Strongly Correlated Electron Systems a DMFT Perspective Gabriel
Strongly Correlated Electron Systems: a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University
REVIEW OF SOLID STATE THEORY. n Chapter 1. The Standard Model of Solids. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
The electron in a solid: wave picture Momentum Space (Sommerfeld) Maximum metallic resistivity 200 mohm cm Standard model of solids Periodic potential, waves form bands , k in Brillouin zone Landau: Interactions renormalize away THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Standard Model of Solids RIGID BAND PICTURE. Optical response, transitions between bands. Quantitative tools: DFT, LDA, GGA, total energies, good starting point for spectra, GW, and transport THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Density functional and Ko • Kohn Sham spectra, proved to be an excelent starting point for doing perturbatio theory in screened Coulomb interactions GW. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
LDA+GW: semiconducting gaps THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Solid State Physics n Chapter 2. Mott insulators. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
The electron in a solid: particle picture. n Ni. O, Mn. O, …Array of atoms is insulating if a>>a. B. Mott: correlations localize the electron e_ e_ • Superexchange • Think in real space , solid collection of atoms • High T : local moments, Low T spin-orbital order THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Mott : Correlations localize the electron Low densities, electron behaves as a particle, use atomic physics, work in real space. • One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bands. Ni. O, Co. O Mn. O…. ) • H H H+ H H H motion of H+ forms the lower Hubbard band • H H- H H motion of H_ forms the upper Hubbard band • Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic RUTGERS Physics. THE STATE UNIVERSITY OF NEW JERSEY
n Solid State Physics Chapter 3, strongly correlated electrons. n Status: unfinished. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Strong Correlation Problem • A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem. • These systems display anomalous behavior (departure from the standard model of solids). • Neither LDA –GW or LDA+U or Hartree Fock work well. • Need approach which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands. New reference point, to replace the Kohn Sham system. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Failure of the standard model n DMFT is a new reference frame to approach strongly correlated phenomena, and describes naturally , NON RIGID BAND picture, highly resistive states, treats quasiparticle excitations and Hubbard bands on the same footing. . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Correlated Materials do big things n n Mott transition. Huge resistivity changes V 2 O 3. Copper Oxides. . (La 2 -x Bax) Cu. O 4 High Temperature Superconductivity. 150 K in the Ca 2 Ba 2 Cu 3 Hg. O 8. Uranium and Cerium Based Compounds. Heavy Fermion Systems, Ce. Cu 6, m*/m=1000 (La 1 -x. Srx)Mn. O 3 Colossal Magneto-resistance. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Pressure Driven Mott transiti THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
V 2 O 3 resistivity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Cuprate Superconductors THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
n Correlated Electron Materials are based on different physical principles outside the “standard model”, exciting perspectives for technological applications (e. g. high Tc). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Strongly Correlated Materials. n n n Large thermoelectric response in Ce. Fe 4 P 12 (H. Sato et al. cond-mat 0010017). Ando et. al. Na. Co 2 -x. Cux. O 4 Phys. Rev. B 60, 10580 (1999). Large and ultrafast optical nonlinearities Sr 2 Cu. O 3 (T Ogasawara et. a Phys. Rev. Lett. 85, 2204 (2000) ) Huge volume collapses, Ce, Pu. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Large thermoelectric power in a metal with a large number of carriers Na. Co 2 O 4 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Large and ultrafast optical nonlinearities Sr 2 Cu. O 3 (T Ogasawara et. a Phys. Rev. Lett. 85, 2204 (2000) ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
More examples n n Li. Co. O 2 Used in batteries, laptops, cell phones THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Breakdown of standard model n n Many Qualitative Failures Large metallic resistivities exceeding the Mott limit. [Anderson, Emery and Kivelson] Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics. [G. Sawatzki] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Failure of the standard model : Anomalous Resistivity: Li. V 2 O 4 Takagi et. al. PRL 2000 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Failure of the Standard Model: Anomalous Spectral Weight Transfer Optical Conductivity Schlesinger et. al (1993) Neff depends on T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Breakdown of the standard model : Anomalous transfer of optical weight [D. Van der Marel group ] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Breakdown of the Standard Model n n The LDA+GW program fails badly, Qualitatively incorrect predictions. Incorrect phase diagrams. Physical Reason: The one electron spectra, contains both Hubbard Bands and Quasiparticle featurs. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Basic Difficulties n n Lack of a small parameter. Kinetic energy is comparable to Coulomb energies. Relevant degrees of freedom change their form in different energy scales, challenge for traditional RG methods. WANTED: a simple picture of the physical phenomena, and the physics underlying a given material. WANTED: a computational tool to replace LDA+GW THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
n Breakthru: Development of Dynamical Mean Field Theory. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Dynamical Mean Field Theory n n Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition. Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
