Comment on "Heat-Flow Induced Anomalies in Superfluid 4He near T lambda "

Comment on " Heat-Flow Induced Anomalies in Superfluid 4 He near T l " In a recent Letter [1], Haussmann and Dohm (HD) presented a renormalization group treatment of the 4 He lambda transition in a heat current Q. In this Comment, we use simple arguments that yield the same critical point exponent for the depressed T l , and nearly the same critical velocity, but indicate that HD may not have calculated the proper specific heat anomaly. Near T l , the heat current is given by Q ෇ 2r s y s ST in standard notations of the two-fluid model. Of the terms in Q, only r s and y s may be singular, so for the purpose of computing exponents, we write Q c ϳ ͑r sc y 2 sc ͞2͒͞y sc. The numerator is a singular term in the free energy density, and every such term goes to zero inversely as the correlation volume, i.e., r sc y 2 sc ϳ j 2d. The denominator is given by [2] y sc ෇ 2i͑"͞m͒ j=cj͞c ϳ j=cj͞c , (1) where m is the atomic mass of 4 He. Thus y sc has the character of an inverse length. Since the correlation length is the only relevant length at a critical point, y sc ϳ j 21 ϳ t n , where t ෇ ͑T l 2 T ͒͞T l. Thus Q c ϳ j 2d t 2n , or T l ͑0͒ 2 T l ͑Q͒ ϳ Q 1͞n͑d21͒ , (2) which is the same result arrived at by HD. Equation (1) envisions a wave-function-like order parameter which, in uniform flow has the form c ෇ c 0 e i ៬ k?៬ r , where ៬ r is a space vector and ៬ k is related to y s by ៬ y s ෇ " ៬ k͞m. The order parameter is governed by a differential equation [3] j 2 = 2 c ෇ ͑jcj 2 2 1͒c , (3) which has a solution jcj 2 ෇ 1 2 ͑kj͒ 2. Thus jcj 2 is driven to zero at superfluid velocity. y sc ෇ "͞mj ෇ 112t n ͓m͞sec͔. (4) This justifies the argument in Eq. (1) that y sc ϳ j 21. Fluctuations are taken into account by using the experimental value of n rather than that predicted by mean field theory. Equation (4) may be compared …