Trapping effect of periodic structures on the thermodynamic properties of Fermi and Bose gases
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Trapping effect of periodic structures on the thermodynamicproperties of Fermi and Bose gases
P. Salas and M. A. Sol´ıs
Instituto de F´ısica, UNAM, Apdo. Postal 20-364, 01000 M´exico D.F., M´exico
We report the thermodynamic properties of Bose and Fermi ideal gases immersed in periodicstructures such as penetrable multilayers or multitubes simulated by one (planes) or two perpen-dicular (tubes) external Dirac comb potentials, while the particles are allowed to move freely inthe remaining directions. Although the bosonic chemical potential is a constant for
T < T c , a nondecreasing with temperature anomalous behavior of the fermionic chemical potential is confirmedand monitored as the tube bundle goes from 2D to 1D when the wall impenetrability overcomes acritical value. In the specific heat curves dimensional crossovers are very noticeable at high temper-atures for both gases, where the system behavior goes from 3D to 2D and latter to 1D as the wallimpenetrability is increased. PACS numbers: 05.30.Fk; 05.30.Jp; 03.75.Hh; 67.85.BcKeywords: Anomalous chemical potential, dimensional crossover, periodic multilayers
I. INTRODUCTION
Non-relativistic quantum fluids (fermions or bosons) constrained by periodic structures, such as layered or tubular,are found in many real or man-made physical systems. For example, we find electrons in layered structures such ascuprate high temperature superconductors or semiconductor superlattices, or in tubular structures like organo-metalicsuperconductors.On the experimental side, there are a lot of experiments around bosonic gases in low dimensions, such as: BECin 2D hydrogen atoms , 2D bosonic clouds of rubidium , superfluidity in 2D He films , while for in 1D we have theconfinement of sodium , to mention a few.Meanwhile, for non-interacting fermions there are only a few experiments, for example, interferometry probes whichhave led to observe Bloch oscillations .To describe the behavior of fermion and boson gases inside this symmetries, several works have been published.For a review of a boson gas in optical lattices see , and for fermions is very complete. Most of this theoretical worksuse parabolic , sinusoidal and biparabolic potentials, with good results only in the low particle energy limit,where the tight-binding approximation is valid.Although in most of the articles mentioned above the interactions between particles and the periodic constrictionsare taken simultaneously in the system description, the complexity of the many-body problem leads to only anapproximate solution. So that the effects of interactions and constrictions in the properties of the system, are mixedand indistinguishable.In this work we are interested in analyzing the effect of the structure on the properties of the quantum gasesregardless of the effect of the interactions between the elements of the gas, which we do as precisely as the accuracyof the machines allows us to do.This paper unfolds as follows: in Sec. 2 we describe our model which consists of quantum particles gas in aninfinitely large box where we introduce layers of null width separated by intervals of periodicity a . In Sec. 3 we obtainthe grand potential for a boson and for a fermion gas either inside a multilayer or a multitube structure. From thesegrand potentials we calculate the chemical potential and specific heat, which are compared with the properties of theinfinite ideal gas. In Sec. 4 we discuss results, and give our conclusions. II. QUANTUM GASES WHITHIN MULTILAYERS AND MULTITUBES
We consider a system of N non-interacting particles, either fermions or bosons, with mass m b for bosons or m f forfermions respectively, within layers or tubes of separation a i , i = x or y , and width b , which we model as periodicarrays of delta potentials either in the z -direction and free in the other two directions for planes, and two perpendiculardelta potentials in the x and y directions and free in the z one for tubes. The procedure used here is described indetail in Refs. and for a boson gas, where we model walls in all the constrained directions using “Dirac comb”potentials. In every case, the Schr¨odinger equation for the particles is separable in x , y and z so that the single-particleenergy as a function of the momentum k = ( k x , k y , k z ) is ε k = ε k x + ε k y + ε k z . For the directions where the particlesmove freely we have the customary dispertion relation ε k i = ~ k i / m i , with k i = 2 πn i /L , n i = ± , ± , ... , and weare assuming periodic boundary conditions in a box of size L . Meanwhile, in the constrained directions, z for planesand x, y for tubes, the energies are implicitly obtained through the transcendental equation ( P i /α i a i ) sin( α i a i ) + cos( α i a i ) = cos( k i a i ) , (1)with α i = 2 m i ε i / ~ , and the dimensionless parameter P i = m i V i a i / ~ represents the layer impenetrability interms of the strength of the delta potential V i . We redefine P i = ( m i V i λ F / ~ )( a i /λ F ) ≡ P i ( a i /λ F ), where λ F ≡ h/ √ πm i k B T F is the thermal wave length of an ideal gas inside an infinite box, with k B T F = E F =(3 π ) / ( ~ / m i ) ρ / the Fermi energy and ρ = ( k B T F ) / / π a ( ~ / ma ) / is the density of the gas.The energy solution of Eq. (1) for has been extensively analized in Refs. and , where the allowed and forbiddenenergy-band structure is shown, and the importance of taking the full band spectrum has been demonstrated. III. THERMODYNAMIC PROPERTIES OF QUANTUM GASES IN MULTILAYERS AND INMULTITUBES
Every thermodynamic property may be obtained starting from the grand potential of the system under study,whose generalized form is Ω( T, L , µ ) = U − T S − µN = δ a, − Ω − k B Ta X k =0 ln (cid:8) a exp[ − β ( ε k − µ )] (cid:9) , (2)where a = − δ is the Kronecker delta function and β = 1 /k B T .The ground state contribution Ω , which is representative of the Bose gas, is not present when we analyze the Fermigas.For a boson gas inside multilayers we go through the algebra described in , and taking the thermodynamic limitone arrives to Ω (cid:0) T, L , µ (cid:1) = k B T ln (cid:0) − exp[ − β ( ε − µ )] (cid:1) − β L m (2 π ) ~ Z ∞−∞ dk z g (cid:8) exp[ − β ( ε k z − µ )] (cid:9) . (3)Meanwhile, for a fermion gas we getΩ (cid:0) T, L , µ (cid:1) = − L (2 π ) m ~ β Z ∞−∞ dk z f (cid:8) exp[ − β ( ε k z − µ )] (cid:9) , (4)where g σ ( t ) and f σ ( t ) are the Bose and Fermi-Dirac functions . The spin degeneracy has been taken into account forthe development of Eq. (4).On the other hand, for a multitube structure we haveΩ (cid:0) T, L , µ (cid:1) = k B T ln[1 − e − β ( ε − µ ) ] − L m / (2 π ) / ~ β / Z ∞−∞ Z ∞−∞ dk x dk y g / ( e − β ( ε kx + ε ky − µ ) ) (5)for a boson gas, andΩ (cid:0) T, L , µ (cid:1) = − L m / (2 π ) / ~ β / Z ∞−∞ Z ∞−∞ dk x dk y f / (cid:8) exp[ − β ( ε k x + ε k y − µ )] (cid:9) (6)for a fermion gas.For calculation matters, it is useful to split the infinite integrals into an number J of integrals running over theenergy bands, taking J as large as necessary to acquire convergence. A. Chemical potential and specific heat
For a gas inside a multilayer structure, the particle number N is directly obtained from Eqs. (3) and (4). Importantcharacteristics can be extracted, such as the critical temperature for a condensating boson gas and the influence ofthe system parameters on it, a and P , already reported in Refs. . But for the case of a fermion gas, we focus onthe chemical potential since it is closely related to the Fermi energy of the system. In this case the number equationis N = 2 L (2 π ) m ~ β Z ∞−∞ dk z ln (cid:8) − β ( ε k z − µ )] (cid:9) , (7)from which we are able to numerically extract the Fermi energy of the system, which corresponds to the chemicalpotential for T = 0, over the Fermi energy of the IFG E F = ( ~ / m ) k F = ( ~ / m )4 π /λ F , namely E F /E F , asa function of the impenetrability parameter P , whose behavior corresponds to a monotonically increasing curve as P increases, being more evident for smaller values of a/λ F . Another important feature is the chemical potential ofthe system over its Fermi energy, µ/E F as a function of the temperature in units of the Fermi temperature T /T F ,Fig III A, for a given value of P . There is a special interest in Fig III A, since for certain geometrical configurationsthe chemical potential shows an anomalous behavior, as will be shown later. Also, in this last figure one may noticethat for P = 0 the 3D IFG behavior for the chemical potential is recovered, and that the curve crosses the x axisin T /T F = 0 . . Meanwhile, as P → ∞ , we have a fermion gas inside a two dimensionstructure, giving a zero chemical potential at T /T F = 1 .
