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Gauge-field interaction and superconductivity in the t1−J model
T ~
J
C 1~$- 189 (1991) 14"/9-14~0 North-Holland
GAUGE-FIELD
INTERACTION
AND
SUPERGONDUGTIVII"Y
tN I"tt~ # - d M O D E L
B.Normand and P.A.Ime, Dept. of Ph~nties, M.I.T., Cambridge, MA 02139, U.S.A., ~ d N . N ~ Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo.ku, Tokyo 113' Japan.
Dept. o¢
In a theory of dilute fermionic holes in a 2-dimenfional a n t i f e ~ m a g n e t in the d i ~ e r ~ ante, the interparti¢~ interactions can be c u t in terms of a gauge field. We have calculated the g a u ~ f l e l d prop~ator D a ( q , , } at zero temperature, taking into account both fermloni¢ and bosoni¢ degrees of freedom. Numerical ~ t a t ~ of the interaction spectral function show it to be dominated by the fermion ~ t r i b u t i o n , and to have an unconventional power-law dependence a~F(~) .~ ~ - ~ , wlth x dose to ~. S u p e ~ n d u e t i ~ i t y in the rrax~!d i.~ expressed in the formalism of Eiiaahber$ theory, with the unus,a! cJF{~). In the t' - J model of an antiferromagneti¢ sys-
must be included.
Dt,,,(q ) = ll~,~{q) aeparat~ into
tem with short-range order in 2-dimensions, the holes
scalar and vector parts according to the v ~ e x co~apfing~
may hop only on A or B sublattices separately so
in (1), yielding an effective coupling g i ~
that the number difference na - nB is conserved) A
e~
X-~
2flV p~p,,
Schwinger boson/slave fermion decomposition gives a
by B,,,t =
! (P+ ~q)~. ~ ,(P, - 1~q)~ Do(q) - ~ v , ~ t q l . . . . . .AI
bosons (represented by staggered spinor field z), with
t t , fA.V + fa,~,_¢fB,r+¢fB.pfa~p°] [/A,p+q/s.,-/s~ ~ (2}
interactions mediated by a gauge field ~ at, , whose U(1 )
in Matsubara and momentum spa,:e (q = (q, i~,,n }), with
gauge symmetry is a consequence of the conservation
propagators Do(q) = (AoAo) =-- 7l II~'~l{q) and D,~z =
law. The Hamiltonian
(A~A~) ~
system of spinless fermions ( f . 4 , f B ) and spin-carrying
×
Summing ~he polarization bubbles yields the zero>
H=ff dr f d~z~l(O.-iea.)zl 2
~emperature boson part R~.(q) and, separately, the thermal fermion parts I]ge(q) and It~a(q ).
-
2M
(a,
-
-
I . 2M
alytic continuation and taking the T ~ +
(~)
By an-
0 limit of
the latter functions, one obtains Mgebraic expresmons
b,n
~, R
is that of the C P a model, which is taken to be in the
for
disordered phase, with a gap m for spin excitation; M is
y.R ( q , w ) ( 6 ~ - q~q'a), each of which has several forms II,,,¢
the fermion mass in a small energy expansion. The low-
in different regions of (q,w) space. By coding t he_~e expressions in all re~mes of (q, ~ }.
previously i, while in this paper the analysis is extended
one has the exact zero-~empera~ure gauge boson prop-
to include frequencies up to an energy scale of order d.
agator Da(q,,v) for the lowest order interaction. The
To find the lowest order effective interaction be-
interaction spectral function is given by a complete q
tween f a and fB particles required for superconductiv-
integration as
ity, both fermion and boson polarization contributions to the effective gauge propagator Do. = (a~,(q)a,,(-q))
c~2F("v) -
0921-4534/91/'$03.50 © 1991 - Elsevier Science Publishers B.V. All fights r'e~erve&
1 [ d2q 1 e : p } l ~ D a ( q , ¢ t , (2i 2~r J (2~} 2 vFq 2312 ......
R Nonnand et at / Superconductivity in the t'-J model
1480
t# 2
which is valid at all frequencies w < J (the spin-spin interaction setting the upper validity limit of the original decomposition). The function has a power-law dependence at low frequencies aaF(w) ~ ¢0-" (Fig.1.), with 0.25 < z < 0.4 for all reasonable ranges of the external parameters t', J , m and doping $ (which sets pF). In the limit of low frequency and when Iq[ is small relative to m, it can be shown 1'3 that II~'~ << II~-e~ and IIt,R ec "., x d q ~ - -
.ta z~q-~ -, where Xd is a diamagnetic suscep-
tibility. In the approximation that this form is valid for all q < 2pF, one may evaluate the spectral function as 4=(Mxa)]
,
i.e. z = ~. This implies
that the low-q form of II~e¢ dominates the exact integral also, but the effects of averaging over all q <_ 2pg cause deviations from z = ~. The change in form at w > 1pF (Fig.l) is the result of denominator terms in D
of 0 ( q-r), outside the above approximation, becoming significant.. This result, that the fermion part is dominant and the boson mass m plays an insignificant role at zero temperature, is in contrast to the assumption made previously I that m is the upper limit of validity, and marks a crossover in behavior of the effective interaction. The order parameter for pairing in the superconducting phase is (.fA,pf~,-p/, which suggests a N a m b u formulation 4 with two-component field operators ~pT _ (f1,: ' f/L-p) and ~p. The important difference from the spin problem is that the B vertex has "gauge charge" - e (1), which alters the sign of the B fermion contribution to the density p(q) = ~"]~p~ + q l q p j(q) = X-~p~ ( p + ½ q ) ~ + q l " , ~ p .