The Mott transition Electronically driven MIT. n Forces to face directly the localization delocalization problem. Central issue in correlated electron systems. n Relevant to many systems, eg V 2 O 3 n Techniques applicable to a very broad range or problems. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Mott transition in V 2 O 3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Pressure Driven Mott transiti THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Insight n n n Phase diagram in the T, U plane of a frustrated ((the magnetic order is supressed)) correlated system at integer filling. At high temperatures, the phase diagram is generic, insensitive to microscopid details. At low temperatures, every detail matters. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et. al PRL (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Pressure Driven Mott transiti THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Insight, in the strongly correlated region the one particle density of states has a three peak structure Low energy Quasiparticle Peak plus Hubbard bands. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
DMFT has bridged the gap be n Delocalized picture, it should resemble the density of states, (perhaps with some additional shifts and satellites). Localized picture. Two peaks at the ionization and affinity energy of the atom. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
One electron spectra near the Mott transition. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Insights from DMFT q. The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase q. Control parameters: doping, temperature, pressure… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
ARPES measurements on Ni. S 2 -x. Sex . Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
QP in V 2 O 3 was recently found Mo et. al THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Anomalous metallic resistivities In the “ in between region “ anomalous resistivities are the rule rather than the exception. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Failure of the Standard and Model: Ni. Se. Miyasaka S 2 -x x (2000) Takagi THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Anomalous Resistivity and Mott transition (Rozenberg et. Al. ) Ni Se 2 -x Sx Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined RUTGERS THE STATE UNIVERSITY OF NEW JERSEY
More recent work, organics, Limelette et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
RUTGERS THE STATE UNIVERSITY OF NEW JERSEY other types of printers. cript printer, but not to Comment: was not saved a preview included in it. Preview: Creator: gnuplot Title: Anomalous Resistivities when wave picture does not apply. Doped Hubbard model
Qualitative single site DMFT predictions: Optics n n Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is drive by transfer of spectral weight. Consequences for optics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Anomalous transfer of spectral weight heavy fermions Rozenberg Kajueter Kotliar (1996) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Anomalous Spectral Weight Transfer: Optics Below energy Apreciable. T dependence found. Schlesinger et. al (Fe. Si) PRL 71 , 1748 , (1993) B Bucher et. al. Ce 2 Bi 4 Pt 3 PRL 72, 522 (1994), Rozenberg et. al. PRB 54, 8452, (1996). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Ising critical endpoint! In V 2 O 3 P. Limelette et. al. Science Vol 302 (2003). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Conclusion. n n An electronic model accounts for all the qualitative features of the finite temperature of a frustrated system at integer occupancy. The observation of the spinodal lines and the wide classical critical region where mean field holds indicate the coupling to the lattice is quantitatively very important. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Formations of structures in k space. n Cluster dynamical mean field study. n Parcollet Biroli and Kotliar Cond-Matt 0308577 n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
• Evolution of the distribution in k space of the low energy spectral intensity as the Mott transition is approached. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
U/D=2, U/D=2. 25 (Parcollet et. al. ) Uc=2. 35+-. 05, Tc/D=1/44 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Conjecture n n Formation of hot regions is a more general phenomena due to the proximity to the Mott point. Plaquette reference system is good enough to contain the essential features of momentum space differentiation. Application to cuprates. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Lattice and cluster self energies THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Mechanism for hot spot formation THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Deviations from single site DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
System specific application : Pu THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Pu in the periodic table actinides THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Electronic Physics of Pu THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
DFT studies. n n Underestimates the volume by 35 % Predicts Pu to be magnetic. Largest quantitative failure of DFT-LDA-GA Fails to predict a stable delta phase. (negative shear) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Alpha and delta Pu THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Phonon Spectra n n n Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure. Phonon spectra reveals instablities, via soft modes. Phonon spectrum of Pu had not been measured until recently. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Expts’ Wong et. al. Science 301. 1078 (2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Plutonium is just one correlated element, there are many more strongly correlated COMPOUNDS which can be studies with this method. n Worldwide activity. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
n n Test the idea that the crucial physics of strongly correlated materials can be captured from a local reference set. Test worst case scenario, one dimension. [Kancharla and Bolech] [Capone and Civelli]. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMF T vs Nc=1 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
N vs mu in one dimension. Comparaison of 2+8 vs exact Bethe Anzats, [M. Capone and M. Civelli] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
