.134
5 1.057 1.016 1.303 1.353 1.222 -0.675
6 1.047 1.029 1.230 1.260 0.785 1.094
7 1.073 1.053 1.123 1.145 1.052 0.826
8 1.104 1.095 1.112 1.124 0.958 1.054
9 1.180 1.170 1.066 1.079 1.020 0.959
10 1.322 1.318 1.067 1.075 0.988 1.034
11 1.672 1.675 1.042 1.051 1.010 0.988
12 3.173 3.225 1.046 1.053 0.996 1.023
## (3) Held spin R1, retarded vertex (Step 12); J S E/(|g'1 g'2| s1 s2), sgn sigma1 = sgn sigma2, so vs = eps1 eps2 = (-1)^r
continuum crossover function F(beta) (equal held spins), with its small-beta form 1 - pi (3+vs) beta/4 and large-beta form -2 vs/beta^2 + 24 (1+2 vs)/beta^4
vs=+1 beta= 0.01: F = 9.699956e-01 small-beta form 9.685841e-01 large-beta form 7.199980e+09
vs=+1 beta= 0.03: F = 9.155432e-01 small-beta form 9.057522e-01 large-beta form 8.888667e+07
vs=+1 beta= 0.10: F = 7.604598e-01 small-beta form 6.858407e-01 large-beta form 7.198000e+05
vs=+1 beta= 0.30: F = 4.754809e-01 small-beta form 5.752220e-02 large-beta form 8.866667e+03
vs=+1 beta= 1.00: F = 1.004787e-01 small-beta form -2.141593e+00 large-beta form 7.000000e+01
vs=+1 beta= 3.00: F = -3.875257e-02 small-beta form -8.424778e+00 large-beta form 6.666667e-01
vs=+1 beta= 10.00: F = -1.496680e-02 small-beta form -3.041593e+01 large-beta form -1.280000e-02
vs=+1 beta= 30.00: F = -2.137889e-03 small-beta form -9.324778e+01 large-beta form -2.133333e-03
vs=+1 beta= 100.00: F = -1.992836e-04 small-beta form -3.131593e+02 large-beta form -1.992800e-04
vs=-1 beta= 0.01: F = 9.847956e-01 small-beta form 9.842920e-01 large-beta form -2.399980e+09
vs=-1 beta= 0.03: F = 9.564331e-01 small-beta form 9.528761e-01 large-beta form -2.962741e+07
vs=-1 beta= 0.10: F = 8.708995e-01 small-beta form 8.429204e-01 large-beta form -2.398000e+05
vs=-1 beta= 0.30: F = 6.929129e-01 small-beta form 5.287611e-01 large-beta form -2.940741e+03
vs=-1 beta= 1.00: F = 3.785504e-01 small-beta form -5.707963e-01 large-beta form -2.200000e+01
vs=-1 beta= 3.00: F = 1.241269e-01 small-beta form -3.712389e+00 large-beta form -7.407407e-02
vs=-1 beta= 10.00: F = 1.808965e-02 small-beta form -1.470796e+01 large-beta form 1.760000e-02
vs=-1 beta= 30.00: F = 2.193524e-03 small-beta form -4.612389e+01 large-beta form 2.192593e-03
vs=-1 beta= 100.00: F = 1.997607e-04 small-beta form -1.560796e+02 large-beta form 1.997600e-04
vs=+1: F changes sign at beta_0 = 1.69838 (r_0 = beta_0 l_b / 2); vs=-1: F > 0 at every beta
