The imbalanced antiferromagnet in an optical lattice

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The imbalanced antiferromagnet in an optical lattice

  1. 1. The imbalanced antiferromagnet in an optical lattice Arnaud Henk Stoof Koetsier Floris van Liere
  2. 2. Introduction • Fermions in an optical lattice • Described by the Hubbard model • Realised experimentally [Esslinger ’05] • Fermionic Mott insulator recently seen [Esslinger ’08, Bloch ’08] • There is currently a race to create the Néel state • Imbalanced Fermi gases • Experimentally realised [Ketterle ’06, Hulet ’06] • High relevance to other areas of physics (particle physics, neutron stars, etc.) • Imbalanced Fermi gases in an optical lattice ?? 2
  3. 3. Fermi-Hubbard Model P P P H = −t c† cj 0 ,σ j,σ +U c† c† cj,↓ cj,↑ j,↑ j,↓ σ hjj 0 i j Sums depend on: Filling N Dimensionality (d=3) On-site interaction: U Tunneling: t Consider nearest-neighbor tunneling only. The positive-U (repulsive) Fermi-Hubbard Model, relevant to High-Tc SC 3
  4. 4. Quantum Phases of the Fermi-Hubbard Model • Positive U (repulsive on-site interaction): Conductor 1 Filling Fraction Band Insulator Conductor 0.5 Conductor Mott Insulator (need large U) 0 • Negative U: Pairing occurs — BEC/BCS superfluid at all fillings. 4
  5. 5. Mott insulator: Heisenberg Model (no imbalance yet) • At half filling, when U À t and kB T ¿ U we are deep in the Mott phase. Hopping is energetically supressed Model simplifies: only spin degrees of freedom remain (no transport) • Integrate out the hopping fluctuations, then the Hubbard model reduces to the Heisenberg model: J X H= S j · Sk 2 hjki Spin ½ operators: S = 1 σ z 1³ † † ´ 2 Si = ci,↑ ci,↑ − ci,↓ ci,↓ 2 Si =c† ci,↓ + i,↑ Si =c† ci,↑ − i,↓ 4t2 Superexchange constant (virtual hops): J= U 5
  6. 6. Néel State (no imbalance yet) • The Néel state is the antiferromagnetic ground state for J > 0 • Néel order parameter 0 ≤ h|n|i ≤ 0.5 measures amount of “anti-alignment”: 0.5 nj = (−1)j hSj i 〈n〉 h|n|i • Below some critical temperature Tc, we enter the Néel state and h|n|i becomes non-zero. 0 0 T Tc 6
  7. 7. Heisenberg Model with imbalance • Until now, N↑ = N↓ = N/2 • Now take N↑ 6= N↓ — spin population imbalance. • This gives rise to an overal magnetization m = (0, 0, mz ) N↑ − N↓ (fermions: S = 1 ) mz = S 2 N↑ + N↓ • Add a constraint to the Heisenberg model that enforces hSi = m J X X H= S j · Sk − B · (Si − m) 2 hjki i Effective magnetic field (Lagrange multiplier): B 7
  8. 8. Mean field analysis • J > 0 ⇒ ground state is antiferromagnetic (Néel state) Two sublattices: A, B A(B) A(B) A(B) • Linearize Hamiltonian: Si = hSi i + δSi B A B hSA i + hSB i • Magnetization: m= A B A 2 B A B A B hS i − hS i • Néel order parameter: n= 2 A B A • Obtain the on-site free energy f (n, m; B) subject to the constraint ∇B f = 0 (eliminates B) 8
  9. 9. Phase Diagram in three dimensions 1.5 n=0 m Canted: 1 n kB T/J 0.5 n 6= 0 m Ising: n 0 0 0.1 0.2 0.3 0.4 0.5 mz 0.5 0.4 n 〈n〉 0.2 0.0 Add imbalance 0.0 0.2 0 mz 0.3 0.6 0.4 0 0.9 0 Tc kB T J 1.2 T 9
  10. 10. Spin waves (magnons) dS i • Spin dynamics can be found from: = [H, S] dt ~ No imbalance: Doubly 0.5 degenerate antiferromagnetic dispersion 0.4 • Imbalance splits the ¯ ω/J z 0.3 degeneracy: 0.2 Gap: h Ferromagnetic (Larmor magnons: ω ∝ k2 0.1 precession of n) 0 π π Antiferromagnetic − 0 magnons: ω ∝ |k| 2 kd 2 10
  11. 11. Long-wavelength dynamics: NLσM • Dynamics are summarised a non-linear sigma model with an action Z Z ½ µ ¶2 dx 1 ∂n(x, t) S[n(x, t)] = dt ~ − 2Jzm × n(x, t) dD 4Jzn2 ∂t 2 ¾ Jd − [∇n(x, t)]2 • lattice spacing: d = λ/2 2 • number of nearest neighbours: z = 2D • local staggered magnetization: n(x, t) • The equilibrium value of n(x, t) is found from the Landau free energy: Z ½ 2 ¾ dx Jd 2 F [n(x), m] = [∇n(x)] + f [n(x), m] dD 2 • NLσM admits spin waves but also topologically stable excitations in the local staggered magnetisation n(x, t). 11
