arXiv:1006.4705v1 [cond-mat.stat-mech] 24 Jun 2010 ttetaos uhas such atoms, stituent lzto fsio oecnest notclytrapped optically in condensate atoms alkali Bose spinor of alization 96 g rpsdta h E fbsncelectron- superconductivity. bosonic in of BEC result the might that pairs proposed Ogg 1946, oos aeas ensuidtheoretically. studied been also have bosons) Schafroth urtsuigteCGmodel. CBG superconducting the of using cuprates diamagnetism normal-state en- hanced the for accounted quantitatively Alexandrov recently, osn ogroccur. longer BEC no changes: does qualitative introduces field value magnetic small field. the or- arbitrarily of magnetic an the that in indicated of particles clearly quantization Schafroth charged the exter- of to the motion by owing bital affected field, nontriv- strongly magnetic condensation is nal exhibits the CBGs it example, in For Especially, phenomenon properties. right. magnetic own ial its in terest research gases. stimulated Bose it charged in importantly interest more supercon- of and understanding the ductivity to helpful Schafroth-Blatt- is the theory pairs, Butler real-space to Cooper- opposed form as superconductor pairs, theory a BCS in the electrons equilibrium that Although essential revealed superconductor. the a exhibits of features bosons charged of gas ae ndffrn methods. different on based iesoa B n on hti a odneol for only condense > can it d that found and CBG dimensional ipasMise ffc tlwtemperatures. low CBG 3-dimensional at the effect that Meissner out displays pointed diamagnetism was It Landau large CBGs. in extremely in results motion BC eo rtcltmeaue n13,London 1938, the In to BEC temperature. connected condensation critical first Bose-Einstein a the below the exhibits (BEC) that gas known well Bose super- is ideal including It phenomena, superconductivity. quantum and fluidity exotic of series a nteohrhn,nurlbsn ihsi (spinor spin with bosons neutral hand, other the On h hre oegs(B)i oeyo cdmcin- academic of solely is (CBG) gas Bose charged The h oegspasasgicn oei understanding in role significant a plays gas Bose The 4. 7 hspitwsreaie yohrresearchers other by reexamined was point This 3 n lt n Butler and Blatt and 16 ASnmes 53.p 52.g 51.p 74.20.Mn 75.10.Lp, c 75.20.-g, the 05.30.Jp, of numbers: details PACS some demonstrate to order in contribution calculated diamagnetic also and paramagnetis from paramagnetic shift The a decreases. shows between in Lande-factor the taltmeaue,wieagswith gas a while temperatures, all at inbtentedaants n aaants,uigthe using paramagnetism, and externa b diamagnetism in the gases to between Bose solely tion spin-1 appears charged respectively, of properties freedom magnetic of degrees spin or h tegho aaantceet epooeta a wit gas a that propose We effect. paramagnetic of strength the imgeimvru aaants ncagdsi- oeg Bose spin-1 charged in Paramagnetism versus Diamagnetism tmltsnwrsac neet h con- The interest. research new stimulates eateto hsc,Uiest fSineadTechnolog and Science of University Physics, of Department thsbe ugse htete imgeimo paramagne or diamagnetism either that suggested been has It .INTRODUCTION I. 3 a xeddti dat a to idea this extended May 87 Rb, 10,11 λ rniini Helium. in transition 23 14 4 enhl,teorbital the Meanwhile, a and Na, hwdta nideal an that showed ioigJa,Jhn i,adQagGu Qiang and Qin, Jihong Jian, Xiaoling 2 7 Furthermore, ihv (hy- have Li 3,12,13 15 Dtd coe ,2018) 2, October (Dated: > g h re- The 1 3,6–12 More / 1 lasehbt aaants.Mroe,agswith gas a Moreover, paramagnetism. exhibits always 2 In d 5 - E ae lc,temgeiainrmisfiieeven finite remains once that magnetization found if the