and current
Taking the vector part
to be dominant, the effective interaction (2)then has the same vertex structure (~'3.r3) as the Eliashberg phonon problem.
10
¸
The coupled Eliashberg equations 4 are expressed 8
in terms of the gap function A(w) and the quasipar-
6
tide renormalization Z(w) using the spectra.! function
4
a2F(w). At zero temperature with a simple power-law
2
form of c~2 F(w), the equations are analytically tractable
0
to lowest order by scaling arguments if one assumes also
!
0
100
50 w (in units of F.E./100)
l0 t
e power-law variation of A(~o) = A 1 ~"
. At low fre-
quencies w < A0 it is found that the only consistent solution is y = 0, i.e. A(w) = A0 is a real constant. Nu-
I
merical solutions currently being refined confirm the existence of a conventional order parameter with ~ ,
_'2 5
if the same divergent spectral function is assumed valid at finite temperature.
At high frequencies w >> A0,
A(w) = A0 remains the only solution in this approxi10
I0 o
Fig.l:
I
,
,
,
I
, It
I
t
I
10 ~ w (in units of F.E./100)
I
I
I
I
I
10 ~-
The interactio- spectral function a 2F'(w), which diverges as w -+ 0. The logarithmic plot has gradient -~c for 0 < w < ½¢F, illustrating persistence of the power law for the entire range.
mation. 1 P.A.Lee, Phys. Rev. Lett. 03, (1989) 680. 2 See for example X.G.Wen, Phys. Rev. B 30, (1989) 7223. 3 N.Nagaosaand P.A.Lee, Phys. Rev. LetL 04, (1989) 2450. 4 D.J.Scalapino, in Superconductivity, ed. R.D. Parks (Marcel Dekker, New York, 1969) pp. 449-560.
J
C 1~$- 189 (1991) 14"/9-14~0 North-Holland
GAUGE-FIELD
INTERACTION
AND
SUPERGONDUGTIVII"Y
tN I"tt~ # - d M O D E L
B.Normand and P.A.Ime, Dept. of Ph~nties, M.I.T., Cambridge, MA 02139, U.S.A., ~ d N . N ~ Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo.ku, Tokyo 113' Japan.
Dept. o¢
In a theory of dilute fermionic holes in a 2-dimenfional a n t i f e ~ m a g n e t in the d i ~ e r ~ ante, the interparti¢~ interactions can be c u t in terms of a gauge field. We have calculated the g a u ~ f l e l d prop~ator D a ( q , , } at zero temperature, taking into account both fermloni¢ and bosoni¢ degrees of freedom. Numerical ~ t a t ~ of the interaction spectral function show it to be dominated by the fermion ~ t r i b u t i o n , and to have an unconventional power-law dependence a~F(~) .~ ~ - ~ , wlth x dose to ~. S u p e ~ n d u e t i ~ i t y in the rrax~!d i.~ expressed in the formalism of Eiiaahber$ theory, with the unus,a! cJF{~). In the t' - J model of an antiferromagneti¢ sys-
must be included.
Dt,,,(q ) = ll~,~{q) aeparat~ into
tem with short-range order in 2-dimensions, the holes
scalar and vector parts according to the v ~ e x co~apfing~
may hop only on A or B sublattices separately so
in (1), yielding an effective coupling g i ~
that the number difference na - nB is conserved) A
e~
X-~
2flV p~p,,
Schwinger boson/slave fermion decomposition gives a
by B,,,t =
! (P+ ~q)~. ~ ,(P, - 1~q)~ Do(q) - ~ v , ~ t q l . . . . . .AI
bosons (represented by staggered spinor field z), with
t t , fA.V + fa,~,_¢fB,r+¢fB.pfa~p°] [/A,p+q/s.,-/s~ ~ (2}
interactions mediated by a gauge field ~ at, , whose U(1 )
in Matsubara and momentum spa,:e (q = (q, i~,,n }), with
gauge symmetry is a consequence of the conservation
propagators Do(q) = (AoAo) =-- 7l II~'~l{q) and D,~z =
law. The Hamiltonian
(A~A~) ~
system of spinless fermions ( f . 4 , f B ) and spin-carrying
×
Summing ~he polarization bubbles yields the zero>
H=ff dr f d~z~l(O.-iea.)zl 2
~emperature boson part R~.(q) and, separately, the thermal fermion parts I]ge(q) and It~a(q ).
