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Long-range two-hole bound states in t-J model
Physica C 235-240 (I 994) 2219-2220 North-Holland
PHYSICA
Long-range two-hole bound states in t-J model M.Yu. Kuchiev and O.P.Sushkov School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia Several series ,~f shallow long-range (l 7~, lattice spacing) two-hole bound states on the Need background axe found in the two-dimensional t - J model. Their binding energ'~ depend exponentially on the inverse value of the hole-spin-wave coupling constant. The possible role of these states in the formation of superconducting pairing is considered.
The problem of mobile holes in the t-d model is closely related to high-To superconductivity. The
t-J model with less than half-filling is defined by the Hamiltonian
( at
H=-ta
where d~o is the creation operator of a hoie w~,th spin cr (or =~, ~) at site n on a two-dimensio~ ::i square lattice. T h e d~a operator acts in th" Hilbert space where there is no double electron occupancy. The spin operator is S , =
it ~dna~ra~dnp. are neighbour sites on the lattice. Below we set J = 1. At half-filling (one hole per site) the t - J model is equivalent to the Heisenberg antiferromagnet model [1,2] which has the long-range antiferromagnetic order in the ground state [3,4]. Single particle properties in the t-J model are by now well established (see e.g. Refs.[5-12]). Considering much more complicated dynamics in the two-hole problem we follow Ref.[13]. In the two-hole problem the long-range antiferromagnetic order is preserved. It is known that at a finite concentra~,:ion of holes t.he ~_nti_ferromagnctic order is destroyed, but here we consider the pure problem of two holes. For the long-range excitations the Hamiltonian (1) is reduced to the form H = H0 + H i . ,
(2)
where H0 describes the single particle excitations which are holes and spin-waves, and Hi,~, describes the interaction of holes with spin-waves. 0921-4534/94/$07.00 © 1994 SSDI 0921-4534(94)01675-5
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In the long-rang region a hole is a composed object with strong admixture of short-range spinwaves. The dispersion ck of a composite hole is defined in the magnetic Brillouin zone 7k > 0. The band bottom is at the face of Brillouin zone: k0 = (+Tr/2,+~r/2). Near the band b o t t o m the hole dispersion can be presented as the usual quadratic form c k ~ ~1 t l k 21 + l fl2k~, f12 <'( ill, where k is a deviation of the momentum from the m i n i m u m of the zone k0 and indexes 1 and 2 here and later label the projections of a vector on the axes orthogonal and parallel to the face of the Brillouin zone. Due to works[7-12,14] fll ~. 0.7t for t >> 0.33. The curvature along the face of Brillouin zone is rather small and can be estimated as i2 "" 0.1t if t >> 0.33. The dispersion of the hole is wq ~ x/~q for q < 1. The effective interaction Hi,t describes the creation (or annihilation) of spin-waves by a hole. The corresponding vertex is 9(k, q) = (k - q , - t y ; q, slH,.,Ik, o), where [k, a) and [ k - q , - c r ) are the states of the hole and I% s} dei~,~es the state of the spin-wave having the momentum q and the projection of spin s = :kl. In the vicinity of the band bottom k -- (+7r/2, +7r/2) 9(k, q) is of the form (see, e.g. Refs.[7,10,11,14,15]) 9( k, q) ~ g(q) -- 2 3 / 4 f q l / ~ ,
(3)
The coupling constant f was shown to be tindependent: f ~ 2, if t > 1, see Ref.[14]. Consider a system of two holes with total mom e n t u m P = 0 and the projection of spin S.. = 0. The interaction between them caused by the exchange of a soft spin-wave results in the effective
Elsevier Science B.V. All rights rcsc~'cd.
M. Yu. Kuchieu O.R Sushkou/Physica C 235-240 (1994) 2219-2220
2220
potential V(q) = 2g2(q)/wq ~-, 4f2q2/q 2
(4)
In the denominator in (4) only wq plays a role because for long-range states wq ::~ (q. This property makes V(q) be a static potential. We find from (4) the coordinate representation
V(r) = ~(2f~/~r)(z~ - z~)/r 4
(5)
Here P = 4-1 is parity. It is convenient to perform rescaling and introduce the polar coordinates p and ~: zl = vf~'pcos~, z2 = V ~ p s i n ~. Then the variables p and ~ in the Schrodinger equation are separated. The equation for the radial wave function R(p) is - -~ -~p p -~p + - ~
R (p ) = E R ( p ) .
(6)
There is a long-range potential in (6)
(7)
V(p) =
in which the coefficient A should be found from the angular equation. It depends on the mass ratio (fll/fl2), coupling constant f and parity P. It is important that ,~ may be negative. The reason for this one can deduce from (5): there are the regions where V(r) is negative. For negative A there is the long-range attractive potential in Eq.(6). As a result for every such A there appears the infinite series of bound states for the two holes. We can find their energies using the semiclassical approximation E, = -
exp -2
x/~(n + 3/4) .
