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Towards an exact solution of the Anderson model
24 November 1980
PHYSICS LETTERS
Volume 80A, number 2,3
TOWARDS AN EXACT SOLUTION OF THE ANDERSON MODEL
P.B. WIEGMANN L.D. Landau Institute for Theoretical Physics, Academy of Sciences of the
lJS,TR, MOSCOW,
USSR
Received 2 September 1980
It is shown that the Anderson model, which describes the formation of a localized moment in a metal, is completely integrable.
1. In this letter we report, that under certain assumptions the Anderson of a localized moment in a metal, is a completely integrable model. The Anderson hamiltonian has the form [l] :
91, =ed(n+ t n$) t Un+nl
model, which describes the formation
(n, = d,+d,),
(1)
and describes the interaction of a conduction band with a localized level. The simplifying assumptions adopted while solving (1) are the same as those used previously to construct an exact solution of the Kondo problem (s-d-exchange model) [2,3]. We suppose that: (1) V does not depend on k, (2) the amplitudes U, ed, V2 are small compared with the Fermi energy. These assumptions imply that frrst of all the problem is a spherically symmetric one and, consequently, reduces to a one-dimensional problem. Secondly, one canneglect electron states far from the Fermi surface, so that in the vicinity of k = k, the spectrum may be considered as a linear one: e(k) = (lk I - kF)VF. Thus one can reformulate the problem in terms of the coordinate along an arbitrary straight line intersecting the impurity position [2,3] : 9to = Jdx
{-icz(x)(d/dx)c,(x)
where x is the corresponding
t V~(X)[C~(X)~, + d~c,(x)]}
,
(2)
coordinate.
2. Obviously the total number of particles N = X0 [n, t Jdxc~(x)c,(x)] of N = 2 and Sz = 0, the eigenvector of the hamiltonian is of the form: I$) = JdxI
dx2 g(xl,x2)c~(x1)c:(x2)10)
This function L-i@,,
satisfies the Schrodinger
+ ax*) --EldXl,X2)
(-iaX1 -E
+ &)e(x)
+ ldx
- cf(x)dT]
IO) tf$dflO)
.
(3)
equation:
+ W(Xlk(X2)
+ @(0,X)
e(x)[cT(x)d:
is conserved. For instance in the case
+ vti(X)f=
+ NX2MXl)l 0,
(u+
=0 , hd)f+
2ve(o)
=
Ef.
(4)
For U = 0 the solution is 163
‘dXl>X2)‘g&&&2)
+g&kp(q>?
e(x)=epg/Jx)+ q&w
where gk(x) and ek are the solutions of the single-particle
g,dx)= eikx[l
24 November 1980
PHYSICS LETTERS
Volume 80A, number 2,3
-iisgnx*
V2/(k-Ed)],
(5)
1
equations
ck = v/(k - ed) ,
(6)
where the energy E equals k t p. For U # 0 we search for a solution of the form: &1,x2)
=gk(x&p(x2)z,Q(x1
Substituting
px2)
+gk(x2kp(x1)zkp(x2
-x1)
(7)
.
(7) into eqs. (4) we obtain:
c(x) =g&)ckz,(x)
+ &(x)cpZkp(-X).
I tisgnx.
Z,(x)=
~2U/(k+p-U-2Ed)(k~p).
