Sectional Curvature on SU(3)

Mathematical Foundations, numerical solution.

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Sectional Curvature on SU(3)

Background

For the bi-invariant metric \(g(A,B) = -\mathrm{Tr}(AB)\) on skew-Hermitian matrices, the sectional curvature of the plane of two orthonormal \(X, Y \in \mathfrak{u}(D)\) takes the closed form (Milnor, 1976)

\[ K(X,Y) = \frac{1}{4}\|[X,Y]\|^2 \geq 0 . \]

It vanishes for commuting generators and grows with the failure of the two directions to commute. On \(SU(2)\) it equals \(\tfrac12\) for every plane, the value \(1/R^2\) of the three-sphere of radius \(\sqrt2\). This exercise compares planes of different curvature in \(SU(3)\).

Exercise

With \(E_{jk} = |j\rangle\langle k|\) on \(\mathbb{C}^3\), consider the generators

\[ X_1 = \tfrac{-i}{\sqrt2}\,\mathrm{diag}(1,-1,0), \quad Y_1 = \tfrac{-i}{\sqrt6}\,\mathrm{diag}(1,1,-2), \quad X_2 = \tfrac{-i}{\sqrt2}\,(E_{12}+E_{21}), \quad Y_2 = \tfrac{1}{\sqrt2}\,(E_{21}-E_{12}), \quad Y_3 = \tfrac{-i}{\sqrt2}\,(E_{23}+E_{32}). \]

(a) Show that each generator lies in \(\mathfrak{su}(3)\) and that each of the pairs \((X_1,Y_1)\), \((X_2,Y_2)\) and \((X_2,Y_3)\) is orthonormal.

(b) Compute \([X,Y]\) and \(K(X,Y)\) for the three pairs.

(c) Two Hamiltonian evolutions leave the identity in the directions \(X\) and \(X+\epsilon Y\). For which pair do they stay parallel, and for which do they reconverge fastest? Relate the answer to the basis states the two generators act on.

Solution

Part (a)

All five matrices are skew-Hermitian and traceless. For the norms, \(g(X_1,X_1) = \tfrac12\mathrm{Tr}\,\mathrm{diag}(1,1,0) = 1\), \(g(Y_1,Y_1) = \tfrac16(1+1+4) = 1\), and \(g(X_2,X_2) = g(Y_2,Y_2) = g(Y_3,Y_3) = \tfrac12\cdot 2 = 1\). The cross terms vanish, since \(\mathrm{Tr}(X_1Y_1) \propto 1\cdot1 + (-1)\cdot1 = 0\), \(X_2Y_2 = -\tfrac{i}{2}(E_{11}-E_{22})\) is traceless, and \(X_2Y_3 = -\tfrac12 E_{13}\) has no diagonal entries.

Part (b)

Diagonal matrices commute, so \([X_1,Y_1] = 0\) and \(K = 0\). For the second pair \([X_2,Y_2] = -i\,\mathrm{diag}(1,-1,0)\), so \(\|[X_2,Y_2]\|^2 = 2\) and \(K = \tfrac12\), the value of \(SU(2)\), since \(X_2\) and \(Y_2\) generate an \(\mathfrak{su}(2)\) acting on \(|1\rangle, |2\rangle\) only. For the third pair \([X_2,Y_3] = \tfrac12(E_{31}-E_{13})\), so \(\|[X_2,Y_3]\|^2 = \tfrac12\) and \(K = \tfrac18\).

Part (c)

The diagonal pair commutes, the two evolutions generate a flat torus of diagonal unitaries, and the geodesics stay parallel. The pair acting on the same two states, \(|1\rangle\) and \(|2\rangle\), has the largest curvature and reconverges fastest. The pair \((X_2,Y_3)\) shares only the state \(|2\rangle\), its commutator is smaller, and the curvature is a quarter of the same-block value.

Numerical verification

Each pair is checked to be skew-Hermitian, traceless and orthonormal, and \(K = \tfrac14\|[X,Y]\|^2\) is evaluated.

Code 
import numpy as np

g = lambda A, B: np.real(np.trace(A.conj().T @ B))   # Hilbert-Schmidt metric, = -Tr(AB) on skew-Hermitian
comm = lambda A, B: A @ B - B @ A
E = lambda i, j: np.outer(np.eye(3)[i], np.eye(3)[j])

X1 = -1j*np.diag([1, -1, 0])/np.sqrt(2)
Y1 = -1j*np.diag([1, 1, -2])/np.sqrt(6)
X2 = -1j*(E(0,1) + E(1,0))/np.sqrt(2)
Y2 = (E(1,0) - E(0,1))/np.sqrt(2)
Y3 = -1j*(E(1,2) + E(2,1))/np.sqrt(2)

for name, (X, Y) in {"(X1,Y1)": (X1, Y1), "(X2,Y2)": (X2, Y2), "(X2,Y3)": (X2, Y3)}.items():
    assert np.allclose(X.conj().T, -X) and np.allclose(Y.conj().T, -Y)          # skew-Hermitian
    assert np.isclose(np.trace(X), 0) and np.isclose(np.trace(Y), 0)            # traceless
    assert np.isclose(g(X, X), 1) and np.isclose(g(Y, Y), 1) and np.isclose(g(X, Y), 0)
    C = comm(X, Y)
    K = g(C, C)/4
    print(f"{name}: |[X,Y]|^2 = {g(C,C):.4f},  K = {K:.4f}")

assert np.isclose(g(comm(X1,Y1), comm(X1,Y1))/4, 0)
assert np.isclose(g(comm(X2,Y2), comm(X2,Y2))/4, 1/2)
assert np.isclose(g(comm(X2,Y3), comm(X2,Y3))/4, 1/8)
print("K = 0, 1/2, 1/8 VERIFIED")
(X1,Y1): |[X,Y]|^2 = 0.0000,  K = 0.0000
(X2,Y2): |[X,Y]|^2 = 2.0000,  K = 0.5000
(X2,Y3): |[X,Y]|^2 = 0.5000,  K = 0.1250
K = 0, 1/2, 1/8 VERIFIED

Takeaway

The curvature of a plane in \(\mathfrak{su}(3)\) measures how much the two generators overlap in the states they couple: \(K=0\) for commuting generators, \(\tfrac12\) within one \(\mathfrak{su}(2)\) block, \(\tfrac18\) for blocks sharing one state.

References

  • Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Mathematics 21, 293–329 (1976). doi:10.1016/S0001-8708(76)80002-3
  • Płodzień, Information Scrambling, Quantum Chaos, and Haar-Random States, lecture notes (2025). arXiv:2511.14397