Hans Bethe solved the linear chain Heisenberg model Born July 2, 1906 Strassburg, Germany Died March 6, 2005 Ithaca, NY, USA Residence USA Nationality American Field Physicist Institution University of Tübingen Cornell University Alma mater University of Frankfurt University of Munich Academic advisor Arnold Sommerfeld Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette Zeitschrift für Physik A, Vol. 71, pp. 205-226 (1931) 1 1

Symmetries 2 2

1 reversed spin in the chain Ground state for J>0: all spins up, no spin can be raised, and so no shift can occur 1 reversed spin in the chain 3 3

We must insert the periodic boundary condition for the N-spin chain One- Magnon solution: the spin wave is a boson (spin 1) Two- Magnon solution: scattering, bound states 4 4

Two Magnons a configuration is denoted by |n1,n2>, n1<n2 reversed spins provided that adiacent sites are up spins: if the reversed spins are close the energy is different and this is an effective interaction. 5 5

Two independent magnons? NO! New k1 and k2 must be found since PBC do not separately apply to n1 and n2 (you cannot translate one magnon across the other). 6 6

The Schrödinger equation must fix q The Schrödinger equation must fix q . This phase shift must contain the magnon interaction.

Consider the solution with eigenvalue E. 9 9

The two contributions must enter with the same probability, A and B The degenerate solution with the k1 and k2 exchanged and must be included for generality and also to satisfy the boundary conditions, as I show in a moment; so one should write f(n1, n2) = A e i(k1n1+k2n2) + B e i(k2n1+k1n2) , 0 < n1 < n2 < N. The two contributions must enter with the same probability, A and B differ by a phase a and therefore Two phases are too many. Next we show that a = q 10 10

We must be free to decide that the numbering of spins runs in the range n2,…n2+N-1 rather than 1,…N; when we do, n2<n1+N but the wave function does not care: 11

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Both equations must be true! In summary, Both equations must be true! 14 14

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Before putting this equation in real form we check that this is really a phase factor. 18 18

Real form of Bethe Ansatz equation for q 19 19

Real form of Bethe Ansatz equation for q

The Bethe Ansatz equations can be solved for k1, k2 and θ by combined analytic and numerical techniques for N = some tens. Most solutions are scattering states with real momenta and a finite θ; however when one of the Bethe quantum numbers λ vanishes, the corresponding k and θ also vanish, and there is no interaction. Other solutions with |λ1 - λ2|=1 have complex momenta with k1 = k2∗ and represent magnon bound states in which the two flipped spins propagate at a close distance. 21

Rossi: uno dei l nullo : non c’e’ interazione Bianchi l2 – l1 >= 2 scattering states Blu: alcuni sono stati legati Da Karbach et al. Cond-mat9809162 22 22

Three Magnons: here the real fun begins! Basis: a configuration is denoted by |n1, n2, n3>, n1<n2<n3 provided that adiacent sites are up spins: this is an effective interaction 23 23

and all must enter with same probability  mutual phase shifts: But we must account for 3! permutations P of 3 objects: (123),(231),(312),(213),(132),(321) and all must enter with same probability  mutual phase shifts: 24 24

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Let us go on with the same change of the numbering

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3 independent magnons? NO! Interaction produces phase shift and E is not the sum of independent magnon energies 29

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The special recurrence formula for the amplitude f accounting for the hindered jumps becomes: 31 31

With 3 magnons, many more terms, but direct extension of rules. 33

Many Magnons (many reversed spins) One tries with a product of magnons and everything generalizes! Basis : |n1,n2,..nr> in growing order :sites with down spins

to be solved with

However, without the ingenuity of Hans Bethe, the solution The Bethe equations are quite hard to solve for many flipped spins in large systems. In recent years solutions have been obtained which are far from trivial; among others, bound states of several magnons have been reported. The Bethe Ansatz is a rare example of exact solution of an interacting many-body problem; it keeps the same form independent of the size of the system. The root of the (relative) simplicity that allows this solution is one-dimensionality: the evolution does not allow overtakings, and spins always keep a fixed order. However, without the ingenuity of Hans Bethe, the solution could have remained undetected; who knows how many important model problems are solvable, but still unsolved.

How magnons arise Holstein-Primakov transformation to boson operators

This holds on every site of the solid:

AF phase only Predicted spin flip excitation in NiO exchange field of 0.25 eV,

Some Developments: theoretical physics in 1d Hultén (1938) modeled 1d antiferromagnet Lieb-Liniger (1963) solved exactly 1d spin 0 Bose gas Lieb-Wu (1968) 1d Hubbard model (exact solution, no magnetism) Wiegmann-Tsvelik solution of Anderson model (with assumptions that V does not depend on k, U is small, level close to Fermi level): as long as the Anderson model is taken to be spherically symmetric, it is 1d. Integrable systems-statistical mechanics, string theory…