With a high-performance Monte Carlo algorithm we study the
interaction-induced shift of the critical point in weakly interacting
three-dimensional $|\psi|^4$-theory (which includes quantum Bose gas). In terms
of critical density, $n_c$, mass, $m$, interaction, $U$, and temperature, $T$,
this shift is universal: $\Delta n_c(T) = - C m^3 T^2 U$, the constant $C$
found to be equal to $0.0140 \pm 0.0005$. For quantum Bose gas with the
scattering length $a$ this implies $\Delta T_c/T_c = C_0 a n^{1/3}$, with
$C_0=1.29 \pm 0.05$.Comment: 4 pages, latex, 3 figure

Kashurnikov, V. A.

Prokof'ev, Nikolay

Svistunov, Boris

English

arXiv

With a high-performance Monte Carlo algorithm we study the interaction-induced shift of the critical point in weakly interacting three-dimensional |ψ|4 theory (which includes quantum Bose gas). In terms of critical density, nc, mass, m, interaction, U, and temperature, T, this shift is universal: Δnc(T) = -Cm3T2U, the constant C found to be equal to 0.0140±0.0005. For quantum Bose gas with the scattering length a this implies ΔTc/Tc = C0an1/3, with C0 = 1.29±0.05

Kashurnikov, V

Prokof\u27ev, N

Svistunov, B

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics2001Critical Temperature Shift in Weakly InteractingBose GasV KashurnikovN Prokof 'evprokofev@physics.umass.eduB Svistunovsvistunov@physics.umass.eduFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physical Sciences and Mathematics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationKashurnikov, V; Prokof 'ev, N; and Svistunov, B, "Critical Temperature Shift in Weakly Interacting Bose Gas" (2001). Physics ReviewLetters. 1114.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/1114arXiv:cond-mat/0103149v1  6 Mar 2001Critical Temperature Shift in Weakly Interacting Bose GasV. A. Kashurnikov1, N. V. Prokof’ev2, and B. V. Svistunov 2,31Moscow State Engineering Physics Institute, 115409 Moscow, Russia2 Department of Physics, University of Massachusetts, Amherst, MA 01003, USA3 Russian Research Center “Kurchatov Institute”, 123182 Moscow, RussiaWith a high-performance Monte Carlo algorithm we study the interaction-induced shift of thecritical point in weakly interacting three-dimensional |ψ|4-theory (which includes quantum Bosegas). In terms of critical density, nc, mass, m, interaction, U , and temperature, T , this shift isuniversal: ∆nc(T ) = −Cm3T 2U , the constant C found to be equal to 0.0140±0.0005. For quantumBose gas with the scattering length a this implies ∆Tc/Tc = C0an1/3, with C0 = 1.29 ± 0.05.The problem of interaction-induced shift of the crit-ical temperature in a weakly interacting Bose gas hasbeen attracting a lot of theoretical effort during recentyears [1–5]. Apart from an apparent fundamental inter-est, the study of this problem has been strongly stimu-lated by the experimental achievement of Bose-Einsteincondensation in ultra-cold gases [6] and, correspondingly,the perspective of experimental verification of theoreti-cal predictions. The first experimental study of the crit-ical temperature of interacting Bose gas was reportedrecently for the 4He-Vycor system [7].Despite a considerable number of papers dealing withthe issue, the answer for the critical temperature shiftstill remains a matter of controversy. As it follows fromsimple analysis of dimensions, the result should have theform∆TcTc= C0an1/3 , (1)where C0 is a dimensionless constant. There are reason-able physical arguments for C0 to be positive [2], andmost studies predict that C0 > 0. However, there isa profound discrepancy between different approaches inthe value of C0. The range of variation of different predic-tions is an order-of-magnitude large: 0.34 [2], 0.7 [3], 2.3[5], 2.9 [4], 4.66 [1]. As far as the experiment of Ref. [7]is concerned, where the law (1) was clearly observed, itshould be realized that the only quantity that was reallymeasured was the product C0a. Therefore, until the ac-curate value of the scattering length a is available, it isimpossible to make a reliable conclusion about C0 [8].Though Eq. (1) formally looks like a perturbative cor-rection in terms of the gas parameter an1/3, physicallyit is clear that this is not the case (cf., however, [5]), be-cause in the thermodynamic limit any finite interaction,no matter how small, changes the universality class ofthe phase transition: while the ideal Bose gas belongsto the universality class of the Gaussian complex-fieldmodel, the interacting system pertains to the XY -modeluniversality class. Due to this fact the first-principle ana-lytic description of the