Mott Transition and Superconductivity in Twodimensional ttU Hubbard

Mott Transition and Superconductivity in Two-dimensional t-t’-U Hubbard model Masao Ogata (Univ. of Tokyo) H. Yokoyama (Tohoku Univ. ) Y. Tanaka (Nagoya Univ. ) Mott transition (Brinkman-Rice transition) Superconductivity in Hubbard model Doped case Variational Monte Carlo （VMC) study

Mott transition at half-filling “Doped Mott insulator” ----- What is Mott insulator? t-J model Heisenberg model (d 0) We need to study Hubbard model. Brinkman-Rice transition doublon number 0 at Uc (second-order phase transition) <d> However this Brinkman-Rice transition is not observed in VMC. Yokoyama-Shiba, JPSJ (1987) Uc

Mott transition at half-filling Brinkman-Rice transition is not observed in VMC. <d> E=0 for U >Uc Uc (Brinkman-Rice) Yokoyama-Shiba, JPSJ (1987) Doublon number is always finite. We modify variational states. Mott transition as a first-order (like a liquid-gas phase transition)

Superconductivity at half-filling 1) Organic conductors k-(BEDT-TTF)2 X Φ X 1 - , BEDT-TTF 0. 5+ = quarter filling BEDT-TTF layer X layer κ-ET t’ If 2 BEDT-TTF molecules form a dimer, it can be regarded as a single site. Cu. O 2 t t’ H. Kino and H. Fukuyama, J. Phys. Soc. Japan 64, 2726 (1995). t

Superconductivity in κ-(BEDT-TTF)2 X AF insulator d-wave 　　pressure Substitution of X 　 First order SC K. Kanoda U/t is controled 　　 (not filling control) 　 insulator SC metal Nonmagnetic SC Fisrt order insulator Nonmagnetic down to T=32 m. K Y. Shimizu et al (NMR)

Superconductivity at half-filling 1) Organic conductors k-(BEDT-TTF)2 X κ-ET t’ Cu. O 2 t’ t t Superconductivity first-order Mott transition k-(BEDT-TTF)2 Cu[N(CN)2]Br Also Spin-Liquid state k-(BEDT-TTF)2 Cu 2(CN)3 2) High-Tc cuprates T’-La 2 -x. Rx. Cu. O 4 （R = Sm, Eu, Gd, Tb, Lu, Y） Tc=21. 4 K T’ -structure Tsukada et al. , SSC 133, 427 (2005)

Motivation for t-t’-U Hubbard model Yokoyama-Ogata-Tanaka: Cond-mat/0607470 Mott transition as a liquid-gas phase transition Effects of t’ : Frustration (AF superconductivity, RVB-Insulator) Superconductivity at half-filling We have to study the Hubbard model. So far, quantum MC is negative, but FLEX gives SC. Doped case Weak coupling U<W Strong coupling U>W BCS-like t-J like = doped Mott insulator T=0 Variational Monte Carlo （VMC) study

Modified variational states Usual Gutzwiller factor nearest-neighbor doublon-holon correlation Doublon-holon bound states are favored in wave functions

PQ is important Mott Transition free （μ→ 0 ） bound （μ→ 1 ） conductive insulating variational states: Fermi sea, d-wave SC, mean-field AF First-order Mott transition is realized.

I. Phase diagram half filling (d=0) t-t’-U Hubbard model RVB insulator (Para-insulator with d-wave order parameter) d-wave SC × L = 10 ＋ L = 12

First-order Mott transition half filling (d=0) Energy crossing Doublon density order parameter of Mott transition (similar to gas-liquid transition) d-wave SC Uc RVB insulator

d-wave to RVB insulator half filling (d=0) d-wave pair correlation function d-wave is enhanced at U / t < 6. 5 t’ / t ~ －0. 25 Wave function has d-wave order parameter, but Pd vanishes. Uc “RVB insulator”

Momentum distribution function nodal point U < Uc : Fermi surface (metallic) U > Uc : no Fermi surface (insulator)

I. Phase diagram half filling (d=0) t-t’-U Hubbard model Similar results for triangular lattice RVB insulator (Para-insulator with d-wave order parameter) d-wave SC × L = 10 ＋ L = 12 Gan et al. , PRL 94, 067005 (2005) d-wave SC AF (Gutzwiller approx. )

II. Doped　Case less-than-half filling t-t’-U Hubbard model RVB insulator (Para-insulator with d-wave order parameter) d-wave SC × L = 10 ＋ L = 12 doping

II. Doped　Case less-than-half filling d = 0. 12 d-wave pair correlation function for various values of t’ Large U (U > Uco) “Doped Mott insulator” doublon-holon bound state = n. n. doublon-holon = virtual process inducing J-term t-J region USI (Excess holes are mobile) Small U (U < Uco) weak-coupling region

Condensation energy DE = Enorm - ESC PQ|FS PQ|BCS 1) Small U (U < USI) very small DE DE ~ 10 - 4 t at U=4 t ~ 0. 4 K DE ~ e - t / U consistent with QMC, and weak-coupling theory 2) Intermediate U (USI < UCO) abrupt increase of DE 3) Large U (UCO < U) large DE DE ~ e- a U / t = e- 4 a t / J consistent with t-J model

Result 2: t’-dependence for various values of U / t d = 0. 12, 0. 14 Large U (U > Uco) U/t=8 U / t = 12 U / t = 16 van Hove singularity t’ = - 0. 15 t’ > 0 -- unfavorable Small U (U < Uco) U/t=4 (electron doped) High-Tc should be in　U>Uco

Result 3: doping-dependence t-J region U/t=8 t’ = -0. 1 weak-coupling region U/t=4 t’ = -0. 1 t’ = -0. 3 High-Tc should be in U>Uco (t-J region) Consistent with Uemura’s plot (Tc ∝ d)

Conclusions • Modified variational state doublon-holon bound state is important. • Mott transition (half-filling)(Brinkman-Rice) “RVB insulator” • Doped case “Doped Mott insulator” large U (non-BCS) (t-J region) small U (BCS-like) (weak-coupling region) d-wave SC is enhanced for t’< 0 for U > Uco ~ 6. 5 t High-TC cuprates belong to the large U region (non-BCS) Experimental check: t’ -dependences (Uemura plot) Kinetic energy gain (optical sum rule) large U ----- kinetic energy gain in SC (non-BCS) (t-J region) small U ------ potential energy gain in SC (BCS-like)