The de Haas van Alphen Oscillation in Two-Dimensional QED at Finite Temperature and Density

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1 Commun. Theor. Phys. (Beijing, China) 35 (21) pp c International Academic Publishers Vol. 35, No. 6, June 15, 21 The de Haas van Alphen Oscillation in Two-Dimensional QED at Finite Temperature and Density CHAI Kang-Min and XU Hong-Hua Department of Physics, Shanghai Jiao Tong University, Shanghai 23, China (Received March 21, 2; Revised May 22, 2) Abstract The de Haas van Alphen oscillations in two-dimensional QED at finite temperature and density are investigated. It is shown that for a given particle density, besides the oscillation of magnetization, the chemical potential is also oscillating with the same period. Different from the earlier work (J.O. Andersen and T. Haugset, Phys. Rev. D51 (1995) 373), the magnetization oscillations we studied have a correct nonrelativistic limit at zero temperature. PACS numbers: 11.1.Wx, 12.2.Ds, 5.3.-d Key words: QED, thermodynamic potential, de Haas van Alphen effect 1 Introduction QED at finite temperature and density in external magnetic fields has been studied recently for both 3D and 2D QED. [1,2] For 3D QED it has a background in astrophysics, while for 2D QED it is related to the quantum Hall effect. In Ref. [1] magnetization oscillations, known as the de Haas van Alphen oscillations, were demonstrated indirectly in terms of the effective action oscillations for 3D QED. In Ref. [2] magnetization oscillations in 2D QED were studied for a given particle density at zero temperature with a result not in agreement with the wellknown nonrelativistic limit, while for low temperatures and weak magnetic fields the oscillations were studied for a fixed chemical potential which is less realistic. The purpose of the present paper is to give a further study for 2D QED. We consider the situation with a fixed particle density which is nearly the electron density, because at low temperatures the particle density is dominated by electrons and the contribution from positrons is negligible. With the particle density being fixed, the de Haas van Alphen oscillations in 2D QED can give, as will be shown in the context, a correct nonrelativistic limit at zero temperature. [3] At finite temperature an analytical approach following Lifshitz and Pitaevskii [4] is applied with a result in agreement with numerical calculations for weak magnetic fields and low temperatures. It is found also that the chemical potential displays oscillations for weak magnetic fields, similar to magnetization oscillations, and they have the same period. To our knowledge, the behavior of the chemical potential in external fields has not been paid much attention. In fact, as will be shown in the present paper, the de Haas van Alphen effect is demonstrated not only through the magnetization oscillations, but also through the chemical potential oscillations. 2 Oscillation of the Chemical Potential In our previous paper [5] the thermodynamic potential density of the 2D QED including interaction was derived where the free part, or the effective action in Ref. [2], reads = eb {ln[1 + e β(en µ) ] β + λ n ln[1 + e β(en+µ) ]} n, (1) in which F n (T) is the contribution from the nth Landau level, E n = m 2 + 2neB is the Landau energy, T = 1/β is the temperature, and λ n = 1 δ n. In this paper we take c = h = k B = 1. The particle density is given by ρ (T) (T) F = µ = eb [f(e n µ) λ n f(e n + µ)] ρ (T) n, (2) where f(e n ±µ) = [1+ e β(en±µ) ] 1 is the Fermi function and ρ n (T) the particle density on the nth Landau level. At T = it becomes ρ () = eb j θ(µ E n ), (3) in which the integer j = int[(µ 2 m 2 )/2eB] denotes the number of occupied Landau levels. It should be pointed out here that there is an uncertainty at µ = E n when ρ () is fixed which will be discussed later. For a given magnetic induction B, and hence the degeneracy eb/, the chemical potential has a jump when the electron density takes the value neb/, n = 1,2,3,, where all the n lower levels have been completely filled and the (n + 1)th level just begins to be occupied. This yields a step-wise relation between the electron density and the chemical potential which is shown by the solid line in Fig. 1. The dashed line in Fig. 1 is a corresponding relation at finite temperature resulting from Eq. (2). It is The project supported by National Natural Science Foundation of China