interacting Bose gas at the criticalpoint is rather difficult. To the best of our knowledge,up to now there was done only one ab initio Monte Carlosimulation of the interacting gas in the context of ∆Tc(Gru¨ter, Ceperley, and Laloe¨, Ref. [2]). An alternativeMonte Carlo approach by Holzmann and Krauth [5] wasbased on a hypothesis that Eq. (1) can be obtained ina (rather sophisticated) perturbative way, by simulatingan ideal gas.The goal of this Letter is to develop a numeric schemefor describing weakly interacting Bose gas in the fluc-tuation region, which, in particular, could produce anaccurate result for C0. The crucial point that rendersthis aim attainable is the universality of the long-wavebehavior of weakly interacting |ψ|4-theories in the fluc-tuation region at transition point: All these theories, nomatter quantum or classical, continuous or discrete, leadto a generic long-wave Hamiltonian (cf., e.g., [4])H =∫ {12m|∇ψ|2 + U2|ψ|4}dr . (2)From this universality it follows that in any such systemthe shift of the critical density (which is not sensitiveto the ultra-violet cutoff of the theory) is given by theformula (from now on h¯ = 1)∆nc(T ) = −Cm3T 2U , (3)where C is a universal constant. Then, the critical tem-perature shift (which is not universal, being sensitive tothe short-wave physics) can be obtained for each partic-ular system from the obvious relation∆Tc∆nc= −dT(0)c (n)dn, (4)where T(0)c is the critical temperature of the correspond-ing ideal system. From Eq. (4) one readily obtains forthe quantum Bose gasC0 ≈ 91.8C . (5)The universality of the density shift (3) suggests thatit can actually be calculated not in a continuous quan-tum system, where simulations are computationally ex-pensive, but in a discrete classical model1H = −∑<ij>[ψ∗i ψj + c.c.] +U2∑i|ψi|4 , (6)where i and j stand for the sites of 3D cubic lattice,< ij > denotes nearest-neighbor sites, and ψi is a com-plex variable. The long-wave behavior of the discretesystem (6) is described by the Hamiltonian (2) withm = 1/2. The advantage of utilizing model (6) for nu-meric analysis is associated with the existence of a verypowerful Worm algorithm for simulating discrete classicalsystems of this type [10].Analysis of dimensions. We start with renderinggeneral considerations leading to Eq. (3). Consider thegrand canonical counterpart of the Hamiltonian (2)H ′ =∫ {− 12m|∇ψ|2 + U2|ψ|4 − µ˜|ψ|2}dr . (7)Here µ˜ is an effective chemical potential [which, generallyspeaking, does not coincide with the bare chemical po-tential of the original system for which H ′ plays the roleof the effective Hamiltonian]. As the field ψ describesonly the long-wave modes of the original system, it issupposed to have some ultra-violet cutoff:ψ(r) =∑k<k0akeikr . (8)The choice of k0 is arbitrary as long as k0 is larger thanthe momentum kc which characterizes the momentum re-gion where harmonics are strongly coupled to each other(k ≫ kc harmonics are almost free). If k0 ∼ kc, then allthree terms in Eq. (7) are of the same order, and we havethe following relationsk2c/m ∼ |µ˜| ∼ n˜U , (9)n˜ ∼ 〈|ψ|2〉 ∼∑k<kcnk , (10)where nk = 〈|ak|2〉 and 〈. . .〉 means statistical averaging.According to Eq. (10), n˜ ∼ k3cnkc . Since kc is a mo-mentum separating strongly coupled long-wave harmon-ics from ideal short-wave ones, the order-of-magnitudeestimate for nkc may be obtained from the ideal gas for-mula :nkc ∼Tk2c/2m− µ˜∼ Tµ˜. (11)Substituting this back to Eqs. (9)-(10) we see thatkc ∼ m2TU , n˜ ∼ m3T 2U . (12)Consider now the shift of the critical density∆nc(T ) =∑k[n(crit)k − n(crit,0)k ] , (13)where n(crit)k and n(crit,0)k are the occupation numbers atthe critical density (corresponding to a given tempera-ture T ) for interacting and non-interacting system, re-spectively. The main contribution to the sum in Eq. (13)comes from k ∼ kc, since at k ≫ kc the terms n(crit)k andn(crit,0)k are almost equal and compensate each other. Atk ∼ kc one can use the estimate (11), which means that∆nc ∼ n˜, and we arrive at Eq. (3).Numeric procedure and results. Our Monte Carloscheme is based on the Worm-algorithm simulation ofthe high-temperature expansions for the two-point cor-relator 〈ψ∗i ψj〉 in the grand canonical ensemble. (Thedescription of this approach and performance tests see inRef. [10]).To proceed, we need a formal definition of the criticalpoint for finite system with linear dimension L, which,on one hand, is consistent with the thermodynamic limitL → ∞, and, on the other hand, is convenient from thecomputational viewpoint. We adopt the following defi-nition. By critical