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.

In condensed matter physics, various physical phenomena can be effectively described using Green’s functions, typically corresponding to non-Hermitian Hamiltonians. Recent advancements in non-Hermitian physics have offered a fresh perspective to condensed matter physics, leading to exploration of non-Hermitian self-energies with intricate structures. One intriguing non-Hermitian phenomenon is the non-Hermitian skin effect, characterized by the nonreciprocal propagation of wave packets within the system and the accumulation of bulk states at open boundaries. In this talk, I will discuss the implementation and control of the non-Hermitian skin effect in mesoscopic electronic systems. Specifically, the conventional, spin-, and valley-resolved non-Hermitian skin effect can be realized through complex band engineering in mesoscopic heterostructures, resulting in nonreciprocal electron transport phenomena. This effect holds promise for the development of robust electronic devices, such as valley filters. Additionally, the application of a magnetic field can suppress the skin effect, providing an effective means for its control. In the second part of the talk, we will introduce a general framework for non-Hermitian spontaneous symmetry breaking. The transition of energy spectra from real to complex values, together with the accompanying spontaneous symmetry breaking of eigenstates, is one of the central topics in non-Hermitian physics. However, a complete and universal theory that characterizes the properties of individual energy levels has been lacking. By employing the complex path integral formalism and developing a generalized Gutzwiller trace formula, we establish a universal quantum-classical correspondence that precisely connects the properties of single energy levels to the symmetries of their corresponding semiclassical orbits. This physical mechanism applies broadly to systems with pseudo-Hermitian symmetries and provides practical insights for the precise control of non-Hermitian phenomena.