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The exact solution of the s-d exchange model with arbitrary impurity spin S (Kondo problem)
Volume 81A, number 2,3
PHYSICS LETTERS
12 January 1981
THE EXACT SOLUTION OF THE s - d EXCHANGE MODEL WITH ARBITRARY IMPURITY SPIN S (KONDO PROBLEM) V.A. FATEEV 1
CERN, Geneva, Switzerland and P.B. WIEGMANN
Landau Institute for Theoretical Physics, Moscow, USSR Received 31 October 1980
An exact solution o f the s - d exchange model with arbitrary impurity spin S (Kondo problem) is proposed. On the basis of this solution, the magnetic susceptibility as a function of arbitrary magnetic field is calculated in explicit form.
1
Recently, an exact solution of the s - d exchange model with localized spin S = ~ was proposed [1]. In this paper, we extend this approach to the case of arbitrary impurity spin S. The canonical hamiltonian which describes a magnetic atom in a non-magnetic host metal, the so-called " s - d exchange model", has the form: -
=
e k C ; , u G , . +J k, ~=±1
k,k',t~,l~'
G,.%.'
(1)
G',u 's ,
where ~ = (o 1, o 2 , a 3) are the usual Pauli matrices, J > 0 and S = ( S 1 , S 2 , $3) are the spin operators o f the impurity. The conduction bound states are to be decomposed according to their orbital angular momentum relative to the impurity, and it is supposed that only l = 0 (s-wave) components can interact with a point-like impurity. The s-wave electrons obey a one-dimensional wave equation on an arbitrary line intersecting the impurity position at x = 0. Furthermore, we suppose that J ' ~ 1, that the temperature and magnetic field are much smaller than the Fermi energy e F, and that we may consider only the linear expansion e k = e F + VF(Ik[ - kF) for the electron spectrum. Under these assumptions, the hamiltonian (1) will be diagonalized exactly. The method employed goes back to Bethe. Its application to the solution of the Kondo problem was exposed in ref. [ 1 ]. For this reason, only the basic points will be outlined below. On the basis of this solution, the magnetic susceptibility as a function of arbitrary magnetic field will be calculated in explicit form. 1. The wave function of a system consisting o f N one-dimensional particles, the spins and the coordinates of which are (~tj} and ( x j } , and of a localized magnetic moment with spin S, satisfies the Schr6dinger equation: N --
j=l
N xIdgI ..... I~N,S(Xl
..... XN) + J G ~ t j # ) S s s ' ~ % ) x I / ~ l ..... #j. ..... #N, S'I(X1 . . . . . XN) = O. j=l
(2)
1 Permanent address: Landau Institute for Theoretical Physics, Moscow, USSR.
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
! 79
Volume 81 A, number 2,3
PHYSICS LETTERS
12 January 1981
The solution of this equation is given by the Bethe ansatz. Let Q = {q0, '", qN} be a certain permutation of the integers {0, 1 ..... N}. Then in the region XQ = {Xqo < Xql < ... < XqN } AT
where k 1.... , k N is a set of unequal numbers, Q' is the remainder of Q with qi = 0 deleted, P is a permutation of the integers {1 ..... N}, PQ' is a product of permutations, and we put x 0 = 0. The eigenvalues of the hamiltonian are N
E : ~ k]. j=l
(4)
2. Not all the coefficients A(Q) in (3) are independent. The SchrSdinger equation determines the discontinuities of A on the boundary of the region XQ. IfXQ and X~) only differ by permutation of the particle x~ and the impurity x 0 --- 0, then:
Aul ..... ,N,S(Q) = (R/o)sisU'SAttx ..... u} ..... ttN,s'(Q) ,
(5)
where R/0 = exp(iJ a" S). If two regions differ by permutation of the particles x i and x/, then the coefficients A are connected by the permutation operator:
uiui"
, ui uj
,
=SuiSu{
(6)
These rules enable one to express an arbitrary A(Q) in terms of, say, A(1 ..... N, 0) = Cui ..... uN,S " As was pointed out by Yang [2], for the Bethe hypothesis (3) to hold, the necessary and sufficient conditions are the unitarity and factorization relations, which in our case have a form:
RjoRoj = 1,
PijPji = 1,
RjoRioPi] = PijRioRjo .
