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Bethe-ansatz solutions of 1D correlated electron systems with long-range exchange
PHYSICA
Physica B 186-188 (1993) 828-830 North-Holland
Bethe-ansatz solutions of 1D correlated electron systems with long-range exchange Norio Kawakami 1 Max-Planck-lnstitut fiir festk6rperforschung, 7000 Stuttgart 80, Germany Asymptotic Bethe-ansatz solutions are obtained for the SU(N) spin model and the supersymmetric t-J model with 1/r2 exchange interaction in one dimension. We classify low-energy excitations and determine the critical exponents of correlation functions using conformal field theory.
1. Introduction
One-dimensional quantum systems with 1/r 2 interactions have been studied extensively in connection with basic notions in solid state physics such as the Jastrow wavefunction, the random matrix, etc. It is known that the ground-state wavefunction in these systems is given exactly by the Jastrow wavefunction, i.e. the product of two-body functions [1-6]. Many interesting aspects of these interacting quantum systems have been demonstrated for the continuum boson system [1], the SU(2) spin chain [2], the supersymmetric t - J model [3] and SU(N) generalized models ]4-6]. More recently the asymptotic bethe-ansatz (ABA) solution has been proposed to treat the bulk quantities and excitation spectra of these systems systematically [7]. In this paper we discuss the results obtained by the A B A for the critical behavior of low-energy excitations in SU(N) electron systems with i / r 2 interactions.
2. SU(N) electron systems with l / r 2 interactions
Let us consider N-component interacting particles on a lattice of circumference L, which can be described by the Hamiltonain [7]:
(1)
with 1/r2-type exchange interaction Jo=('II2/L 2) × sin 2(~x(x; - x j ) / L ) , where hT~(a,/3 = 1,2 . . . . . N) is the Hubbard operator which interchanges the states from/3 to a at the ith site. Each particle component is specified by a fermion number G(/3). For example, the two-component system with G(/3)= (0, 0) corresponds to the SU(2) spin chain [2], and the threecomponent system with G(/3) = (1, 0, 0) to the supersymmetric t - J model [3]. Similarly, N-component and ( N + 1)-component systems with G(/3) = (0, 0 . . . . . 0) and G(/3) = (1,0 . . . . . 0), respectively, coincide with the SU(N) spin chain and SU(N) supersymmetric t - J models [4-7]. Explicit expressions for the Hamiltonians can be found in refs. [2-7]. Henceforth we refer to the N-component internal degrees of freedom as the spin.
3. Asymptotic Bethe ansatz
A remarkable point in the above class of Hamiltonians is that the ground state is exactly given by the Gutzwiller state Iq'c3) = P~lFermi sea), where Pc projects out the configurations with more than one electron at each lattice site [2,3,5]. The corresponding Gutzwiller wave function for the SU(N) supersymmetric t-J model is written in a Jastrow form by taking the reference state full of Nth-spin particles as the vacuum state: 0 ~ ; = e x p -i~r 2 x l "~
Correspondence to: N. Kawakami, Max-Planck-lnstitut fiir Festk6rperforschung, 7000 Stuttgart 80, Germany. Present address: Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606, Japan.
l-I ~I f ~ , ( x ~ -
~,
where xl ~) (1 <~ a ~< N - 1 ) is the coordinate of particles with the spin-index a and x~N) is the coordinate of the hole [2-6]. Here the two-body function f,,, is defined as f~,(x) = [(L/'rr) sin(~rx/L)] p, with p : 1 +
0921-4526/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
829
N. Kawakami / 1 D electron systems with long-range exchange
6,t3 -- 6,,N613,U" The above Gutzwiller wavefunction in Jastrow form suggests that the two-body scattering is essential in 1/r 2 systems. This leads to the conjecture of the A B A [7]: the many-body S-matrix may be factorizable into two-body matrices in the above class of quantum systems. The two-body S-matrix with rapidities k~ and kj has a form s~aib = lim,~,[(k~ - kj)P,~ + it/]/(k~ - kj - i t / ) , which results in the step-wise phase shift function. Here the permutation operator P,o, which exchanges the coordinate of particles with indices a and b, characterizes the underlying symmetry specified by the fermion number G(/3). Based on the factorized S-matrix we can diagonalize the many-body scattering problem, and then obtain the A B A equations [7]. The bulk quantities and excitation spectra are calculated systematically using the A B A equations.
