Intertwined Vestigial Order in Quantum Materials: Nematicity and Beyond

Abstract

A hallmark of the phase diagrams of quantum materials is the existence of multiple electronic ordered states, which, in many cases, are not independent competing phases, but instead display a complex intertwinement. In this review, we focus on a particular realization of intertwined orders: a primary phase characterized by a multi-component order parameter and a fluctuation-driven vestigial phase characterized by a composite order parameter. This concept has been widely employed to elucidate nematicity in iron-based and cuprate superconductors. Here we present a group-theoretical framework that extends this notion to a variety of phases, providing a classification of vestigial orders of unconventional superconductors and density-waves. Electronic states with scalar and vector chiral order, spin-nematic order, Ising-nematic order, time-reversal symmetry-breaking order, and algebraic vestigial order emerge from one underlying principle. The formalism provides a framework to understand the complexity of quantum materials based on symmetry, largely without resorting to microscopic models. arXiv:1804.00818v1 [cond-mat.str-el]

Intertwined spin-orbital coupled orders in the iron-based superconductors

“…As discussed in Ref. 7, the six elements of M transform as the product of irreps E 5 ⊗ Γ S=1 , where E 5 is a two-dimensional irreducible representation of C 4v and Γ S=1 is the standard 3-dimensional irreducible representation of SO(3). This warrants writing M = (M 1 , M 2 ) T and identifying the appropriate order parameters as the three-dimensional vectors M 1 and M 2 , instead of M. The resulting free energy, which can be constructed by imposing that its elements transform trivially under the group symmetry operations, becomes:…”

“…In the simple approximation where the As/Se atoms are neglected, and the crystal is described as a single-Fe square lattice, the six components of M do not transform according to an irreducible representation of the SO(6) group, but instead according to an irrep of the C 4v ⊗ SO(3) group, as discussed in Ref. 7. Here, C 4v is the extended point group corresponding to the point group C 4v supplemented by three translations alongx, y, andx +ŷ.…”

“…A vestigial order parameter is a composite order parameter that can order even in the absence of the primary order, and that breaks a subset of the symmetries broken by the primary order parameter (for a review, see Ref. 7). Vestigial order can only appear if the primary order parameter transforms as a multi-dimensional irrep, i.e.…”

“…In this scenario, the various broken-symmetry phases can be interpreted as vestigial orders that break only a subset of the symmetries of this "mother" state. To contrast with the standard case of competing orders and fine-tuned multicritical points, they have been dubbed intertwined orders [2,7].…”

Strain tuning and anisotropic spin correlations in iron-based systems

“…We perform a continuum's description for the coordinates within the planes, but keep the discrete lattice structure for the third dimension, with layer index l. Thus, we express the three-dimensional coordinates R = (r, la z ) in terms of the two-dimensional vector r = (x, y) and the discrete layer index l. Following Ref. [25] we combine the two vectors into m = (m x , m y ). and obtain the effective action of the problem…”