N.21 Su­per­flu­id­ity ver­sus BEC

Many texts and most web sources sug­gest quite strongly, with­out ex­plic­itly say­ing so, that the so-called lambda phase tran­si­tion at 2.17 K from nor­mal he­lium I to su­per­fluid he­lium II in­di­cates Bose-Ein­stein con­den­sa­tion.

One rea­son given that is that the tem­per­a­ture at which it oc­curs is com­pa­ra­ble in mag­ni­tude to the tem­per­a­ture for Bose-Ein­stein con­den­sa­tion in a cor­re­spond­ing sys­tem of non­in­ter­act­ing par­ti­cles. How­ever, that ar­gu­ment is very weak; the sim­i­lar­ity in tem­per­a­tures merely sug­gests that the main en­ergy scales in­volved are the clas­si­cal en­ergy ${k_{\rm B}}T$ and the quan­tum en­ergy scale formed from $\hbar^2$$\raisebox{.5pt}{$/$}$$2m$ and the num­ber of par­ti­cles per unit vol­ume. There are likely to be other processes that scale with those quan­ti­ties be­sides macro­scopic amounts of atoms get­ting dumped into the ground state.

Still, there is not much doubt that the tran­si­tion is due to the fact that he­lium atoms are bosons. The iso­tope $\fourIdx{3}{}{}{}{\rm {He}}$ that is miss­ing a neu­tron in its nu­cleus does not show a tran­si­tion to a su­per­fluid un­til 2.5 mK. The three or­ders of mag­ni­tude dif­fer­ence can hardly be due to the mi­nor dif­fer­ence in mass; the iso­tope does con­dense into a nor­mal liq­uid at a com­pa­ra­ble tem­per­a­ture as plain he­lium, 3.2 K ver­sus 4.2 K. Surely, the vast dif­fer­ence in tran­si­tion tem­per­a­ture to a su­per­fluid is due to the fact that nor­mal he­lium atoms are bosons, while the miss­ing spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ neu­tron in $\fourIdx{3}{}{}{}{\rm {He}}$ atoms makes them fermi­ons. (The even­tual su­per­fluid tran­si­tion of $\fourIdx{3}{}{}{}{\rm {He}}$ at 2.5 mK oc­curs be­cause at ex­tremely low tem­per­a­tures very small ef­fects al­low the atoms to com­bine into pairs that act as bosons with net spin one.)

While the fact that the he­lium atoms are bosons is ap­par­ently es­sen­tial to the lambda tran­si­tion, the con­clu­sion that the tran­si­tion should there­fore be Bose-Ein­stein con­den­sa­tion is sim­ply not jus­ti­fied. For ex­am­ple, Feyn­man [18, p. 324] shows that the bo­son char­ac­ter has a dra­matic ef­fect on the ex­cited states. (Dis­tin­guish­able par­ti­cles and spin­less bosons have the same ground state; how­ever, Feyn­man shows that the ex­is­tence of low en­ergy ex­cited states that are not phonons is pro­hib­ited by the sym­metriza­tion re­quire­ment.) And this ef­fect on the ex­cited states is a key part of su­per­flu­id­ity: it re­quires a fi­nite amount of en­ergy to ex­cite these states and thus mess up the mo­tion of he­lium.

An­other ar­gu­ment that is usu­ally given is that the spe­cific heat varies with tem­per­a­ture near the lambda point just like the one for Bose-Ein­stein con­den­sa­tion in a sys­tem of non­in­ter­act­ing bosons. This is cer­tainly a good point if you pre­tend not to see the dra­matic, glar­ing, dif­fer­ences. In par­tic­u­lar, the Bose-Ein­stein spe­cific heat is fi­nite at the Bose-Ein­stein tem­per­a­ture, while the one at the lambda point is in­fi­nite.. How much more dif­fer­ent can you get? In ad­di­tion, the spe­cific heat curve of he­lium be­low the lambda point has a log­a­rith­mic sin­gu­lar­ity at the lambda point. The spe­cific heat curve of Bose-Ein­stein con­den­sa­tion for a sys­tem with a unique ground state stays an­a­lyt­i­cal un­til the con­den­sa­tion ter­mi­nates, since at that point, out of the blue, na­ture starts en­forc­ing the re­quire­ment that the num­ber of par­ti­cles in the ground state can­not be neg­a­tive, {D.57}.