N
x = ¼mtFm + d'F 'd + ~
(n + + n ~ ) ,
(4)
which determines the long-distance (or low-energy) behavior of the correlation function as (~p(r)O(0))-~ r 2x Here each element of the N x N matrix F is given by F ~ = 2 - - 6 , , u , F~(,~+l)=--l, and F~t~=0 otherwise. Note that the diagonal elements of the matrix correspond to the scaling dimensions of spinon and holon excitations. A remarkable point is that this expression for the scaling dimension depends neither on the magnetic field and the electron concentration, in contrast to the ordinary SU(N) electron systems with short-range interactions for which the correlation exponents may vary continuously. The above scaling dimension therefore implies that the present system has a specific fixed point (non-interacting fixed point, see below [3,7]) on the critical line of the SU(N) Luttinger liquid (see ref. [5] for the definition).
4. Elementary excitations and conformal properties Let us now consider the critical behavior of the SU(N) supersymmetric t - J model with 1/r 2 interactions [7]. In this case we have N - 1 kinds of spin excitations (spinons) and the charge excitation (holon). The velocity of spinons in magnetic fields H is computed by the A B A equations as v, = ½[-n2n(n - 2) - 2 a ( ~ + 1)H]'/20(H(f ) - H ) , with H ~ ~ = Ir2n(n - 2) / [ 2 a ( a + 1)], and the velocity of the holon is given by v~ = r r ( 1 - n ) / 2 with the electron concentration n. The low-temperature specific heat is expressed in terms of these velocities as C / T = (~r/ 3)[1/v~ + ZN-~ ( 1 / % ) ] , which implies that holons and spinons are massless and described by c = 1 conformal field theories. The excitation spectrum is classified as AE=Z] ~(2~rv / L ) x in terms of the scaling dimension
2 + +n~, x ~ = ~ ,1 (Z 'm), ~2 + , (Ztd),,~+n~
(3)
where the dressed charge matrix is Z,~ = ~[/3(/3 + 1)] 1/2 for a ~
5. Correlation functions Critical exponents for correlation functions are now easily computed with the help of conformal field theory. For example, we discuss the long-distance behavior of the charge correlation function,
(p(r)p(O)) = ~
b r-'" cos(2akvr ).
(5)
a-I
The corresponding critical exponent r/,(=2x) is obtained by setting the quantum numbers as m = 0, dt~ = 6,t3, and n 2 = 0, rt~ = 2a
for a = 1, 2 . . . . .
N.
(6)
Similarly, the spin correlation exponents are easily computed; they are given by the same expression as (6) without r/N. It should be noted that the above formula for the critical exponent is characteristic of the non-interacting SU(N) electron systems. This noninteracting critical behavior is seen clearly in the momentum distribution function. In order to study this function, we first determine the long-distance behavior of the electron field correlator ( c * ( r ) c , ( O ) ) =cos(kFr)r -~, with ~: = 1. Fourier transformation of this correlator gives the momentum distribution function near the Fermi point as n k ~-- nkr qconst.lk - kF] ° sgn(k - kF), with 0 = ~ - 1 = 0. Therefore for the SU(N) supersymmetric t - J model there is a discontinuity in the momentum distribution at the Fermi point kF, as firstly pointed out for the SU(2) case [3]. This non-interacting behavior is stable against
830
N. Kawakami / 1 D electron systems with long-range exchange
the changes in the electron density and the magnetic field, but unstable against other perturbations such as the anisotropy of the exchange interaction. In the latter case the system shows SU(N) Luttinger liquid properties characterized by the universal scaling relations among correlation exponents [5]: for example, n~ = 2 a ( N - a ) / N + a2"qu/N2, 0 = ('qN - - 2 N ) 2/ (4N2~N), etc. Finally let us briefly mention the critical behavior of the SU(N) spin chain with 1 / r 2 interactions. In this case the holon becomes massive and remaining ( N 1) kinds of massless spinons determine the universal low-energy behavior completely. As a result the spin correlation exponent is modified from (6) to ~ = 2a(N-a)/N, We stress here again that this formula for the critical exponent does not depend on the magnetic field.
