Jinwu Ye Penn State University

1. Jinwu Ye Penn State University Quantum phase transitions from Solids to Supersolids in bi-partite lattices Outline of the talk: Introduction to Boson Hubbard Model and supersolids on lattices Boson -Vortex duality in boson Hubbard Model 3. Extended Hubbard Model at honeycomb and square lattice near half filling Charge Density Wave supersolid Valence Bond Supersolid Implications to QMC 7. Conclusions

2. References: 1. The mobility of dual vortices in square, honeycomb, triangular, Kagome and Dice lattices. Longhua Jiang and Jinwu Ye, J. Phys, Condensed Matter. 18 (2006) 6907-6922 2. Quantum phase transitions from solids to supersolids in bipartite lattices Jinwu Ye, cond-mat/0503113. Revised version, submitted to Phys. Rev. Lett. 3. Supersolids and solids to Supersolids transitions in frustrated lattices , Jinwu Ye, cond-mat/0612009, 4. The Extended Bose Hubbard Model on the Two Dimensional Honeycomb Lattice Jing Yu Gan, Yu Chuan Wen, Jinwu Ye, Tao Li, Shi-Jie Yang, Yue Yu, Phys. Rev. B 75, 214509 (2007)

3. Introduction to extended boson Hubbard model and supersolid on a lattice Extended Boson Hubbard Model (EBHM): t : nearest neighbor hopping : onsite, nn and nnn interactions Ring exchange interaction: Ultra-cold alkali gases: Superfluid to Mott insulator transition in optical lattice, Greiner et al, 2002

4. (1) Superfluid: Bose-Einstein condensation (BEC) Off-diagonal long range order: (2) Charge density wave: Diagonal order in boson density: (3) Supersolid: Both off-diagonal and diagonal long range order

5. Supersolids on lattices are very different than possible supersolids in Helium 4 Kim and Chan, 2005 Rather complete quantum phenomenology was worked out in Jinwu Ye, Phys. Rev. Lett. 97, 125302 (2006). Jinwu Ye, cond-mat/0705.0770, submitted to Nature Physics Jinwu Ye, cond-mat/0603269, submitted to Phys. Rev. B. See my seminar at KITP-C on Aug. 8, 2007

6. 2. Boson-Vortex duality in EBHM at filling factor • At Integer fillings , Superconductor to Mott insulator transition ( SIT ) was studied in: • Onsite interaction only: • Fisher, Weichman, Grinstein and Fisher, 1989 • Long-range Coulomb interaction: • Fisher and Grinstein, 1988; Jinwu Ye, 1998

7. Boson-Vortex duality at Particles ~ Holes ~ Number of particles= number of holes relativistic field theory for Insulator: < > =0; Superfluid < > 0

8. Excitations of the superfluid: • Gapless Phonons: represented by a dual U(1) gauge field • (2) Gapful Vortices: vortex operator Number of vortices = number of antivortices relativistic field theory for Superfluid: < > =0; Insulator: < > 0

9. Boson-Vortex duality and SIT at SIT at cold alkali gas: Greiner et al, 2002 Boson quasi-particle excitations ~ in the insulator : is dual to Vortex excitations ~ + phonon excitations in the superfluid: Insulator: < > =0 < > 0 Superfluid: < > 0 < > = 0 Dasgupta and Halperin, 1981

10. The boson acquires a phase when it encircles a vortex A vortex in the vortex field is the original boson, the vortex acquires a phase when it encircles a boson The average strength of the dual magnetic field on the vortex field = density of boson= flux quanta per plaquette

11. Balents, Bartosch, Burkov, Sachdev, Sengupta, 2005 Bosons at filling factor f=p/q hopping on a direct lattice is dual to Vortices hopping on the dual lattice subject to a fluctuating dual " magnetic field“ whose average strength is f=p/q flux quanta per dual plaquette. This is similar to Hofstadter problem of electrons moving in a lattice in the presence of f=p/q flux quanta per plaquette. Hofstadter , 1976 The spectrum can be classified by Magnetic Space Group ( MSG) MSG dictates that there are at least q -fold degenerate minima which forms a q dimensional representation of the MSG. The effective theory in terms of these q vortices should be invariant under this MSG.

12. Balents, Bartosch, Burkov, Sachdev, Sengupta, 2005 Taking and apply to study CDW state in high temperature superconductors Two holes in the DVM developed by Balents et.al 2005 (1) It can NOT be applied to the CDW side: In the CDW side, the boson density takes, for example, a checkboard ( or stripe ) Order. The average strength of the dual magnetic field per dual plaquette should also be taken the same pattern ! Not as uniform as The average strength of the dual magnetic field on the vortex field = density of boson

13. (2) As shown in: Longhua Jiang and Jinwu Ye, 2006 The bandwidth of the dual vortex band The DVM completely breaks down when 1. I take a completely different route: Taking , so the DVM works, then moving slightly away from , the return is interesting and important: Supersolids could exist only at in-commensurate fillings ! 2. We will also extend the DVM also to the CDW side

