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| -rw-r--r-- | bezout.tex | 2 |
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@@ -62,7 +62,7 @@ whose elements are complex normal with $\overline{|J|^2}=p!/2N^{p-1}$ and $\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The constraint becomes $z^2=N$. The motivations for this paper are of two types. On the practical side, there are situations in which complex variables -have in a disorder problem appear naturally: such is the case in which they are {\em phases}, as in random laser problems \cite{Antenucci_2015_Complex, etc}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians \cite{Anninos_2016_Disordered} modeling Black Hole horizons in the zero-temperature limit. +have in a disorder problem appear naturally: such is the case in which they are {\em phases}, as in random laser problems \cite{Antenucci_2015_Complex}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians \cite{Anninos_2016_Disordered} modeling Black Hole horizons in the zero-temperature limit. There is however a more fundamental reason for this study: we know from experience that extending a problem to the complex plane often uncovers an underlying simplicity that is hidden in the purely real case. Consider, for example, the procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying $\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ is a polynomial of degree $p$ chosen to have simple, known roots. Because we are working in complex variables, and the roots are simple all the way (we shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. With real |