(8)
3. The Bethe ansatz for the solution of the Schrbdinger equation for N particles with an arbitrary fixed spin is as follows. Let Q and P be permutations of the integers (1, ....N}. In each domain XQ ={x4r
N (9)
where {9} are the spins of the particles, {kj} is a set of unequal numbers and gk(x) is given by eq. (6). One can write down a similar expressions for e (Y ...&N_&xl .,.X&l) as well. The underlying hypothesis is that in each domain the wave function is described &y a unique set of Q-independent parameters. 4. There are a number of constraints imposed on the N! X N! matrixA(Q, P) by the Schrodinger equation, the continuity conditions at the boundaries of the domains and by antisymmetry. The number of these constraints greatly exceeds the number of elements of the matrixA(Q, P). The consistency of these constraints proves the Bethe hypothesis. Let I = { 1, ....N}. Then A(Q, P) is related to A(I, I) by the matrix S(Q, I): XQ
A
,,.,.,,ce~~ =s~~.::.~~(Q,~)A,;,.,,~(~,~), 4,...,,(~3)
= (-1)%,
.,.,aN(~~p~. 1
(10)
Insert the wave function into the Schrijdinger equation. The resulting equations for the discontinuities of the coefficients A(Q) across the boundaries xi = xi take the form of a two-particle problem. Consequently the matrix S is a two-particle factorizable matrix. To construct this matrix it is necessary to decompose the permutation Q into a product of pair perturbations and relate each permutation to a two-particle matrix S,(ki, kj), which acts on the spin indices cui, OIiof the matrix A(Q, P). Of course there are many different decompositions into pair permutations. For the Bethe hypothesis to hold all choices of permutations should lead to identical results. It can easily be checked that a necessary and sufficient condition for this is the equivalence of two possible permutations in the three-particle system [4-61. This equivalence can be expressed in terms of factorizability and unitarity equations :
(11) The total spin of the system is conserved, so that the matrix Sij has the form Sii=Wo+Waiaj=b+CPij,
164
(12)
PHYSICS LETTERS
Volume 80A, number 2,3
w. and w are functions of ki,j of a special form;
where @ = f (1 + u * O) is the permutation operator. The quantities they satisfy eqs. (1 l), whichin terms of 6 and c take the form: b(k, p)lc(k, p) = o(k, p),
o(k,p)+o(p,r)=o(k,r),
24 November 1980
lb12 t lc12= 1.
(13)
The general solution has the form o(k, P) = q(k) - q(p) 9
(14)
where q(k) is an arbitrary function,
b(k, P>= 4, P)
1
+$$),
c@>p)= @,p)
1
3
+
;(kp>,lal=l.
AS noted above, the matrix Sij is given by the two-particle using (7) and (8), we have g(k) = (UY2)-l
(15)
3
[(k - U - ed)(k - cd) - ed(U t Ed)],
solution of the Schrbdinger
equation.
That is why,
a=1,
(16)
and the Bethe hypothesis is valid. The validity of the Bethe ansatz and consequently the complete integrability a hidden symmetry and an infinite set of conservation laws.
of the system is connected
with
5 To study the spectrum of the system, we must impose boundary conditions on the wave functions. It is convenient to choose periodic boundary conditions in a spherical box of radius L/2. In the same way as in ref. [4] they lead to the eigenvalue problem of the operators T(j), which give the cyclic permutation of particle j: T(j) =x~+~,%~+~
...SflSjl ...Sjj_l,
7’(#+%4ffi...ah a1...cx~
=
expW#)Aorl...aN .
The vector A is simultaneously an eigenvector of N operators. These operators commute stitute a particular set from the parametric family of matrixes T(u) [8] :
(17) with each other and con-
(18) The factorizability and the unitarity conditions (11) are necessary and sufficient for the family T(u) to be selfcommuting: [T(u), T(u’)] = 0 [6]. Yang was the first to solve a similar eigenvalue problem [4]. An elegant method of diagonalizing the family T(u) was recently proposed by Faddeev et al. [7]. The eigenvalues of the operators T(j) with total spin projection N/2-M are given by the equations M ig(k.) - iXp - l/2 exp(ikjL) = p=l II Ig(k:~)_ ids + 1/2’ where the quantity
P=l,
‘..,M,
i=‘,...,N,
(19)
A, satisfies the conditions:
(20) In terms of logarithms we have - 70(2X,
E=/$
kj,
- 2g(kj)) = 2nJ, - 5 6(k)=2
arctg[V2/2(k-
B(h, - Ap),
ed)] ,
Lkj = 2n$ + F
8(2g(kj) - 2h,) ,
(21) (22) 16.5
Volume 80A,
PHYSICS LETTERS
number 2,3
where the integers J, and Ii are quantum
24 November 1980
numbers of the system and 0(x) = 2 arctg x.
