Dec. 5, 2018 Tensor Network States: Algorithms and Applications (TNSAA) 2018-2019 Synthetic triangular antiferromagnets with ultracold fermions in optical lattices Daisuke Yamamoto Aoyama-Gakuin Univ.
Introduction - Revival Interest in triangular-lattice antiferromagnets (TLAFs) - Optical-lattice quantum simulations
Recent “revival” of interest in TLAF - Properties and mechanism of ET, dmit, CAT, YbMgGaO 4 , Cs 2 CuCl 4 , “spin - liquid (like)” behavior ZnCu 3 (OH) 6 Cl 2 ,… - Fractional excitations S =1/2? S =1/2? Magnon ( S =1)? or spinon ( S =1/2)? Bosonic? or Fermionic? S =1 Ba 3 CoSb 2 O 9 Ito et al . Nat. Commun. (2017) Continuous ( S =1/2)? Paddison et al . Nat. Phys. (2016) YbMgGaO 4 - Exotic magnetic orders Multiple- Q orders Multipolar orders
Greiner Lab. (Harvard Univ.) optical lattice Two hyperfine states of 6 Li ⇒ The Hubbard model ☑ Quantum gas “Long - range” Neel order on microscope (QGM) square lattice was realized. ⇒ Frustrated magnetism (artificial TLAFs) must be a next target! Becker et al ., New J. Phys. (2010)
Notice: No TRUE long-range order in 2D with SU(2). Greiner Lab. (Harvard Univ.) (Even BKT phase is absent.) optical lattice Two hyperfine states of 6 Li ⇒ The Hubbard model ☑ Quantum gas “Long - range” Neel order on microscope (QGM) square lattice was realized. The correlation length is much longer than the system size. ⇒ Frustrated magnetism (artificial TLAFs) must be a next target! Becker et al ., New J. Phys. (2010)
Model considered here - Relevance to real and “artificial” materials - Motivation and problem formulation
Triangular XXZ model w/o U(1) ( S =1/2) AF interactions: J, J z >0, transverse magnetic field: H
Triangular XXZ model w/o U(1) ( S =1/2) AF interactions: J, J z >0, transverse magnetic field: H ☑ J=J z — The Heisenberg model with magnetic fields ⇒ Quantum stabilization of 1/3 magnetization plateau Chubokov and Golosov, J. Phys.: Cond. Matt. (1991)
Triangular XXZ model w/o U(1) ( S =1/2) AF interactions: J, J z >0, transverse magnetic field: H ☑ J=J z — The Heisenberg model with magnetic fields ⇒ Quantum stabilization of 1/3 magnetization plateau Chubokov and Golosov, J. Phys.: Cond. Matt. (1991) □ J ≠ J z — No U(1) spin-rotational symmetry in the presence of transverse field H ⇒ Magnetization is no longer good quantum number.
Triangular XXZ model w/o U(1) ( S =1/2) AF interactions: J, J z >0, transverse magnetic field: H ☑ J=J z — The Heisenberg model with magnetic fields ⇒ Quantum stabilization of 1/3 magnetization plateau Chubokov and Golosov, J. Phys.: Cond. Matt. (1991) □ J ≠ J z — No U(1) spin-rotational symmetry in the presence of transverse field H ⇒ Magnetization is no longer good quantum number. ☑ J= 0 — The transverse Ising model on triangular lattice ⇒ Studied by QMC Isakov and Moessner, PRB 68 , 104409 (2003).
Co-based layered TLAF material - Small (~ 5%) interlayer antiferromagnetic coupling ( ⋍ 2D) - No DM interaction - The structure is very simple. (well-separated layers of Ba 3 CoSb 2 O 9 equilateral triangular lattice)
Co-based layered TLAF material - Small (~ 5%) interlayer antiferromagnetic coupling ( ⋍ 2D) - No DM interaction - The structure is very simple. (well-separated layers of Ba 3 CoSb 2 O 9 equilateral triangular lattice) Heisenberg uniaxial crystal spin-orbit Zeeman term interaction field interaction
Co-based layered TLAF material - Small (~ 5%) interlayer antiferromagnetic coupling ( ⋍ 2D) - No DM interaction - The structure is very simple. (well-separated layers of Ba 3 CoSb 2 O 9 equilateral triangular lattice) Heisenberg uniaxial crystal spin-orbit Zeeman term interaction field interaction ⇒ Only the lowest Kramers doublet is essential. *Anisotropy of “easy - plane” type
Single-crystal magnetization curves Ba 3 CoSb 2 O 9 A. Sera et al , PRB 94 (2016). T. Susuki et al , PRL 110 (2013). ☑ The magnetization curve for transverse field || ab had basically not changed by easy-plane anisotropy * We have successfully explained the jump in the curve for H//c: DY, G. Marmorini, I. Danshita, PRL 112 , 127203 (2014).