44, as expected.
T/T F m / E F a/ l F0 = 0.5 a/ l F0 = 1.0 a/ l F0 = 0.1 P = FIG. 1: (Color online) Chemical potential as a function of
T /T F for planes with P = 100 and different values of a/λ F . We make a similar procedure for the boson an fermi gases inside a multitube structure, the first one being reportedin . But for a fermion gas we start from the equation N = 2 L (2 π ) / m / ~ β / Z ∞−∞ dk z f / (cid:8) exp[ − β ( ε k x + ε k y − µ )] (cid:9) (8)and extract the chemical potential over the Fermi energy of the system, µ/E F , which is probably the feature thatattracts greater attention due to its anomalous behavior shown in Fig. 2, which shows up as an unexpected smallhump.Another interesting characteristic is that the chemical potential over the IFG Fermi energy in every case is liftedas P increases due to the presence of the layers, in the same way as the chemical potential of the boson gas startedabove zero.The specific heat of a boson gas has been reported in Ref. where we can observe a transition from a 3D systemto a 2D one, which becomes evident for certain parameter values and sufficiently high temperatures. At this point iswhere the advantages of summing over a great amount of allowed energy bands shows its relevance.Meanwhile, the specific heat for a fermion gas inside layered arrays is obtained going through the derivatives of T/T F P = x / l F0 = a y / l F0 = 0.7 m / E F P = 10 P = P = FIG. 2: (Color online) Chemical potential as a function of
T /T F for multitubes with a x /λ F = a y /λ F = 0 . P . Eqs. (3) and (6), leading, after some algebra, to C V N k B = L N (2 π ) m ~ (cid:8) β Z ∞−∞ dk z f (cid:8) exp[ − β ( ε k z − µ )] (cid:9) +2 Z ∞−∞ dk z ln (cid:8) − β ( ε k z − µ )] (cid:9) { ε k z − µ + T dµdT } +2 β Z ∞−∞ dk z ε k z (cid:8) ε k z − µ + T dµdT (cid:9) exp (cid:8) β ( ε k z − µ ) (cid:9) + 1 (cid:9) (9)for multiplanes, and C V N k B = 2 L N (2 π ) / m / ~ (cid:8) β / Z ∞−∞ Z ∞−∞ dk x dk y × f / ( e − β ( ε kx + ε ky − µ ) )(2 ε k x + 2 ε k y − µ + T dµdT ) + β / Z ∞−∞ Z ∞−∞ dk x dk y ( ε k x + ε k y ) × f − / ( e − β ( ε kx + ε ky − µ ) ) { ε k x + ε k y − µ + T dµdT } + 34 β / Z ∞−∞ Z ∞−∞ dk x dk y f / ( e − β ( ε kx + ε ky − µ ) ) (cid:9) (10)for multitubes.In Figs. 3 and 4 we show the behavior of the specific heat of layers with a separation among layers of a/λ = 0 . P , as a function of the temperature over the system’s Fermi temperature. It maybe observed that, as the barriers disappear ( P = 0), one recovers the IFG specific heat classical value, 3 /
2. It is alsonoticeable that as the value of a/λ diminishes, a dimensional crossover signature from 3D to 2D becomes evident,since the the first minimum (going from right to left) in the specific heat deepens towards a value C V /N k B = 1 . P increases. In Fig. 4 the dimensional crossover from 3D to 1D is very noticeable as the mentionedminimum drops down to the value 1 / -2 -1 T/T F0 P = = 10 P = P = C V / N k B a/ λ F0 = 0.2 FIG. 3: (Color online) Specific heat as a function of
