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One-dimensional spin–orbital model perturbed by Hund coupling
Physica B 329–333 (2003) 946–947
One-dimensional spin–orbital model perturbed by Hund coupling Hyun C. Leea,b,*, P. Azariac, E. Boulatc a
Department of Physics, BK21 Physics Research Division and Institute of Basic Science, Sung Kyun Kwan University, Suwon 440-746, South Korea b Department of Physics, Sogang University, Seoul, South Korea c Laboratorie de Physique Th!eorique des Liquides, Universit!e Pierre et Marie Curie, 4 Place Jussieu, Paris 75252, France
Abstract The one-dimensional spin–orbital model perturbed by Hund coupling is studied by renormalization group and bosonization methods. The Hund coupling breaks the SU(4) spin–orbital symmetry into SUð2Þspin Uð1Þorbital at weak coupling fixed point. The one-loop renormalization group analysis shows that the Hund coupling is relevant irrespective of Coulomb repulsion. When Coulomb repulsion is larger than Hund coupling, the spin–orbital physics in strong coupling regime is described by SO(6) Gross–Neveu model, where the spin and orbital excitations are gapped. When the Hund coupling is much larger than the Coulomb repulsion, the strong coupling regime is described by the two coupled SOð3Þspin SOð3Þorbital Gross–Neveu model, where again the spin and orbital excitations are gapped. r 2003 Elsevier Science B.V. All rights reserved. PACS: 71.10.Hf; 75.10.Jm; 78.67.Lt Keywords: Hund coupling; Spin–orbital model; Renormalization group
The interplay of spin and orbital degrees of freedom plays an important role in diverse correlated electron systems [1]. The studies of spin–orbital system usually start from a quarter-filled two-band Hubbard model [2–5], and in the limit of strong Coulomb repulsion, the model can be mapped to the following coupled spinchain model X H¼K ðx þ Si Siþ1 Þðy þ Ti Tiþ1 Þ; ð1Þ i
where Si and Ti are the SU(2) spin and orbital operators at site i: At low energy the nontrivial dynamics of the Hamiltonian (1) reside in spin–orbital sector. Hamiltonian (1) has an obvious SUð2Þspin SUð2Þorbital symmetry for generic values of x and y: For ðx; yÞ ¼ ð14; 14Þ; the SUð2Þspin SUð2Þorbital symmetry is enhanced to SU(4) symmetry. At the SU(4) symmetric point, Hamiltonian (1) becomes critical, and it can be described by SU(4) *Corresponding author. E-mail address: [email protected] (H.C. Lee).
level 1 ðk ¼ 1Þ Wess-Zumino–Witten (WZW) model [3–5]. Hamiltonian (1) for generic values of x and y can be most naturally studied as a perturbation with respect to the SU(4) symmetric Hamiltonian [3,5]. The notable features of the obtained phase diagram of Eq. (1) are the existence of extended critical region in the vicinity of the symmetric point ðx; yÞ ¼ ð14; 14Þ and the existence of the massive phase with an approximate SO(6) symmetry with a dimerization of spin and orbital singlets [3–5]. In this paper, we study the breaking of SU(4) spin– orbital symmetry by Hund coupling between two bands [2] at quarter filling. The Hund coupling is expected to be present in more realistic description of spin–orbital systems [2]. If the bandwidth t is larger than the Hubbard repulsion U and the Hund coupling J (the weak coupling case), the bosonization and the perturbative renormalization group (RG) method can be employed. By expressing Hamiltonian in terms of charge Uð1Þ and spin–orbital SU(4) currents, the Uð1Þ charge degrees of freedom are shown to decouple from the SU(4) spin–orbital degrees of freedom, and the SU(4)
0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)02616-9