6. Summary We have discussed the critical behavior of SU(N) electron systems with 1 / r z interactions by combining the A B A solution with conformal field theory. The analysis for scaling dimensions clearly shows that the
present model is classified as the SU(N) Luttinger liquid with the special (non-interacting) fixed point. Although the A B A solution may certainly give the exact solution of the whole class of 1 / r 2 systems, it is desirable to conform this conjecture by microscopic treatment [9].
References [1] B. Sutherland, Phys. Rev. A 4 (1971) 2019; A 5 (1971) 1372. [2] F.D.M. Haldane, Phys., Rev. Lett. 60 (1988) 635; 66 (1991) 1529; B.S. Shastry, Phys. Rev. Lett. 60 (1988) 639. [3] Y. Kuramoto and H. Yokoyama, Phys. Rev. Lett. 67 (1991) 1338; Physica C 185-189 (1991) 1557; preprint. [4] Z.N.C. Ha and F.D.M. Haldane, preprint. [5] N. Kawakami, Phys. Rev. B 46 (1992) 3192. [6] H. Kiwata and Y. Akutsu, J. Phys. Soc. Jpn. 61 (1992) 1441. [7] N. Kawakami, Phys. Rev. B 45 (1992) 7525; 46 (1992) 1005. [8] N. Kawakami and S.-K. Yang, Phys. Rev. Lett. 67 (1991) 2493. [9] D.F. Wang, J.T. Liu and P. Coleman, Phys. Rev. B 46 (1992) 6639.
Physica B 186-188 (1993) 828-830 North-Holland
Bethe-ansatz solutions of 1D correlated electron systems with long-range exchange Norio Kawakami 1 Max-Planck-lnstitut fiir festk6rperforschung, 7000 Stuttgart 80, Germany Asymptotic Bethe-ansatz solutions are obtained for the SU(N) spin model and the supersymmetric t-J model with 1/r2 exchange interaction in one dimension. We classify low-energy excitations and determine the critical exponents of correlation functions using conformal field theory.
1. Introduction
One-dimensional quantum systems with 1/r 2 interactions have been studied extensively in connection with basic notions in solid state physics such as the Jastrow wavefunction, the random matrix, etc. It is known that the ground-state wavefunction in these systems is given exactly by the Jastrow wavefunction, i.e. the product of two-body functions [1-6]. Many interesting aspects of these interacting quantum systems have been demonstrated for the continuum boson system [1], the SU(2) spin chain [2], the supersymmetric t - J model [3] and SU(N) generalized models ]4-6]. More recently the asymptotic bethe-ansatz (ABA) solution has been proposed to treat the bulk quantities and excitation spectra of these systems systematically [7]. In this paper we discuss the results obtained by the A B A for the critical behavior of low-energy excitations in SU(N) electron systems with i / r 2 interactions.
2. SU(N) electron systems with l / r 2 interactions
Let us consider N-component interacting particles on a lattice of circumference L, which can be described by the Hamiltonain [7]:
(1)
with 1/r2-type exchange interaction Jo=('II2/L 2) × sin 2(~x(x; - x j ) / L ) , where hT~(a,/3 = 1,2 . . . . . N) is the Hubbard operator which interchanges the states from/3 to a at the ith site. Each particle component is specified by a fermion number G(/3). For example, the two-component system with G(/3)= (0, 0) corresponds to the SU(2) spin chain [2], and the threecomponent system with G(/3) = (1, 0, 0) to the supersymmetric t - J model [3]. Similarly, N-component and ( N + 1)-component systems with G(/3) = (0, 0 . . . . . 0) and G(/3) = (1,0 . . . . . 0), respectively, coincide with the SU(N) spin chain and SU(N) supersymmetric t - J models [4-7]. Explicit expressions for the Hamiltonians can be found in refs. [2-7]. Henceforth we refer to the N-component internal degrees of freedom as the spin.