lattice against continuum: D = |g'|/J = 0, 0.2, 0.5, 2; l_b = sqrt(2z)/D = inf, 17.32, 6.93, 1.73
continuum A = (8.18): -kappa_par F(beta)/r^3; continuum B = (8.18) with the off-diagonal part of the covariance (nonzero for eps1 = eps2 only)
columns per D: lattice value, ratio lattice/A, ratio lattice/B
L = 32
2 +1 -4.6238e-04 1.059 1.106 -3.1747e-04 1.309 1.206 -1.9941e-04 1.753 1.269 -3.9503e-05 -3.139 1.516
3 -1 -1.1983e-04 0.927 0.927 -8.6954e-05 1.018 1.018 -5.8491e-05 1.080 1.080 -1.5967e-05 1.208 1.208
4 +1 -3.9216e-05 0.719 0.726 -1.9221e-05 1.055 0.911 -7.4366e-06 2.036 1.054 5.3250e-07 0.271 0.662
5 -1 -1.6403e-05 0.587 0.587 -1.0329e-05 0.696 0.696 -5.9281e-06 0.758 0.758 -1.0994e-06 0.837 0.837
6 +1 -7.1698e-06 0.444 0.446 -2.6543e-06 0.809 0.669 -5.9044e-07 -14.772 1.047 1.6996e-07 0.427 0.596
7 -1 -3.3509e-06 0.329 0.329 -1.8867e-06 0.423 0.423 -9.6157e-07 0.473 0.473 -1.4012e-07 0.519 0.519
8 +1 -1.5096e-06 0.221 0.222 -4.2363e-07 0.515 0.404 -3.3492e-08 -0.170 -0.903 3.2348e-08 0.282 0.340
9 -1 -6.6588e-07 0.139 0.139 -3.3407e-07 0.190 0.190 -1.5174e-07 0.213 0.213 -1.8610e-08 0.231 0.231
10 +1 -2.5853e-07 0.074 0.074 -4.9564e-08 0.212 0.152 4.0323e-09 0.030 0.051 3.1615e-09 0.076 0.085
11 -1 -8.8087e-08 0.034 0.034 -3.8477e-08 0.047 0.047 -1.6624e-08 0.055 0.055 -3.0898e-09 0.101 0.101
12 +1 -2.4670e-08 0.012 0.012 -8.1078e-09 0.130 0.077 -5.5636e-09 -0.069 -0.093 -1.1010e-09 -0.062 -0.067
cross-check D = 0 (frequency domain) against the along-n single column (time domain), L = 32: max relative difference 1.31e-04
L = 64
2 +1 -4.9703e-04 1.139 1.188 -3.2820e-04 1.353 1.247 -2.0223e-04 1.778 1.287 -3.9294e-05 -3.123 1.508
3 -1 -1.3928e-04 1.077 1.077 -9.7057e-05 1.136 1.136 -6.3638e-05 1.175 1.175 -1.6878e-05 1.277 1.277
4 +1 -5.1458e-05 0.943 0.953 -2.2335e-05 1.226 1.058 -7.9038e-06 2.164 1.120 7.1046e-07 0.362 0.883
5 -1 -2.4828e-05 0.889 0.889 -1.4413e-05 0.971 0.971 -7.8602e-06 1.005 1.005 -1.3845e-06 1.053 1.053
6 +1 -1.3124e-05 0.812 0.816 -3.9294e-06 1.197 0.991 -6.6556e-07 -16.652 1.181 2.6497e-07 0.665 0.930
7 -1 -7.6949e-06 0.756 0.756 -3.8731e-06 0.868 0.868 -1.8478e-06 0.909 0.909 -2.5545e-07 0.946 0.946
8 +1 -4.6787e-06 0.686 0.688 -9.9690e-07 1.211 0.950 -1.7632e-08 -0.090 -0.475 8.1739e-08 0.712 0.859
9 -1 -2.9979e-06 0.626 0.626 -1.3398e-06 0.760 0.760 -5.7561e-07 0.806 0.806 -6.7748e-08 0.839 0.839
