What’s New

This lecture series provides a systematic introduction to the Algebraic Bethe Ansatz (ABA), also known as the Quantum Inverse Scattering Method (QISM): a powerful framework for solving quantum integrable models exactly. We will begin with its basic construction, demonstrating how to obtain the exact spectra of paradigmatic models such as the Heisenberg spin chain, the Lieb-Liniger model, and the 6-vertex model. We will then explore the rich mathematical structure underlying the method, focusing on the central role of the Yang-Baxter equation and its profound connection to quantum group theory. If time permits, we will discuss extensions of the formalism to compute dynamical quantities, such as form factors and correlation functions, illustrating the full power of ABA as a tool for non-perturbative analysis in quantum theory.

The spectral gap of quantum many-body Hamiltonians is an important but difficult concept in condensed matter. Its identification is complicated and, quite often, controversial because the gap, together with the ground-state degeneracy is a thermdynamic limit notion rather than any finite-size energy splitting. In this talk, we will discuss a gaplessness indicator. Specifically, we prove that the ground state(s) of an SO(3)-symmetric gapped spin chain must be spin singlet(s), and the expectation value of a twisting operator asymptotically approaches unity in the thermodynamic limit, where finite-size corrections are inversely proportional to the system size. This theorem provides (i) supporting evidence for various conjectured gapped phases, and, contrapositively, (ii) a sufficient criterion for identifying gapless spin chains. We test the efficiency of our theorem by numerical simulations for a variety of spin models and show that it indeed offers a novel efficient way to identify gapless phases in spin chains with spin-rotation symmetry.