and place, field takes magnetic BEC in bosons spinor tral Cohen and Simkin re umr sgiven. IV, is Section summary a In brief presents results. a obtained III the Section of discussion diamagnetic respectively. detailed and calculated magnetiza- paramagnetic are its total as parts The well as Pauli density and proposed. tion effect is diamagnetic con- effect Landau model paramagnetic the a II, both Section Bose of In spin-1 sisting field. charged magnetic external a in of gas diamagnetism and magnetism problem. compe- interesting their more large, a paramagnetism is extremely tition the be can both diamagnetism since and gas, For Bose paramagnetic. is spinor gas zero-field charged para- the altogether the the so of of part, value) magnetic absolute part (in diamagnetic For one-third is the susceptibility weak. gas, para- relatively electron the are the both diamagnetism where freedom? gas, and electron of magnetism the degrees e.g., charged for gases, spin answered Fermi been and has is: question charge this bosons raised Analogously, the the be if both itself must manifest possess will which magnetism of question kind which immediate one gas, magnetic bosons in spinor effect neutral paramagnetic the field. extraordinary that paramagnet. on pure indicate take a results of the that All show fi- exceed at Lieb temperatures susceptibility zero-field and nite and Eisenberg magnetization the by Moreover, that presented temperature. proofs BEC tem- the rigorous the to as down diverge goes to perature tends susceptibility zero-field The nprl pia rp n hsivsiaino their Yamada of investigation possible. thus active becomes and magnetic become properties traps freedom a magnetic optical of thus purely degrees and spin in freedom Their of moment. degrees spin perfine) H nti ae,w td h optto ewe para- between competition the study we paper, this In Bose the of features contrary two the mind in Bearing ,a ftesse a antzdspontaneously. magnetized was system the if as 0, = antcfil,fcsn ntecompeti- the on focusing field, magnetic l xrodnrl tog einvestigate We strong. extraordinarily e otettlmgeiaindniyare density magnetization total the to s odaants stetemperature the as diamagnetism to m ompetition. Lande-factor imo oegss u otecharge the to due gases, Bose of tism ejn,Biig108,China 100083, Beijing Beijing, y h < g 1 17 / √ ∗ acltdtemgeiaino neu- of magnetization the calculated xiisdiamagnetism exhibits 8 g fprilst evaluate to particles of ases 15 and , 18 2
II. THE MODEL where V is the volume of the system. Then using the Taylor expansion and performing the integral over kz, The orbital motion of a charged boson with charge q we have ∗ and mass m in a constant magnetic field B is quantized ∗ 3/2 into the Landau levels, ωV m Ω 6 = T =0 − ~2 2πβ 2 2 ~ 3 ~ω ~q l 1 kz ∞ − −lβ( −g ∗ σB−µ) ǫ = ( + j)~ω + , (1) l 2 e 2 m c jkz 2 2m∗ . (7) × 1 e−lβ~ω Xl=1 Xσ where j = 0, 1, 2,... labels different Laudau levels and − ∗ ω = qB/(m c) is the gyromagnetic frequency. Since This treatment has been used by Standen and Toms11 ∝ ω B, ω can be used to indicate the magnitude of the to deal with scalar Bose gases. As they mentioned, this magnetic field in the following discussions. We assume theory is more reliable at high temperature. Similarly, we the magnetic field is in the z direction. Each Landau introduce some compact notation for the class of sums, level is degenerate with degeneracy equal to ∞ lα/2e−lx(ε+δ) qBLxLy Σκ[α, δ]= , (8) DL = . (2) (1 e−lx)κ 2π~c Xl=1 −