(7)
3. In order to find the spectrum of the hamiltonian (1), we put the system in a sphere of radius }L around the impurity and impose periodic boundary conditions• This leads in the standard way to an eigenvalue problem: _ u i ..... .jv, s '
exp(ik/L )cbul ..... . N , S = ~/'Ul ..... pN, S
dp u i .....
,N,S, '
(8)
where T = P12P12 --- P1N R 10 "
(9)
To diagonalize the operator T we construct, following Baxter [3], a parametric family of self-commuting opera. tors T(u). Namely, the operator T belongs to:
T(u) =
~
L a l c~N+l ( U ) ,
a l = a N + l = +l
where the 2 X 2 operator-valued m a t r i x LC~laN+I(u) is represented in the form:
180
(10)
Volume 81A, number 2,3
L ( u ; Vl,
.... 0N+I ) _
PHYSICS LETTERS
C
=
D
~ ct2, or3..... aN =± 1
aja./+l. rujuj" tu - v / .) K~scst ,N ~ N + I .tu - V N + l ) .
12 January 1981
(11)
TM
Here r and R are operator matrices of the special form: Ot~'
1
~
bt
1
- % , , = ~ [a(u) + b ( u ) ] ~ , ~,, + ~ [a(u) -
b(u)]a,~,~,o~,
u, ,
n .s~s '' _ - ~1 [a(u) + b ( u / ] ~ , ~ s + ~1 [a(u) - b ( u ) l . . ¢ ( 2 S s s , ) ,
(12) (13)
and a(u) = (u - i : ) / ( u + i J ) ,
(14)
b(u) = u/(u + i S ) .
For o/= --~N+I,/and u = 0, the operators (11) and (9) coincide. Baxter [3] has shown that the self-commutativity [T(u), T(u')] = 0 is equivalent to the factorization relation and holds for all values of the spectral parameter u: (15)
ri/(u)Rio(U + u')R/o(U' ) = R/o(u')Rio(U + u')ri/(u ) .
This relation generalizes eqs. (7). The requirement that conditions (15) are valid determines the functional form of a(u) and b(u). Note that these functions do not depend on the spin S ,1 4. It is convenient to diagonalize T(u), following the quantum inverse scattering method developed in ref. [5]. Eq. (15) defines the communication properties of the elements L~I~N÷I(u), which also do not depend on the spin S. For example, the commutation relations which we will use to diagonalize T ( u ) = A ( u ) + D ( u ) have the form: A ( o ) B ( u ) - a(u - v) B ( u ) A ( o ) - a(u - o) - b(u - v) B ( u ) A ( u ) b(u v) a(u - o) ' D(o)B(u)
a(v - u) B ( u ) D ( v ) - a(u - u) - b(o - u) B(u)D(u) ~(o-u-) b ( v - u) '
_
(16)
[B(v), B(ut] = 0 .
Consider now the state ~0 with the maximal value of the total spin projection S z = ~ N + S: (17)
q~O = ~1 ..... 1,S " It is an eigenstate of the operators A ( u ) , D(u I and C(u) with the eigenvalues N A 0 = [a(u
-
ON+I)(S + 1/21 -- b(u
-
ON+II(S
-
1/2)] l-'I a(u - o/), /'=1 N
(18)
AO = [ - a ( u - VN+I)(S -- 1/2) + b(u - VN+I)(S + 1/2/1 ~ b(u - v / l , /=1
and A O = 0, respectively. Other eigenstates can be constructed by successive applications on ¢b0 of the "annihilation" operator B ( u a ) which decreases the total spin projection by one :
,1 The operators T(u) [eqs. (10)-(14)] belong to the more general family of self-commuting operators constructed in ref. [4]. 181
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PHYSICS LETTERS
12 January 1981
M
{I}M(U;U1..... UM) = I I B(us){I} 0 .