6. Eqs. (21) and (22) give a solution of the spectrum problem, for the Anderson hamiltonian in the case that u, v2,e/eE‘. Let us show that, for instance, in the symmetric case U = -2~~ eqs. (2 1) and (22) in the Schrieffer-Wolff limit U >> V2 tend to similar equations for the s-d-exchange model [2,3]. In this limit we can consider all particles as concentrated near the Fermi surface, except one which is on the localized level. In other words all ki = 0, except the one which is equal to ed. Then eqs. (21) take the form M
Lki = 219 - c
or=1
8(2X,) + U/4V2 ,
(23) M
-
N8(2h,)
- 8(2X, - u/2vq
= 27rJ, - Dgl e(x, ~ A@)
(24)
In the limit of V2/U < 1 the energy of the ground state as a function of the total spin of the system can explicitly be calculated. The resulting magnetic field dependence of the impurity contribution to the magnetic susceptibility is of the form found in refs. [2,3] :
Ximp(H)
= i
HI@
jm r($ - 0) (zr _ i_
(25)
exp [2o ln(H/7’k)] dw ;
(26) ifH%
Tk In In H/Tk
x(H)=
’ 2H(ln H/Tk)2
”
lnH/Tk
“”
(27)
’
where Tk =
const
'
(N/L) CrlJ,
(28)
and J = 4V2/U is the Schrieffer-Wolff exchange coupling [9]. Let us calculate the leading pre-exponential factor in the J dependence of Tk. To do so one must take into account the k dependence of q(k) = k2/UV2, so that 5
j=l
0(2h,
- 2q(ki)) + L J0(2
A, - 2k2/UV2)p(k)
dk ,
where p(k) is the densit of the k distribution. The steepest-descent (24) and (28) by L d-+UV . One gets a well-known result [lo] :
calculation
is, in fact, the substitution
of N in
Tk = const - (UV2)l12 exp(-nU/4V2). Eqs. (21) and (22) are pretty complicated and difficult to analyse. Nonetheless studied also in other interesting cases like the limit ed + 0.
166
we hope that they can be
Volume 80A, number 2,3
PHYSICS LETTERS
24 November 1980
References [l] [2] 13) [4] (51 [6] [7] [8] [9] [lo]
P.W. Anderson, Phys. Rev. 124 (1961) 41. P.B. Wiegmann, Pisma Zh. Eksp. Teor. Fiz. 31 (1980) 392. P.B. Wiegmann. L.D. Landau Institute preprint (1980), to be published in J. Phys. C. C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. A.B. Zamolodchikov, Sov. Sci. Rev. 2 (1980). R.J. Baxter, Ann. Phys. 70 (1972) 193. L.D. Faddeev, E.K. Sklyanin and L.A. Takhtajan, preprint LOMI P-I-79 (1979). A.A. Belavin, Phys. Lett. 87B (1979) 117. J.R. Schrieffer and P.A. Wolff, Phys. Rev. 149 (1966) 491. H.R. Krishna-Murthy, KG. Wilson and J.W. Wilkins, Phys. Rev. Lett. 35 (1975) 1101.
167
PHYSICS LETTERS
Volume 80A, number 2,3
TOWARDS AN EXACT SOLUTION OF THE ANDERSON MODEL
P.B. WIEGMANN L.D. Landau Institute for Theoretical Physics, Academy of Sciences of the
lJS,TR, MOSCOW,
USSR
Received 2 September 1980
It is shown that the Anderson model, which describes the formation of a localized moment in a metal, is completely integrable.
1. In this letter we report, that under certain assumptions the Anderson of a localized moment in a metal, is a completely integrable model. The Anderson hamiltonian has the form [l] :
91, =ed(n+ t n$) t Un+nl
model, which describes the formation
(n, = d,+d,),
(1)
and describes the interaction of a conduction band with a localized level. The simplifying assumptions adopted while solving (1) are the same as those used previously to construct an exact solution of the Kondo problem (s-d-exchange model) [2,3]. We suppose that: (1) V does not depend on k, (2) the amplitudes U, ed, V2 are small compared with the Fermi energy. These assumptions imply that frrst of all the problem is a spherically symmetric one and, consequently, reduces to a one-dimensional problem. Secondly, one canneglect electron states far from the Fermi surface, so that in the vicinity of k = k, the spectrum may be considered as a linear one: e(k) = (lk I - kF)VF. Thus one can reformulate the problem in terms of the coordinate along an arbitrary straight line intersecting the impurity position [2,3] : 9to = Jdx
{-icz(x)(d/dx)c,(x)
where x is the corresponding
t V~(X)[C~(X)~, + d~c,(x)]}
,
(2)
coordinate.
2. Obviously the total number of particles N = X0 [n, t Jdxc~(x)c,(x)] of N = 2 and Sz = 0, the eigenvector of the hamiltonian is of the form: I$) = JdxI
dx2 g(xl,x2)c~(x1)c:(x2)10)
This function L-i@,,
satisfies the Schrodinger
+ ax*) --EldXl,X2)
(-iaX1 -E
+ &)e(x)
+ ldx
- cf(x)dT]
IO) tf$dflO)
.
(3)
equation:
+ W(Xlk(X2)
+ @(0,X)
e(x)[cT(x)d:
is conserved. For instance in the case
+ vti(X)f=
+ NX2MXl)l 0,
(u+
=0 , hd)f+
2ve(o)
=
Ef.
