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The anisotropic spin-$\frac{1}{2}$ Heisenberg chain $$H = \sum_n S^x_n S^x_{n+1} + S^y_n S^y_{n+1} + \Delta S^z_n S^z_{n+1}$$ is known to have the same physics as the two-dimensional classical XY model. More concretely, at $\Delta = 1$ 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 $\Delta = 1$?

Ruben Verresen
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1 Answers1

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To directly answer the question: Fradkin and Susskind offer an interpretation of quantum critical point of a spin chain as the proliferation of domain walls. (Creation of a domain wall is an instanton.)

But the main goal of F&S in fact the opposite of OP's idea: they map a finite-temperature statistical system in $D$ space dimension to a quantum Hamiltonian in $(D-1)$ space dimension at zero temperature, and do meanfield theory on the quantum spin model to infer the phase diagram. It is arguably easier to infer that the quantum XXZ chain has at least two distinct phases (in the limits of $\Delta \rightarrow 0$ and $\Delta \rightarrow \infty$), than to do the same with the classical XY model.

But of course if you want to get analytical results regarding the quantum critical point, you need high-power methods.

T.P. Ho
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