(19)
c~=1
Using (16) and (18) it is easy to see that the vector cI,M is an eigenvector of T(u) only provided that "wrong" terms coming from the second terms on the right-hand side of eqs. (16) vanish. This condition leads to the equations for the values of ua.:
HN a(Us _ vj) [ct(u~ - VN+I)(S + 1/2) -- b(u s - VN+l)(S- 1/2)] = f i a(ua - ug) b(u~ - us) f : l b(Us vj) [ - - - ~ ; 2£VN~I)(S--- l ~ + b - ( u £ - ~ ~ + 1~-)] B=I a(uB - u~) b(u s - u~) "
(20)
If these equations hold, the eigenvalue t(u) of the operator T(u)is:
u~) ' t(u)= AOA(u) ~=l a(ut3 b ~ - u -) u) + A°(u)[-I a(u b(u - u¢)
(21)
where a ° ( u ) and AO(u) are given by (18). Taking the log of (20) and (21) and putting u s = J ( - X a + :1) ~" and v/= --S],N+1 we have M
k/L=27@+~
O(Xs)-J'S,
j = l .... , N ,
(22)
{~=1
M NO 1/2(Xs) + OS(X{~ - 1/J) = 27rJa + ~ 01 (Xs - ~.~) ,
o~= 1..... M ,
(23)
where Oo(x ) = 2 arctgx/o, and the integers 21s and 2J s are the quantum numbers of the system. The energy and the total spin projection are: N E:~k/, j=l
1
SZ::N-M+S.
(24)
5. The further analysis of eqs. (22)-(24) in the limit of N, L -+ ~ (TrN/L = eV) is rather similar to ref. [1]. In the ground state with the fixed spin projection S Z the quantum numbers Js = - ~1N , - ~ - N + 1, ..., :1N + 2S - 2S Z. Then in the thermodynamic limit, in the leading order in l/N, eqs. (23) take the form: b p(X) -
4 1 +24X ~
2 p(X') dX' f_~ 1 + (X - X')2
(25)
The impurity contributions of the energy and the total spin projection are: b
AE=-- /
O S ( X - I/J) p(X)dX,
(26)
b 1
SZ] N - ~ -- / where p(X)
=
D(X) d~.,
(27)
limN~= 1/N(Xs - ks+l) is the X distribution density, and eq. (27) determines b as a function of
Sz/N. In leading order in 1/N, the total spin of the system is determined by the magnetism of the conduction band. 182
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Therefore S z / N = 4Hie F, where H is a magnetic field. Eq. (25) is of the Wiener-Hopf type and can be solved analytically. Then, using (26) and (27), we can obtain the explicit expression for the impurity part of the energy as a function of the ratio of the arbitrary magnetic field and the Kondo temperature T K = e F exp(-21r/J). (For details, see ref. [1].) The explicit formula for the magnetic moment of the impurity for arbitrary spin S seems similar to the S = 1/2 case. Nevertheless, the infra-red properties (H ~ TK) are quite different in these cases: M(H) = S + 4n3/2i _?~ P(1/2 + ico)f2S(co) f2s-_ 1 (co) exp [-2ico In(HIT K)] codco-i0 '
(28)
where f± (co) = (e ±iTr/2co/e)± ito are analytic in the upper and lower hal(planes, respectively, and factorize the function exp(-Trl col) = f + ( c o ) f (co)
(29)
and T K = e F exp(-27r/J). In the lower (upper) half-plane re(cO) has a cut which we draw along the lower (upper) imaginary half-axis. The discontinuities on these axes are disc f±n(co) = +2i e-nlto I log lw/el sin 7rn [co [ .