we understand the point where thecondensate density equals ton(crit)0 (L) = T/(3.75L) . (14)This definition is optimized for the ideal gas (minimalfinite-size corrections for the total density). Neverthe-less, it proved to be quite satisfactory for the interactingsystem as well, including the limit of strong interaction.The classical model (6) possesses an obvious similar-ity property: The transformation ψ →√Tψ, U → U/Treduces the problem to the corresponding T = 1 case.Hence, without loss of generality, we set T = 1.0.25 0.251 0.2520.0060.0070.2514 0.2516nn0n0(crit)nn0(crit)n0FIG. 1. Co-variance of n0 and n for U = 1, L = 40, andtwo different values of µ. Data points correspond to successiveMonte Carlo outputs (n, n0). The crucial observation here isthat fluctuations of n0 and n are strongly correlated, and fol-low the line: δn0/δn ≈ dn¯0/dn¯, where bars above the symbolsdenote statistic limits. With changing chemical potential thewhole linear-shape pattern just shifts along its axis. We thusobtain nc(U = 1, L = 40) = 0.25143(3) from this plot.2The procedure of determining the critical density is asfollows. We iteratively tune the chemical potential µ un-til (14) is satisfied with a required accuracy, and thencalculate the corresponding density. For small U , wherea high precision is required to resolve a small effect [11],we radically improve the efficiency of the pinpointing pro-cedure by utilizing the covariance (see Ref. [12]) in thestatistical fluctuations of Monte Carlo data for n0 and n.The existence of strong covariance between fluctuationsof n0 and n (Fig. 1) allows one to extract the depen-dence n0(n) with significantly higher accuracy than theaccuracy of separate calculations of n0(µ) and n(µ).0 20 40 60 80 1000.230.240.250 5 100.25ncUFIG. 2. Critical ratio (n/T )c for the model (6), as a func-tion of interaction U . The upper bound of the box corre-sponds to (n/T )c = 0.25274 of the ideal system, and thelower bound corresponds to the U → ∞ limit, which is theXY -model with (n/T )c ≈ 0.22708. Insert: The critical den-sity at small U . The dashed line corresponds to Eq. (3) withC = 0.0140. The solid line corresponds to more accurate ap-proximation Eq. (17) at L→∞: ∆nc = −Cm3T 2U/(1+b0U)with b0 = 0.123.Prior to inspecting the region of small U , it is instruc-tive to consider the behavior of the critical density for themodel (6) at finite interactions, in view of a remarkablecontrast to the quantum case. In a quantum gas withthe parameter an1/3 of order unity, the shift of the crit-ical temperature is determined by a competition of twoeffects [2]: (i) the suppression of the long-wave fluctu-ations of density (which stimulates condensation), and(ii) the depletion of the long-wave components of thefield (which effectively reduces density, and thus drivesthe system away from the phase transition). The firstcircumstance dominates at small interaction, while forlarger an1/3 the second effect takes over. Following thiscompetition, the shift of the critical temperature goesthrough a maximum around an1/3 <∼ 1, and then goesdown finally becoming negative [2,7]. While the first ef-fect is of classical-field nature and thus is relevant to (6),the second effect is of quantum nature, and does not haveany classical analog. That is why in the classical systemwe expect that the (negative) shift of the critical densitywill be a monotonous function of U . This is preciselywhat is revealed by our simulation (Fig. 2). The criticaldensity is monotonously decreasing with interaction, sat-urating at U →∞ to the known value (n/T )c ≈ 0.22708corresponding to the XY -model (see, e.g., Ref. [13]). [Asan illustration of the fact that the criterion (14) worksreasonably well for the strongly interacting system, we re-produced the above-mentioned critical point of the XY -model with the accuracy of 5 significant digits by thebrut-force simulation of large systems with L = 160, andchecked that finite-size systematic errors are almost ir-relevant already for L = 40.]Now we turn to the region of small U . As the accuracyof calculations can be significantly improved by a properfinite-size analysis of the data, we begin with general-izing relation (3) to the finite-size case. We note thatin the limit U → 0 the only relevant lengthscale is themode-coupling radiusrc =2pikc∼ 1m2TU, (15)Hence, rc defines the characteristic size of the systemthat can be considered as macroscopic with respect to(small) interaction: The system is in the proper thermo-dynamic limit if, and only if L≫ rc. In the limit L≪ rcinteractions can be treated