Composite order-higher-charge-condensation-charge-4e-vestigial-order-early-and-recent-works-overview
BRIEF LITERATURE OVERVIEW UNDER CONSTRUCTION
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 work
on superconducting systems [6,7]
have shown that higher-order condensates 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 <c_ic_i>, which still preserve higher-than-
bilinear broken composite symmetries such as
<c_ic_ic_jc_j> or <c_ic_ic_j^\dagger c_j^\dagger> . Thus, in
this regime, the only symmetries spontaneously broken
are composite symmetries constructed from quartic
or higher-order electronic operators.
This type of order have been discussed under
different names: composite orders, charge-4e condensates, vestigial
order, quadrupling condensates.
In connection with high-Tc
superconductors and nematic superconductors
such phases are often termed "vestigial order" [35,36,37]
In related but distinct microscopic settings, analogous
phases have been labeled symmetric mass generation
[3] and paired phases [4, 5].
Different microscopic mechanisms have been proposed for stabilization of this type of order
including gauge-field–mediated intercomponent
coupling [6, 7] and partial melting of pair-density wave
order [8–11]. Recent works [1, 12–14] 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 has been demonstrated numerically in
several classes of superconductors with broken symmetries in large-scale Monte-Carlo simulations.
These include (i) U(1)xU(1) → U(1)
[4, 7, 15, 16], (ii) U(1)xZ2 → Z2 as in s + is,
s+id, d+id, and p+ip superconductors [14, 17–21],
(iii)SU(2) → O(3) [5, 22, 23] (iv) SU(N) → SUn(N) [24–28].
Here, the remaining broken symmetries
U(1),Z2,O(3) are associated with a four-electron
composite order. The trend observed is that it
is typically harder to stabilize these phases when
associated to higher broken symmetries.
Recently a microscopic
demonstration of electron quadrupling states was published [49]
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[49] Albert Samoilenka, Egor Babaev Microscopic theory of electron quadrupling condensates arXiv:2505.12542