(30)
6. I f H ~ T K the integration contour could be shifted to encircle the cut in the upper half-plane. Then 1 ? sin 27rcoS P(I/2 +co e-2toln(H/TK ) 09 e M(H) = S - 27r3/----~ 0 ~ ) ( / ) to dco
(31)
and the well known perturbation theory expansion
M(H) = S {1 - 1/ln(H/TK) 2 - In [ln(H/TK) 2 ]/ln;(H/TK)2 + ...}
(32)
is obtained. I f H < T K, i.e., in the region inaccessible for perturbation theory, the integration contour could be shifted to encircle the cut in the lower half-plane. As a result: 1
M(/-/) = S - 1/2 + 2zr3/2 0
F(1/2 - w) e -2w Iln(H/TK)l(w/e)to dw
oo
+~
~
(n~1/-------~2)(n+l/2)
n=0 @
(--1)n cos[21r(S-1/2)(n+ 1/2)][H/TK] 2n+l (n + 1/2)n!
(33)
For H ' ~ T K and S :~ 1/2, the contribution of the poles of the I~ function in (33) is exponentially small in comparison with the integral around the cut. Then M(H) = (S - 1/2)[1 + 1/ln(TK/H)2 - In ln(TK/H)2/ln2(TK/H) 2 ] .
(34)
However, i f S = 1/2, the only singularities of the integrand in the lower half-plane are the poles, leading to a power series in (H/TK) [1]:
M(H)=~vl-~(n+I/----~2) n+1/2 n--0 ~
(-1)n
n!(n + 1/2)
(H/TK) 2n+1
(35) "
183
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7. Eqs. ( 2 8 ) - ( 3 5 ) imply that the ground state of a magnetic impurity (for H = 0) is (2S - 1)-fold degenerate. Screening occurs only for S = 1/2 when the ground state is a singlet. This result confirms what was conjectured, on qualitative grounds, by Mattis [6]. The "duality" of the leading log approximations of H >> T K and H ~ T K is, apparently, a general property. Elsewhere [7] we will show that for T ~ 0, the coefficients of the leading and next-to-leading logs for T >> T K and T ~ T K coincide up to the substitution S ~ (S - 1/2). For example, for the magnetic susceptibility we will have
×(T) = ~S(S + 1)[1 - 1/ln(T/TK) + lnlln(T/TK)[/ln2(T/TK) +...] ,
T>> T K , (36)
= ~(S 2 - 1/4)[1 - 1/ln(T/TK) + lnIln(T/TK)I/ln2(T/TK) + ...] ,
T~ TK
8. The hamiltonian (1) has been studied quite extensively (see, for instance, reviews in ref. [8]). Still, it can hardly correspond to any real alloy. The exchange amplitude J(k, k') is generally non-spherical; rather it contains 2S first-spherical harmonics. A more realistic 2S-fold degenerate model, in which screening occurs can also be solved exactly. This solution will be published elsewhere [9]. This paper was completed when one of the authors (V.A.F.) was at the CERN Theory Division. He is grateful to the Theory Division staff for the hospitality extended to him. Thanks are also due to Professor N.N. Nikolaev, whose generous help advanced the publication of this paper by at least six months.
References [ 1] P.B. Wiegmann, Pis'ma Zh. Eksp. Teor. Fiz. 31 (1980) 392; J. Phys. C (1980), to be published; Landau Institute preprint 18 (1980). [2] C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. [3] R.J. Baxter, Ann. Phys. 70 (1972) 193. [4] V.A. Fateev and A.B. Zamolodchikov, Zh. Eksp. Teor. Fiz. (1980), to be published. [5] L.D. Faddeev, E.K. Sklyanin and L.A. Takhtadjan, Teor. Mat. Fiz. 40 (1979) 199; preprint LOMI P-2-79 (1979). [6] D. Mattis, Phys. Rev. Lett. 19 (1967) 1478. [7] P.B. Wiegmann (1980), to be published. [8] J. Rado and H. Shull, eds., Magnetism, Vol. 5 (Academic Press, 1973). [9] A.M. Tsvelik and P.B. Wiegmann (1980), to be published.
