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848d4b7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | from mpmath import mp
def lieb_liniger_e(gamma, n_nodes=160, dps=140):
mp.dps = dps
gamma = mp.mpf(gamma)
# Gauss–Legendre nodes/weights on [-1,1]
X, W = mp.gauss_quadrature(n_nodes, "legendre")
D = [[(X[j] - X[i])**2 for j in range(n_nodes)] for i in range(n_nodes)]
two_pi = 2 * mp.pi
rhs = mp.mpf(1) / two_pi
def gamma_and_e_from_alpha(alpha):
alpha = mp.mpf(alpha)
alpha2 = alpha * alpha
coef = (mp.mpf(1) / two_pi) * (2 * alpha)
A = mp.matrix(n_nodes)
b = mp.matrix(n_nodes, 1)
for i in range(n_nodes):
b[i] = rhs
for i in range(n_nodes):
for j in range(n_nodes):
val = mp.mpf(1) if i == j else mp.mpf(0)
val -= coef * W[j] / (alpha2 + D[i][j])
A[i, j] = val
g = mp.lu_solve(A, b)
I0 = mp.mpf(0)
I2 = mp.mpf(0)
for i in range(n_nodes):
I0 += W[i] * g[i]
I2 += W[i] * g[i] * (X[i] ** 2)
gam = alpha / I0
e = I2 / (I0 ** 3)
return gam, e
# Secant inversion for alpha(gamma)
# Two decent initial guesses: weak-coupling and strong-coupling heuristics
a0 = mp.sqrt(gamma) / 2
a1 = gamma / mp.pi + mp.mpf("0.2")
f0 = gamma_and_e_from_alpha(a0)[0] - gamma
f1 = gamma_and_e_from_alpha(a1)[0] - gamma
for _ in range(8):
a2 = a1 - f1 * (a1 - a0) / (f1 - f0)
a0, f0, a1, f1 = a1, f1, a2, gamma_and_e_from_alpha(a2)[0] - gamma
gam, e = gamma_and_e_from_alpha(a1)
return e
if __name__ == "__main__":
for g in ["0.5", "1.0", "2.0", "5.0", "10.0"]:
val = lieb_liniger_e(g, n_nodes=160, dps=140)
print(g, mp.nstr(val, 90))
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