Optical Conductivity of Cuprates Superconductors: a Dynamical RVB perspective Work with K. Haule (Rutgers) K. Haule, G. Kotliar, Europhys Lett. 77, (2007).

Optics and RESTRICTED SUM RULES Low energy sum rule can have T and doping dependence. For nearest neighbor it gives the kinetic energy. Use it to extract changes in KE in superconducing state Below energy

J. Rozenberg, G. Kotliar, H. Kajueter, G. A. Thomas, D. H. Rapkine, J. M. Honig, and P. Metcalf, Phys. Rev. Lett. 75, 105 (1995). L. Baldassarre Poster P1 this conference. Hubbard model single site DMFT. [ W(T) is T dependent near Mott trans.

Point of view Study simple [“unrealistic”] models of the doped Mott insulator (RVB) Capture local physics. Reference frame is a plaquette in a medium. No vertex corrections yet! (but work in progress).

Optics and RESTRICTED SUM RULES n is only defined for T> Tc, while s exists only for T<Tc Experiment: use of this equation implies extrapolation. Theory : use of this equation implies of mean field picture to continue the normal state below Tc.

Optical Spectral Weight Can be Used to infer the mechanism of superconductivity.

electrons gain energy due to exchange energy holes gain kinetic energy (move faster) underdoped electrons gain energy due to exchange energy hole loose kinetic energy (move slower) overdoped BCS like same as RVB (see P.W. Anderson Physica C, 341, 9 (2000), or slave boson mean field (P. Lee, Physica C, 317, 194 (1999) Kinetic energy upon condensation J J J J

Treatement needs refinement The kinetic energy of the Hubbard model contains both the kinetic energy of the holes, and the superexchange energy of the spins. Physically they are very different. Experimentally only measures the kinetic energy of the holes.

Hubbard model U t-J model J-t Drude no-U Experiments intraband interband transitions ~1eV Excitations into upper Hubbard band Kinetic energy in t-J model Only moving of holes Optical Conductivity

E Energy difference between the normal and superconducing state of the t-J model. K. Haule GK

. Spectral weight integrated up to 1 eV of the three BSCCO films. a) under- doped, Tc=70 K; b) ∼ optimally doped, Tc=80 K; c) overdoped, Tc=63 K; the full symbols are above Tc (integration from 0+), the open symbols below Tc, (integrationfrom 0, including th weight of the superfuid). H.J.A. Molegraaf et al., Science 295, 2239 (2002). A.F. Santander-Syro et al., Europhys. Lett. 62, 568 (2003). Cond-mat G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 72, (2005).

Superexchange Mechanism Coherent Quasiparticles Re Slave Boson Mean Field Theory Phase Diagram. Formation of Singlets

CDMFT optics t-J model

At very low doping, one can separate two components. [Coherent and Incoherent] At large they merge into one “Drude-like” broad frequency range. Expected temperature dependence in overdoped region. [Narrowing of Drude peak]. Anomalous temperature dependence at low doping.

Cuttoff and temperature dependence of integrated optial spectral weight

At which frequency do we recover all the spectral weight ?

Optical weight increases as temperature decreases. The magnitude is approximately given by single site DMFT [as first computed by Toschi et.al, PRL (2005). ]. Substantial new physics is brought by the cluster effects. Existence of d wave superconductivity and pseudogap. Avoided criticality. Notice that in spite of the opening of a pseudogap. The spectral weight does not decrease with decreasing temperature for reasonable cuttoffs.!!!

At very high frequencies. Of the order of 3t. (t, ev) It is due to the anomalous greens function. Not visible in photoemission.

Optical Mass at low doping

Optical mass and plasma frequency

Padilla et.al.

Conclusion Optical anomalies, do NOT rule out the proximity to a Mott transition as a basis for a theoretical approach to describe the cuprates. a) temperature and doping dependence of the optical spectral weight. CDMFT on a plaquette, is a substantial improvement over the earlier slave boson approach, to describe the optics, and many other key experiments. [ My talk on Wendesday]. Further work to improve: a) our understanding of the plaquette CDMFT equations, b) to make the models more realistics c) to make CDMFT more flexible and d) to incorporate vertex corrections are warranted.

Power laws in optics. A. El Azrak,et.al. PR B 49, 9846 (1994). D. van der Marel, Nature 425, 271 (2003).

Avoided Quantum Criticality Intermediate physics phenomena. No analytic understanding of the dimension 2/3.

Stephan and Horsch Int. Jour Mod Phys B6, 141 (1992) Eskes Oles Meinders and Stephan PRB 50 (1994) 17980

RVB phase diagram of the Cuprate Superconductors. Superexchange. The approach to the Mott insulator renormalizes the kinetic energy Trvb increases. The proximity to the Mott insulator reduce the charge stiffness, TBE goes to zero. Superconducting dome. Pseudogap evolves continously into the superconducting state. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

RVB phase diagram of the Cuprate Superconductors P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987) Connection between the anomalous normal state of a doped Mott insulator and high Tc. Slave boson approach. coherence order parameter.  singlet formation order parameters.

U/t=4. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69, (2004) ]

Finite T, DMFT and the Energy Landscape of Correlated Materials T

Impurity Model-----Lattice Model  Weiss Field

Conclusion More quantitative comparison with experiments On the theory side. Investigate effects of t’ t’’ and more realistic electronic structure. Effects of vertex corrections, periodization. More extreme underdoping and overdoping. Better impurity solvers.

RVB phase diagram of the Cuprate Superconductors. Superexchange. Proximity to Mott insulator renormalizes the kinetic energy Trvb increases. Proximity to the Mott insulator reduce the charge stiffness, and QPcoherence scale. T BE goes to zero. Superconducting dome. Pseudogap evolves continuously into the superconducting state. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) Related approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria N. Trivedi, A. Paramenkanti PRL 87, (2001)

Hubbard vs t-J Drude Transition from uper to lower Hubbard band at U Incoherent part of the spectra

RESTRICTED SUM RULES Low energy sum rule can have T and doping dependence. For nearest neighbor it gives the kinetic energy. Below energy

Optical Spectral Weight Can be Used to infer the mechanism of superconductivity.

RESTRICTED SUM RULES Low energy sum rule can have T and doping dependence. For nearest neighbor it gives the kinetic energy. Below energy

RESTRICTED SUM RULES Low energy sum rule can have T and doping dependence. For nearest neighbor it gives the kinetic energy. Below energy

RVB phase diagram of the Cuprate Superconductors. Superexchange. Proximity to Mott insulator renormalizes the kinetic energy Trvb increases. Proximity to the Mott insulator reduce the charge stiffness, and QPcoherence scale. T BE goes to zero. Superconducting dome. Pseudogap evolves continuously into the superconducting state. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) Related approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria N. Trivedi, A. Paramenkanti PRL 87, (2001)

For reviews of cluster methods see: Georges et.al. RMP (1996) Maier et.al RMP (2005), Kotliar et.al RMP (2006)  Weiss Field Alternative (T. Stanescu and G. K. ) periodize the cumulants rather than the self energies. Parametrizes the physics in terms of a few functions. Impurity solver, NCA, ED, CTQMC

Superexchange mechanism?

Near the Mott transition the optical weight has a surprising large T dependence. M. J. Rozenberg et al., Phys. Rev. Lett. 75, 105 (1995).Phys. Rev. Lett. 75, 105 (1995) This phenomena of buildup of spectral weight with reducing temperature was found in cuprates, and was well accounted by single site DMFT. Toschi et. al. Phys. Rev. Lett. 95, (2005)