Numerical String Hypothesis and Wronskian Bethe Equations for su(N) Spin Chains

The Bethe ansatz provides an exact solution framework for integrable quantum spin chains, yet the detailed structure of its solutions remains subtle. In this work, we solve the Bethe ansatz with SU(N) global symmetry numerically from two perspectives. First, we systematically investigate approximate solutions known as strings, derived from the so-called string hypothesis. These solutions can be labeled by rigged configurations. Second, we compute physically admissible Bethe roots using the Wronskian Bethe equations, whose solutions admit a natural labeling by standard Young tableaux (SYT). These two labeling schemes, SYT and rigged configurations, arise independently in combinatorics. A key combinatorial result, the Kerov-Kirillov-Reshetikhin (KKR) bijection, establishes a one-to-one correspondence between them. While this bijection is well understood in a purely combinatorial setting, its role within the physics of integrable systems has not been systematically examined. To investigate the KKR bijection within integrable spin chains, we perform numerical computations of Bethe roots for su(N) chains up to length L=20. Remarkably, our numerical results show that the SYT labeling of solutions obtained from the Wronskian Bethe equations closely matches the rigged configuration labeling prescribed by the KKR bijection when compared using a nearest neighbor distance measure. Since the two labeling schemes were developed independently and, in particular, the SYT labeling does not rely on the string hypothesis, this correspondence reveals a deep structural relation between the combinatorial and analytic descriptions of integrable systems.