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

Abstract

Whether in the thermodynamic limit, vanishing magnetic field $h\to 0$, 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 $h\to 0$ in the thermodynamic limit of chain length $L\to \infty$, at high temperatures $T\to \infty$. 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 $M_{\mathrm{sp}}=L/2-S$ 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 $h\to 0$ and within the grand-canonical ensemble.

References in corpus (6)

• Open XXZ spin chain: Nonequilibrium steady state and strict bound on ballistic transport
• Spin transport in a one-dimensional anisotropic Heisenberg model
• Conservation laws, integrability and transport in one-dimensional quantum systems
• Quasilocal conservation laws in XXZ spin-1/2 chains: open, periodic and twisted boundary conditions
• Coexistence of diffusive and ballistic transport in a simple spin ladder
• Charge and Spin Transport in the One-dimensional Hubbard Model

Cited by in corpus (5)

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• Finite-temperature charge and spin transport in the one-dimensional Hubbard model accounting for its global [SU (2) X SU(2) X U(1)]/Z_2^2$ symmetry
• The role of q-spin singlet pairs of physical spins in the dynamical properties of the spin-1/2 Heisenberg-Ising XXZ chain
• Ising spin-1/2 XXZ chain's quantum problems beyond the spinon paradigm