2 674 CHAI Kang-Min and XU Hong-Hua Vol. 35 seen in Fig. 1 that at zero temperature the chemical potential has discontinuities at ρ () = neb/, n = 1,2,3,, and can be expressed as µ = m 2 + 2neB, neb < (n + 1)eB ρ() <. (4) completely polarized. It can also be seen that the peaks in Fig. 2 correspond to the situation where electrons are just to occupy a higher level, and the valleys correspond to the one where all lower Landau levels have been entirely occupied. Denoting eb n = ρ/n then it is easy to see that the period with respect to the reciprocal change of the B field is a constant = 1/eB n+1 1/eB n = 1/ ρ, (7) which can be rewritten as = /S, (8) where S is the extremal Fermi surface, a well-known fact for de Haas van Alphen oscillations, as generally investigated by Lifshitz and Pitaevskii. [4] Fig. 1 The chemical potential (in unit of m) versus the particle density ρ (T) (in unit of eb/), the solid line is for T = and the dotted line is for T = m/4. Similar relation was given in Fig. 2 in Ref. [2]. In the present paper we consider only the case with a given particle density, denoted by ρ (T) = ρ, then equation (4) can be transformed to µ = m 2 ρ ρ + 2neB, < eb < n + 1 n, (5) if we let the magnetic induction B, instead of the particle density, vary. In each interval the chemical potential has a maximum µ = m 2 + 4π ρ and a minimum µ min,n = m 2 + 4π ρn/(n + 1), meaning that the chemical potential oscillates when the magnetic field changes as shown in Fig. 2. Here a few explanations are in order. At zero temperature and for a high enough magnetic field satisfying eb = ρ, all electrons are on the ground level and then the chemical potential can be inferred from the expression (2) which becomes now eb 1 ρ = lim, (6) β 1 + eβ(m µ) which tells us that µ m + +. The chemical potential keeps this value (actually µ m + ) if the magnetic field continues to increase. For weak magnetic fields satisfying eb µ 2 m 2, the integer j = int[(µ 2 m 2 )/eb] is large and µ min,j µ, meaning that the oscillation converges at the point µ. This indicates that µ is the unique maximum of the chemical potential and therefore, the Fermi momentum is k F = 4π ρ, instead of k F = ρ. This is understandable, because the system under investigation is Fig. 2 The chemical potential oscillates at T = for a given particle density ρ as the magnetic field changes. When ρ is given the filling of electrons on occupied Landau levels given in Eq. (3) needs to be studied carefully, since, as mentioned before, there is an uncertainty at µ = E n which was ignored in Ref. [2]. We rewrite Eq. (3) as ρ = j then it is easy to see that { eb/, eb/ ρ, ρ () = ρ, ρ eb/. ρ () n, (9) (1) For eb ρ, we consider first the case of eb ρ/(n + 1), then equation (5) tells us that the chemical potential satisfies µ E n+1 > E n and we obtain from Eq. (3), ρ () n = eb/, eb ρ/(n + 1). (11) We then consider the case of ρ/(n + 1) eb ρ/n. It is shown in Fig. 2 that in this case the (n 1) lower

3 No. 6 The de Haas van Alphen Oscillation in Two-Dimensional QED at Finite Temperature and Density 675 levels have been fully filled and the nth level is partially filled with a density ρ () n = ρ neb/, since the total density is ρ. Thus we are led to ρ () n = lim β eb f(e n µ) eb =, ρ n eb ρ, ρ eb n + 1, ρ ρ eb n + 1 n, (12) which is plotted in Fig. 3. It is clearly shown in Fig. 3 that when the magnetic field takes the value, say eb = ρ/2, the ground and the first levels have been completely filled and the second level (curve 2) begins to be occupied. This is just the repeated story said above for the chemical potential. It is easy to see that figure 3 is closely related to the integer quantum Hall effect. higher level begins only when the lower ones are entirely occupied. Besides, the peaks are also depressed as compared