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The ground state energy of an electron gas in a high magnetic field
Volume 39A, number 1
PHYSICS LETTERS
10 April 1972
THE GROUND STATE ENERGY OF AN ELECTRON GAS IN A HIGH MAGNETIC FIELD* A. ISIHARA and J.T. TSAI Department of Physics, State University of New York. Buffalo 14214, USA Received 17 February 1972 The ground state energy of an electron gas is evaluated in the ring diagram approximation in the so-called quantum strong magnetic field limit.
We consider an interacting electron gas at very low temperatures and in a high magnetic field. In this so-called quantum strong magnetic field limit, all the electrons will be localized in the lowest Landau state, with their spins antiparallel to the magnetic field. Therefore, we expect that the exchange energy will be small for high magnetic fields [1]. In view of the growing interest in the behavior of electrons in such a limit [2], we have evaluated the ground state energy rigorously up to the ring diagram approximation. That is, the contributions from the free particles, first order exchange graphs and the ring diagrams are obtained. As customary, we shall use the units such that h = 1 and 2m = 1 where m is the electron mass. We shall also use notations such as a2 _ eH 2c'
z = exp03p2) '
d2 =p2F - a 2 +½ga2
(1)
where a 2 represents the energy due to the magnetic field H which is assumed to be uniform, c is the light velocity, z is the absolute activity, p2F represents the Fermi energy, 3 = 1/kT and g is Lande's g-factor. In addition to d 2 defined in eq. (1), it is convenient to introduce
d 2 = p20 - a 2 + ~ga 2 = 21t2n/a 2
(2)
where p2 is the Fermi energy which corresponds to the Fermi energy in the absence of the Coulomb interactions between the electrons. Note that for the present case, p20 is a function of the magnetic field. In the usual case without a magnetic field, the energy is expressed in terms of r s which represents the density. In the present case, the density, being a singlet distribution function, is related to the magnetic field. Therefore, it is appropriate to use a new parameter R s defined by n - 1 = ~ ¢r(2R s/e 2)3 .
(3)
The density n and energy U can be evaluated from the grand partition function by a method reported elsewhere [3]. The density relation n = (2 ln~/a lnz)3,u yields d=d0[l_2_~d0(2_C_lna~)
2n2d__~(2_C_ln2d2~+
e4a2 . ( 8 _
c
e2
]
(4,
where C is Euler's constant and the right side is correct to order e 4. This relation determines the Fermi energy p2 and therefore the absolute activity as a function of the density and magnetic field. For this determination we need the condition
2u2n/e 2 > a 2 > p2F * This work was supported by the National Science Foundation. 25
Volume 39A, number 1
PHYSICS LETTERS.
10 April 1972
where the first inequality limits the magnitude of the field to prevent unphysical situation where the Fermi distribution becomes meaningless and the second inequality represents the quantum strong magnetic field limit. The energy is evaluated for/3 ~ co. The energy per particle, U/N is found to be
2nR R 2a4 U=a2-~ga 2 4n4n2 +--a2~S(~nn)'/3 (~ C - ] + ~ln8n4n2~ + ~ (~nn) 2' 3 ~
N
3a 4
-a
211 5f 5 a2 1 494 I/2 (2 288 + i-92 + ]9? In - ~ - t,4~ n ~xl/3 + a4 \
a6
-
-
C
/t
_
In
2rr 4 n 2
8rr4 r/2~ {5 ~
a6 )k
-
-
(C
17 ~7r4'12~ - +~ln . . . . . . .
a6 /
~
ln8ff4n2"~ a6 /
(5)
Note that for g = 2 and fixed density, the energy approaches a very small value when the field is very high. This is in conformity with the physical situation where the electrons are localized. On the right side of eq. (5), the first three terms correspond to free electrons, the second term to the first order exchange diagrams and the last term in the curly bracket represents the ring diagram contribution.
Referetlces [11 R.W. Danz and M.L. Glasser, Phys. Rev. B4 (1971) 94. [2] E,D.Haidemenakis, Physics of solids in intense magnetic fields (Plenum Press, New York, 1969). [3] A, lsihara, J. Tsai and M. Wadati, Phys. Rev. A3 (1971) 990; J.T. Tsai, M, Wadati and A. lsihara, Phys. Rev. A4 (1971) 1219.
