Introduction to Recent Developments on p-band Physics in Optical Lattices

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1 Introduction to Recent Developments on p-band Physics in Optical Lattices Gaoyong Sun Institut für Theoretische Physik, Leibniz Universität Hannover Supervisors: Prof. Luis Santos Prof. Temo Vekua Lüneburg, Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

2 Outline 1 Introduction to p-band 2 Tunable Optical Latices 3 Superfluidity in p-band Lattices 4 Haldane Phase in p-band Models 5 Conclusion and Outlook Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

3 Introduction to p-band 1. What is p-band? the excited states of 3-D harmonic oscillator, as seen in Fig. 1 : Figure 1: Spherical harmonic function Y m l (θ,φ) for l=0, 1, 2, 3 original coming from is unknown) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

4 Introduction to p-band 2. Anisotropic hopping for p-band Anisotropic hopping matrix elements, as seen in Fig. 2 : Figure 2: Anisotropic hopping matrix elements of p-orbital bosons on a cubic lattice. The longitudinal t is in general far greater than t because the overlap integral for the latter is exponentially suppressed. The ± symbols indicate the sign of two lobes of p-orbital wave function. So, tunneling t = drp x ( 2 2 2m +V L)P x > 0, t = drp y ( 2 2 2m +V L)P y < 0 and the interaction H int = U 2 r (n2 r 1 3 L2 z). Maximizing orbital angular momentum(oam) can reduce the interaction energy. (W. V. Liu and C. Wu Phys. Rev. A 74, ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

5 Introduction to p-band 3. Why is p-band interesting? Exotic phases exist: 1 Unconventional Superfuild 2 Unconventional BEC beyond the No-node Theorem 3 Topological Phases, e.g. Haldane phases 4... G. Wirth,et al. Nature Physics 7, (2011) M. Ölschläger, et al. New Journal of Physics 15, (2013) C. Wu, Mod. Phys. Lett. B, 23, 1 (2009). W. V. Liu and C. Wu Phys. Rev. A 74, F. Hbert,et al. Phys. Rev. B 87, K. Kobayashi, et al. Phys. Rev. Lett. 109, , Phys. Rev. A 89, G. Sun, G. Jackeli, L. Santos, and T. Vekua. Phys. Rev. B 86, G. Sun, T. Vekua, arxiv: Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

Tunable Optical Latices 1. Optical Lattices A periodic potential is generated by overlapping two counterpropagating laser beams V p (x,y,z) = V 0 (sin 2 kx +sin 2 ky +sin 2 kz), as seen in Fig. 3.

6 Tunable Optical Latices 1. Optical Lattices A periodic potential is generated by overlapping two counterpropagating laser beams V p (x,y,z) = V 0 (sin 2 kx +sin 2 ky +sin 2 kz), as seen in Fig. 3. Figure 3: (a) Two- and (b) three- dimensional optical lattice potentials formed by superimposing two or three orthogonal standing waves. For a two- dimensional optical lattice, the atoms are confined to an array of tightly confining one-dimensional potential tubes, whereas in the three-dimensional case the optical lattice can be approximated by a three-dimensional simple cubic array. ( I. Bloch, et al. Rev. Mod. Phys. 80, 885 ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

7 Tunable Optical Latices 2. Optical lattice with adjustable geometry, as seen in Fig. 4. The optical potential: V p = V X cos2 (kx + θ 2 ) V Xcos 2 (kx) V Y cos 2 (ky) 2α V X V Y cos(kx)cos(ky)cos(ϕ) Figure 4: a, Three retro-reflected laser beams of wavelength λ = 1, 064nm create the 2D lattice potential. Beams X and Y interfere, and beam X creates an independent standing wave. Their relative position is controlled by the detuning δ. b, Top: different lattice potentials can be realized depending on the intensities of the lattice beams. Bottom: diagram showing the accessible lattice geometries as a function of the lattice depths V X and V X. c, Honeycomb lattice has a two-site unit cell (sites A and B). d, Left: first and second Brillouin zones (BZs) of the honeycomb lattice, indicating the positions of the Dirac points. Right: energy spectrum showing two Dirac points, bandwidth W; the gap E G. (L. Tarruell, et al. Nature 483, ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

Tunable Optical Latices 3. Probing the Dirac points., as seen in Fig. 5. Figure 5: a, Quasi-momentum distribution of the atoms before and after one Bloch oscillation, of period T B.