symmetry of spin–orbital degrees of freedom are shown to be broken into SUð2Þspin Uð1Þorbital : The one-loop renormalization group equations (RGE) which is valid in the weak coupling case can be obtained using current algebra techniques [6–8]. The analysis of RGE indicates that the Hund coupling is relevant irrespective of Coulomb repulsion, and it drives system to strong coupling regimes. The nature of the strong coupling regimes crucially depends on the relative magnitude of the Coulomb repulsion and the Hund coupling. When the Coulomb repulsion is larger than the Hund coupling, the RG flows of coupling constants strongly indicate the restoration of SUð4ÞBSOð6Þ symmetry [3,8] at the strong coupling regime. By expressing the spin–orbital current operators in terms of spin, orbital Majorana fermions it can be shown that the strong coupling regime can be described by SO(6) Gross–Neveu (GN) model, where the spin and orbital excitations are gapped. The detailed investigations reveal that the SO(6) symmetric strong coupling regime belongs to the same universality class of massive phase found by Azaria et al. and Itoi et al. apart from some inessential factors [3,5]. From the exactly known excitation spectrum of SO(6) GN model, the orbital excitation is found to be coherent. When the Hund coupling is much larger than the Coulomb repulsion, the full restoration of SO(6) symmetry does not occur, but the orbital Uð1Þorbital symmetry is enhanced to SOð3Þorbital : This region is characterized by a hierarchy of coupling constants. The coupling constants in orbital sector turn out to be the strongest one, while those in spin sector are almost negligible. The strong coupling regime is described by the two coupled SOð3Þspin SOð3Þorbital GN model. Using the aforementioned hierarchy we can reduce the coupled GN model into a free massive triplet spin Majorana fermion theory plus SO(3) GN model in the orbital sector. The free massive spin Majorana fermion theory describes the spin 1 Haldane gap system. This can be checked explicitly by performing a strong coupling expansion. The important difference between SO(3) and SO(6) GN models is the absence of fundamental fermions for SO(3) GN model. The elementary excitations of SO(3) GN model is the so-called ‘‘kinks’’ which are the soliton excitations which cannot be built from a finite superposition of fundamental fermions. The absence of fundamental fermions in the orbital sector implies the incoherency of orbital excitations. Even if the spin and orbital excitations are gapped in both regimes, the properties of spin–spin and orbital– orbital correlation functions are very different from each other, which is essentially due to the differences in the excitation spectrum between SO(6) and SO(3) GN model. Our results are summarized as a phase diagram in Fig. 1.
J/t
H.C. Lee et al. / Physica B 329–333 (2003) 946–947
947
gapped S=1 AF spin chain
Gapped 1
SO(3)_spin free Majorana x SO(3)_orbital GN Gapped SO(6) GN
gapless SO(6) WZW
1
U/t
Fig. 1. The phase diagram based on the properties spin–orbital degrees of freedom only. The symmetry and the effective model at the strong coupling regime is indicated. Every boundary represents smooth crossover rather than critical quantum phase transitions.
The studies of gapless charge excitations, the strong coupling case, and the details of calculations will be published elsewhere [9]. Hyun C. Lee is grateful to Prof. K. Ueda for the suggestion of this problem. Hyun C. Lee was supported by the Korea Science and Engineering Foundation (KOSEF) through the grant No. 1999-2-11400-005-5, and by the Ministry of Education through Brain Korea 21 SNU-SKKU Program.
References [1] T. Tokura, N. Nagaosa, Science 288 (2000) 462. [2] Y. Yamashita, N. Shibata, K. Ueda, Phys. Rev. B 58 (1998) 9114. [3] P. Azaria, A.O. Gogolin, P. Lecheminant, A.A. Nersesyan, Phys. Rev. Lett. 83 (1999) 624. [4] P. Azaria, E. Boulat, P. Lecheminant, Phys. Rev. B 61 (2000) 12112. [5] C. Itoi, S. Qin, I. Affleck, Phys. Rev. B 61 (2000) 6747. [6] I. Affleck, Nucl. Phys. B 265 (1986) 409. [7] I. Affleck, in: E. Brezin, J. Zinn-Justin (Eds.), Fields, Strings, and Critical Phenomena, Les Houches 1988, NorthHolland, Amsterdam, 1990. [8] H. Lin, L. Balents, M.P.A. Fisher, Phys. Rev. B 58 (1998) 1794. [9] H.C. Lee, P. Azaria, E. Boulat, Phys. Rev. B, submitted for publication.
