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of bosons with several degrees of freedom Condensación de bosones con varios grados de libertad

Trabajo presentado por Rafael Delgado López1 para optar al título de Máster en Física Fundamental bajo la dirección del Dr. Pedro Bargueño de Retes2 y del Prof. Fernando Sols Lucia3

Universidad Complutense de Madrid Junio de 2013 Calificación obtenida: 10 (MH)

1 [email protected], Dep. Física Teórica I, Universidad Complutense de Madrid

2 [email protected], Dep. Física de Materiales, Universidad Complutense de Madrid

3 [email protected], Dep. Física de Materiales, Universidad Complutense de Madrid Abstract The condensation of the spinless ideal charged Bose in the presence of a is revisited as a first step to tackle the more complex case of a molecular condensate, where several degrees of freedom have to be taken into account.

In the charged , the conventional approach is extended to include the macroscopic occupation of excited kinetic states lying in the lowest Landau level, which plays an essential role in the case of large magnetic fields. In that limit, signatures of two diffuse transitions (crossovers) appear in the specific heat. In particular, at lower than the cyclotron frequency, the system behaves as an effectively one-dimensional free boson system, with the specific heat equal to (1/2) NkB and a gradual condensation at lower temperatures.

In the molecular case, which is currently in progress, we have studied the condensation of rotational levels in a two–dimensional trap within the Bogoliubov approximation, showing that multi–step condensation also occurs. statistical mechanics, macrocanonical ensemble, Bose Einstein condensation, , degenerate , Bogoliubov interactions

Resumen En este trabajo se analiza la condensación de un gas de Bose ideal de partículas cargadas y sin espín en presencia de un campo magnético como un primer paso para tratar el caso más complejo de un gas de Bose molecular, en el que se han de considerar varios grados de libertad.

Respecto al gas de Bose cargado, el tratamiento tradicional se extiende para incluir un término de niveles cinéticos excitados en el estado fundamental de niveles de Landau. Dicho término juega un papel especial en el límite de grandes campos magnéticos, en el que aparecen señales de dos transiciones de fase difusas (crossovers) en el calor específico. En particular, a temperaturas menores que la frecuencia ciclotrón, el sistema se comporta como un sistema unidimensional de bosones libres, con calor específico (1/2) NkB, y presenta una condensación gradual a temperaturas más bajas.

En el caso molecular, todavía en estudio, hemos estudiado la condensación de niveles rotacionales en una trampa bidimensional dentro de la aproximación de Bogoliubov, mostrando que también se da la condensación múltiple. mecánica estadística, distribución macrocanónica, condensación de Bose Einstein, densidad de estados, materia degenerada, interacciones de Bogoliubov

-i- Index

I. Introduction...... 1

II. Numerical approaches...... 3

III. Structure of the level population...... 5

IV. Numerical results for N0 and Nk...... 6

V. Specific heat...... 7

VI. Conclusions...... 8

-ii- Condensation of bosons with several degrees of freedom

R. L. Delgado∗ Departamento de F´ısica Te´orica I, Universidad Complutense de Madrid, E-28040, Madrid, Spain

The condensation of the spinless ideal charged Bose gas in the presence of a magnetic field is revisited as a first step to tackle the more complex case of a molecular condensate, in which several degrees of freedom have to be taken into account. In the charged Bose gas, the conventional approach is extended to include the macroscopic occupation of excited kinetic states lying in the lowest Landau level, which plays an essential role in the case of large magnetic fields. In that limit, signatures of two diffuse phase transitions (crossovers) appear in the specific heat. In particular, at temperatures lower than the cyclotron frequency, the system behaves as an effectively one-dimensional free boson 1 system, with the specific heat equal to 2 NkB and a gradual condensation at lower temperatures. In the molecular case, which is currently in progress, we have studied the condensation of rotational levels in a two–dimensional trap within the Bogoliubov approximation, showing that multi–step condensation also occurs.

