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arXiv:1407.0732

Vanishing spin stiffness in the spin-1/2 Heisenberg chain for any nonzero temperature

J. M. P. Carmelo, Tomaž Prosen, David K Campbell

Published Jul 2, 2014• 3 citations
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Abstract

Whether at zero spin density $m=0$ and finite temperatures $T>0$ the spin stiffness of the spin-$1/2$ $XXX$ chain is finite or vanishes remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an exact upper bound on the stiffness within a canonical ensemble at any fixed value of spin density $m$ and show that it is proportional to $m^2 L$ in the thermodynamic limit of chain length $L\to\infty$, for any finite, nonzero temperature. Moreover, we explicitly compute the stiffness at $m=0$ and confirm that it vanishes. This allows us to exactly exclude the possibility of ballistic transport within the canonical ensemble for $T>0$.

DOI
10.48550/arxiv.1407.0732

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Whether in the thermodynamic limit, vanishing magnetic field h→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→0 in the thermodynamic limit of chain length L→∞, at high temperatures T→∞. 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 Msp=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→∞, vanishing magnetic field h→0 and within the grand-canonical ensemble.
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We present evidence suggesting that spin transport in the gapless phase of the S=1/2 XXZ model is ballistic rather than diffusive. We map the model onto a spinless fermion model whose charge stiffness determines the spin transport of the original model. By means of exact numerical diagonalisation and finite size scaling we study both the stiffness and the level statistics. We show that the stiffness is non-zero at low temperatures so that the transport is ballistic. Our results suggest that the non-zero stiffness is due to the fact that even in the presence of Umklapp scattering a non-zero fraction of states remain degenerate in the thermodynamic limit.