(8)
Here n -- 0, 1,2... and p0 is a short-range cutoff parameter. The point is that thc interaction - l i p 2 is too singular at the origin and thercfore it gives the "fall" of a particle to the center. To stabilize the solution we introduce infinite repulsion at p _< p0. It is obvious that the cut-off parameter is due ~-" ~he ~ ;*- ~*~'~^ " ~ ~'~ as well as to the contact hole-hole interaction. We get several infinite series of bound states for two holes with S~ - O. One of them possesses the symmetry of the d-wave (or g-wave) with 7~ = 1, the other the symmetry of the p-wave witi~ P =
- 1 . The energies (8) are governed by the only parameter ,~. For a realistic value of t ~ 3 - 4 we find that ~ is not large: I,~01 _ 0.7 for 7) = 1 and IAII <_ 1.5 for :P = - 1 . Therefore the bound states (8) are shallow and long-range ones. There are physical reasons which show that the considered states could be relevant to high-To superconductivity in spite of their tiny binding energies. We considered the two holes separately. If there is an ensemble of holes then the attraction between a pair of holes should be enhanced. The point is that a quite moderate enhancement of the interaction results in the very strong, exponential, increase of the binding energy. This question is discussed in ([16]) REFERENCES 1. J . E . Hirsch, Phys. Rev. Lett. 54, 1317 (1985). 2. C. Gros, R. Joynt, and T. M. Rice, Phys. Rev. B 3 6 , 3 8 1 (1987). 3. J. Oitmaa and D. D. Betts, Can. J. Phys. 56, 897 (1971). 4. E./danousakis, Rev. Mod. Phys. 63, 1 (1991). 5. S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein, Phys. Rev. Lett. 60, 2793 (1988). 6. B. Shraiman and E. Siggia, Phys. Rev. Lett. 61,467 (1988). 7. C. L. Kane, P. A. Lee, and N. Read, Phys. Rev. B 39, 6880 (1989). 8. J. Bonca, P. Prelovsck, and I. Scga, Phys. Rev. B 39, 7074 (1989). 9. E. Dagotto, R. Joynt, S. Bacci, and E. Cagliano, Phys. Rev. B 41, 9049 (1990). 10. G. Martinez and P. Horsch, Phys. Rev. B 44,
q17 (1991). 11 .... Liu and E. Manousakis, Phys. Rev. B 45. ~42~ (1992). 12. O. P. Sushkov, Solid State Communications 83,303 (1992). 13. M. Yu. Kuchiev and O. P. Sushkov, Physica C 218, 197 (1993). 160, 424 (1989). 14. O. P. Sushkov, Phys. Rev. B 49, (1994). 15. O. P. Sushkov, Physica C 206,264 (1993). 16. V. V. Flambaum, M. Yu. Kuchiev and O. P. Sushkov, Proceedings of the present ConferCllCe.
PHYSICA
Long-range two-hole bound states in t-J model M.Yu. Kuchiev and O.P.Sushkov School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia Several series ,~f shallow long-range (l 7~, lattice spacing) two-hole bound states on the Need background axe found in the two-dimensional t - J model. Their binding energ'~ depend exponentially on the inverse value of the hole-spin-wave coupling constant. The possible role of these states in the formation of superconducting pairing is considered.
The problem of mobile holes in the t-d model is closely related to high-To superconductivity. The
t-J model with less than half-filling is defined by the Hamiltonian
( at
H=-t
where d~o is the creation operator of a hoie w~,th spin cr (or =~, ~) at site n on a two-dimensio~ ::i square lattice. T h e d~a operator acts in th" Hilbert space where there is no double electron occupancy. The spin operator is S , =
it ~dna~ra~dnp.
(2)
where H0 describes the single particle excitations which are holes and spin-waves, and Hi,~, describes the interaction of holes with spin-waves. 0921-4534/94/$07.00 © 1994 SSDI 0921-4534(94)01675-5
-
In the long-rang region a hole is a composed object with strong admixture of short-range spinwaves. The dispersion ck of a composite hole is defined in the magnetic Brillouin zone 7k > 0. The band bottom is at the face of Brillouin zone: k0 = (+Tr/2,+~r/2). Near the band b o t t o m the hole dispersion can be presented as the usual quadratic form c k ~ ~1 t l k 21 + l fl2k~, f12 <'( ill, where k is a deviation of the momentum from the m i n i m u m of the zone k0 and indexes 1 and 2 here and later label the projections of a vector on the axes orthogonal and parallel to the face of the Brillouin zone. Due to works[7-12,14] fll ~. 0.7t for t >> 0.33. The curvature along the face of Brillouin zone is rather small and can be estimated as i2 "" 0.1t if t >> 0.33. The dispersion of the hole is wq ~ x/~q for q < 1. The effective interaction Hi,t describes the creation (or annihilation) of spin-waves by a hole. The corresponding vertex is 9(k, q) = (k - q , - t y ; q, slH,.,Ik, o), where [k, a) and [ k - q , - c r ) are the states of the hole and I% s} dei~,~es the state of the spin-wave having the momentum q and the projection of spin s = :kl. In the vicinity of the band bottom k -- (+7r/2, +7r/2) 9(k, q) is of the form (see, e.g. Refs.[7,10,11,14,15]) 9( k, q) ~ g(q) -- 2 3 / 4 f q l / ~ ,
(3)
The coupling constant f was shown to be tindependent: f ~ 2, if t > 1, see Ref.[14]. Consider a system of two holes with total mom e n t u m P = 0 and the projection of spin S.. = 0. The interaction between them caused by the exchange of a soft spin-wave results in the effective
Elsevier Science B.V. All rights rcsc~'cd.
