Loop current order in optical latticesLoop current order in optical latticesXiaopeng Li JQI/CMTC

JQI Summer SchoolJune 13, 2014

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 Hamiltonians1. Current operator in continuous models2. 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

BandstructureBloch function

* 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. Greiner et al., Nature 415, 39, 2002

M. Fisher, et al., PRB (1989)

Loop current order? 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

M. P. Zaletel, et al., PRB (2013)

-phase pattern of a condensate wavefunction

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

S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, PRB(2001) R. B. Laughlin, PRB (2013)

-Topological Mott insulator

S. Raghu, X.L. Qi, S.C. Zhang, PRL (2008)

spin loop currents

Experimental evidence of current order in Experimental evidence of current order in optical latticesoptical lattices

P-band condensation in a checkerboard lattice

-super lattice

• 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)

Hamburg/ A. Hemmerich groupFirst observation of p-band BEC with C4 symmetry and hence orbital degeneracy

Direct probe of the local loop currents is propsed, XL, A. Paramekanti, A. Hemmerich, W. Vincent Liu, Nat Comm (2014)

Pi-Flux triangular lattice

Orientation of arrows denotes the phase angle of the condensate wavefunction.

Condensation at the momentum points leads to loop current order.

J. Struck, K. Sengstock et al., Science (2010) These dots are not random!!!

Spin loop current in lattice spinor bosonsSpin 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

* assumed the exchange mechanism holds here. See XL, et al., arXiv:1405.6715 (2014) for details.

This relies on density-density interactions.

(a)(b)

(c) (d)

Second order perturbation theory

TRS:

an anti-unitary transformation

Universal quantum “order-by-disorder”

The spin loop current state has lower fluctuation energy. This universal quantum “order by disorder” selection rule only relies on the “Time-reversal” symmetry.

the universal winner!

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

k

[C. Parker et al., Nat Phys (2013)]

-one component boson -two component boson

k

This is then very similar to SOC Bose gases

Y. J. Lin, I. Spielman, et al., Nature (2011)

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.

[Our expectation for spinor bosons]

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 time-reversal 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 two spin components move in opposite transverse directions in response to an external force (or a potential gradient).

-The relative motion of the two spin components

F

D. Xiao et al., RMP (2010)

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 Hamiltonians1. Current operator in continuous models2. 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