High T c Superconductors & QED 3 theory of the cuprates Tami Pereg-Barnea

outline High T c – Known and unknown Some experimental facts and phenomenology Some experimental facts and phenomenologyModels Attempts to solve the problem Attempts to solve the problem The inverted approach The inverted approach QED 3 – formulations and consequences

Facts The parent compounds are AF insulators. 2D layers of CuO 2 Superconductivity is the condensation of Cupper pairs with a D-wave pairing potential. The cuprates are superconductors of type II The cuprates are superconductors of type II The “normal state” is a non-Fermi liquid, strange metal.

YBCO microwave conductivity BSCCO ARPES

Underdoped Bi2212 Neutron scattering – (  ) resonance in YBCO

Phenomenology The superconducting state is a D-wave BCS superconductors with a Fermi liquid of nodal quasiparticles. The AF state is well described by a Mott- Hubbard model with large U repulsion. The pseudogap is strange! Gap in the excitation of D-wave symmetry but no superconductivity Gap in the excitation of D-wave symmetry but no superconductivity Non Fermi liquid behaviour – anomalous power laws in verious observables. Non Fermi liquid behaviour – anomalous power laws in verious observables.

Phase diagram AF

Theoretical approaches Starting from the Hubbard model at ½ filling.  Slave bosons SU(2) gauge theories  Spin and charge separation  Stripes Phenomenological Phenomenological  SO(5) theory  DDW competing order

The inverted approach Use the phenomenology of d-SC as a starting point. “Destroy” superconductivity without closing the gap and march backwards along the doping axis. The superconductivity is lost due to quantum/thermal fluctuations in the phase of the order parameter.

Vortex Antivortex unbinding Emery & Kivelson Nature 374, 434 (1995) Franz & Millis PRB 58, (1998)

Phase fluctuations Assume  0 = |  |= const. Treat exp{i  (r)} as a quantum number – sum over all paths. Fluctuations in  are smooth (spin waves) or singular (vortices). Perform the Franz Tešanović transformation - a singular gauge transformation. The phase information is encoded in the dressed fermions and two new gauge fields.

Formalism Start with the BdG Hamiltonian FT transformation – in order to avoid branch cuts.

The transformed Hamiltonian The gauge field a  couples minimally    a  The resulting partition function is averaged over all A, B configurations and the two gauge fields are coarse grained.

The physical picture RG arguments show that v  is massive and therefore it’s interaction with the Toplogical fermions is irrelevant. The a  field is massive in the dSC phase (irrelevant at low E) and massless at the pseudogap. The kinetic energy of a  is Maxwell - like.

Quantum “Electro” Dynamics Linearization of the theory around the nodes. Construction of 2 4-component Dirac spinors.

Dressed QP’s QED 3 Spectral function Optimally doped BSCCO Above T c T.Valla et al. PRL (’00)

Chiral Symmetry Breaking  AF order The theory of Quantum electro dynamics has an additional symmetry, that does not exist in the original theory. The Lagrangian is invariant under the global transformation where  is a linear combination of

The symmetry is broken spontaneously through the interaction of the fermions and the gauge field. The symmetry breaking (mass) terms that are added to the action, written in the original nodal QP operators represent: Subdominant d+is SC order parameter Subdominant d+is SC order parameter Subdominant d+ip SC order parameter Subdominant d+ip SC order parameter Charge density waves Charge density waves Spin density waves Spin density waves

Antiferromagnetism The spin density wave is described by: The spin density wave is described by: where ,  labels denote nodes. where ,  labels denote nodes. The momentum transfer is Q, which spans two antipodal nodes. At ½ filling, Q → ( ,  ) – commensurate Antiferromagnetism. _

Summary Inverted approach: dSC → PSG → AF View the pseudogap as a phase disordered superconductor. View the pseudogap as a phase disordered superconductor. Use a singular gauge transformation to encode the phase fluctuation in a gauge field and get QED 3 effective theory. Chirally symmetric QED 3  Pseudogap Broken symmetry  Antiferromagnetism