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1. **Title:** Quantum Phase Transitions in the Bose-Hubbard Model Explained 2. **Summary:** Quantum phase transitions i

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1. **Title:** Quantum Phase Transitions in the Bose-Hubbard Model Explained 2. **Summary:** Quantum phase transitions i

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  1. Physics 471 – Introduction to Modern Atomic Physics The Bose-Hubbard Model Lecturer: Lev Khaykovich

  2. Physics 471 – Introductoin to Modern Atomic Physics Quantum phase transitions S. Sachdev and B. Keimer, Physics Today 64 (2), 29 (2011) • Quantum phase transitions occur even at zero temperature. • They are a result of competition between different energies, and are initiated by quantum fluctuations. • Superfluid to Mott insulator transition is a (text book) example of the quantum phase transition. The two competing energies are kinetic and (on-site) interaction energies. • Bose-Hubbard model describes this phase transition and it is realizable with ultracold atoms. LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  3. Physics 471 – Introductoin to Modern Atomic Physics Optical lattices and Bloch bands Optical potential: R. Grimm, M. Weidemüller and Y. B. Ovchinnikov, Adv. At., Mol., Opt. Phys. 42, 95 (2000) Periodic potential is generated by overlapping two counterpropagating laser beams. −2?2 For a gaussian profile: ?2?sin ??2 ? ?,? = −?0? ? = 2?/? LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  4. Physics 471 – Introductoin to Modern Atomic Physics Optical lattices and Bloch bands Optical potential: R. Grimm, M. Weidemüller and Y. B. Ovchinnikov, Adv. At., Mol., Opt. Phys. 42, 95 (2000) Periodic potential is generated by overlapping two counterpropagating laser beams. −2?2 For a gaussian profile: ?2?sin ??2 ? ?,? = −?0? ? = 2?/? 2D potential is created by overlapping two optical standing waves: 3D potential is created by overlapping three optical standing waves: Bloch bands for different potneial depths: ℏ?2 2? I. Bloch, J. Dalibard, W. Zwerger, Rev. of Mod. Phys. 80, 885 (2008) ??= LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  5. Physics 471 – Introductoin to Modern Atomic Physics Bose Hubbard model † ??+? ??? ?? ? = − Hamiltonian: ( ?,? are nearest neighbour sites) 2 ?? ??− 1 − ? ?? ?? ? ? In the limit of tight-binding approximation (???= 0): the Hamoltonian is diagonal in occupation number basis |?⟩. 0=? Eigenvalues: ?? 2? ? − 1 − ?? 0 0= ??+1 0= ?? ?? requiring ?? Macroscopic degeneracy in the tight-binding limit: The nature of ground state is, thus, defined by the chemical potential ? : 1) If ? ∈ ](? − 1)?,??[ , the ground state is nondegenerate. 2) If ? = ??, the ground state is 2??degenerate (??is the number of sites) – having ? or ? − 1 particles are equally probable. LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  6. Physics 471 – Introductoin to Modern Atomic Physics Lifting the macroscopic degeneracy Staircase form of the ground state in the tight-binding limit: The first excited state is separated from the ground state by a gap: 0 0 0= ??+1 0= ? ∆? + ??−1 − 2?? ? − 1 ? + 1 vacuum When ???> 0 (small perturbation): 1) At ? = ?? points the macroscopic degeneracy is removed. 2) The obtained state is a superfluid (a superposition of all Each plateau shows vanishing compressibility: −1 possible degenerate states with reduced phase fluctuations). ?2? ??2 =?? ??≈ 0 ? = LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  7. Physics 471 – Introductoin to Modern Atomic Physics Mean-field theory of the Hubbard model †??⟶ ? + ?? †??+ ?? †?? + ⋯ The idea of mean-field approach: ?? †= ?? †− ?? †(???= ??− ??) and neglecting these fluctuations ??? †???≈ 0). (expressing the fluctuation field ??? ????= −???†+ ? +? 2 ? ? − 1 − ? ? Then, the effective one-site Hamiltonian is: ??= ????? where ? For uniform lattice with connectivity ? and nearest-neighbour hopping: ? = ?? ? where ? is the order paramter associated with Bose condensate. Solution: 1) Self-consistent iterative diagonalization while satisfying equation for ? 2) Perturbation theory analysis LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  8. Physics 471 – Introductoin to Modern Atomic Physics Mean-field theory: perturbative analysis ? ? + 1 ? − ?? First order correction to the eigenstates: | ⟩ ? = ⟩ ⟩ ⟩ |? − ? |? − 1 + |? + 1 ? ? − 1 − ? ? ? + 1 ? − ?? leads to the second order correction to the energies: ? ? ? = −? ? ? − 1 − ?+ ? ? + 1 ? − ?? ? = −??? ? ? − 1 − ?+ The self-consistency condition: ? −? ? ?− ? + 1 1 +? ? ?? ? leads to the critical boundary condition: ?= LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  9. Physics 471 – Introductoin to Modern Atomic Physics Superfluid to Mott insulator transition ? −? ? ?− ? + 1 1 +? ? ?? ? ?= Critical boundary condition (CBC): Phase diagram SF MI ? = 3 Mott gap MI Δ?? = ?+? − ?−(?) ? = 2 From the solution of CBC : MI ? = 1 2 ?? ? ?? ?+ 1 Δ?? = ? − 2 2? + 1 ?? ?→ 0 Δ?? ≈ ? ? ??? ≈ 5.8 LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  10. Physics 471 – Introductoin to Modern Atomic Physics Experimental demonstration M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch and I. Bloch, Nature 415, 39 (2002) 1) BEC of Rb is loaded into a 3D optical lattice potential (150,000 lattice sites are occupied) with 2.5 atoms/site in the center. 2) At ? = 0 BEC is released for a TOF. 3D interference pattern for different potneital depths: 3D interference pattern from a periodic array of phase coherent sources located at each site of optical lattice. ?) ?? ??????? b) 3 ?? c) 7 ?? d) 10 ?? e) 13 ?? f) 14 ?? g) 16 ?? h) 20 ?? LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  11. Physics 471 – Introductoin to Modern Atomic Physics Experimental demonstration I. Bloch, Phys. World 17, 25 (2004) superfluid Mott insulator LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  12. Physics 471 – Introductoin to Modern Atomic Physics Excitation spectrum: observation of the Mott gap M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch and I. Bloch, Nature 415, 39 (2002) Excitation gap with 1 atom/site Interferometric peak at different potential gradients. Mott insulator phase is largerly protected from excitation and is characterized by a sharp peak. Tunneling is allowed when potential gradient matches the on-site interaction energy. LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  13. Physics 471 – Introductoin to Modern Atomic Physics Experimental demonstration J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch and S. Kuhr, Nature 467, 68 (2010) Single-site resolution imaging of Mott Insulator phase: “wedding-cake” type structure. Only odd number of atoms survive. LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

  14. Physics 471 – Introductoin to Modern Atomic Physics Bose-Hubbard model related phenomena • Coherence, critical momentum (dynamics), number statistics, particle-hole admixture. • Low-dimensional systems: superfluid to Mott insulator transition in 1D and 2D systems • Fermi-Hubbard model • Magnetic phases of bosonic and fermionic mixtures on lattices • etc… LKh ultracool LKh ultracool Li LiLab Lab group @BIU group @BIU

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