Composite-order-charge-4e-order-vestigial-order-quadrupling-order-historical-overview

States with order parameters made of four- and higher-electron composites

These composite orders can emerge in systems with a superconducting ground state, resulting in an order parameter formed by four (or more) rather than two electrons. While standard Bardeen–Cooper–Schrieffer (BCS) theory forbids their formation, early works on superconducting systems [1,2] have shown that higher-order condensates can form under certain conditions when two key assumptions of BCS theory are violated: (i) the superconducting order parameter breaks multiple symmetries, and (ii)  strong fluctuations invalidate the BCS mean-field approximation, producing a regime of incoherent Cooper pairs, i.e. no order in bilinear fields ⟨cᵢ cᵢ⟩ = 0, while higher-than-bilinear composite symmetries such as ⟨cᵢ cᵢ cⱼ cⱼ⟩ ≠ 0 or ⟨cᵢ cᵢ cⱼ† cⱼ†⟩ ≠ 0 remain broken

Thus, in this regime, the only symmetries that are spontaneously broken are composite symmetries constructed from quartic or higher-order electronic operators. This type of order has been discussed under different names: composite orders and metallic superfluids [1,2], charge-4e condensates, vestigial order, and quadrupling condensates. In connection with high-T_c superconductors and nematic superconductors, such phases, this type of order concepts and mechanism are often termed “vestigial order” [3–5]. In related but distinct microscopic settings, analogous phases have been labeled symmetric mass generation [6] and paired phases [7,8]. Also, occasionally, the term "preemptive transitions" or "preemptive order" introduced in [31] is used for these phases.

Different microscopic mechanisms have been proposed for the stabilization of this type of order, including gauge-field–mediated intercomponent coupling [1,2] and partial melting of pair-density-wave order [9–12]. Recent works [13–15] have demonstrated that simplified analytical approaches may lead to false positives, predicting composite order in systems where it does not actually occur. However, in two- and three-dimensional London and Ginzburg–Landau models, large-scale Monte Carlo simulations have demonstrated numerically that such composite order can appear in several classes of superconductors with broken symmetries. These include
(i) U(1)×U(1) → U(1) [7,16,17],
(ii) U(1)×Z_2 → Z_2, as in s+is, s+id, d+id, and p+ip superconductors [18–22],
(iii) SU(2) → O(3) [8,23,24], and
(iv) SU(N) → SUn(N) [25–29].

Here, the remaining broken symmetries U(1), Z_2, and O(3) are associated with a four-electron composite order. The trend observed is that it is typically harder to stabilize these phases when they are associated with higher broken symmetries.

Recently, a microscopic demonstration of electron-quadrupling states was published in Ref. [30].

Reference list

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