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Journal of Physics A Mathematical and General

The pair correlation function in the ground state of the two-dimensional spin-1/2 XY model

S B Kelland, Donald D. Betts, J. Oitmaa

Published Mar 1, 1981• 2 citations
Publisher

Abstract

The decay to a constant value of the correlation function (S 0 x S r x ) in the ground state of the spin- 1 / 2 XY ferromagnet in two dimensions is found to be exponential.
DOI
10.1088/0305-4470/14/3/004

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The spin-spin correlation function of the two-dimensional (2D) XYXYXY model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature-dependent exponent $\ensuremath{\eta}=\ensuremath{\eta}(T/J)$ (JJJ is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D XYXYXY model with site or bond dilution this exponent depends on the concentration ppp of removed sites or bonds as well. Knowing the slope $\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p$ at point p=0p=0p=0, one can predict the value of the exponent for small dilution concentrations: ${\ensuremath{\eta}(p)\ensuremath{\simeq}\ensuremath{\eta}(0)+p(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}$. As shown in this paper, the spin-wave Hamiltonian allows one to obtain exact results for this slope: ${(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}=T/(2J)+O({(T/J)}^{2})$ and $T/(\ensuremath{\pi}J)+O({(T/J)}^{2})$ for site and for bond dilution, respectively.
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The spin-correlation functions of the one-dimensional XYXYXY model are studied in the presence of a constant magnetic field. We find that the asymptotic behavior of these correlation functions depends strongly on the various parameters of the Hamiltonian.
We study an asymptotic behaviour of a special correlator known as the emptiness formation probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field. This correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models. For the XY spin chain, the correlator can be expressed as the determinant of a Toeplitz matrix and its asymptotical behaviours for n → ∞ throughout the phase diagram are obtained using known theorems and conjectures on Toeplitz determinants. We find that the decay is exponential everywhere in the phase diagram of the XY model except on the critical lines, i.e. where the spectrum is gapless. In these cases, a power-law prefactor with a universal exponent arises in addition to an exponential or Gaussian decay. The latter Gaussian behaviour holds on the critical line corresponding to the isotropic XY model, while at the critical value of the magnetic field the EFP decays exponentially. At small anisotropy one has a crossover from the Gaussian to the exponential behaviour. We study this crossover using the bosonization approach.