Synthetic triangular antiferromagnets with ultracold fermions in - - PowerPoint PPT Presentation
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
“spin-liquid (like)” behavior
ET, dmit, CAT, YbMgGaO4, Cs2CuCl4, ZnCu3(OH)6Cl2,…
Magnon (S=1)? or spinon (S=1/2)? Bosonic? or Fermionic? Multiple-Q orders Multipolar orders Continuous (S=1/2)? S=1 S=1/2? S=1/2?
Ito et al. Nat. Commun. (2017) Paddison et al. Nat. Phys. (2016)
Ba3CoSb2O9 YbMgGaO4
“Long-range” Neel order on square lattice was realized.
Two hyperfine states of 6Li Greiner Lab. (Harvard Univ.) ⇒The Hubbard model
⇒ Frustrated magnetism (artificial TLAFs) must be a next target!
☑ Quantum gas microscope (QGM)
Becker et al., New J. Phys. (2010)
“Long-range” Neel order on square lattice was realized.
Two hyperfine states of 6Li Greiner Lab. (Harvard Univ.) ⇒The Hubbard model
⇒ Frustrated magnetism (artificial TLAFs) must be a next target!
☑ Quantum gas microscope (QGM)
Becker et al., New J. Phys. (2010)
The correlation length is much longer than the system size.
Notice: No TRUE long-range order in 2D with SU(2). (Even BKT phase is absent.)
(S=1/2) AF interactions: J, Jz>0, transverse magnetic field: H
(S=1/2) ☑ J=Jz — The Heisenberg model with magnetic fields AF interactions: J, Jz>0, transverse magnetic field: H ⇒ Quantum stabilization of 1/3 magnetization plateau
Chubokov and Golosov, J. Phys.: Cond. Matt. (1991)
(S=1/2) ☑ J=Jz — The Heisenberg model with magnetic fields AF interactions: J, Jz>0, transverse magnetic field: H
□ J≠Jz — No U(1) spin-rotational symmetry in the presence
⇒ Magnetization is no longer good quantum number. ⇒ Quantum stabilization of 1/3 magnetization plateau
Chubokov and Golosov, J. Phys.: Cond. Matt. (1991)
(S=1/2) ☑ J=Jz — The Heisenberg model with magnetic fields AF interactions: J, Jz>0, transverse magnetic field: H
□ J≠Jz — No U(1) spin-rotational symmetry in the presence
⇒ Magnetization is no longer good quantum number. ⇒ Quantum stabilization of 1/3 magnetization plateau ☑ J=0 — The transverse Ising model on triangular lattice ⇒ Studied by QMC
Chubokov and Golosov, J. Phys.: Cond. Matt. (1991) Isakov and Moessner, PRB 68, 104409 (2003).
Ba3CoSb2O9
antiferromagnetic coupling (⋍2D)
(well-separated layers of equilateral triangular lattice)
Ba3CoSb2O9
antiferromagnetic coupling (⋍2D)
(well-separated layers of equilateral triangular lattice)
Heisenberg interaction uniaxial crystal field spin-orbit interaction Zeeman term
Ba3CoSb2O9
antiferromagnetic coupling (⋍2D)
(well-separated layers of equilateral triangular lattice)
Heisenberg interaction uniaxial crystal field spin-orbit interaction Zeeman term ⇒ Only the lowest Kramers doublet is essential.
*Anisotropy of “easy-plane” type
Ba3CoSb2O9
* We have successfully explained the jump in the curve for H//c:
☑ The magnetization curve for transverse field ||ab had basically not changed by easy-plane anisotropy
DY, G. Marmorini, I. Danshita, PRL 112, 127203 (2014).
“Easy-axis” anisotropy
DY et al., arXiv:1808.08916
triangular optical lattices
Goldman et al., PRL (2010)
⇒ Classical approximation
⇒ Classical approximation
Continuous degeneracy (coplanar) Ψ
Inverted-Y
saturation
three sublattices ←easy-axis easy-plane→
⇒ Classical approximation
Continuous degeneracy (coplanar) Ψ
Inverted-Y
saturation
three sublattices ←easy-axis easy-plane→ etc.
Sheng-Henley, J. Phys. (1992) Miyashita-Kawamura, JPSJ (1985)
・coplanar ・nontrivial U(1) remains in the relative angles
Classical
Continuous degeneracy (coplanar) Ψ
Inverted-Y
Quantum fluctuations Order-by-disorder?
□ Spin-wave － Only can guess the selected orders □ No U(1) symmetry ⇒ The Hilbert space cannot be divided. □ Frustration ⇒ Suffer from the minus-sign problem □ 2D ⇒ Conventional DMRG is not efficient.
How can we obtain a “sufficiently quantitative” ground-state phase diagram (w/ full quantum effects)?
