The Quantum many-body problem:
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The Quantum many-body problem: Less is more and more is different. Jorn Mossel University of Amsterdam, ITFA Supervisor: Jean- Sébastien Caux
description
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 - PowerPoint PPT Presentation
Transcript of The Quantum many-body problem:
The Quantum many-body problem:Less is more and more is different.
Jorn MosselUniversity of Amsterdam, ITFA
Supervisor: Jean-Sébastien Caux
Talk outlineIntroduction
More is differentLess is more
Spin chainHeisenberg modelExact solutions with the Bethe AnsatzLow energy behavior
More is different*2-body problem solved:
with Newton’s gravitation law3-body problem: no general
solution is known.
Weak interactions: approximate methodsBose Einstein CondensationLow Temperature Superconductivity
Strong interactions: Problem!High Temperature Superconductivity not
understood
?
*Philip Anderson (theoretical physicist)
Less is more*Low dimensional systems are usually strongly
interacting:In 1+1 dim: particles always interact when
interchanging positions.New phenomenaOften exactly solvable!
*Robert Browning (English poet)
Dynamics in 1+1 dimensionsClassical 2-body scattering:
Elastic scatteringsConservation of total energy
and momentum
Momenta are interchanged
Quantum 2-body scattering: wavefunctions can gain a phase shift!
The Spin Chain
Spin-spin interaction
Pauli exclusion principle
Effective spin-flip
Coulomb repulsion
Heisenberg model
zj
zjjjj
N
jj SSSSSSJH 111
1
)(
Kinetic part
Potential partAnti-aligned spins are preferred
Down/up spins can move
11 SSN
Werner Heisenberg
Three cases
111
Bethe AnsatzWavefunction for downspins onlyN-body scatterings are products of 2-body
scatteringsBethe Ansatz:
),()(1
1
!
1 iP
M
ii
M
PPMM kxAxx
Wavefunction for M downspins
Sum of all M! permutations of the momenta.
Coefficient related to the scattering phases.
Free particle wavefunctions
Hans Bethe
Bethe Ansatz equationsPeriodic boundary conditions:
momenta are restricted
),()( 121 Nxxxxx MMMM
),(21
2 lj
M
jljj kkIkN
)2
cos()2
cos(
)2
sin(arctan2),(
2121
21
212 kkkk
kk
kk
Quantum numbers: half-odd integers/ integers
Scattering phase
Low energy excitations
Excitations are Solitons:Localizable objectsPermanent shapeEmerge unchanged after scattering
k1 k2
Groundstate
Spin flip
Artist’s Impression
Low Energy spectrum: N=1002 4
Algebraic Bethe AnsatzProblems with the Bethe Ansatz
Wavefunctions can not be normalizedinconvenient for further calculations
Solution: Algebraic Bethe AnsatzWavefunctions in terms of operators:
)()()( 21 MM kBkBkB
Creates a downspin with momentum k1.
State with all spins up.
From theory to experimentCorrelation function:
Use a computer to calculate this.Inelastic neutron scattering data corresponds
with the correlation functions.
)(2),(2
GSq EEGSSqS
Probability: GS -> M-1 downspins
Summary and ConclusionQuasi-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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