1. Less is more and more is different. Jorn Mossel University of Amsterdam, ITFA Supervisor: Jean-Sébastien Caux

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 general solution 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 energy and momentum Momenta are interchanged Quantum 2-body scattering: wavefunctions can gain a phase shift!

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

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

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

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 k1k1 k2k2 Groundstate 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 k 1. 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

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