8 Tunable Optical Latices 3. Probing the Dirac points., as seen in Fig. 5. Figure 5: a, Quasi-momentum distribution of the atoms before and after one Bloch oscillation, of period T B. For trajectory 1 (blue filled circle), the atoms remain in the first energy band. In contrast, trajectory 2 (green open circle) passes through a Dirac point at t = T B /2. There the energy splitting between the bands vanishes and the atoms are transferred to the second band. b, Dependence of the total fraction of atoms transferred to the second band, ξ, on the detuning, δ, sublattice energy offset,. The maximum indicates the point of inversion symmetry, where = 0. (L. Tarruell, et al. Nature 483, ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

9 Superfluidity in p-band Lattices 1. Anisotropy and p-band energy minima, as seen in Fig. 6. The optical potential used to realize the p-band: V p = V 2z 2 0 w 4 e 0 2 η[(ẑcos(α)+ŷsin(α))e ikx +ǫẑe ikx ]+e iθ ẑ(e iky +ǫe iky ) 2 Figure 6: a, Energy surface E(q 1,q 2 ) of the p-band plotted for α = α iso. The red and blue discs indicate the local minima. Discs indicating equivalent minima share the same colour. b, The δe of the two inequivalent local energy minima is plotted versus δα and δη. (G. Wirth, et al. Nature Physics 7, (2011) ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

Superfluidity in p-band Lattices 2. Population of excited bands, as seen in Fig. 7. Figure 7: a, The lattice comprises two classes of lattice sites denoted by A and B.

10 Superfluidity in p-band Lattices 2. Population of excited bands, as seen in Fig. 7. Figure 7: a, The lattice comprises two classes of lattice sites denoted by A and B. The grey area denotes the unit cell of the A-sublattice. b, Experimental sequence used to populate excited bands. c, The upper panel shows illustrations of the first, second and fourth Brillouin zones (top row) and their observed populations for values of θ f within the grey bar and at the dashed lines (bottom row). (G. Wirth, et al. Nature Physics 7, (2011) ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

Superfluidity in p-band Lattices 3.Nature of superfluid order(as seen in Fig. 8. ) Figure 8: a, The Brillouin zones and the energy minima. b, Momentum spectrum for α = α iso.

11 Superfluidity in p-band Lattices 3.Nature of superfluid order(as seen in Fig. 8. ) Figure 8: a, The Brillouin zones and the energy minima. b, Momentum spectrum for α = α iso. c, Orbital ordering for α = α iso. d, Orbital alignment for α < α iso. The dark, grey show unit cell. e, Schematic representation of the phase ordering in d. The black lines show the orientation of plane wavefronts. f, Positions of the Bragg resonances (red discs) in momentum space corresponding to the wavefronts in e. For g and h, Lattice potential and Bloch function. (G. Wirth, et al. Nature Physics 7, (2011) ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

to p x and p y orbitals, as seen in Fig. 9.

Schematic diagram of fermionic gases on an optical lattice, with multiple bands.

12 Haldane Phase in p-band Models 1. P-band in 1-D lattice The p-orbitals in a 1D chain along z-axis lead to double degeneracy with respect to p x and p y orbitals, as seen in Fig. 9. t unpaired fermion partially filled t fermion pair 1-D optical lattices fully occupied S Figure 9: Schematic diagram of fermionic gases on an optical lattice, with multiple bands. The intra-orbital interaction (U pp) forms fermion pairs. The fermion pairs hop between different orbitals, by the pair-hopping interaction (U pxp y ) (K. Kobayashi, et al. Phys. Rev. A 89, ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

