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. Uc Yokoyama-Shiba, JPSJ (1987)

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 Organic conductors k-(BEDT-TTF)2X X1- , BEDT-TTF 0.5+ = quarter filling Φ BEDT-TTF layer X layer κ-ET CuO2 t’ t’ t t If 2 BEDT-TTF molecules form a dimer, it can be regarded as a single site. H. Kino and H. Fukuyama, J.Phys.Soc.Japan 64, 2726 (1995).

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

Superconductivity at half-filling t’ t’ t t κ-ET CuO2 t’ t’ t t Organic conductors k-(BEDT-TTF)2X Superconductivity first-order Mott transition k-(BEDT-TTF)2Cu[N(CN)2]Br Also Spin-Liquid state k-(BEDT-TTF)2Cu2(CN)3 2) High-Tc cuprates T’-La2-xRxCuO4 （R = Sm, Eu, Gd, Tb, Lu, Y） Tc=21.4K T’ -structure Tsukada et al., SSC 133, 427 (2005)

Mott transition as a liquid-gas phase transition Motivation for t-t’-U Hubbard model Yokoyama-Ogata-Tanaka: Cond-mat/0607470 Mott transition as a liquid-gas phase transition Superconductivity at half-filling Effects of t’ : Frustration (AF  superconductivity, RVB-Insulator) We have to study the Hubbard model. So far, quantum MC is negative, but FLEX gives SC. Doped case Weak coupling U<W BCS-like Strong coupling U>W 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 conductive insulating Mott Transition bound （μ→ 1 ） 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 RVB insulator Uc

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. “RVB insulator” Uc

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.)

less-than-half filling 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

less-than-half filling 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 (Excess holes are mobile) USI Small U (U < Uco) weak-coupling region

DE = Enorm - ESC DE ~ 10- 4 t at U=4t DE ~ e- t / U Condensation energy 1) Small U (U < USI) very small DE DE ~ 10- 4 t at U=4t ~ 0.4K DE = Enorm - ESC PQ|FS PQ|BCS DE ~ e- t / U consistent with QMC, and weak-coupling theory 2) Intermediate U (USI < U < UCO) abrupt increase of DE 3) Large U (UCO < U) large DE DE ~ e- a U / t = e- 4a 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 Small U (U < Uco) U / t = 4 t’ > 0 -- unfavorable van Hove singularity (electron doped) t’ = - 0.15 High-Tc should be in　U>Uco

Result 3: doping-dependence t-J region weak-coupling region t’ = -0.1 t’ = -0.1 U / t = 8 U / t = 4 t’ = -0.3 High-Tc should be in U>Uco (t-J region) Consistent with Uemura’s plot (Tc ∝ d)

“Doped Mott insulator” Conclusions Modified variational state doublon-holon bound state is important. Mott transition (half-filling)(Brinkman-Rice) Doped case “RVB insulator” “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)