The Bethe ansatz after 75 years

Abstract

Bethe ansatz, which is Hans Bethe's solution to a quantum mechanical model, is finding application in various subjects ranging from superconductors to string theory after 75 years of its discovery. The Bethe ansatz began as a solution to the Heisenberg model, which is a 1D array of quantum mechanical spin-1/2 particles at fixed locations. The Bethe ansatz provides complete solution to the 1D model of interacting spin-less bosons in which the interaction strength, parameterization of the relative magnitudes of the kinetic and potential energy terms are variable. Applications of the Bethe Ansatz extend to systems such as ice model, a 2D model in statistical mechanics, which are seemingly unrelated to the 1D problem in quantum mechanics that Bethe originally considered. The Bethe ansatz has led to some developments in the field of combinatorics such as components of the ground-state wave-function for a variant of the Heisenberg spin chain can be used to count combinatorial objects called alternating sign matrices.