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Less is more and more is different. Jorn Mossel University of Amsterdam, ITFA Supervisor: Jean-Sébastien Caux
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Talk outline Introduction More is different Less is more Spin chain Heisenberg model Exact solutions with the Bethe Ansatz Low energy behavior
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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)
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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)
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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!
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Spin-spin interaction Pauli exclusion principle Effective spin-flip Coulomb repulsion
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Heisenberg model Kinetic part Potential part Anti-aligned spins are preferred Down/up spins can move Werner Heisenberg Three cases
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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
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Bethe Ansatz equations Periodic boundary conditions: momenta are restricted Quantum numbers: half- odd integers/ integers Scattering phase
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Low energy excitations Excitations are Solitons: Localizable objects Permanent shape Emerge unchanged after scattering k1k1 k2k2 Groundstate Spin flip
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Artist’s Impression
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Low Energy spectrum: N=100
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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.
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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
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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
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