Infinite temperature spin dc conductivity of the spin-1/2 XXZ chain

Abstract Using the Bethe ansatz method and the thermodynamic Bethe ansatz equations for the higher-spin integrable XXZ chain, the regular zero frequency contribution to the spin current correlation (spin dc conductivity) is analyzed for the spin-1/2 XXZ chain with the anisotropy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>⩽</mml:mtext> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly evaluate the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit. We find that the high temperature proportionality constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>0</mml:mn> <mml:mrow> <mml:mi>reg</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>π</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>cos</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>⁡</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> point is exceptional. Close to this point, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>0</mml:mn> <mml:mrow> <mml:mi>reg</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> slowly increases in proportion to the first power of the one quasi-particle magnetization. On the other hand, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>0</mml:mn> <mml:mrow> <mml:mi>reg</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> is proportional to the second power of the same when p 0 approaches irrational numbers.