The Mott Transition: a CDMFT study G. Kotliar Physics Department and Center for Materials Theory Rutgers Sherbrook July 2005

oModel for kappa organics. [O. Parcollet, G. Biroli and G. Kotliar PRL, 92, (2004)) ] oModel for cuprates [O. Parcollet (Saclay), M. Capone (U. Rome) M. Civelli (Rutgers) V. Kancharla (Sherbrooke) GK(2005). Cluster Dynamical Mean Field Theories a Strong Coupling Perspective. T. Stanescu and G. Kotliar (in preparation 2005) Talk by B. Kyung et. al. Tomorrow. cond- mat/ Short-Range Correlation Induced Pseudogap in Doped Mott Insulators Talk by V. Kancharla Sarma (this morning) References

Outline Motivation and Objectives.Schematic Phase Diagram(s) of the Mott Transition. Finite temperature study of very frustrated anisotropic model. [O. Parcollet ] Low temperature study of the normal state of the isotropic Hubbard model. [M. Civelli, T. Stanescu ] [See also B. Kyung’s talk] Superconducting state near the Mott transition. [ M. Capone. V. Kancharla Sarma ] Conclusions.

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.

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) 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)

Problems with the approach. Neel order Stability of the pseudogap state at finite temperature. [Ubbens and Lee] Missing incoherent spectra. [ fluctuations of slave bosons ] Dynamical Mean Field Methods are ideal to remove address these difficulties.

T/W Phase diagram of a Hubbard model with partial frustration at integer filling. M. Rozenberg et.al., Phys. Rev. Lett. 75, (1995)..Phys. Rev. Lett. 75, (1995). COHERENCE INCOHERENCE CROSSOVER

Focus of this work Generalize and extend these early mean field approaches to systems near the Mott transition. Obtain the solution of the 2X 2 plaquette and gain physical understanding of the different CDMFT states. Even if the results are changed by going to larger clusters, the short range physics is general and will teach us important lessons. Follow states as a function of parameters. Adiabatic continuity. Furthermore the results can be stabilized by adding further interactions.

Finite T Mott tranisiton in CDMFT Parcollet Biroli and GK PRL, 92, (2004))

Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

Evolution of the k resolved Spectral Function at zero frequency. (QMC study Parcollet Biroli and GK PRL, 92, (2004)) ) Uc= , Tc/D=1/44. Tmott~.01 W U/D=2 U/D=2.25

Momentum Space Differentiation the high temperature story T/W=1/88

Physical Interpretation Momentum space differentiation. The Fermi liquid –Bad Metal, and the Bad Insulator - Mott Insulator regime are realized in two different regions of momentum space. Cluster of impurities can have different characteristic temperatures. Coherence along the diagonal incoherence along x and y directions. Connection with slave Boson theory divergence of Sigma13. Connections with RVB (Schmalian and Trivedi)

Cuprate superconductors and the Hubbard Model. PW Anderson 1987

. Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition. Earlier studies (Katsnelson and Lichtenstein, M. Jarrell, M Hettler et. al. Phys. Rev. B 58, 7475 (1998). T. Maier et. al. Phys. Rev. Lett 85, 1524 (2000). ) used QMC as an impurity solver and DCA as cluster scheme. We use exact diag ( Krauth Caffarel 1995 with effective temperature 32/t=124/D ) as a solver and Cellular DMFT as the mean field scheme. Connect the solution of the 2X2 plaquette to simpler mean field theories. CDMFT study of cuprates

Follow the “normal state” with doping. Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

Hole doped case t’=-.3t, U=16 t n=

K.M. Shen et. al. Science (2005). For a review Damascelli et. al. RMP (2003)

Approaching the Mott transition: CDMFT Picture Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! D wave gapping of the single particle spectra as the Mott transition is approached. Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, (2004) and Senechal and tremblay for cuprates with VCPT.

Results of many numerical studies of electron hole asymmetry in t-t’ Hubbard models Tohyama Maekawa Phys. Rev. B 67, (2003) Senechal and Tremblay. PRL (2004) Kusko et. al. Phys. Rev 66, (2002)

Experiments. Armitage et. al. PRL (2001). Momentum dependence of the low-energy Photoemission spectra of NCCO

Approaching the Mott transition: CDMFT picture. Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! General phenomena, BUT the location of the cold regions depends on parameters. Quasiparticles are now generated from the Mott insulator at ( , 0). Results of many l studies of electron hole asymmetry in t-t’ Hubbard models Tohyama Maekawa Phys. Rev. B 67, (2003) Senechal and Tremblay. PRL (2004) Kusko et. al. Phys. Rev 66, (2002). Kusunose and Rice PRL 91, (2003).

Comparison with Experiments in Cuprates: Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k hole dopedelectron doped K.M. Shen et.al P. Armitage et.al. 2001

To test if the formation of the hot and cold regions is the result of the proximity to Antiferromagnetism, we studied various values of t’/t, U=16.