184
PHYSICS LETTERS
12 January 1981
THE EXACT SOLUTION OF THE s - d EXCHANGE MODEL WITH ARBITRARY IMPURITY SPIN S (KONDO PROBLEM) V.A. FATEEV 1
CERN, Geneva, Switzerland and P.B. WIEGMANN
Landau Institute for Theoretical Physics, Moscow, USSR Received 31 October 1980
An exact solution o f the s - d exchange model with arbitrary impurity spin S (Kondo problem) is proposed. On the basis of this solution, the magnetic susceptibility as a function of arbitrary magnetic field is calculated in explicit form.
1
Recently, an exact solution of the s - d exchange model with localized spin S = ~ was proposed [1]. In this paper, we extend this approach to the case of arbitrary impurity spin S. The canonical hamiltonian which describes a magnetic atom in a non-magnetic host metal, the so-called " s - d exchange model", has the form: -
=
e k C ; , u G , . +J k, ~=±1
k,k',t~,l~'
G,.%.'
(1)
G',u 's ,
where ~ = (o 1, o 2 , a 3) are the usual Pauli matrices, J > 0 and S = ( S 1 , S 2 , $3) are the spin operators o f the impurity. The conduction bound states are to be decomposed according to their orbital angular momentum relative to the impurity, and it is supposed that only l = 0 (s-wave) components can interact with a point-like impurity. The s-wave electrons obey a one-dimensional wave equation on an arbitrary line intersecting the impurity position at x = 0. Furthermore, we suppose that J ' ~ 1, that the temperature and magnetic field are much smaller than the Fermi energy e F, and that we may consider only the linear expansion e k = e F + VF(Ik[ - kF) for the electron spectrum. Under these assumptions, the hamiltonian (1) will be diagonalized exactly. The method employed goes back to Bethe. Its application to the solution of the Kondo problem was exposed in ref. [ 1 ]. For this reason, only the basic points will be outlined below. On the basis of this solution, the magnetic susceptibility as a function of arbitrary magnetic field will be calculated in explicit form. 1. The wave function of a system consisting o f N one-dimensional particles, the spins and the coordinates of which are (~tj} and ( x j } , and of a localized magnetic moment with spin S, satisfies the Schr6dinger equation: N --
j=l
N xIdgI ..... I~N,S(Xl
..... XN) + J G ~ t j # ) S s s ' ~ % ) x I / ~ l ..... #j. ..... #N, S'I(X1 . . . . . XN) = O. j=l
(2)
1 Permanent address: Landau Institute for Theoretical Physics, Moscow, USSR.
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
! 79
Volume 81 A, number 2,3
PHYSICS LETTERS
12 January 1981
The solution of this equation is given by the Bethe ansatz. Let Q = {q0, '", qN} be a certain permutation of the integers {0, 1 ..... N}. Then in the region XQ = {Xqo < Xql < ... < XqN } AT
where k 1.... , k N is a set of unequal numbers, Q' is the remainder of Q with qi = 0 deleted, P is a permutation of the integers {1 ..... N}, PQ' is a product of permutations, and we put x 0 = 0. The eigenvalues of the hamiltonian are N
E : ~ k]. j=l
(4)
2. Not all the coefficients A(Q) in (3) are independent. The SchrSdinger equation determines the discontinuities of A on the boundary of the region XQ. IfXQ and X~) only differ by permutation of the particle x~ and the impurity x 0 --- 0, then:
Aul ..... ,N,S(Q) = (R/o)sisU'SAttx ..... u} ..... ttN,s'(Q) ,
(5)
where R/0 = exp(iJ a" S). If two regions differ by permutation of the particles x i and x/, then the coefficients A are connected by the permutation operator:
uiui"
, ui uj
,
=SuiSu{
(6)
These rules enable one to express an arbitrary A(Q) in terms of, say, A(1 ..... N, 0) = Cui ..... uN,S " As was pointed out by Yang [2], for the Bethe hypothesis (3) to hold, the necessary and sufficient conditions are the unitarity and factorization relations, which in our case have a form:
RjoRoj = 1,
PijPji = 1,
RjoRioPi] = PijRioRjo .
