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Superconductivity From Confinement of Singlets in Metal Oxides
J. M. Booth1, ∗
1ARC Centre of Excellence in Exciton Science, RMIT University, Melbourne, Australia
(Dated: October 15, 2018)
The Yang-Mills description of phonons and the consequent structure of electron liquids in strongly
anharmonic crystals such as metal oxides is shown to yield an attractive electron-phonon interac-
tion, and thus an instability towards the formation of bound states, which can condense to form a
superconductor. This mechanism differs significantly from the pairing mechanism of conventional
superconductivity: the ground state from which superconductivity emerges is a complicated many-
body state of paired electrons and holes which is not amenable to a quasiparticle description, and
whose properties are similar to those seen in the Cuprate high temperature superconductors. Con-
finement arises because the complicated electron liquid structure acts as a source for Yang-Mills
bosons, and not the traditional longitudinal density waves of BCS pairing.
The discovery of high temperature superconductivity
in the cuprates by Bednorz and M¨uller1generated sig-
nificantly renewed interest in the question of how super-
conductivity can arise in crystalline materials. In the
ensuing three or so decades a huge number of studies
have explored both the experimental properties of the
cuprates, and their theoretical implications, along the
way finding another class of materials which exhibit un-
conventional superconductivity at elevated temperatures:
the pnictides.2
However, despite all of this attention, the cuprate
phase diagram still contains many mysterious phases, not
least of which is the superconducting phase itself,10 while
the strange-metal3and psuedogap4phases are also foci of
intense curiosity for condensed matter physics due to the
unusual properties they exhibit. The goal of all of this
attention is the determination of how high temperature
superconductors form, and what sort of “normal” state
they form out of, such that new materials can be engi-
neered which can hopefully produce superconductivity at
room-temperature.
There have been many proposals for the underlying
mechanism of the emergence of superconductivity in
cuprates, beginning with the Resonating Valence Bond5
and spin-wave mediated pairing,6,7 and we cannot be-
gin to summarize them here. For thorough reviews of i)
the Mott Insulating perspective see Lee et al.8, and for a
more general review see Keimer et al.9What is obvious
from the literature is that no clear consensus on the pair-
ing mechanism exists apart from i) the recognition that
magnetic fluctuations are important, ii) as are strong
interactions between the electrons in the normal state,
which must be mediated by doping to give a supercon-
ducting state, and iii) the fact that whatever the pairing
mechanism is, it cannot be the same mechanism which
gives rise to pairing in conventional superconductors.10
In two recent studies the application of Yang-Mills the-
ory to condensed matter systems was studied first in the
context of electron-phonon interactions,11 and then a de-
scription of anharmonic phonons in terms of pure Yang-
Mills theory was presented.12 In this work, the scattering
of Cooper pairs is investigated in the context of a pure
Yang-Mills theory coupled to a Fermi surface, which gives
an electron phonon mechanism of the type explored in
the first paper.11 It is found that the Yang-Mills scatter-
ing cross-section has an attractive potential some spinors,
while other scattering events for different spinor combi-
nations are repulsive. The attractive potential occurs for
scattering between Cooper pairs.