Introduce much larger frustration: t’=.9t U=16t n=

Approaching the Mott transition: Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! General phenomena, but the location of the cold regions depends on parameters. With the present resolution, t’ =.9 and.3 are similar. However it is perfectly possible that at lower energies further refinements and differentiation will result from the proximity to different ordered states.

Understanding the result in terms of cluster self energies (eigenvalues)

Cluster Eigenvalues

Evolution of the real part of the self energies.

Fermi Surface Shape Renormalization ( t eff ) ij =t ij + Re(  ij 

Fermi Surface Shape Renormalization Photoemission measured the low energy renormalized Fermi surface. If the high energy (bare ) parameters are doping independent, then the low energy hopping parameters are doping dependent. Another failure of the rigid band picture. Electron doped case, the Fermi surface renormalizes TOWARDS nesting, the hole doped case the Fermi surface renormalizes AWAY from nesting. Enhanced magnetism in the electron doped side.

Understanding the location of the hot and cold regions. Interplay of lifetime and fermi surface.

Comparison of periodization methods for A(  =0,k)   

Pseudogap: insights from cumulant periodization. Qualitative Difference between the periodization methods. The cumulant periodization, is a non linear interpolation of the self energies (linear interpolation of the cumulants). When the off diagonal elements of the self energy get large, it gives rise to lines of poles of the self energy, in addition to the Fermi lines. Quasi-one d, Essler and Tsvelik.

Subtle topological phase transition at intermediate doping ? Is there a quantum critical point related to the change of topology of the Fermi surface ? Is there a quantum critical point associated to the emergence of lines of zeros. This is NOT a Fermi surface instability, invisible to weak coupling analysis. We checked with PCMDFT that the results survive, unlike the corresponding quasi-1d case (Arrigoni et. al. Giamarchi Georges and Biermann).

Large Doping

Small Doping

How is the Mott insulator approached from the superconducting state ? Work in collaboration with M. Capone, see also V. Kancharla’s talk.

Superconductivity in the Hubbard model role of the Mott transition and influence of the super- exchange. (M. Capone, V. Kancharla. CDMFT+ED, 4+ 8 sites t’=0).

The superconductivity scales with J, as in the RVB approach. Qualitative difference between large and small U. The superconductivity goes to zero at half filling ONLY above the Mott transition.

Order Parameter and Superconducting Gap.

In BCS theory the order parameter is tied to the superconducting gap. This is seen at U=4t, but not at large U. How is superconductivity destroyed as one approaches half filling ?

Evolution of the low energy tunneling density of state with doping. Decrease of spectral weight as the insulator is approached. Low energy particle hole symmetry.

Superconductivity is destroyed at half filling due to a reduction of the one electron weight. Just like in the slave boson. High energy ph asymmetry. Low energy ph symmetry.

Alternative view

DMFT is a useful mean field tool to study correlated electrons. Provide a zeroth order picture of a physical phenomena. Provide a link between a simple system (“mean field reference frame”) and the physical system of interest. [Sites, Links, and Plaquettes] Formulate the problem in terms of local quantities (which we can usually compute better). Allows to perform quantitative studies and predictions. Focus on the discrepancies between experiments and mean field predictions. Generate useful language and concepts. Follow mean field states as a function of parameters. K dependence gets strong as we approach the Mott transition. Fermi surfaces and lines of zeros of G. Conclusions

Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! General phenomena, but the location of the cold regions depends on parameters. Study the “normal state” of the Hubbard model is useful. Character of the superconductivity is different for small and large U.

the Hubbard model does not capture the trend of supra with t’. Need augmentation.(Venky Kancharla Sarma).Same trend observed in DCA Maier and Jarrell.

D wave Superconductivity and Antiferromagnetism t’=0 M. Capone V. Kancharla (see also VCPT Senechal and Tremblay ). Antiferromagnetic (left) and d wave superconductor (right) Order Parameters

Estimates of upper bound for Tc exact diag. M. Capone. U=16t, t’=0, ( t~.35 ev, Tc ~140 K~.005W)

Site  Cell. Cellular DMFT. C-DMFT. G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001) tˆ(K) hopping expressed in the superlattice notations. Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998) Katsnelson and Lichtenstein periodized scheme, Nested Cluster Schemes, causality issues, O. Parcollet, G. Biroli and GK Phys. Rev. B 69, (2004)Phys. Rev. B 69, (2004)

Understanding in terms of cluster self-energies. Civelli et. al.

Insulating anion layer  -(ET) 2 X are across Mott transition ET = X -1 [(ET) 2 ] +1 conducting ET layer t’ t modeled to triangular lattice X-X- Ground State U/tt’/t Cu 2 (CN) 3 Mott insulator Cu[N(CN) 2 ]Cl Mott insulator Cu[N(CN) 2 ]BrSC Cu(NCS) 2 SC Cu(CN)[N(CN) 2 ] SC Ag(CN) 2 H 2 OSC I3I3 SC

Electron doped case t’=.9t U=16t n= Color scale x=.9,.32,.22

Two paths for calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions DMFT ideas can be used in both cases.