13 Haldane Phase in p-band Models 1. P-band in 1-D lattice The Hamiltonian for fermions: H = p,σ <i,j> tc p,σ,i c p,σ,j + pp +U pxp y )/2)n p,σ,i + p,σ,i (µ (U p,p,i with h (U) ( p,p =p,i = U pp np,,i 1 2)( np,,i 1 ) ( ) 2 ρ p,i ρ p,i S p,i S p,i h (U) p,p p,i = U pp S p,i = 1 2 ρ (+) p,j = c p,,j c p,,j σ,σ c p,i,σ τ σ,σ c p,i,σ (K. Kobayashi, et al. Phys. Rev. A 89, ) h (U) p,p,i (1) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

Haldane Phase in p-band Models 1. P-band in 1-D lattice Results of Attractive interaction, as seen in Fig. 10. 0.06 (a) 0.05 0.04 3.

5 3 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 1.55 0.50 (b)-3 0.2 0.35 0 50 100 i Figure 10: (a) Charge gap E c vs filling ñ, gapless is found at ñ = 1.

(c) Fourier-transformed density fluctuations n(k), peak (k, ñ) = (3.14, 1.0) near Haldane phase.

14 Haldane Phase in p-band Models 1. P-band in 1-D lattice Results of Attractive interaction, as seen in Fig (a) (b) (b) (c) (b) i Figure 10: (a) Charge gap E c vs filling ñ, gapless is found at ñ = 1.0 due to edge states (b) Particle density n(i), with ñ, similar with S=1 haldane Chain at n = 1. (c) Fourier-transformed density fluctuations n(k), peak (k, ñ) = (3.14, 1.0) near Haldane phase. Population imbalance is P = 0, the coupling U pp = 10 and U pxpy = (4/9)U pp, and L = 100. (K. Kobayashi, et al. Phys. Rev. A 89, ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

15 Haldane Phase in p-band Models 1. P-band in 1-D lattice Results with trap, as seen in Fig (a1) m(i) n(i) (b1) m(i) n(i) (c1) m(i) n(i) 1.5 P=0 P=20 P= V/t=0.4 V/t=0.4 V/t= i i i (a2) m(i) n(i) (b2) m(i) n(i) (c2) m(i) n(i) 1.5 P=0 P=20 P= V/t=0.4 V/t=0.4 V/t= i i i Figure 11: Particle density n(i) (solid red) and the spin density m(i) (dashed blue) in harmonic trap V/t = 0.4, with different population imbalances P. Here, L = 140,N = 180. The upper panels (a1) (c1) show the emergence of the Mott core which is considered to be a magnetization plateau associated with the opening of Haldane gap for p-orbital 1D chain. For (U pxp y = 0), there is no Mott-Core which behaivor as a single-band attractive Hubbard chain. (K. Kobayashi, et al. Phys. Rev. A 89, ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

16 Haldane Phase in p-band Models 2. P-band in zig-zag lattice The p-orbitals in a zig-zag chain, as seen in Fig. 12. t 2i 1 2i+1 λ t y x p x 2i p y Figure 12: Geometry of the orbitally degenerate zig-zag lattice with the nearest neighbour intersite hopping t (between similar orbitals) and onsite hopping λ between the orthogonal orbitals. (G. Sun, T. Vekua. arxiv: ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

17 Haldane Phase in p-band Models 2. P-band in zig-zag lattice The Hamiltonian for bosons: In the strong coupling limit U ±J H,U ±J H t and with one particle per site, and in second order perturbation theory in t the spin-orbital model (SOM) Hamiltonian for bosons is, H = (P i,i+1 +1 α) [ 1+( 1) i σi z ][1+( 1)i σ z ] i+1 i + i (P i,i+1 1/2) [ 1 σ z i σ z i+1], (2) where P i,i+1 = 2S i S i+1 +1/2. The S i spin- 1 2 operators and σz i is a Pauli matrix, with Eigenvalue +1( 1) corresponding to p x (p y ) orbital. Fixing units of t 2 /2Ũ = 1, with Ũ = (U2 J2 H )/U, α = Ũ(U J H /2)/(U 2 J2 H ) U /U > 0 and = J H Ũ/(U 2 J2 H ) > 0. (G. Sun, T. Vekua. arxiv: ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

