Strongly interacting fermions
      in optical lattices
         PhD thesis talk




        Arnaud Koetsier
Outline


• Topics covered in this talk:
    Chapter 2: introduction to Hubbard and Heisenberg models
    Chapter 3: cooli...
Introduction

• Fermions in an optical lattice
      • Described by the Hubbard model
      • Realised experimentally [Ess...
Fermi-Hubbard Model


                   P P                            P
      H = −t                    c† cj 0 ,σ
     ...
Quantum Phases of the Fermi-Hubbard Model


          • Positive U (repulsive on-site interaction):


                    ...
Mott insulator: Heisenberg Model

• At half filling, when U À t and kB T ¿ U we are deep in the Mott phase.
      hopping ...
Néel State


• The Néel state is the antiferromagnetic ground state for J > 0




• Néel order parameter 0 ≤ h|n|i ≤ 0.5
 ...
How to reach the Néel state: Step 1


 Start with trapped 2-component fermi gas of cold atoms. The entropy is:

          ...
How to reach the Néel state: Step 2


Adiabatically turn on the optical lattice. We enter the Mott phase: 1 particle per s...
How to reach the Néel state: Step 3

Final temperature is below Tc : we are in the Néel state. The entropy remained
consta...
Mean-field theory in the lattice

• Mott insulator:
                                  J X
                               H...
Mean-field theory results

• Entropy in the lattice:

                ln(2)                             Mott
             ...
Improved 2-site mean-field theory

• Improve on “single-site” mean-field theory by including the interaction between two
 ...
Heisenberg Model with imbalance

• Until now, N↑ = N↓ = N/2
• What happens if N↑ 6= N↓ — spin population imbalance?
• This...
Mean field analysis

• J > 0 ⇒ ground state is antiferromagnetic (Néel state)
    Two sublattices: A, B

                 ...
Phase Diagram in three dimensions

         1.5
                                    n=0
                                  ...
Spin waves (magnons)
                                                   dS  i
  • Spin dynamics can be found from:        ...
Long-wavelength dynamics: NLσM

• Dynamics are summarised a non-linear sigma model with an action
            Z     Z     ...
Topological excitations

• The topological excitaitons are vortices; Néel vector has an out-of-
  easy-plane component in ...
Meron size

• Core size λ of meron found by plugging the spin texture into F [n(x), m]
  and minimizing (below Tc):
      ...
Meron pairs

Low temperatures:
     A pair of merons with opposite vorticity, has a finite energy since
     the deformati...
Kosterlitz-Thouless transition

• The unbinding of meron pairs in 2D signals a KT transition. This
  drives down Tc compar...
Experimental feasibility

• Experimental realisation:
    Néel state in optical lattice: adiabatic cooling
    Imbalance: ...
Conclusion
• Néel state appears to be is experimentally achievable (just) by adiabatically
  ramping up the optical lattic...
Free energy and Tc for imbalanced AFM

• On-site free energy:
                Jz 2
 f (n, m; B) =     (n − m2 ) + m · B
  ...
Finding KT transition: Anisotropic O(3) model

• Dimensionless free energy of the anisotropic O(3) model [Klomfass et al, ...
Finding KT transition: Analogy with the anisotropic O(3) model


• Landau free energy:
                βJ X              X...
2-Site Mean-Field Theory


• Improve on standard mean-field approach by including 2 sites exactly:

            H = JS1 · ...
2-site mean-field theory: Néel order parameter

                                               0.5
Comparison with 1-site ...
Entropy in the lattice: Three temperature regimes

• High T: 2-site mean field theory result
                             ...
Entropy in the lattice, T>Tc


•Function with the correct properties above Tc:

                               ∙µ         ...
Entropy in the lattice, T<Tc


• Function for with the correct properties below Tc:

                    ∙µ            ¶κ ...
Entropy in the lattice: Coefficients for T<Tc


• Result:
                                 µ                              ...
Maximum number of particles in the trap

For smooth traps, tunneling is not site-dependent, overfilling leads to
         ...
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Strongly interacting fermions in optical lattices

