- Email: [email protected]
Two-hole states in square lattice antiferromagnet
Journal of Magnetism and Magnetic Materials 226}230 (2001) 444}445
Two-hole states in square lattice antiferromagnet Janusz Morkowski* Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan& , Poland
Abstract Two-holes states in two-dimensional square lattice antiferromagnets described by the t}J model are considered. In the self-consistent Born approximation for J/t(0.5 the energy of two-hole state E (k, k) is higher than the sum of single hole energies, E (k)#E (k) for any k, k implying non-existence of bound states. 2001 Elsevier Science B.V. All rights reserved. Keywords: t}J model; Two-dimensional antiferromagnets; Spin-polarons
Motion of charge carriers in two-dimensional weakly doped antiferromagnetic CuO planes is usually de scribed by the t}J model. The t}J Hamiltonian is mapped into a space of spinless holon and spin variables and is reduced to [1,2] H" > #N\ (P h>h #h.c.), (1) O O O IO I I\O O O IO where and h are the magnon and holon operators (in O O the k-space), "2J(1! is the antiferromagnetic O O magnon energy (for the square lattice " O (cos qx#cos qy)), P "Zt( u # v ), (Z"4), IO I\O O I O where u "[(1/(1!#1)] , v "!sgn( ); O O O O [(1/(1!!1)] comes from the well-known O Bogoliubov transformation. The magnon}holon interaction in Eq. (1) is scaled by the hopping integral t which for real systems is much larger than the exchange integral J. Thus, in contrast to the lattice polaron problem, the interaction dominates over the unperturbed part and there is no holon kinetic energy term in H. It was shown [2] that even in the strong coupling limit the quasiparticle holon spectrum can be calculated with good accuracy in the self-consistent Born approximation (SCBA). The energy spectrum of a single hole calculated in SCBA from Eq. (1) compares very well with exact numerical studies on small systems and some features of the spectrum are believed to be con"rmed by experi-
* Corresponding author. Fax: #48-61-8684-524. E-mail address: [email protected] (J. Morkowski).
ments. Still there is an open question whether the dressed holes or the spin polarons can be treated as a gas of, perhaps, weakly interacting quasiparticles. Form the very form of H that is not obvious and from the numerical analysis of the single-particle Green's function it follows that only rather small part of the single-particle spectral weight is associated with the peak identi"ed with the quasiparticle. The problem of bound states of two holes was studied numerically for small clusters [3}5] and by solving the Bethe}Salpeter equation for zero total momentum [6}8]. Here, we shall try to provide an answer to the question whether the quasi-particle energies calculated in SCBA for two holes are really additive. In mapping exactly the t}J model into the form (1) two contributions are usually neglected. First is the small holon kinetic energy (Eq. (52) of Ref. [9]), second the direct interaction of holes at nearest-neighbour sites (the last term in Eq. (14) of Ref. [9], see also Ref. [8]). Both contributions originate physically from spin #uctuations in the quantum anti-ferromagnetic ground state. The small holon kinetic energy has negligible e!ect on the single hole spectrum. For the present two}hole problem, the direct hole}hole interaction < "( e>e (S ) S !)e>e (see e.g. Eq. (14) of 6G H7 G G G H H H Ref. [9]) is taken into account approximately replacing S ) S ! by its mean "eld value 0(S ) S !)0 $ G H $ G H over the quantum antiferromagnetic ground state 0 . $ Thus in calculations to follow H, Eq. (1), is replaced by H I "H#< with < "N\ = h> h> h h IIYO O I\O IY>O IY I and ="ZJ 0(S ) S !)0 !1.16J for the G H $ $ AF-ground state.
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 2 9 2 - 0
J. Morkowski / Journal of Magnetism and Magnetic Materials 226}230 (2001) 444}445
445
Fig. 1. E (k, k) (full lines) and E (k)#E (k) (broken lines) for (from top to bottom) k"(0, 0), (0, ) and (/2, /2), respectively.
