Two-dimensional Bose-Einstein condensate under pressure

Evading the Mermin-Wagner-Hohenberg no-go theorem and revisiting with rigor the ideal Bose gas confined in a square box, we explore a discrete phase transition in two spatial dimensions. Through both analytic and numerical methods we verify that thermodynamic instability emerges if the number of particles is sufficiently yet finitely large: specifically $N\geq 35131$. The instability implies that the isobar of the gas zigzags on the temperature-volume plane, featuring supercooling and superheating phenomena. The Bose-Einstein condensation then can persist from absolute zero to the superheating temperature. Without necessarily taking the large $N$ limit, under constant pressure condition, the condensation takes place discretely both in the momentum and in the position spaces. Our result is applicable to a harmonic trap. We assert that experimentally observed Bose-Einstein condensations of harmonically trapped atomic gases are a first-order phase transition which involves a discrete change of the density at the center of the trap.

Introduction. The existence of Bose-Einstein condensate (BEC) in two spatial dimensions (2D) is a longstanding controversial issue, attracting a wide range of interests from both theoretical and experimental perspectives. While the Mermin-Wagner-Hohenberg (MWH) theorem [1,2] prohibits free bosons from condensing on a homogeneous infinite plane via long-range thermal fluctuations, 2D BEC has been experimentally realized for harmonically trapped atomic gases, c.f. [3] and references therein, in support of some analytic estimations [4][5][6].
In this work, we intend to revisit with rigor a textbook 2D system of an ideal Bose gas which is confined in a square box, with the area (or 2D volume), V , and the number of particles, N . Experimental realization of such a system has just begun this year [7]. In contrast to the homogeneous infinite plane, the presence of the box breaks the translational symmetry and converts the energy spectrum from continuous uncountable infinite to discrete countable infinite. This leads to fundamental differences between the two systems, effectively classical v.s. quantum, which should persist even in the large V limit of the box. As the harmonic potential allows 2D BEC, it is physically natural to expect that a "small" box should do so as well.
It is the purpose of this Letter to report that, indeed the ideal Bose gas in the 2D box features BEC at a priori finite temperature which is given by the pressure, P , and the mass of the particle, m, as T BEC = kB 12 πm P . More precisely, including the leading order correction by finite N , BEC occurs whenever [c.f. Eq.(38) and FIGs. 2, 3], and further can be 'superheated' [8] up to [c.f. Eq.(39)], T ≤ T * * = kB 12 πm P 1 + 9 (2) * Electronic correspondence: park@sogang.ac.kr At the superheating point, the ground state occupancy rate is close to unity as N0

