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Mean field theory of hole pairs in three dimensions
Physica C 185-189 (1991) 1515-1516
North-Holland
M E A N FIELD T H E O R Y OF HOLE PAIILS IN T H R E E D I M E N S I O N S Kazuhiko SAKAKIBARA," Ikuo ICHINOSE, and Tetsuo MATSUI
• Department of Physics, Kanazawa University, Kanazawa. 920 Japan institute of Physics. University of Tokyo, Komaba. Tokyo. 153 Japan Taking the t-J model as the model of cupper-oxide high-To superconductors, we study the mechanism of superconductivity. The antiferromagnetic coupling generates attractive force between nearestneighbor holes. We review the derivation of mean-field theory of hole-pairs. The resulting gap equation is analyzed at zero temperature in three spatial dimensions, showing the in-plane superconductivity.
Since the earlier consideration by Anderson z, the t-J model of electrons in two dimensions (with a weak threedimensionality) has been studied extensively with the hope
~'~t~z+a~az = I foreach z. (a~az=-~,a~aze etc.)It is solved by introducing CP I spin operator (Schwinger boson) : ~ satisfying zt:,~ = 1, asa~ = (1 - ¢,~=):~. By
in the normal state. However, there still remains another
substituting this representation, the last J-term produces the four-fermi interaction H4F = -(J/2) ~.=p. ~ M ~ . t t is the creation operator of gaugewhere Mt~, - ~zVx~x+.
important issue, i.e., whether the t-J model does exhibit the high-T~ superconductivity, and if so, how its mecha-
invariant hole-pair at (z, z + it). (V~, - zzzr.~+l,.~:x2:z+i,.l.) H~r favors the condensation of such pairs.
nism looks like ? To answer these various questions in a coherent picture, we developed a path-integral formalism ~ in slave-fermion
signaling the superconductivity. We decouple H4r by introducing a complex hole-pair field A ~ , the MF of Mz,,
to describe the high-T~ superconductivity of cupper-oxides. Recent interest is mainly taken in its anomalous behaviors
as the order parameter. Our MF Hamiltonian then reads
representation. After solving the local constraints, we obtained a four-fermi attractive interaction of holes from
t ...l"
the antiferromagnetic (AF) exchange coupling. This force may induce formation of pairs of holes at nearest-neighbor (N N) sites, t heir condensation giving rise to superconductivity. This description in the level of effective action is a manifestation of the naive explanation 3 of attractive forces by counting the destructed AF NN bonds of spins due to holes. In Ref.4 a mean field (MF) theory of such hole pairs is developed in Hamiltonian formulation. The Hamiltonian H,.j of the t-J mode! in d-dimensions is given by tt, l = t~-".( "'tu.'~+.,,x~tax+,,,w~'.t_ l' h.c.) _/z= ~-~(ata)= X,/~
X
+~.J ~'~[(at~a)=(at~a)=+~, - (ata)=(ata)=+~,]
(1)
x#
-
"llC.:r~ ~
~(£..vt~ , .~. +,,~x
".tx + ~ ""x
~
h.c.)
,7.,/,/
9
+ b.c.) + 7"]~la.,d'. ~,,
(z)
and the mass m enforces < !b~qJ~ > = 6.
In Ref.4 this Hamiltonian is solved for d - 2 b y (i) imposing the periodicity of A ~ with the period of V/2a, i.e., four indepedent Al.2.a,~ starting from each odd site into the right, up, left, down directions; and (ii) treating the spin variables by the spiral-state MF theory s. i.e., V'l.z.a.4 = (1, 1, 1, 1), ,~1.z.3., = 6(1, 1 , - 1 , - 1 ) . The so,uuv,, ,.,, ~,ov ~uation generates * " " aaTo.,,m kint~ nf the ground state. (See Fig.1.) If the ratio t/J is relatively small, the dimer-like state is realized, in which one ~, has a bigger amplitude than other three.
As men-
tioned in Ref.4 this dimer-like state should be interpreted
where ~r is the site index, it(= 1, .., d) the direction index;
as the phase-separation statc (3 in this periodic space. For
and ~,t the fermionic hole operator, al~o the bosonic spin
larger values of t/J, the t-term favors uniform (nonvanish-
operator ( tr = 1, 2 spin index). They are constrained as
ing) amplitudes, resulting the state ~.2.3.~ = ~(1. i. 1. i).
0921-4534/91/5('3.50 © 1991 - Elsevier Science Publishers B.V. All fights reserved.