perturbatively, and thus areessentially irrelevant. From this consideration we arriveat the following generalization of Eq. (3):∆nc(T, L)m3T 2U= −f(rc/L) +O(U) , (16)where the particular form of the dimensionless function fdepends on the definition of nc(L), apart from universallimits f(0) = C and f(∞) = 0. On the basis of Eq. (16),we employ the following scaling function which shouldadequately describe the limit U → 0, L→∞− ∆nc(T, L)m3T 2U=C1 + b0U + (a1 + b1U)xη, (17)where x = 1/(Lm2TU) and η = 1.038 [13] in accor-dance with expected asymptotic behavior of the finite-size corrections for XY -type models. Fitting parametersC, b0, a1, and b1 were obtained by stochastic optimiza-tion (C = 0.0140, a1 = 1.29, b0 = 0.123, b1 = 0.744).As is seen from Fig. 3, the scaling function (17) worksextremely well. With errorbars defined from the uncer-tainty of the fitting procedure, for the constant C wehaveC = 0.0140± 0.0005 , (18)which, according to Eq. (5), meansC0 = 1.29± 0.05 . (19)30 1 2 30y1x-  0.125-  0.25-  0.5-  1.0-  2.5-  5.0FIG. 3. Scaling plot of y = 102 ∆nc/(m3T 2U) vsx = 1/(Lm2TU) at small U . The upper curve is given bythe scaling function at U = 0. Note that while this functionis not supposed to fit the data at x ≥ 1, actually it is ratheraccurate up to x ∼ 3.While the particular form of f(x) in the region 0 <x < ∞ is sensitive to the definition of nc(L), the typi-cal scale x ∼ xc of significant variation of f(x) from itslimiting value f(0) depends only on the mode-couplingradius: xc ∼ 1/(rcm2TU). From Fig. 3 we deducerc ≈ 1/(m2TU) . (20)Using Eq. (20), one can readily estimate how small shouldbe the interaction for a particular weakly interacting sys-tem to demonstrate the universal long-wave behavior.The generic requirement is that rc be much larger thanother microscopic lengthscales at which the behavior ofthe system becomes non-universal (due to the discrete-ness of the system, or/and quantization of occupationnumbers). The most severe requirement for the quantumBose gas is rc ≫ λT , where λT ≈ n−1/3 is the thermalde Broglie wavelength at the Bose condensation point.In terms of the gas parameter this means an1/3 ≪ 0.025,and we immediately realize that neither Monte Carlo sim-ulation of Ref. [2], nor the experiment of Ref. [7] havereached the universality region: In both cases the mini-mal value of na3 was ∼ 10−5, while our estimate showsthat one may expect universality only at na3 <∼ 10−7.In conclusion, we have developed a Monte Carlo ap-proach for the study of weakly interacting |ψ|4-theories inthe fluctuation region. With this technique we obtainedan accurate result for the critical temperature of weaklyinteracting three-dimensional Bose gas, and establisheda criterion of its applicability. We note that the methodcan be also applied to weakly interacting Bose gas in twodimensions. The CPU time required for collecting high-precision data reported above is equivalent to two yearson PIII-733 processor.We are grateful to J. Machta for his interest and discus-sions. This work was supported by the National ScienceFoundation under Grant DMR-0071767.[1] H.T.C. Stoof, Phys. Rev. A 45, 8398 (1992); M. Bijlsmaand H.T.C. Stoof, Phys. Rev. A 54, 5085 (1996).[2] P. Gru¨ter, D. Ceperley, and F. Laloe¨, Phys. Rev. Lett.79, 3549 (1997).[3] M. Holzmann, P. Gru¨ter, and F. Laloe¨, Euro. Phys. J. B10, 739 (1999).[4] G. Baym et al., Phys. Rev. Lett. 83, 1703 (1999).[5] M. Holzmann and W. Krauth, Phys. Rev. Lett. 83, 2687(1999).[6] E.A. Cornell et al., Science 269, 198 (1995); K.B. Daviset al., Phys. Rev. Lett. 75, 3969 (1995); C.C. Bradley,C.A. Sackett, and R.G. Hulet, Phys. Rev. Lett. 78, 985(1997).[7] J.D. Reppy et al., Phys. Rev. Lett. 84, 2060 (2000).[8] The authors of Ref. [7] estimate C0 by adopting a =0.22 nm, following Ref. [9], which is appropriate for de-scribing properties of solid 4He. The reliability of this as-sumption for the 4He-Vycor system is questionable, givencompletely different kinematics at the atomic lengthscale.[9] K.J. Runge and G.V. Chester, Phys. Rev. B 38, 135(1988).[10] N. Prokof’ev and B. Svistunov, cond-mat/0103146.[11] The difference in critical density between the Gaussianand XY-models is only ∼ 10%, and for U ∼ 1 the densityshift (see Figs. 2,3) is of order 0.001. For the reliable esti-mate of C one has to perform calculations with accuracybetter than 10−4.[12] A.W. Sandvik, Phys. Rev. B 54, 5334 (1996).[13] H.G. Ballesteros et al., Phys. Lett. B 387 125 (1996).4

Critical Temperature Shift in Weakly Interacting Bose Gas

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