arXiv:1910.00634v4 [cond-mat.stat-mech] 31 Dec 2019 in epnil o h superfluid the for responsible tions h eclto eprtr fteloscnb smc as much as that be [8] can recently loops made the claim of a address percolation to the result this use We fu 5.Ti hoyrsle na3 abatp from difference lambda-type crit temperature a 3D the as a of zero law to in power went cal resulted density superfluid theory the This where transition one [5]. by proposed us was loops of vortex circular renormalization using 3D theory first group The loops. vortex th (3D) of equivalents dimensional 2D the superfluid pairs (2D) vortex-antivortex two-dimensional with vortex- for films, Kosterli a 4] by [3, of (KT) out calculations Thouless carried and actual were transition first vanish superfluid The of mediated “ghosts rings”. the vortex excited, being ing loops superfluids excitatio smallest on roton Landau’s the paper that were review proposed them 1955 of by his both proposed and of [2], was end idea the speculative at same Feynman s the conference in Much same circulation of the perfluids. quantization the in proposed he [1], where paper Onsager by suggested tuitively udnjm ozr ftespruddniyat density superfluid the of is quite zero there was to where This jump transition, KT sudden 2D values. the known from behavior match crit different not the though did temperature, exponents same ical the at peaked heat specific hnei h pcfi eta htpoint. that at heat specific the in change o eryalo h nrp on eycoet the to close very found entropy the accoun easily of can all loops heat nearly the specific for that critical shows the universal, theory not of the though magnitude see that The pressure find with experiments. and scaling the pressure, universal the of predicts function correctly a as loops tex uni- O(2) 3D the loop of class. the values versality of known with exponents agreement critical [7 into form the theory circular brought the now basi- from advance was fluctuations This that the size of extent core the renormalized circular- cally a the random-walkin employing actual that the by show to loops, ther- to generalized be able spin-wave could also ideas and was loop He loop excitations. vortex mal into which exactly Hamiltonian, transfor diagonalizes duality Landau-Ginzburg-Wilson a the using of [6] Shenoy mation by verified later were ory h data otxloscudb h hra excita- thermal the be could loops vortex that idea The h euso eain eutn rmtecrua-opt circular-loop the from resulting relations recursion The nti ae ecluaeteseicha rmtevor- the from heat specific the calculate we paper this In h pcfi etaefudt ei odareetwt experi with agreement 5 good about in within be to to dependence found pressure are t heat of specific the parameters are the non-universal parameters the input determine The which bars. density, 26 to up pressures various ei esnbeareet atro orlre hntee the than larger four of factor a agreement, reasonable in be eprtr ftesprudpaetransition. phase superfluid vortex-loo the the of temperature claimed that simulations Gross-Pitaevskii otxlo aclto fteseicha fsuperfluid of heat specific the of calculation Vortex-loop otxlo eomlzto sue ocmuetespecific the compute to used is renormalization Vortex-loop eateto hsc n srnm,Uiest fCalifor of University Astronomy, and Physics of Department .Introduction 1. λ tasto a rtin- first was -transition nrwFretradGr .Williams A. Gary and Forrester Andrew T % T KT h o-nvra rtclapiueo h pcfi eti heat specific the of amplitude critical non-universal The . λ Dtd aur ,2020) 3, January (Dated: n the and , n no and , λ -point. in n ree- he- ns u- tz i- ]. g a - - - - t , eclto eprtr i o ac h critical the match not did temperature percolation p with h iei nryo h oighelium. flowing from the is term of log energy the while kinetic core, the vortex the of energy the gives tmcms,and mass, atomic where %lwrta h rtcltemperature critical the than lower 6% oesz,adfrtebr aei rprinlto proportional is case bare the for and size, core aetesmltossol aeseen have should simulations the case ry hr h nryo ag opi cendb the by screened is loop scr “dielectric” large the following a and it, of around loops energy smaller the where ergy, n tteproaintmeaueadoeat one and temperature percolation the at one n ec ti eue yteplrzbiiyo h dipoles the of polarizablility the as by varies flow, reduced net which applied is the the it to is response hence density in and flowing superfluid volume The unit per backflow mass flow. their dipolar that applied such are the flow loops applied opposes Vortex an in 2D. orient in and b objects, [3] initially Thouless used ideas and fundamental Kosterlitz two on builds theory loop rleuto o h cl-eedn eomlzdsuperfl renormalized density scale-dependent the for equation gral e nyasnl ek h eclto on and point percolation the peak, single a only see o h nryo igecrua op fdiameter of loops circular single of energy the for ytetemlenergy thermal the by dnia,a es owti h rcso ftesimulatio the of precision the within to least at identical, where evre oeeeg n oesz.Terslsfor results The size. core and energy core vortex he o h aeo ayitrciglos h renormalized the loops, interacting many of case the For h opter trsfo h yrdnmcexpression hydrodynamic the from starts theory loop The h rsuedpnec fT of dependence pressure the prmns epitotpolm ihrecent with problems out point We xperiments. K eto superfluid of heat ρ etldt,mthn h xetduniversal expected the matching data, mental s 0 r K 1 ( ρ h neomlzd(br” uefli density, superfluid (“bare”) unrenormalized the U a r s r 0 ) /K ( / a = k ) i,LsAgls A90095 CA Angeles, Los nia, B , 0 K T = 1 a o = 4 a ρ + ierrsos hoylast ninte- an to leads theory response Linear . 0 s r π .Vre-optheory Vortex-loop 2. h aecr imtr h aaee C parameter The diameter. core bare the /ρ 4 4 K 4 3 2 ena h abapitat point lambda the near He eudrpressure under He k K π a s 0 0 B o 7 3 . 0 = T Z ( U , a/a a m ( ¯ h o a a λ 2 a 2 ) ρ 0 n h superfluid the and k 4 (ln( ) s 0 sternraie open- loop renormalized the is B e a − T 0 U ( a/a two a on to found s , ) /k T c c pcfi etpeaks, heat specific B + ) fta eethe were that If . T a c a T C sa effective an is 2 c a, da ) ic they Since . a T 0 a . c divided , , utbe must m n. een- uid the (2) (3) (1) y , 2

