Emergent Matter

Cuprates and the Mott Problem

Why high-temperature superconductivity forced condensed-matter physics to take strong electron correlation seriously — and why the story begins with an insulator that band theory says should be a metal.

By Abdulrahman Anter Université Paris-Saclay · LPS, Orsay
graduate evergreen Experiment Theory updated 2026-06-21

This subject is a Master’s thesis — Mott Physics in Correlated Lattices: A Dynamical Mean Field Theory Study (Université Paris-Saclay / LPS) — rendered in full as a browsable set of topics: every derivation, figure, and result lives on these pages. (The compiled report is also embedded on the Report page if you prefer a single PDF.) We start with the experimental puzzle that motivates everything.

From conventional superconductors to a puzzle

In 1911 Kamerlingh Onnes found that the DC resistivity of mercury drops abruptly to zero below : a current induced in a superconducting ring persists essentially forever. Two decades later Meissner and Ochsenfeld showed that superconductors are perfect diamagnets — they expel applied magnetic fields entirely (the Meissner effect).

Onnes (1911): the resistivity of a mercury wire falls abruptly to zero at T_c ≈ 4.2 K.
Onnes (1911): the resistivity of a mercury wire falls abruptly to zero at T_c ≈ 4.2 K.
The Meissner effect: a superconductor expels magnetic flux from its interior.
The Meissner effect: a superconductor expels magnetic flux from its interior.
Experimental signatures of conventional superconductivity: below the critical temperature, a material shows zero electrical resistance and expels magnetic fields from its interior (the Meissner effect) — perfect diamagnetism.

The distinction between bosons and fermions is central here. Particles are classified by the symmetry of the wavefunction under exchange of two identical particles: symmetric for bosons, antisymmetric for fermions. As Pauli first anticipated, half-integer-spin particles are fermions (obeying the exclusion principle), integer-spin particles bosons. Bosons may occupy the same quantum state, so a gas of non-interacting bosons can condense into one ground state — a Bose–Einstein condensate (BEC) — which, with weak interactions, becomes a superfluid that flows without friction. Electrons are fermions and do not condense in this way; but if pairs of electrons form, each composite pair behaves effectively as a boson and can condense. The puzzle is that electrons repel through the Coulomb interaction — so how do they pair at all? The answer lies in the crystal lattice we have so far neglected.

These conventional superconductors are explained by BCS theory. Its phenomenological core: although electrons repel through the Coulomb interaction, the crystal lattice mediates an effective attraction between them via phonons. The isotope effect — depending on the lattice mass (Maxwell; Reynolds et al., 1950) — was the smoking gun for phonons, and Cooper showed that an arbitrarily weak attraction within a Fermi sea binds electrons into Cooper pairs. These bosonic pairs condense, and that condensation is superconductivity.

High- and the arrival of strong correlation

In 1986 Bednorz and Müller found superconductivity near in — a layered perovskite cuprate obtained by substituting some La with Ba in . Modest as the excess over the BCS ceiling was, it cast serious doubt on whether BCS applied at all here. The decisive point: is strongly correlated — electron–electron interactions dominate — and the superconducting order parameter has an unconventional symmetry, making these both high- and unconventional superconductors.

Crucially, the parent compound is not a metal but an insulator — a Mott insulator — in flat contradiction with the conventional band theory of solids. High- superconductivity emerges only when this Mott insulator is doped. The widely held view follows immediately:

Cuprates: structure and phase diagram

Cuprates share a layered structure: planes separated by insulating spacer layers that act as charge reservoirs for doping. Because the in-plane Cu–O distances are far shorter than the spacing to the insulating layers, the essential physics is believed to live within the planes.

Layered structure: CuO₂ planes separated by insulating spacers; the active states are a Cu 3d(x²−y²) hole hybridised with planar O 2p orbitals (Keimer et al., 2015).
Layered structure: CuO₂ planes separated by insulating spacers; the active states are a Cu 3d(x²−y²) hole hybridised with planar O 2p orbitals (Keimer et al., 2015).
The CuO₂ plane — the arena where the correlated physics happens.
The CuO₂ plane — the arena where the correlated physics happens.
The layered copper-oxide architecture and the CuO₂ plane where the correlated physics lives.

Each copper hosts a shell whose ten-fold near-degeneracy is split by the crystal field, so the system is usually reduced to a single active orbital per Cu site. Hopping between coppers is actually mediated by the bridging oxygens, but is conventionally described as direct Cu–Cu hopping. Tuning temperature , interaction strength , or doping then traces out a famously rich phase diagram.

Cuprate phase diagram: temperature vs. hole doping, showing the antiferromagnetic Mott insulator at zero doping, the superconducting dome, pseudogap, and strange-metal regions (Keimer et al., 2015).
Cuprate phase diagram: temperature vs. hole doping, showing the antiferromagnetic Mott insulator at zero doping, the superconducting dome, pseudogap, and strange-metal regions (Keimer et al., 2015).

This work zooms in on one corner of that diagram: the parent state at zero doping (), and the correlation-driven metal–insulator transition there.

Where this subject goes next

The roadmap mirrors the thesis:

  1. The Hubbard model — the minimal model of reference for these materials, and its symmetries and Green’s functions.
  2. Its analytic limits — weak coupling (mean-field / Slater) and strong coupling (perturbation theory and superexchange).
  3. Dynamical Mean-Field Theory (DMFT) — a method that genuinely handles strong correlation — and a numerical study of the MIT, compared against the classic Mott system .

Prerequisites

None — this is a good starting point.

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