The Bethe Ansatz

These online lecture notes concern models of many-body quantum mechanics which have the peculiarity of allowing for an exact solution of the Schrödinger problem, at least in principle. The toys are spins, bosons and fermions, the rules are the postulates of quantum mechanics, and the playground is either a chain (one-dimensional lattice) or a line.

Various sections cover the main general branches of the field:

• the Coordinate Bethe Ansatz (CBA) for solving the Schrödinger problem
• the Thermodynamic Bethe Ansatz (TBA) for computing equilibrium properties
• the Algebraic Bethe Ansatz (ABA) as a unifying, higher-level framework
• methods to extract quantitative results from otherwise intractable expressions for e.g. correlation functions
• some emerging ideas like the Quench Action, as a variational extension to treat out-of-equilibrium situations

The material is organized by (class of) model, with the bosonic Lieb-Liniger gas and the isotropic Heisenberg spin-\(1/2\) antiferromagnetic chain at the forefront, especially in the CBA and TBA parts. Later sections are arranged in a more ad hoc opportune fashion.

Why should you care?

Exactly solvable models have been around since shortly after the birth of quantum mechanics. The Bethe Ansatz appeared in 1931 in Bethe's paper 1931.Bethe.ZP.71 describing the eigenstates of Heisenberg's model of ferromagnetism. Ironically, Bloch had earlier (in 1930.Bloch.ZP.61) proposed a correct form for the Heisenberg model wavefunctions. He however succumbed to some (lazy) oversimplifications and failed to provide a correct basis of eigenstates; the Bloch Ansatz was thus stillborn. Undeterred by logarithms and arctangents, Bethe carefully constructed and counted the eigenstates, showing that their number coincided with the dimensionality of Hilbert space. The devil, but also eternal recognition, was indeed hidden in the details, and Bethe here arguably got his first lasting result as a midwife.

Bethe's solution however quickly obtained a reputation for being too complicated: already in 1938, Fritz Sauter complained in 1938.Sauter.AP.425 that

This approximately translates to:

This sentiment persists to this day. When I was a young postdoc, a senior researcher confronted me in the corridor of the institute, displeased that I was focusing on (and perhaps too often expressing my predilection for) so-called "integrable models" of many-body physics. To him, I was simply wasting my time: these models were clearly exceptional (pathological even), fine-tuned, and most importantly totally divorced from physical reality. Studying them was a misdirected waste of effort, like medieval scholars asking how many angels can dance on the head of a pin.

There is no doubt that the study of the Bethe Ansatz requires handling complications which are rarely encountered in other simpler/simplified models. Sauter's assessment (and by extension the closed mind of my senior colleague) has however arguably been proven to be spectacularly wrong in the decades since it was made.

Today integrable models are used by a wider variety of researchers than ever. Their richness provides lasting delight to mathematicians. They provide much-needed confidence to theoretical physicists when exploring nonperturbative effects. They have also journeyed into the laboratories of condensed matter and atomic physics experimentalists, where some of their most spectacular features have been directly observed. Numerical methods have been battle-hardened by stringent tests from integrable models; the future development of out-of-equilibrium physics and of quantum information processing protocols and devices is certainly going to rely on exact solutions for orientation and certification.

Who knows what will come next; but as long as quantum mechanics stands (and there is no current indication that we are anywhere near abandoning that), integrable models will remain trustworthy lighthouses guiding the explorers of the stormy seas of many-body physics. This site is thus devoted to helping people see the treasures hidden in pinheads, and to befriend the many angel civilizations which are thriving in these seemingly boundless universes.