(7)
3. In order to find the spectrum of the hamiltonian (1), we put the system in a sphere of radius }L around the impurity and impose periodic boundary conditions• This leads in the standard way to an eigenvalue problem: _ u i ..... .jv, s '
exp(ik/L )cbul ..... . N , S = ~/'Ul ..... pN, S
dp u i .....
,N,S, '
(8)
where T = P12P12 --- P1N R 10 "
(9)
To diagonalize the operator T we construct, following Baxter [3], a parametric family of self-commuting opera. tors T(u). Namely, the operator T belongs to:
T(u) =
~
L a l c~N+l ( U ) ,
a l = a N + l = +l
where the 2 X 2 operator-valued m a t r i x LC~laN+I(u) is represented in the form:
180
(10)
Volume 81A, number 2,3
L ( u ; Vl,
.... 0N+I ) _
PHYSICS LETTERS
C
=
D
~ ct2, or3..... aN =± 1
aja./+l. rujuj" tu - v / .) K~scst ,N ~ N + I .tu - V N + l ) .
12 January 1981
(11)
TM
Here r and R are operator matrices of the special form: Ot~'
1
~
bt
1
- % , , = ~ [a(u) + b ( u ) ] ~ , ~,, + ~ [a(u) -
b(u)]a,~,~,o~,
u, ,
n .s~s '' _ - ~1 [a(u) + b ( u / ] ~ , ~ s + ~1 [a(u) - b ( u ) l . . ¢ ( 2 S s s , ) ,
(12) (13)
and a(u) = (u - i : ) / ( u + i J ) ,
(14)
b(u) = u/(u + i S ) .
For o/= --~N+I,/and u = 0, the operators (11) and (9) coincide. Baxter [3] has shown that the self-commutativity [T(u), T(u')] = 0 is equivalent to the factorization relation and holds for all values of the spectral parameter u: (15)
ri/(u)Rio(U + u')R/o(U' ) = R/o(u')Rio(U + u')ri/(u ) .
This relation generalizes eqs. (7). The requirement that conditions (15) are valid determines the functional form of a(u) and b(u). Note that these functions do not depend on the spin S ,1 4. It is convenient to diagonalize T(u), following the quantum inverse scattering method developed in ref. [5]. Eq. (15) defines the communication properties of the elements L~I~N÷I(u), which also do not depend on the spin S. For example, the commutation relations which we will use to diagonalize T ( u ) = A ( u ) + D ( u ) have the form: A ( o ) B ( u ) - a(u - v) B ( u ) A ( o ) - a(u - o) - b(u - v) B ( u ) A ( u ) b(u v) a(u - o) ' D(o)B(u)
a(v - u) B ( u ) D ( v ) - a(u - u) - b(o - u) B(u)D(u) ~(o-u-) b ( v - u) '
_
(16)
[B(v), B(ut] = 0 .
Consider now the state ~0 with the maximal value of the total spin projection S z = ~ N + S: (17)
q~O = ~1 ..... 1,S " It is an eigenstate of the operators A ( u ) , D(u I and C(u) with the eigenvalues N A 0 = [a(u
-
ON+I)(S + 1/21 -- b(u
-
ON+II(S
-
1/2)] l-'I a(u - o/), /'=1 N
(18)
AO = [ - a ( u - VN+I)(S -- 1/2) + b(u - VN+I)(S + 1/2/1 ~ b(u - v / l , /=1
and A O = 0, respectively. Other eigenstates can be constructed by successive applications on ¢b0 of the "annihilation" operator B ( u a ) which decreases the total spin projection by one :
,1 The operators T(u) [eqs. (10)-(14)] belong to the more general family of self-commuting operators constructed in ref. [4]. 181
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12 January 1981
M
{I}M(U;U1..... UM) = I I B(us){I} 0 .
