A characteristic of ferroic materials is the emergence of a temporally static finite expectation value of an order parameter. Here, we introduce a new mechanism [1] for ferroic order, in which a non-zero quasi-static magnetoelectric quadrupolar order appears, mediated by a strong coupling of spin and phonon fluctuations. We show that our proposed mechanism is consistent, to our knowledge, with many experimental observations for the onset of the pseudo-gap phase in cuprate superconductors and therefore propose the quasi-static magnetoelectric quadrupole as a possible pseudo-gap order parameter. By using first-principles calculations in combination with our recent developed formalism [2,3], to calculate multipole moments within a Berry phase approach, we calculate the magnitude of the effect for the the prototypical cuprate superconductor, HgBa2CuO4+δ. Using these results we finally show that our mechanism embraces several key findings of experimental reports and in addition also aspects of previous theoretical models.
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