18 Haldane Phase in p-band Models 2. P-band in zig-zag lattice The phase diagram, as seen in Fig α 2 λ (F,F) S 2i+1 2 λ (P,F) (Q, ) α c1 2 (H,F) αc2 4 S 2i (ih,af) Figure 13: Exact analytical ground state phase diagram of spin-orbital model Eq. (2) obtained in thermodynamic limit. (spin,orbital) notation of different phases is employed. There are three ground states (P,F), (Q, ) and (ih,af) phases. Inset shows Haldane phase (H,F) is formed for infinitezimal quantum fluctuations in orbitals λ towards (P,F) phase - which is called Topological order-by-disorder. Dotted contours encircle 2 sites forming effective spins-1 T i = S 2i +S 2i+1. (G. Sun, T. Vekua. arxiv: ) 6 Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

19 Haldane Phase in p-band Models 2. P-band in zig-zag lattice Spin correlation functions, as seen in Fig α=1 =0.75 S 2i S 2i+1 > < S 2i+1 S 2i+2 > << S 2i S 2i+2 S 2i S 2i+3 > λ Figure 14: Bulk short-range spin correlation functions dependence on λ in (H,F) phase. Due to extensive degeneracy of ground states in (P,F) phase, numerically we can not approach arbitrary close to λ = 0, but the tendency is evident. Symmetry with respect to translations on 2 sites of (H,F) state imposes: S 2i S 2i+1 = S 2i+2 S 2i+3 and S 2i S 2i+2 = S 2i+1 S 2i+3. (G. Sun, T. Vekua. arxiv: ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

20 Haldane Phase in p-band Models 2. P-band in zig-zag lattice Néel order and string order, as seen in Fig (a) String Order(Haldane chain) Neel Order(Haldane chain) String Order(SOM) Neel Order(SOM) i Figure 15: Blue symbols: bulk Néel order and string order of spin-orbital model (SOM) in (H,F) phase for λ 0 (here λ = 0.1, α = 1 and = 0.1 ). Red symbols: corresponding order parameters of spin-1 Haldane chain. (G. Sun, T. Vekua. arxiv: ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

21 Haldane Phase in p-band Models 2. P-band in zig-zag lattice Magnetization profile, as seen in Fig i Haldane chain SOM (b) i Figure 16: Magnetization profile in S z = 1 Kennedy-Tasaki ground state of spin-1 Haldane chain on L = 48 sites (red symbols) is nearly identical to magnetization profile of Ti z = S2i z +S2i+1 z of SOM on L = 96 sites (blue symbols) in (H,F) phase for λ 0 (here λ = 0.1, α = 1 and = 0.1). Inset (green circles) shows site resolved magnetization profile of SOM S2i z Sz 2i+1 Tz i /2. (G. Sun, T. Vekua. arxiv: ) Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

22 Conclusion and Outlook Conclusion and Outlook 1 Adjacent to the topological state in zigzag lattices locally correlated exact ground state with spontaneously quadrupoled lattice constant is realized for the broad parameter regime. (See: G. Sun, T. Vekua, arxiv: ) 2 For repulsive interaction with half filling in 1-D lattices, there is also Haldane phase. (See: K. Kobayashi, et al. Phys. Rev. Lett. 109, ) 3 For repulsive interaction with half filling in zigzag lattices, there is also Haldane phase. (See: G. Sun, G. Jackeli, L. Santos, and T. Vekua. Phys. Rev. B 86, ) 4 We show the some interesting (exotic) phases on p-band physics. e.g. Superfluidity, Haldane Phase..., is there exotic states in d-band, f-band...??? Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

23 Thank you very much! Gaoyong Sun (LUH) P-band Physics Lüneburg, / 23

Design and realization of exotic quantum phases in atomic gases

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