with Fig. 2, meaning that the degeneracy is lowered. These changes in filling at finite temperatures should affect destructively the integer quantum Hall effect. Fig. 4 The chemical potential oscillates at finite temperatures for a given particle density ρ =.1m 2 as the magnetic field changes. The solid line is for T = m/4, the dashed line is for T = m/2 and the dotted line is for T = m/1. Fig. 3 The change in filling of particles on occupied Landau levels at T = for a given density ρ () = ρ as the magnetic field is reduced. Where the curve indicates the particle density on the ground level and the curves 1, 2, 3, (see insertions) represent the particle densities on higher occupied Landau levels. At finite temperatures we examine first the behavior of the chemical potential. For a given particle density, the chemical potential is a function of the magnetic field at a fixed temperature, as has been seen from Eq. (2). The numerical solution is shown in Fig. 4. It is seen that the higher the temperature is, the weaker the oscillation is. For sufficiently high temperatures oscillations disappear. With the chemical potential being determined by Eq. (2), the particle density on occupied Landau levels at finite temperature can be calculated numerically and the result is displayed in Fig. 5 which shows clearly that the filling of occupied levels is crossed due to thermal excitations, contrary to zero-temperature case where the filling of a Fig. 5 The filling of particles on occupied Landau levels at temperature T = m/2 with a fixed particle density ρ =.1m 2. 3 Oscillations of the Magnetization Oscillations of the magnetization, known as the de Haas van Alphen oscillations, are more familiar and are easier to be measured experimentally, though this kind of oscillation is intimately related to the oscillation of the

4 676 CHAI Kang-Min and XU Hong-Hua Vol. 35 chemical potential studied in the previous section. A partial derivative of the thermodynamic potential (2) with respect to the magnetic induction B yields where M (T) = F B = M n (T), (13) M (T) n = e β {ln[1 + e β(en µ) ] + λ n ln[1 + e β(en+µ) ]} ne2 B [f(e n µ) + f(e n + µ)]. (14) E n With the chemical potential being numerically solved according to Eq. (2), the magnetization M (T) can be calculated numerically and the resulting curves are shown in Fig. 6. It is seen that the magnetization tends to zero when the B field near ρ/e which is understandable by the reason that in strong magnetic fields almost all particles are on the ground level and as a consequence, the thermodynamic potential is nearly independent of B. In Ref. [2] similar curves were plotted in their Fig. 4 with the chemical potential being fixed and as a result, the magnetization seems to increase linearly for strong B fields. We wonder whether this kind of oscillations is really related to the de Haas van Alphen oscillations. Figure 6 shows that the de Haas van Alphen oscillations are present for low temperatures and weak magnetic fields and disappear gradually for vanishing B fields and finally converge at some point which can be determined analytically. Following Lifshitz and Pitaevskii [4] and with the help of Poisson formula 1 2 f() + f(n) = n=1 f(x)dx + 2 Re F(x)e ılx dx, (15) the thermodynamic potential (1) can be separated into two parts, the non-oscillatory part F non (T) = eb 4πβ {ln[1 + e β(m µ) ] ln[1 + e β(m+µ) ]} eb β dn{ln[1 + e β(en µ) ] + ln[1 + e β(en+µ) ]} (16) and the oscillatory one osc = eb β 2Re dn{ln[1 + e β(en µ) ] + ln[1 + e β(en+µ) ]}e ıln. (17) After a change of integral variable non can be expressed by non = eb 4πβ {ln[1 + e β(m µ) ] ln[1 + e β(m+µ) ]} 1 dee{ln[1 + e β(e µ) ] β m + ln[1 + e β(e+µ) ]}. (18) Then the non-oscillatory part of the magnetization is obtained after a partial derivative with respect to B, M non (T) = e 4πβ {ln[1+ e β(m µ) ] ln[1+ e β(m+µ) ]}. (19) This term, which was ignored in Ref. [4] and was lost in Ref. [2], determines the convergence point of the de Haas van Alphen oscillations in Fig. 6 as will be shown below. At low temperatures T µ and for small magnetic fields µ B µ, where