Correlation functions of the XY model (abstract)
Date of Original Version
Recently an elegant and quite powerful finite-system approach to determine the exponents ηx and ηz from simple spectral properties has been proposed. For critical systems, the two exponents can be expressed in terms of finite-size spectral gaps as follows: η(N) x=2ΔE01(N)/ΔE(N), η(N)z=2ΔE00(N)/ΔE(N) . Here ΔE(N) is the finite-size gap between the ground state (S zT=0,k=0) and the lowest excitation at k=2π/N; ΔE01(N) is the gap to the lowest ∥S zT∥=1 excitations (at k=π), and ΔE 00(N) is the gap to the next lowest SzT=0 excited state. The η(N) sequence is then extrapolated to N→∞. For XY models, differences between s=1/2 and s≥1 appear. For s=1/2, the excitations which determine ΔE 00(N) and ΔE(N) are degenerate, which implies that ηz=1/2, in agreement with the exact analytic result. For spin-1, however, the next lowest SzT=0 state is located at k=2π/N instead of k=π, and is therefore identical to the state which determines the gap ΔE. The resulting equality ΔE=ΔE 00 implies ηz=2, as in the spin-1/2 case. In fact, our result corresponds to power-law decay for all s, and hence we differ from Schulz and Ziman, who claim the out-of-plane correlation function decays exponentially for s>1/2. For the in-plane correlation function, the spectral gap method again agrees with the exact result ηx=0.5 for s=1/2. The consensus of this and other numerical methods for s=1 gives a value ηx≅0.20, considerably different from the case of s=1/2. Hence it is tempting to conjecture that ηx is s dependent, implying that XY models belong to different universality classes for different s. However, a finite-size study of the conformal anomaly produces the result that c=1, independent of s. This situation is further discussed.
Publication Title, e.g., Journal
Journal of Applied Physics
Bonner, J. C., G. Müller, and J. B. Parkinson. "Correlation functions of the XY model (abstract)." Journal of Applied Physics 67, 9 (1990): 5457. doi: 10.1063/1.345843.