Proposition. The logarithm on the accretive cone [fdip-009L]
✍️source

  • November 1, 2026

On the open convex cone \(\mathcal C_n\) of strictly accretive complex matrices, there is a unique holomorphic function \(\mathcal L\) such that \[e^{\mathcal L(K)}=\det K,\qquad \mathcal L(I)=0.\] For \(K_t=I+t(K-I)\), it is given by \[\mathcal L(K)=\int _0^1 \operatorname {tr}\left [K_t^{-1}(K-I)\right ]dt.\] On Hermitian positive matrices it equals the real \(\log \det K\). Thus the fermionic logarithmic contribution \(W_F(K)=-\mathcal L(K)\) is unambiguous on this cone.

Proof. 

  • November 1, 2026

Write \(K_t=I+t(K-I)\). Convexity makes \(K_t\) accretive and invertible for every \(0\le t\le 1\). In a neighborhood of each \(K\in \mathcal C_n\), the Hermitian parts of all these matrices have a common positive lower bound. Their inverses and derivatives in the matrix entries are therefore locally bounded uniformly in \(t\). The integral of the holomorphic rational integrand is holomorphic in \(K\).

Put \(g(t)=\det K_t\). The determinant derivative, as in the real log-determinant calculation, gives \[\frac {g'(t)}{g(t)} =\operatorname {tr}(K_t^{-1}(K-I)).\] This complex version follows directly from \(\det (K_t+hB)=\det K_t\det (I+hK_t^{-1}B)\) and \(\det (I+hM)=1+h\operatorname {tr}M+O(h^2)\), obtained by the permutation expansion. Therefore the derivative of \(g(t)\exp (-\int _0^t g'(s)/g(s)\,ds)\) is zero. Its value at zero is one, proving \(\exp \mathcal L(K)=\det K\); the identity matrix gives \(\mathcal L(I)=0\).

Two such logarithms differ by a continuous function with values in \(2\pi i\mathbb Z\). The cone is connected, so this function is constant, and normalization makes it zero. On Hermitian positive \(K\), all \(K_t\) are positive and \(g(t)>0\); ordinary real logarithmic differentiation identifies the integral with \(\log \det K\).

This branch is not always the principal scalar logarithm of \(\det K\). For \(K=e^{i\pi /3}I_4\), the Hermitian part is \(I_4/2>0\). Along \(K(s)=e^{is}I_4\), \(0\le s\le \pi /3\), continuity and normalization give \(\mathcal L(K(s))=4is\). Hence \[\mathcal L(e^{i\pi /3}I_4)=\frac {4\pi i}{3},\] whereas the principal scalar logarithm of its determinant is \(-2\pi i/3\). The determinant can even reach the scalar principal branch cut while the matrix stays accretive.

For a general invertible complex matrix, writing \(-\log \det K\) requires a declared branch on the domain under consideration; there is no single-valued continuous logarithm on all invertible matrices when \(n\ge 1\). The loop \(\operatorname {diag}(e^{it},1,\ldots ,1)\), \(0\le t\le 2\pi \), already has determinant winding once. If \(\det K=0\), no logarithm exists, although the Grassmann determinant identity still does. The boson-fermion cancellation itself uses no logarithm branch.