Here we suppose the gas is in a box with Li , where ~ ~ ∗ 1 ~ → ∞ where x = β ω and µ+g qσB/(m c) = ( 2 ε) ω. With Li is the length of the box in the ith direction. For a spin- this notation we may rewrite Eq. (7) as − 1 boson, the Zeeman energy levels split in the magnetic field due to the intrinsic magnetic moment associated 3/2 ωV m∗ with the spin degree of freedom, Ω 6 = Σ [ D, 0], (9) T =0 − ~2 2πβ 1 − Xσ ~q ǫze = g σB, (3) σ − m∗c where D = 3. Then the density of particles n = N/V can be derived from the thermodynamic potential, where σ refers to the spin-z index of Zeeman state F =1,mF = σ (σ = +1, 0, 1) and g is the Lande- 1 ∂Ω | i − n = factor. The quantization of the orbital motion and the −V ∂µ Zeeman effect give rise to the Landau diamagnetism and T,V Pauli paramagnetism, respectively. m∗ 3/2 = x Σ [2 D, 0]. (10) We consider an assembly of N bosons, whose effective 2πβ~2 1 − Hamiltonian reads Xσ The magnetization density M is written as H¯ µN = D ǫl + ǫze µ n , (4) − L jkz σ − jkz σ j,kXz,σ 1 ∂Ω MT =06 = where µ is the chemical potential. The charged spinor −V ∂B T,V bosons have been discussed theoretically in the context ~ ∗ 3/2 19 q m of relativistic pair creation . However, magnetism of = Σ1[ D, 0] m∗c 2πβ~2 − charged spinor Bose gases is less studied and of major Xσ interest in the present work. 1 + x(gσ )Σ [2 D, 0] xΣ [2 D, 1] . (11) The grand thermodynamic potential is formally ex- − 2 1 − − 2 − pressed as It is convenient to introduce some dimensionless param- 1 −β(H¯ −µN) ∗ ~ ~ ∗ ΩT =06 = ln Tre eters, such as M = m cM/(n q), ω = ω/(kBT ), −β t = T/T ∗ and x′ = ω/t, to re-express equations (10) 1 −β(ǫl +ǫze−µ) and (11), = D ln[1 e jkz σ ], (5) β L − j,kXz ,σ 1= ωt1/2 Σ′ [2 D, 0], (12) 1 − −1 σ where β = (kB T ) . Converting the sum over kz to X continuum integral, we get
∗ ∞ 3/2 ′ ′ 1 ωm V M T =06 = t Σ1[ D, 0]+ x (gσ ) ΩT =06 = dkz − − 2 (2π)2~β Z Xσ Xj=0 Xσ 2 2 1 ~ k ~q ′ ′ ′ −β[(j+ )~ω+ ∗z −g ∗ σB−µ] Σ [2 D, 0] x Σ [2 D, 1] . (13) ln 1 e 2 2m m c , (6) × 1 − − 2 − × { − } 3
0.50
0.4
0.45 0.0
M c g
-0.4
(a)
0.40
1
1.0
m
0.35 p
0 5 10 15 20
0
M
0 1
0.5 g
1/t
(b)
FIG. 2: Plots of the critical value of Lande-factor, gc as a
0.0 function of 1/t for fixed values of ω. The field is chosen as: ω = 10 (dash-dot-dotted line), 3 (dotted line), 0.5 (short dotted (c) line), 0.3 (dashed line), 0.1 (short-dashed line), and 0.05 (solid line). d
-0.4
M ze energy term ǫσ in the Hamiltonian (4). In the case that the Lande-factor g tends to zero, the model degenerates into a charged scalar boson model which exhibits strong diamagnetism as already intensively discussed.7–12 As g 0.0 0.5 1.0 becomes larger, the paramagnetic effect is strengthened. We calculate the dimensionless magnetization density g M as a function of g, as shown in Fig. 1(a). M is neg- ative in the small g region, which means that the dia- FIG. 1: (a) The total magnetization density (M), (b) the magnetism dominates. The absolute value of M is larger paramagnetization density (M p), and (c) the diamagnetiza- in the stronger field ω. For each given value of ω, M tion density (M d) as a function of g for fixed magnetic fields at t = 0.1. Here the solid line, dashed line and dotted line increases monotonically with g. M changes its sign from correspond to ω = 0.05, 0.3, and 3, respectively. Inset: m as negative to positive at a critical value of g, noted as gc a function of g. hereinafter, reflecting that the paramagnetism becomes dominant as g increases. Note that the slope of the M curve is dependent on g. When g is near to zero, M in- Here the characteristic temperature T ∗ is given by creases slowly but the slope rises quickly with g. It means ∗ 2 2/3 ∗ kBT = 2π~ n /m . The Bose-Einstein condensa- that the interplay between diamagnetism and paramag- tion