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Exact expression of the ground-state energy for the symmetric anderson model
Volume 86A, number 9
PHYSICS LETTERS
14 December 1981
EXACT EXPRESSION OF THE GROUND-STATE ENERGY FOR THE SYMMETRIC ANDERSON MODEL Norio KAWAKAMI and Ayao OKIJI Department of AppliedPhysics, Osaka University, Suita, Osaka 565, Japan Received 14 July 1981
The exact expression of the ground-state energy for the symmetric Anderson model is obtained with the use of the Wiegmann approach. It is found that some of the quasi-momenta appearing in Wiegmann’s paper are necessarily complex to obtain the expression of the ground-state energy.
It was shown by Wiegmann [1] that the Anderson hamiltonian [2], which describes the formation of a localized moment in a metal, is completely soluble under certain assumptions. In this paper, the exact expression of the ground-state energy for the symmetric Anderson harniltonian is given with the use of Wiegmann’s approach. Firstly, let us summarize Wiegmann’s work briefly. The Anderson hamiltonian can be written as
where 6(x) is the Dirac 6-function and the energy is measured from the Fermi energy. It was shown that the solution of this hamiltoriian (H0 + H’) can be obtamed with the use of the Bethe ansatz [1]. In order to diagonalize it, it is convenient to make use of Yang’s method [3] which was applied to the one-dimensional fermion system with the interaction of 6-function type. The following results can be deduced,
H=H°-f-H’,
exp(ik•L)=II
a=1
x
I~I0=~ekckcka+~ Vk(c~ada+d:cko),
iB(k.)—iA —UV2/2 __~~!
~
iB(k
2/2 1)
—
iA~+ UV
l+v2I(ki~2,
/1
(3a)
N,
1—~iv2I(k/—ed)
H’=ed(nt +n~)+Untn~(n
2/2
0=dd0). (1) Suppose that Vk does not depend on k and that the values of U, ~d and V2 are small compared with the Fermi energy. With these assumptions the following deductions can be made: the system reduces to a onedimensional one; the impurity effect of electron states far from the Fermi surface on the physical quantities can be neglected; then the energy spectrum, 6k~may be expressed as a linear function of k. Furthermore, assuming that thebeimpurity at x = 0,form, the hamil0 can rewrittenis located in the following tonian H = J’dx { —ic(x) (d/dx)c 0(x)
N iB(k)_iAa+UV
—
II
~~1~__________ /=1 iB(k~) UV2/2 —
—
M =
~ ~=i
iA iA0
2
— —
iA(3 + uv2 UV iA —
=
a
1,
...
,M,
(3b)
a
where B(k) = k(k U 2ed). Here, L is the length of the system, k 1 denotes a quasi-momentum, Aa is a(up-spin) variable electron related tonumber. the spinTaking and Nthe (M)logarithm is the total of (3), Wiegmann calculated the magnetic susceptibility 2/U 0) [1]. In that in the case of the s—d limit (Vthat all the electrons are calculation it was considered —
—
+ V6(x)[c(x)d
0 + dc0(x)]}
(2)
0031-9163/81/0000—0000/s 02.75 © 1981 North-Holland
concentrated near the Fermi surface except for one 483
Volume 86A, number 9
PHYSICS LETTERS
which is on the localized level (i.e. all k1 = 0 except for ~d)~ However, if one intends to calculate the groundstate-energy other quantities for arbitrary U, Ed and V2, one mustorconsider the electron states belonging
14 December 1981
half integers. Taking the thermodynamic limit of eqs. (4e) and (4d), one can obtain the following equations, 2a(A) dA 2irp(k) = 1 + k B (UV2)2+4(k2—A)2 8UV
f
to k
1 ‘ 0. Hereafter, the case of U ~ 0 and U = is treated. Supposing all the values of k and A are real, it follows that in the case of U~+0or V2 ~ +0, all the electron states above the Fermi energy are occupied and those below the Fermi energy are unoccupied. This is an unphysical result. For the ground state with the magnetization S~= ~(N 2M), it is found that one must consider 2M complex k’s, N 2M real k’s and M real A’s to obtain the solution of eq. (3). Such a treatment of complex k appeared previously in the study of the one-dimensional fermion system with attractive interaction [4,5]. For large L, 2M complex k’s may be expressed with the use ofM real A’s as —
2 V
_______________
~
f ~
~2
~
(5a)
~
4UV~~(k) dk +4(k2 — A)2
(uV2)2 +
—
j’ B
2UV~o(A’)dA’ (UV2)2
+
(A
A(A) +L~Z(A)
+
2iro(A)
A’)2
—
(5b)
,
where
k 0
=
X(A0) ±iy(A0),
a = 1, 2,
...
