Quantum criticality is associated with a system composed of a nearly infinite number of interacting quantum degrees of freedom at zero temperature, and it implies that the system looks on average the same regardless of the time- and length scale on which it is observed. Electrons on the atomic scale do not exhibit such symmetry, which can only be generated as a collective phenomenon through the interactions between a large number of electrons. In materials with strong electron correlations a quantum phase transition at zero temperature can occur, and a quantum critical state has been predicted, which manifests itself through universal power-law behaviours of the response functions. Candidates have been found both in heavy-fermion systems and in the high-transition temperature (high-Tc) copper oxide superconductors, but the reality and the physical nature of such a phase transition are still debated. Here we report a universal behaviour that is characteristic of the quantum critical region. We demonstrate that the experimentally measured phase angle agrees precisely with the exponent of the optical conductivity. This points towards a quantum phase transition of an unconventional kind in the high-Tc superconductors.
Bibliographical noteFunding Information:
Acknowledgements We gratefully acknowledge interactions with K. Birnbaum, C.-W. Chou, A. C. Doherty, L.-M. Duan, T. Lynn, T. Northup, S. Polyakov and D. M. Stamper-Kurn. This work was supported by the National Science Foundation, by the Caltech MURI Center for Quantum Networks, and by the Office of Naval Research.
Acknowledgements We thank C. M. Varma, P. Prelovsek, C. Pepin, S. Sachdev and A. Tsvelik for comments during the preparation of this work, and N. Kaneko for technical assistance. This investigation was supported by the Netherlands Foundation for Fundamental Research on Matter (FOM) with financial aid from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). The crystal growth work at Stanford University was supported by the Department of Energy’s Office of Basic Energy Sciences, Division of Materials Science.