1516
K Sakakibara et aL / Hole pairs in three dimensiostv
FLUX
0
.Y0
6
FIGURE 2 Assignment of six A{s in 3D
FIGURE 1 2D Phase Diagram (T=0) Their phases generate a half unit of flux for each plaquette irrespective of the doping 6. The gap now has the s+id symmetry. This "flux" state may be viewed as a super version of the flux state discussed for the normal chan-
tlJ
6
nel in the Hubbard model T. We note that the effective field theory derived in Ref.2 is based upon this kind of flux
SUPER
state. We believe that the study of MF Hamiltonian (2) including the three dimen.~ionality (3D) should be done for various important reasons. Here we report some results obtained for the ground state on a symmetric three dimen-
0
I
I
0.05
0.10
6
FIGURE 3 3D Phase Diagram (T=O)
sional lattice. Corresponding to the ansatz (i,ii) above, we
introduce six A,(i = 1, ..., 6) starting from each odd site, and put l,; = 1, :~i = 6 ( 1 , 1 , 1 , - 1 , - 1 , - 1 ) .
The ground
state is realized in two typical states; (i) Planar flux state (PFS) and (ii) Normal state. PFS is the state in which the flux state for d = 2 above is realized in every xy-plane, and every vertical Ai's along the z-axis vanishes. One might guess that the 3D flux state may be realized. However it is impossible to assign six phases of Ai such that all six plaquettes support the half unit of flux. (See Fig.2. The xy and xz plaquettes may carry flux, then yz one cannot.) The normal state is the state with all A, vanishing. Note that the MFT for d = 2 at T = 0 can produce the flux state whatever large
L/J
is (due to the 2D momentum in-
tegration in the gap equation). In 3D, PFS cannot survive for larger
t/J.
The phase diagram is sketched in Fig.3.
We feel that the inclusion of three dimensionality in MF theory is not a "straightforward" extension of d = 2 case at all. It brings various interesting and new problems that we should face. s
ACKNOWLEDGEMENTS We thank G.Tatara for valuable comments. REFERENCES 1. P. W. Anderson, Science 235. 1196 (1987). 2. I.Ichinose and T.Matsui, Mod. Phys. Lett. ]B4, 995
(1990). 3. J.E.Hirsch, Phys.Rev.Lett.59,2287(1987). 4. G.Tatara and T.Matsui, Phys.Rev.B(1991)in press. 5. B.Schraimanand E.Siggia, Phys. Rev. Left. 62, 1564 (1989); B.Chakraborty, N.Read, C.Kane and P.A.Lee, Phys. Rev. B42, 4819(1990). 6. V.J.Emery, S.A.Kivelson, an H.Q.Lin, Phys.Rev.Lett.64,475(1990). 7. l.Affleck and J.B.Marston, Phys. Rev. B37, 3774 (1988). 8. K.Sakakibara, {.Ichinoseand T.Matsui, unpublished.
North-Holland
M E A N FIELD T H E O R Y OF HOLE PAIILS IN T H R E E D I M E N S I O N S Kazuhiko SAKAKIBARA," Ikuo ICHINOSE, and Tetsuo MATSUI
• Department of Physics, Kanazawa University, Kanazawa. 920 Japan institute of Physics. University of Tokyo, Komaba. Tokyo. 153 Japan Taking the t-J model as the model of cupper-oxide high-To superconductors, we study the mechanism of superconductivity. The antiferromagnetic coupling generates attractive force between nearestneighbor holes. We review the derivation of mean-field theory of hole-pairs. The resulting gap equation is analyzed at zero temperature in three spatial dimensions, showing the in-plane superconductivity.
Since the earlier consideration by Anderson z, the t-J model of electrons in two dimensions (with a weak threedimensionality) has been studied extensively with the hope
~'~t~z+a~az = I foreach z. (a~az=-~,a~aze etc.)It is solved by introducing CP I spin operator (Schwinger boson) : ~ satisfying zt:,~ = 1, asa~ = (1 - ¢,~=):~. By
in the normal state. However, there still remains another
substituting this representation, the last J-term produces the four-fermi interaction H4F = -(J/2) ~.=p. ~ M ~ . t t is the creation operator of gaugewhere Mt~, - ~zVx~x+.
important issue, i.e., whether the t-J model does exhibit the high-T~ superconductivity, and if so, how its mecha-
invariant hole-pair at (z, z + it). (V~, - zzzr.~+l,.~:x2:z+i,.l.) H~r favors the condensation of such pairs.
nism looks like ? To answer these various questions in a coherent picture, we developed a path-integral formalism ~ in slave-fermion
signaling the superconductivity. We decouple H4r by introducing a complex hole-pair field A ~ , the MF of Mz,,
to describe the high-T~ superconductivity of cupper-oxides. Recent interest is mainly taken in its anomalous behaviors
as the order parameter. Our MF Hamiltonian then reads
representation. After solving the local constraints, we obtained a four-fermi attractive interaction of holes from
t ...l"
the antiferromagnetic (AF) exchange coupling. This force may induce formation of pairs of holes at nearest-neighbor (N N) sites, t heir condensation giving rise to superconductivity. This description in the level of effective action is a manifestation of the naive explanation 3 of attractive forces by counting the destructed AF NN bonds of spins due to holes. In Ref.4 a mean field (MF) theory of such hole pairs is developed in Hamiltonian formulation. The Hamiltonian H,.j of the t-J mode! in d-dimensions is given by tt, l = t~-".( "'tu.'~+.,,x~tax+,,,w~'.t_ l' h.c.) _/z= ~-~(ata)= X,/~
X
+~.J ~'~[(at~a)=(at~a)=+~, - (ata)=(ata)=+~,]
(1)
x#
-
"llC.:r~ ~
~(£..vt~ , .~. +,,~x
".tx + ~ ""x
~
h.c.)