3. Asymptotic Bethe ansatz
A remarkable point in the above class of Hamiltonians is that the ground state is exactly given by the Gutzwiller state Iq'c3) = P~lFermi sea), where Pc projects out the configurations with more than one electron at each lattice site [2,3,5]. The corresponding Gutzwiller wave function for the SU(N) supersymmetric t-J model is written in a Jastrow form by taking the reference state full of Nth-spin particles as the vacuum state: 0 ~ ; = e x p -i~r 2 x l "~
Correspondence to: N. Kawakami, Max-Planck-lnstitut fiir Festk6rperforschung, 7000 Stuttgart 80, Germany. Present address: Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606, Japan.
l-I ~I f ~ , ( x ~ -
~,
where xl ~) (1 <~ a ~< N - 1 ) is the coordinate of particles with the spin-index a and x~N) is the coordinate of the hole [2-6]. Here the two-body function f,,, is defined as f~,(x) = [(L/'rr) sin(~rx/L)] p, with p : 1 +
0921-4526/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
829
N. Kawakami / 1 D electron systems with long-range exchange
6,t3 -- 6,,N613,U" The above Gutzwiller wavefunction in Jastrow form suggests that the two-body scattering is essential in 1/r 2 systems. This leads to the conjecture of the A B A [7]: the many-body S-matrix may be factorizable into two-body matrices in the above class of quantum systems. The two-body S-matrix with rapidities k~ and kj has a form s~aib = lim,~,[(k~ - kj)P,~ + it/]/(k~ - kj - i t / ) , which results in the step-wise phase shift function. Here the permutation operator P,o, which exchanges the coordinate of particles with indices a and b, characterizes the underlying symmetry specified by the fermion number G(/3). Based on the factorized S-matrix we can diagonalize the many-body scattering problem, and then obtain the A B A equations [7]. The bulk quantities and excitation spectra are calculated systematically using the A B A equations.
N
x = ¼mtFm + d'F 'd + ~
(n + + n ~ ) ,
(4)
which determines the long-distance (or low-energy) behavior of the correlation function as (~p(r)O(0))-~ r 2x Here each element of the N x N matrix F is given by F ~ = 2 - - 6 , , u , F~(,~+l)=--l, and F~t~=0 otherwise. Note that the diagonal elements of the matrix correspond to the scaling dimensions of spinon and holon excitations. A remarkable point is that this expression for the scaling dimension depends neither on the magnetic field and the electron concentration, in contrast to the ordinary SU(N) electron systems with short-range interactions for which the correlation exponents may vary continuously. The above scaling dimension therefore implies that the present system has a specific fixed point (non-interacting fixed point, see below [3,7]) on the critical line of the SU(N) Luttinger liquid (see ref. [5] for the definition).
4. Elementary excitations and conformal properties Let us now consider the critical behavior of the SU(N) supersymmetric t - J model with 1/r 2 interactions [7]. In this case we have N - 1 kinds of spin excitations (spinons) and the charge excitation (holon). The velocity of spinons in magnetic fields H is computed by the A B A equations as v, = ½[-n2n(n - 2) - 2 a ( ~ + 1)H]'/20(H(f ) - H ) , with H ~ ~ = Ir2n(n - 2) / [ 2 a ( a + 1)], and the velocity of the holon is given by v~ = r r ( 1 - n ) / 2 with the electron concentration n. The low-temperature specific heat is expressed in terms of these velocities as C / T = (~r/ 3)[1/v~ + ZN-~ ( 1 / % ) ] , which implies that holons and spinons are massless and described by c = 1 conformal field theories. The excitation spectrum is classified as AE=Z] ~(2~rv / L ) x in terms of the scaling dimension
2 + +n~, x ~ = ~ ,1 (Z 'm), ~2 + , (Ztd),,~+n~
(3)
where the dressed charge matrix is Z,~ = ~[/3(/3 + 1)] 1/2 for a ~
5. Correlation functions Critical exponents for correlation functions are now easily computed with the help of conformal field theory. For example, we discuss the long-distance behavior of the charge correlation function,
(p(r)p(O)) = ~
b r-'" cos(2akvr ).