µ is the Bohr magneton, F osc (T) can be approximately simplified, [4] up to (eb) 2, osc = (eb)2 4π 3 µ 1 λ ( l 2 sinh λ cos πl µ2 m 2 eb ). (2) Then the dominant term of the oscillating part can be approximated by [4] M (T) osc = e(µ2 m 2 ) 4π 2 µ 1 l λ ( sinh λ sin πl µ2 m 2 eb ), (21) where λ = 2 lµt/eb. It is easy to see that equation (21) exponentially converges to zero when B due to the presence of the sinh function in the denominator and therefore, M non (T) gives the convergence point of the de Haas van Alphen oscillations at B =. Contrary to zero-temperature case, the oscillation period in Eq. (21) is not a constant, because µ depends on both the magnetic field and the temperature as shown in Fig. 4. However the oscillation periods can be formally written as [4] (1/eB) = 2/(µ 2 m 2 ) = /S ex, (22) where S ex denotes the extremal areas of the Fermi surface. The curve of M osc (T) versus eb is plotted at T = m/4 according to Eq. (21) with the chemical potential being read from Eq. (2) and is shown by the dashed line in Fig. 7 which is up shifted by an amount according to Eq. (19) in which we set approximately µ = µ. For the sake of comparison the dashed line in Fig. 6 resulting from numerical calculations is replotted in Fig. 7 by a solid line. It is seen in Fig. 7 that the dashed line and the solid one agree well for (eb/ ρ) <.1, showing that the analytical expression (21) can give a good description of the de Haas van Alphen oscillations for weak magnetic fields.

5 No. 6 The de Haas van Alphen Oscillation in Two-Dimensional QED at Finite Temperature and Density 677 Fig. 6 The de Haas van Alphen oscillations at finite temperatures for a fixed particle density ρ =.1m 2 where the solid line is for T = m/1 and the dashed line for T = m/4. Fig. 7 The de Haas van Alphen oscillations at finite temperature (T = m/4), where the density ρ =.1m 2. The solid line results from numerical calculations and the dashed line comes from the analytical expression (22). At zero temperature equation (13) is reduced to M () = e j ] [µ E n nρ() n, E n ρ ρ < eb <, (23) j + 1 j where µ is given in Eq. (5) and the uncertainty of the step function at µ = E n has been implemented with the help of Eq. (12), i.e., the contribution from the partially filled level has been taken into account. If all occupied levels happen to be completely filled, then, and only then, equation (23) can be expressed by M () = e = e j [ µ E n neb ] E n j [µ 3 ] 2 E n + m2, (24) 2E n which is just the expression (42) in Ref. [2]. The de Haas van Alphen oscillation at zero temperature is shown in Fig. 8 according to Eq. (23). It is seen in Fig. 8 that peaks are located at eb j = ρ/j, j = 1,2,3..., and therefore the oscillation period is = 1/eB j+1 1/eB j = 1/ ρ, (25) identical to that of the chemical potential given in Eq. (7). Contrary to the finite temperature case where the de Haas van Alphen oscillations die out when eb, at zero temperature the de Haas van Alphen oscillations survive for small B fields and suddenly cease at the point M when B = as shown in Fig. 8, where M = lim M non (T) = e ( µ m). (26) T 4π The magnetization reaches its maximum, denoted by M max, at eb = ρ where all particles are on the ground level, then we obtain from Eq. (24), M max = (e/)( µ m). (27) One could ask why the non-oscillatory part M can be a zero-temperature limit of M non (T) while the oscillatory part cannot be a corresponding limit of M osc (T) in Eq. (21). The physics behind it is that the oscillatory part M osc (T) is determined by the particles near the Fermi surface, as pointed out by Lifshitz and Pitaevskii, [4] which is not continuous when T due to the discontinuous change of the Fermi function at T =. While the non-oscillatory part depends on all the particles and therefore is continuous when T. The oscillatory amplitude of the magnetization for small B fields can be calculated by using the expression (24), because the peaks and valleys in Fig. 8 are at the points eb = ρ/j = ( µ 2 m 2 )/2j, j = 1,2,3,..., where the occupied levels are completely filled. For small B fields j = ( µ 2 m 2 )/eb is a large integer, then the following Poisson s formula [4] N ( f n + 1 ) 2 N f(x)dx 1 24 f (x) N (28) is applicable and equation (24) can be reduced to j M () = dx [µ 3 ] m2 E(x) + + O(eB), (29) 2 2E(x) where E(x) = m 2 eb + 2eBx and j = ( µ 2 m 2 )/2eB. Under the condition eb m 2, the magnetization at a peak, where µ = µ, can be calculated to give M p = e { µ µ2 m 2 2eB 1 2eB [ µ 3 m ( µ m)eb ]

