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4.2 The thermodynamic limit of the spin chain

In order to make contact with the spinning string solution discussed in Section  3, we will now consider the thermodynamic limit of the spin chain, in which the length L and the number of magnons M become large. This is necessary, as the classical string solutions only limit to the true quantum result in the BMN type limit J2,J --> oo with the filling fraction J2/J held fixed (here J2 = M and J = L). This thermodynamic groundstate solution of the gauge theory Bethe equations was found in [2216854], which we closely follow.

For this, it is useful to reexpress the Bethe equations (64View Equation) in terms of the Bethe roots uk related to the momenta via

1- pk- uk = 2 cot 2 (70)
for which the Bethe equations (64View Equation) and the momentum constraint (66View Equation) become
( ) ui + i/2 L M prod ui - uk + i M prod ui + i/2 -------- = -----------, -------- = 1 . (71) ui- i/2 k/=iui - uk - i i=1ui - i/2
The energy then is
sum M Q2 = ---1---. (72) i=1 ui2 + 14
The momentum constraint can be satisfied by considering symmetric root distributions of the form (ui,- ui,u*i,- u*i). The thermodynamic limit is now obtained by first taking the logarithm of (71View Equation)
(u + i/2 ) sum M (u - u + i) L ln -i------ = ln -i----k---- - 2 pi ni, (73) ui- i/2 k=1(k/=i) ui- uk- i
where ni is an arbitrary integer associated to every root ui. One self-consistently assumes that the momenta scale as pi ~ 1/L for L --> oo implying that the Bethe roots scale as ui ~ L. Therefore, in the L --> oo limit the above equation reduces to
1 2 M sum 1 -- = 2pnj + -- --------. (74) ui L k=1(k/=i)uj - uk
In the thermodynamic limit, the roots ui accumulate on smooth contours in the complex plane known as “Bethe strings,” which turn the set of algebraic Bethe equations into an integral equation. To see this, introduce the Bethe root density
M integral r(u) := 1-- sum d(u - u ) with du r(u) = 1, (75) M j C j=1
where C is the support of the density, i.e. the union of all Bethe string contours. Multiplying (74View Equation) with u i and introducing r(u), one arrives at the singular integral equation7
integral - dv r(v)u-= - 1--+ pnC(u)-u- where u (- C and a := M--. (76) C v- u 2a a L
The mode numbers n C(u) are integers, which are assumed to be constant on each smooth component C n of the density support C = U Cn in the complex plane. These integers and the distribution of components Cn select the numerous solutions to the continuum Bethe equations (76View Equation). Furthermore, the continuum energy now becomes
integral r(u) Q2 = M --2-. (77) C u
As was shown in [22], the gauge dual to the folded string solution of Section 3.2 corresponds to a two-cut support * C = C1 U C 1, with nC1 = - 1 and nC*1 = 1, sketched in Figure 5View Image.
View Image

Figure 5: Bethe root distribution for the gauge dual of the folded string. For large L the roots condense into two cuts in the complex plane.
The key trick to obtain analytical expressions for r(u) is to consider the analytic continuation to negative filling fraction b := -a: Then the two cuts * C1 U C 1 are mapped to intervals on the real line (C*1-- > [- b,-a] and C1 --> [a, b]) [22]. Then (76View Equation) may be brought into the compact form
integral b ~r(v)u2 1 p u integral b b - dv --------= --- ---- with dv ~r(v) = --, (78) a v2 - u2 4 2 a 2
using r(- v) = r(v) and defining ~r(v) := b r(v). In order to proceed, one introduces the resolvent
integral oo b ---v2--- a- sum 2k H(u) := dvr~(v) v2 - u2 = - 2 + Q2k u (79) a k=1
which gives rise to the infinite tower of conserved even charges Q2k with the energy E2 = -12 Q2 8p [8]. Across the cut u (- [a,b] the resolvent H(u) behaves as
a- 1- p- u- H(u ± ie) = - 2 + 4 - 2 u± ip 2 ~r(u), u (- [a,b], (80)
which one shows using the distributional identity --1- 1 x±ie = P (x)± ip d(x) and Equation (78View Equation). From this, one obtains an integral expression for the resolvent
integral V~ ------------------- a 1 b v2 (b2- u2)(a2- u2) H(u) = - 2-+ 4-- dv v2---u2- (b2---v2)(v2---a2) , (81) a
which in turn self-consistently yields the density
V~ ------------------- 2 integral b v2 (b2 - u2)(u2 - a2) r~(u) = ----- dv -------- ------------------ , (82) p u a v2 - u2 (b2 - v2)(v2 - a2)
Finally, the interval boundaries a and b are implicitly determined through the normalization and positivity conditions on ~r(u) (see [168]) by the relations
1 1 V~ ----- b2 - a2 --= 4K(q) , --= 4 1 - q K(q) , q := ----2-- . (83) a b b
The resolvent and the density may be expressed in closed forms using the elliptic integral of the third kind8
V~ -------- a 1 p 1 a2- u2[b2 (b2 - u2 )] H(u) = - --+ --- --u - --- -2----2- --- 4 u2 TT ---2---,q , 2 4 V~ --2-----4b b - u a b 1 u2 - a2 [b2 2 (b2 - u2 )] ~r(u) = 2-p-bu b2---u2- -a - 4 u TT --b2---,q . (84)
From this, it is straightforward to (finally) extract the energy eigenvalue Q2 of the two cut solution in the parametric form
[ ] --1- E2 = 2p2 K(q) 2E(q) - (2 - q) K(q) (85)
with
a = J2-= 1-- - V~ -1----E(q)-. (86) J 2 2 1 - q K(q)
This final result for the one-loop gauge theory anomalous scaling dimension can now be compared to the folded string energies of Section 3.2, Equations (34View Equation, 35View Equation). They do not manifestly agree, however, if one relates the auxiliary parameters q0 and q (see [16]) through
V~ ----- 2 q0 = - (1-- V~ -1---q)- (87) 4 1- q
one may show that
K(q ) = (1 - q)1/4K(q) , E(q ) = 1(1 - q)-1/4E(q) + 1(1- q)1/4 K(q) (88) 0 0 2 2
using elliptic integral modular transformations. Using these relations, the gauge theory result (85View Equation, 86View Equation) may be transformed into the string result of (34View Equation, 35View Equation). Hence, the one-loop gauge theory scaling dimensions indeed agree with the string prediction! The analysis of the circular string configuration goes along the same lines. Here, the Bethe roots turn out to condense on the imaginary axis. The root density is then symmetric along the imaginary axis, r(u) = r(- u), and remains constant along a segment [-c,c]. For u > c and u < - c it falls off towards zero. We shall not go through the detailed construction of the density for this configuration but refer the reader to the original papers [2216], being best explained in [8]. The outcome of this analysis is again a perfect matching of the energy eigenvalue of the spin chain with the circular string energy of Equations (42View Equation, 43View Equation).

As a matter of fact, one can go beyond this and match all the higher charges of gauge and string theory, as was shown for the first time in [8], by using an approach based on the Bäcklund transform.


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