Quantum Phases in Bose-Hubbard Models with Spin-orbit Interactions

Quantum Phases in Bose-Hubbard Models with Spin-orbit Interactions Shizhong Zhang The University of Hong Kong Institute for Advanced Study, Tsinghua 24 October 2012

The plan 1. Introduction to Bose-Hubbard model (BHM) 2. BHM with spin-orbit coupling Weak interaction superfluid Strong coupling Mott insulator; 1D & 2D magnetic models Phase diagram - magnetic structure in strongly interacting superfluid 3. Slave boson theory Construction Some consequences 4. Conclusions and outlook

Optical lattice: Bose-Hubbard model hopping on-site interaction Theory: M. Fisher et al, PRB 40 546 (1989) D. Jaksch et al, PRL 81 3108 (1998) Experiment: M.Greiner et al., Nature 415 39 (2002)

Bose-Hubbard Model: mean field theory a i a j!ha i ia j + a i ha ji ha i iha ji order parameter H mft = U n i(n i 1) 2 µn i t X j2nn i (ha j ia i + ha j ia i) superfluid Mott insulator M. Fisher et al, PRB 40 546 (1989) K.Sheshadri et al, Europhys.Lett. 22 257 (1993) Credit: Bloch@Munich

Bose-Hubbard Model: excitations, RPA Superfluid Mott insulator M X M X K.Sheshadri et al, Europhys.Lett. 22 257 (1993)

Bose-Hubbard Model: summary Mott Superfluid Normal Order Parameters zero nonzero (uniform) zero Compressibility zero nonzero nonzero Excitations gapped gapless gapless (?) Charge Transport (DC...vities) zero nonzero + superfluid nonzero Considerations in terms of many-body wave functions and density matrices can be carried out for these different phases (Yang, Kohn, Bloch, Leggett).

What are the effects of spin-orbit couplings in Bose-Hubbard model? W.Cole et al. PRL 109 085302 (2012) William Cole Arun Paramekanti Nandini Trivedi

Bose-Hubbard Model: with spin-orbit interactions hopping two internal states on-site interaction H hop = ta i R 0 ˆ a i+ˆ 0, 0 =", # ˆ =ˆx, ŷ e i y e i x U

Bose-Hubbard Model: non-interacting band structure hopping two internal states on-site interaction Lattice version of the Rashba spin-orbit coupling U

Non-interacting band structure Non-trivial winding (Chern number) around the Γ point due to existence of Dirac points: Lattice version of the Rashba spin-orbit coupling ky 0 0 k x

Weak coupling superfluid Four degenerate states: (±k 0, ±k 0 ) p 2 tan k0 = tan Spins lie in the x-y plane. U int / 1+ 2 (n " + n # ) 2 + 1 2 (n " n # ) 2 < 1 no polarization only one state is occupied; uniform spin and number density > 1 polarization two opposite states are occupied; strip spin and uniform number density Cf. Considerations of Y.Li et al, PRL 108 225301 (2012) W.Cole et al. PRL 109 085302 (2012)

Strong coupling Mott insulator Consider the case in which on average, there is one boson per site. Standard perturbation theory gives low energy effective magnetic Hamiltonian x-direction: cos(2 ) S x i S x l 1 S y i Sy l 2 1 cos(2 )S z i S z l S j sin(2 )ŷ (S i S l ) S l y-direction: S i 1 S x i S x j cos(2 ) S y i Sy j 2 1 cos(2 )S z i S z j +sin(2 )ˆx (S i S j ) Dzyaloshinskii-Moriya coupling Cf. DM term in superfluid, X. Xu and J.Han PRL 108 185301 (2012) W.Cole et al. PRL 109 085302 (2012) Z.Cai et al. PRA 85 061606R (2012) J.Radic et al. PRL 109 085303 (2012) M.Gong et al, arxiv:1205.6211

1D magnetic Hamiltonian For example, 1D Hamiltonian along x-direction. Rotate spins around x by π/2, such that DM vector is along z cos(2 ) S x i S x l cos(2 ) (2 1)S y i Sy l 1 S z i S z l sin(2 )(S y i Sx l S x i S y l ) Some 1D AFM system with DM System DM/Exchange Cooper Benzoate 0.05 Yb4As3? BaCu2Si2O7 0.02? CsCuCl3 0.18 XY-exchange and DM couplings can be tuned by changing α and λ, in particular, DM can be made as large as exchange coupling; Various limits of the model can be solved exactly.

1D magnetic Hamiltonian: special cases Case I! 0 rotation around x by π every other site 1 (cos(2 )S x i S x l + cos(2 )S y i Sy l S z i S z l ) Z-ferromagnetic 0 < < /2; cos(2 ) < 1 critical points: =0, /2; cos(2 ) =1 whkl 2 1 0-1 -2 -p - p 2 a=pê6 a=0 a=pê2 a=pê3 0 k p 2 p 0 Z-FM /2

1D magnetic Hamiltonian: special cases Case II =1 XXZ+DM S x i S x l S y i Sy l 1 cos(2 ) Sz i S z l tan(2 )(S y i Sx l S x i S y l ) Rotate each spin around z by φ=2α. ẑ ŷ ˆx Can be mapped to XXZ model with a new twisted boundary condition. It can be solved with Bethe ansatz and turns out to be always critical in bulk. 1 critical line ( e S x i e S x l + e S y i e S y l + e S z i e S z l ) 0 Z-FM /2