10 +1 -1.9500e-06 0.559 0.559 -2.9846e-07 1.277 0.918 3.8749e-08 0.292 0.489 2.7983e-08 0.670 0.757
11 -1 -1.3056e-06 0.498 0.498 -5.2592e-07 0.640 0.640 -2.0720e-07 0.686 0.686 -2.1722e-08 0.714 0.714
12 +1 -8.7709e-07 0.434 0.435 -9.5241e-08 1.522 0.908 2.8430e-08 0.353 0.478 1.0335e-08 0.579 0.630
L = 128
2 +1 -5.0712e-04 1.162 1.213 -3.2958e-04 1.359 1.252 -2.0238e-04 1.779 1.288 -3.9235e-05 -3.118 1.506
3 -1 -1.4529e-04 1.124 1.124 -9.9216e-05 1.161 1.161 -6.4553e-05 1.192 1.192 -1.7015e-05 1.287 1.287
4 +1 -5.5538e-05 1.018 1.029 -2.2735e-05 1.248 1.077 -7.8727e-06 2.156 1.116 7.4691e-07 0.381 0.928
5 -1 -2.7844e-05 0.997 0.997 -1.5411e-05 1.038 1.038 -8.2500e-06 1.055 1.055 -1.4330e-06 1.090 1.090
6 +1 -1.5445e-05 0.956 0.960 -4.0969e-06 1.248 1.033 -6.2131e-07 -15.545 1.102 2.8560e-07 0.717 1.002
7 -1 -9.5458e-06 0.938 0.938 -4.4457e-06 0.997 0.997 -2.0583e-06 1.013 1.013 -2.7876e-07 1.033 1.033
8 +1 -6.1799e-06 0.906 0.909 -1.0765e-06 1.307 1.026 2.0663e-08 0.105 0.557 9.4103e-08 0.820 0.989
9 -1 -4.2391e-06 0.885 0.885 -1.7021e-06 0.966 0.966 -7.0242e-07 0.984 0.984 -8.0674e-08 0.999 0.999
10 +1 -2.9846e-06 0.855 0.856 -3.3780e-07 1.445 1.039 6.9143e-08 0.521 0.873 3.5771e-08 0.857 0.967
11 -1 -2.1783e-06 0.830 0.830 -7.6770e-07 0.934 0.934 -2.8830e-07 0.955 0.955 -2.9477e-08 0.968 0.968
12 +1 -1.6157e-06 0.800 0.801 -1.1424e-07 1.825 1.089 5.1893e-08 0.645 0.872 1.5410e-08 0.863 0.940
L = 256
2 +1 -5.0987e-04 1.168 1.219 -3.2969e-04 1.359 1.252 -2.0237e-04 1.779 1.288 -3.9225e-05 -3.117 1.505
3 -1 -1.4698e-04 1.137 1.137 -9.9587e-05 1.166 1.166 -6.4690e-05 1.195 1.195 -1.7033e-05 1.289 1.289
4 +1 -5.6727e-05 1.040 1.051 -2.2756e-05 1.249 1.078 -7.8563e-06 2.151 1.113 7.5255e-07 0.384 0.935
5 -1 -2.8751e-05 1.029 1.029 -1.5592e-05 1.051 1.051 -8.3112e-06 1.063 1.063 -1.4399e-06 1.096 1.096
6 +1 -1.6168e-05 1.000 1.005 -4.0997e-06 1.249 1.034 -6.0795e-07 -15.210 1.079 2.8886e-07 0.725 1.013
7 -1 -1.0142e-05 0.996 0.996 -4.5556e-06 1.021 1.021 -2.0932e-06 1.030 1.030 -2.8229e-07 1.046 1.046
8 +1 -6.6809e-06 0.980 0.982 -1.0734e-06 1.304 1.023 3.1052e-08 0.158 0.837 9.6145e-08 0.838 1.010
9 -1 -4.6684e-06 0.975 0.975 -1.7762e-06 1.008 1.008 -7.2475e-07 1.015 1.015 -8.2754e-08 1.025 1.025
10 +1 -3.3566e-06 0.961 0.963 -3.3280e-07 1.424 1.023 7.7243e-08 0.582 0.975 3.7132e-08 0.889 1.004
