89 lines
3 KiB
Python
89 lines
3 KiB
Python
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"""
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Exact SO(3) isotypic projectors on (R^3)^{\otimes k} via the Casimir
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operator J^2, instead of Monte-Carlo character averaging.
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The spin-1 (vector) generators in the real Cartesian basis are
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(J_a)_{bc} = -i * epsilon_{abc} (standard so(3) generators)
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Built exactly with sympy, then verified to satisfy [J_a,J_b] = i eps_abc J_c
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and J^2 = J_x^2+J_y^2+J_z^2 = 2*I_3 (i.e. j=1, j(j+1)=2) -- symbolically exact.
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For k copies, total J_a = sum_{l=1}^k I x ... x J_a^{(l)} x ... x I,
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J^2_total is Hermitian on (C^3)^{\otimes k}; its eigenspaces are EXACTLY
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the isotypic components (eigenvalue j(j+1)). No integration needed.
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"""
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import numpy as np
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import sympy as sp
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i = sp.I
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eps = lambda a,b,c: sp.LeviCivita(a,b,c)
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def J_component(a):
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# a in {0,1,2} = x,y,z ; (J_a)_{bc} = -i * eps(a,b,c)
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M = sp.zeros(3,3)
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for b in range(3):
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for c in range(3):
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M[b,c] = -i*eps(a,b,c)
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return M
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Jx, Jy, Jz = J_component(0), J_component(1), J_component(2)
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# --- symbolic sanity checks ---
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comm = Jx*Jy - Jy*Jx
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print("[Jx,Jy] - i*Jz == 0 ?", sp.simplify(comm - i*Jz) == sp.zeros(3,3))
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J2_single = sp.simplify(Jx*Jx + Jy*Jy + Jz*Jz)
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print("J^2 (single spin-1 leg), should be 2*I_3:")
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sp.pprint(J2_single)
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# convert to numpy (complex) for fast Kronecker-sum construction at larger k
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Jx_np = np.array(Jx.tolist(), dtype=complex)
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Jy_np = np.array(Jy.tolist(), dtype=complex)
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Jz_np = np.array(Jz.tolist(), dtype=complex)
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def total_J2(k):
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dim = 3**k
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Jtot = {a: np.zeros((dim,dim), dtype=complex) for a in range(3)}
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comps = [Jx_np, Jy_np, Jz_np]
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for leg in range(k):
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for a in range(3):
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mats = [np.eye(3, dtype=complex)]*k
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mats[leg] = comps[a]
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M = mats[0]
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for m in mats[1:]:
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M = np.kron(M, m)
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Jtot[a] += M
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return Jtot[0]@Jtot[0] + Jtot[1]@Jtot[1] + Jtot[2]@Jtot[2]
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def exact_projectors(k, jmax):
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J2 = total_J2(k)
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assert np.abs(J2 - J2.conj().T).max() < 1e-10, "J^2 not Hermitian!"
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evals, evecs = np.linalg.eigh(J2)
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Ps = {}
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for j in range(jmax+1):
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target = j*(j+1)
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mask = np.abs(evals - target) < 1e-6
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if not np.any(mask):
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Ps[j] = np.zeros((3**k,3**k))
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continue
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V = evecs[:, mask]
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P = (V @ V.conj().T).real
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Ps[j] = P
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# sanity: eigenvalues actually cluster near integers j(j+1)
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return Ps, evals
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print("\nBuilding exact projectors for k=2,3,4 via Casimir diagonalization...")
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Ps2, ev2 = exact_projectors(2, 2)
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Ps3, ev3 = exact_projectors(3, 3)
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Ps4, ev4 = exact_projectors(4, 4)
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for k,Ps,jmax in [(2,Ps2,2),(3,Ps3,3),(4,Ps4,4)]:
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print(f"\nk={k}:")
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for j in range(jmax+1):
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tr = np.trace(Ps[j]).real
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print(f" j={j}: trace(P_j) = {tr:.10f} (expect (2j+1)*m_j)")
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np.savez("projectors_exact.npz",
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P2_0=Ps2[0],P2_1=Ps2[1],P2_2=Ps2[2],
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P3_0=Ps3[0],P3_1=Ps3[1],P3_2=Ps3[2],P3_3=Ps3[3],
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P4_0=Ps4[0],P4_1=Ps4[1],P4_2=Ps4[2],P4_3=Ps4[3],P4_4=Ps4[4])
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print("\nsaved projectors_exact.npz")
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