  12. 12. Topological excitations • The topological excitaitons are vortices; Néel vector has an out-of- easy-plane component in the core • In two dimensions, these are merons: • Spin texture of a meron: ⎛ p ⎞ n 2 − [n (r)]2 cos φ p z n = ⎝nv n2 − [nz (r)]2 sin φ⎠ nz (r) nv = 1 n • Ansatz: nz (r) = [(r/λ)2 + 1]2 • Merons characterised by: Pontryagin index ±½ Vorticity nv = ±1 Core size λ nv = −1 12
  13. 13. Meron size • Core size λ of meron found by plugging the spin texture into F [n(x), m] and minimizing (below Tc): 1.5 Meron core size 6 1 Λ kB T/J Merons 4 d 2 0.5 present 1.5 1.2 0 0.9 0 0 kB T J 0 0.1 0.2 0.3 0.4 0.5 0.1 0.6 mz 0.2 0.3 0.3 mz 0.4 • The energy of a single meron diverges logarithmically with the system area A at low temperature as Jn2 π A ln 2 2 πλ merons must be created in pairs. 13
  14. 14. Meron pairs Low temperatures: A pair of merons with opposite vorticity, has a finite energy since the deformation of the spin texture cancels at infinity: Higher temperatures: Entropy contributions overcome the divergent energy of a single meron The system can lower its free energy through the proliferation of single merons 14
  15. 15. Kosterlitz-Thouless transition • The unbinding of meron pairs in 2D signals a KT transition. This drives down Tc compared with MFT: 1 MFT in 2D 0.8 0.06 0.6 kB T/J kB T/J 0.04 n 6= 0 0.4 KT transition 0.02 0.2 0 0 0 0.05 0.1 0 0.2 0.4 mz mz • New Tc obtained by analogy to an anisotropic O(3) model (Monte Carlo results: [Klomfass et al, Nucl. Phys. B360, 264 (1991)] ) 15
  16. 16. Experimental feasibility • Experimental realisation: Imbalance: drive spin transitions with RF field Néel state in optical lattice: adiabatic cooling [AK et al. PRA77, 023623 (2008)] • Observation of Néel state Correlations in atom shot noise Bragg reflection (also probes spin waves) • Observation of KT transition Interference experiment [Hadzibabic et al. Nature 441, 1118 (2006)] In situ imaging [Gericke et al. Nat. Phys. 4, 949 (2008); Würtz et al. arXiv:0903.4837] 16
  17. 17. Conclusion • Tc calculated for entering an antiferromagnetically ordered state in mean field theory • Topological excitations give rise to a KT transition in 2D which significantly lowers Tc compared to MFT. • The imbalanced antiferrromagnet is a rich system ferro- and antiferromagnetic properties contains topological excitations models quantum magnetism, bilayers, etc. merons possess an internal Ising degree of freedom associated to Pontryagin index — possible application to topological quantum computation • Future work: include fluctuations beyond MFT for better accuracy in three dimensions investigate topological excitations in 3D (vortex rings) incorporate equilibrium in the NLσM gradient of n gives rise to a magnetization 17
  18. 18. Results • On-site free energy: Jz 2 f (n, m; B) = (n − m2 ) + m · B 2 ∙ µ ¶ µ ¶¸ 1 |BA | |BB | − kB T ln 4 cosh cosh 2 2kB T 2kB T where BA (B) = B − Jzm ± Jzn • Constraint equation: ∙ µ ¶ µ ¶¸ 1 BA |BA | BB |BB | m= tanh + tanh 4 |BA | 2kB T |BB | 2kB T • Critical temperature: Jzmz Tc = 2kB arctanh(2mz ) • Effective magnetic field below the critical temperature: B = 2Jzm 18
  19. 19. Anisotropic O(3) model • Dimensionless free energy of the anisotropic O(3) model [Klomfass et al, Nucl. Phys. B360, 264 (1991)] : X X βf3 = −β3 Si · Sj + γ3 (Si )2 z hi,ji i • KT transition: 1.0 0.8 g3êH1+g3L 0.6 0.4 0.2 0.0 1.0 1.2 1.4 1.6 1.8 2.0 b3 β3 • Numerical fit: γ3 (β3 ) = exp[−5.6(β3 − 1.085)]. β3 − 1.06 19
  20. 20. Analogy with the anisotropic O(3) model • Landau free energy: βJ X X βF = − ni · nj + β f (m, ni , β) 2 i hI,ji βJn2 X 2 X '− Si · Sj + βn γ(m, β) (Si )2 z 2 i hI,ji 3.0 Numerical fit parameter 2.5 2 Mapping of our model to 0.0 Anisotropic O(3) model: 2.0 β = 1/J 0.2 g Hm, bLêJ 1.5 Jβn2 0.4 β3 = 2 1.0 0.6 2β3 0.8 γ3 = γ(m, β) 0.5 Tc J 0.0 0.0 0.1 0.2 0.3 0.4 0.5 m 20

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