JQI Summer School June 13, 2014 Loop current order in optical lattices Xiaopeng Li JQI/CMTC
Outline Ultracold atoms confined in optical lattices 1. Why we care about lattice? 2. Band structures and Berry phases 3. Tight binding models and correlated states (Loop) current order in model Hamiltonians 1. Current operator in continuous models 2. Current in lattice models 3. Symmetry requirement for finite current order 4. Relevance to hight Tc, topological Mott,... Experimental evidence in optical lattices 1. Checkerboard lattice (A. Hemmerich) 2. Pi-flux triangular lattice (K. Sengstock) Spin loop current in lattice spinor bosons 1. Systems to support spin loop current 2. Spontaneous spin Hall effects
Ultracold gases More is different: Molecules, polaritons,
Bose-Einstein condensation M. H. Anderson et al., Science 269, 198 (1995) Extremly dilute---five orders of magnitude less than the density of the air.
Strongly correlated physics with optical lattices
Bandstructures Bloch function Bandstructure * n is the band index, k the lattice momentum * for optical lattices, plane-wave basis is usually a good basis for Bloch functions.
Berry phase in momentum space The flux density of Berry phase defines Berry curvature Time-Reversal symmetry Inversion symmetry Both bandstructures and Bloch functions are important.
Tight binding model and Mott-superfluid transition M. Fisher, et al., PRB (1989) M. Greiner et al., Nature 415, 39, 2002
Loop current order?
Current operator in continuum -Other approaches to derive current Noether current from Langrangian Couple to auxiliary gauge fields *continuous symmetry is the key to define current
Example of loop current in continuum -vortex in BEC
Current operator in lattice Hamiltonians *charge U(1) symmetry -Other approaches to derive current Noether current from Langrangian Couple to auxiliary gauge fields
Symmetry requirement for finite current *T is time-reversal transformation (anti-unitary) We need to break time-reversal symmetry. Rotating the cold gas Creating synthetic gauge fields Interaction induced spontaneous symmetry breaking More interesting to me!!! Interesting excitations due to spontaneous symmetry breaking could sometimes be more important than the order itself.
Example: Pi-Flux triangular lattice -phase pattern of a condensate wavefunction M. P. Zaletel, et al., PRB (2013) Other examples: Complex p-band condensate in a square lattice Excited band condensate in Kagome lattice
Relevance to high Tc, TMI,... -orbital current order in D-density wave -Topological Mott insulator spin loop currents S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, PRB(2001) R. B. Laughlin, PRB (2013) S. Raghu, X.L. Qi, S.C. Zhang, PRL (2008)
Experimental evidence of current order in optical lattices
P-band condensation in a checkerboard lattice -super lattice Hamburg/ A. Hemmerich group First observation of p-band BEC with C4 symmetry and hence orbital degeneracy Early observation: finite momentum BEC, single p-band by [Mueller, Bloch, et al, PRL, 2007] Even earlier p-band fermion observed in Feshbach crossing accidentally M. Köhl et al, PRL 94, 080403 (2005) Direct probe of the local loop currents is propsed, XL, A. Paramekanti, A. Hemmerich, W. Vincent Liu, Nat Comm (2014)
Pi-Flux triangular lattice J. Struck, K. Sengstock et al., Science (2010) These dots are not random!!! Orientation of arrows denotes the phase angle of the condensate wavefunction. Condensation at the momentum points leads to loop current order.
Spin loop current in lattice spinor bosons XL, S. Natu, A. Paramekanti, S. Das Sarma, arxiv (2014)
Spin-dependent honeycomb lattice Each spin component sees a pi-flux triangular lattice Spin loop current Charge loop current
Spinor Bosons in a double-valley band (b) (a) (c) (d) This relies on density-density interactions. * assumed the exchange mechanism holds here. See XL, et al., arxiv:1405.6715 (2014) for details.
Second order perturbation theory TRS: an anti-unitary transformation
Universal quantum order-by-disorder the universal winner! The spin loop current state has lower fluctuation energy. This universal quantum order by disorder selection rule only relies on the Time-reversal symmetry.
Double-valley bands in experiments C. Chin group (Chicago) [C. Parker et al., Nat Phys (2013)] [Related theory work: XL, E. Zhao, W.Vincent Liu, Nat Comm (2013)] T. Esslinger group (ETH) [L. Tarruell et al., Nature (2012)] Sengstock group (Hamburg) [P. Soltan-Panahi et al., Nat Phys (2011)]
Application to Chin's shaken lattice -two component boson -one component boson k k [Our expectation for spinor bosons] This is then very similar to SOC Bose gases [C. Parker et al., Nat Phys (2013)] The crucial difference is the spontaneous nature. Chiral spin superfluid in Chin's lattice would behave like SOC Bose gases with SOC of a spontaneous chosen sign. Y. J. Lin, I. Spielman, et al., Nature (2011)
Berry curvatures and spin Hall effect Time-Reversal symmetry Inversion symmetry Berry curvature is finite, if we break either of the two symmetries. Optical lattices with timereversal but lacking inversion symmetry are recently obtained in many experiments [by C. Chin's group, T. Esslinger's group and K. Sengstock's group]. Such lattices have finite Berry curvatures at finite momentum.
Berry curvatures and spin Hall effect -The relative motion of the two spin components D. Xiao et al., RMP (2010) The two spin components move in opposite transverse directions in response to an external force (or a potential gradient). F XL, S. Natu, A. Paramekanti, S. Das Sarma, arxiv (2014)
Summary Ultracold atoms confined in optical lattices 1. Why we care about lattice? 2. Band structures and Berry phases 3. Tight binding models and correlated states (Loop) current order in model Hamiltonians 1. Current operator in continuous models 2. Current in tight binding models 3. Symmetry requirement for finite current order 4. relevance to hight Tc, topological Mott,... Experimental evidence in optical lattices 1. Checkerboard lattice (A. Hemmerich) 2. Pi-flux triangular lattice (K. Sengstock) Spin loop current in lattice spinor bosons 1. Systems to support spin loop current 2. Spontaneous spin Hall effects