(19)
c~=1
Using (16) and (18) it is easy to see that the vector cI,M is an eigenvector of T(u) only provided that "wrong" terms coming from the second terms on the right-hand side of eqs. (16) vanish. This condition leads to the equations for the values of ua.:
HN a(Us _ vj) [ct(u~ - VN+I)(S + 1/2) -- b(u s - VN+l)(S- 1/2)] = f i a(ua - ug) b(u~ - us) f : l b(Us vj) [ - - - ~ ; 2£VN~I)(S--- l ~ + b - ( u £ - ~ ~ + 1~-)] B=I a(uB - u~) b(u s - u~) "
(20)
If these equations hold, the eigenvalue t(u) of the operator T(u)is:
u~) ' t(u)= AOA(u) ~=l a(ut3 b ~ - u -) u) + A°(u)[-I a(u b(u - u¢)
(21)
where a ° ( u ) and AO(u) are given by (18). Taking the log of (20) and (21) and putting u s = J ( - X a + :1) ~" and v/= --S],N+1 we have M
k/L=27@+~
O(Xs)-J'S,
j = l .... , N ,
(22)
{~=1
M NO 1/2(Xs) + OS(X{~ - 1/J) = 27rJa + ~ 01 (Xs - ~.~) ,
o~= 1..... M ,
(23)
where Oo(x ) = 2 arctgx/o, and the integers 21s and 2J s are the quantum numbers of the system. The energy and the total spin projection are: N E:~k/, j=l
1
SZ::N-M+S.
(24)
5. The further analysis of eqs. (22)-(24) in the limit of N, L -+ ~ (TrN/L = eV) is rather similar to ref. [1]. In the ground state with the fixed spin projection S Z the quantum numbers Js = - ~1N , - ~ - N + 1, ..., :1N + 2S - 2S Z. Then in the thermodynamic limit, in the leading order in l/N, eqs. (23) take the form: b p(X) -
4 1 +24X ~
2 p(X') dX' f_~ 1 + (X - X')2
(25)
The impurity contributions of the energy and the total spin projection are: b
AE=-- /
O S ( X - I/J) p(X)dX,
(26)
b 1
SZ] N - ~ -- / where p(X)
=
D(X) d~.,
(27)
limN~= 1/N(Xs - ks+l) is the X distribution density, and eq. (27) determines b as a function of
Sz/N. In leading order in 1/N, the total spin of the system is determined by the magnetism of the conduction band. 182
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12 January 1981
Therefore S z / N = 4Hie F, where H is a magnetic field. Eq. (25) is of the Wiener-Hopf type and can be solved analytically. Then, using (26) and (27), we can obtain the explicit expression for the impurity part of the energy as a function of the ratio of the arbitrary magnetic field and the Kondo temperature T K = e F exp(-21r/J). (For details, see ref. [1].) The explicit formula for the magnetic moment of the impurity for arbitrary spin S seems similar to the S = 1/2 case. Nevertheless, the infra-red properties (H ~ TK) are quite different in these cases: M(H) = S + 4n3/2i _?~ P(1/2 + ico)f2S(co) f2s-_ 1 (co) exp [-2ico In(HIT K)] codco-i0 '
(28)
where f± (co) = (e ±iTr/2co/e)± ito are analytic in the upper and lower hal(planes, respectively, and factorize the function exp(-Trl col) = f + ( c o ) f (co)
(29)
and T K = e F exp(-27r/J). In the lower (upper) half-plane re(cO) has a cut which we draw along the lower (upper) imaginary half-axis. The discontinuities on these axes are disc f±n(co) = +2i e-nlto I log lw/el sin 7rn [co [ .
