1. Wu-Ming Liu （刘伍明） Institute of Physics, Chinese Academy of Sciences e-mail: wmliu@iphy.ac.cnwmliu@iphy.ac.cn Novel states of cold atoms in gauge field and optical lattices Critical behavior of lattice models in atomic and molecular, condensed matter and particle

2. Collaborators Yao-Hua Chen （陈耀桦） Ren-Yuan Liao （廖任远） Chao-Fei Liu （刘超飞） Hong-Shuai Tao （陶红帅） Wei Wu （吴为） Yi-Xiang Yu （喻益湘） Dao-Xin Yao （姚道新，中山大学）

3. Outline 1.Introduction 2.Cold atoms in gauge field: Half-skyrmion 3. Cold atoms in optical lattices: Bose metal 4.Cold atoms in gauge field and optical lattices: Spin Hall effect, Topological insulator 5. Summary

4. Bose-Einstein condensation Bose-Einstein statistics (1924) Bose enhancement Fermi-Dirac statistics (1926) Fermi sea Pauli Exclusion 1 、 Introduction ： Bose-Einstein condensate

5. 1 、 Introduction ： Cold atoms 19241987 Nobel Prize (1997)... 1995 Nobel Prize (2001)

6. Cold atoms and molecules are nano- or micro-kelvin temperature laboratory New quantum states Quantum simulation Quantum information Nonequilibrium dynamics Application in high technology 1 、 Introduction ： Low temperature laboratory

7. High technology Space clock Gyroscope Interferometer Optical clock 1 、 Introduction ： Application

8. 1 、 Introduction ： 200 Experiment group JILA,NIST,Har vard,Rice,Duk e,MIT,Purde USA （ 60 ） ANU,UQ,Swinburne Australia （ 10 ） Toronta,UBC, York Univ Canada （ 15 ） TUD,LMU,MPQ, US,UD,UH Germany （ 25 ） SIOM,PKU,WIPM,SXU,IOP,USTC China （ 15 ） Come from ： CNRS,LCAR,ENS Paris, PhLAM France （ 20 ） Niels Bohr, Aarhus Univ Danmark （ 5 ） Oxford,Cambri dge,Imperial,U CL,Durham England （ 15 ） JST,Tokyo,NTT Japan （ 15 ） IITK,SSMRV,ARSD India （ 10 ）

9. 2.1. Spin polarized fermi gas in gauge field: Magnetized superfluid 2.2. Spinor BEC in gauge field: Half-skyrmion 2 、 Cold atoms in gauge field

10. Spielman, Nature 462, 628 (2009) 2 、 Cold atoms in gauge field

11. R.Y. Liao, Y.X. Yu, W.M. Liu, Phys. Rev. Lett. 108, 080406 (2012) 2. Cold atoms in gauge field ： Phase diagram

12. FIG. 1. Isoenergy surface (Ek=0.8EF) for quasiparticle excitation spectrum at unitarity where 1/(kFas)=0 at T=0: (a) h=0,λ=0.125vF; (b) h=0,λ= 0.25vF; (c) h=0.1EF,λ=0.125vF; (d) h=0.1EF,λ=0.25vF. Red dashed line is plotted for Ek-, blue solid line is for Ek+, green dash-dotted circle is for a spherical isoenergy surface, plotted for comparison.

13. FIG. 2 Upper panel: Finite-temperature phase diagram as a function of T and h at 1/(kFas)= -1 (BCS side). There are four different phases: N state, PS state, SF state, magnetized superfluid (SFM). Above tricritical point, transition line separating broken-symmetry state (SFM) and symmetric state (N) is of second order. Below tricritical point (TP), it changes to first order. Lower panel: Evolution of tricritical point (Ttri/TF, htri/EF) as a function of SOC strength λ.

14. FIG. 3. Finite-temperature phase diagram in plane of T and P at 1/(kFas)=-1. The inset shows corresponding polarization Ptri for tricritical point as a function of SOC strength. The phase SF is along line of P=0. The notation is the same as in Fig. 2.