-
2M
(a,
-
-
I . 2M
alytic continuation and taking the T ~ +
(~)
By an-
0 limit of
the latter functions, one obtains Mgebraic expresmons
b,n
~, R
is that of the C P a model, which is taken to be in the
for
disordered phase, with a gap m for spin excitation; M is
y.R ( q , w ) ( 6 ~ - q~q'a), each of which has several forms II,,,¢
the fermion mass in a small energy expansion. The low-
in different regions of (q,w) space. By coding t he_~e expressions in all re~mes of (q, ~ }.
previously i, while in this paper the analysis is extended
one has the exact zero-~empera~ure gauge boson prop-
to include frequencies up to an energy scale of order d.
agator Da(q,,v) for the lowest order interaction. The
To find the lowest order effective interaction be-
interaction spectral function is given by a complete q
tween f a and fB particles required for superconductiv-
integration as
ity, both fermion and boson polarization contributions to the effective gauge propagator Do. = (a~,(q)a,,(-q))
c~2F("v) -
0921-4534/91/'$03.50 © 1991 - Elsevier Science Publishers B.V. All fights r'e~erve&
1 [ d2q 1 e : p } l ~ D a ( q , ¢ t , (2i 2~r J (2~} 2 vFq 2312 ......
R Nonnand et at / Superconductivity in the t'-J model
1480
t# 2
which is valid at all frequencies w < J (the spin-spin interaction setting the upper validity limit of the original decomposition). The function has a power-law dependence at low frequencies aaF(w) ~ ¢0-" (Fig.1.), with 0.25 < z < 0.4 for all reasonable ranges of the external parameters t', J , m and doping $ (which sets pF). In the limit of low frequency and when Iq[ is small relative to m, it can be shown 1'3 that II~'~ << II~-e~ and IIt,R ec "., x d q ~ - -
.ta z~q-~ -, where Xd is a diamagnetic suscep-
tibility. In the approximation that this form is valid for all q < 2pF, one may evaluate the spectral function as 4=(Mxa)]
,
i.e. z = ~. This implies
that the low-q form of II~e¢ dominates the exact integral also, but the effects of averaging over all q <_ 2pg cause deviations from z = ~. The change in form at w > 1pF (Fig.l) is the result of denominator terms in D
of 0 ( q-r), outside the above approximation, becoming significant.. This result, that the fermion part is dominant and the boson mass m plays an insignificant role at zero temperature, is in contrast to the assumption made previously I that m is the upper limit of validity, and marks a crossover in behavior of the effective interaction. The order parameter for pairing in the superconducting phase is (.fA,pf~,-p/, which suggests a N a m b u formulation 4 with two-component field operators ~pT _ (f1,: ' f/L-p) and ~p. The important difference from the spin problem is that the B vertex has "gauge charge" - e (1), which alters the sign of the B fermion contribution to the density p(q) = ~"]~p~ + q l q p j(q) = X-~p~ ( p + ½ q ) ~ + q l " , ~ p .
and current
Taking the vector part
to be dominant, the effective interaction (2)then has the same vertex structure (~'3.r3) as the Eliashberg phonon problem.
10
¸
The coupled Eliashberg equations 4 are expressed 8
in terms of the gap function A(w) and the quasipar-
6
tide renormalization Z(w) using the spectra.! function
4
a2F(w). At zero temperature with a simple power-law
2
form of c~2 F(w), the equations are analytically tractable
0
to lowest order by scaling arguments if one assumes also
!
0
100
50 w (in units of F.E./100)
l0 t
e power-law variation of A(~o) = A 1 ~"
. At low fre-
quencies w < A0 it is found that the only consistent solution is y = 0, i.e. A(w) = A0 is a real constant. Nu-
I
merical solutions currently being refined confirm the existence of a conventional order parameter with ~ ,
_'2 5
if the same divergent spectral function is assumed valid at finite temperature.
At high frequencies w >> A0,
A(w) = A0 remains the only solution in this approxi10
I0 o
Fig.l:
I
,
,
,
I
, It
I
t
I
10 ~ w (in units of F.E./100)
I
I
I
I
I
10 ~-
The interactio- spectral function a 2F'(w), which diverges as w -+ 0. The logarithmic plot has gradient -~c for 0 < w < ½¢F, illustrating persistence of the power law for the entire range.
mation. 1 P.A.Lee, Phys. Rev. Lett. 03, (1989) 680. 2 See for example X.G.Wen, Phys. Rev. B 30, (1989) 7223. 3 N.Nagaosaand P.A.Lee, Phys. Rev. LetL 04, (1989) 2450. 4 D.J.Scalapino, in Superconductivity, ed. R.D. Parks (Marcel Dekker, New York, 1969) pp. 449-560.