(5)
a-I
The corresponding critical exponent r/,(=2x) is obtained by setting the quantum numbers as m = 0, dt~ = 6,t3, and n 2 = 0, rt~ = 2a
for a = 1, 2 . . . . .
N.
(6)
Similarly, the spin correlation exponents are easily computed; they are given by the same expression as (6) without r/N. It should be noted that the above formula for the critical exponent is characteristic of the non-interacting SU(N) electron systems. This noninteracting critical behavior is seen clearly in the momentum distribution function. In order to study this function, we first determine the long-distance behavior of the electron field correlator ( c * ( r ) c , ( O ) ) =cos(kFr)r -~, with ~: = 1. Fourier transformation of this correlator gives the momentum distribution function near the Fermi point as n k ~-- nkr qconst.lk - kF] ° sgn(k - kF), with 0 = ~ - 1 = 0. Therefore for the SU(N) supersymmetric t - J model there is a discontinuity in the momentum distribution at the Fermi point kF, as firstly pointed out for the SU(2) case [3]. This non-interacting behavior is stable against
830
N. Kawakami / 1 D electron systems with long-range exchange
the changes in the electron density and the magnetic field, but unstable against other perturbations such as the anisotropy of the exchange interaction. In the latter case the system shows SU(N) Luttinger liquid properties characterized by the universal scaling relations among correlation exponents [5]: for example, n~ = 2 a ( N - a ) / N + a2"qu/N2, 0 = ('qN - - 2 N ) 2/ (4N2~N), etc. Finally let us briefly mention the critical behavior of the SU(N) spin chain with 1 / r 2 interactions. In this case the holon becomes massive and remaining ( N 1) kinds of massless spinons determine the universal low-energy behavior completely. As a result the spin correlation exponent is modified from (6) to ~ = 2a(N-a)/N, We stress here again that this formula for the critical exponent does not depend on the magnetic field.
6. Summary We have discussed the critical behavior of SU(N) electron systems with 1 / r z interactions by combining the A B A solution with conformal field theory. The analysis for scaling dimensions clearly shows that the
present model is classified as the SU(N) Luttinger liquid with the special (non-interacting) fixed point. Although the A B A solution may certainly give the exact solution of the whole class of 1 / r 2 systems, it is desirable to conform this conjecture by microscopic treatment [9].
References [1] B. Sutherland, Phys. Rev. A 4 (1971) 2019; A 5 (1971) 1372. [2] F.D.M. Haldane, Phys., Rev. Lett. 60 (1988) 635; 66 (1991) 1529; B.S. Shastry, Phys. Rev. Lett. 60 (1988) 639. [3] Y. Kuramoto and H. Yokoyama, Phys. Rev. Lett. 67 (1991) 1338; Physica C 185-189 (1991) 1557; preprint. [4] Z.N.C. Ha and F.D.M. Haldane, preprint. [5] N. Kawakami, Phys. Rev. B 46 (1992) 3192. [6] H. Kiwata and Y. Akutsu, J. Phys. Soc. Jpn. 61 (1992) 1441. [7] N. Kawakami, Phys. Rev. B 45 (1992) 7525; 46 (1992) 1005. [8] N. Kawakami and S.-K. Yang, Phys. Rev. Lett. 67 (1991) 2493. [9] D.F. Wang, J.T. Liu and P. Coleman, Phys. Rev. B 46 (1992) 6639.