%& ing ideas of Kosterlitz and Thouless gives !"#$ !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!>? !@A a !$;$

kBT ao Ko/Kr(a) ∂a :8 ' !# ;$ #;E<'!!!!!!!!!#;E('<(   9 aZ !)(;#$ #; '

These two coupled integral equations can be iterated to macroscopic scales, but it is more convenient to convert them )

to differential recursion relations using a length scale ℓ = *+,-./0+12!/.345167!!8 6 ln(a/a0) and fugacity y = (a/a0) exp(−U(a)/kBT ):

3 ∂K 4π 2 $ = K − K y (5) $;$ $;< #;$ #;< );$ );< %& ∂ℓ 3 &;$!"#$ = 5 ∂y = 6 − π2K (ln(a/a ) + 1) y (6) ∂ℓ c FIG. 1. (Color online) Fits to the data of Ref. [18], showing the val-  ′ ues used for Tλ and the resulting fit values of the amplitude A −ℓ Kr = Ke . (7)

These can be iterated starting from the ℓ = 0 values K0 and exactly satisfies the Josephson hyperscaling relation [14] for 2 y0 = exp(−π K0C), and the macroscopic superfluid density the of the specific heat, α = 2 − dν, giving is found from Kr as ℓ → ∞. The correlation length of the α = −0.015065 ... for d =3. theory is given by The critical exponent η in the loop theory is effectively η

2 = 0 [6], due to the neglect of small logarithmic terms in the a0 m kB T duality transformation. It is known to be small in 3D; the most ξ = = 2 , (8) Kr ¯h ρs accurate simulation [11] gives η = 0.0381(2). In the loop theory the Hausdorff fractal dimension of the wandering loops the known value [9]. The effective core parameter ac is found [15] is given by D =1/(1 − θ)=2.50, in exact agreement using a free-energy minimization of the fluctuations about the H with the scaling theory [16] prediction DH = (5 − η)/2 with minimum-energy circular configuration [7], and is given by η =0. θ ac/a = K , (9) with the exponent θ = d/((d + 2)(d − 2)) in d dimensions 3. Specific heat calculation [10] near d = 3. The fixed points of the recursion equations ∗ ∗ using θ = 0.6 in d = 3 are K = 0.387508 ... and y = To compare with experiments in superfluid 4He, it is neces- 0.062421 ... . The temperature is varied by changing K0, and sary to determine the non-universal parameters of the core di- the critical value K is determined when K and y scale to 0c ameter a0 and the core energy constant C in the critical region their fixed-point values at large ℓ; y blows up and K scales near Tλ. We use the experimental values of the critical ampli- rapidly to zero at values of K0 < K0c. The value of K0c is tude of the superfluid density with pressure, and the pressure completely dependent on the value of C, through the initial dependence of Tλ. For the superfluid density we postulate value of y0. 0 that the starting value ρs of the superfluid density at ℓ = 0 Expanding the recursion relations about the fixed points is equal to the total density ρ. It is well known that phonons gives an analytic expression for the correlation-length expo- contribute a negligible amount to the thermodynamics at the nent [6], lambda point, particularly at higher pressures where the sound velocity increases, and in any event there are already larger ∗ 1 1 π2θK noncritical contributions to the background specific heat from = 1+24 1 − − 1 , (10) ν 2 s 6 ! other sources, such as chemical-bond effects [17] that would   be present even in the normal . We also do not include ∗ 0 and putting in the d = 3 values of K and θ gives ν = any “roton” contributions to ρs, as we identify these excita- 0.67168835 ... . This is in complete agreement with the tions as the smallest vortex loops in the theory, as will be best high-temperature expansion [11] value (0.6717(1)), the apparent shortly. The recursion relations are iterated using best Monte Carlo simulation [12] value (0.6717(3)), and is standard fourth-order Runge-Kutta techniques to values of ℓ within the bounds of the recent conformal bootstrap calcula- between 10 and 100, depending on the closeness to Tλ. tions [13]. We note that Eq.(7) guaranteesthat the loop theory Figure 1 shows fits to the superfluid density measurements 3