M. Yu. Kuchieu O.R Sushkou/Physica C 235-240 (1994) 2219-2220
2220
potential V(q) = 2g2(q)/wq ~-, 4f2q2/q 2
(4)
In the denominator in (4) only wq plays a role because for long-range states wq ::~ (q. This property makes V(q) be a static potential. We find from (4) the coordinate representation
V(r) = ~(2f~/~r)(z~ - z~)/r 4
(5)
Here P = 4-1 is parity. It is convenient to perform rescaling and introduce the polar coordinates p and ~: zl = vf~'pcos~, z2 = V ~ p s i n ~. Then the variables p and ~ in the Schrodinger equation are separated. The equation for the radial wave function R(p) is - -~ -~p p -~p + - ~
R (p ) = E R ( p ) .
(6)
There is a long-range potential in (6)
(7)
V(p) =
in which the coefficient A should be found from the angular equation. It depends on the mass ratio (fll/fl2), coupling constant f and parity P. It is important that ,~ may be negative. The reason for this one can deduce from (5): there are the regions where V(r) is negative. For negative A there is the long-range attractive potential in Eq.(6). As a result for every such A there appears the infinite series of bound states for the two holes. We can find their energies using the semiclassical approximation E, = -
exp -2
x/~(n + 3/4) .
(8)
Here n -- 0, 1,2... and p0 is a short-range cutoff parameter. The point is that thc interaction - l i p 2 is too singular at the origin and thercfore it gives the "fall" of a particle to the center. To stabilize the solution we introduce infinite repulsion at p _< p0. It is obvious that the cut-off parameter is due ~-" ~he ~ ;*- ~*~'~^ " ~ ~'~ as well as to the contact hole-hole interaction. We get several infinite series of bound states for two holes with S~ - O. One of them possesses the symmetry of the d-wave (or g-wave) with 7~ = 1, the other the symmetry of the p-wave witi~ P =
- 1 . The energies (8) are governed by the only parameter ,~. For a realistic value of t ~ 3 - 4 we find that ~ is not large: I,~01 _ 0.7 for 7) = 1 and IAII <_ 1.5 for :P = - 1 . Therefore the bound states (8) are shallow and long-range ones. There are physical reasons which show that the considered states could be relevant to high-To superconductivity in spite of their tiny binding energies. We considered the two holes separately. If there is an ensemble of holes then the attraction between a pair of holes should be enhanced. The point is that a quite moderate enhancement of the interaction results in the very strong, exponential, increase of the binding energy. This question is discussed in ([16]) REFERENCES 1. J . E . Hirsch, Phys. Rev. Lett. 54, 1317 (1985). 2. C. Gros, R. Joynt, and T. M. Rice, Phys. Rev. B 3 6 , 3 8 1 (1987). 3. J. Oitmaa and D. D. Betts, Can. J. Phys. 56, 897 (1971). 4. E./danousakis, Rev. Mod. Phys. 63, 1 (1991). 5. S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein, Phys. Rev. Lett. 60, 2793 (1988). 6. B. Shraiman and E. Siggia, Phys. Rev. Lett. 61,467 (1988). 7. C. L. Kane, P. A. Lee, and N. Read, Phys. Rev. B 39, 6880 (1989). 8. J. Bonca, P. Prelovsck, and I. Scga, Phys. Rev. B 39, 7074 (1989). 9. E. Dagotto, R. Joynt, S. Bacci, and E. Cagliano, Phys. Rev. B 41, 9049 (1990). 10. G. Martinez and P. Horsch, Phys. Rev. B 44,
q17 (1991). 11 .... Liu and E. Manousakis, Phys. Rev. B 45. ~42~ (1992). 12. O. P. Sushkov, Solid State Communications 83,303 (1992). 13. M. Yu. Kuchiev and O. P. Sushkov, Physica C 218, 197 (1993). 160, 424 (1989). 14. O. P. Sushkov, Phys. Rev. B 49, (1994). 15. O. P. Sushkov, Physica C 206,264 (1993). 16. V. V. Flambaum, M. Yu. Kuchiev and O. P. Sushkov, Proceedings of the present ConferCllCe.