Superfluid density

PI3.2 The phase transition of 4He into its superfluid phase from the normal liquid is continuous. The order parameter, rj, of the transition has been found to vary as the square root of the ratio of the superfluid density, ps, to the total density, px. Some data for rj as a function of (Tc — T) are given below. Obtain the critical exponent, /3, for the order parameter from this data. 

A + B v[/ =6, which results in an explicit relation between the homogeneous system superfluid density p and the coefficients A and B, so that p = A /B. At this stage the phenomenological theory of Ginzburg and Sobyanin can be adopted, representing the bulk order parameter and its superfluid density p in terms of a critical exponent of the macroscopic system... 

At this stage finite-size scaling theory [155, 193-197, 199] is applicable for the description of the specific heat maximum and of the onset of the superfluid density (see the beginning of Section 11), which characterize the rounded-off X transition. The singular free energy density, /, of the finite system (in the absence of external fields) can be described in terms of a universal function (T()) in the form [194—197] f = where is a metric factor,... 

Superfluid density data in ( He) clusters N = 8-64), obtained from quantum simulation data with periodic boundary conditions [155] (Eig. 8), obey Eq. (46),... 

From the analysis of the cluster size dependence of the superfluid density (or order parameter) the following conclusions emerge ... 

A surprising result emerging from the quantum simulations [65, 66] of small ( He)jv clusters and the analyses in Sections II.B and ll.C is the manifestation of a well-characterized, broadened, high-order phase transition for small (" He) y clusters (i.e., N = 8) for the superfluid density [155], and N = 32 for the appearance of the lambda transition [65]. An open question pertains to the threshold size of these equations What is the system s smallest size for the exhibition of superfluidity and what are the corresponding phase transitions ... 

Tt o- Taking the short correlation length 2 A, we roughly estimate that 77 6 A, so that the smallest " He cluster will consist of a central atom and its first coordination layer. Thus the threshold size domain for the realization of the lambda transition is TVmin " 5-13. Such a low value of Amin is consistent with the value Amin < 8 for the exhibition of the superfluid density in finite systems [155]. Finally, the threshold size for the appearance of rotons in the elementary excitation spectra of ( He) y clusters [128] is realized for 20 < Amin < 70 (Section l.D). 

We now recall that the classical planar rotator model may be used as a model of superfluid He4, 0 being the phase of the condensate wave function, S being related to the superfluid density ps as S = ps(hjm)2, m being the mass of a He4 atom. Thus one can have superfluid-normal fluid transition in d = 2 dimensions, despite the lack of conventional long range order This conclusion seems to be corroborated by experiments on He4 films (Bishop and Reppy, 1978). 

Figure 10 presents a simplified phase diagram based on the experimental results discussed above. The characteristic temperatures T and are plotted as functions of the carrier density in the Cu02 planes. We have chosen the superfluid density o)ps for the horizontal axis in this particular plot because the magnitude of (O s can be determined unambiguously from infrared results (see sect. 5.1). Evaluation of the ffee-carrier density... 

Fig. 10. Schematic phase diagram of Y123 and Y124 cuprate superconductors. In the underdoped regime, a pseudogap state forms below a temperature T > 7. The curves for 7 and 7 cross at optimal doping where the pseudogap and the superconductivity develop at the same tempeiature. T is determined from the c-axis conductivity the doping level is determined from the superfluid density = n/ni in the CUO2 planes. From Puchkov et al. (1996a).

It should be noted that within the BCS theory one does not expect to observe any explicit correlation between and the superfluid density in a clean-limit superconductor. The linear dependence between the transition temperature and tis/m is expected if... 

Some time ago, Uemura et al. (1991) proposed a universal Unear relationship Tc = ps between the superconducting transition temperature and the superfluid density ps of charge carriers for a group of superconducting compounds. Although the Uemura law describes underdoped cuprates well, it does not work for optimal or overdoped cuprates. 

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