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.py7 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.
Far-site decomposition (v6): dS(far) = dS_3 + B_L + coll * w_far + rest, with the collective term
coll = sqrt(2z/(N eta_eff))/2, the relaxed restoring coefficient of the torus eta_eff = (1/eta + ghat_L(0))^-1
(ghat_L the pseudo-inverse of the Laplacian on the torus), the far-site weight of the soft eigenvector
v0 ~ 1 - eta_eff ghat_L(. - h), w_far = N v0(far)^2 = (1 - eta_eff ghat_L(far))^2 / (1 + eta_eff^2 |ghat_L|^2 / N),
and the bulk finite-size term B_L = (1/2)[(1/N) sum_{k not 0, pi} (1-gamma_k^2)^(-1/2) - 1] - dS_3
(torus without push and zero modes). The soft mode of V (lowest eigenvalue and eigenvector) is also
computed exactly and compared with N lambda_0 = eta_eff and with w_far.
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, :]
wV, UV = np.linalg.eigh(V)
Vh = (UV * np.sqrt(wV)) @ UV.T
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)
soft = (n * wV[0], n * UV[far, 0] ** 2) # exact soft mode of V: N lambda_0 and N v0(far)^2
return dS[far], dS[0], np.max(np.abs(np.diag(covQ) - np.diag(covP))), soft
def ghat_torus(L):
"""Cubic torus: ghat_L(0), ghat_L(far) with far = (L/2,L/2,L/2), and |ghat_L|^2 = sum_x ghat_L(x)^2."""
k = 2 * np.pi * np.arange(L) / L
c = np.cos(k)
lam = (6.0 - 2.0 * (c[:, None, None] + c[None, :, None] + c[None, None, :])).ravel()
sgn = ((-1.0) ** np.arange(L))
phase = (sgn[:, None, None] * sgn[None, :, None] * sgn[None, None, :]).ravel() # e^{i k.far}
keep = lam > 1e-12
n = L ** 3
return (np.sum(1.0 / lam[keep]) / n, np.sum(phase[keep] / lam[keep]) / n,
np.sum(1.0 / lam[keep] ** 2) / n)
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 dS_torus_nozero(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).ravel()
keep = np.abs(np.abs(g) - 1.0) > 1e-12
return 0.5 * (np.sum(1.0 / np.sqrt(1.0 - g[keep] ** 2)) / L ** 3 - 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>|")
far, soft, est = {}, {}, {}
for L in (8, 12, 16):
for eta in (0.1, 1.0, 10.0):
f, h, sym, sm = dS_push(L, 3, eta)
far[(L, eta)], soft[(L, eta)] = f, sm
est[(L, eta)] = 0.5 * np.sqrt(12.0 / (L ** 3 * eta))
out.append(f"{L:3d} {eta:6.2f} {f:.5f} {h:.5f} {est[(L, eta)]:.5f} {sym:.1e}")
out.append("\n## Far site: torus Green function and bulk finite-size term B_L (torus without push and zero modes, minus dS_3)")
out.append("L ghat_L(0) ghat_L(far) |ghat_L|^2 B_L")
gt, BL = {}, {}
for L in (8, 12, 16):
gt[L] = ghat_torus(L)
BL[L] = dS_torus_nozero(L) - rich
out.append(f"{L:3d} {gt[L][0]:.5f} {gt[L][1]:+.5f} {gt[L][2]:.5f} {BL[L]:+.5f}")
out.append("\n## Far site: soft mode of V, exact against the rank-one formulas (eta_eff = (1/eta + ghat_L(0))^-1)")