T /T F for a fermion gas in multilayers. -2 -1 T/T F0 P = = 10 P = P = C V / N k B a x / λ F0 = a y / λ F0 = 0.1 FIG. 4: (Color online) Specific heat as a function of
T /T F for a fermion gas in multitubes. IV. CONCLUSIONS
In summary, we have calculated the thermodynamic properties of ideal boson and fermion gases inside periodicalstructures. In our model the multilayers and multitubes are generated with Dirac-comb potentials in either one or twodirections, while the particles are free in the remaining directions. Just by introducing the planes, the translationalsymmetry of the particles is broken. This fact reflects in every thermodynamic property of the constrained system.In particular, fermions in multi-tubes progress from a 3D behavior to that 2D and finally to 1D as the wallimpenetrability is increased, which is observed in the curves of the specific heat as a function of temperature. Thereis a critical value of the wall impenetrability for which the system begins to behave in dimensions less than two, whichis signaled by the appearance of an anomalous chemical potential. Bosons in multitubes show similar dimensionalcrossover like that expressed by fermions, in addition to the Bose-Einstein condensation at temperatures below thecritical temperature of a ideal Bose-gas with the same particle density.
Acknowledgements.
We acknowledge partial support from UNAM-DGAPA-PAPIIT (M´exico) A. I. Safonov, S. A. Vasilyev, I. S. Yasnikov, I. I. Lukashevich, and S. Jaakkola, Phys. Rev. Lett. , 4545 (1998). Z. Hadzibabic, P. Kr¨uger, M. Cheneau, B. Battelier and J. Dalibard, Nature , 1118 (2006). D. J. Bishop and J. D. Reppy, Phys. Rev. Lett. , 1727 (1978). A. G¨orlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S.Inouye, T. Rosenband, and W. Ketterle, Phys. Rev. Lett. , 130402 (2001). G. Roati, E. de Mirandes, F. Ferlaino, H. Ott, G. Modugno, and M. Inguscio, Phys. Rev. Lett. , 230402-1 (2004). F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Rev. Mod. Phys. ,463 (1999). S. Giorgini, L. P. Pitaevskii and S. Stringari, Rev. Mod. Phys. , 1215 (2008). V. Bagnato and D. Kleppner, Phys. Rev. A , 7439 (1991). D. S. Petrov, M. Holzmann, and G.V. Shlyapnikov, Phys. Rev. Lett , 2551 (2000). X-G Wen and R. Kan, Phys. Rev. B , 595 (1988). A. Hærdig and F. Ravndal, Eur. J. Phys. , 171 (1993). A. Zh. Muradyan and G. A. Muradyan, arXive: cond-mat /0302108. P. Salas, M. Fortes, M. de Llano, F. J. Sevilla, and M. A. Sol´ıs, J. of Low Temp. Phys. , 540 (2010). P. Salas, F. J. Sevilla, M. Fortes, M. de Llano, A. Camacho, and M. A. Sol´ıs, Phy. Rev. A , 033632 (2010). P. Salas, F. J. Sevilla, and M. A. Sol´ıs, J. of Low Temp. Phys. , 258 (2012). Kronig R. de L. and Penney W.G., Proc. Roy. Soc. (London), A , 499 (1930). R.K. Pathria,
Statistical Mechanics , 2nd Ed. (Pergamon, Oxford, 1996) pag. 134. M. Grether, M. de Llano and M. A. Sol´ıs, Eur. Phys. J. D25