! 1.2 C 1.0

0c 0.8 "

0.6

and K K0c

C 0.4 # 0.2

0.0 0 5 10 15 20 25 30 $ Pressure

% *:''1*/;)'(-<*='()*)+)(,-*>%3 &*0/*? &'()*)+)(,-.('/'+*,01*234 ! @ FIG. 2. (Color online) The parameters C and K0c versus pressure. *AB997*)/*0:*2 CCC4*+)8/('+*7=0//)(B+,*('/'+*,01*0/*?@ *D'++)::-*/09:)7*('/'+*,01*0/*:'E*? Additional data at low pressures from Ref. [18] were not shown in * Fig. 1. The lines are polynomial fits to guide the eye. ! ! 5 ! 5 %! %5 6()778()*290(4 of Ref.[18]. We fit to the amplitude A′ in the form

ρs ′ ν Kr = A t = , (11) FIG. 3. Vortex core energy at Tλ, compared with the neutron scatter- ρ K0 ing roton gap from Ref. [19]. The solid fit line is a guide to the eye, while the dashed curve shows the roton gap energy at low tempera- where t =1−T/Tλ, and in 3D the critical superfluid exponent ture from Ref.[20]. is the same as the correlation-length exponent, from Eqs.(8) and (10). We assume that ν = 0.67168835 is universal and independent of pressure, and only allow A′ to vary, with re- !" sults shown in the table in Fig. 1. This is different from the %$&$(@#!B )C assumption in Ref. [18], who allowed also ν to vary, finding values close to 0.6717 but varying slightly with pressure. Our ′ values for A turn out to be consistently about 2-3% higher !# &$( than those of Ref.[18]. $ Once A′ is known in the critical region, the correspond- ing values of C can be found from iterating the recursion

&'()*)% ! relations. At a given pressure trial values of C and K0 are $ % inserted in y0, and Kr is calculated at long length scales from the recursion. K0 is then incremented to sweep the temperature, finding the critical value K0c (where Kr goes !$ )23%1.-/45)%56)78.95:%-8)&$!");( )344')8<.4-=>)?)@)? to zero) and the superfluid fraction Kr/K0 as a function of A T/Tλ = K0c/K0. The resulting superfluid amplitude is then $ + $ + #$ #+ "$ ′ compared with A , and if it does not match, the value of C ,-.//0-.)&1%-( is incremented, and the process is repeated until a match is found. Figure 2 shows the resulting values of C and K0c as a function of pressure. The core energy in the critical region 2 FIG. 4. (Color online) Vortex core diameter versus pressure. The from Eq.(1), U0/kB = π K0cCTλ, is plotted in Fig. 3 as solid line is a fit to guide the eye. The low-temperature measurements a function of pressure, and also shown are the roton energy are from Ref.[21]. gaps at T = Tλ from precision neutron scattering measure- ments under pressure [19]. The agreement is seen to be quite good, further justifying our claim that the smallest loops can point; it is the effective core size ac which becomes diver- be treated as the roton excitations seen in experiments. gent there. The pressure dependence of a0 is shown in Fig. 4, The value of a0(p) near the lambda point can now be found where it increases with pressure, though not quite as rapidly from Tλ(p), as found in low-temperature vortex-ring measurements.

