Competition Between Paramagnetism and Diamagnetism in Charged

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K, 2 σ ǫ en h yoantcfeunyo a of frequency gyromagnetic the being ) − = jk . . . , = β z 1 D I H MODEL THE II. − D ( = e L ∗ q aeigdffrn adulvl and levels Landau different labeling g L g j,k n mass and m X fco hc separates which -factor 2 j,k 1 ~ | X z ,σ F, ∗ q + ,σ g z c c ,σ σB ( j fBs n Fermi and Bose of ǫ i n1+ ln[1 ) jk . g ~ gdFrigases Fermi rged ω z = ein h anpurpose main The region. m + + − ∗ ǫ e σ gσ . ~ 2 σ − 2 m D − β eest h spin- the to refers k ~ ∗ L ( z 2 ω, ǫ µ N jk ǫ = ) jk z n + atce,with particles, qBS z B jk ǫ σ stequan- the is , z − σ xy µ , ) g ] / S , region, (2 g xy We . π the ~ (4) (3) (2) (1) ǫ c σ z ) 2

−1 0.50 where β = (kBT ) . Performing Taylor expansions and

(a) then integrating out kz, Eq. (4) becomes

ωV m∗ 3/2 Ω = T =06 2 0.00

− ~ 2πβ M ∞ 3 ~ω ~ ( 1)l+1l− 2 e−lβ( 2 −gσ ω−µ) − , (5) × 1 e−lβ~ω Xl=1 Xσ −

-0.50 where V is the volume of the system. For simplicity, the (b) following notation is introduced, 0.80

∞ p l+1 α/2 −lx(ησ+δ) σ ( 1) l e

Fτ [α, δ]= − , (6) M (1 e−lx)τ 0.40 Xl=1 − ~ ~ ~ where x = β ω and η = ( ω/2 µ + ǫ )/( ω). Then σ − σ Eq. (5) is rewritten as 0.00

(c)

∗ 3/2 ωV m -0.44 Ω = F σ[ 3, 0]. (7) T =06 ~2 1 − 2πβ − d Xσ

M For a system with the given particle density n = N/V , the chemical potential is obtained according to the fol- -0.48 lowing equation

∗ 3/2 0.0 0.4 0.8 1 ∂Ω m σ n = = x 2 F1 [ 1, 0]. (8) −V ∂µ 2πβ~ − g T,V Xσ

A similar treatment has been employed to study diamag- FIG. 1: (a) The total magnetization density (M), (b) the netism of the charged spinless Bose gas[13] and extended paramagnetization density (M p), and (c) the diamagnetiza- to the study of competition of diamagnetism and param- tion density (M d) as a function of g at t = 0.2. The dotted agnetism in charged spin-1 Bose gases[12]. This method and dashed lines correspond to ω = 5 and 10, respectively. is more applicable at relatively high temperatures. To determine gc, we need calculate the magnetization and density M as a function of the magnetic ﬁeld B and temperature T . The system is paramagnetic when M > 0 3/2 σ 1 M 6 =t F [ 3, 0]+ x(gσ ) while diamagnetic when M < 0. M can be derived from T =0 1 − − 2 Xσ the thermodynamic potential by the standard procedure 1 ∂Ω M 6 = , which yields σ σ T =0 − V ∂B T,V F1 [ 1, 0] xF2 [ 1, 1] . (11) × − − − ~ ∗ 3/2 q m σ In following discussions, ω is used to indicate the magni- M 6 = F [ 3, 0] T =0 m∗c 2πβ~2 1 − tude of magnetic ﬁeld since it is proportional to B. Xσ 1 + x(gσ )F σ[ 1, 0] xF σ[ 1, 1] . (9) − 2 1 − − 2 − III. RESULTS AND DISCUSSIONS

For carrying out numerical calculations, it is useful 1 First, we look at a spin- 2 Fermi gas, setting σ = 1 to make the parameters dimensionless, such as M = to present the two Zeeman levels. The dimensionless± ∗ ~ ~ ∗ ∗ ∗ m cM/(n q), ω = ω/(kBT ), and t = T/T . Here T magnetization density M as a function of g is shown in is the characteristic temperature of the system, which is 2 Fig. 1(a). As expected, M is negative in the small g ∗ 2 ∗ given by kB T = 2π~ n 3 /m . Then we have x = ω/t, ′ ′ ∗ region, which means that the diamagnetism dominates. ησ =1/2 µ /ω gσ and µ = µ/(kBT ) in the notation For each given value of ω, M grows with g and changes σ − − Fτ [α, δ]. Equations (8) and (9) can be re-expressed as its sign from negative to positive in the larger g region, indicating that the paramagnetic eﬀect is enhanced due 1= ωt1/2 F σ[ 1, 0] (10) to increase of g. This phenomenon is also observed in the 1 − Xσ Bose system[12]. gc is just the value of g where M = 0. 3