I. INTRODUCTION are those not having a definite critical , but an interval of temperatures along which the transition The equilibrium behavior of a charged Bose gas (CBG) occurs gradually. The concept was already introduced in in the presence of a magnetic field is a fundamental prob- the study of phase transitions which occur in certain fer- lem in quantum statistical mechanics. It was first ad- roelectric materials [12]. In this work we treat the terms dressed by Schafroth [1] at a time where the BCS the- diffuse transition and crossover as synonyms. The no- ory of had not yet been discovered. tion of BEC of a CBG in the presence of a magnetic field Although it was soon recognized that a model of non- in 3D continued to be studied by Toms and coworkers interacting bosons had little predictive power for super- [13, 14], who showed that, although there is no BEC in conductivity, the CBG problem has continued to attract the sense of a sharp , the specific heat considerable attention because of its appealing simplicity exhibits a clear maximum that can be used to define a and as a model to simulate astrophysical scenarios where critical temperature. Using that definition, the authors charged particles are subject to extremely large magnetic inferred that the critical temperature increases with the fields, or neutral particles move in a fast rotating back- magnetic field [14], reaching the usual value for the 3D ground. It is also interesting as a preliminary step in free Bose gas when no magnetic field is present. The ex- the study of the condensation of interacting bosons in a tension of these studies to the case of vector bosons in magnetic field or under rotation. strong magnetic fields and at high temperatures was per- For the non–interacting case, Schafroth noticed that, formed by Khalilov and coworkers [15–17]. Specifically, in the presence of a magnetic field, Bose–Einstein con- the condensation and effective of a charged densation (BEC) no longer occurs in a strict sense, since vector boson gas were studied, showing that there is no at low enough temperatures, the system behaves as one- true BEC as well, although a significant amount of bosons dimensional (1D) and a non-interacting 1D boson system can accumulate in the at low temperatures. is known to exhibit not a sharp but a gradual (diffuse) More recently, the magnetic properties of charged –1 BEC transition. Later, May [2, 3] investigated the con- Bose in an external magnetic field have been re- densation in the non–relativistic case and showed that a visited, with focus on the competition between diamag- sharp transition occurs for dimension 5. Daicic and netism and paramagnetism [18]. coworkers [4, 5] extended May’s findings≥ to the relativis- In the particular case of a large magnetic field, the tic high-temperature case. Using a different definition of main feature of a CBG is the coexistence of highly de- BEC, Toms [6–8] proved that BEC in the presence of a generate Landau levels and a fine structure of levels with uniform magnetic field does not occur in any number of nonzero momentum parallel to the magnetic field. We spatial dimensions and Elmfors and coworkers [9] consid- propose that a CBG under a large magnetic field can also ered that, in the three–dimensional (3D) case, although a be used as a crude model to understand basic features true condensate is not formed, the Landau ground state of systems where a fine sublevel structure coexists with can accommodate a large boson number. In this spirit, sectors characterized by highly discrete quantum num- Rojas [10, 11] found that BEC may occur in the pres- bers. Specifically, we have in mind the condensation of ence of a homogeneous magnetic field, but there is no bosonic molecules, whose level structure is hierarchical, critical temperature at which condensation starts, the with their rotational states structured into vibrational phase transition being diffuse. Diffuse phase transitions sectors which in turn can be grouped into largely spaced electronic levels. Thus we are motivated by the study of molecular BECs, where a complex molecular level structure may ∗ [email protected] give rise to multi-step BEC. Molecular condensates are 2 expected to display a wealth of fundamental phenom- the energy of each individual boson in a non–interacting ena. For instance, it has been proposed that molecular gas is given by a function like condensates may permit the experimental study of low- ǫ = ǫ(k , k , ), (1) energy parity violation [19]. Specifically, the fact that k 1 2 recently both homonuclear [20], heteronuclear [21, 22] where the values (k , k , ) are the quantum numbers ultracold molecules and heteronuclear molecular 1 2 [23, 24] have been produced in the rovibrational ground of the problem. The number of particles N is given by state, suggests that molecular BEC may not lie too far in dk the future. The conclusion is that the study of multi-step N = −1 . (2) z exp( βǫk) 1 condensation of the CBG is interesting not only from an kX∈K − − academic point of view, but it may provide a qualitative understanding of the condensation of molecules with a In this expression, K represents the space of all possi- hierarchical level structure. ble combinations of non–degenerate quantum numbers (k , k , ); d , the possible remaining degeneracy of k We consider a wide range of magnetic fields because we 1 2 k are interested in understanding the general trends. This state; z, the fugacity; and β, the inverse temperature means that in our analysis we include magnetic fields so (kB = 1). Thus, the challenging task is: given the func- tions d and ǫ , to recover the value of the fugacity z large that would be unrealistic in some contexts [25–27]. k k Comparing our results with the previous related litera- from Eq. (2) for a given value of β and N. The func- tions d and ǫ together with the space K are, actually, ture on the CBG (see, for example, Refs. 13 and 14), k k we notice the existence of a term that has so far been the solution of the eigenvalues problem for an individual neglected. This is the occupation of states in the low- boson. est Landau level but with arbitrary kinetic energy in the Once the value of z for a given β and N has been obtained, the population of a certain subset of levels K˜ direction parallel to the magnetic field. This term is par- ⊂ ticularly important at large magnetic fields and in fact K is given by is responsible for the two-step condensation we refer to dk in the title and which we shall discuss in detail. We NK˜ = −1 , (3) z exp( βǫk) 1 advance here that, as the temperature descends, a first kX∈K˜ − − step is defined by the onset of a preferential occupation of states in the lowest Landau level but without any of the total internal energy U, by them absorbing a large fraction of the bosons. Then a d ǫ plateau is reached where the system behaves as effectively U = k k (4) z−1 exp( βǫ ) 1 one-dimensional. The second condensation step occurs k kX∈K − − at even lower temperatures, when the true one-particle ground state becomes host to a macroscopic fraction of and the heat capacity at constant volume, by the total boson number. By the time this occurs, the sys- ∂U tems behaves as one-dimensional, which translates into Cv = −1 . (5) a diffuse Bose-Einstein condensation, as opposed to the ∂(β ) conventional, sharp phase transition which is character- In practice, the space of quantum numbers K will be of istic of three dimensions. Interestingly, these two dis- the form Nn, where n is the number of degrees of freedom. tinct, both gradual condensation steps have been noted Therefore, the summations k∈K˜ are finally numerical by Ketterle and coworkers [28, 29] in a different but some- series in n variables. Thus, obtaining z requires the so- what analogous context, namely, that of condensation in lution of an equation which includesP in a non–trivial way a strongly anisotropic harmonic trap. As compared to such a series. a CBG in a large magnetic field, the role of the Landau The usual (i.e., textbook) approach for dealing with levels is played there by the transverse subbands char- Eq. (2) is replacing the summations over all quantum acterizing the motion in the most confined direction. In numbers by integrals. For Hamiltonians such harmonic Refs. 28 and 29 the above mentioned term of the oc- or hard wall boundaries ones, there is a change of vari- cupation of the states with the lowest value of the most ables which allows to reduce the number of variables of discrete quantum number, was correctly included, which integration. And the resulting unidimensional integral resulted in the prediction of a two-step condensation. result to be a function, whose properties Although the macrocanonical ensemble of a non– are well known [30, 32]. interacting system is conceptually very simple, the calcu- Lastly, a numerical treatment, a truncated series or lations which it involves are challenging, specifically when the properties of the resulting polylogarithm function is several quantum numbers are considered. In addition, as used. Anyway, the crucial point is the reduction of the the Bogoliubov approximation, when valid, transforms number of variables. Even if a computational library interacting to non–interacting problems [30, 32], it would is used to evaluate the resulting polylogarithm function, be eventually analyzed through the same formalism. such a library may use a (conveniently tuned) numerical As it is well known (see, for example, Refs. 30 and 32), series instead of a quadrature method. 3

