Kosterlitz-Thouless in the XXZ chain: instanton condensation?
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The anisotropic spin-1212 Heisenberg chain
H=∑nSxnSxn+1+SynSyn+1+ΔSznSzn+1H=∑nSnxSn+1x+SnySn+1y+ΔSnzSn+1z
is known to have the same physics as the two-dimensional classical XY model. More concretely, at Δ=1Δ=1 it undergoes the (topological) Kosterlitz-Thouless transition, below which it is has algebraic decay of correlations and above which it is has exponential decay. Usually this is shown by using the quite sophisticated methods of bosonization to show that its field theory description is given by the sine-Gordon model, which is also the field theory describing the standard KT transition.
The intuitive picture behind the 2D classical KT transition is that there is an entropic gain when adding a vortex, which eventually beats its energetic cost at the KT transition, leading to a condensation of vortices. My question is then: is there a similar `intuitive' picture for the 1D quantum spin system (without having to resort to bosonized field theories etc)? In particular, can I in the spin language have a simple picture of something (presumably instantons) condensing at Δ=1Δ=1?
H=∑nSxnSxn+1+SynSyn+1+ΔSznSzn+1H=∑nSnxSn+1x+SnySn+1y+ΔSnzSn+1z
is known to have the same physics as the two-dimensional classical XY model. More concretely, at Δ=1Δ=1 it undergoes the (topological) Kosterlitz-Thouless transition, below which it is has algebraic decay of correlations and above which it is has exponential decay. Usually this is shown by using the quite sophisticated methods of bosonization to show that its field theory description is given by the sine-Gordon model, which is also the field theory describing the standard KT transition.
The intuitive picture behind the 2D classical KT transition is that there is an entropic gain when adding a vortex, which eventually beats its energetic cost at the KT transition, leading to a condensation of vortices. My question is then: is there a similar `intuitive' picture for the 1D quantum spin system (without having to resort to bosonized field theories etc)? In particular, can I in the spin language have a simple picture of something (presumably instantons) condensing at Δ=1Δ=1?
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