Absence of high-temperature ballistic transport in the spin-$1/2$ $XXX$ chain within the grand-canonical ensemble

TL;DR

This paper demonstrates that the spin-$1/2$ $XXX$ Heisenberg chain exhibits no high-temperature ballistic transport in the thermodynamic limit when the magnetic field approaches zero, within the grand-canonical ensemble, resolving a longstanding controversy.

Contribution

The authors establish an upper bound on the spin stiffness, showing it vanishes at high temperatures and zero magnetic field in the thermodynamic limit, using a spin-$1/2$ representation approach.

Findings

Spin stiffness vanishes as $h o 0$ in the thermodynamic limit.

Ballistic transport is absent at high temperatures in the grand-canonical ensemble.

The spin-$1/2$ representation provides physical insights into the transport properties.

Abstract

Whether in the thermodynamic limit, vanishing magnetic field $h\rightarrow 0$, and nonzero temperature the spin stiffness of the spin-$1/2$ $XXX$ Heisenberg chain is finite or vanishes within the grand-canonical ensemble remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an upper bound on the stiffness and show that within that ensemble it vanishes for $h\rightarrow 0$ in the thermodynamic limit of chain length $L\to\infty$, at high temperatures $T\rightarrow\infty$. Our approach uses a representation in terms of the $L$ physical spins $1/2$. For all configurations that generate the exact spin-$S$ energy and momentum eigenstates such a configuration involves a number $2S$ of unpaired spins $1/2$ in multiplet configurations and $L-2S$ spins $1/2$ that are paired within $M_{\rm sp}=L/2-S$ spin-singlet pairs. The Bethe-ansatz strings of length $n=1$ and $n>1$ describe a single unbound spin-singlet pair and a configuration within which $n$ pairs are bound, respectively. In the case of $n>1$ pairs this holds both for ideal and deformed strings associated with $n$ complex rapidities with the same real part. The use of such a spin $1/2$ representation provides useful physical information on the problem under investigation in contrast to often less controllable numerical studies. Our results provide strong evidence for the absence of ballistic transport in the spin-$1/2$ $XXX$ Heisenberg chain in the thermodynamic limit, for high temperatures $T\to\infty$, vanishing magnetic field $h\rightarrow 0$ and within the grand-canonical ensemble.