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.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.

arXiv:cond-mat/0502015Journal of Physics A Mathematical and GeneralMay 25, 200577 citations

Exact solution of a one-dimensional XY model in a random field

Robert A. Pelcovits, David Mukamel

We present an exact solution of a one-dimensional $\mathrm{XY}$ model in a random magnetic field in the limit of strong-field pinning. The structure factor exhibits Lorentzian-squared behavior at nonzero temperatures. The scaling behavior of the correlation length as a function of randomness is different from that obtained in the weak-field pinning regime. We also study the dynamical behavior of the system using a simple relaxational model. The magnetization decays anomalously with time as ${t}^{\frac{7}{6}}{e}^{{\ensuremath{-}ct}^{\frac{1}{3}}}$.We present an exact solution of a one-dimensional $\mathrm{XY}$ model in a random magnetic field in the limit of strong-field pinning. The structure factor exhibits Lorentzian-squared behavior at nonzero temperatures. The scaling behavior of the correlation length as a function of randomness is different from that obtained in the weak-field pinning regime. We also study the dynamical behavior of the system using a simple relaxational model. The magnetization decays anomalously with time as ${t}^{\frac{7}{6}}{e}^{{\ensuremath{-}ct}^{\frac{1}{3}}}$.

Physical review. B, Condensed matterNov 1, 198310 citations

We present an exact solution of a one-dimensional

\mathrm{XY}

model in a random magnetic field in the limit of strong-field pinning. The structure factor exhibits Lorentzian-squared behavior at nonzero temperatures. The scaling behavior of the correlation length as a function of randomness is different from that obtained in the weak-field pinning regime. We also study the dynamical behavior of the system using a simple relaxational model. The magnetization decays anomalously with time as ${t}^{\frac{7}{6}}{e}^{{\ensuremath{-}ct}^{\frac{1}{3}}}$.