Competing insulating phases in onedimensional extended Hubbard models
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) Oct. 26, 2005 KIAS 1
Contents One-dimensional models of interacting electrons at half filling n n n 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, …. Weak-coupling approach, Bosonization Oct. 26, 2005 KIAS 2
Extended Hubbard model at half filling t Oct. 26, 2005 U V KIAS 3
Standard phase diagram (before 1999) Emery (1979) Hirsch (1984) Cannon, Scalettar, Fradkin (1991) ………. Oct. 26, 2005 KIAS 4
Weak-coupling theory (g-ology) L R 1 -loop RG charge sector Spinsector spin Charge gap if Oct. 26, 2005 Spin gap if KIAS 5
Phase diagram since 1999 Discovery of Bond-charge-density wave (BCDW) phase or Bond-Order-Wave (BOW) Found numerically Nakamura (1999, 2000) Sengupta, Sandvik, Campbell (2002) …. . Oct. 26, 2005 KIAS 6
Vertex correction Degeneracy of zeros of and are lifted Separate transitions in charge & spin sectors In the strong-coupling regime 1 st order SDW-CDW transition Oct. 26, 2005 KIAS 7
Tam, Tsai, & Campbell, cond-mat/0505396 Oct. 26, 2005 KIAS 8
Bosonization charge spin Order parameters Oct. 26, 2005 KIAS 9
Bosonized form of the Hamiltonian density kinetic energy marginal perturbation relevant perturbation irrelevant perturbation SU(2) symmetry etc Oct. 26, 2005 KIAS 10
Order parameters Classical analysis Oct. 26, 2005 KIAS 11
Phase transitions SDW-BCDW transition: 2 nd order Oct. 26, 2005 CDW-BCDW transition: 2 nd order KIAS CDW-SDW transition: 1 st order 12
Ground-state phase diagram from bosonization approach 1 -loop RG + classical analysis M. Tsuchiizu and A. F. , Phys. Rev. Lett. 88, 056402 (2002) Oct. 26, 2005 KIAS 13
Numerical Results Quantum Monte Carlo DMRG Sengupta, Sandvik, & Campbell, Phys. Rev. B 65, 155113 (2002) Y. G. Zhang, PRL 92, 246404 (2004) Oct. 26, 2005 KIAS 14
Sandvik, Balents & Campbell, PRL 92, 236401 (2004) n Tricritical point on the CDW-BCDW phase boundary SSE QMC Luttinger liquid parameter at the continuous transition Oct. 26, 2005 KIAS 15
umklapp scattering becomes relevant for Oct. 26, 2005 KIAS 16
Phase diagram (schematic) 1 st order transition CDW-BCDW c=1 Gaussian SDW-BCDW c=1 SU(2)1 Oct. 26, 2005 KIAS 17
Extended Ionic Hubbard model at half filling and Mott insulator Quantum Phase Transition ? and n Band insulator Ionic Hubbard model Nagaosa & Takimoto (1986), Egami, Ishihara, & Tachiki (1993) Fabrizio, Gogolin, & Nersesyan, PRL 83, 2014 (1999) Oct. 26, 2005 KIAS 18
n Spontaneously Dimerized Insulating Phase (SDI) (= BCDW Phase) Fabrizio, Gogolin, & Nersesyan, PRL 83, 2014 (1999) Ising 0 Oct. 26, 2005 BI KT SDI KIAS MI 19
Bosonization perturbative RG + classical analysis n Extended ionic Hubbard model nearest-neighbor repulsion V Oct. 26, 2005 KIAS 20
Bosonized form of the Hamiltonian density Kinetic energy marginal perturbation relevant perturbation irrelevant perturbation Oct. 26, 2005 KIAS 21
Classical analysis Gaussian Oct. 26, 2005 KIAS Ising 22
1 st order transition Ground-state phase diagram cf. Oct. 26, 2005 KIAS 23
Schematic phase diagrams Oct. 26, 2005 KIAS 24
Generalized Hubbard ladder at half filling tpair Oct. 26, 2005 t⊥ V⊥ , J⊥ KIAS 25
Various Insulating Ground States that can appear in half-filled ladders rung singlet state (D-Mott) Ex. SO(5) ladder model charge density wave (CDW) ground-state phase diagram singlet paring state (S-Mott) staggered flux state (SF) d-density wave orbital antiferromagnet Oct. 26, 2005 KIAS Lin, Balents & Fisher (1998) Fjaerestad & Marston (2002) 26
Strong-coupling approach 4 basis states Oct. 26, 2005 KIAS 27
degenerate perturbation theory Oct. 26, 2005 KIAS 28
ordered state Ising model in a transverse field disordered state n CDW—S-Mott transition XXZ model in a magnetic field gapless (c=1) n D-Mott—S-Mott transition Oct. 26, 2005 Gaussian transition KIAS 29
Weak-coupling approach s-wave density wave order p-wave staggered dimerization p-wave d-wave f-wave d-density-wave =SF f-wave These states break Z 2 symmetry Oct. 26, 2005 KIAS 30
Bosonization charge spin Hamiltonian density pinning potential Order parameters Oct. 26, 2005 KIAS 31
n Ising transitions order disorder Disorder parameters Oct. 26, 2005 KIAS 32
Universality class of quantum phase transitions Ising transition (c=1/2) Gaussian transition (c=1) SU(2)2 criticality (c=3/2) or 1 st order transition M. Tsuchiizu and A. Furusaki Phys. Rev. B 66, 245106 (2002) Oct. 26, 2005 KIAS 33
Duality transformation Momoi & Hikihara, PRL (2003) Oct. 26, 2005 KIAS 34
model V’ Oct. 26, 2005 KIAS 35
Summary n Competing interactions competing phases exotic order Various (density) ordered phases Various Mott insulating phases n 2 D systems ? n n M. T. & A. F. , PRL 88, 056402 (2002) PRB 66, 245106 (2002) PRB 69, 035103 (2004) Oct. 26, 2005 KIAS 36
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