Anomalous Fluctuations in Phases with a Broken Continuous Symmetry

Abstract

It is shown that the Goldstone modes associated with a broken continuous symmetry lead to anomalously large fluctuations of the zero field order parameter at any temperature below Tc. In dimensions 2 < d < 4, the variance of the extensive spontaneous magnetization scales as L 4 with the system size L, independent of the order parameter dynamics. The anomalous scaling is a consequence of the 1/q 4−d divergence of the longitudinal susceptibility. For ground states in two dimensions with Goldstone modes vanishing linearly with momentum, the dynamical susceptibility contains a singular contribution (q 2 − ω 2 /c 2 ) −1/2 . The dynamic structure factor thus exhibits a critical continuum above the undamped spin wave pole, which may be detected by neutron scattering in the Néel-phase of 2D quantum antiferromagnets.It is one of the basic properties of any thermodynamic system that the fluctuations Var =  2 −  2 ∼ V of an extensive variable scale linearly with the system volume V . This property guarantees that the intensive variablesÂ/V are self-averaging, with rms fluctuations vanishing proportional to V −1/2 . Physically, the extensive nature of VarÂ∼ V is due to the existence of a finite, microscopic correlation length ξ. A large system can thus be partitioned into an extensive number of V /ξ d subvolumes which are statistically independent. Since the variance in each subvolume is finite, the central limit theorem then quite generally implies a Gaussian distribution for with a variance of order V , in agreement with the standard Einstein theory of fluctuations in macroscopic thermodynamics.The above argument indicates that the linear scaling Var ∼ V may break down only at a critical point of a continuous phase transition where the correlation length ξ diverges. In fact, from the standard relation VarM = V · T χ between the fluctuations of the total 'magnetization'M and the corresponding linear susceptibility, the finite size scaling χ(T c , L) ∼ L 2−η of the susceptibilty right at T c [1] implies a nontrivial dependence VarM ∼ L d+2−η on system size L at the critical point. Below, it will be shown that anomalous fluctuations of the order parameter are present not only at T c but in fact at any temperature below T c provided the broken symmetry is continuous. This is a result of the presence of Goldstone modes, which imply that correlations decay algebraically below T c with exponents that are independent of temperature. A specific example of recent interest are fluctuations of the condensate numberN 0 in a Bose Einstein condensate. Using a weak coupling Bogoliubov approach, Giorgini et.al have shown [2] that VarN 0 ∼ T 2 L 4 scale anomalously at low temperatures. In fact, this result also applies to strongly interacting superfluids, being essentially a consequence of the long wavelength phase fluctuations [3]. In the present Letter, we will show, that the anomalous fluctuations in a BoseEinstein condensate are just one particular example of a rather general phenomenon which appears in any phase with a broken co...

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