Download 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
 Low energy behavior
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)
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)
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!
Spin-spin interaction
Pauli exclusion principle
Coulomb repulsion
Effective spin-flip
Heisenberg model
Werner Heisenberg
N
H  J  (S S
j 1

j

j 1
Kinetic part
Down/up spins can move

j
S S

j 1
)  S S
z
j
z
j 1
Potential part
Anti-aligned spins are preferred
S N 1  S1
Three cases
 1
 1
 1
Bethe Ansatz
 Wavefunction for downspins only
Hans Bethe
 N-body scatterings are products of 2-body scatterings
 Bethe Ansatz:
M!
M
 M ( x1  xM )   AP  1 ( xi , k P )
P
Wavefunction
for M downspins
Sum of all M!
permutations of
the momenta.
i 1
Coefficient
related to the
scattering
phases.
i
Free particle
wavefunctions
Bethe Ansatz equations
 Periodic boundary conditions:
momenta are restricted
 M ( x1  xM )   M ( x2  xM , x1  N )
M 1
N k j  2 I j  2 (k j , kl )
l j
Quantum numbers: halfodd integers/ integers
k k


 sin( 1 2 )


2

2 (k1 , k2 )  2 arctan 
k

k
k

k
 cos( 1 2 )   cos( 1 2 ) 


2
2


Scattering phase
Low energy excitations
Groundstate
k1
Spin flip
 Excitations are Solitons:
 Localizable objects
 Permanent shape
 Emerge unchanged after scattering
k2
Artist’s Impression
Low Energy spectrum: N=100
2
4
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:
M  B(k1 ) B(k 2 )  B(k M )   
Creates a downspin
with momentum k1.
State with all
spins up.
From theory to experiment
 Correlation function:
S (q,  )  2   S GS  (  E  EGS )



q
2
Probability: GS -> M-1 downspins
 Use a computer to calculate this.
 Inelastic neutron scattering data corresponds with the
correlation functions.
Summary and Conclusion
 Quasi-one dimensional system
Spin-spin interaction
 Heisenberg model
Algebraic Bethe Ansatz
Bethe Ansatz
 Low energy spectrum
Correlation functions
Computer

Quantitative predictions for experiments