,M,
(4a)
A(A) = —2x’, Z(A) -2
where 2 x(A) = _W~[A + (A y(A) = ~.,/~[—A+ (A2
+
U2V4/4)~2]1/2,
(4b)
+
U~V4/4)1/2]
(4c)
1/2
~
~
-
L (x+i~U)2+(y~~V2)2
(x +~U)y’—(y +~V2)x’ and A 0 can be determined by N
—
x = dx/dA andy’ dy/dA. Here, u(A) and p(k) are the distribution functions of A 1, AM and k2M÷l, respectively. integration regions equations, B and Q are kN, determined by theThe following conditional
2] O[2(A0
—
I—
ki)/UV
...
M
2] + Ii [(A0
+ ~ I =
—
...
2)/(x
+0[(y0
0 [(y0 ~ V 0 + 2)/(x +~V 0÷~U)] —
—
2M real k’s can be
—
MIL
E/L
...
=
+ fp(k)
Q
dk
fu(A) dA.
(6a)
(6b)
A~)/UV
1+~U)] +2irJ~,
(4e)
,N.
Here, 0(x) = tan~(x)and 484
B
Further, the ground-state energy with magnetization S~ = ~(N — 2M) can be written as
+20[V2/2(k
/ = 2M + 1,
2 fo(A) dA
B
2] 0 [2(k/
=
(4d)
.
M
—2
N/L
—
On the other hand, from (3a) N expressed as =
,
A~/UV
—x 0L
k1L
,
=
f2x(A) a(A) dA + fk p(k) dk. B
(7)
Q
Now, let us calculate the energy of the singlet ground and 10 are integers or
state. From (6), region Q disappears in the exptession for the singlet ground state. Furthermore, in this case
Volume 86A, number 9
PHYSICS LETTERS
region B can be shown to extend from ~00 to +00 and the integral equation (Sb) for the distribution function can be solved. The explicit expression for cr5(A)
14 December 1981 0
1.0
2.0
3.0
U%r~
proportional to ilL is —0.5
2~) r~(k 1 os(A)~f1214 2 x LUV Note that
(8)
J
2
f
El -1.0 ____________
Fig. 1. The ground-state energy E
1 = EilL — EU—0/L as a function of U. The solid line corresponds to the 2 = 0.1, while thenumerical broken calculation with a fixed value V
a 5(A) dA = l/L
(9)
.
line corresponds tothe perturbation calculation by Yamada. The band width of the virtual bound state ~ corresponds to ~V2 in this calculation.
The energy of the singlet ground state is expressed with the use of (8) as B1
2x(A)u5(A)dA.
E5/Lf
(10)
______________________________ S
For U
0 expression (10) reduces to
EUO/L=~f 2
=
k(~~4)dk
0
and for V 0
EV2...o/L
=~-
f
L~_
______________________________________________________
k6(k
+
~U)dk.
-0.4
HereV~(~U, V2)is the half band width of ~ The numerical calculation of E 5/L Eu~,o/Lis shown in 4 by [6]the is also shown. Furthermore, in to figs. fig. Yamada 1 in which perturbation calculation up U
-0.2
X
-0.1
1’
(b)
—
2a and 2b the distribution function for the singlet state, D(x), is shown as a function of x ( Re k). Fi. nally, it is interesting to point out that when U ~ V2 the anomalous energy gain proportional to TK (Kondo temperature) [7] can be obtained analytically by calculating the integral in expression (10) within the re gion near the position of the arrow in fig. 2b. Note that the Fermi surface of the U = 0 or V2 = 0 cases is located at this arrow.
-0.3
0
0.015
100
0.01
o.~5 -0.15
-0.05 1’
-0.1
50
0
-_____________________________________________ 04
0.3
-0.2
0
X Fig. 2. The distribution function of x 3X)0
2
= 0.3, (b)
=
Re k, D(x)
U=
0.8,
The 5(A). position (a)of U the = 0.02, arrow V corresponds to x
V2
(aA/
= 0.005.
=
485
Volume 86A, number 9
PHYSICS LETTERS
The authors would like to express their sincere thanks to Professor A. Yoshimori for his valuable discussions and reading of manuscript and also to Professor I. Syozi for encouragement. Numerical calculations were carried out with the aid of ACOS-77NEAC-SYSTEM-900 in the Computation Center of Osaka University.
486
14 December 1981
References Wiegmann, Phys. Lett. 80A (1980) 163. Anderson, Phys. Rev. 124 (1961) 41. [3] C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. [4] M. Gaudin, Phys. Lett. 24A (1967) 55. [51M. Takahashi, Prog. Theor. Phys. 42 (1969) 1098. [6] K. Yamada, Prog. Theor. Phys. 53(1975) 970. [7] K. Yosida and A. Yoshimori, in: Magnetism, Vol. 5, ed. H. Suhl (Academic Press, New York, 1973) p. 253. [1]
P.B.