(4)
For U = 0 the solution is 163
‘dXl>X2)‘g&&&2)
+g&kp(q>?
e(x)=epg/Jx)+ q&w
where gk(x) and ek are the solutions of the single-particle
g,dx)= eikx[l
24 November 1980
PHYSICS LETTERS
Volume 80A, number 2,3
-iisgnx*
V2/(k-Ed)],
(5)
1
equations
ck = v/(k - ed) ,
(6)
where the energy E equals k t p. For U # 0 we search for a solution of the form: &1,x2)
=gk(x&p(x2)z,Q(x1
Substituting
px2)
+gk(x2kp(x1)zkp(x2
-x1)
(7)
.
(7) into eqs. (4) we obtain:
c(x) =g&)ckz,(x)
+ &(x)cpZkp(-X).
I tisgnx.
Z,(x)=
~2U/(k+p-U-2Ed)(k~p).
(8)
3. The Bethe ansatz for the solution of the Schrbdinger equation for N particles with an arbitrary fixed spin is as follows. Let Q and P be permutations of the integers (1, ....N}. In each domain XQ ={x4r
N (9)
where {9} are the spins of the particles, {kj} is a set of unequal numbers and gk(x) is given by eq. (6). One can write down a similar expressions for e (Y ...&N_&xl .,.X&l) as well. The underlying hypothesis is that in each domain the wave function is described &y a unique set of Q-independent parameters. 4. There are a number of constraints imposed on the N! X N! matrixA(Q, P) by the Schrodinger equation, the continuity conditions at the boundaries of the domains and by antisymmetry. The number of these constraints greatly exceeds the number of elements of the matrixA(Q, P). The consistency of these constraints proves the Bethe hypothesis. Let I = { 1, ....N}. Then A(Q, P) is related to A(I, I) by the matrix S(Q, I): XQ
A
,,.,.,,ce~~ =s~~.::.~~(Q,~)A,;,.,,~(~,~), 4,...,,(~3)
= (-1)%,
.,.,aN(~~p~. 1
(10)
Insert the wave function into the Schrijdinger equation. The resulting equations for the discontinuities of the coefficients A(Q) across the boundaries xi = xi take the form of a two-particle problem. Consequently the matrix S is a two-particle factorizable matrix. To construct this matrix it is necessary to decompose the permutation Q into a product of pair perturbations and relate each permutation to a two-particle matrix S,(ki, kj), which acts on the spin indices cui, OIiof the matrix A(Q, P). Of course there are many different decompositions into pair permutations. For the Bethe hypothesis to hold all choices of permutations should lead to identical results. It can easily be checked that a necessary and sufficient condition for this is the equivalence of two possible permutations in the three-particle system [4-61. This equivalence can be expressed in terms of factorizability and unitarity equations :
(11) The total spin of the system is conserved, so that the matrix Sij has the form Sii=Wo+Waiaj=b+CPij,
164
(12)
PHYSICS LETTERS
Volume 80A, number 2,3
w. and w are functions of ki,j of a special form;
where @ = f (1 + u * O) is the permutation operator. The quantities they satisfy eqs. (1 l), whichin terms of 6 and c take the form: b(k, p)lc(k, p) = o(k, p),
o(k,p)+o(p,r)=o(k,r),
24 November 1980
lb12 t lc12= 1.
(13)
The general solution has the form o(k, P) = q(k) - q(p) 9
(14)
where q(k) is an arbitrary function,
b(k, P>= 4, P)
1
+$$),
c@>p)= @,p)
1
3
+
;(kp>,lal=l.
AS noted above, the matrix Sij is given by the two-particle using (7) and (8), we have g(k) = (UY2)-l
(15)
3
[(k - U - ed)(k - cd) - ed(U t Ed)],
solution of the Schrbdinger
equation.
That is why,
a=1,
(16)
and the Bethe hypothesis is valid. The validity of the Bethe ansatz and consequently the complete integrability a hidden symmetry and an infinite set of conservation laws.
of the system is connected
with
5 To study the spectrum of the system, we must impose boundary conditions on the wave functions. It is convenient to choose periodic boundary conditions in a spherical box of radius L/2. In the same way as in ref. [4] they lead to the eigenvalue problem of the operators T(j), which give the cyclic permutation of particle j: T(j) =x~+~,%~+~
...SflSjl ...Sjj_l,
7’(#+%4ffi...ah a1...cx~
=
expW#)Aorl...aN .
