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Magnetization curve for a one-dimensional interacting fermi system with magnetic impurities
Volume 89A, number 9
PHYSICS LETTERS
14 June 1982
MAGNETIZATION CURVE FOR A ONE-DIMENSIONAL INTERACTING FERMI SYSTEM WITH MAGNETIC IMPURITIES E.H. REZAYI Center for Pure and AppliedPhysical Sciences, University of California, San Diego, La Jolla, CA 92093, USA
and J. SAK Serin Physics Laboratory, Rutgers University, New Brunswick, NJ 08903, USA
Received 11 March 1982
A model of a one-dimensional metal with an attractive electron—electron and an antiferromagnetic electron—impurity interaction is studied. The gap in the triplet excitation spectrum and the shape of the magnetization curve near the onset of the magnetic moment are calculated by the Bethe Ansatz.
We consider a one-dimensional fermion system with the hamiltonian H= —i f’,t4c~~
+
uf
~
~ +
~
if
%~‘2~ ~
‘P20
clv
4’y’ya ‘Pin
+~f
~7~3’POI3•
[‘1
+ ‘P2,y6ya 1P
ip~dx.
2&] ~
(1)
Here the dispersion law for the right-going electrons (field ‘P~~) and the left-going electrons (field i~213)are linearized. The second subscript on the field denotes the z-component of spin. The field ~ corresponds to a static magnetic impurity with spin equal to one half. The interaction J> 0 (assumed weak, which is the interesting limit) is an antiferromagnetic Kondo coupling between electrons and impurities and the spin-exchange U> 0 (also weak) between electrons simulates an attractive potential backscattenng. In the previous paper [1] we studied in general the integrabiity of this system by the Bethe Ansatz [2—7].We came to the conclusion that the system (1) was integrable by the Bethe Ansatz only if the following relation between the coupling constants is satisfied: 2)2U/(1 ~iigU2)_c. (2) 2J/(1 —~J
In this case the problem can be reduced to an integral equation. In this paper we explore some physical properties of the model. In particular we shall calculate the gap in the triplet excitation spectrum and the shape of the magnetization curve for small values of the magnetization. In ref. [1] we derived (eq. 24) the following integral equation for the density of “spin momenta” p(X): 2
(
~
(X) Ne 2c + 2c + 2N~ N\c2+4(x+1)2 c2÷4(X_l)21 N c2+4X2 —
~
—
r
2cp(X’) dX’ IX~I>Bc2+O~~)2
(3)
B is a parameter to be determined from the expression for the spin (eq. (23) of ref. [11)
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
451
Volume 89A, number 9
PHYSICS LETTERS
14 June 1982
S~fp(X)dX.
(4)
The total number of electrons is Ne (Ne /2
=
number of right-going electrons = number of left-going electrons). N0
is the number of impurities, N = Ne + N0. The energy (apart from spin independent term) is given by (eq. (22) of ref. [1]) NNe E = —j~—
f
dA p(X) (arctg
2X+2 ~
—
arctg
2X—2 c
XI>B
)
NeMIT —
—
J.LHS,
(5)
where L is the length of the system (introduced through periodic boundary conditions), M = N/2 s, ~uis the magnetic moment of the fermions (assumed to be the same for the electrons and the impurities). The last term in (5), containing the magnetic field H is the Zeeman energy. Mathematically the problem is the following: solving (3) and (4) for p(X) andB, and substituting into (5) we obtain E as a function of H and 5; then we minimize E with respect to S; this gives S as a function of H. Our calculation is similar to that of Takahashi [8] for the halffified Hubbard model. Japaridze and Nersesyan [9] studied the magnetic properties of a model without spin impurities. We start by rewriting the integral equation (3) in a form which can be easily iterated: —
p(X) = p0(X) _!
f R(X
-
X’)/c)p(X’) dX’,
R(x) =
-
~
(6)
~
where p0(X) is the solution of (3) for B = 0 (S = 0). In this case the integral in (3) is over the whole real axis and the equation can be solved easily by Fourier transformation. Introducing -
~(p)
f
e~p0(X)dX
we find ~‘o(P) = (NeI2N) [cos(p)/cosh(cp/2)]
+ (N0 /2N) [1/cosh(cp/2)] We also rewrite (5) in a form suitable for our purposes. Substituting (6) into (5) we have 2N
N~[E(S)
—~
—
E(0)]
=
— —~-~
f dA p(X) f
(7)
.
f dA p(X) (arctg
2X+2
~
—
arctg
2X—2 c
)
dA’ [R((X’ A)/c) + R((X’ + X)/c)] (arctg 2X’+ 2 —
—
arctg 2X’— 2) + (Nelr/L
—
~H)S/N. (8)
0
The integral over X’ can be evaluated and the result is N~[E(S)
-
E(0)]
=
-
f~
dX + (Neir/L
-
~zH)S/N.