  1. 1. Strongly interacting fermions in optical lattices PhD thesis talk Arnaud Koetsier
  2. 2. Outline • Topics covered in this talk: Chapter 2: introduction to Hubbard and Heisenberg models Chapter 3: cooling into the Néel state Chapter 4: imbalanced antiferromagnets in optical lattices. • Topics in the thesis but not covered in this talk: Chapter 1: Introduction (mainly historical) Chapter 6: BEC-BCS crossover for fermions in an optical lattice Chaoter 7: Analogy of the BEC-BCS crossover for bosons Juicy details 2
  3. 3. 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 — How to achieve the Néel state in an optical lattice? • Imbalanced Fermi gases • Experimentally realised [Ketterle ’06, Hulet ’06] • High relevance to other areas of physics (particle physics, neutron stars, etc.) — How does imbalance affect the Néel state? 3
  4. 4. 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 4
  5. 5. 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. 5
  6. 6. Mott insulator: Heisenberg Model • At half filling, when U À t and kB T ¿ U we are deep in the Mott phase. hopping is energetically supressed only spin degrees of freedom remain (no transport) • Integrate out the hopping fluctuations, then the Hubbard model reduces to the simpler 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 6
  7. 7. Néel State • 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 7
  8. 8. How to reach the Néel state: Step 1 Start with trapped 2-component fermi gas of cold atoms. The entropy is: T 2 SFG = N kB π TF Total number of particles: N Fermi temperature in the harmonic trap: kB TF = (3N )1/3 ~ω Trapping potential: 1 V = mω 2 r2 2 8
  9. 9. How to reach the Néel state: Step 2 Adiabatically turn on the optical lattice. We enter the Mott phase: 1 particle per site. Entropy remains constant: temperature changes! 9
  10. 10. How to reach the Néel state: Step 3 Final temperature is below Tc : we are in the Néel state. The entropy remained constant throughout. To reach the Néel state: Prepare the system so that the initial entropy in the trap equals the final entropy below Tc in the lattice: SFG (Tini ) = SLat (T ≤ Tc ) What is the entropy of the Néel state in the lattice? 10
  11. 11. Mean-field theory in the lattice • Mott insulator: J X H= S j · Sk 2 hjki • Perform a mean-field analysis about the equilibrium value of the staggered magnetisation hni : J X½ ¾ H' (−1)i nSj + (−1)j nSi − J n2 2 hiji ∙ ¸ Jz 2 1 β|n|Jz 1 • Landau free energy: fL (n) = n − ln cosh( ) − ln(2). 2 β 2 β Self-Consistency: Entropy: ¯ ∂fL (n) ¯ ¯ hni ∂fL (hni) =0 S = −N ∂n ¯n=hni ∂T 11
  12. 12. Mean-field theory results • Entropy in the lattice: ln(2) Mott Lattice depth 6ER , 0.6 Tc = 0.036 TF Cooling 0.4 S/NkB l Né e 0.2 Lattice Entropy Trap Entropy 0.0 0.02 0.04 0.06 0.08 T/TF Heating kB Tc = 3J/2 12
  13. 13. Improved 2-site mean-field theory • Improve on “single-site” mean-field theory by including the interaction between two sites exactly: ln(2) • Incorporate correct Mott 0.6 low-temperature and critical behaviour Example: B 0.4 N éel S/Nk 40 K atoms with a lattice depth of 8ER : 0.2 Lattice, MFT Tc = 0.012 TF Lattice, fluc. Trap 0.0 Tini = 0.059 TF 0.02 0.04 0.06 0.08 T/T F 13
  14. 14. Heisenberg Model with imbalance • Until now, N↑ = N↓ = N/2 • What happens if 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 14
  15. 15. Mean field analysis • J > 0 ⇒ ground state is antiferromagnetic (Néel state) Two sublattices: A, B A(B) A(B) A(B) • Linearize the 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) 15
  16. 16. 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 16
  17. 17. 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 17
  18. 18. 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). 18
  19. 19. 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 19
  20. 20. 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 • At low temperatures, the energy of a single meron diverges logarithmically with the system area A as Jn2 π A ln 2 2 πλ merons must be created in pairs. 20
  21. 21. 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 21