One simple, but significant assumption is made in this
work: the effect of strong correlations in metal oxides is
simply to give a tendency towards half-filling, i.e. each
electron state is correlated with a hole state, and thus
operations on electrons must include the effect on the
associated hole. To this end the spinor states which enter
the interaction vertex must consist stacked electrons and
holes, i.e. Nambu spinors, which are 4-component Weyl
spinors. Thus the electrons and holes in a spinor have
opposite helicities, and the 4 possible combinations are
grouped like so:11
ˆc†
k↑
ˆc−k↓= up,ˆc†
−k↓
ˆck↑= down,
ˆck↓
ˆc†
−k↑= top,ˆc−k↑
ˆc†
k↓= bottom (1)
and it is the dynamics of these which gives rise to the
interesting phenomena exhibited. The Yang-Mills inter-
action vertices are given by:11
g(+,−,3)γµˆ
Wµ(x)ψ=
γµW3
µ(x)γµW−
µ(x)
γµW+
µ(x)−γµW3
µ(x)
ˆc†
k↑
ˆc−k↓
ˆc†
−k↓
ˆck↑
g(+,−,3)γµˆ
Wµ(x)ψ=
γµW3
µ(x)γµW−
µ(x)
γµW+
µ(x)−γµW3
µ(x)
ˆck↓
ˆc†
−k↑
ˆc−k↑
ˆc†
k↓
(2)
The incoming spinor states can be grouped using spin as
a “gauge charge”, i.e. a colour index, which gives the
following structure:11
arXiv:1810.05339v1 [cond-mat.supr-con] 12 Oct 2018
2
a)
b)
FIG. 1: Schematic representation of the transformations
enacted by the a) Neutral boson W3
µand b) the
Charged bosons W±
µ.
Colour Flavour
1 2
aup top
bdown bottom
The actions of the bosons on the spinors is presented
schematically in Figure (1), with the ˆ
W3
µfield the neutral
boson which doesn’t couple to spin, while the ˆ
W±
µraising
and lowering the spins. The most significant difference in
this formalism is that making the ansatz that strong cor-
relations result in the requirement of grouping the elec-
tron and hole states into 4-component Weyl spinors, and
including phonon anharmonicity12 means that the cur-
rents which generate bosons are now comprised of double-
stacked 4-component spinors with the SU(2) generators
providing the outgoing boson structure.11 Thus the cur-
rent becomes:
ˆ
Jµ=¯
ψγµψ→¯
ψiˆ
Ta
ij γµψj(3)
where ˆ
Ta
ij is a generator of the group SU(2), i.e. a Pauli
matrix. Thus the interaction between spinors in the con-
text of the Yang-Mills interaction vertex to leading order
goes as:13
−iM=Ta
ji Ta
kl(iga)2¯
ψjγµψi
−i(gµν −kνkµ
k2)
k2¯
ψkγνψl(4)
In this formalism, colour-anticolour pairs are Cooper
pairs:
u¯u=ˆc†
k↑ˆc†
−k↓
ˆc−k↓ˆck↑,¯
tt =ˆck↓ˆc−k↑
ˆc†
−k↑ˆc†
k↓(5)
Plugging a Cooper pair in, for example i= 1, k = 1, or
i= 2, k = 2 the colour prefactor can be calculated by the
Fierz completeness relation:13
3
X
a=1
Ta
ji Ta
kl =1
2(δilδj k −1
2δij δkl) = 1
40
01
2jl
(6)
and is positive. For states with different colours, for ex-
ample i= 2, k = 1, the colour factor is −1
4. Therefore,
different combinations of colours give different signs for
the scattering effective potential. Since the scattering
factor for colour-anticolour pairs is positive, the poten-
tial between them is attractive, while for different colours
the sign is negative, and therefore the potential is repul-
sive.
Therefore since the scattering of colour singlets by
phonons in a Yang-Mills theory gives an attractive in-
teraction, the Bethe-Salpeter equation will give an in-
stability for the formation of bound states (Cooper
Pairs) similar to the Fr¨ohlich interaction in conven-
tional superconductors,14 but without the requirement
of the propagators being off-shell. Thus the fairly tortu-
ous derivation of Bardeen and Pines15 is unnecessary in
Yang-Mills theory, colour singlet states are automatically
confined, and this interaction does not require the exis-
tence of quasiparticles, but does require the existence of
a many-body ground state of electron and hole pairs to
act as a source for the Yang-Mills bosons.
The author acknowledges the support of the ARC Cen-
tre of Excellence in Exciton Science (CE170100026). Cor-
respondence and requests for materials should be ad-
dressed to JMB, email: jamie.booth@rmit.edu.au
∗jamie.booth@rmit.edu.au
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