(9)
To find the threshold for spin excitations we expand the right-hand side of (9) to first order in S using the connection (4) between S and B which is, to first order in B, S = NBp0(0) + O(B 3). For the spin dependence of energy
weget 452
Volume 89A, number 9 N~[E(S)
E(0)]
=
PHYSICS LETTERS [(2NeIL)
arctg sinh(ir/c) +Ne7T/L ~zH]S/N
14 June 1982 (10)
+ O(S~).
Since S > 0, the minimum of (10) is at S = 0 unless the magnetic field excedes the critical value H~= (2NeIPL) arctg[sinh(ir/c)] —1
(11)
The gap in the triplet excitation spectrum is pH~.The remarkable feature of expression (11) is that it does not depend on the impurity concentration. The gap is caused only by attractive electron—electron interaction (simulated here by an antiferromagnetic exchange). It is known that in the Kondo problem [6] there is no gap even with an arbitrary concentration of impurities (and only one kind of electrons), so that it is perhaps not surprising that Kondo exchange in our model does not modify the gap of the electronic subsystem. In the weak coupling limit (11) becomes Hc
(4NeII.LL)e’~.
(12)
It is noteworthy that the expected prefactor ~ is missing from (12). This is probably due to the cutoff procedure which has been employed here and which is necessary to make the problem soluble by the Bethe Ansatz. It could be called “cutoff after the solution” in contrast to the usual procedure of building the cutoff into the hamitonian. The square root of the coupling constant is also missing from the mass of the color spin ~ particles in the Gross— Neveu model [5] and from the Kondo temperature [6] when this peculiar procedure is used. On the other hand, if cutoff had been introduced into the original hamiltonian, the square root, in all probability, would be present in (12). Such cutoff could be implemented either by writing the hamiltonian (1) in k-space and restricting the range of the allowed values of the momenta [10] or by putting the system on the lattice. The Hubbard model is one of the few systems on a lattice which can be solved by the Bethe Ansatz [11]. Indeed, the expression for the gap contains the square root of the coupling constant. To calculate the shape of the magnetization curve for H larger than but close to Hc (H Hc ~ Hc) we need terms of order S3 in the energy: N1 [E(S) E(0)] = ii(H Hc)S/N + (~2/6c2 ) [sinh(ir/c)/cosh2(iiic)]p~2(0)(2Ne/L)(S/N)3 + O(S~). (13) —
—
—
The value of p 0(0) can be obtained by integrating (7), using formulas from the paper of Yang and Yang [12] p0(0) =
~—
f
p0(p) dp
=
[c cosh(ir/c)]~[Ne/N
+ (N0/N)
cosh(ir/c)].
(14)
Minimizing (13) and using (14) we get the shape of the magnetization curve near the onset 2[izW Hc)Iir2shlhØiic)] 1/2 S/N = [Ne/N + (N0/N) cosh(nic)}(L/N~)’/ The susceptibility near Hc is given by —
~2 [Ne +N
(15)
2/2[ir2p(H Hc)sinhØr/c)] 1/2 (16) 0 cosh(ir/c)](L/N~)’I This expression is infinite for H = H~and has qualitatively similar behavior to that found by Takahashi [8] in the Hubbard model. In the scaling limit c 0, NeIL oo, keeping Hc fixed we obtain the universal functions X
= ~UdS/dH =
.
—~
—
.
-+
S/N = (L/N + 2N 0 INIZHc)(1 /ir.f2)p.H~~/H/H~1 ,
(17)
—
and 2(L + 2N X ~
0 /IzHc)12\filr\/H/Hc
—
I
(18)
.
In the absence of impurities, N0 = 0, and setting p Nersesyan [9].
1 we obtain, as a special case the results of Japaridze and
453
Volume 89A, number 9
PHYSICS LETTERS
We thank N. Andrei for discussions and comments. References [1] E.H. Rezayi and J. Sak, preprint. [2] M. Gaudin, Phys. Lett. 24A (1967) 55. [3] C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. [4] A.A. Belavin, Phys. Lett. 87B (1979) 117. [5] N. Andrei and J.H. Lowenstein, Phys. Rev. Lett. 43 (1979) 1698. [6] N. Andrei, Phys. Rev. Lett. 45 (1980) 379. [7] P.B. Vigman, Pis’ma Zh. Eksp. Teor. Fiz. 31(1980) 392. [8] M. Takahashi, Prog. Theor. Phys. 42 (1969) 1098. [9] G.I. Japaridze and A.A. Nersesyan, Phys. Lett. 85A (1981) 23. [10] E.H. Rezayi, J. Sak and I. Solyom, Phys. Rev. B20 (1979) 1129. [11] E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20(1968)1445. [12] C.N. Yang and C.P. Yang, Phys. Rev. 150 (1966) 327.
