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Exact theory for superconductivity in a doped Mott insulator


Because the cuprate superconductors are doped Mott insulators, it would be advantageous to solve even a toy model that exhibits both Mottness and superconductivity. We consider the Hatsugai–Kohmoto model1,2, an exactly solvable system that is a prototypical Mott insulator. Upon either doping or reducing the interaction strength, our exact calculations show that the system becomes a non-Fermi liquid metal with a superconducting instability. In the presence of a weak pairing interaction, the instability produces a thermal transition to a superconducting phase, which is distinct from the traditional state described by Bardeen–Cooper–Schrieffer (BCS) theory, as evidenced by a gap-to-transition temperature ratio exceeding the universal BCS limit. The elementary excitations of this superconductor are not Bogoliubov quasiparticles but rather superpositions of doublons and holons, composite excitations that show that the superconducting ground state of the doped Mott insulator inherits the non-Fermi liquid character of the normal state. An unexpected feature of this model is that it exhibits a superconductivity-induced transfer of spectral weight from high to low energies, as seen in the cuprates3, as well as a suppression of the superfluid density relative to that in BCS theory.

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Fig. 1: Single-particle Green functions and phase diagram of the HK model.
Fig. 2: Superconducting energy scales in two dimensions.
Fig. 3: Spectral function in the superconducting ground state.
Fig. 4: Ground-state superfluid stiffness.

Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


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We thank C. Setty for extensive discussions on his work that ultimately motivated the search for an exactly solvable model with Mottness and superconductivity, G. La Nave for a critical analysis of the results, and C. Boyd for many helpful discussions. P.W.P. acknowledges DMR19-19143 for partial funding of this project and E.W.H. was supported by the Gordon and Betty Moore Foundation EPiQS Initiative through grant no. GBMF 4305.

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P.W.P. initiated the problem and computed the pair instability. L.Y. computed the susceptibility with the numerics on the binding energy and E.W.H. computed the wave function, elementary excitations and the superfluid density.

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Correspondence to Philip W. Phillips.

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Peer review information Nature Physics thanks Jan Zaanen and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Phillips, P.W., Yeo, L. & Huang, E.W. Exact theory for superconductivity in a doped Mott insulator. Nat. Phys. 16, 1175–1180 (2020).

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