DY et al., arXiv:1808.08916
(i) Approximate the Hamiltonian by an effective cluster Hamiltonian with mean-field boundary conditions (ii) Solve the cluster Hamiltonian with DMRG:
Sublattice mean fields
(efficient for 1D) (infinite dimensions)
(iii) Self-consistently determine the mean fields:
“Large-size cluster mean-field (CMF) with DMRG solver”
Equivalent 1D problem
Iterate (i-iii) till convergence of mean fields (typically, within 10-8)
Continuous degeneracy (coplanar) Ψ
Inverted-Y
Classical phase diagram
←easy-axis easy-plane→
Saturated
Continuous degeneracy (coplanar) Ψ
Inverted-Y
Classical phase diagram
←easy-axis easy-plane→
Saturated Classical
Continuous degeneracy (coplanar) Ψ
Inverted-Y
Classical phase diagram
←easy-axis easy-plane→
Saturated
Continuous degeneracy (coplanar) Ψ
Inverted-Y
Classical phase diagram
←easy-axis easy-plane→
Saturated
Continuous degeneracy (coplanar) Ψ
Inverted-Y
Classical phase diagram
←easy-axis easy-plane→
Saturated
Continuous degeneracy (coplanar) Ψ
Inverted-Y
Classical phase diagram
←easy-axis easy-plane→
Saturated
Continuous degeneracy (coplanar) Ψ
Inverted-Y
Classical phase diagram
←easy-axis easy-plane→
Saturated
Extrapolate the results with scaling parameter: → 1 (NC →∞)
Continuous degeneracy (coplanar) Ψ
Inverted-Y
Classical phase diagram
←easy-axis easy-plane→
Saturated “CMF+S” H
Classical
Continuous degeneracy (coplanar) Ψ
Inverted-Y
CMF+S
Classical CMF+S
Continuous degeneracy (coplanar) Ψ
Inverted-Y
S =1/2
Classical CMF+S
Continuous degeneracy (coplanar) Ψ
Inverted-Y
S =1/2
Easy-plane side
Not sensitive to the anisotropy
PRL 110 (2013).
Ba3CoSb2O9 Consistent to the experiment Rescaled by Hs
Classical CMF+S
Continuous degeneracy (coplanar) Ψ
Inverted-Y
S =1/2 Significant quantum fluctuation effects!
Ising limit
The extrapolated Hs/Jz⋍ 0.85, which is consistent to QMC: 0.825±0.025
Isakov-Moessner, PRB (2003).
Classical CMF+S
Continuous degeneracy (coplanar) Ψ
Inverted-Y
S =1/2 Significant quantum fluctuation effects!
Easy-axis side
[ ] 1st-order, [ ] 2nd-order
DY et al., arXiv:1808.08916
[ ] 1st-order, [ ] 2nd-order
DY et al., arXiv:1808.08916
Extrapolation of the phase boundary
[ ] 1st-order, [ ] 2nd-order
[ ] 1st-order, [ ] 2nd-order ・coplanar ・nontrivial U(1) remains in the relative angles
[ ] 1st-order, [ ] 2nd-order ・coplanar ・nontrivial U(1) remains in the relative angles
☑ At H=Hr, an emergent Ising transition takes place from an intermediate (reorienting) state to the classical inverted-Y state.
Low-field region
Selected by quantum fluctuations
Inverted-Y (=Ψ) breaks discrete symmetry. The Hamiltonian has no U(1) symmetry. ⇒“Standard” second-order phase transition?
No! DY et al., arXiv:1808.08916.
Inverted-Y (=Ψ) breaks discrete symmetry. The Hamiltonian has no U(1) symmetry. ⇒“Standard” second-order phase transition?
Classical Monte Carlo
No! DY et al., arXiv:1808.08916.
L×L rhombic clusters (L=24,48,72,96)
correlation length
A finite critical range Tc1 <T < Tc2 for Sz Berezinskii-Kosterlitz- Thouless (BKT) behavior
Sx Sz Szcomponent:
(+m, 0, -m) ⇒ 3! = 6 elements
Correlation function exponent
T=Tc1 ⇒ η⋍1/9 T=Tc2 ⇒ η⋍1/4
consistent with the ℤ6 clock model
Jose et al., PRB 16 (1977)
H
correlation length
Isolated critical points T=Tc1 for κ and T=Tc2 for Sx, respectively Long-range order (LRO) ・Vector chirality ( y component )
Sx Sz H
Berezinskii-Kosterlitz- Thouless (BKT) behavior
cf.
DY et al., arXiv:1808.08916.
* The transition at Tc2 shows a similar behavior with almost identical exponents Critical exponents at Tc1
correlation length
・Cyclic group: ℤ6
(g0=g6=e)
・Symmetric group: S3
(the present case)
two generators
A→B→C→A A ⇔ B B ⇔ C C ⇔ A A→C→B→A *Further investigation is required.
☑ The CMF+S scheme with the use of DMRG solver significantly expands the scope of its applications to broader areas of quantum frustrated systems. ☑ The extension of the tractable cluster size NC enables us to construct a good extrapolation series.
Continuous degeneracy (coplanar)
☑ Using this scheme, one can get a quantitative phase diagram for quantum frustrated systems.
□ Unconventional criticality
related to symmetric group?
DY et al., arXiv:1808.08916.