454
14 June 1982
PHYSICS LETTERS
14 June 1982
MAGNETIZATION CURVE FOR A ONE-DIMENSIONAL INTERACTING FERMI SYSTEM WITH MAGNETIC IMPURITIES E.H. REZAYI Center for Pure and AppliedPhysical Sciences, University of California, San Diego, La Jolla, CA 92093, USA
and J. SAK Serin Physics Laboratory, Rutgers University, New Brunswick, NJ 08903, USA
Received 11 March 1982
A model of a one-dimensional metal with an attractive electron—electron and an antiferromagnetic electron—impurity interaction is studied. The gap in the triplet excitation spectrum and the shape of the magnetization curve near the onset of the magnetic moment are calculated by the Bethe Ansatz.
We consider a one-dimensional fermion system with the hamiltonian H= —i f’,t4c~~
+
uf
~
~ +
~
if
%~‘2~ ~
‘P20
clv
4’y’ya ‘Pin
+~f
~7~3’POI3•
[‘1
+ ‘P2,y6ya 1P
ip~dx.
2&] ~
(1)
Here the dispersion law for the right-going electrons (field ‘P~~) and the left-going electrons (field i~213)are linearized. The second subscript on the field denotes the z-component of spin. The field ~ corresponds to a static magnetic impurity with spin equal to one half. The interaction J> 0 (assumed weak, which is the interesting limit) is an antiferromagnetic Kondo coupling between electrons and impurities and the spin-exchange U> 0 (also weak) between electrons simulates an attractive potential backscattenng. In the previous paper [1] we studied in general the integrabiity of this system by the Bethe Ansatz [2—7].We came to the conclusion that the system (1) was integrable by the Bethe Ansatz only if the following relation between the coupling constants is satisfied: 2)2U/(1 ~iigU2)_c. (2) 2J/(1 —~J
In this case the problem can be reduced to an integral equation. In this paper we explore some physical properties of the model. In particular we shall calculate the gap in the triplet excitation spectrum and the shape of the magnetization curve for small values of the magnetization. In ref. [1] we derived (eq. 24) the following integral equation for the density of “spin momenta” p(X): 2
(
~
(X) Ne 2c + 2c + 2N~ N\c2+4(x+1)2 c2÷4(X_l)21 N c2+4X2 —
~
—
r
2cp(X’) dX’ IX~I>Bc2+O~~)2
(3)
B is a parameter to be determined from the expression for the spin (eq. (23) of ref. [11)
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
451
Volume 89A, number 9
PHYSICS LETTERS
14 June 1982
S~fp(X)dX.
(4)
The total number of electrons is Ne (Ne /2
=
number of right-going electrons = number of left-going electrons). N0
is the number of impurities, N = Ne + N0. The energy (apart from spin independent term) is given by (eq. (22) of ref. [1]) NNe E = —j~—
f
dA p(X) (arctg
2X+2 ~
—
arctg
2X—2 c
XI>B
)
NeMIT —
—
J.LHS,
(5)
where L is the length of the system (introduced through periodic boundary conditions), M = N/2 s, ~uis the magnetic moment of the fermions (assumed to be the same for the electrons and the impurities). The last term in (5), containing the magnetic field H is the Zeeman energy. Mathematically the problem is the following: solving (3) and (4) for p(X) andB, and substituting into (5) we obtain E as a function of H and 5; then we minimize E with respect to S; this gives S as a function of H. Our calculation is similar to that of Takahashi [8] for the halffified Hubbard model. Japaridze and Nersesyan [9] studied the magnetic properties of a model without spin impurities. We start by rewriting the integral equation (3) in a form which can be easily iterated: —
p(X) = p0(X) _!
f R(X
-
X’)/c)p(X’) dX’,
R(x) =
-
~
(6)
~
where p0(X) is the solution of (3) for B = 0 (S = 0). In this case the integral in (3) is over the whole real axis and the equation can be solved easily by Fourier transformation. Introducing -
~(p)
f
e~p0(X)dX
we find ~‘o(P) = (NeI2N) [cos(p)/cosh(cp/2)]
+ (N0 /2N) [1/cosh(cp/2)] We also rewrite (5) in a form suitable for our purposes. Substituting (6) into (5) we have 2N
N~[E(S)
—~
—
E(0)]
=
— —~-~
f dA p(X) f
(7)
.
f dA p(X) (arctg
2X+2
~
—
arctg
2X—2 c
)
dA’ [R((X’ A)/c) + R((X’ + X)/c)] (arctg 2X’+ 2 —
—
arctg 2X’— 2) + (Nelr/L
—
~H)S/N. (8)
0
The integral over X’ can be evaluated and the result is N~[E(S)
-
E(0)]
=
-
f~
dX + (Neir/L
-
~zH)S/N.