6 678 CHAI Kang-Min and XU Hong-Hua Vol. 35 [ + m2 µ m + 1 ( 1 1 µ) ]} 2eB 2 m eb + O(eB), (3) which becomes a constant when eb, M p M b = e (3 µ 2m m2 ). (31) 8π µ The magnetization at a valley can be calculated similarly to yield M v = e {µ µ2 m 2 2eB 1 [ µ 3 m 3 3 ] 2eB 2 ( µ m)eb [ + m2 µ m + 1 ( 1 1 µ) ]} 2eB 2 m eb + O(eB). (32) For a large j the chemical potential at the valley can be approximated by µ = m 2 + 2jeB µ eb/ µ. (33) Substituting the above expression into Eq. (32) and then taking the limit eb we obtain M v M a = (e/8π)( µ 2m + m 2 / µ). (34) It is obvious that M a M b, meaning that the magnetization does oscillate for small B fields around the point (M b + M a )/2 = M with an amplitude (M b M a )/2 = e ρ/2 µ. Corresponding curves were given in Fig. 3 in Ref. [2] which shows that the de Haas van Alphen oscillations converge to M. However this figure was plotted according to their expressions (42) and (46) which, as mentioned above, are valid only when all occupied levels are completely filled and therefore is not correct. To confirm our conclusion, we take the limit ρ m 2 /4π, or after restoring the dimension, ρ m 2 c 2 /4π h 2 = cm 2, which holds for real systems, then equations (26), (31) and (34) are reduced to M a =, M b = M () max = e ρ/m = 2µ ρ, M = e ρ/2m = µ ρ, (35) respectively. This means that for small magnetic fields the de Haas van Alphen oscillation becomes an equiamplitude one as shown in Fig. 8, in agreement with the one for an electron system [3] if the magnetization M is shifted down by an amount of µ ρ which arises due to a total polarization of the system under investigation. However two points need to be mentioned here. First, only at zero temperature the de Haas van Alphen oscillation in the free 2D QED tends to be an equi-amplitude one as the magnetic field is reduced, but at finite temperature it disappears as shown in Fig. 6. Besides, if interactions are considered the thermodynamic potential becomes complicated as shown in our previous paper, [5] then the magnetization oscillation may be damped for small fields even at zero temperature. Second, for the 3D QED the de Haas van Alphen oscillation disappears for small fields even without interactions. This can be demonstrated exactly along the same line in the present paper, but can be seen intuitively by the fact that the de Haas van Alphen effect arises due to the rearrangement of electrons in magnetic subbands which will be smoothed out by the third freedom in the direction of magnetic field if the strength is weak enough. Fig. 8 The de Hass-van Alphen oscillation at zero temperature with a fixed particle density ρ =.1m 2. Since the free 2D electron system at zero temperature cannot be strictly realized, we suppose that the aforementioned equi-amplitude de Haas van Alphen oscillations can be hardly observed for real systems and what could be probably observed might be the oscillations similar to those shown in Fig. 6. To conclude, the de Haas van Alphen oscillations in 2D QED at finite temperature and density are studied for a given particle density which are demonstrated not only through the magnetization oscillations, but also through the chemical potential oscillations. Our results, as compared with earlier works, have a clearer picture in physics and can give a correct nonrelativistic limit. References [1] P. Elmfors, D. Persson and B.S. Skagersta, Phys. Rev. Lett. 71 (1993) 48. [2] Jens O. Andersen and Tor Haugset, Phys. Rev. D51 (1995) 373. [3] K. Huang, Statistical Mechanics, (Second Edition), John Wiley & Sons, Inc. (1987). [4] E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics, Part 2, Pergamon Press (198). [5] CHAI Kang-Min and XU Hong-Hua, Phys. Lett. A259 (1999) 43.