Outline ____________________________________________________________ INTUITIVE NOTIONS OF DMFT AND WEISS FIELD CAVITY CONSTRUCTION. Mapping of lattice onto a cluster in a medium. With a prescription for building the medium from the computation of the cluster quantities. Prescription for reconstructing lattice quantities. Weiss field describe the medium. Cavity Construction is highly desireable. Delta is non zero on the boundary EFFECTIVE ACTION CONSTRUCTION. Show how it is done. It is not perturbative. It is general. It includes everything you want to know. It gives you a reference system. Examples. APPROXIMATE WEISS FIELD SUBTLETIES WITH THE WEISS FIELD, WHAT ABOUT GETTING NOT ONLY Gii but also Gij, and admiting Delta_ij, is the unique solution t_ij ? How to go from lattice to impurity model. CDMFT construction. Picture giving simga_c, M_cluster chi_cluster Gamma_cluster OTHER CLUSTER SCHEMES. CDMFT-PCDMFT. Truncation using U---> and G----> CPT and VCPT from the point of view of a Weiss field functional. cal{G-0} ----> cal{G_0} atom cal{G_0} -----Cal{G_0}+Anomalous static self energy List of pair schemes. Bethe Peiersls Pair Schemes. DCA. PCDMFT.

a) Baym Kadanoff Functional b) Self energy functional. c) DFT. Want to generate good approximations. and their hybrids WEIS FIELD. I NEED TO GET THIS RIGHT HOW I DO THE SEPARATION, INTO PIECES WHAT IS THE EXACT AND WHA IS TEH

Nested Cluster Schemes. Explicit Cavity constructions CAUSALITY PROBLEMS. Parcollet Biorli KOtliar clausing CONVERGENCE. 1/L vs e-L. DCA converges as 1/L2 Classical limits. See what olivier ahs done on w-cdmft Impurity Solvers. OCA-QMC-ED. Lessons form the past. Combinations of methods ED discretization Role of distance. Role of functionals. Why not variational is usesless. Few sites Things in practice. Examples from one dimension. How to extract physical quantities CONCLUSIONS:

Medium of free electrons : impurity model. Solve for the medium using Self Consistency G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001)

Mott transition in layered organic conductors S Lefebvre et al. cond-mat/ , Phys. Rev. Lett. 85, 5420 (2000)

Insulating anion layer  -(ET) 2 X are across Mott transition ET = X -1 [(ET) 2 ] +1 conducting ET layer t’ t modeled to triangular lattice t’ t modeled to triangular lattice

References

CDMFT one electron spectra n=.96 t’/t=.-.3 U=16 t i

Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme. Causality issues O. Parcollet, G. Biroli and GK Phys. Rev. B 69, (2004)Phys. Rev. B 69, (2004)

Medium of free electrons : impurity model. Solve for the medium using Self Consistency G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001)

Difference in Periodizations

Advantages of the Weiss field functional. Simpler analytic structure near the Mott transition.

How to determine the parameters of the bath ? Extremize Potthoff’s self energy functional. It is hard to find saddles using conjugate gradients. Extremize the Weiss field functional.Analytic for saddle point equations are available Minimize some distance.

Convergence of Cluster Schemes as a function of cluster size. Aryamanpour et. al. DCA observables converge as 1/L^2. cond-mat/ [abs, ps, pdf, other] :abspspdfother –Title: Two Quantum Cluster Approximations Authors: Th. A. Maier (1), O. Gonzalez (1 and 2), M. Jarrell (1), Th. Schulthess (2) ((1) University of Cincinnati, Cincinnati, USA, (2) Oak Ridge National Laboratory, Oak Ridge, USA)Th. A. MaierO. GonzalezM. JarrellTh. Schulthess Aryamampour et. al. The Weiss field in CDMFT converges as 1/L. Title: The Dynamical Cluster Approximation (DCA) versus the Cellular Dynamical Mean Field Theory (CDMFT) in strongly correlated electrons systems Authors: K. Aryanpour, Th. A. Maier, M. Jarrell Comments: Comment on Phys. Rev. B 65, (2002). 3 pages, 2 figures Subj-class: Strongly Correlated Electrons Journal-ref: Phys. Rev. B 71, (2005)K. AryanpourTh. A. MaierM. Jarrell Biroli and Kotliar. Phys. Rev. B 71, (2005);cond-mat/ Local observables (i.e. observables contained in the cluster ) converge EXPONENTIALLY at finite temperatures away from critical points.Phys. Rev. B 71, (2005)cond-mat/

Hole doped case t’=-.3t, U=16 t n= Color scale x=

oQualitative Difference between the hole doped and the electron doped phase diagram is due to the underlying normal state.” In the hole doped, it has nodal quasiparticles near ( ,  /2) which are ready “to become the superconducting quasiparticles”. Therefore the superconducing state can evolve continuously to the normal state. The superconductivity can appear at very small doping. oElectron doped case, has in the underlying normal state quasiparticles leave in the (  0) region, there is no direct road to the superconducting state (or at least the road is tortuous) since the latter has QP at (  /2,  /2).

ED and QMC

Systematic Evolution