Gap-closing failure

Let ${\sigma }_z=\mathrm{diag}(1,-1)$ on ${\mathbb{C}}^2$, with orthonormal basis $e_{\uparrow }=(1,0)$, $e_{\downarrow }=(0,1)$. Let $H(s)=s{\sigma }_z$ for $s\in [-1,1]$, and let $U_T$ solve $i{U}_T^{\mathrm{\prime }}=TH(s)U_T$, $U_T(-1)=I$, for $T>0$. For $s\mathrm{\ne }0$ define $P_{+}(s)$ to be the spectral projection onto the ordered upper eigenvalue $E_{+}(s)=\mathrm{\mid }s\mathrm{\mid }$.

(a) The eigenvalues are $s$ and $-s$, so their separation $2\mathrm{\mid }s\mathrm{\mid }$ closes at $0$. The uniform positive-gap hypothesis of Finite gapped adiabatic estimate fails.

(b) The ordered upper projection is $P_{+}(s)=\mathrm{diag}(0,1)$ for $s<0$ and $P_{+}(s)=\mathrm{diag}(1,0)$ for $s>0$. Its distinct one-sided limits preclude a continuous extension through $0$. At the crossing $H(0)=0$, the spectral projection for its only eigenvalue is $I$, not a distinguished rank-one projection. Assigning any rank-one projection at the crossing does not repair the unequal one-sided limits.

(c) The exact propagator is $U_T(s)=\exp [-iT{\sigma }_z(s^2-1)\mathrm{/}2]$, so $U_T(1)=I$ and

$\mathrm{\parallel }(I-P_{+}(1))U_T(1)P_{+}(-1)\mathrm{\parallel }=1(T>0)\mathrm{.}$

Thus upper-energy tracking fails maximally at the endpoints. This is the ordered branch for which the leakage assertion is made.

(d) There is nevertheless a continuous constant rank-one branch $P_{\downarrow }=\mathrm{diag}(0,1)$ whose vectors obey $H(s)e_{\downarrow }=-s{e}_{\downarrow }$ for every $s$. It starts at energy $+1$ at $s=-1$, ends at energy $-1$ at $s=1$, and satisfies $(I-P_{\downarrow })U_T(s)P_{\downarrow }=0$ for every $s,T$. At $s=0$ this is a subprojection of the degenerate eigenspace; it is not the rank-two full spectral projection $E_{H(0)}(\{0\})$. The original claim that no continuous rank-one eigenbranch can pass through this crossing was false.