temperature of spin-1 Bose gas with density n is netism is complex and nonlinear. However, in the strong 2 2/3 ∗ 2/3 just defined as kBTc = 2π~ n / m [3ζ(3/2)] paramagnetic region, M grows almost linearly with g. ∗ { ′ } ≈ Figure 1(b) plots the paramagnetic contribution to M kBT /3.945. The dimensionless notation Σκ[α, δ] should be (named as the paramagnetization density), M p = gm, with an inset showing m = n1 n−1, and Figure 1(c) ′ ∞ α/2 −lx (ε+δ) − ′ l e shows the diamagnetic contribution to M (named as the Σ [α, δ]= ′ , (14) κ (1 e−lx )κ diamagnetization density), M d = M M p. The increas- Xl=1 − − ing tendency of M p is similar to that of M. It is notewor- ′ ∗ ′ 1 thy that the diamagnetization density is not suppressed, where µ = µ/(kBT ) and µ + gσω = ( 2 ε)ω. The ′ − but enhanced as g becomes larger. Comparing Figs 1(a), two variables µ and M are determined by Eqs. (12) and 1(b) and 1(c), it can be seen that the increase in M comes (13). from the paramagnetic effect. In the small g region, both M p and M d are strengthened nonlinearly with increas- ing g. As shown in the inset of Fig. 1(b), m grows very III. RESULTS AND DISCUSSIONS quickly. Nevertheless, both m and M d flatten out in the large g region. So the slope of M curve is mainly due to In our model, both the charge and spin degrees of free- the paramagnetization. dom are taken into account, which are described respec- According to discussions above, the critical value of the l tively by the Landau energy term ǫjkz and the Zeeman Lande-factor, gc, is an important parameter to describe 4
0.50
(a)
0.0
0.45
M c g
-0.1
0.40
(b)
0.1
0.35
0 5 10 15 20
M
1/t
0.0
FIG. 3: Curves of gc 1/t obtained respectively from the Bose-Einstein (BE) and− Maxwell-Boltzmann (MB) statistics with fixed values of ω. Where ω = 10 (dash-dot-dotted line, (c)
BE; solid line, MB), 3 ( dotted line, BE; short dotted line, 0.2 MB), and 0.3 (dashed line, BE; short dashed line, MB).
M the competition between the diamagnetism and param- 0.1 agnetism. gc is a function of the temperature t and the magnetic field ω. Figure 2 shows gc as a function of 1/t in different magnetic fields ω = 10, 3, 0.5, 0.3, 0.1 and 0.0
0.05. Obviously, gc varies monotonically with the tem- 0 5 10 perature t, while its dependence on the field ω is not simple. For example, in the low temperature region, gc 1/t decreases with decreasing ω at a given temperature as ω is still larger than 0.3, then it rises up as ω goes down FIG. 4: Shown are plots of dimensionless M as a function of further from ω 0.3. 1/t for each given value of ω at a fixed g. The value of ω ≈ were in sequence as 10 (dash-dot-dotted line), 3 (dotted line), As already mentioned, the results of our theory are 0.3 (dashed line), and 0.05 (solid line). (a) corresponds to more credible at high temperature. In the high temper- g = 0.35. (b) corresponds to g = 0.45. (c) corresponds to ature limit, gc seems universal with respect to different g = 0.5. choices of magnetic field, g →∞ 0.35356. For a given c|t ≈ magnetic field, gc increases as the temperature falls down. This suggests that the diamagnetic region is larger at low and temperatures than at high temperatures. Although the 1 ω ′ − t ( 2 −gσω−µ ) B 3/2 e exact value of gc can not be obtained at very low temper- M = t T =06 − ω atures, its variation trend can be estimated from Fig. 2. 