- Email: [email protected]
More upon the properties of correlated electrons
Physica B 259—261 (1999) 777—778
More upon the properties of correlated electrons J. Szeftel* Laboratoire Le& on Brillouin(CEA-CNRS), CE-Saclay, 91191 Gif-sur-Yvette Ce& dex, France
Abstract The set of eigenvectors of the most general interacting electron Hamiltonian has been shown recently to consist of two classes t and t . The t -like eigenvectors have off-diagonal long-range order while the t -like ones do not have. Further properties of the eigenvectors are discussed. 1999 Elsevier Science B.V. All rights reserved. PACS: 3.65; 71.45; 74.20 Keywords: Off-diagonal long-range order; Electron correlation; Hubbard model
1. Introduction There is much current interest for the study of electron correlations in metals because of their significance in magnetism and superconductivity [1,2]. Some progress was recently achieved [3—5] for 2n electrons coupled through the most general Hamiltonian H comprising a two-body force within the one-electron band of a periodic crystal in arbitrary dimension. It is then convenient to introduce the pair creation and annihilation operators b>(k, k ), b (k, k ): D D b> (k, k )"c> c> , b (k, k )"c c , ! I !N IY !N ! IY !N I !N b>(k, k )"c> c> , b (k, k )"c c . (1) I N IY \N IY \N I N The Fermi operators c> and c describe electron creI N I N ation and annihilation on the Bloch state k, p. The subscript f"0,$1 stands for the projection of the total spin of the pair. The Hamiltonian H can then be recast in terms of the subsidiary Hamiltonians H , H as " ) D H"H # H , where H is diagonal in " " ) D ! ) D a Slater determinant basis and H operates within the ) D subspace S characterised by all pairs having the same ) D K, f.
* Tel.: #33169085577; [email protected].
fax:
#33169088261;
Two theorems have been demonstrated elsewhere [3—5]. They characterise the two classes of eigensolutions t, e of the Schro¨dinger equation (H!e)t"0, designated, respectively, as t , e and t , e . Theorem 1. To each eigensolution t , e where ) D (H #H !e )t "0, there corresponds an eigensolu" ) D ) D tion t , e of H such that (H!e )t "0. Each t belongs to a subspace S spanned by Slater ) D ) D determinants " L b>(k ,K!k )"02, where "02 ) D H D H H designates the no-electron state. This feature ensures that t has off-diagonal long-range order. Theorem 2. For every t , e , the equation (H!e )t "0 implies that (H !e )t "0. " It follows from Theorem 2 that t has no off-diagonal long-range order. The eigenvectors t subspan the subspace S which proves to be orthogonal to every S so that the t , t ’s ) D make up a basis. The purpose of this work is threefold:
e-mail:
1. a slight inaccuracy will be corrected in Theorem 1;
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 8 7 6 - X
778
J. Szeftel / Physica B 259—261 (1999) 777—778
2. additional properties of the eigenvectors t and t with respect to the pair number operator N " b>(k, K!k)b (k, K!k) will be given; ) D D D 3. Theorem I1 has enabled us to make a definite statement [3] upon the groundstate of the Hubbard model having off-diagonal long-range order in one dimension which is seemingly at odds with common wisdom [1,2]. This issue will be discussed. Theorem 1 (revisited). It is inferred from Theorem 1 that t Ot . It will be shown by contradiction that actually ) D the contrary is true, i.e. t "t . Suppose then that ) D t "t #t and t O0. Since any two eigenvectors ) D t and t of a Hermitian operator H, are orthogonal and t 3S and t 3S are orthogonal too, it entails ) D ) D that t 3S is orthogonal to t . Thence t 3S cannot be expanded over the set of t vectors, which is equivalent to the eigenvectors of H not making up a basis and thus contradicts H being Hermitian. Q.E.D. In case of the Hubbard Hamiltonian, the validity of the property t "t is supported by numerous eigenstates ) D built with the help of the g-pair mechanism [6].
two-point correlation function decreases as a power law of the interparticle distance. This seems to conflict with a previous statement [3] that the ground state of the Hubbard model does have indeed off-diagonal longrange order in one dimension. Actually, the conclusion that there is no long-range order is reached via a renormalisation group based argument taking advantage of the same statement made for the Tomonaga—Luttinger model [8]. In case of the Tomonaga model this result is achieved by a mean-field approximation wherein some operator is replaced by its expectation value reckoned for the Fermi sea of independent electrons whereas it turns out to be exact for the Luttinger model. However, the price to be paid for this achievement in the latter case is that every observable quantity becomes infinite, which makes the model unrealistic and deprives the proof of any mathematical rigour in marked contrast with the proofs of Theorems 1 and 2. The contradiction is thus resolved. Noteworthy is the issue that the low-temperature properties of the one-dimensional Hubbard model reflect a t character although the whole contribution of the t eigenvectors to the eigenbasis becomes negligible with respect to that of the t ’s in the thermodynamic limit.
