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4.1 The coordinate Bethe ansatz

We now discuss the ansatz that enabled Bethe to diagonalize the Heisenberg model in 1931 [37Jump To The Next Citation Point]6. For this, we will drop the cyclicity constraint imposed on us from the underlying trace structure of the gauge theory operators for the moment, and treat a general non-cyclic, but periodic, spin chain. The vacuum state of the Heisenberg chain is then given by | |, ..., | >. Let |x1,x2,...,xJ > with x1 < x2 < ...< xJ denote a state of the chain with up-spins (magnons) located at sites xi, i.e. |1,3,4>L=5 = | |, |^ |, |, |^ >. It is useful to think of these spin flips as particles located at the sites xi. Note that the Hamiltonian preserves the magnon or particle number. The one magnon sector is then trivially diagonalized by Fourier transformation
L sum ip1x 2 p1- |y(p1)> := e |x>, with Q2 |y(p1)> = 4 sin ( 2 )|y(p1)> (52) x=1 sum L where Q2 = (1i,i+1 - Pi,i+1) (53) i=1
as ip -ip 2 2 - e - e = 4 sin (p/2). The periodic boundary conditions require the one-magnon momenta to be quantized p1 = 2p k/L with k (- Z.

Next, consider a general two-magnon state of the form

sum |y(p ,p )> = y(x ,x )| x ,x >. (54) 1 2 1 2 1 2 1<x1<x2<L
with a two-particle wave-function y(x1,x2). The “position space” Schrödinger equation following from sum L (1 - Pi,i+1) |y(p1, p2)> = E2 |y(p1,p2)> i=1 then leads to two sets of equations, depending on whether the particles lie next to each other or not:
x2 > x1 + 1 E2 y(x1, x2) = 2 y(x1,x2) - y(x1 - 1,x2)- y(x1 + 1,x2) + 2 y(x1,x2) - y(x1, x2- 1)- y(x1, x2 + 1) (55) x2 = x1 + 1 E2 y(x1, x2) = 2 y(x1,x2) - y(x1 - 1,x2)- y(x1,x2 - 1). (56)
E2 is the eigenvalue of Q2 and related to the gauge theory scaling dimensions as D = L + c2-E2 + O(c2) 8p. The above equations can be fulfilled by a superposition ansatz with an incoming and outgoing plane wave (Bethe’s ansatz)
y(x1, x2) = ei(p1x1+p2x2) + S(p2, p1)ei(p2x1+p1x2), (57)
where S(p1,p2) denotes the S-matrix of the scattered particles. Note that in the second term describing the scattered contribution, the two particles have simply exchanged their momenta. One easily sees that (55View Equation) is fulfilled for an arbitrary S(p2, p1) yielding the energy as a sum of one-particle energies
2 p1 2 p2 E2 = 4 sin (2-) + 4 sin (-2-). (58)
Equation (56View Equation) then determines the S-matrix to be of the form
f(p1) - f(p2) + i p S(p1, p2) = ----------------- with f(p) = 12 cot(2). (59) f(p1)- f(p2) - i
Note that S(p1, p2)-1 = S(p2,p2). This solves the infinite length chain. For a finite chain, the momenta p i are no longer arbitrary continuous quantities, but become discrete through the periodic boundary condition
y(x1, x2) = y(x2,x1 + L) . (60)
This, in turn, leads to the Bethe equations for the two magnon problem
ip1L ip2L e = S(p1,p2) and e = S(p2, p1) (61)
implying p1 + p2 = 2pm with an arbitrary integer m. The solutions of the algebraic equations (61View Equation) for p1 and p2 then determine the corresponding energies by plugging the resulting quasi-momenta pi into (58View Equation).

The magic of integrability now is that this information is all that is needed to solve the general M-body problem! This phenomenon is known as factorized scattering: The multi-body scattering process factorizes into a sequence of two-body interactions under which two incoming particles of momenta pi and pj scatter off each other elastically with the S-matrix S(pj,pi), thereby simply exchanging their momenta. That is, the M-body wave-function takes the form [3773]

M sum [ sum -i sum ] y(x1,...,xM ) = exp i pP(i) xi + 2 hP(i)P(j) (62) P (- Perm(M) i=1 i<j
where the sum is over all M ! permutations of the labels {1, 2,...,M } and the phase shifts h = - h ij ji are related to the S-matrix (59View Equation) by
S(pi,pj) = exp[ihij]. (63)
The M-magnon Bethe ansatz then yields the set of M Bethe equations
prod M eipkL = S(pk,pi) (64) i=1,i/=k
with the two-body S-matrix of (59View Equation) and the additive energy expression
sum M 2 pi E2 = 4 sin (2 ). (65) i=1
In order to reinstate the cyclicity of the trace condition, one needs to further impose the constraint of a total vanishing momentum
sum M p = 0 . (66) i i=1

As an example, let us diagonalize the two magnon problem exactly. Due to (66View Equation), we have p := p1 = - p2 and the Bethe equations (61View Equation) reduce to the single equation

p eipL = cot-2p-+-i= eip ==> eip(L- 1) = 1 ==> p = 2p-n--. (67) cot 2 - i L - 1
The energy eigenvalue then reads
( ) 2 2 -p-n-- L-->o o 2 n- E2 = 8 sin L - 1 - --> 8p L2 , (68)
which upon reinserting the dropped prefactor of c 8p2- yields the one-loop scaling dimension D(1) = c2 sin2(pn22-) p L of the two-magnon operators [1092]
J ( ) O(J,2)= sum cos p n 2p +-1- Tr(W Zp W ZJ -p). (69) n J + 1 p=0
In the BMN limit N, J --> oo with c/J 2 fixed, the scaling dimension takes the famous value D(1) = n2 c/J 2, corresponding to the first term in the expansion of the level-two energy spectrum of the plane-wave superstring V~ -----2-----2 Elight- cone = 1 + n c/J [33].

Hence, from the viewpoint of the spin chain, the plane-wave limit corresponds to a chain of diverging length L » 1 carrying a finite number of magnons M, which are nothing but the gauge duals of the oscillator excitations of the plane-wave superstring.

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