[21P.W.
PHYSICS LETTERS
14 December 1981
EXACT EXPRESSION OF THE GROUND-STATE ENERGY FOR THE SYMMETRIC ANDERSON MODEL Norio KAWAKAMI and Ayao OKIJI Department of AppliedPhysics, Osaka University, Suita, Osaka 565, Japan Received 14 July 1981
The exact expression of the ground-state energy for the symmetric Anderson model is obtained with the use of the Wiegmann approach. It is found that some of the quasi-momenta appearing in Wiegmann’s paper are necessarily complex to obtain the expression of the ground-state energy.
It was shown by Wiegmann [1] that the Anderson hamiltonian [2], which describes the formation of a localized moment in a metal, is completely soluble under certain assumptions. In this paper, the exact expression of the ground-state energy for the symmetric Anderson harniltonian is given with the use of Wiegmann’s approach. Firstly, let us summarize Wiegmann’s work briefly. The Anderson hamiltonian can be written as
where 6(x) is the Dirac 6-function and the energy is measured from the Fermi energy. It was shown that the solution of this hamiltoriian (H0 + H’) can be obtamed with the use of the Bethe ansatz [1]. In order to diagonalize it, it is convenient to make use of Yang’s method [3] which was applied to the one-dimensional fermion system with the interaction of 6-function type. The following results can be deduced,
H=H°-f-H’,
exp(ik•L)=II
a=1
x
I~I0=~ekckcka+~ Vk(c~ada+d:cko),
iB(k.)—iA —UV2/2 __~~!
~
iB(k
2/2 1)
—
iA~+ UV
l+v2I(ki~2,
/1
(3a)
N,
1—~iv2I(k/—ed)
H’=ed(nt +n~)+Untn~(n
2/2
0=dd0). (1) Suppose that Vk does not depend on k and that the values of U, ~d and V2 are small compared with the Fermi energy. With these assumptions the following deductions can be made: the system reduces to a onedimensional one; the impurity effect of electron states far from the Fermi surface on the physical quantities can be neglected; then the energy spectrum, 6k~may be expressed as a linear function of k. Furthermore, assuming that thebeimpurity at x = 0,form, the hamil0 can rewrittenis located in the following tonian H = J’dx { —ic(x) (d/dx)c 0(x)
N iB(k)_iAa+UV
—
II
~~1~__________ /=1 iB(k~) UV2/2 —
—
M =
~ ~=i
iA iA0
2
— —
iA(3 + uv2 UV iA —
=
a
1,
...
,M,
(3b)
a
where B(k) = k(k U 2ed). Here, L is the length of the system, k 1 denotes a quasi-momentum, Aa is a(up-spin) variable electron related tonumber. the spinTaking and Nthe (M)logarithm is the total of (3), Wiegmann calculated the magnetic susceptibility 2/U 0) [1]. In that in the case of the s—d limit (Vthat all the electrons are calculation it was considered —
—
+ V6(x)[c(x)d
0 + dc0(x)]}
(2)
0031-9163/81/0000—0000/s 02.75 © 1981 North-Holland
concentrated near the Fermi surface except for one 483
Volume 86A, number 9
PHYSICS LETTERS
which is on the localized level (i.e. all k1 = 0 except for ~d)~ However, if one intends to calculate the groundstate-energy other quantities for arbitrary U, Ed and V2, one mustorconsider the electron states belonging
14 December 1981
half integers. Taking the thermodynamic limit of eqs. (4e) and (4d), one can obtain the following equations, 2a(A) dA 2irp(k) = 1 + k B (UV2)2+4(k2—A)2 8UV
f
to k
1 ‘ 0. Hereafter, the case of U ~ 0 and U = is treated. Supposing all the values of k and A are real, it follows that in the case of U~+0or V2 ~ +0, all the electron states above the Fermi energy are occupied and those below the Fermi energy are unoccupied. This is an unphysical result. For the ground state with the magnetization S~= ~(N 2M), it is found that one must consider 2M complex k’s, N 2M real k’s and M real A’s to obtain the solution of eq. (3). Such a treatment of complex k appeared previously in the study of the one-dimensional fermion system with attractive interaction [4,5]. For large L, 2M complex k’s may be expressed with the use ofM real A’s as —
2 V
_______________
~
f ~
~2
~
(5a)
~
4UV~~(k) dk +4(k2 — A)2
(uV2)2 +
—
j’ B
2UV~o(A’)dA’ (UV2)2
+
(A
A(A) +L~Z(A)
+
2iro(A)
A’)2
—
(5b)
,
where
k 0
=
X(A0) ±iy(A0),
a = 1, 2,
...