What do we want from materials theory? New concepts , qualitative ideas n Understanding, explanation of existent experiments, and predictions of new ones. n Quantitative capabilities with predictive power. n Notoriously difficult to achieve in strongly correlated materials. DMFT is delivering on both counts. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Outlook n n Local approach to strongly correlated electrons offers a new starting point or reference frame to describe new physics. Breakdown of rigid band picture. Many extensions, make the approach suitable for getting insights and quantitative results in many correlated materials. RESEARCH OPPORTUNITY. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Networks. n n Psik f electrons. European Research and training network, with nodes in Denmark, UK, France, Germany, and Holland NJ. Computational Material Science Network. CMSN, with nodes at Brookhaven, UC Davis, ORNL, NJIT, Rutgers, NRL, Cornell, THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Students and Postdocs n n n Marcelo Rozenberg. Development of DMFT. Goetz Moeller. Theory of the Mott transition. Henrik Kajueter. Development of techniques for solving DMFT equations. Indranil Paul. Thermal Transport in Correlated Materials. Sergej Pankov. Extensions of DMFT and studies of disordered system and electron phonon interactions. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Students and Postdocs n n n Vlad Dobrosavlevic. Studies of disordered systems with DMFT. Metal Insulator Transition in two dimensions. Ping Sun. Combinations of EDMFT and GW. Studies of Heavy fermion critical points. Sahana Murthy. Study of the Mott transition in americium. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Students and Postdocs n n n Sergej Savrasov: DMFT study of the volume collapse and photoemission in plutonium. Viktor Udovenko: Thermoelectric power of correlated materials. Optical studies of correlated materials. Christjan Haule: New techniques for solving the DMFT equations. Optical studies of Cerium. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Postdocs Students. n n Harald Jeschke, development of DMFT solvers, and molecular dynamics using DMFT. Qimiao Si. Non Fermi liquid states using DMFT and its extensions. Ekke Lange. Magneto-transport studies of correlated materials. Landau theory of the Mott transition. Michael Sindel Andrea Perali and Marcello Civelli hot spots in cuprates. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Students and Postdocs n n n Chris Marianetti, DMFT studies of materials for battery applications. Olivier Parcollet. and Giulio Biroli Extensions of DMFT to clusters. High temperature superconductitivity and organic materials. . Venky Kancharla and Carlos Bolech, development of DMFT-DMRG for clusters. Applications to charge density wave materials THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Students and Postdoc n n Marcello Civelli and Massimo Capone. High temperature superconductivity using CDMFT. Antonina Toropova Cr. O 2, a high temperature half metallic systems. Tudor Stanescu. Recent improvement of DMFT Xi Dai. Phonons in plutonium. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Acknowledgements: Development of DMFT Collaborators: V. Anisimov, G. Biroli, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, K. Haujle, W. Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, O. Parcollet , G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X. Y. Zhang Support: NSF DMR 0096462 Support: Instrumentation. NSF DMR-0116068 Work on Fe and Ni: ONR 4 -2650 Work on Pu: DOE DE-FG 02 -99 ER 45761 and LANL subcontract No. 03737 -001 -02 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Materials Science n n n New concepts. Techniques. Analytical. Quantum Field Theory and the Renormalization Group. Numerical. New algoritms. Hardware. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
High Performance Computing http: //beowulf. rutgers. edu High Performance Computing Project Department of Physics and Astronomy National Science Foundation - NSF 0116068: Acquisition of a Network Cluster of Advanced Workstations for First Principles Electronic Structure Calculations of Complex Materials THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
TOP 500 (ICL-UT) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
TOP 500 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Epsilon Plutonium. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Anomalous transfer of spectral weight heavy fermions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Anomalous transfer of spectral weight THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Anomalous transfer of spectral weigth heavy fermions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Example: DMFT for lattice model (e. g. single band Hubbard). Muller Hartman 89, Chitra and Kotliar 99. n n n Observable: Local Greens function Gii (w). Exact functional G [Gii (w) ]. DMFT Approximation to the functional. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Spectral Density Functional : effective action construction (Chitra and GK). n n Introduce local orbitals, ca. R(r-R)orbitals, and local GF G(R, R)(i w) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for r(r) and G and performing a Legendre transformation, G[r(r), G(R, R)(iw)] Approximate functional using DMFT insights. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Mott transition and superexchange THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
How good is the LOCAL approximation? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMF T vs Nc=1 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
N vs mu in one dimension. Compare 2+8 vs exact Bethe Anzats, [M. Capone and M. Civelli] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Strongly correlated systems are usually treated with model Hamiltonians THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Mean-Field : Classical vs Quantum Classical case Phys. Rev. B 45, 6497 Quantum case THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A. Georges, G. Kotliar (1992)
view. R. Chitra and G. Kotliar Phys Rev. B. n (2000). Identify observable, A. Construct an exact functional of n n n <A>=a, G [a] which is stationary at the physical value of a. Example, density in DFT theory. (Fukuda et. al. ) When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. DMFT, build functionals of the LOCAL spectral function. Exact functionals exist. We also have good approximations! Extension to an ab initio method. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Realistic applications of DMFT References n n V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 -7367 (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov and G. Kotliar and Abrahams funcional formulation for full self consistent Nature {bf 410}, 793(2001). Reviews: Held et. al. , Psi-k Newsletter #{bf 56} (April 2003), p. 65 Lichtenstein Katsnelson and Kotliar cond-mat/0211076: THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Double Occupancy vs U n CDMFT Parcollet, Biroli GK THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Compare with single site results Rozenberg Chitra Kotliar PRL 2002 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Mott transition in cluster THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Evolution of the Spectral Function. U/D=2, U/D=2. 25 (Parcollet et. al. ) Uc=2. 35+-. 05, Tc/D=1/44 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
- Slides: 108