Current Operators in Bethe Ansatz and Generalized Hydrodynamics: An Exact Quantum-Classical Correspondence

Generalized hydrodynamics is a recent theory that describes large-scale transport properties of one-dimensional integrable models. It is built on the (typically infinitely many) local conservation laws present in these systems and leads to a generalized Euler-type hydrodynamic equation. Despite the successes of the theory, one of its cornerstones, namely, a conjectured expression for the currents of the conserved charges in local equilibrium, has not yet been proven for interacting lattice models. Here, we fill this gap and compute an exact result for the mean values of current operators in Bethe ansatz solvable systems valid in arbitrary finite volume. Our exact formula has a simple semiclassical interpretation: The currents can be computed by summing over the charge eigenvalues carried by the individual bare particles, multiplied with an effective velocity describing their propagation in the presence of the other particles. Remarkably, the semiclassical formula remains exact in the interacting quantum theory for any finite number of particles and also in the thermodynamic limit. Our proof is built on a form-factor expansion, and it is applicable to a large class of quantum integrable models.

Popular Summary

The world of elementary particles is governed by the laws of quantum mechanics. Nevertheless, Newtonian mechanics (along with statistical physics and hydrodynamics built on it) are correct theories for large systems. How can we connect the two frameworks, and how can we derive the classical theories from the underlying quantum laws? We contribute to this subject by looking at special quantum models, which are 1D exactly solvable models. We show that in these models the flows of certain physical quantities can be computed using a simple classical physical picture, despite the theory being quantum mechanical and interacting.

We investigate current mean values in what are known as Bethe ansatz solvable models. These models possess a large number of conservation laws, which pose strong restrictions on the dynamics. The large-scale transport properties can be described by the essentially classical theory of generalized hydrodynamics. It is a central conjecture of generalized hydrodynamics that the currents of the charges can be computed by semiclassical arguments: They are given by a sum of the individual charges carried by the particles in the system, each multiplied with an effective velocity. We prove that this statement is exact in the interacting quantum theory.

The underlying reason for this classical-quantum correspondence is the two-particle irreducibility of the wave function: In these special models all multiparticle interactions are given by a succession of two-particle scattering events.

Figure 1

The semiclassical picture for the effective velocities. There are $N$ particles moving on a circle of circumference $L$. The only effect of the interaction is the time delays after the scattering events. These delays accumulate and lead to well-defined effective speeds in the long time limit.