2 The loop recursion relation for the free energy was derived m kBTλ(p) a0(p)= K0c(p) . (12) by Shenoy [6] from the duality transformation: ¯h2ρ(p) ∂f ˚ = −π exp(−3ℓ)y (13) Atp= 0barthisgivesa valueof a0(0) = 2.41 A, in reasonable ∂ℓ agreement with the value of 1.6 A˚ deduced from vortex-ring 3 measurements [21] at low (0.2 K). The bare core where f = (F/kB T )(a0/V ) with F the free energy in the 3 diameter remains microscopic in the loop theory at the lambda volume V , normalized by a0. At constant pressure, the total 4 heat capacity is derivatives of V and a0 can be rewritten in terms of derivatives of ρ, and we convert to the molar specific heat using cp = ∂2G ∂2F ∂2V , where is the molar volume, with (14) (Vm/V )Cp Vm = NAm/ρ Cp = −T 2 = −T 2 − Tp 2 , ∂T p ∂T p ∂T p NA Avogadro’s constant. With the constant R = kB NA,       the specific heat becomes (neglecting very small terms) where G = F +pV is the Gibbs free energy. The temperature

m R ∂2f ∂f 4T ∂ρ 2T 2 ∂2ρ 8T ∂ρ c = −K2 + K − f + . (15) p,loop ρa3 0 ∂K2 0 ∂K ρ ∂T ρ2 ∂T 2 ρ ∂T 0 "  0 p  0 p    p (   p  p)#

9:;<<=:;)*>?:2)))) TABLE I. Specific heat fit parameters, in units J/(mole K). ! )!@!$))))))))))))) ′ ′ ′ ′ ′ ′ ′′ Pressure (bar) Bloop Dloop Bexp Dexp Bexp/Bloop D )0@#$))))))))))))) 0.05 -1397 1372 -388.0 400.1 0.2777 19.14 )"@&&)))))))))))) )0$@!&))))))))) "! 1.65 -1406 1381 -382.1 394.0 0.2717 18.68 )0 @0 ))))))))))) 7.33 -1401 1377 -355.0 365.8 0.2534 16.76 )3$@ #)))))))))) 15.03 -1328 1308 -337.5 346.6 0.2542 13.99 )

18.18 -1286 1269 -322.7 331.9 0.2510 13.48 2 ) /0 #! 22.53 -1223 1210 -299.6 309.5 0.2449 13.16 1 /0 25.87 -1173 1164 -306.5 315.8 0.2613 11.76

$! )*+),-. ( ' By iterating Eqns.(5-7) and (13,15) for seven different val- ues of pressure between 0.05 and 25.87 bars, we are able to %! get the calculated loop specific heat cp,loop versus temperature and pressure. To analyze the temperature behavior we employ a fitting function &!