Proposal for quantum simulations DY et al ., arXiv:1808.08916 - Raman laser beams - State-dependent triangular optical lattices - Rf field by atom chip Goldman et al ., PRL (2010) “Easy -axis ” anisotropy
Classical ground state of the model ⇒ Classical approximation
Classical ground state of the model ⇒ Classical approximation saturation three sublattices Continuous degeneracy Ψ (coplanar) Inverted-Y ← easy-axis easy-plane →
Classical ground state of the model ・ coplanar ⇒ Classical approximation ・ nontrivial U (1) remains in the relative angles saturation etc. three sublattices Continuous degeneracy Ψ Miyashita-Kawamura, JPSJ (coplanar) (1985) Sheng-Henley, J. Phys. (1992) Inverted-Y ← easy-axis easy-plane →
Problems Classical Quantum fluctuations Continuous Order-by-disorder? degeneracy Ψ (coplanar) Inverted-Y How can we obtain a “sufficiently quantitative” ground-state phase diagram (w/ full quantum effects)? □ Spin-wave － Only can guess the selected orders □ Frustration ⇒ Suffer from the minus-sign problem □ 2D ⇒ Conventional DMRG is not efficient. □ No U(1) symmetry ⇒ The Hilbert space cannot be divided.
Calculations and Results - Combine DMRG with mean-field for T =0 - Monte-Carlo simulations for T ≠ 0
DY et al., arXiv:1808.08916 Combine DMRG with mean-field (efficient for 1D) (infinite dimensions) (i) Approximate the Hamiltonian by an effective cluster Hamiltonian with mean-field boundary conditions Sublattice mean fields (ii) Solve the cluster Hamiltonian with DMRG: Equivalent 1D problem (iii) Self-consistently determine the mean fields: Iterate (i-iii) till convergence of mean fields (typically, within 10 -8 ) “Large -size cluster mean- field (CMF) with DMRG solver”
Cluster-size dependence Classical phase diagram Saturated Continuous degeneracy Ψ (coplanar) Inverted-Y ← easy-axis easy-plane →
Cluster-size dependence Classical Classical phase diagram Saturated Continuous degeneracy Ψ (coplanar) Inverted-Y ← easy-axis easy-plane →
Cluster-size dependence Classical phase diagram Saturated Continuous degeneracy Ψ (coplanar) Inverted-Y ← easy-axis easy-plane →
Cluster-size dependence Classical phase diagram Saturated Continuous degeneracy Ψ (coplanar) Inverted-Y ← easy-axis easy-plane →
Cluster-size dependence Classical phase diagram Saturated Continuous degeneracy Ψ (coplanar) Inverted-Y ← easy-axis easy-plane →
Cluster-size dependence Classical phase diagram Saturated Continuous degeneracy Ψ (coplanar) Inverted-Y ← easy-axis easy-plane →
Cluster-size dependence Classical phase diagram Saturated Extrapolate the results with scaling parameter: → 1 ( N C → ∞ ) Continuous degeneracy Ψ (coplanar) Inverted-Y ← easy-axis easy-plane →
Cluster-size dependence “CMF+S” Classical phase diagram H Saturated Continuous degeneracy Ψ (coplanar) Inverted-Y ← easy-axis easy-plane →
Quantum phase diagram Classical Continuous degeneracy Ψ (coplanar) Inverted-Y CMF+S
Quantum phase diagram Classical Continuous degeneracy Ψ (coplanar) Inverted-Y S =1/2 CMF+S
Quantum phase diagram Rescaled by H s Classical Continuous Not sensitive to degeneracy Ψ the anisotropy (coplanar) Easy-plane side Inverted-Y Ba 3 CoSb 2 O 9 S =1/2 CMF+S Consistent to the experiment T. Susuki et al , PRL 110 (2013).
Quantum phase diagram Ising limit Classical Significant quantum Continuous fluctuation effects! degeneracy Ψ (coplanar) Inverted-Y The extrapolated H s / J z ⋍ 0.85, S =1/2 CMF+S which is consistent to QMC: 0.825 ± 0.025 Isakov-Moessner, PRB (2003).
Quantum phase diagram Classical Continuous degeneracy Ψ (coplanar) Inverted-Y S =1/2 CMF+S Easy-axis side Significant quantum fluctuation effects!
DY et al., arXiv:1808.08916 Easy-axis quantum ground state [ ] 1 st -order, [ ] 2 nd -order
DY et al., arXiv:1808.08916 Easy-axis quantum ground state Extrapolation of the phase boundary [ ] 1 st -order, [ ] 2 nd -order
Low-field region [ ] 1 st -order, [ ] 2 nd -order
Low-field region ・ coplanar ・ nontrivial U (1) remains in the relative angles [ ] 1 st -order, [ ] 2 nd -order
Low-field region ・ coplanar ・ nontrivial U (1) remains in the relative angles [ ] 1 st -order, [ ] 2 nd -order
Spin reorientation driven by fluctuations Low-field region Selected by quantum fluctuations ☑ At H = H r , an emergent Ising transition takes place from an intermediate (reorienting) state to the classical inverted-Y state.
Thermal phase transitions The Hamiltonian has no U(1) symmetry. Inverted- Y (=Ψ) breaks discrete symmetry. ⇒ “Standard” second -order phase transition? No! DY et al., arXiv:1808.08916.
Thermal phase transitions Classical Monte Carlo L × L rhombic clusters ( L =24,48,72,96) The Hamiltonian has no U(1) symmetry. Inverted- Y (=Ψ) breaks discrete symmetry. ⇒ “Standard” second -order phase transition? No! DY et al., arXiv:1808.08916.
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