26
PHYSICS LETTERS
10 April 1972
THE GROUND STATE ENERGY OF AN ELECTRON GAS IN A HIGH MAGNETIC FIELD* A. ISIHARA and J.T. TSAI Department of Physics, State University of New York. Buffalo 14214, USA Received 17 February 1972 The ground state energy of an electron gas is evaluated in the ring diagram approximation in the so-called quantum strong magnetic field limit.
We consider an interacting electron gas at very low temperatures and in a high magnetic field. In this so-called quantum strong magnetic field limit, all the electrons will be localized in the lowest Landau state, with their spins antiparallel to the magnetic field. Therefore, we expect that the exchange energy will be small for high magnetic fields [1]. In view of the growing interest in the behavior of electrons in such a limit [2], we have evaluated the ground state energy rigorously up to the ring diagram approximation. That is, the contributions from the free particles, first order exchange graphs and the ring diagrams are obtained. As customary, we shall use the units such that h = 1 and 2m = 1 where m is the electron mass. We shall also use notations such as a2 _ eH 2c'
z = exp03p2) '
d2 =p2F - a 2 +½ga2
(1)
where a 2 represents the energy due to the magnetic field H which is assumed to be uniform, c is the light velocity, z is the absolute activity, p2F represents the Fermi energy, 3 = 1/kT and g is Lande's g-factor. In addition to d 2 defined in eq. (1), it is convenient to introduce
d 2 = p20 - a 2 + ~ga 2 = 21t2n/a 2
(2)
where p2 is the Fermi energy which corresponds to the Fermi energy in the absence of the Coulomb interactions between the electrons. Note that for the present case, p20 is a function of the magnetic field. In the usual case without a magnetic field, the energy is expressed in terms of r s which represents the density. In the present case, the density, being a singlet distribution function, is related to the magnetic field. Therefore, it is appropriate to use a new parameter R s defined by n - 1 = ~ ¢r(2R s/e 2)3 .
(3)
The density n and energy U can be evaluated from the grand partition function by a method reported elsewhere [3]. The density relation n = (2 ln~/a lnz)3,u yields d=d0[l_2_~d0(2_C_lna~)
2n2d__~(2_C_ln2d2~+
e4a2 . ( 8 _
c
e2
]
(4,
where C is Euler's constant and the right side is correct to order e 4. This relation determines the Fermi energy p2 and therefore the absolute activity as a function of the density and magnetic field. For this determination we need the condition
2u2n/e 2 > a 2 > p2F * This work was supported by the National Science Foundation. 25
Volume 39A, number 1
PHYSICS LETTERS.
10 April 1972
where the first inequality limits the magnitude of the field to prevent unphysical situation where the Fermi distribution becomes meaningless and the second inequality represents the quantum strong magnetic field limit. The energy is evaluated for/3 ~ co. The energy per particle, U/N is found to be
2nR R 2a4 U=a2-~ga 2 4n4n2 +--a2~S(~nn)'/3 (~ C - ] + ~ln8n4n2~ + ~ (~nn) 2' 3 ~
N
3a 4
-a
211 5f 5 a2 1 494 I/2 (2 288 + i-92 + ]9? In - ~ - t,4~ n ~xl/3 + a4 \
a6
-
-
C
/t
_
In
2rr 4 n 2
8rr4 r/2~ {5 ~
a6 )k
-
-
(C
17 ~7r4'12~ - +~ln . . . . . . .
a6 /
~
ln8ff4n2"~ a6 /
(5)
Note that for g = 2 and fixed density, the energy approaches a very small value when the field is very high. This is in conformity with the physical situation where the electrons are localized. On the right side of eq. (5), the first three terms correspond to free electrons, the second term to the first order exchange diagrams and the last term in the curly bracket represents the ring diagram contribution.
Referetlces [11 R.W. Danz and M.L. Glasser, Phys. Rev. B4 (1971) 94. [2] E,D.Haidemenakis, Physics of solids in intense magnetic fields (Plenum Press, New York, 1969). [3] A, lsihara, J. Tsai and M. Wadati, Phys. Rev. A3 (1971) 990; J.T. Tsai, M, Wadati and A. lsihara, Phys. Rev. A4 (1971) 1219.
26