# Correlation functions of the XY model (abstract)

## Document Type

Article

## Date of Original Version

12-1-1990

## Abstract

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

## Volume

67

## Issue

9

## Citation/Publisher Attribution

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.