The problem is that this classical approach fails if a Taking βǫr2 = x, the integral term remains as large fraction of bosons is concentrated in a reduce subset 1 2π ∞ x1/2 of K or, eventually, in a unique value of k. Of course, N = dx (9) as we are dealing with Bose-Einstein condensation, it is − z−1e3βǫ 1 (βǫ)3/2 z−1ex 1 Zx=0 expected that, for sufficiently low temperatures (high β), − − the population of bosons tends to concentrate in the level Now, defining the polylogarithm function as of minimum energy [30, 32]. This fact is usually taken 1 ∞ xn−1 into account by adding a term of the form g (z)= dx , (10) n Γ(n) z−1ex 1 Z0 − d0 Eq. (9) can be written as N0 = −1 (6) z exp( βǫ0) 1 − − 1 2πΓ(3/2) N = g (z). (11) − z−1e3βǫ 1 (βǫ)3/2 3/2 to the resulting polylogarithm function, and the corre- − sponding correction to the internal energy U. We notice that the complexity of evaluating the three– For instance, the usual case of a 3D Bose variable series of Eq. (7) has been reduced to that of trap is described by evaluating a polylogarithm function g3/2(z), which actu- ally can be written as ∞ ∞ ∞ 1 ∞ j N = 2 2 2 , (7) z −1 βǫ(kx+ky +kz ) z e 1 g3/2(z)= . (12) kx=1 ky =1 kz =1 − j3/2 X X X j=1 X where ǫ = 2/(2mL2) is a constant. This expression is Therefore, we arrive at a single–variable series. In the approximated by process, the only approximation made is that of substi- tuting the numerical series by integrals but maintain- 1 ∞ 4πr2 ing the term corresponding to the fundamental level N = 2 2 2 + dr 2 (8) k = k = k = 1, that is z−1eβǫ(1 +1 +1 ) z−1eβǫr 1 x y z Z0 −

∞ ∞ ∞ 1 1 ∞ ∞ ∞ 1 N = 2 2 2 + dx dy dz 2 2 2 , (13) −1 βǫ(k +k +k ) −1 3βǫ −1 βǫ(x +y +z ) z e x y z 1 ≃ z e 1 0 0 0 z e 1 kXx=1 kXy =1 kXz =1 − − Z Z Z −