The two}hole Green's function is G(k, k, )" 0h h (!H I )\h>h>0, where 0 is the product of IY I I IY the holon vacuum and the magnon AF ground state. In the approximation equivalent to widely used in the single}hole problem SCBA or non}crossing approxima ![G(k, k, )]\ is tion the selfenergy (k, k, )" calculated from the equation (k, k, )"N\ P G (! ) IO I\O IY O O #P G (! ) IY O I IY\O O #2=N\( ! )G () . I\O IY\O O O\k\k (2) Like in the single}hole problem, Eq. (2) is solved numerically for (k, k ) by an iteration procedure and the two}hole energy E (k, k) is found from the low}energy peak in the two}hole Green's function. Calculations were done for the value J/t"0.3 relevant to real systems. For that value 12 iteration steps were su$cient to compute E (k, k) within accuracy of $0.005t. In Fig. 1, E (k, k) for "xed k are plotted vs. k for high symmetry directions in the Brillouin zone and compared with the sum [E (k)#E (k)] of single quasiparticle energies (all energies measured in units of t). Similar plots of E (k, k!s) and [E (k)#E (k!s)] vs. k for a few s are presented in Fig. 2. For all the values of (k, k) presented in Figs. 1 and 2, the energy di!erence "E (k, k)![E (k)#E (k)]'0 so there is no bound
Fig. 2. E (!k, k!s) and E (k)#E (k!s) for s"(0, 0) ( and *), (/8, /8) (䉫 and #) and (/2, /2) (䊐 and ;), respectively.
two-hole states for J/t"0.3. The direct hole}hole interaction term favouring attraction of holes leads to negligible contributions. Computations for a range of values of J/t show that for J/t((J/t) 0.5 for any pair (k, k) within the Brillouin zone '0 so stable two}hole bound states do not exist. For J/t*0.6 there are regions in the k}space where (0, but such high values of J/t are not relevant to real systems. This research was supported by the KBN Grant No. 2 P03 B 118 14
References [1] S. Schmitt-Rink, C.M. Varma, A.E. Ruckenstein, Phys. Rev. Lett. 60 (1988) 2793. [2] G. Martinez, P. Horsch, Phys. Rev. B 44 (1991) 317. [3] Y. Hasegawa, D. Poilblanc, Phys. Rev. B 40 (1989) 9035. [4] T. Barnes, A.E. Jacobs, M.P. Kovarik, W.G. Macready, Phys. Rev. B 45 (1991) 256. [5] M. Bonisegni, E. Manousakis, Phys. Rev. B 47 (1993) 11897. [6] M.I. Kuchiev, O.P. Sushkov, Physica C 218 (1993) 197. [7] V.I. Belinicher, A.L. Chernyshev, V.A. Shubin, Phys. Rev. B 56 (1997) 3381. [8] A.V. Dotsenko, Phys. Rev. B 57 (1998) 6917. [9] J. Loos, J.A. Morkowski, Physica C 298 (1998) 17.
Two-hole states in square lattice antiferromagnet Janusz Morkowski* Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan& , Poland
Abstract Two-holes states in two-dimensional square lattice antiferromagnets described by the t}J model are considered. In the self-consistent Born approximation for J/t(0.5 the energy of two-hole state E (k, k) is higher than the sum of single hole energies, E (k)#E (k) for any k, k implying non-existence of bound states. 2001 Elsevier Science B.V. All rights reserved. Keywords: t}J model; Two-dimensional antiferromagnets; Spin-polarons
Motion of charge carriers in two-dimensional weakly doped antiferromagnetic CuO planes is usually de scribed by the t}J model. The t}J Hamiltonian is mapped into a space of spinless holon and spin variables and is reduced to [1,2] H" > #N\ (P h>h #h.c.), (1) O O O IO I I\O O O IO where and h are the magnon and holon operators (in O O the k-space), "2J(1! is the antiferromagnetic O O magnon energy (for the square lattice " O (cos qx#cos qy)), P "Zt( u # v ), (Z"4), IO I\O O I O where u "[(1/(1!#1)] , v "!sgn( ); O O O O [(1/(1!!1)] comes from the well-known O Bogoliubov transformation. The magnon}holon interaction in Eq. (1) is scaled by the hopping integral t which for real systems is much larger than the exchange integral J. Thus, in contrast to the lattice polaron problem, the interaction dominates over the unperturbed part and there is no holon kinetic energy term in H. It was shown [2] that even in the strong coupling limit the quasiparticle holon spectrum can be calculated with good accuracy in the self-consistent Born approximation (SCBA). The energy spectrum of a single hole calculated in SCBA from Eq. (1) compares very well with exact numerical studies on small systems and some features of the spectrum are believed to be con"rmed by experi-
* Corresponding author. Fax: #48-61-8684-524. E-mail address: [email protected] (J. Morkowski).