,M,
(4a)
A(A) = —2x’, Z(A) -2
where 2 x(A) = _W~[A + (A y(A) = ~.,/~[—A+ (A2
+
U2V4/4)~2]1/2,
(4b)
+
U~V4/4)1/2]
(4c)
1/2
~
~
-
L (x+i~U)2+(y~~V2)2
(x +~U)y’—(y +~V2)x’ and A 0 can be determined by N
—
x = dx/dA andy’ dy/dA. Here, u(A) and p(k) are the distribution functions of A 1, AM and k2M÷l, respectively. integration regions equations, B and Q are kN, determined by theThe following conditional
2] O[2(A0
—
I—
ki)/UV
...
M
2] + Ii [(A0
+ ~ I =
—
...
2)/(x
+0[(y0
0 [(y0 ~ V 0 + 2)/(x +~V 0÷~U)] —
—
2M real k’s can be
—
MIL
E/L
...
=
+ fp(k)
Q
dk
fu(A) dA.
(6a)
(6b)
A~)/UV
1+~U)] +2irJ~,
(4e)
,N.
Here, 0(x) = tan~(x)and 484
B
Further, the ground-state energy with magnetization S~ = ~(N — 2M) can be written as
+20[V2/2(k
/ = 2M + 1,
2 fo(A) dA
B
2] 0 [2(k/
=
(4d)
.
M
—2
N/L
—
On the other hand, from (3a) N expressed as =
,
A~/UV
—x 0L
k1L
,
=
f2x(A) a(A) dA + fk p(k) dk. B
(7)
Q
Now, let us calculate the energy of the singlet ground and 10 are integers or
state. From (6), region Q disappears in the exptession for the singlet ground state. Furthermore, in this case
Volume 86A, number 9
PHYSICS LETTERS
region B can be shown to extend from ~00 to +00 and the integral equation (Sb) for the distribution function can be solved. The explicit expression for cr5(A)
14 December 1981 0
1.0
2.0
3.0
U%r~
proportional to ilL is —0.5
2~) r~(k 1 os(A)~f1214 2 x LUV Note that
(8)
J
2
f
El -1.0 ____________
Fig. 1. The ground-state energy E
1 = EilL — EU—0/L as a function of U. The solid line corresponds to the 2 = 0.1, while thenumerical broken calculation with a fixed value V
a 5(A) dA = l/L
(9)
.
line corresponds tothe perturbation calculation by Yamada. The band width of the virtual bound state ~ corresponds to ~V2 in this calculation.
The energy of the singlet ground state is expressed with the use of (8) as B1
2x(A)u5(A)dA.
E5/Lf
(10)
______________________________ S
For U
0 expression (10) reduces to
EUO/L=~f 2
=
k(~~4)dk
0
and for V 0
EV2...o/L
=~-
f
L~_
______________________________________________________
k6(k
+
~U)dk.
-0.4
HereV~(~U, V2)is the half band width of ~ The numerical calculation of E 5/L Eu~,o/Lis shown in 4 by [6]the is also shown. Furthermore, in to figs. fig. Yamada 1 in which perturbation calculation up U
-0.2
X
-0.1
1’
(b)
—
2a and 2b the distribution function for the singlet state, D(x), is shown as a function of x ( Re k). Fi. nally, it is interesting to point out that when U ~ V2 the anomalous energy gain proportional to TK (Kondo temperature) [7] can be obtained analytically by calculating the integral in expression (10) within the re gion near the position of the arrow in fig. 2b. Note that the Fermi surface of the U = 0 or V2 = 0 cases is located at this arrow.
-0.3
0
0.015
100
0.01
o.~5 -0.15
-0.05 1’
-0.1
50
0
-_____________________________________________ 04
0.3
-0.2
0
X Fig. 2. The distribution function of x 3X)0
2
= 0.3, (b)
=
Re k, D(x)
U=
0.8,
The 5(A). position (a)of U the = 0.02, arrow V corresponds to x
V2
(aA/
= 0.005.
=
485
Volume 86A, number 9
PHYSICS LETTERS
The authors would like to express their sincere thanks to Professor A. Yoshimori for his valuable discussions and reading of manuscript and also to Professor I. Syozi for encouragement. Numerical calculations were carried out with the aid of ACOS-77NEAC-SYSTEM-900 in the Computation Center of Osaka University.
486
14 December 1981
References Wiegmann, Phys. Lett. 80A (1980) 163. Anderson, Phys. Rev. 124 (1961) 41. [3] C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. [4] M. Gaudin, Phys. Lett. 24A (1967) 55. [51M. Takahashi, Prog. Theor. Phys. 42 (1969) 1098. [6] K. Yamada, Prog. Theor. Phys. 53(1975) 970. [7] K. Yosida and A. Yoshimori, in: Magnetism, Vol. 5, ed. H. Suhl (Academic Press, New York, 1973) p. 253. [1]
P.B.
[21P.W.