At low temperatures, the condensate term will be even and computes numerically the values of z (actually, the β greater than the integral which approximates all the , where z = e ), U and Cv by nu- other terms of the series. If it were neglected, the ef- merically solving Eq. (2). The Newton-Raphson algo- fect would be the absence of such a condensed phase, rithm is used and the derivative ∂N/∂z was explicitly thus qualitatively modifying the physics of the problem. hard–coded. Thus, the f function whose zeros are sought Merging all the other terms of the original series into through the Newton-Raphson algorithm is the integral assumes that each of such terms is negligi- ble compared to the whole series. Even more, it also N 1 f()= (14) assumes that the series summing over a given ki for cer- eβ(ǫi−) 1 i=1 tain i = x, y, z and evaluating the series over the other X − variables is also negligible. Although critical for some and its derivative, purposes, this is seldom explicitly noticed, the articles of N Ketterle and coworkers [28, 29] and our own [34] being β(ǫi−) df() diβe an exception. = 2 (15) d β(ǫi−) i=1 e 1 X − Then, one of the tasks is that of computing a list of II. NUMERICAL APPROACHES energetic levels which has to be large enough to include all the relevant levels, so that the effect of the finiteness In this work, we have followed three numerical ap- of such a list can be neglected. This is computationally proaches. The first one is based on the development of a cheap when we are dealing with one degree of freedom, C++ computer program which accepts a finite list of ener- but becomes demanding when we have to sum contribu- getic levels ǫi and their degeneracies di (i = 1, 2,...,N) tions from several degrees of freedom, to sort the levels 4 by energies and to sum the degeneracies where the same energy levels are found as a result of the sum. And, at 108 PARI/GP for ǫ =0 the same time, special care must be taken in order not 0 107 PARI/GP for ǫ =3 to forget levels with less energy than a certain cutoff. 0 π ǫ3/2 This has been successfully solved by using the program 106 6

PARI/GP [33]. Actually, it is a Computer Algebra System 105 (CAS) developed by Henri Cohen and coworkers for fast calculations in the field of number theory, which can deal 104 with finite but extremely large series. When multiplying 103 and powering the series, PARI/GP computes the order of the resulting series correctly, without computing or keep- 102 ing unnecessary terms. 101 Thus, for each degree of freedom, a list of energy levels 0 is computed. The energies are coded in the power of 10 the monomials and the associated degeneracies in the 10-1 100 101 102 103 104 105 multiplicative terms. That is, if the have a list of N levels with energies ǫi and degeneracies di, the associated truncated series will be Figure 1. Exact solutions for the cumulative density of states N versus energy computed with PARI/GP [33] for the cases of ǫi ǫN +1 ǫ0 = 0 and ǫ0 = 3, compared with the approximation of the S[N, ǫi, di]= dix + O(x ) (16) ǫ ′ ′ π 3/2 continuous limit R0 ρ(ǫ ) dǫ = 6 ǫ . i=1 ! X PARI/GP does not accept series with rational coefficients, but this can be bypassed by multiplying the column of 101 energies by a common multiple of the denominators, and adding constants if needed (i.e., for dealing with a con- stant +1/2). This transformation must be reversed be- fore using the list, although the addition of a finite con- 100 stant c only adds a constant cN to the total internal en- ergy U and a rescaling factor to the fugacity z, which is not directly measurable. Of course, this approach cannot 10-1 deal with irrational energetic levels. But all the energetic levels considered in this work are integer or multiples of 2 1/2. They come from potential wells (ǫ k + C), har- -2 k ∝ 10 monic traps (ǫk k + C), and vibro–rotational levels ∝ Error |1−(y /y )| for ǫ =0 (ǫν,J = Aν + BJ(J +1)+ C). In the vibro–rotational contin PARI 0 Error |1−(y /y )| for ǫ =3 case, ν = 0, 1, ... and J = 0, 1, ... are, respectively, the contin PARI 0 10-3 quantum number associated with the vibrational and ro- 100 101 102 103 104 105 tational levels. And A, B and C are constants which depend on the properties of the molecule. Thus, it can be proved that the task of merging several Figure 2. Relative error of the approximation of the contin- lists of energetic levels is reduced to that of multiplying uous limit given two models (ǫ0 = 0 and ǫ0 = 3). The exact the series associated to such lists. This is actually a non– solutions have been computed with PARI/GP [33]. trivial computation, but PARI/GP can deal efficiently with it, both in terms of processor and memory usage. The only handicap is that PARI/GP does not provide a method with the exact (and discrete) solution for both cases to parallelize this computation, and that the available ki = 0, 1, 2, ... (ǫ0 = 0) and ki = 1, 2, 3, ... (ǫ0 = 3). RAM memory on a single computer is a practical limit. Actually, the true solution would be that corresponding 2 2 2 This could be problematic if several energy hierarchies to ǫk = kx + ky + kz with ki = 1, 2, 3, ... . separated by some orders of magnitude were taken into It is remarkable that these results apply to the case account. of both bosons and , if the respective statistics To test this approach, we have compared in figs. 1 are used. However, when dealing with a large number and 2 the approximate cumulative density of states of of particles (thermodynamic limit), the effect is negligi- a 3D gas confined in a potential well (eigenstates ǫk = ble. Specially in the case of fermions, where the so called 2 2 2 kx + ky + kz , normalization ǫ = 1), i.e., (ǫF ) tends to be far away from the ground state of a single . This effect is likely to be dom- ǫ ǫ π π inant in the regime of very few particles and even with ρ(ǫ′) dǫ′ = ǫ1/2 dǫ′ = ǫ3/2, (17) 4 6 an isolated particle in a well trap. Z0 Z0 5