ments. Still there is an open question whether the dressed holes or the spin polarons can be treated as a gas of, perhaps, weakly interacting quasiparticles. Form the very form of H that is not obvious and from the numerical analysis of the single-particle Green's function it follows that only rather small part of the single-particle spectral weight is associated with the peak identi"ed with the quasiparticle. The problem of bound states of two holes was studied numerically for small clusters [3}5] and by solving the Bethe}Salpeter equation for zero total momentum [6}8]. Here, we shall try to provide an answer to the question whether the quasi-particle energies calculated in SCBA for two holes are really additive. In mapping exactly the t}J model into the form (1) two contributions are usually neglected. First is the small holon kinetic energy (Eq. (52) of Ref. [9]), second the direct interaction of holes at nearest-neighbour sites (the last term in Eq. (14) of Ref. [9], see also Ref. [8]). Both contributions originate physically from spin #uctuations in the quantum anti-ferromagnetic ground state. The small holon kinetic energy has negligible e!ect on the single hole spectrum. For the present two}hole problem, the direct hole}hole interaction < "( e>e (S ) S !)e>e (see e.g. Eq. (14) of 6G H7 G G G H H H Ref. [9]) is taken into account approximately replacing S ) S ! by its mean "eld value 0(S ) S !)0 $ G H $ G H over the quantum antiferromagnetic ground state 0 . $ Thus in calculations to follow H, Eq. (1), is replaced by H I "H#< with < "N\ = h> h> h h IIYO O I\O IY>O IY I and ="ZJ 0(S ) S !)0 !1.16J for the G H $ $ AF-ground state.
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 2 9 2 - 0
J. Morkowski / Journal of Magnetism and Magnetic Materials 226}230 (2001) 444}445
445
Fig. 1. E (k, k) (full lines) and E (k)#E (k) (broken lines) for (from top to bottom) k"(0, 0), (0, ) and (/2, /2), respectively.
The two}hole Green's function is G(k, k, )" 0h h (!H I )\h>h>0, where 0 is the product of IY I I IY the holon vacuum and the magnon AF ground state. In the approximation equivalent to widely used in the single}hole problem SCBA or non}crossing approxima ![G(k, k, )]\ is tion the selfenergy (k, k, )" calculated from the equation (k, k, )"N\ P G (! ) IO I\O IY O O #P G (! ) IY O I IY\O O #2=N\( ! )G () . I\O IY\O O O\k\k (2) Like in the single}hole problem, Eq. (2) is solved numerically for (k, k ) by an iteration procedure and the two}hole energy E (k, k) is found from the low}energy peak in the two}hole Green's function. Calculations were done for the value J/t"0.3 relevant to real systems. For that value 12 iteration steps were su$cient to compute E (k, k) within accuracy of $0.005t. In Fig. 1, E (k, k) for "xed k are plotted vs. k for high symmetry directions in the Brillouin zone and compared with the sum [E (k)#E (k)] of single quasiparticle energies (all energies measured in units of t). Similar plots of E (k, k!s) and [E (k)#E (k!s)] vs. k for a few s are presented in Fig. 2. For all the values of (k, k) presented in Figs. 1 and 2, the energy di!erence "E (k, k)![E (k)#E (k)]'0 so there is no bound
Fig. 2. E (!k, k!s) and E (k)#E (k!s) for s"(0, 0) ( and *), (/8, /8) (䉫 and #) and (/2, /2) (䊐 and ;), respectively.
two-hole states for J/t"0.3. The direct hole}hole interaction term favouring attraction of holes leads to negligible contributions. Computations for a range of values of J/t show that for J/t((J/t) 0.5 for any pair (k, k) within the Brillouin zone '0 so stable two}hole bound states do not exist. For J/t*0.6 there are regions in the k}space where (0, but such high values of J/t are not relevant to real systems. This research was supported by the KBN Grant No. 2 P03 B 118 14
References [1] S. Schmitt-Rink, C.M. Varma, A.E. Ruckenstein, Phys. Rev. Lett. 60 (1988) 2793. [2] G. Martinez, P. Horsch, Phys. Rev. B 44 (1991) 317. [3] Y. Hasegawa, D. Poilblanc, Phys. Rev. B 40 (1989) 9035. [4] T. Barnes, A.E. Jacobs, M.P. Kovarik, W.G. Macready, Phys. Rev. B 45 (1991) 256. [5] M. Bonisegni, E. Manousakis, Phys. Rev. B 47 (1993) 11897. [6] M.I. Kuchiev, O.P. Sushkov, Physica C 218 (1993) 197. [7] V.I. Belinicher, A.L. Chernyshev, V.A. Shubin, Phys. Rev. B 56 (1997) 3381. [8] A.V. Dotsenko, Phys. Rev. B 57 (1998) 6917. [9] J. Loos, J.A. Morkowski, Physica C 298 (1998) 17.