### 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 and the number of magnons become large. This is necessary, as the classical string solutions only limit to the true quantum result in the BMN type limit with the filling fraction held fixed (here and ). 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 (64) in terms of the Bethe roots related to the momenta via

for which the Bethe equations (64) and the momentum constraint (66) become
The energy then is
The momentum constraint can be satisfied by considering symmetric root distributions of the form . The thermodynamic limit is now obtained by first taking the logarithm of (71)
where is an arbitrary integer associated to every root . One self-consistently assumes that the momenta scale as for implying that the Bethe roots scale as . Therefore, in the limit the above equation reduces to
In the thermodynamic limit, the roots 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
where is the support of the density, i.e. the union of all Bethe string contours. Multiplying (74) with and introducing , one arrives at the singular integral equation
The mode numbers are integers, which are assumed to be constant on each smooth component of the density support in the complex plane. These integers and the distribution of components select the numerous solutions to the continuum Bethe equations (76). Furthermore, the continuum energy now becomes
As was shown in [22], the gauge dual to the folded string solution of Section 3.2 corresponds to a two-cut support , with and , sketched in Figure 5.
The key trick to obtain analytical expressions for is to consider the analytic continuation to negative filling fraction : Then the two cuts are mapped to intervals on the real line ( and ) [22]. Then (76) may be brought into the compact form
using and defining . In order to proceed, one introduces the resolvent
which gives rise to the infinite tower of conserved even charges with the energy [8]. Across the cut the resolvent behaves as
which one shows using the distributional identity and Equation (78). From this, one obtains an integral expression for the resolvent
which in turn self-consistently yields the density
Finally, the interval boundaries and are implicitly determined through the normalization and positivity conditions on (see [168]) by the relations
The resolvent and the density may be expressed in closed forms using the elliptic integral of the third kind
From this, it is straightforward to (finally) extract the energy eigenvalue of the two cut solution in the parametric form
with
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 (34, 35). They do not manifestly agree, however, if one relates the auxiliary parameters and (see [16]) through
one may show that
using elliptic integral modular transformations. Using these relations, the gauge theory result (85, 86) may be transformed into the string result of (34, 35). 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, , and remains constant along a segment . For and 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 (42, 43).

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.