Thus, a mixed formalism could be developed where are expected over a certain energy level [35]. several degrees of freedom could be fitted by a contin- Nevertheless, the most general case would be that of a uous function while the others should be kept discrete. Bose Einstein condensate with strong interactions, which Actually, those quasi–continuous levels would be approx- would invalidate the hypothesis of non–interacting par- imated by a discrete structure of levels separated by a ticles without giving a feasible way of recovering it via a small distance compared with the other quantum num- change of variables. In either case, there is a bruteforce bers but much larger than the exact separation. This method, namely lattice Monte Carlo, which would allow procedure allow us to perform the computations with to extract the internal energy and almost any thermody- limited resources. Special care should be taken on the namical variable (spending enough computer time). The validity of such approximations in the various regions of price to be paid would be the loss of simplicity of our the spectrum. Such an analysis has not been performed semi–analytical approaches and the clear identification because it is beyond the scope of our present work. of the physical interactions which are responsible for the The second approach, followed by us in Ref. [34], was thermodynamical behavior of the Bose gas. Such method based on the use of the usual analysis which involves have indeed been widely used to study phase transitions, the approximation of substituting series by integrals and and constitutes a wide field of current research. See, for using the properties of polylogarithm functions to re- example, Refs. [36–39]. duce the integration variables to just one. However, for Finally, we note that the described methods give us solving Eq. (2), a numerical method for extracting the U(β), but the specific heat at constant volume, CV = zero of a function has been used. Furthermore, like in β2 (∂U/∂β) , must be obtained numerically from such − V Refs. [28, 29], several terms have been treated separately expression. That is, through a numerical derivation. from the integral over all variables. These terms cor- When using Maple, its numerical libraries are used. But respond to the ground level of a certain variable while when using either PARI/GP [33] or the finite series ap- summing over the rest. This approach was explained in proach, we have to compute with a numerical deriva- section I, when describing the usual approaches to the tion method implemented by ourselves. The simplest macrocanonical problem. Furthermore, as this approach one has been selected. That is, simply taking df/dx ≈ was chosen to deal with the case of a charged Bose gas (f(x + h) f(x))/h + O(h). − in a large magnetic field [34], it will be explained in more detail in section III. III. STRUCTURE OF THE LEVEL And the third approach is the numerical evaluation of POPULATION the original multi–variable series, by using the program PARI/GP [33], which also has very fast methods for di- rectly evaluating numerical series of different kinds. No Let us consider a charged gas of bosonic particles con- ad hoc cutoff is needed except for the working precision. fined in a 3D box of volume AL, where A is the area It can handle two–variable series easily and it can com- in the x–y plane and L is the length in the z direction. pute up to three–variable series in a reasonable time, al- Away from the edges of the box, the energy of a charged though some optimization has to be manually performed. particle in a uniform magnetic field B pointing in the z Specifically, the optimal method to sum the series de- direction is quantized as [30] pends on the space of parameters (i.e., β and the different 1 parameters which define the hierarchy of energy levels). E (n )= ω n + + εn2, (18) n z 2 z This method, although overlaps with the first one (that   of the truncated series), has been used to deal with the where ω = qB/mc is the cyclotron frequency, ε case of a molecular condensate with rotational levels 2π2/2mL2 characterizes the level spacing in the z di-≡ within the Bogoliubov approximation. It is remarkable rection, and both n and nz run over natural numbers that in that case there is no clear possibility of reducing (we assume hard wall boundary conditions at z = 0 and degrees of freedom via a change of variable in the integral z = L). The index n is said to characterize the Landau approximation, thus making it impossible to follow the level. usual approach of using either polylogarithm functions or The area A is assumed to be large enough for the role integrals. Of course, changing the series by multivariate of edge states in the x y plane to be negligible. Specif- integrals still remains possible (within the limits of such ically, l2 A, where l−2 = c/qB is the magnetic length approximations). But this does not allow to reduce the squared. ≪ number of variables. We start by writing the total number of particles N as However, if we were to deal with the more general case ( = 1) of an experimental rovibrational spectrum, or a more de- ∞ ∞ tailed model, it would be necessary to use the first ap- z N = d 2 , (19) proach of summing over a finite (but large) set of energy exp(βεnz + βωn) z nz =0 n=0 − levels. Even more, including a limited set of levels is un- X X −1 2 avoidable, since the theory itself is limited in energies. where β (kBT ) , d = A/l is the Landau degeneracy, For instance, in the molecular case, chemical reactions and z ≡exp(β) is the fugacity, with the chemical po- ≡ 6 tential [which absorbs the 1/2 term in (18)]. We partition The factor d appearing in all the occupation numbers, the total particle number into four different groups: Eqs. (21)-(28), reveals the existence of Landau degen- eracy, which in particular reflects a multiplicity of one- N = N0 + Nk + Nm + Nmk . (20) particle ground states where bosons are in both the ki- netic and magnetic ground state (n = nz = 0). N0 is the number of particles in the one-particle ground The second term, Nk, represents the occupation of par- state (n = nz = 0), ticles lying in the ground Landau manifold but in ex- z cited kinetic states (with motion in the z direction). We N = d ; (21) 0 1 z point out that this term has often been neglected in stud- − ies of the CBG. We show below that Nk plays a most Nk is the number of particles in the ground state of the important role in the thermodynamic properties of the magnetic level but in excited kinetic levels (n = 0 and CBG under large magnetic fields. The third term, Nm, n = 0), which accounts for those bosons which are in the kinetic z ∞ ground state but in excited magnetic states, will not be z studied here because it acquires importance only for very N = d ; (22) k exp(βεn2) z small magnetic fields, that is, when the energy of the first n =1 z Xz − magnetic excited state is much lower than the energy of the first kinetic state (ω ε). This would describe a Nm is the number of particles in the kinetic ground state ≪ but in excited magnetic states, two-dimensional boson gas which will not be considered here. The last term, Nmk, corresponds to the conven- ∞ z tional term in which the bosons are in both magnetic N = d ; (23) m exp(βωn) z and kinetic excited states. It is the dominant term at n=1 X − high enough temperature, where the 3D limit is recov- ered (ηm,ηk 1). and Nmk is the number of particles in excited states of ≫ both magnetic and kinetic degrees of freedom, Finally, we note that the order of the polylogarithm functions appearing in Eq. (28), gd/2(z), corresponds ∞ ∞ z to the population of excited states of free particles in d Nmk = d . (24) dimensions. exp(βεn2 + βωn) z n=1 n =1 z X Xz − The term N can be approximated by k IV. NUMERICAL RESULTS FOR N0 AND Nk

