1. Oct. 26, 2005KIAS1 Competing insulating phases in one-dimensional extended Hubbard models Akira Furusaki (RIKEN) Collaborator: M. Tsuchiizu (Nagoya) M.T. & A.F., PRL 88, 056402 (2002) PRB 66, 245106 (2002) PRB 69, 035103 (2004)

2. Oct. 26, 2005 KIAS 2 Contents Extended Hubbard model Ionic Hubbard model Generalized Hubbard ladder Various types of insulators: Mott insulator, Charge-Density Wave, Peierls insulator, Band insulator, staggered-flux state, …. One-dimensional models of interacting electrons at half filling Weak-coupling approach, Bosonization

3. Oct. 26, 2005 KIAS 3 Extended Hubbard model at half filling t U V

4. Oct. 26, 2005 KIAS 4 Standard phase diagram (before 1999) Emery (1979) Hirsch (1984) Cannon, Scalettar, Fradkin (1991) ……….

5. Oct. 26, 2005 KIAS 5 Weak-coupling theory (g-ology) 1-loop RG charge sector Charge gap if Spin sector Spin gap if spin sector RL

6. Oct. 26, 2005 KIAS 6 Phase diagram since 1999 Discovery of Bond-charge-density wave (BCDW) phase Nakamura (1999, 2000) Sengupta, Sandvik, Campbell (2002) ….. or Bond-Order-Wave (BOW) Found numerically

7. Oct. 26, 2005 KIAS 7 Vertex correction Separate transitions in charge & spin sectors In the strong-coupling regime 1 st order SDW-CDW transition Degeneracy of zeros of and are lifted

8. Oct. 26, 2005 KIAS 8 Tam, Tsai, & Campbell, cond-mat/0505396

9. Oct. 26, 2005 KIAS 9 Bosonization charge spin Order parameters

10. Oct. 26, 2005 KIAS 10 Bosonized form of the Hamiltonian density kinetic energy marginal perturbation relevant perturbation irrelevant perturbation SU(2) symmetry etc

11. Oct. 26, 2005 KIAS 11 Order parameters Classical analysis

12. Oct. 26, 2005 KIAS 12 Phase transitions SDW-BCDW transition: 2 nd orderCDW-BCDW transition: 2 nd orderCDW-SDW transition:1 st order

13. Oct. 26, 2005 KIAS 13 Ground-state phase diagram from bosonization approach 1-loop RG + classical analysis M. Tsuchiizu and A.F., Phys. Rev. Lett. 88, 056402 (2002)

14. Oct. 26, 2005 KIAS 14 Numerical Results Quantum Monte Carlo Sengupta, Sandvik, & Campbell, Phys. Rev. B 65, 155113 (2002) DMRG Y.G. Zhang, PRL 92, 246404 (2004)

15. Oct. 26, 2005 KIAS 15 Tricritical point on the CDW-BCDW phase boundary SSE QMC Luttinger liquid parameter at the continuous transition Sandvik, Balents & Campbell, PRL 92, 236401 (2004)

16. Oct. 26, 2005 KIAS 16 umklapp scattering becomes relevant for

17. Oct. 26, 2005 KIAS 17 Phase diagram (schematic) 1 st order transition CDW-BCDW c=1 Gaussian SDW-BCDW c=1 SU(2) 1

18. Oct. 26, 2005 KIAS 18 Extended Ionic Hubbard model at half filling Ionic Hubbard model Nagaosa & Takimoto (1986), Egami, Ishihara, & Tachiki (1993) Fabrizio, Gogolin, & Nersesyan, PRL 83, 2014 (1999) and Mott insulator and Band insulator Quantum Phase Transition ?

19. Oct. 26, 2005 KIAS 19 Spontaneously Dimerized Insulating Phase (SDI) (= BCDW Phase) 0 IsingKT BISDI MI Fabrizio, Gogolin, & Nersesyan, PRL 83, 2014 (1999)

20. Oct. 26, 2005 KIAS 20 Extended ionic Hubbard model nearest-neighbor repulsion V Bosonization perturbative RG + classical analysis

21. Oct. 26, 2005 KIAS 21 Bosonized form of the Hamiltonian density Kinetic energy marginal perturbation irrelevant perturbation relevant perturbation

22. Oct. 26, 2005 KIAS 22 Classical analysis GaussianIsing

23. Oct. 26, 2005 KIAS 23 Ground-state phase diagram cf. 1 st order transition

24. Oct. 26, 2005 KIAS 24 Schematic phase diagrams

25. Oct. 26, 2005 KIAS 25 Generalized Hubbard ladder at half filling t⊥t⊥ V ⊥, J ⊥ t pair

26. Oct. 26, 2005 KIAS 26 rung singlet state (D-Mott) charge density wave (CDW) Various Insulating Ground States that can appear in half-filled ladders singlet paring state (S-Mott) staggered flux state (SF) d-density wave orbital antiferromagnet Ex. SO(5) ladder model ground-state phase diagram Lin, Balents & Fisher (1998) Fjaerestad & Marston (2002)

27. Oct. 26, 2005 KIAS 27 Strong-coupling approach 4 basis states

28. Oct. 26, 2005 KIAS 28 degenerate perturbation theory

29. Oct. 26, 2005 KIAS 29 CDW—S-Mott transition D-Mott—S-Mott transition Ising model in a transverse field ordered state disordered state XXZ model in a magnetic field gapless (c=1) Gaussian transition

30. Oct. 26, 2005 KIAS 30 density wave order s-wave p-wave staggered dimerization d-wave d-density-wave =SF f-wave Weak-coupling approach s-wave p-wave d-wave f-wave These states break Z 2 symmetry

31. Oct. 26, 2005 KIAS 31 Bosonization Hamiltonian density chargespin Order parameters pinning potential

32. Oct. 26, 2005 KIAS 32 Ising transitions order disorder Disorder parameters

33. Oct. 26, 2005 KIAS 33 Universality class of quantum phase transitions Gaussian transition (c=1) Ising transition (c=1/2) SU(2) criticality (c=3/2) 2 or 1 st order transition M. Tsuchiizu and A. Furusaki Phys. Rev. B 66, 245106 (2002)

34. Oct. 26, 2005 KIAS 34 Duality transformation Momoi & Hikihara, PRL (2003)

35. Oct. 26, 2005 KIAS 35 model V’

36. Oct. 26, 2005 KIAS 36 Summary Competing interactions competing phases exotic order Various (density) ordered phases Various Mott insulating phases 2D systems ? M.T. & A.F., PRL 88, 056402 (2002) PRB 66, 245106 (2002) PRB 69, 035103 (2004)