2. Pair number related properties As the pair number operator N commutes with H , ) D ) D i.e. [H , N ]"0, every t , eigenvector of H, fulfils ) D ) D ) D N t "nt . Inversely note that [H , N ]O0 ) D ) D ) D ) D )Y DY for any (K , f )O(K, f). Due to Theorem 2, every eigenvector t of H is also an eigenvector of H and there is [H , N ]"0, ∀(K, f). It " " ) D implies that every t satisfies N t "n t , ∀(K, f), ) D ) D where n (n is an integer. Notice that [H,N ]" ) D ) D 0, ∀(K, f) within S while [H, N ]O0 in general in the ) D whole Hilbert space. These properties will be used to devise an iteration procedure intended to diagonalise H at finite electron concentration and in arbitrary dimension. It generalises the g-pair mechanism [6] and will only be outlined here. The key idea consists of introducing the promotion operator P such that t "Pt , where t and t desigL> L L L> nate eigenvectors of H comprising n and n#1 pairs, respectively. After showing the existence of the unknown P, the task to work it out proves to be equivalent to solving two kinds of n"1 problems [7], renormalised by the finite electron concentration and corresponding, respectively, to t - and t -like eigenstates. 3. The ground state of the Hubbard model It is widely believed [1,2] that the ground state of the one-dimensional Hubbard model does not have any type of long-range order at all and hence every associated
4. Conclusion In conclusion, several consequences of Theorems 1 and 2 have been worked out which should prove useful in extending the treatment used to diagonalise H in the n"1 case [7] to the n<1 one. The controversy about the presence of off-diagonal long-range order in the groundstate of the one-dimensional Hubbard model has been settled.
Acknowledgements I dedicate this work to the memory of Abraham Szeftel, Aaron Silberman, Zelman Nisenbaum and Gilbert Dreyfus. I also thank my wife Rachel and children Je´re´mie and Judith for providing encouragement.
References [1] [2] [3] [4] [5] [6] [7] [8]
J. Solyom, Adv. Phys. 28 (1979) 201. J. Voit, Rep. Prog. Phys. 58 (1995) 977. J. Szeftel, A. Khater, Phys. Rev. B 54 (1996) 13 581. J. Szeftel, Act. Phys. Polon. A 91 (1997) 341. J. Szeftel, Physica B 230—232 (1997) 482. C.N. Yang, Phys. Rev. Lett. 63 (1989) 2144. J. Hubbard, Proc. Roy. Soc. A 276 (1963) 238. G.D. Mahan, Many-Particle Physics, Plenum Press, New York.
More upon the properties of correlated electrons J. Szeftel* Laboratoire Le& on Brillouin(CEA-CNRS), CE-Saclay, 91191 Gif-sur-Yvette Ce& dex, France
Abstract The set of eigenvectors of the most general interacting electron Hamiltonian has been shown recently to consist of two classes t and t . The t -like eigenvectors have off-diagonal long-range order while the t -like ones do not have. Further properties of the eigenvectors are discussed. 1999 Elsevier Science B.V. All rights reserved. PACS: 3.65; 71.45; 74.20 Keywords: Off-diagonal long-range order; Electron correlation; Hubbard model
1. Introduction There is much current interest for the study of electron correlations in metals because of their significance in magnetism and superconductivity [1,2]. Some progress was recently achieved [3—5] for 2n electrons coupled through the most general Hamiltonian H comprising a two-body force within the one-electron band of a periodic crystal in arbitrary dimension. It is then convenient to introduce the pair creation and annihilation operators b>(k, k ), b (k, k ): D D b> (k, k )"c> c> , b (k, k )"c c , ! I !N IY !N ! IY !N I !N b>(k, k )"c> c> , b (k, k )"c c . (1) I N IY \N IY \N I N The Fermi operators c> and c describe electron creI N I N ation and annihilation on the Bloch state k, p. The subscript f"0,$1 stands for the projection of the total spin of the pair. The Hamiltonian H can then be recast in terms of the subsidiary Hamiltonians H , H as " ) D H"H # H , where H is diagonal in " " ) D ! ) D a Slater determinant basis and H operates within the ) D subspace S characterised by all pairs having the same ) D K, f.
* Tel.: #33169085577; [email protected].