1 e t Xσ − It seems that gc ranges from 0.475 to 0.50 in various − ω ω 1 e t magnetic fields. [1 + (gσ )] . (17) − ω × t − 2 − 1 e t It is useful to reexamine the high temperature behav- − iors of gc by generalizing above calculations to a spin-1 Substituting Eq. (16) into Eq. (17), yields Boltzmann gas, since the Bose-Einstein statistics reduces ′ 2gx to Maxwell-Boltzmann statistics in the high temperature B 1 1 1 g(e 1) M = ′ + ′ −′ . (18) limit. The grand thermodynamic potential based on the T =06 x′ − 2 − ex 1 e2gx + egx +1 Maxwell-Boltzmann statistics reads − An analytical formula for gc can be obtained, 1 l ze −β(ǫjkz +ǫσ −µ) ′ ΩT =06 = DLe . (15) 2gcx −β 1 1 1 gc(e 1) j,kXz ,σ = ′ + ′ −′ . (19) 2 x′ − ex 1 e2gcx + egcx +1 Then equations of the dimensionless chemical potential µ′ − B The value of gc can be derived from Eq. (19) exactly and magnetization density M are derived respectively, in two limit cases: gc t→∞ = 1/√8 0.35355 and 1 ′ | ≈ − ( ω −gσω−µ ) e t 2 gc t→0 = 1/2. The value of gc for a Boltzmann gas is 1= ωt1/2 (16) | − ω reasonably equal to that of a Bose gas in the high tem- 1 e t Xσ − perature limit. Figure 3 plots the numerical solutions of 5
0.4 p
M
0.2
(a) (b) (c)
0.0
(d) (e) (f)
-0.2 d
M
-0.4
0 4 8 0 4 8 0 4 8
1/t 1/t 1/t
FIG. 5: The dimensionless M p and M d as a function of 1/t for fixed values of ω. From left to right g = 0.35, 0.45 and 0.5, respectively. For each given value of g, the field is chosen as: ω = 10 (dash-dot-dotted line), 3 (dotted line), 0.3 (dashed line), and 0.05 (solid line), respectively.
Eq. (19) and compares with the Bose gas. An interesting with g in the intermediate region, which looks quite sim- point is that the low-temperature-limit value of gc is also ilar to Fig. 4(c). The key difference is that M can change consistent with that of a Bose gas at low temperature but its sign from positive to negative as the temperature de- in high magnetic field, although the Maxwell-Boltzmann creases, indicating that the system undergoes a shift from statistics is just valid in the high temperature region. paramagnetism to diamagnetism. The stronger the field, the better the accordance. Figures 4 demonstrate the total magnetic performance
The gc 1/t curves in Figs. 2 and 3 mark the bound- of the charged spin-1 Bose gas. We now turn to exam- ary between− the diamagnetic and paramagnetic regions. ine the underlying paramagnetic and diamagnetic effects The gas exhibits diamagnetism at all temperatures and for each corresponding case. As shown in Figs. 5, both in all magnetic field when g Figure 4(a) denotes the case of g = 0.35 < gc t→∞. can go beyond M d. M p thoroughly exceeds M d when The temperature-dependence of M is very similar to| that g rises to 0.5. If g increases further, the paramagnetic of charged scalar Bose gases,11 as if the paramagnetic ef- effect can be so strong as to cover up the diamagnetic fect associated with the spin degree of freedom is thor- effect completely. Then the total magnetization density oughly hidden. The diamagnetism is even stronger at M becomes monotonously increasing with lowering the lower temperatures. As the external field tends to be temperature and finally reaches a plateau, instead of a weak, a sharp bend appears gradually on the curve, which peak, at low temperatures. located at the point corresponding to the BEC tempera- ture in zero field.11 In our model, the BEC temperature IV. SUMMARY for a spin-1 gas is (1/t)c 3.945. Figure 4(c) shows M for a paramagnetic case to≈ the contrary when g = 0.5. M is always positive in the field at all temperatures. An This paper has studied the interplay between param- interesting phenomenon is that the M 1/t curve shows agnetism and diamagnetism of the ideal charged spin-1 up a peak in this case. The decline in− M at low tem- Bose gas. The Lande-factor g is introduced to describe peratures is attributed to the diamagnetic effect. When the