,7.,/,/
9
+ b.c.) + 7"]~la.,d'. ~,,
(z)
and the mass m enforces < !b~qJ~ > = 6.
In Ref.4 this Hamiltonian is solved for d - 2 b y (i) imposing the periodicity of A ~ with the period of V/2a, i.e., four indepedent Al.2.a,~ starting from each odd site into the right, up, left, down directions; and (ii) treating the spin variables by the spiral-state MF theory s. i.e., V'l.z.a.4 = (1, 1, 1, 1), ,~1.z.3., = 6(1, 1 , - 1 , - 1 ) . The so,uuv,, ,.,, ~,ov ~uation generates * " " aaTo.,,m kint~ nf the ground state. (See Fig.1.) If the ratio t/J is relatively small, the dimer-like state is realized, in which one ~, has a bigger amplitude than other three.
As men-
tioned in Ref.4 this dimer-like state should be interpreted
where ~r is the site index, it(= 1, .., d) the direction index;
as the phase-separation statc (3 in this periodic space. For
and ~,t the fermionic hole operator, al~o the bosonic spin
larger values of t/J, the t-term favors uniform (nonvanish-
operator ( tr = 1, 2 spin index). They are constrained as
ing) amplitudes, resulting the state ~.2.3.~ = ~(1. i. 1. i).
0921-4534/91/5('3.50 © 1991 - Elsevier Science Publishers B.V. All fights reserved.
1516
K Sakakibara et aL / Hole pairs in three dimensiostv
FLUX
0
.Y0
6
FIGURE 2 Assignment of six A{s in 3D
FIGURE 1 2D Phase Diagram (T=0) Their phases generate a half unit of flux for each plaquette irrespective of the doping 6. The gap now has the s+id symmetry. This "flux" state may be viewed as a super version of the flux state discussed for the normal chan-
tlJ
6
nel in the Hubbard model T. We note that the effective field theory derived in Ref.2 is based upon this kind of flux
SUPER
state. We believe that the study of MF Hamiltonian (2) including the three dimen.~ionality (3D) should be done for various important reasons. Here we report some results obtained for the ground state on a symmetric three dimen-
0
I
I
0.05
0.10
6
FIGURE 3 3D Phase Diagram (T=O)
sional lattice. Corresponding to the ansatz (i,ii) above, we
introduce six A,(i = 1, ..., 6) starting from each odd site, and put l,; = 1, :~i = 6 ( 1 , 1 , 1 , - 1 , - 1 , - 1 ) .
The ground
state is realized in two typical states; (i) Planar flux state (PFS) and (ii) Normal state. PFS is the state in which the flux state for d = 2 above is realized in every xy-plane, and every vertical Ai's along the z-axis vanishes. One might guess that the 3D flux state may be realized. However it is impossible to assign six phases of Ai such that all six plaquettes support the half unit of flux. (See Fig.2. The xy and xz plaquettes may carry flux, then yz one cannot.) The normal state is the state with all A, vanishing. Note that the MFT for d = 2 at T = 0 can produce the flux state whatever large
L/J
is (due to the 2D momentum in-
tegration in the gap equation). In 3D, PFS cannot survive for larger
t/J.
The phase diagram is sketched in Fig.3.
We feel that the inclusion of three dimensionality in MF theory is not a "straightforward" extension of d = 2 case at all. It brings various interesting and new problems that we should face. s
ACKNOWLEDGEMENTS We thank G.Tatara for valuable comments. REFERENCES 1. P. W. Anderson, Science 235. 1196 (1987). 2. I.Ichinose and T.Matsui, Mod. Phys. Lett. ]B4, 995
(1990). 3. J.E.Hirsch, Phys.Rev.Lett.59,2287(1987). 4. G.Tatara and T.Matsui, Phys.Rev.B(1991)in press. 5. B.Schraimanand E.Siggia, Phys. Rev. Left. 62, 1564 (1989); B.Chakraborty, N.Read, C.Kane and P.A.Lee, Phys. Rev. B42, 4819(1990). 6. V.J.Emery, S.A.Kivelson, an H.Q.Lin, Phys.Rev.Lett.64,475(1990). 7. l.Affleck and J.B.Marston, Phys. Rev. B37, 3774 (1988). 8. K.Sakakibara, {.Ichinoseand T.Matsui, unpublished.