Bosons in optical lattices Subroto Mukerjee Department of Physics - PowerPoint PPT Presentation

Bosons in optical lattices Subroto Mukerjee Department of Physics Indian Institute of Science Bangalore Current trends in frustrated magnetism, JNU 2015

Collaborators • Arya Dhar • Maheswar Maji • Sayonee Ray • Tapan Mishra • Ramesh Pai • Arun Paramekanti Funding: Department of Science and Technology, Govt. of India

Outline • Frustration and bosons • Frustrated Bose-Hubbard ladder and the Chiral Mott phase • Bond ordered and supersolid phases

Bose-Hubbard model h ij i c † i c j + h . c . + U H = − t P P i n i ( n i − 1) − µ P i n i 2 Interaction strength tunes quantum phase transition from a superfluid to a Mott insulator SF - superfluid MI-Mott insulator

Experiments Josephson junction arrays Cold atoms Mooij group (1992) Greiner et. al. (2002)

Frustration ? Why is it interesting? • Many classical ground states • Quantum effects pick particular states • Interesting states like spin liquids

Bosons with frustration h ij i t ij c † i c j + h . c . + U H = − P P i n i ( n i − 1) − µ P i n i 2 Flux φ = p q q bands with degenerate minima Kinetically frustrated bosons have many ways to condense Interactions relieve frustration Phase diagram?

Frustrated bosons φ = 1 / 2 h a j i = | ψ j | e − i θ j h a † j a k + a k a † j i ⇠ cos ( θ k � θ j ) Frustration of the superfluid phase

Experiments Optical lattices with frustration using artificial gauge fields Struck et. al. (2011) Aidelsberger et. al. (2011)

Frustrated Bose-Hubbard model + - 2-Leg Ladder i a i +1 + h . c . + U h i n ( a ) X X a † n ( a ) − 1 H = − t i 2 i i i b i +1 + h . c . + U h i n ( b ) X X b † n ( b ) − 1 + t i 2 i i Filling: one boson per site φ = 1 / 2 per plaquette. Frustration

Band structure + - 2-Leg Ladder U = 0 band structure Multiple minima t ⊥ = 0 Single boson condensate wavefunction t ⊥ 6 = 0 | ψ i = a 0 | 0 i + a π | π i

Weak interactions + - 2-Leg Ladder | ψ i = a 0 | 0 i + a π | π i a ( b ) | 0 i ( | π i ) has the character of the leg a ( b ) leg ∝ | a 0 | 2 ( | a π | 2 ) Avg. density in Filling and n = 1 U > 0 ⇒ | a 0 | = | a π |

Chiral order | ψ i = Ae i θ � | 0 i + e i φ | π i � ⌘i N h e i θ ⇣ a † 1 0 + e i φ a † | ψ i = | 0 i π √ N ! Ginzburg-Landau theory | ϕ i | 2 + U X � X E mft u 4 0 + v 4 | ϕ i | 4 � = ( − E 0 − µ ) low 0 i =0 , π i =0 , π 0 | ϕ 0 | 2 | ϕ π | 2 + 2 Uu 2 +8 Uu 2 0 v 2 0 v 2 ϕ ∗ 2 0 ϕ 2 π + ϕ ∗ 2 π ϕ 2 � � 0 0 Z 2 symmetry φ = ± π / 2 Favours | ϕ 0 | = | ϕ π | Dhar, Maji, Mishra, Pai, Mukerjee & Paramekanti (2013)

Chiral SuperFluid (CSF) Breaks U (1) × Z 2 symmetry Polini et. al. (2005), Lim et. al. (2008), Powell et. al. (2010), Moller and Cooper (2010), Sinha and Sengupta (2011)

Increasing interaction strength Single site mean-field theory gives single transition from CSF to MI Beyond mean-field theory • Monte Carlo on classical 1+1D model • DMRG on quantum model

Cut to the chase As U is increased a novel Chiral Mott Insulator (CMI) forms which has a charge gap but retains the chiral order of the CSF. Upon further increase of U , a regular MI develops. Dhar, Maji, Mishra, Pai, Mukerjee & Paramekanti (2013)

1+1D classical model BKT transition from CSF to CMI Ising transition from CMI to MI

DMRG on the quantum model BKT transition from CSF to CMI Ising transition from CMI to MI

Phase diagram Monte-Carlo DMRG • CSF- Algebraic SF, long-ranged loop current order • CMI - Charge gap, long-ranged loop current order • MI - Charge gap, no loop current order

Chiral Mott Insulator Physical Pictures • Vortex-antivortex supersolid • Indirect excitonic condensate Variational wavefunction ψ ( r 1 , r 2 , . . . r n ) = e − P ij v ( r i − r j ) ψ CSF ( r 1 , r 2 , . . . r n )

Sine-Gordon model + - 2-Leg Ladder Also Tokuno & Georges (2014)

Sine-Gordon model

Chiral Mott states elsewhere • Two component boson system, Petrescu & Le Hur (2013) • Frustrated triangular lattice, Zalatel, Parameswaran, Ruegg & Altman (2014) • Chiral Bose liquid at finite temperature, Li, Paramekanti, Hemmerich & Liu (2014) • Field theoretical study of ladders with flux, Tokuno & Georges (2014)

Extended Bose-Hubbard model Extended Bose Hubbard model  � j b j +1 + h . c . + U X b † H = − t 2 n j ( n j − 1) + V n j n j +1 j MI - Mott Insulator SF - Superfluid DW - Density Wave HI - Haldane Insulator Kurdestany, Pai, Mukerjee & Pandit (2014)

Other types of chiral states Unfrustrated coupled extended Bose-Hubbard ladders Essentially the same phases Dela Torre, Berg, Altman and Giamarchi (2011) Frustrated ladders, chiral version of DW and HI phase? Roy, Mukerjee & Paramekanti (in progress)

t-t’-V model Hard core bosons b † b † X i b i +1 + h . c . − t 0 X X H = − t i b i +2 + h . c . + V n i n i +1 i i i t > 0 , t 0 < 0 Frustration Mishra, Pai, Mukerjee & Paramekanti (2013 & 2014)

t-t’-V model Effective spin 1/2 model X i +1 + h . c . − 2 t 0 X S + S + i S � i S � H = − 2 t i +2 + h . c . i i X + V n i n i +1 i V = 0 & t 0 = t/ 2: Easy plane Majumdar Ghosh model Y Ground state: | ψ i G ⇠ ( | "i j | #i j +1 + | #i j | "i j +1 ) j ∈ even Bond ordering

t-t’-V model t 0 = 0 can be mapped on the XXZ model X X j +1 + S y j S y S x j S x S z j S z � � H = − 2 t + V j +1 j +1 j j Bethe anstaz solvable V ≤ 2 t spin Luttinger liquid (superfluid) V > 2 t gapped Ising antiferromagnet (CDW)

t-t’-V model DMRG calculation Mishra, Pai, Mukerjee & Paramekanti (2013) • Re-entrant phase transition • Continuously varying Luttinger parameter along phase boundaries

t-t’-V model DMRG calculation Incommensurate filling • SS - supersolid phase • No bond-ordered supersolid found

Summary Frustration in bosonic systems can produce interesting phases like the Chiral Mott state, bond ordered solid, supersolid etc.