strength of paramagnetic effect caused by the spin weakening the magnetic field, the peak is lowered and degree of freedom. The gas exhibits a shift from diamag- moves to low temperatures. Fig. 4(b) depicts the case netism to paramagnetism as g increases. The critical 6 value of g, gc, is determined by evaluating the dimen- bly relevant to the present work. Although the charged sionless magnetization density M. Our results show that spinor Bose gas has not been realized so far, the up- gc increases monotonically as t decreases. In the high to-date achievement in experiments makes it attainable 20 temperature limit, gc goes to a universal value in all dif- perhaps in the near future. For example, it is already ferent magnetic fields, gc t→∞ = 1/√8. At low temper- possible to create ultracold plasmas by photoionization of | 21 atures, our results indicate that gc ranges from 0.475 to laser-cooled neutron atoms. The ions can be regarded 0.50 as the magnetic field varies. Therefore, a gas with as charged bosons if their spin is an integer. The Lande- g < 1/√8 exhibits diamagnetism at all temperatures, factor for different magnetic ions could be different. As but a gas with g > 1/2 always exhibits paramagnetism. reported by Killian et al.21, the temperatures of elec- In cases where 1/√8 ∗ Corresponding author: [email protected] 16 D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. 1 F. London, Phys. Rev. 54, 947 (1938). Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle, Phys. 2 R. A. Ogg, Jr., Phys. Rev. 69, 243 (1946). Rev. Lett. 80, 2027 (1998); J. Stenger, S. Inouye, D. M. 3 M. R. Schafroth, Phys. Rev. 100, 463 (1955). Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and W. 4 J. M. Blatt and S. T. Butler, Phys. Rev. 100, 476 (1955). Ketterle, Nature (London) 396, 345 (1998). 5 J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 17 M. V. Simkin and E. G. D. Cohen, Phys. Rev. A 59, 1528 108, 1175 (1957). (1999). 6 M. F. M. Osborne, Phys. Rev. 76, 400 (1949). 18 E. Eisenberg and E. H. Lieb, Phys. Rev. Lett. 89, 220403 7 R. M. May, J. Math. Phys. 6, 1462 (1965). (2002). 8 T. A. Arias and J. D. Joannopoulos, Phys. Rev. B 39, 4071 19 J. Daicic and N. E. Frankel, Phys. Rev. D 52, 7174 (1995). (1989). 20 V. V. Kabanov and A. S. Alexandrov, Phys. Rev. B 71, 9 A. S. Alexandrov, Phys. Rev. B 48, 10571 (1993) 132511 (2005). 10 J. Daicic, N. E. Frankel, and V. Kowalenko, Phys. Rev. 21 T. C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Lett. 71, 1779 (1993); J. Daicic , N. E. Frankel, R. M. Orzel, and S. L. Rolston, Phys. Rev. Lett. 83, 4776 (1999). Gailis, and V. Kowalenko, Phys. Rep. 237, 63 (1994); J. 22 S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. Daicic and N. E. Frankel, Phys. Rev. D 53, 5745 (1996). K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, 11 D. J. Toms, Phys. Rev. B 50, 3120 (1994); D. J. Toms, S. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, I. Phys. Rev. D 51, 1886 (1995); G. B. Standen and D. J. Sheikin, D. Braithwaite, and J. Flouquet, Nature (London) Toms, Phys. Rev. E 60, 5275 (1999). 406, 587 (2000); D. Aoki, A. Huxley, E. Ressouche, D. 12 H. P. Rojas, Phys. Lett. B 379, 148 (1996). Braithwaite, J. Flouquet, J.-P. Brison, E. Lhotel, and C. 13 K. Kirsten and D. J. Toms, Phys. Rev. D 55, 7797 (1997). Paulsen, Nature (London) 413, 613 (2001); E. Slooten, T. 14 A. S. Alexandrov, Phys. Rev. Lett. 96, 147003 (2006). Naka, A. Gasparini, Y. K. Huang, and A. de Visser, Phys. 15 K. Yamada, Prog. Theor. Phys. 67, 443 (1982); A. Caram- Rev. Lett. 103, 097003 (2009). ico D’Auria, L. De Cesare, and I. Rabuffo, Physica A 225, 363 (1996).