Zeros of the dispersion relation of the elementary excitation and the correlation length of strongly correlated quantum systems

We argue that the imaginary part of a zero of the dispersion relation of the elementary excitation of quantum systems is equal to the inverse correlation length. We confirm the relation for the Hubbard model[1] in the half-filled case; it has been confirmed only for the S=1/2 antiferromagnetic XXZ chain[2]. In order to search zeros of the dispersion relation in the complex momentum space efficiently, we introduce a non-Hermitian generalization of quantum systems by adding an imaginary vector potential ig to the momentum operator[3]. We also show for the half-filled Hubbard model the reason why the non-Hermitian critical point[4] is equal to the inverse correlation length[5] by noting the dispersion relation of the charge excitation. \newline [1] Y. Nakamura and N. Hatano, in preparation. \newline [2] K. Okunishi, Y. Akutsu, N. Akutsu and T. Yamamoto, Phys. Rev. B 64 (2001) 104432. \newline [3] Y. Nakamura and N. Hatano, Physica B 378-380 (2006) 292; J. Phys. Soc. Jpn. 75 (2006) 114001. \newline [4] T. Fukui and N. Kawakami, Phys. Rev. B 58 (1998) 16051. \newline [5] C. A. Stafford and A. J. Millis, Phys. Rev. B 48 (1993) 1409.