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The Quantum many-body problem:

The Quantum many-body problem:. Less is more and more is different. Jorn Mossel University of Amsterdam, ITFA. Supervisor: Jean- Sébastien Caux. Talk outline. Introduction More is different Less is more Spin chain Heisenberg model Exact solutions with the Bethe Ansatz

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The Quantum many-body problem:

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  1. The Quantum many-body problem: Less is more and more is different. JornMossel University of Amsterdam, ITFA Supervisor: Jean-SébastienCaux

  2. Talk outline • Introduction • More is different • Less is more • Spin chain • Heisenberg model • Exact solutions with the Bethe Ansatz • Low energy behavior

  3. More is different* • 2-body problem solved:with Newton’s gravitation law • 3-body problem: no generalsolution is known. • Weak interactions: approximate methods • Bose Einstein Condensation • Low Temperature Superconductivity • Strong interactions: Problem! • High Temperature Superconductivity not understood ? *Philip Anderson (theoretical physicist)

  4. Less is more* • Low dimensional systems are usually strongly interacting: • In 1+1 dim: particles always interact when interchanging positions. • New phenomena • Often exactly solvable! *Robert Browning (English poet)

  5. Dynamics in 1+1 dimensions • Classical 2-body scattering: • Elastic scatterings • Conservation of total energyand momentum • Momenta are interchanged • Quantum 2-body scattering: wavefunctions can gain a phase shift!

  6. The Spin Chain

  7. Spin-spin interaction Pauli exclusion principle Coulomb repulsion Effective spin-flip

  8. Heisenberg model Werner Heisenberg Kinetic part Potential part Three cases Anti-aligned spins are preferred Down/up spins can move

  9. Bethe Ansatz • Wavefunction for downspins only • N-body scatterings are products of 2-body scatterings • Bethe Ansatz: Hans Bethe Free particle wavefunctions Sum of all M! permutations of the momenta. Coefficient related to the scattering phases. Wavefunction for M downspins

  10. Bethe Ansatz equations • Periodic boundary conditions:momenta are restricted Quantum numbers: half-odd integers/ integers Scattering phase

  11. Low energy excitations • Excitations are Solitons: • Localizable objects • Permanent shape • Emerge unchanged after scattering Groundstate k2 k1 Spin flip

  12. Artist’s Impression

  13. Low Energy spectrum: N=100

  14. Algebraic Bethe Ansatz • Problems with the Bethe Ansatz • Wavefunctions can not be normalized • inconvenient for further calculations • Solution: Algebraic Bethe Ansatz • Wavefunctions in terms of operators: Creates a downspin with momentum k1. State with all spins up.

  15. From theory to experiment • Correlation function: • Use a computer to calculate this. • Inelastic neutron scattering data corresponds with the correlation functions. Probability: GS -> M-1 downspins

  16. Summary and Conclusion • Quasi-one dimensional system • Heisenberg model • Low energy spectrum Correlation functions • Quantitative predictions for experiments Spin-spin interaction Algebraic Bethe Ansatz Bethe Ansatz Computer

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