Nk dηkg1/2(z). (25) ≃ For zero magnetic field and vanishing ε the CBG re- 1/2 −1/2 ∞ j α sembles a 3D free Bose gas, showing BEC at the usual where ηk = (π/4) (βε) and gα(z) z /j ≡ j=1 critical temperature. In this case, a discontinuous phase stands for the polylogarithm function of order n [31]. ∞ P∞ transition occurs [32]. However, for large values of the Here, the identity lim → λf(λj) = f(x) has λ 0 j=1 0 magnetic field (ω ε), the transition becomes diffuse been used assuming that λ = √βε is sufficiently small. ≫ P R [10, 11] and only crossover temperatures can be defined. Similarly, Nm and Nmk can be approximated as: Specifically, due to the presence of the Nk term, two dif- ferent crossover temperatures will be identified. It is quite remarkable that, for these high magnetic Nm dηmg1(z) (26) ≃ fields, Nk gives a significant contribution to N, as re- and vealed in Fig. 3, where we show the relative weight of N0(T ) described by both the conventional and extended Nmk dηmηkg3/2(z), (27) approaches (Nk = 0 and Nk = 0, respectively) for sev- ≃ eral values of the magnetic field. From this figure it can −1 where ηm (βω) . We can group these approximations be noted not only that the phase transition becomes dif- to write the≡ total particle number as fuse, but also that the macroscopic occupation of bosons with kinetic excited states lying in the ground Landau N manifold is very important at large magnetic fields. = g0(z)+ ηkg1/2(z)+ ηmg1(z)+ ηkηmg3/2(z). (28) d We note that, within this limit of large magnetic fields and for z 1 (i.e. β = β 1, high tempera- As noted by the authors of Ref. 29 in a similar con- ture limit for≪ the effective| | 1D− system),≫ the temperature text (with harmonic instead of hard-wall confinement), dependence of N /N tends to the expression Eq. (28) is an excellent and controlled 0 approximation for the whole temperature regime. We N0 1 note that, in this equation, the second, third, and fourth = , (29) N 1+ ηk terms become dominant in the respective cases ηk 1, η 1 and both η ,η 1. ≫ as shown in Fig. 3. In fact, this limit function cannot be m ≫ k m ≫ 7

