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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: 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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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Strongly interacting fermions in optical lattices

  • 1. Strongly interacting fermions in optical lattices PhD thesis talk Arnaud Koetsier
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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