15. FIG. 4. Left: polarization P=(n↑-n↓)/ (n↑+n↓) as a function of magnetic field h for various SOC strength at zero temperature at unitarity. Right: The critical temperature for balanced superfluid at unitarity; Tc0 is calculated from mean field theory and Tcg is calculated by taking account of Nozieres–Schmitt-Rind correction.

16. FIG. 5. The momentum distribution nkσ and correlation function C ↑↓ (k) at unitarity at zero temperature with SO coupling strength λ=0.2vF for two typical polarizations: P=0.7 (left) and P=0.9 (right).

17. C. F. Liu, W. M. Liu, Phys. Rev. A 86, in press (2012) Combination SOC and rotation 2 、 Cold atoms in gauge field ： Dynamics

18. C. F. Liu, W.M. Liu, Phys. Rev. A 86, in press (2012) Densities and phases of spinor BEC of 87 Rb with SOC when system reaches equilibrium state. (a) К=0.1; (b) К=0.2; (c) К=0.5; (d) К=0.7; (e) К=1.0. Ω=0.5ω. 2 、 Cold atoms in gauge field ： SO effect

19. C.F. Liu, W.M. Liu, Phys. Rev. A 86, in press (2012) (a) Spin texture, К=0.5, Ω=0.5ω. Color of each arrow indicates magnitude of Sz. Black pane points out a Skyrmion, blue pane indicates a half-Skyrmion. (b) Position of vortices and spin texture. Green, blue, red spots are center of vortices formed by m F =- 1, m F =0, m F =1. (c) Topological charge density. (d) Position of vortices and spin texture, К=0.1, Ω=0.5ω. (e) Scheme of three vortices structure. 2. Cold atoms in gauge field ： Half-s kyrmion

20. C.F. Liu, W.M. Liu, Phys. Rev. A 86, in press (2012) Effect of rotation frequency for spinor BEC of 23 Na,К=1, (a) Ω=0; (b) Ω=0.2ω; (c) Ω=0.5ω. Fourth column shows corresponding spin textures and position of vortices in region of y>0. 2. Cold atoms in gauge field ： Rotation effect

21. C.F. Liu, W.M. Liu, Phys. Rev. A 86, in press (2012) Phase diagrams of products in spin-1 BEC with rotation frequency Ω and SOC strength К: (a) 87 Rb; (b) 23 Na. 2. Cold atoms in gauge field ： Phase diagram

22. 3.1. Quantum phase transition 3.2. Super-counter-fluidity 3.3 Spin liquid 3.4 Kondo metal and plaquette insulator 3 、 Cold atoms in optical lattices

23. Tuning ： interaction, component, structure, dimension v0v0 3.1. Superfluid-Mott insulator transition

24. one component Superfluid-Mott insulator transition （ PRA03 ） super-counter-fluidity （ PRL07 ） Nematic phase (PRA08) Two componentThree component Tuning ： interaction, component, structure, dimension 3.2. Super-counter-fluidity

25. Dirac fermion （ PRB 2010 ） Frustrated system （ PRA 2010 ） Supersolid (PRA 2011) Triangular lattice Square latticeHoneycomb lattice Tuning ： interaction, component, structure, dimension 3.3. Spin liquid Spin liquid

26. Y.H. Chen, H.S. Tao, D.X. Yao, W.M. Liu, Phys. Rev. Lett. 108, 246402 (2012) 3.4. Kondo metal and plaquette insulator

27. FIG. 1: (a1) Unit cell of triangular Kagome lattice (TKL) without asymmetry (λ=1.0). Open circles denote A-sites and Solid circles denote B-sites. Blue lines represent hopping between A-sites and B-sites. Red lines denote hopping between B-sites. (a2) λ>1.0, TKL is similar with Kagome lattice. (a3) λ<1.0, TKL is transformed into a system composed of many triangular plaquettes. (b) Thick red lines show first Brillouin zone of triangular Kagome lattice. Thin black lines correspond to Fermi surface for non-interacting case. Γ, K, M, K’, M points denote points in first Brillouin zone with different symmetry.