5 Figure 3. Computed relative weight of N0 for N = 10 and Figure 4. Computed relative weights of N0 ( line), Nk 2 5 5 5 ω = 10 (black lines), ω = 10 (red -light gray- lines) and (dashed line) and Nmk (dotted line) for ω = 10 and N = 10 . ω = 108 (blue -gray- lines), in units of ε. Temperature is given See text for details. in units of ε/kB . The conventional approach (which ignores Nk) is plotted with dashed lines. The effect of including the Nk in the relative weight of N0 is plotted with solid lines. The −1 phase transition remains diffuse in the presence of a mag- limiting function (1 + ηk) is represented with black dotted lines (it overlaps with the solid blue (gray) line) [see Eq. (29)]. netic field, as previously discussed, one can still associate a critical temperature to the maximum of the specific heat, as in the free gas case [14]. In Fig. 5 it can also 2 distinguished from the case ω = 108 depicted in Fig. 3. be seen that, for ω = 10 , no significant deviations from The origin of the discrepancy shown between the con- the conventional approach are found except for one: the ventional and the extended approach (which increases inclusion of the Nk terms tends to round off the rela- with the magnetic field) can be traced to the relative tively sharp maximum shown in the specific heat calcu- weights of the different terms entering Eq. (28), which lated from the conventional approach. On the contrary, for ω = 108, significant differences between the conven- explicitly includes the contribution of Nk. Take for ex- ample ω = 105 and N = 105. In this case, as shown in tional (dashed line) and extended (solid line) persist in a wide range of temperatures. Specifically, for large mag- Fig. 4, Nk represents a significant (and occasionally dom- inant) amount of the total particle number. Moreover, netic fields (ω ε) and high temperatures (βω 1, or η 1), C ≫/N approaches the classical value of≪ 3/2. two different crossover temperatures appear when either m ≫ V N or N begin to grow substantially. In particular, the Within this regime, the classical limit can be obtained by k 0 taking N dη η z and U 3 dη η β−1z, which results phase transition that occurs at a lower temperature re- ≃ k ω ≃ 2 k ω mains somewhat more diffuse than the high–temperature in U = 3N/2β. This is the classical limit for a system in one, because of its effective 1D nature. three dimensions. It is interesting to note that signatures of a double phase transition appear in the specific heat. In particu- V. SPECIFIC HEAT lar, Fig. 5 shows a plateau–like structure in the specific heat when the effective number of particles, N/d, is of the order of 10−3 (solid blue -gray- line). This plateau spans The essential features of BEC as a phase transition are the temperature range ε k T ω. As T decreases, clearly exhibited in the behavior of the specific heat. In B the first phase transition,≪ observed≪ at a high tempera- particular, we consider the specific heat at constant vol- ture, reveals the transition from states which have both ume, defined by C = β2 (∂U/∂β) . In the extended V V kinetic and magnetic excitations to those which are in approach, the internal energy− can be expressed as excited kinetic states but in the ground Landau mani- d 3d fold. The second phase transition shows how the abso- U(β,z)= η β−1g (z)+ η η β−1g (z). (30) 2 k 3/2 2 k ω 5/2 lute ground state (kinetic and magnetic) is reached. In fact, the existence of two crossovers can be understood When no magnetic field is present, CV tends to the 3D from Fig. 4. This is in accordance with the large slope free Bose gas, displaying a sharp maximum and a dis- of the CV (T ) curve observed for the high–temperature continuous first derivative. The effect of a nonzero mag- crossover, compared to the low–temperature one. On the netic field is that of smoothing out that maximum [14]. contrary, the specific heat calculated within the conven- This can be clearly appreciated in Fig. 5. Although the tional approach (that which ignores Nk) does not show 8

2500 Cv

2000

1500

1000

500

0 0.1 1 10 100 1000 10000 100000 1e+06

Figure 5. Specific heat at constant volume as a function of Figure 6. Specific heat of a molecular Bose gas in a 2D trap temperature. The extended and conventional approaches are as a function of T . We have taken ǫ = 1, B = 105, C = 100. plotted with solid and dashed lines, respectively. We have taken N = 105 and ω = 102 (black lines), ω = 105 (red - 8 2500 light gray- lines) and ω = 10 (blue -gray- lines). See text for Cv details.

