High Temperature Superconductivity, Long-range Order and Broken Symmetries in Strongly Correlated Electronic Systems Lawrence J. Dunne , Erkki J. Brändas, Hazel Cox a School of Engineering , London South Bank University, London SE1 0AA, UK, b Department of Materials, Imperial College London, London SW7 2AZ,UK, C Department of Chemistry, University of Sussex, Falmer, Brighton, BN1 9QJ, UK d Institute of Theoretical Chemistry, Ångström Laboratory, University of Uppsala, Box 518, Uppsala, S-75120, Sweden My name is Lawrence Dunne. Thank organising committee. I was a postdoc in POLs group. Very fruitful time in scientific life and I am honoured to be asked to speak here today. The title indicates that the talk is about Sc when electronic repulsions are strong. Presented at International Society of Theoretical Chemical Physics 2016 Conference July 17-22, 2016 Grand Forks, ND - USA

WHAT IS SUPERCONDUCTIVITY? For some materials, the electrical resistivity vanishes at a transition temperature Tc: they become superconducting. Highest Tc in Cuprates at about 164K which is focus here. Spontaneously broken global gauge (phase) symmetry occurs at Tc –related to the onset of Off-diagonal Long-range order (ODLRO) -SUPERSELECTION .

Macroscopic Wavefunctions and Off-Diagonal long-range order (ODLRO) Superconductivity requires macroscopically large eigenvalue of the second order reduced electronic density matrix (Chen Ning Yang and Sasaki in Löwdin group) The macroscopic condensate wave function is the ‘large’ eigenvector of . In cuprates the condensate wave function has a d-wave symmetry

Forbidden Longitudinal photon modes turned on. Photons become massive. Anderson-Higgs Mechanism for Coupling Condensate to Electromagnetic Field. Forbidden Longitudinal photon modes turned on. Photons become massive. Superconducting electrodynamics obeys Proca equations.

Left side: Structure of Lanthanum Strontium Cuprate. Right side-Alternant Cuprate layer • - Copper O –Oxygen . Electron-hole doping Symmetry

Accepted Bardeen-Cooper-Schrieffer (BCS) Theory of Low Tc Superconductors Electron pairs attract each other via virtual phonon exchange which is dominant.

The standard Hamiltonian is Brief summary of Bohm-Pines –Bardeen Method For Conventional Superconductors The standard Hamiltonian is A is an irrotational ( ) magnetic vector potential . Uses Quantum Field Theory. H is approximately transformed into short range (sr) ,long-range (lr) plasmon and phonon induced electron-electron terms

The total energy of system is The last 2 plasmon terms are assumed to be the same in metallic normal state and the superconducting state. These can be dropped. The eigenfunctions of the Hamiltonian matrix are key and determined by configuration interaction as The ground state must exhibit ODLRO to be superconducting . We will focus on two extremes where either electron repulsions or phonon induced electron-electron interactions dominate.

Effective Bohm-Pines electron-electron potential in system is heavily screened given by shown below for several values of the screening length kc. ( shows Friedel oscillations). Coulomb repulsions give very small effect in a good metal.

CASE 1 . Hamiltonian Matrix for Cooper’s famous problem showing k-fold stabilisation for attractive matrix elements due to dominant phonon induced electron-electron interactions Conventional superconductivity –state splits off from degenerate manifold

CASE 2 . Hamiltonian Matrix showing k-fold stabilisation for repulsive matrix elements due to dominant electronic repulsions. Cuprate superconductivity –state splits off from degenerate manifold

Case(2) Cuprates The most likely active combination formed of time-reversed pairs of localised molecular orbitals consistent with direct product analysis of condensate wavefunction with 1B1 symmetry , a short coherence length and with ‘strong superconductivity’ is a pair of e-localised orbitals See L.J Dunne, E.J Brändas, J.N Murrell, V. Coropceanu, Solid State Comm. 108 619 (1998)

For the superconducting state to be the ground state must be negative. u is the pair Coulomb repulsion. Last term is a coherent correlation term which ‘eats up’ the short range Coulombic repulsion in the Bohm-Pines potential.

Second order reduced electronic density matrix pair subspace P for attractive (BCS) and repulsive cases showing identical macroscopic condensation with ODLRO in both scenarios. M pairs , N states.

Theoretical Shape of d-wave Cuprate condensate wavefunction in k-space and real-space.

Plot of experimental condensate density (Uemura) against theoretical prediction for a series of cuprates (LSCO and YBCO). For full details see L J. Dunne, Int. J. Quant. Chem. 2015, 115, 1443-1458

Mean-Field Theory of Thermal Properties and Phase Diagram for electron doping for hole doping Theoretical Phase diagram for doping of a cuprate layer showing superconducting (SC) domes.

An averaged energy gap can also be obtained and shown below for electron and hole doping respectively. These gap rations embrace the ranges found experimentally for cuprates and are typically higher than the BCS ratio

CONCLUSIONS High Temperature cuprate superconductivity is an electron correlation effect. Remarkable Hamiltonian matrix structure allows electron correlations to produce off-diagonal range order (ODLRO) giving rise to superconductivity There in no exchange boson. Time reversed electron pairs undergo a superconducting condensation with dx2-y2 symmetry on alternant Cuprate lattices with many calculated properties following experimental trends .

The End Thank you for listening. Still a lot to do! Model has many limitations. For More details see (1) Dunne, L. J.; Brändas. E. J. Superconductivity from Repulsive Electronic Correlations on Alternant Cuprate And Iron-Based Lattices; Int. J. Quantum Chem. 2013, 113, 2053-2059 (2) Dunne, L J. High-temperature superconductivity and long-range order in strongly correlated electronic systems Int. J. Quant. Chem. 2015, 115, 1443-1458 (3) High Temperature Superconductivity in Strongly Correlated Electronic Systems Lawrence J. Dunne, Erkki J. Brändas , Hazel Cox, Advances in Quantum Chemistry, 2016