fax:
#33169088261;
Two theorems have been demonstrated elsewhere [3—5]. They characterise the two classes of eigensolutions t, e of the Schro¨dinger equation (H!e)t"0, designated, respectively, as t , e and t , e . Theorem 1. To each eigensolution t , e where ) D (H #H !e )t "0, there corresponds an eigensolu" ) D ) D tion t , e of H such that (H!e )t "0. Each t belongs to a subspace S spanned by Slater ) D ) D determinants " L b>(k ,K!k )"02, where "02 ) D H D H H designates the no-electron state. This feature ensures that t has off-diagonal long-range order. Theorem 2. For every t , e , the equation (H!e )t "0 implies that (H !e )t "0. " It follows from Theorem 2 that t has no off-diagonal long-range order. The eigenvectors t subspan the subspace S which proves to be orthogonal to every S so that the t , t ’s ) D make up a basis. The purpose of this work is threefold:
e-mail:
1. a slight inaccuracy will be corrected in Theorem 1;
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 8 7 6 - X
778
J. Szeftel / Physica B 259—261 (1999) 777—778
2. additional properties of the eigenvectors t and t with respect to the pair number operator N " b>(k, K!k)b (k, K!k) will be given; ) D D D 3. Theorem I1 has enabled us to make a definite statement [3] upon the groundstate of the Hubbard model having off-diagonal long-range order in one dimension which is seemingly at odds with common wisdom [1,2]. This issue will be discussed. Theorem 1 (revisited). It is inferred from Theorem 1 that t Ot . It will be shown by contradiction that actually ) D the contrary is true, i.e. t "t . Suppose then that ) D t "t #t and t O0. Since any two eigenvectors ) D t and t of a Hermitian operator H, are orthogonal and t 3S and t 3S are orthogonal too, it entails ) D ) D that t 3S is orthogonal to t . Thence t 3S cannot be expanded over the set of t vectors, which is equivalent to the eigenvectors of H not making up a basis and thus contradicts H being Hermitian. Q.E.D. In case of the Hubbard Hamiltonian, the validity of the property t "t is supported by numerous eigenstates ) D built with the help of the g-pair mechanism [6].
two-point correlation function decreases as a power law of the interparticle distance. This seems to conflict with a previous statement [3] that the ground state of the Hubbard model does have indeed off-diagonal longrange order in one dimension. Actually, the conclusion that there is no long-range order is reached via a renormalisation group based argument taking advantage of the same statement made for the Tomonaga—Luttinger model [8]. In case of the Tomonaga model this result is achieved by a mean-field approximation wherein some operator is replaced by its expectation value reckoned for the Fermi sea of independent electrons whereas it turns out to be exact for the Luttinger model. However, the price to be paid for this achievement in the latter case is that every observable quantity becomes infinite, which makes the model unrealistic and deprives the proof of any mathematical rigour in marked contrast with the proofs of Theorems 1 and 2. The contradiction is thus resolved. Noteworthy is the issue that the low-temperature properties of the one-dimensional Hubbard model reflect a t character although the whole contribution of the t eigenvectors to the eigenbasis becomes negligible with respect to that of the t ’s in the thermodynamic limit.
2. Pair number related properties As the pair number operator N commutes with H , ) D ) D i.e. [H , N ]"0, every t , eigenvector of H, fulfils ) D ) D ) D N t "nt . Inversely note that [H , N ]O0 ) D ) D ) D ) D )Y DY for any (K , f )O(K, f). Due to Theorem 2, every eigenvector t of H is also an eigenvector of H and there is [H , N ]"0, ∀(K, f). It " " ) D implies that every t satisfies N t "n t , ∀(K, f), ) D ) D where n (n is an integer. Notice that [H,N ]" ) D ) D 0, ∀(K, f) within S while [H, N ]O0 in general in the ) D whole Hilbert space. These properties will be used to devise an iteration procedure intended to diagonalise H at finite electron concentration and in arbitrary dimension. It generalises the g-pair mechanism [6] and will only be outlined here. The key idea consists of introducing the promotion operator P such that t "Pt , where t and t desigL> L L L> nate eigenvectors of H comprising n and n#1 pairs, respectively. After showing the existence of the unknown P, the task to work it out proves to be equivalent to solving two kinds of n"1 problems [7], renormalised by the finite electron concentration and corresponding, respectively, to t - and t -like eigenstates. 3. The ground state of the Hubbard model It is widely believed [1,2] that the ground state of the one-dimensional Hubbard model does not have any type of long-range order at all and hence every associated
4. Conclusion In conclusion, several consequences of Theorems 1 and 2 have been worked out which should prove useful in extending the treatment used to diagonalise H in the n"1 case [7] to the n<1 one. The controversy about the presence of off-diagonal long-range order in the groundstate of the one-dimensional Hubbard model has been settled.
Acknowledgements I dedicate this work to the memory of Abraham Szeftel, Aaron Silberman, Zelman Nisenbaum and Gilbert Dreyfus. I also thank my wife Rachel and children Je´re´mie and Judith for providing encouragement.
References [1] [2] [3] [4] [5] [6] [7] [8]
J. Solyom, Adv. Phys. 28 (1979) 201. J. Voit, Rep. Prog. Phys. 58 (1995) 977. J. Szeftel, A. Khater, Phys. Rev. B 54 (1996) 13 581. J. Szeftel, Act. Phys. Polon. A 91 (1997) 341. J. Szeftel, Physica B 230—232 (1997) 482. C.N. Yang, Phys. Rev. Lett. 63 (1989) 2144. J. Hubbard, Proc. Roy. Soc. A 276 (1963) 238. G.D. Mahan, Many-Particle Physics, Plenum Press, New York.