′ −α ′ cp,fit = B t + D , (16) /3 /& /% /$ /# 0! 0! 0! 0! 0! using the value of α given above in accordance with Joseph- 4)5)*67/62867 son hyperscaling. Fits to the calculated results in the critical −3 ′ regime at t< 4×10 give values of the amplitude Bloop and ′ the offset Dloop, shown in Table I for the different pressures. FIG. 5. Specific heat data from Ref. [23] (0.05 bar) and all other data The experimental data shown in Fig. 5 is also known to follow points from Ref. [22], compared with the adjusted loop theory cp,adj a similar form[22], and we have fit the data to this form(again (solid lines). for t< 4 × 10−3 and using our value α = −0.015065), with ′ ′ the results for Bexp and Dexp shown in Table I. ′ [17], and any non-loop excitations such as phonons. The ad- Since the amplitude Bloop is a non-universal quantity that depends on other non-universal values at the bare scale a0, we justed loop theory curves are seen to completely match the ′ ′ experimental data for −3. We note that this type of would not expect Bloop and Bexp to be the same, and indeed t< 4 × 10 their ratio shown in Table I averages to a value of nearly 0.26. scaling adjustment of the amplitudes is also used in perturba- What we do expect, however, is the pressure dependence of tive renormalization group calculations of the lambda specific both amplitudes to match, due to universality arguments given heat [26], where again there is uncertainty in the precise val- by Rudnick and Jasnow [24] and Ferer [25]. Their ratio is ues of the parameters at the microscopic bare length scale. nearly independent of pressure, within fluctuations of at most A further test of universality is the parameter X which was 5% that are likely due to the uncertainties in the experimental proposed by Ferer to be a universal constant independent of data. To compare more carefully with the data we plot as the pressure [25], solid lines in Fig. 5 the theoretical specific heat adjusted by ′ 3 the amplitude ratios, |B | TλVm X = . (18) RV A′ ′ ′ ′′ m   cp,adj = (Bexp/Bloop) cp, loop + D , (17) In Table II we compare values of X from the above values of ′′ ′ ′ ′ ′ ′ ′ where D = Dexp − (Bexp/Bloop)Dloop is the non-critical B and A from the loop theory and the experiments, and with background probably coming from the chemical-bond effects Ferer’s values using the experimental data fit to α = −0.02 5 and ν = 0.669. It is seen that the loop theory values are out over several lattice points. Near the transition the vor- almost completely independent of pressure, while the exper- tex density becomes very high, and it becomes very difficult imental values are nearly constant, but fluctuate on the order to follow the trajectory of a loop when it comes close to an- of 5%. The loop values are larger by a factor of about 4, since other loop. To resolve which loop is which they employed ′ ′ Bloop/Bexp ≈ 4. two different guesses in this situation: a “stochastic” scenario in which the loop directions near a crossing are randomly cho- sen (which gave the percolation point 2% below the superfluid − TABLE II. Universal parameter X, in units 10 4R(mole/cm3)(K Tc), and a second scenario where the configuration resulting 3 3 cm /mole) . Xexp values from our fits to Refs. [22] and [18]; in the longest loop is chosen (percolation point 6% below Tc). XF erer values from Ref. [25]. The fact that the two different scenarios give quite different percolation points is already evidence that something is amiss Pressure (bar) Xloop Xexp XF erer 0.05 8.68 2.41 2.06 with the procedures for determining the loop distributions. 1.65 8.68 2.36 2.02 We disagree with the conclusion that the two temperatures 7.33 8.68 2.20 1.99 are different, and use their simulation of the GP specific heat 15.03 8.67 2.21 2.02 and superfluid density as evidence this is not the case. Their 18.18 8.67 2.18 1.99 specific heat data (Figs.3b and 4c of Ref.[8]) shows only a 22.53 8.66 2.12 1.96 single peak that is coincident with their superfluid transition. 25.87 8.66 2.26 2.19 If the loop percolationtemperaturewas as much as 6% smaller than Tc, our calculation shows that there should have been a large specific heat peak right at that point, and yet nothing at all is visible there in the data. There is only the sharp peak right at Tc, which we and others [28] would argue is actually 4. Conclusions also the percolation point. It is also impossible that the super- fluid density would be unaffected by the loop percolation, and These results show that the vortex-loop theory provides a yet it shows no change at all at the purported lower percola- complete description of the rapid rise of the specific heat of tion temperatures(Figs.6 and 7 of Ref.[8]). In the loop theory superfluid very close to Tλ. The theory is entirely the specific heat and superfluid density are only driven by the consistent with universality, and the critical exponents agree very longest loops, and are completely independent of the re- with the best simulation values. Even the non-universal am- connection guesses discussed above. It is only in computing plitude of the loop specific heat is the correct order of magni- aspects of the the loop distribution function that the inablility tude, only a factor of about four larger than the experimental to accurately trace the loops will become troublesome, and we value. This difference likely stems from our use of the hydro- postulate this is the source of the claim in Ref. [8] that the loop dynamic form of Eq.(1) for the energyof the smallest loops at percolation does not coincide with the superfluid Tc [29]. We the bare scale, since with a0 only a few A,˚ quantum methods point out that recently a new procedure for accurately track- might give a better description [27]. The amplitude of Eq.(15) ing the cores of GP vortices has been developed [30, 31]. This varies rapidly with a0, so only a small change in a0 would be procedure is able to completely track a reconnection event be- needed to better match with the experiments. tween two vortex lines that cross, following the details of the In this work the superfluid transition temperature Tλ is ex- motion even at the point of closest approach. It would be very actly coincident with the temperature at which the loops per- interesting if this procedure could be applied to tracing the colate, e.g. for an infinite system the temperature where in- vortex loops in simulations like those of Ref. [8] to determine finite diameter loops are just nucleated, made possible by whether the loops do in fact percolate at the superfluid transi- the screening of the largest loops by all the loops of smaller tion. size. In a recent paper, Kobayashi and Cugliandolo [8] have claimed from Gross-Pitaevskii (GP) simulations that these two temperatures are different, with the loop percolation tem- ACKNOWLEDGMENTS perature falling below the superfluid transition temperatureby as much as 6%. In coming to this conclusion they needed to This work was supported in part by a grant from the Julian study the loop distribution function in their simulations, the Schwinger Foundation. We thank J. Lipa for sending us his number of loops with a given diameter. The problem with GP experimental data, and acknowledge useful discussions with vortices is that the core size is relatively very large, spread S. Putterman, J. Rudnick, and J. M. Kosterlitz.

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