28. FIG. 2: Phase diagram of triangular kagome lattice atλ=0.6. The black solid lines show transition line of A-sites, red dashed lines show transition line of B-sites. Two kinds of coexisting phases between red lines and black lines are plaquette insulator and Kondo metal. Inset: Phase diagram of symmetric triangular kagome lattice (λ=1), in which there are no coexisting phases.

29. FIG. 3: Momentum resolved spectrum Ak(ω) atλ=0.6. (a) Metallic phase at U =6, T=0.5. (b) Mott insulating phase at U=9, T=0.5. A visible single particle gap shows up around Fermi energy. (c) Plaquette insulating phase at U=7.6, T=0.5, A sites are insulating, B sites are metallic. A small gap shows up. (d) Kondo metallic phase at U=7, T=0.2, A sites are metallic, B sites are insulating. Single particle gap vanishes.

30. FIG. 4: Evolution of double occupancy D on A sites as a function of U with different temperatures at λ=0.6. The inset figure shows evolution of double occupancy on B sites. The arrows with different colors show phase transition points at different temperatures.

31. FIG. 5: The evolution of spectral function on Fermi surface. (a) λ=0:6. (b)λ=1. (c) λ=1:25.

32. FIG. 6: Single particle gap ΔE and ferrimagnetic order parameter m at λ=1.0, T=0.2, tbb=1.0. A paramagnetic metallic phase is found when interaction is weak with ΔE=0, m=0. As interaction U increases, a gap is opened and no magnetic order is formed with ΔE≠0, m=0. This paramagnetic insulating phase can be a short range RVB spin liquid. An obvious magnetic order is formed when interaction is strong enough withΔE≠0, m≠0. Insert picture shows evolution of E at λ=0.6, T=0.5. A plaquette insulator is found when A-sites are insulating, B- sites are metallic.

33. FIG. 7: Phase diagram represents competition between interaction U and asymmetry λfor T=0.2, tbb=1. Region between black lines with square points and red lines with circle points denotes coexisting zone which contains plaquette insulator and Kondo metal. A wide paramagnetic insulating region is found with an intermediate U. Blue lines with triangular points show transition point to ferrimagnetic insulator with a clear magnetic order. Insert: (a) Dimers formed in paramagnetic insulator, which is a candidate for short range RVB spin liquid, (b) Spin configuration of ferrimagnetic insulator.

34. 4.1. Quantum spin Hall effect 4.2. Topological insulator 4 、 Cold atoms in gauge field and optical lattices

35. FIG. 1: (a) Illustration of honeycomb lattice. The dashed line sketches six-site cluster scheme. (b) The first Brillouin zone of honeycomb lattice. The linear low-energy dispersion relation displays conical shapes near Fermi level. 4.1. Quantum spin Hall effect W. Wu, S. Rachel, W. M. Liu, K. Le Hur, Phys. Rev. B 85, 205102 (2012)

36. FIG. 2: Phase diagram of KMH model obtained within CDMFT, including four phases: (i) topological band insulator (TBI); (ii) magnetically ordered spin density wave (SDW); (iii) non-magnetic insulator (SL); (iv) semi-metal (SM) region which is shown (from right to left) for T=0.05, 0.025, 0.0125, 0.005. 4.1. Quantum spin Hall effect

37. FIG. 3: Temperature dependence of phase diagram at SOCλ=0.02. Inset: Single- particle gap Δsp and magnetization m vs. U is shown forλ=0.02 and T=0.025. 4.1. Quantum spin Hall effect

38. FIG. 4: α-λ phase diagram of plaquette honeycomb model at U=0. Theα=1 line corresponds to KM model. Spectra for armchair ribbons (L=96) are shown atλ=0.15,α=1.5 (top, QSH phase) andα=0.48 (bottom, entrance of PI phase). Blue lines correspond to SM. 4.2. Topological insulator

39. Summary 1.Cold atoms in gauge field: Half-skyrmion 2. Cold atoms in optical lattices: Bose metal 3.Cold atoms in gauge field and optical lattices: spin Hall effect, Topological insulator

40. Outlook 1.Fundamental physics: Topological phase transition, Quantum critical phenomena, Strong correlated effect, Non-equilibrium dynamics, … 2. Application in high technology: Optics and interferometry, Precision metrology, …