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In disordered systems, the amplitudes of the localized states will decreaseexponentially away from their centers and the localization lengths arecharacterizing such decreasing. In this article, we find a model in which eacheigenstate is decreasing at two distinct rates. The model is a one-dimensionaldisordered system with a constrained Hilbert space: all eigenstates$|\Psi\rangle$s should be orthogonal to a state $|\Phi \rangle$, $\langle \Phi| \Psi \rangle =0$, where $|\Phi \rangle$ is a given exponentially localizedstate. Although the dimension of the Hilbert space is only reduced by $1$, theamplitude of each state will decrease at one rate near its center and atanother rate in the rest region, as shown in Fig. \ref{fig1}. Depending on $|\Phi \rangle$, it is also possible that all states are changed from localizedstates to extended states. In such a case, the level spacing distribution isdifferent from that of the three well-known ensembles of the random matrices.This indicates that a new ensemble of random matrices exists in this model.Finally we discuss the physics behind such phenomena and propose an experimentto observe them.

Each state in a one-dimensional disordered system has two localization lengths when the Hilbert space is constrained