Exotic magnetic orders for high spin ultracold fermions

Ultracold atoms as simulator of fundamental modelsQuantum simulation offundamental models (properties,phenomena).Novel behavior, completely newphases are expected due to the**high** **spin**.**spin**-3/2 **fermions**UCA experiments :6 Li, 132 Cs, 9 Be, 135 Ba, 137 Ba;Ba 2 CoGe 2 O 7 (square lattice),Bi 3 Mn 4 O 12 (NO 3 ) (double layeredhexagonal);If we have a non-SU(N) symmetric**high** **spin** system the life is much morecomplicated.Workshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**

Ultracold atoms as simulator of fundamental modelsQuantum simulation offundamental models (properties,phenomena).Novel behavior, completely newphases are expected due to the**high** **spin**.5/2 **spin**ű fermionokUCA experiments: 173 Yb, 52 Cr;different Mn-compounds ; NaReO 4 ,NH 4 ReO 4 (with 185 Re or 187 Re);If we have a non-SU(N) symmetric**high** **spin** system the life is much morecomplicated.Workshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**

Ultracold atoms as simulator of fundamental modelsThe Hamiltonian:H = H kin + H intwhere H int contains manytypes of scatteringprocesses.on-site interaction=⇒Pauli’s principleThe only nonzero terms arecompletly antisymmetric **for**the exchange of the **spin** ofthe two scattering particles.**spin**-3/2 **fermions**Classification of the allowedscattering processes:S tot = 0, singletS tot = 2, quintet=⇒[]H = −t ∑ 〈i,j〉 (c † i,σ c j,σ + H.c.) + ∑ i U 0 P (i)0 + U 2P (i)2 ,and P (i)S= c† i,σ 1c † i,σ 2c i,σ3 c i,σ4 ˆPS , where ˆPS projects tothe subspace S tot = S. T. L. Ho PRL (1998)T. Ohmi and K. Machida JPSJ (1998)Workshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**

Strong coupling limit: U S /t → ∞no particle transportIf there is no **magnetic** order and the lattice symmetries are preserved:**spin** liquid state.Workshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**

Strong coupling limit: U S /t → ∞Perturbation theory up to leading orderwith respect to t.t preserves S and m Snearest-neighbor hoppingEffective Hamiltonian[]H = ∑ g 0 P (i,j)0+ g 2 P (i,j)2〈i,j〉wherenearest-neighbor interactionthe same **spin** dependencethat has the original modelP (i,j)S= c † i,σ 1c † j,σ 2c j,σ3 c i,σ4 ˆPS and g S = −4t 2 /U S .H eff = ∑ 〈i,j〉[a0 n i n j + a 1 S i S j + a 2 (S i S j ) 2 + a 3 (S i S j ) 3]n i = c † i,σ c i,σ , and S i = c † i,σ F σ,σ ′c i,σ ′.Workshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**

[]Strong coupling limit: U S /t → ∞ H eff = ∑ 〈i,j〉 g 0 P (i,j)0+ g 2 P (i,j)2The general **for**m of the projectors:ˆP S = ∑ 2Fl=0 a S,l(F 1 F 2 ) l with (F 1 F 2 ) 0 = E 1,2BUTˆP S is antisymmetric in the **spin** indices of the scattering particles!) (s) ) (as)(F 1 F 2 ) l =((F 1 F 2 ) l +((F 1 F 2 ) l) (as)ˆP S = ∑ 2Fl=0 b S,l((F 1 F 2 ) l**spin**-3/2 **fermions**[]H int = ∑ a n E (as)i,j+ a s (F 1 F 2 ) (as)i,jwith a n = (5G 2 − G 0 )/4, a s = (G 2 − G 0 )/3, andE (as)i,j = c i,α † c† j,β c j,δ c i,γ[E (as)] α,βγ,δ , and (F 1F 2 ) (as) = c i,α † c† j,β c j,δ c i,γ[(F 1 F 2 ) (as)] α,βγ,δ ,[E (as)] α,β[= γ,δ δα,γ δ β,δ − δ α,δ δ β,γ , and (F 1 F 2 ) (as)] α,β= γ,δ [F 1] α,γ [F 2 ] β,δ − [F 1 ] α,δ [F 2 ] β,γ .Workshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**

[]Strong coupling limit: U S /t → ∞ H eff = ∑ 〈i,j〉 g 0 P (i,j)0+ g 2 P (i,j)2The new effective Hamiltonian:[ ())H eff = ∑ a n n i n j + χ †i,j χ i,j − n i + a s(S ]i S j + B † i,j B i,j − 154 n iE. Sz. and M. Lewenstein EPL 93 66005 (2011)Site- and bond-centered **orders**n i = c † i,α c i,α (particle number at site i)S i = c † i,α F α,β c i,β (**spin** at site i)χ i,j = c † i,α c j,α (BCDW)B i,j = c † i,α F α,β c j,β (BSDW)|χ i,j | 2 ∝ 〈S i S j 〉 → **spin**-Peierls distorsion B. Marston and J. Affleck PRB (1989)|B i,j | 2 ∝ 〈Q i Q j 〉 → quadrupole-Peierls distorsionWorkshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**

Mean-field phase diagram: F = 3/2, 1/4 filling, square latticeParametersan < 0an = 0−an ≈ 2.6asas > 0as = 0(a)a n = 5g 2 − g 04a s = (g 2 − g 0 )3g S = − 4tU SAFM−g 0−g 2U(1) plaquette(b)Nonzero order parametersAFM: 〈S i 〉U(1) plaquette: 〈χ i,j 〉 (Φ = 0, or ±π; VBS)For the SU(4) line: C. Wu MPL (2006)E. Sz. and M. Lewenstein EPL 93 66005 (2011)Workshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**

In presence of **magnetic** fieldHamiltonian with **magnetic** field:H h = H MF + h∑S i .ienergy−8−12−16−20−24g 0 = −0.01g 2 = −0.4SU(2) plaquette/dimerU(1) plaquetteFM(a)(b)Order parametersSU(2) plaquette:〈S i 〉, 〈χ i,j 〉, 〈B i,j 〉SU(2) dimer:〈S i 〉, 〈χ i,j 〉, 〈B i,j 〉0.2 0.4 0.6 0.8hE. Sz. and M. Lewenstein EPL 93 66005 (2011)Workshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**

SummaryWe considered **high**-**spin** fermion systems on lattice.With a simple treatment a novel effective model was introducedthat able to describe the competition between the site and bondorder (even on mean-field level).This description can be interpreted within the concept of site andbond multipoles.We determined the mean-field phase diagram of the **spin**-3/2fermion system on square lattice at 1/4 filling and we showed thatthe exotic SU(2) plaquette state can have the lowest energy inpresence of external **magnetic** field. This state can becharacterized by quadrupole-Peierls distorsion.Workshop on Correlations and Coherence in Quantum Systems**Exotic** **magnetic** **orders** **for** **high** **spin** **ultracold** **fermions**