Exotic magnetic orders for high spin ultracold fermions

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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 thehigh spin.spin-3/2 fermionsUCA 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) symmetrichigh spin system the life is much morecomplicated.Workshop on Correlations and Coherence in Quantum SystemsExotic 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 thehigh spin.5/2 spinű 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) symmetrichigh spin system the life is much morecomplicated.Workshop on Correlations and Coherence in Quantum SystemsExotic 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 forthe exchange of the spin ofthe two scattering particles.spin-3/2 fermionsClassification 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 SystemsExotic 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 SystemsExotic 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 SystemsExotic 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 form 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 ) lspin-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 SystemsExotic 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 ordersn 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 SystemsExotic 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 SystemsExotic 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 SystemsExotic 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 SystemsExotic magnetic orders for high spin ultracold fermions

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