11 -1 -2.5045e-06 0.955 0.955 -8.2075e-07 0.999 0.999 -3.0360e-07 1.006 1.006 -3.0821e-08 1.012 1.012
12 +1 -1.9039e-06 0.942 0.943 -1.0874e-07 1.737 1.036 5.8278e-08 0.724 0.979 1.6361e-08 0.916 0.998
ratio lattice/B extrapolated to L -> infinity: Aitken from L = 64, 128, 256 (as in block (2))
2 +1 1.222 1.252 1.288 1.505 (continuum B: -4.182e-04 -2.632e-04 -1.572e-04 -2.605e-05)
3 -1 1.142 1.167 1.195 1.289 (continuum B: -1.293e-04 -8.542e-05 -5.415e-05 -1.322e-05)
4 +1 1.060 1.079 1.111 0.937 (continuum B: -5.398e-05 -2.110e-05 -7.056e-06 8.046e-07)
5 -1 1.043 1.053 1.065 1.096 (continuum B: -2.793e-05 -1.484e-05 -7.818e-06 -1.314e-06)
6 +1 1.025 1.034 1.068 1.016 (continuum B: -1.609e-05 -3.966e-06 -5.637e-07 2.850e-07)
7 -1 1.024 1.027 1.033 1.048 (continuum B: -1.018e-05 -4.460e-06 -2.033e-06 -2.699e-07)
8 +1 1.019 1.023 0.941 1.014 (continuum B: -6.801e-06 -1.050e-06 3.710e-08 9.519e-08)
9 -1 1.022 1.019 1.022 1.030 (continuum B: -4.789e-06 -1.762e-06 -7.139e-07 -8.074e-08)
10 +1 1.023 1.023 1.012 1.012 (continuum B: -3.486e-06 -3.253e-07 7.924e-08 3.698e-08)
11 -1 1.029 1.017 1.017 1.022 (continuum B: -2.623e-06 -8.218e-07 -3.019e-07 -3.044e-08)
12 +1 1.035 1.036 1.019 1.011 (continuum B: -2.018e-06 -1.049e-07 5.952e-08 1.639e-08)
output/spinwave_tilt.txt2 KB · raw
# Leading order in 1/S: first correction dS, m = eps_h (S - dS) + O(1/S)
## Cubic torus, L -> infinity, no push (midpoint sums): dS_3 = (W - 1)/2
L= 64: dS_3 = 0.078304
L= 128: dS_3 = 0.078344
L= 256: dS_3 = 0.078354
Richardson (error ~ L^-2): dS_3 = 0.07836
## Cubic torus with push (exact Gaussian), far site (L/2,L/2,L/2)
L eta=h_p/(JS) dS(far) dS(h) zero-mode estimate sqrt(2z/(N eta))/2 max|<Q^2>-<P^2>|
8 0.10 0.31135 0.29896 0.24206 0.0e+00
8 1.00 0.15197 0.11065 0.07655 0.0e+00
8 10.00 0.11169 0.01999 0.02421 0.0e+00
12 0.10 0.20637 0.19853 0.13176 0.0e+00
12 1.00 0.11960 0.08897 0.04167 0.0e+00
12 10.00 0.09770 0.01863 0.01318 0.0e+00
16 0.10 0.16197 0.15613 0.08558 0.0e+00
16 1.00 0.10561 0.07974 0.02706 0.0e+00
16 10.00 0.09139 0.01805 0.00856 0.0e+00
## Ring with push (exact Gaussian), eta = 1, far site L/2
L dS(L/2) dS(L/2) - (1/pi) ln(L/2)
64 1.10378 0.00061
128 1.31488 -0.00893
256 1.53065 -0.01380
512 1.74882 -0.01626
1024 1.96822 -0.01750
slope d dS / d ln L (last three sizes) = 0.31564; 1/pi = 0.31831 (Dirichlet image at the push)
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