Tilley and Tilley [47, p. 37] claim that the qual­i­ta­tive cor­re­spon­dence be­tween the curves for the num­ber of atoms in the ground state in Bose-Ein­stein con­den­sa­tion and the frac­tion of su­per­fluid in a two-fluid de­scrip­tion of liq­uid he­lium “are suf­fi­cient to sug­gest that $T_\lambda$ marks the on­set of Bose-Ein­stein con­den­sa­tion in liq­uid $\strut^4$He.” Sure, if you think that a curve reach­ing a max­i­mum of one ex­po­nen­tially has a sim­i­lar­ity to one that reaches a max­i­mum of one with in­fi­nite cur­va­ture. And note that this com­pares two com­pletely dif­fer­ent quan­ti­ties. It does not com­pare curves for par­ti­cles in the ground state for both sys­tems. It is quite gen­er­ally be­lieved that the con­den­sate frac­tion in liq­uid he­lium, un­like that in true Bose-Ein­stein con­den­sa­tion, does not reach one at zero tem­per­a­ture in the first place, but only about 10% or so, [47, pp. 62-66].

Since the spe­cific heat curves are com­pletely dif­fer­ent, Oc­cam’s ra­zor would sug­gest that he­lium has some sort of dif­fer­ent phase tran­si­tion at the lambda point. How­ever, Tilley and Tilley [47, pp. 62-66] present data, their fig­ure 2.17, that sug­gests that the num­ber of atoms in the ground state does in­deed in­crease from zero at the lambda point, if var­i­ous mod­els are to be be­lieved and one does not de­mand great ac­cu­racy. So, the best avail­able knowl­edge seems to be that Bose-Ein­stein con­den­sa­tion, what­ever that means for liq­uid he­lium, does oc­cur at the lambda point. But the fact that many sources see ev­i­dence of con­den­sa­tion where none ex­ists is wor­ri­some: ob­vi­ously, the de­sire to be­lieve de­spite the ev­i­dence is strong and wide­spread, and might af­fect the ob­jec­tiv­ity of the data.

Snoke & Baym point out (in the in­tro­duc­tion to Bose-Ein­stein Con­den­sa­tion, Grif­fin, A., Snoke, D.W., & Stringari, S., Eds, 1995, Cam­bridge, p. 4), that the ex­per­i­men­tal sig­nal of a Bose-Ein­stein con­den­sate is taken to be a delta func­tion for the oc­cu­pa­tion num­ber of the par­ti­cles [par­ti­cle state?] with zero mo­men­tum, as­so­ci­ated with long-range phase co­her­ence of the wave func­tion. It is not likely to be un­am­bigu­ously ver­i­fied any time soon. The ac­tual ev­i­dence for the oc­cur­rence of Bose-Ein­stein con­den­sa­tion is in the agree­ment of the­o­ret­i­cal mod­els and ex­per­i­men­tal data, in­clud­ing also mod­els for the spe­cific heat anom­aly. How­ever, Sokol points out in the same vol­ume, (p. 52): “At present, how­ever, liq­uid he­lium is the only sys­tem where the ex­is­tence of an ex­per­i­men­tally ob­tain­able Bose con­densed phase is al­most uni­ver­sally ac­cepted” [em­pha­sis added].

The ques­tion whether Bose-Ein­stein con­den­sa­tion oc­curs at the lambda point seems to be aca­d­e­mic any­way. The fol­low­ing points can be dis­tilled from Schmets and Mont­frooij [39]:

Bose-Ein­stein con­den­sa­tion is a prop­erty of the ground state, while su­per­flu­id­ity is a prop­erty of the ex­cited states.
Ideal Bose-Ein­stein con­den­sates are not su­per­fluid.
Be­low 1 K, es­sen­tially 100% of the he­lium atoms flow with­out vis­cos­ity, even though only about 7% is in the ground state.
In fact, there is no rea­son why a sys­tem could not be­come a su­per­fluid even if only a very small frac­tion of the atoms were to form a con­den­sate.

The state­ment that no Bose-Ein­stein con­den­sa­tion oc­curs for pho­tons ap­plies to sys­tems in ther­mal equi­lib­rium. In fact, Snoke & Baym, as men­tioned above, use lasers as an ex­am­ple of a con­den­sate that is not su­per­fluid.