and the density is as high as
. Hence the latter diverges in the thermodynamic limit and the MWH nogo theorem can be circumvented (c.f. discussion around Eqs. (18,19) in Ref. [2]).
Our main results are summarized in Eqs.(38,39) as well as in FIGs. 1, 2, 3 which are methodologically twofold: i) analytical for large N and ii) numerical for arbitrary N . They are in excellent agreement and show us that, like the 3D ideal Bose gas [9][10][11][12], the 2D gas also condenses discretely at the finite temperature, in both the momentum and the position spaces, provided that the pressure is kept fixed and that there are sufficiently-yet-finitely many particles, specifically N ≥ N c = 35131.
Setup. In a parallel manner to Ref. [11], here we focus on the grand canonical ensemble with the fixed average number of particles, N . From the non-relativistic dispersion relation, E = p 2 /(2m), the grand canonical partition function is actually a two-variable function depending on the fugacity, z, and the combination of the temperature and the area, T V . This implies a scaling symmetry: different sizes of the volume can be traded with different scales of the temperature. In particular, the small volume or the 'confining' potential effect should persist for large volume.
Specifically we set, as for the two dimensionless fundamental variables in our analysis, In terms of these, the grand canonical partition function reads With the Dirichlet boundary condition which we deliberately impose, n = (n 1 , n 2 ) ∈ N 2 is a positive integervalued 2D lattice vector, such that the lowest value of n 2 is two and σ is bounded from below as σ > −2ε. The average number of particles is then and the standard expression, P = k B T ∂ V ln Z(T, V, z), of the pressure is equivalent to Being a combination of T and P , this dimensionless quantity, T P , can denote the physical temperature on an arbitrarily given isobar. Similarly we may define a dimensionless "volume", and another dimensionless "temperature", Further, the number of particles on the ground state is As we already denoted, N , N 0 , T P , V P and T ρ are all functions of the two variables, ε, σ only. They satisfy identities: Generically, the superheating (BEC) and the supercooling points correspond to the two turning points of an isobar which zigzags on the (T, V )-plane, satisfying the spinodal curve condition, dN = 0, dP = 0, dT = 0 [10,11,[13][14][15]. In our case, the spinodal curve is to be positioned on the (ε, σ)-plane to satisfy dN (ε, σ) = 0 and dT P (ε, σ) = 0, and hence the following linear equation must admit a nontrivial solution, (10) Consequently the 2 × 2 matrix must be singular, This algebraic equation determines the spinodal curve on the (ε, σ)-plane. For consistency, we note that the determinant, Φ, is proportional to dTP dVP N , as where the denominator on the right hand side can be shown to be positive definite [11]. Hence, the vanishing of the determinant is, as expected, equivalent to the vanishing of dTP Our aims are first to solve (11), second to express the solutions in terms of the more physical variables, N , T P , V P , T ρ using (5), (6), (7), (8), and finally to confirm BEC, i.e. N 0 /N → 1, at the superheating point. Searching for spinodal curves near to the thermodynamic limit, we shall focus on the region, ε → 0 + and σ + 2ε → 0 + . Analytic Approximation. To solve the spinodal condition (11) and to compute the number of particles (5), we focus henceforth on Our computational scheme to obtain the analytic solution of (11) is, based on Ref. [11], as follows.
3. Assume an ansatz with two positive quantities: While we put h to be a constant, we allow g to depend possibly on 'ln ε'. The appearance of logarithmic dependency is a novel feature in 2D compared with 3D [11]. The number of the particles on the ground state is now, We further set h = 1, as it agrees with the numerical results and simplifies our algebraic analysis. (13) in powers of ε. For consistency, we should trust only the singular terms which are insensitive to the cutoff, Λ. We shall see that for each quantity, at least first two leading singular powers are Λ-independent.
We set some constants: and consider associated ε-dependent integrals, For s > 0, t > −1, the constants, a s , b t , are finite. Hence neglecting nonsingular terms we may estimate It is then straightforward to see for h > 1 .
(25) From these, the first two reliable leading singular terms for the number of particles, N (5), are In particular, we note that N 0 ≃ g −1 ε −h is significant in N (indicating BEC) only for h > 1. Now, we turn to the computations of the second order derivatives in (13), which in part requires us to consider For r > 1, an integration by parts with trivial boundary contribution gives a simple relation between ω r and α s , such that It follows then For r = 1 we perform the same partial integration, this time receiving nontrivial yet non-singular boundary contribution. We obtain with (A. 3), For r = 1 2 , using (A.2), (A.6), (A.7), we get Our scheme then gives (34) Here for h > 1, we have chosen Λ → ∞.
The numerical values of the constants are Having the key expressions, (23), (26), (30), (33), (34), we are now ready to solve the spinodal condition of Eq.(11). We consider eight possible cases separately: As our ansatz (16) contains a single unknown term, we demand at least the leading power in Φ should be canceled out in a nontrivial manner. It is straightforward to check that only the two values, h = 1 2 and h = 3 2 , admit solutions, as below. * Supercooling spinodal curve of h = 1 2 , (36) * * Superheating (BEC) spinodal curve of h = 3 2 , Main Results. In terms of the physical variables, N , T P (6), V P (7), T ρ (8), the supercooling and the superheating (BEC) points are as follows. * Supercooling point : (38) * * Superheating (BEC) point : Here T BEC P = 24 π 3 denotes a dimensionless constant which gives the 2D BEC critical temperature in the thermodynamic limit : As we see below, this formula holds universally even for the 2D harmonic potential.   Consistency with harmonic trap. For the ideal Bose gas subject to a 2D harmonic potential, V (r) = 1 2 mω 2 r 2 , let us define N (r) to be the number of particles within the radius r and P (r) to be the radially varying pressure. They assume boundary values, N (0) = 0, N (∞) = N , P (∞) = 0. Further from the equilibrium condition, −2πrP ′ (r) = mN ′ (r)ω 2 r, they satisfy In particular, at the origin where BEC is typically observed we have P (0) = mω 2 2π N . Substituting this into Eq.(40), we recover precisely -and satisfactorily -the known BEC critical temperature for the 2D harmonic trap [3,5,6], Discussion. To summarize, when N ≥ N c = 35131, the ideal Bose gas confined in a 2D box reveals a pair of spinodal curves: supercooling and superheating (BEC). The gas condenses discretely at the finite temperature (40) under constant pressure, in both the momentum and the position spaces. As in 3D [9][10][11][12], this is an emergent phenomenon which finitely many bosonic identical particles can feature, without assuming the thermodynamic limit.
In this work we have focused on the grand canonical ensemble. Analyzing instead the canonical ensemble will require more computational power and most likely suggest a different value of the critical number: for example in 3D N c = 7616 (canonical) [9] or N c = 14393 (grand canonical) [11]. Yet, in our opinion, what matters is the existence of such a definite number, alternative to infinity, which may answer to a question, "How many is different?" [12]. For larger values of N the differences due to different choices of the ensembles are expected to be anyhow negligible (c.f. TABLE I in Ref. [11]).
As computable from our analytic solutions, (38), (39), the gap between the supercooling and the superheating temperatures, T * * P − T * P , becomes maximal when the number of particles is equal to N MAX ≃ 1.43056 × 10 6 . This also agrees with the numerical result shown in FIG.1. In a way, the two numbers, N c and N MAX , enable us to divide the Bose gas system into three quantum realms: i) microscopic for 1 ≤ N < N c , ii) mesoscopic for N c ≤ N ≤ N MAX and iii) macroscopic for N MAX < N ≤ ∞.
Lastly, from the formula, if N = N c , this volume expansion ratio becomes 1.31267 which is for consistency of order unity.
We would like to thank Chuan-Tsung Chan and Yong-il Shin for helpful comments. This work was supported by the National Research Foundation of Korea (NRF) with the Grants, 2012R1A2A2A02046739 and 2013R1A1A1A05005747.