1D magnetic Hamiltonian: special cases Case III!1 Y-FM XY-chiral Y-AFM 2 cos(2 )S y i Sy l sin(2 )(S y i Sx l S x i S y l ) Can be solved using Jordan-Wigner. E ± (k) =sin(2 )sink ± cos(2 ) Critical points: = /8; 3 /8 EHkL 0 a=pê6 a=pê8,3pê8 a=0.1p 1 critical line 0 p 2 k p 0 Z-FM /2

1D magnetic Hamiltonian: special cases Case IV = /4 Y-FM XY-chiral Y-AFM 1 S z i S z l (S y i Sx l S x i S y l ) Ising+DM, can be mapped to XXZ model > 1 XY-chiral < 1 Z-Ferromagnetism Case V XXZ model 1 (±S x i Sl x ± (2 1)S y i Sy l + Si z Sl z ) =0 = /2 1 para Y-FM Z-FM XY-chiral critical line para Y-AFM < 1 Para Para > 1 Y-Ferro Y-Antiferro 0 Z-FM Z-FM /2

Schematic Phase diagram Y-FM Y-AFM What needs to be done: XY-chiral Exact Diagonalization (12 sites) suggests phase diagram as shown left. It confirms the part for λ>1; but for λ<1, not very clear; 1 Calculate phase diagram with DMRG technique; Para Z-FM Para Calculate correlation functions and investigate experimental signatures. 0 /2

2D classical magnetic phases Calculated with classical Monte Carlo annealing procedure + variational ansatz Magnetic structure factors: S q = X i S i exp(iq r i ) 2

What are the implications of magnetic ordering for the superfluid states?

Mean field theory H hop = ta i R 0 ˆ a i+ˆ 0 H int = U 2 (n2 i" + n 2 i# +2 n i" n i# ) H mft hop = t(ha i ir 0 ˆ )a i+ˆ 0 ta i (R 0 ˆ ha i+ˆ 0i) Due to complicated magnetic ordering, we carry out calculations on a finite cluster (8*8) with periodic boundary conditions to attain self-consistency. e i y e i x Local magnetization: m i = ha i 0a i 0i Bond current: apple 0 ˆ = it(r 0 ˆ ha i a i+ˆ, 0i c.c.)

Phase diagram 1.0 0.8 l á1.5 a ápê2 m U 0.6 0.4 n á1 0.2 m U 0.0 1.0 0.8 0.6 0.4 0.02 0.04 0.06 0.08 0.10 0.12 têu l á0.5 a ápê2 Smooth evolution of magnetic order from Mott insulator to superfluid! 0.2 0.0 n á1 0.02 0.04 0.06 0.08 0.10 0.12 têu

How are the current patterns related to magnetic ordering?

Slave boson theory: construction To describe the interplay between magnetism and superfluidity, introduce a = 1 p nb b f n b = b b Both b and f are bosons operators, satisfying commutation relations Single site Hilbert space: m ",n#i [b, b ] = 1; [f,f 0]= 0 Local constraint: X f f = b b m + ni b m ",n#i f The canonical commutation relations of a-operators are preserved in the physical Hilbert space. H hop = ta i R 0 ˆ a i+ˆ 0 H hop = t 1 p nib f i R 0 ˆ f i+ˆ 0 1 p ni+ˆ,b b i b i+ˆ

Slave boson mean field theory Assuming that the magnetic moments are ordered in the ground state, we can then make the classical field approximation and define: z = 1 fpnb z =(z ",z #) z z =1 Hopping Hamiltonian within mean field becomes: H mft hop = t h 2 z i R 0 ˆ z i+ˆ 0 i b i b i+ˆ Thus, the original spin-orbit couplings for a-bosons become abelian gauge fields for the charge degrees of freedom b, within slave boson mean field and if spinons (f) are condensed. The constraint is implemented with U(1) gauge fields, which will be gapped through Higgs mechanism, if spinons are condensed. The suppressed gauge fluctuations may ensure the validity of the slave boson mean field theory.

Slave boson theory: understand current patterns Case I: Ferromagnetic background H mf hop = z i =(1, 0), 8 i (t cos )b i b i+ˆ renormalized hopping. U fm c = U c cos 4 1 2 3 Case II: Anti-ferromagnetic background z i =(1, 0); Sublattice A z i =(0, 1); Sublattice B Y t12 t 23 t 34 t 41 = t 4 sin 4 < 0 π 4 1 2 π π 3 π-flux lattice, Dirac points at (0,±π/2) Two-fold degeneracy of current patterns!

Slave boson theory: understand current patterns Case III: Spin crystal background Y t12 t 23 t 34 t 41 = 1 4 (cos ip 2 sin ) 4 Alternating flux Φ in each plaquette. Fixed current patterns! Φ 4 -Φ 1 2 Φ Φ 3 What needs to be done: Self-consistent determination of the spinon fields z (full slave boson mean field theory); Possibility of an exotic Mott insulator in BHM with spin-orbit coupling; flux f p p 2 0 - p 2 -p 0 p 4 a p 2

Conclusions and outlook We have studied Bose-Hubbard Model with spin-orbit interactions and established its weak coupling superfluid states, magnetic structure in the Mott insulating states and determined the phase diagram using mean field theory. We proposed a new slave boson theory and argued that it was helpful for us to understand certain features of the strongly interacting superfluids close to the Mott transition. Magnetic models in either 1D or 2D are worth investigating in detail. In particular, for 1D, the complete phase diagram with exact diagonalization or density matrix renormalization group calculation; possibility of experimental implementation. For 2D, collective excitations and order from disorder calculations. Understand the phase diagram with slave boson theory. In particular, investigate the possibility of exotic Mott insulating states (e.g. disordered magnetic states close to the Mott boundary).

Thank you!