The vector A is simultaneously an eigenvector of N operators. These operators commute stitute a particular set from the parametric family of matrixes T(u) [8] :
(17) with each other and con-
(18) The factorizability and the unitarity conditions (11) are necessary and sufficient for the family T(u) to be selfcommuting: [T(u), T(u’)] = 0 [6]. Yang was the first to solve a similar eigenvalue problem [4]. An elegant method of diagonalizing the family T(u) was recently proposed by Faddeev et al. [7]. The eigenvalues of the operators T(j) with total spin projection N/2-M are given by the equations M ig(k.) - iXp - l/2 exp(ikjL) = p=l II Ig(k:~)_ ids + 1/2’ where the quantity
P=l,
‘..,M,
i=‘,...,N,
(19)
A, satisfies the conditions:
(20) In terms of logarithms we have - 70(2X,
E=/$
kj,
- 2g(kj)) = 2nJ, - 5 6(k)=2
arctg[V2/2(k-
B(h, - Ap),
ed)] ,
Lkj = 2n$ + F
8(2g(kj) - 2h,) ,
(21) (22) 16.5
Volume 80A,
PHYSICS LETTERS
number 2,3
where the integers J, and Ii are quantum
24 November 1980
numbers of the system and 0(x) = 2 arctg x.
6. Eqs. (21) and (22) give a solution of the spectrum problem, for the Anderson hamiltonian in the case that u, v2,e/eE‘. Let us show that, for instance, in the symmetric case U = -2~~ eqs. (2 1) and (22) in the Schrieffer-Wolff limit U >> V2 tend to similar equations for the s-d-exchange model [2,3]. In this limit we can consider all particles as concentrated near the Fermi surface, except one which is on the localized level. In other words all ki = 0, except the one which is equal to ed. Then eqs. (21) take the form M
Lki = 219 - c
or=1
8(2X,) + U/4V2 ,
(23) M
-
N8(2h,)
- 8(2X, - u/2vq
= 27rJ, - Dgl e(x, ~ A@)
(24)
In the limit of V2/U < 1 the energy of the ground state as a function of the total spin of the system can explicitly be calculated. The resulting magnetic field dependence of the impurity contribution to the magnetic susceptibility is of the form found in refs. [2,3] :
Ximp(H)
= i
HI@
jm r($ - 0) (zr _ i_
(25)
exp [2o ln(H/7’k)] dw ;
(26) ifH%
Tk In In H/Tk
x(H)=
’ 2H(ln H/Tk)2
”
lnH/Tk
“”
(27)
’
where Tk =
const
'
(N/L) CrlJ,
(28)
and J = 4V2/U is the Schrieffer-Wolff exchange coupling [9]. Let us calculate the leading pre-exponential factor in the J dependence of Tk. To do so one must take into account the k dependence of q(k) = k2/UV2, so that 5
j=l
0(2h,
- 2q(ki)) + L J0(2
A, - 2k2/UV2)p(k)
dk ,
where p(k) is the densit of the k distribution. The steepest-descent (24) and (28) by L d-+UV . One gets a well-known result [lo] :
calculation
is, in fact, the substitution
of N in
Tk = const - (UV2)l12 exp(-nU/4V2). Eqs. (21) and (22) are pretty complicated and difficult to analyse. Nonetheless studied also in other interesting cases like the limit ed + 0.
166
we hope that they can be
Volume 80A, number 2,3
PHYSICS LETTERS
24 November 1980
References [l] [2] 13) [4] (51 [6] [7] [8] [9] [lo]
P.W. Anderson, Phys. Rev. 124 (1961) 41. P.B. Wiegmann, Pisma Zh. Eksp. Teor. Fiz. 31 (1980) 392. P.B. Wiegmann. L.D. Landau Institute preprint (1980), to be published in J. Phys. C. C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. A.B. Zamolodchikov, Sov. Sci. Rev. 2 (1980). R.J. Baxter, Ann. Phys. 70 (1972) 193. L.D. Faddeev, E.K. Sklyanin and L.A. Takhtajan, preprint LOMI P-I-79 (1979). A.A. Belavin, Phys. Lett. 87B (1979) 117. J.R. Schrieffer and P.A. Wolff, Phys. Rev. 149 (1966) 491. H.R. Krishna-Murthy, KG. Wilson and J.W. Wilkins, Phys. Rev. Lett. 35 (1975) 1101.
167