One-dimensional spin–orbital model perturbed by Hund coupling Hyun C. Leea,b,*, P. Azariac, E. Boulatc a
Department of Physics, BK21 Physics Research Division and Institute of Basic Science, Sung Kyun Kwan University, Suwon 440-746, South Korea b Department of Physics, Sogang University, Seoul, South Korea c Laboratorie de Physique Th!eorique des Liquides, Universit!e Pierre et Marie Curie, 4 Place Jussieu, Paris 75252, France
Abstract The one-dimensional spin–orbital model perturbed by Hund coupling is studied by renormalization group and bosonization methods. The Hund coupling breaks the SU(4) spin–orbital symmetry into SUð2Þspin Uð1Þorbital at weak coupling fixed point. The one-loop renormalization group analysis shows that the Hund coupling is relevant irrespective of Coulomb repulsion. When Coulomb repulsion is larger than Hund coupling, the spin–orbital physics in strong coupling regime is described by SO(6) Gross–Neveu model, where the spin and orbital excitations are gapped. When the Hund coupling is much larger than the Coulomb repulsion, the strong coupling regime is described by the two coupled SOð3Þspin SOð3Þorbital Gross–Neveu model, where again the spin and orbital excitations are gapped. r 2003 Elsevier Science B.V. All rights reserved. PACS: 71.10.Hf; 75.10.Jm; 78.67.Lt Keywords: Hund coupling; Spin–orbital model; Renormalization group
The interplay of spin and orbital degrees of freedom plays an important role in diverse correlated electron systems [1]. The studies of spin–orbital system usually start from a quarter-filled two-band Hubbard model [2–5], and in the limit of strong Coulomb repulsion, the model can be mapped to the following coupled spinchain model X H¼K ðx þ Si Siþ1 Þðy þ Ti Tiþ1 Þ; ð1Þ i
where Si and Ti are the SU(2) spin and orbital operators at site i: At low energy the nontrivial dynamics of the Hamiltonian (1) reside in spin–orbital sector. Hamiltonian (1) has an obvious SUð2Þspin SUð2Þorbital symmetry for generic values of x and y: For ðx; yÞ ¼ ð14; 14Þ; the SUð2Þspin SUð2Þorbital symmetry is enhanced to SU(4) symmetry. At the SU(4) symmetric point, Hamiltonian (1) becomes critical, and it can be described by SU(4) *Corresponding author. E-mail address: [email protected] (H.C. Lee).
level 1 ðk ¼ 1Þ Wess-Zumino–Witten (WZW) model [3–5]. Hamiltonian (1) for generic values of x and y can be most naturally studied as a perturbation with respect to the SU(4) symmetric Hamiltonian [3,5]. The notable features of the obtained phase diagram of Eq. (1) are the existence of extended critical region in the vicinity of the symmetric point ðx; yÞ ¼ ð14; 14Þ and the existence of the massive phase with an approximate SO(6) symmetry with a dimerization of spin and orbital singlets [3–5]. In this paper, we study the breaking of SU(4) spin– orbital symmetry by Hund coupling between two bands [2] at quarter filling. The Hund coupling is expected to be present in more realistic description of spin–orbital systems [2]. If the bandwidth t is larger than the Hubbard repulsion U and the Hund coupling J (the weak coupling case), the bosonization and the perturbative renormalization group (RG) method can be employed. By expressing Hamiltonian in terms of charge Uð1Þ and spin–orbital SU(4) currents, the Uð1Þ charge degrees of freedom are shown to decouple from the SU(4) spin–orbital degrees of freedom, and the SU(4)
0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)02616-9