out.append("L eta eta_eff N lambda_0 (exact) N v0(far)^2 (exact) w_far (formula) 1 - 2 eta_eff ghat_L(far)")
for L in (8, 12, 16):
g0, gf, g2 = gt[L]
for eta in (0.1, 1.0, 10.0):
ee = 1.0 / (1.0 / eta + g0)
wf = (1 - ee * gf) ** 2 / (1 + ee ** 2 * g2 / L ** 3)
out.append(f"{L:3d} {eta:6.2f} {ee:.5f} {soft[(L, eta)][0]:.5f} {soft[(L, eta)][1]:.5f}"
f" {wf:.5f} {1 - 2 * ee * gf:.5f}")
out.append("\n## Far site: decomposition dS(far) - dS_3 = B_L + coll * w_far + rest, coll = sqrt(2z/(N eta_eff))/2")
out.append("L eta dS(far)-dS_3 coll coll*w_far rest(coll with h_p) rest(coll, w_far=1) rest(coll*w_far)")
for L in (8, 12, 16):
g0, gf, g2 = gt[L]
for eta in (0.1, 1.0, 10.0):
ee = 1.0 / (1.0 / eta + g0)
coll = 0.5 * np.sqrt(12.0 / (L ** 3 * ee))
wf = (1 - ee * gf) ** 2 / (1 + ee ** 2 * g2 / L ** 3)
d = far[(L, eta)] - rich - BL[L]
out.append(f"{L:3d} {eta:6.2f} {far[(L, eta)] - rich:.5f} {coll:.5f} {coll * wf:.5f} "
f"{d - est[(L, eta)]:+.5f} {d - coll:+.5f} {d - coll * wf:+.5f}")
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.txt4 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
## Far site: torus Green function and bulk finite-size term B_L (torus without push and zero modes, minus dS_3)
L ghat_L(0) ghat_L(far) |ghat_L|^2 B_L
8 0.22461 -0.00837 0.09515 -0.01218
12 0.23394 -0.00543 0.13818 -0.00542
16 0.23863 -0.00404 0.18092 -0.00305
## Far site: soft mode of V, exact against the rank-one formulas (eta_eff = (1/eta + ghat_L(0))^-1)
L eta eta_eff N lambda_0 (exact) N v0(far)^2 (exact) w_far (formula) 1 - 2 eta_eff ghat_L(far)
8 0.10 0.09780 0.09780 1.00164 1.00164 1.00164
8 1.00 0.81659 0.81649 1.01365 1.01359 1.01367
8 10.00 3.08066 3.07522 1.05112 1.05039 1.05158
12 0.10 0.09771 0.09771 1.00106 1.00106 1.00106
12 1.00 0.81041 0.81037 1.00880 1.00877 1.00881
12 10.00 2.99453 2.99238 1.03237 1.03207 1.03255
16 0.10 0.09767 0.09767 1.00079 1.00079 1.00079
16 1.00 0.80734 0.80732 1.00651 1.00650 1.00652
16 10.00 2.95307 2.95193 1.02376 1.02359 1.02385
## Far site: decomposition dS(far) - dS_3 = B_L + coll * w_far + rest, coll = sqrt(2z/(N eta_eff))/2
L eta dS(far)-dS_3 coll coll*w_far rest(coll with h_p) rest(coll, w_far=1) rest(coll*w_far)
8 0.10 0.23299 0.24476 0.24517 +0.00310 +0.00040 -0.00000
8 1.00 0.07361 0.08471 0.08586 +0.00924 +0.00108 -0.00008
8 10.00 0.03333 0.04361 0.04581 +0.02130 +0.00190 -0.00030
12 0.10 0.12801 0.13329 0.13344 +0.00167 +0.00014 -0.00000
12 1.00 0.04124 0.04628 0.04669 +0.00500 +0.00038 -0.00002
12 10.00 0.01934 0.02408 0.02485 +0.01158 +0.00068 -0.00009
16 0.10 0.08361 0.08660 0.08667 +0.00108 +0.00007 -0.00000
16 1.00 0.02725 0.03012 0.03032 +0.00324 +0.00019 -0.00001
16 10.00 0.01303 0.01575 0.01612 +0.00752 +0.00033 -0.00004
## 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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