(9)
To find the threshold for spin excitations we expand the right-hand side of (9) to first order in S using the connection (4) between S and B which is, to first order in B, S = NBp0(0) + O(B 3). For the spin dependence of energy
weget 452
Volume 89A, number 9 N~[E(S)
E(0)]
=
PHYSICS LETTERS [(2NeIL)
arctg sinh(ir/c) +Ne7T/L ~zH]S/N
14 June 1982 (10)
+ O(S~).
Since S > 0, the minimum of (10) is at S = 0 unless the magnetic field excedes the critical value H~= (2NeIPL) arctg[sinh(ir/c)] —1
(11)
The gap in the triplet excitation spectrum is pH~.The remarkable feature of expression (11) is that it does not depend on the impurity concentration. The gap is caused only by attractive electron—electron interaction (simulated here by an antiferromagnetic exchange). It is known that in the Kondo problem [6] there is no gap even with an arbitrary concentration of impurities (and only one kind of electrons), so that it is perhaps not surprising that Kondo exchange in our model does not modify the gap of the electronic subsystem. In the weak coupling limit (11) becomes Hc
(4NeII.LL)e’~.
(12)
It is noteworthy that the expected prefactor ~ is missing from (12). This is probably due to the cutoff procedure which has been employed here and which is necessary to make the problem soluble by the Bethe Ansatz. It could be called “cutoff after the solution” in contrast to the usual procedure of building the cutoff into the hamitonian. The square root of the coupling constant is also missing from the mass of the color spin ~ particles in the Gross— Neveu model [5] and from the Kondo temperature [6] when this peculiar procedure is used. On the other hand, if cutoff had been introduced into the original hamiltonian, the square root, in all probability, would be present in (12). Such cutoff could be implemented either by writing the hamiltonian (1) in k-space and restricting the range of the allowed values of the momenta [10] or by putting the system on the lattice. The Hubbard model is one of the few systems on a lattice which can be solved by the Bethe Ansatz [11]. Indeed, the expression for the gap contains the square root of the coupling constant. To calculate the shape of the magnetization curve for H larger than but close to Hc (H Hc ~ Hc) we need terms of order S3 in the energy: N1 [E(S) E(0)] = ii(H Hc)S/N + (~2/6c2 ) [sinh(ir/c)/cosh2(iiic)]p~2(0)(2Ne/L)(S/N)3 + O(S~). (13) —
—
—
The value of p 0(0) can be obtained by integrating (7), using formulas from the paper of Yang and Yang [12] p0(0) =
~—
f
p0(p) dp
=
[c cosh(ir/c)]~[Ne/N
+ (N0/N)
cosh(ir/c)].
(14)
Minimizing (13) and using (14) we get the shape of the magnetization curve near the onset 2[izW Hc)Iir2shlhØiic)] 1/2 S/N = [Ne/N + (N0/N) cosh(nic)}(L/N~)’/ The susceptibility near Hc is given by —
~2 [Ne +N
(15)
2/2[ir2p(H Hc)sinhØr/c)] 1/2 (16) 0 cosh(ir/c)](L/N~)’I This expression is infinite for H = H~and has qualitatively similar behavior to that found by Takahashi [8] in the Hubbard model. In the scaling limit c 0, NeIL oo, keeping Hc fixed we obtain the universal functions X
= ~UdS/dH =
.
—~
—
.
-+
S/N = (L/N + 2N 0 INIZHc)(1 /ir.f2)p.H~~/H/H~1 ,
(17)
—
and 2(L + 2N X ~
0 /IzHc)12\filr\/H/Hc
—
I
(18)
.
In the absence of impurities, N0 = 0, and setting p Nersesyan [9].
1 we obtain, as a special case the results of Japaridze and
453
Volume 89A, number 9
PHYSICS LETTERS
We thank N. Andrei for discussions and comments. References [1] E.H. Rezayi and J. Sak, preprint. [2] M. Gaudin, Phys. Lett. 24A (1967) 55. [3] C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. [4] A.A. Belavin, Phys. Lett. 87B (1979) 117. [5] N. Andrei and J.H. Lowenstein, Phys. Rev. Lett. 43 (1979) 1698. [6] N. Andrei, Phys. Rev. Lett. 45 (1980) 379. [7] P.B. Vigman, Pis’ma Zh. Eksp. Teor. Fiz. 31(1980) 392. [8] M. Takahashi, Prog. Theor. Phys. 42 (1969) 1098. [9] G.I. Japaridze and A.A. Nersesyan, Phys. Lett. 85A (1981) 23. [10] E.H. Rezayi, J. Sak and I. Solyom, Phys. Rev. B20 (1979) 1129. [11] E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20(1968)1445. [12] C.N. Yang and C.P. Yang, Phys. Rev. 150 (1966) 327.
454
14 June 1982