# Transport properties of the classical Toda chain: effect of a pinning potential

## TL;DR

This study examines how adding a pinning potential affects energy transport in the classical Toda chain, revealing that harmonic pinning minimally impacts low-temperature behavior while quartic pinning significantly disrupts integrability.

## Contribution

It provides a detailed analysis of how different pinning potentials influence the breakdown of integrability and energy transport properties in the Toda chain.

## Key findings

- Harmonic pinning has a smaller effect at low temperatures.
- Quartic pinning significantly disrupts integrability.
- Low-temperature Toda potential approximates a harmonic system.

## Abstract

We consider energy transport in the classical Toda chain in the presence of an additional pinning potential. The pinning potential is expected to destroy the integrability of the system and an interesting question is to see the signatures of this breaking of integrability on energy transport. We investigate this by a study of the non-equilibrium steady state of the system connected to heat baths as well as the study of equilibrium correlations. Typical signatures of integrable systems are a size-independent energy current, a flat bulk temperature profile and ballistic scaling of equilibrium dynamical correlations, these results being valid in the thermodynamic limit. We find that, as expected, these properties change drastically on introducing the pinning potential in the Toda model. In particular, we find that the effect of a harmonic pinning potential is drastically smaller at low temperatures, compared to a quartic pinning potential. We explain this by noting that at low temperatures the Toda potential can be approximated by a harmonic inter-particle potential for which the addition of harmonic pinning does not destroy integrability.