symmetry of spin–orbital degrees of freedom are shown to be broken into SUð2Þspin Uð1Þorbital : The one-loop renormalization group equations (RGE) which is valid in the weak coupling case can be obtained using current algebra techniques [6–8]. The analysis of RGE indicates that the Hund coupling is relevant irrespective of Coulomb repulsion, and it drives system to strong coupling regimes. The nature of the strong coupling regimes crucially depends on the relative magnitude of the Coulomb repulsion and the Hund coupling. When the Coulomb repulsion is larger than the Hund coupling, the RG flows of coupling constants strongly indicate the restoration of SUð4ÞBSOð6Þ symmetry [3,8] at the strong coupling regime. By expressing the spin–orbital current operators in terms of spin, orbital Majorana fermions it can be shown that the strong coupling regime can be described by SO(6) Gross–Neveu (GN) model, where the spin and orbital excitations are gapped. The detailed investigations reveal that the SO(6) symmetric strong coupling regime belongs to the same universality class of massive phase found by Azaria et al. and Itoi et al. apart from some inessential factors [3,5]. From the exactly known excitation spectrum of SO(6) GN model, the orbital excitation is found to be coherent. When the Hund coupling is much larger than the Coulomb repulsion, the full restoration of SO(6) symmetry does not occur, but the orbital Uð1Þorbital symmetry is enhanced to SOð3Þorbital : This region is characterized by a hierarchy of coupling constants. The coupling constants in orbital sector turn out to be the strongest one, while those in spin sector are almost negligible. The strong coupling regime is described by the two coupled SOð3Þspin SOð3Þorbital GN model. Using the aforementioned hierarchy we can reduce the coupled GN model into a free massive triplet spin Majorana fermion theory plus SO(3) GN model in the orbital sector. The free massive spin Majorana fermion theory describes the spin 1 Haldane gap system. This can be checked explicitly by performing a strong coupling expansion. The important difference between SO(3) and SO(6) GN models is the absence of fundamental fermions for SO(3) GN model. The elementary excitations of SO(3) GN model is the so-called ‘‘kinks’’ which are the soliton excitations which cannot be built from a finite superposition of fundamental fermions. The absence of fundamental fermions in the orbital sector implies the incoherency of orbital excitations. Even if the spin and orbital excitations are gapped in both regimes, the properties of spin–spin and orbital– orbital correlation functions are very different from each other, which is essentially due to the differences in the excitation spectrum between SO(6) and SO(3) GN model. Our results are summarized as a phase diagram in Fig. 1.
J/t
H.C. Lee et al. / Physica B 329–333 (2003) 946–947
947
gapped S=1 AF spin chain
Gapped 1
SO(3)_spin free Majorana x SO(3)_orbital GN Gapped SO(6) GN
gapless SO(6) WZW
1
U/t
Fig. 1. The phase diagram based on the properties spin–orbital degrees of freedom only. The symmetry and the effective model at the strong coupling regime is indicated. Every boundary represents smooth crossover rather than critical quantum phase transitions.
The studies of gapless charge excitations, the strong coupling case, and the details of calculations will be published elsewhere [9]. Hyun C. Lee is grateful to Prof. K. Ueda for the suggestion of this problem. Hyun C. Lee was supported by the Korea Science and Engineering Foundation (KOSEF) through the grant No. 1999-2-11400-005-5, and by the Ministry of Education through Brain Korea 21 SNU-SKKU Program.
References [1] T. Tokura, N. Nagaosa, Science 288 (2000) 462. [2] Y. Yamashita, N. Shibata, K. Ueda, Phys. Rev. B 58 (1998) 9114. [3] P. Azaria, A.O. Gogolin, P. Lecheminant, A.A. Nersesyan, Phys. Rev. Lett. 83 (1999) 624. [4] P. Azaria, E. Boulat, P. Lecheminant, Phys. Rev. B 61 (2000) 12112. [5] C. Itoi, S. Qin, I. Affleck, Phys. Rev. B 61 (2000) 6747. [6] I. Affleck, Nucl. Phys. B 265 (1986) 409. [7] I. Affleck, in: E. Brezin, J. Zinn-Justin (Eds.), Fields, Strings, and Critical Phenomena, Les Houches 1988, NorthHolland, Amsterdam, 1990. [8] H. Lin, L. Balents, M.P.A. Fisher, Phys. Rev. B 58 (1998) 1794. [9] H.C. Lee, P. Azaria, E. Boulat, Phys. Rev. B, submitted for publication.