0.50

= 10

0.49

0.45

0.42 = 5 c c g

= 20

0.40 = 10 0.35

= 5

0.28

0.35

0 5 10 15 20

1/t

FIG. 2: Plots of the critical value of g-factor as a function of FIG. 3: The gc 1/t curves for charged spin-1 gases obey- 1/t. The magnetic field is chosen as ω = 20 (solid line), 10 ing the Bose-Einstein− (BE, dotted line), Maxwell-Boltzmann (dashed line) and 5 (dotted line), respectively. (MB, solid line) and Fermi-Dirac (FD, dashed line) statistics, respectively. The magnetic field is chosen as ω = 10 and 5. The total magnetization M shown in Fig. 1(a) con- 1 1 sists of both the paramagnetic and diamagnetic contri- where λ = hβ 2 /(2πm∗) 2 , z = eβ(µ−ǫσ) and the Fermi- butions. Figure 1(b) depicts the pure paramagnetic con- Dirac integral fn(z) is normally defined as tribution to M, which is calculated by M p = gm where m = n n− . Figure 1(c) plots the diamagnetic con- ∞ n−1 1 − 1 1 x tribution to M, M d = M M p. For each fixed value fn(z) −1 x dx, (14) − ≡ Γ(n) Z0 z e +1 of ω, the diamagnetization M d is slightly weakened with increasing g. Interestingly, for a charged spin-1 Bose gas where Γ(n) is a usual gamma function and x = βǫ. Then in a constant magnetic field, the diamagnetism is not the magnetization density, M, is obtained from grand suppressed but enhanced as g becomes larger[12]. Com- thermodynamic potential in Eq. (13), paring Figs 1(a), 1(b) and 1(c), it can be seen clearly 1/2 that the increase of M with g is mainly owing to the 3/2 ′ ωt ′ M T =06 =gt σf 3 (z ) f 1 (z ) paramagnetization M p. 2 − 12 2 Xσ Xσ gc is an important parameter to describe the competi- 2 −1/2 tion between the diamagnetism and paramagnetism. Fig- ω t ′ 1 g σf− 1 (z ), (15) ure 2 plots gc for the charged spin- Fermi gas. The tem- − 24 2 2 Xσ perature is described by 1/t and the magnetic fields are ′ chosen to be relatively larger, since the method adopted where z′ = e(µ +gσω)/t. After some algebra, we get from here is more applicable at higher temperatures or in Eq. (15) that gc ω→0 =1/√12 0.28868. So in the weak stronger fields. In the high temperature limit, gc tends | ≈ field limit, gc has the same value both at the high and to a universal value, gc t→∞ =0.28868, regardless of the | low temperature limit. This implies that the gc 1/t strength of the magnetic field. In a fixed magnetic field, curve is likely to flatten out in the weak-field limit.− gc increases monotonically as the temperature falls down. To proceed, we make a comparison between Fermi But the trend of increasing is slowed down if the magnetic and Bose gases. Considering that gc for charged spin- field is weakened. 1 Bose gas has already been studied[12], we discuss a Present method can not produce valid results in the Fermi gas with three sublevels, σ = 1, 0 (a pseudo- low temperature region if a small magnetic field is chosen. ± ′ spin-1 Fermi gas) to ensure the comparability. Results In this case that gω t µ , the grand thermodynamic for the spin-1 Boltzmann gas are also obtained. Fig- potential in Eq. (4)≪ can≪ be calculated using the Euler- ure 3 shows the gc 1/t curves for the three kinds of Maclaurin formula − charged spin-1 gases. In a given value of ω, gc of all the ∞ 1 ∞ 1 three gases displays similar temperature-dependence: gc ψ(j + ) ψ(x)dx + ψ′(0), (12) increases monotonously as t reduces. In the high tem- 2 ≈ Z0 24 Xj=0 perature limit, gc goes to the same value in all magnetic fields, gc t→∞ = 1/√8, reflecting that the Bose-Einstein and then Eq. (4) is transformed into (BE) and| Fermi-Dirac (FD) statistics coincide with the Maxwell-Boltzmann (MB) statistics in this case. V Vβ 2 ΩT =06 = f 5 (z)+ (~ω) f 1 (z), (13) Figure 3 also demonstrates the difference among the −βλ3 2 24λ3 2 Xσ Xσ three kinds of statistics. For each given fixed value of ω, 4 the gc 1/t curves of Bose and Fermi gases always locate critical value of g and gc increases monotonically as the at the− two sides of that of the Boltzmann gas. Given the temperature t decreases in a fixed magnetic field ω, and same temperature and magnetic field, gc of Fermi gas the rise in gc is lowered as ω is reduced. We conjecture is the smallest. According to our previous research[12], that gc holds a constant at all temperatures in the weak 1 g → = 1/2 for the Boltzmann gas regardless of the field limit. For a spin- Fermi gas, g → =1/√12. We c|t 0 2 c|ω 0 magnetic field. This means that gc of the Fermi gas does also briefly compare gc of charged spin-1 gases obeying never exceed 1/2 in the low temperature, no matter how the Fermi-Dirac, Bose-Einstein and Maxwell-Boltzmann strong the field is. statistics. The gc 1/t curves of Boltzmann gases are always between those− of Bose and Fermi gases in the same magnetic field. In the high temperature limit, gc of all IV. SUMMARY the three gases tends to the same value. This work was supported by the Fok Ying Tung Educa- In summary, we study the interplay between paramag- tion Foundation of China (No. 101008), the Key Project netism and diamagnetism of charged Fermi gases suppos- of the Chinese Ministry of Education (No. 109011), and ing that the Lande-factor g is a variable. The gas under- the Fundamental Research Funds for the Central Univer- goes a shift from diamagnetism to paramagnetism at the sities of China.

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