2000 signatures of double phase transitions, as expected. We remark that, in the plateau ε kBT ω, a constant value for the specific heat is obtained≪ between≪ 1500 the two crossover temperatures. Within this temperature 1 −1 range, N dηkz and U 2 dηkβ z. Finally, we get U N/2β≃. Thus, the plateau≃ of the specific heat lies 1000 ≃ in the value CV = NkB/2, as shown in Fig. 5, which corresponds to that of a Bose gas in one dimension. We note that, although the value of the specific heat in the 500 intermediate temperature plateau is independent of the degeneracy d, the details of the low and high temperature 0 behavior depend on the particular value of d. In fact, 0.1 1 10 100 1000 10000 100000 1e+06 we recall that the very existence of the plateau relies essentially on the degeneracy being much greater than Figure 7. As in fig. 6, for C = 102. unity.

clude the occupation of states in the lowest Landau level VI. CONCLUSIONS but with arbitrary kinetic energy in the direction parallel to the magnetic field. In this work we have revisited the free charged Bose gas Furthermore, three different numerical approaches to focusing on the case of a large magnetic field, in princi- the macrocanonical ensemble problem have been stud- ple as a toy model to tackle in the future the more com- ied. In one of them, to solve the case of a free charged plex molecular case. Although a sharp Bose–Einstein Bose gas [34], the usual approximation of polylogarithm condensation does not occur, two diffuse phase transi- functions (which amounts to a change of variable in a tions (or crossovers) are found. As the temperature is multidimensional integration) has been eventually used. decreased, the first crossover reflects the enhanced occu- The approximation really simplifies the numerically chal- pation of states in the lowest Landau level. Then a range lenging task of summing a series over several variables. of intermediate temperature is found where the system But such approximation will not be so useful when behaves has an effectively one-dimensional system in its dealing with the rotational levels or the Bogoliubov cases, high-temperature regime. At even lower temperatures, because the change of variables relays on a specific struc- the system undergoes a 1D Bose condensation, which is ture for the energetic levels. So, to deal with the more known to be a diffuse phase transition. To account for general case of a diatomic Bose gas with rotational lev- this two-step condensation, it has proved essential to in- els and a kinetic term corrected by the Bogoliubov ap- 9

3000 Even more, if an experimental or more realistic model Cv of molecular levels were used, the approach of mixing the molecular levels with the kinetic ones through the for- 2500 malism of multiplying truncated series (eq. 16) would be necessary. And, finally, if a model of strong interactions 2000 were used (where the Bogoliubov approximation would break down), then using a lattice Monte Carlo method (Refs. [36, 37]) could be a good choice, although it does 1500 not allow to perform directly a simple study of the actual energetic level structure. 1000 To sum up, in the considered cases, if we have a com- plex level structure composed by several substructures

500 associated with each quantum number, and those sub- structures have different level separations, several con- densations emerge for each substructure. The corre- 0 0.1 1 10 100 1000 10000 100000 1e+06 sponding critical temperature emerges when the excited states of the associated substructure are macroscopically Figure 8. As in fig. 6, for C = 104. populated. When several substructures have equal or comparable level separations, as in the case of the text- book example of a 3D Bose gas in a box, they give rise to a simple condensation, which tends to be sharper, as proximation, we find more useful the approach of direct if several were occurring at the same tem- summation of the series through PARI/GP [33] or even perature (see fig. 5 for instance). the truncated series. To show the complexity of those In fact, the specific heat of simple molecules at moder- models, we include here a simulation of a case with ate temperatures is nearly constant along a wide range of temperatures, as required by the , but presents steps when the temperature is high enough ǫ = ǫ2 + Cǫ +B J(J +1), ǫ = ǫ(k2 +k2), (31) k kin kin kin x y to excite additional degrees of freedom. However, it is not q so easy to find a case in which several of those steps are where C = 2nV0, with n and V0 standing for the den- degenerate. That is, in which they occur at temperatures sity and interaction energy, respectively; and the degen- which are close enough to observe an overlapping. eration term is given by gk = 2J +1. That is, a 2D To conclude, we note that another interesting task Bose condensation of molecules with a rotational struc- would be the analysis of a Dirac gas with several energetic ture level and a Bogoliubov correction which affects the hierarchies, so that, even if there is no condensation as- kinetic structure. The specific heat is shown in figs. 6 sociated to the lowest energetic excitation, a macroscopic (C = 100,7(C = 102) and 8 (C = 104). In all the cases, fraction of particles can condensate in the ground state ǫ = 1 and B = 105. This work is currently in progress. of a highly spaced quantum number.

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