(30)
6. I f H ~ T K the integration contour could be shifted to encircle the cut in the upper half-plane. Then 1 ? sin 27rcoS P(I/2 +co e-2toln(H/TK ) 09 e M(H) = S - 27r3/----~ 0 ~ ) ( / ) to dco
(31)
and the well known perturbation theory expansion
M(H) = S {1 - 1/ln(H/TK) 2 - In [ln(H/TK) 2 ]/ln;(H/TK)2 + ...}
(32)
is obtained. I f H < T K, i.e., in the region inaccessible for perturbation theory, the integration contour could be shifted to encircle the cut in the lower half-plane. As a result: 1
M(/-/) = S - 1/2 + 2zr3/2 0
F(1/2 - w) e -2w Iln(H/TK)l(w/e)to dw
oo
+~
~
(n~1/-------~2)(n+l/2)
n=0 @
(--1)n cos[21r(S-1/2)(n+ 1/2)][H/TK] 2n+l (n + 1/2)n!
(33)
For H ' ~ T K and S :~ 1/2, the contribution of the poles of the I~ function in (33) is exponentially small in comparison with the integral around the cut. Then M(H) = (S - 1/2)[1 + 1/ln(TK/H)2 - In ln(TK/H)2/ln2(TK/H) 2 ] .
(34)
However, i f S = 1/2, the only singularities of the integrand in the lower half-plane are the poles, leading to a power series in (H/TK) [1]:
M(H)=~vl-~(n+I/----~2) n+1/2 n--0 ~
(-1)n
n!(n + 1/2)
(H/TK) 2n+1
(35) "
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Volume 81A, number 2,3
PHYSICS LETTERS
12 January 1981
7. Eqs. ( 2 8 ) - ( 3 5 ) imply that the ground state of a magnetic impurity (for H = 0) is (2S - 1)-fold degenerate. Screening occurs only for S = 1/2 when the ground state is a singlet. This result confirms what was conjectured, on qualitative grounds, by Mattis [6]. The "duality" of the leading log approximations of H >> T K and H ~ T K is, apparently, a general property. Elsewhere [7] we will show that for T ~ 0, the coefficients of the leading and next-to-leading logs for T >> T K and T ~ T K coincide up to the substitution S ~ (S - 1/2). For example, for the magnetic susceptibility we will have
×(T) = ~S(S + 1)[1 - 1/ln(T/TK) + lnlln(T/TK)[/ln2(T/TK) +...] ,
T>> T K , (36)
= ~(S 2 - 1/4)[1 - 1/ln(T/TK) + lnIln(T/TK)I/ln2(T/TK) + ...] ,
T~ TK
8. The hamiltonian (1) has been studied quite extensively (see, for instance, reviews in ref. [8]). Still, it can hardly correspond to any real alloy. The exchange amplitude J(k, k') is generally non-spherical; rather it contains 2S first-spherical harmonics. A more realistic 2S-fold degenerate model, in which screening occurs can also be solved exactly. This solution will be published elsewhere [9]. This paper was completed when one of the authors (V.A.F.) was at the CERN Theory Division. He is grateful to the Theory Division staff for the hospitality extended to him. Thanks are also due to Professor N.N. Nikolaev, whose generous help advanced the publication of this paper by at least six months.
References [ 1] P.B. Wiegmann, Pis'ma Zh. Eksp. Teor. Fiz. 31 (1980) 392; J. Phys. C (1980), to be published; Landau Institute preprint 18 (1980). [2] C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. [3] R.J. Baxter, Ann. Phys. 70 (1972) 193. [4] V.A. Fateev and A.B. Zamolodchikov, Zh. Eksp. Teor. Fiz. (1980), to be published. [5] L.D. Faddeev, E.K. Sklyanin and L.A. Takhtadjan, Teor. Mat. Fiz. 40 (1979) 199; preprint LOMI P-2-79 (1979). [6] D. Mattis, Phys. Rev. Lett. 19 (1967) 1478. [7] P.B. Wiegmann (1980), to be published. [8] J. Rado and H. Shull, eds., Magnetism, Vol. 5 (Academic Press, 1973). [9] A.M. Tsvelik and P.B. Wiegmann (1980), to be published.
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