  22. 22. 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)] ) 22
  23. 23. Experimental feasibility • Experimental realisation: Néel state in optical lattice: adiabatic cooling Imbalance: drive spin transitions with RF field • 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] 23
  24. 24. Conclusion • Néel state appears to be is experimentally achievable (just) by adiabatically ramping up the optical lattice – accurate determination of initial T is crucial. • The imbalanced antiferrromagnet is a rich system spin-canting below Tc ferro- and antiferromagnetic properties, topological excitations in 2D, KT transition significantly lowers Tc compared to MFT results models quantum magnetism, bilayers, etc. possible application to topological quantum computation and information: merons possess an internal Ising degree of freedom associated to Pontryagin index • Future work: incorporate equilibrium in the NLσM action gradient of n gives rise to a magnetization (can possibly be used to manipulate topological excitations) effect of imbalance on initial temperature needed to achieving the canted Néel state by adiabatic cooling 24
  25. 25. Free energy and Tc for imbalanced AFM • 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 25
  26. 26. Finding KT transition: 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 26
  27. 27. Finding KT transition: 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 27
  28. 28. 2-Site Mean-Field Theory • Improve on standard mean-field approach by including 2 sites exactly: H = JS1 · S2 + J(z − 1)|n|(Sz − Sz ) + J(z − 1)n2 1 2 • First Term: Treats interactions between two neighboring sites exactly, 1 2 • Second Term: Treats interactions between other neighbors within mean-field theory 28
  29. 29. 2-site mean-field theory: Néel order parameter 0.5 Comparison with 1-site mean- field theory: 0.4 • Depletion at zero temperature due to quantum fluctuations 0.3 〈 n〉 0.2 0.1 2-site 1-site 0 0 0.5 1 1.5 k BT/J • Lowering of Tc: kB Tc ' 1.44 J 29
  30. 30. Entropy in the lattice: Three temperature regimes • High T: 2-site mean field theory result ∙ 2 ¸ 3J S(T À Tc ) = N kB ln(2) − 2 64kB T 2 • Low T: entropy of magnon gas µ ¶3 4π 2 k T S(T ¿ Tc ) = N kB √B 45 2 3Jhni • Intermediate T: non-analytic critical behaviour T − Tc S(T = Tc ) = S(Tc ) ± A ± |t|dν−1 t= → 0± Tc Where, from renormalization group theory [Zinn-Justin] d = 3, ν = 0.63, A+/A− ' 0.54 •Tc: from quantum Monte-Carlo [Staudt et al. ’00]: Tc = 0.957J/kB 30
  31. 31. Entropy in the lattice, T>Tc •Function with the correct properties above Tc: ∙µ ¶κ ¸ S(T ≥ Tc ) T − Tc κTc ' α1 −1+ + ln(2) NkB T T First term: Critical behavior Other terms: To retrieve correct high-T limit of 2-site theory. → Found by expanding critical term and subtracting all terms of lower order than in T than high-T expression, which is ∼ 1/T 2 . •Result: 3J 2 κ = 3ν − 1 ' 0.89 α1 = 2 2 (32κ(κ − 1)kB Tc ) 31
  32. 32. Entropy in the lattice, T<Tc • Function for with the correct properties below Tc: ∙µ ¶κ ¸ S(T ≤ Tc ) Tc − T T κ(κ − 1) T 2 T3 T4 = −α2 −1+κ − 2 + β0 3 + β1 4 N kB Tc Tc 2 Tc Tc Tc First term and last term: Critical behavior and continuous interpolation with T>Tc result. Other terms: Retrieve low-T behavior of magons, again found by expanding critical term and subtracting all terms of lower order than T 3. 32
  33. 33. Entropy in the lattice: Coefficients for T<Tc • Result: µ ¶ 6 4π 2 kB Tc 3 3 α2 = √ − α1 (κ − 1) + β1 − ln(2) (κ − 1)(κ − 2)(κ − 3) 135 3J 3 µ 2 3 3 ¶ κ 4π kB Tc β0 = √ + α1 (κ − 1) − β1 + ln(2) (κ − 3) 45 3κJ 3 + − 2 6(A /A + 1) + κ(κ − 5) 4π 2 kB Tc 3 3 β1 = ln 2 − J 2 2 − √ 64κkB Tc A+ /A− 135 3J 3 (same as high-T expression): 3J 2 κ = 3ν − 1 ' 0.89 α1 = 2 2 (32κ(κ − 1)kB Tc ) 33
  34. 34. Maximum number of particles in the trap For smooth traps, tunneling is not site-dependent, overfilling leads to double occupancy: Destroys Mott- insulator state in the centre! The trap limits the number of particles to avoid double occupancy: µ ¶3/2 4π 8U N ≤ Nmax = Example: 3 mω2 λ2 40 K atoms with a lattice depth of 8ER and λ = 755 nm ⇒ Nmax ' 3 × 106 34

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