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Furthermore, we investigate their spectral properties, in particular, to find out whether not only their eigenvalues but also the fluctuations in their spectra and their length spectra differ.Although the nonequilibrium-relaxation (NER) method has been widely used in Monte Carlo studies on phase transitions in classical spin systems, such studies have been quite limited in quantum phase transitions. The reason is that the relaxation process based on cluster-update quantum Monte Carlo (QMC) algorithms, which are now standards in Monte Ca
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