14. 3. EBHM on honeycomb lattice near half filling Bosons hopping on honeycomb lattice at filling factor f=p/q is dual to Vortices hopping on triangular lattice subject to a fluctuating magnetic field whose average strength is f=p/q per triangular plaquette The Hamiltonian of the vortices is: + interaction terms is the vortex hopping amplitude

15. The space group of the triangular lattice: • Point group: , 12 elements with 2 generators • Two translations The Magnetic Space Group (MSG ) are generated by: Reflections: Translations and Rotations: is the complex congugate They all commute with the Hamiltonian H

16. The rotations around the two direct lattice points A and B: Both also commute with H In the following, we focus on q=2 where the energy band is: There are two minima at where the two eigenmodes are labeled by

17. transform under the MSG as: The quadratic terms of the effective action is the scalar electrodynamics: The most general quartic term invariant under all the transformations is:

18. Moving slightly away from half filling f=1/2 corresponds to adding a small dual magnetic field in the action: • r < 0, insulating state, • < > for at least one l • : Ising limit • : Easy-plane limit r > 0, superfluid state, < > =0, It turns out that the action only works in the SF and VBS sides, But breaks down in the CDW side The SF is stable against the changing of chemical potential In the following, we focus on the insulating side.

19. 4. Charge density wave supersolid In the Ising limit, at mean field: Density Wave order parameter in the Ising limit: A uniform saddle point does not work in the CDW side Saddle point for the dual gauge field: for sublattice A for sublattice B A and B could be checkboard or stripe order

20. There is only one vortex minimum in such a staggered dual magnetic field with Slightly away from 1/2 filling, the effective action inside the CDW is: It has a structure identical to q=1 component Ginzburg-Landau model for a type-II superconductor in a magnetic field is always massive, so does not appear in above Eqn.

21. Phase diagram of Type II superconductor in external magnetic field: D. Nelson, 1989 Fisher, Fisher and Huse, 1991

22. IC-CDW C-CDW Fisher, Weichman, Grinstein and Fisher, 1989 Ising limit The CDW could also be stripe order

23. In the CDW state: is massive In the CDW-SS state: becomes gapless which stands for the superfluid mode The transition from the CDW to the CDW-SS is in the same universality class as that from the SF to Mott insulator, Therefore have exact critical exponents: The superfluid density in the CDW-SS scales as: with a logarithmic correction

24. The CDW-SS has the same diagonal density order as the CDW at The transition from the CDW-SS to the IC-CDW is first order The transition from the SF to the CDW-SS is first order The IC-CDW can be stabilized only by very long-range interactions Similar phase diagram also holds for square lattice It is important to compare with microscopic calculations such as quantum monte-carlo (QMC)

25. QMC simulations by: F. Hebert, et.al, Phys. Rev. B, 65, 014513 (2001). Hard core Soft core case: P. Sengupta, et.al. 2005 SS is stable The stripe solid to stripe-SS transition is 2nd order with in the stripe supersolid, but scale in the same way with different coefficients

26. 5.Valence bond supersolid In the easy-plane limit: Valence Bond Solid (VBS) order parameter: At mean field: The uniform saddle point holds in both SF and the VBS Sixth order terms are needed to determine the relative phase

27. Only can fix the relative phase It turns out both signs of are equivalent The other two VBS can be obtained by P.W. Anderson; Science 235, 1196(1987) A ring exchange term : is needed to stabilize the VBS order is relevant, so the SF to VBS transition is 1st order

28. Slightly away from half-filling inside the VBS: sector is massive, so can be integrated out Assuming It is also identical to q=1 component GL model for a type-II superconductor in a magnetic field

29. All the discussions on the Ising case follow In the VBS: is massive In the VB-SS: Gapless superfluid mode:

30. In square lattice: The lowest order term coupling the two phases is VBS has Columnar dimer ( plaquette ) pattern is irrelevant, so the SF to the VBS is a second order transition through so called deconfined QCP discovered by O. Motrunich and A. Vishwanath, 2004 T. Senthil et. al, 2004 There are strong numerical evidence that there is no such deconfined QCP, the SF to the VBS is weakly 1st order Anatoly Kuklov, Nikolay Prokof'ev, Boris Svistunov, M. Troyer, con-mat/0602466

31. 6. Implications on future QMC • In square lattice: Stripe solid to stripe supersolid in the • hard core case, Checkboard solid to checkboard supersolid • in soft case to test the universality class by finite size scaling K: tuning parameter: Static structure factor: Equal time is the ordering wave vector Similar quantities can be defined for a bond (2) Do the same things in the honeycome lattice (3) With ring-exchange term, test the VBS to VB-SS transition

32. Soft core boson in honeycomb lattice Jing Yu Gan, Yu Chuan Wen, Jinwu Ye, Tao Li, Shi-Jie Yang, Yue Yu, Phys. Rev. B 75, 214509 (2007)

33. 7. Conclusions • Map out the global phase diagram in a unified theory: • SF, CDW( VBS), CDW-SS (VB-SS), IC-CDW (IC-VBS) • (2) Discover universality class of solids to supersolids transitions • (3) Propose a novel kind of supersolid: valence bond supersolid • (4) Several important implications on QMC These results should have direct impacts on ongoing experiments of cold atoms loaded on optical lattices. It should also shed some lights on possible supersolid Helium 4.