PHYSICAL REVIEW
8
VOLUME 49, NUMBER
Flux binding and superconductivity
Texas Center for Superconductivity
1
in the
1
JANUARY 1994-I
t-J model
Z. Y. geng, D. N. Sheng, and C. S. Ting
and Department of Physics, University of Houston,
Houston, Texas 77204 55-06
(Received 18 February 1993)
A generalized slave-particle scheme is introduced to study the t-J model with nonlocal effects included. A charge-spin decoupled state is found in this scheme, in which the nonlocal gauge fields play a role
of attaching fictitious quantized fluxes to the particles as an effective way to describe antiferromagnetic
correlation. As a result, the gauge fluctuation is suppressed in the long-wavelength limit, which shows
the stability of flux binding, and the electromagnetic response function of such a state exhibits the Meissner effect. An important consequence of the present approach is that the electron, as a combination of
the charge and spin degrees of freedom, does not show time-reversal and parity-symmetry-broken
behavior, in contrast to the usual anyon theory. This is consistent with experimental results. The
robustness of this conclusion relies on the exactness of the constraint between the charge and spin degrees of freedom as well as the flux binding condition. We also calculate the momentum distribution of
electrons, which shows a characteristic scale equal to the Fermi momentum kf.
I. INTRODUCTION
In recent years, the two-dimensional (2D)t-J model has
attracted quite intensive attention because of its close relation with high-T, copper-oxide superconductivity.
As
the large-U version of the Hubbard model, the t-J Hamiltonian reads
H=
t
H. c. +J g
g c; c. +—
(i,j )o.
(ij )
S;
Si- n,
above charge-spin separation picture to the slave-particle
formulation which has been often used in the literature.
In the slave-boson (fermion) method, the electron operator c; is decomposed into f; h;, where f; is a fermion
(boson) operator carrying the spin label while h, is a boson (fermion) operator creating a hole. The no double occupancy constraint (1.2) is then reduced to an equality
n.
haft
f, +h; h; =1 .
(1.3)
The no double occupancy constraint (1.2) imposes a
strong correlation on electrons. In the one-dimensional
(1D) case, it has been shown that such a constraint will
lead to the charge-spin separation, which exactly decouples the hopping and superexchange processes in the t-J
model (1.1) as J~O . It turns out that the electron can
be properly regarded as a composite particle of a spinon,
describing the spin degree of freedom, and a holon, carrying the charge. But more than that, some nonlocal phase
factors have been found to appear in the decomposition
of the electron. These string fields resemble the Jordan%'igner-like nonlocal phase and are physically related to
the phase-shift concept. These phases have been crucial
in obtaining the correct Luttinger-liquid behaviors of the
single-particle Green's function and various correlation
functions.
It would be instructive to make a comparison of the
Because of this constraint, h; and f; are not decoupled
and in that sense they do not really describe the holon
and spinon, respectively. Nevertheless, in the 1D chain
this local constraint has been solved by introducing the
so-called squeezed spin chain ' such that the charge
(hole) and spin get separated in the t JHamiltonia-n. The
effect would be simply
corresponding long-wavelength
described by some nonlocal phase shifts generated from h
and
while they become decoupled.
An alternative way in dealing with the constraint (1.3}
In the gauge theory
is the gauge-field approach.
could
description, a gauge interaction between h and
be found as a replacement of the constraint (1.3) for both
1D and 2D. However, it would be very hard to get some
delicate nonlocal effect discussed above within the quadratic gauge fluctuation. A plausible way to describe it is
to first build in such a nonlocal effect in the formalism
and then study the gauge fluctuation around it. Unfortunately, in the 1D case there is a more serious diSculty
with the gauge theory; that is, the short-range physics is
especially important which cannot be simply described by
the U(1) degree of freedom of the gauge theory.
In the 2D case, we expect that the similar nonlocal
effect would be still important in the long distance. Thus
we may generalize the slave-particle scheme to build in
the nonlocal effect. ' This nonlocal effect turns out to be
quite nontrivial in the 2D case where the long-range and
short-range processes are more closely related. As will be
discussed in the present paper, a charge and spin separa-
0163-1829/94/49(1)/607(14)/$06. 00
607
S; = 'g
c;t (o ) c;
(& is the Pauli matrix},
the
notation
and
(i, ) refers to two
c;
n;
nearest-neighbor
sites on the square lattice in the 2D
case. The sup erexch ange coupling
is determined
through the hopping integral t and the on-site Coulomb
interaction U by J=4t /U in the strong-coupling limit
t
U. The Hilbert space of the Hamiltonian (1.1) is restricted by a no double occupancy constraint
where
=g
—,
~
c;,
j
J
«
(1.2)
49
f
f
1994
The American Physical Society
608
49
Z. Y. WENG, D. N. SHENG, AND C. S. TING
tion state is found in this scheme and it is an intrinsic superconducting state. Since the nonlocal fields, as generalized from 1D, resemble the statistics-transmutation
operators, the present scheme may be properly regarded
'
A recent study'
as the slave-semion decomposition.
on an assembly of the weakly coupled large-U Hubbard
chains also shows that the charge-spin separation persists
into 2D and the nonlocal phase string turns the holon
into semion. The Meissner effect exhibited in the present
state is similar to that in a single-species semion gas. '
An important consequence obtained from the present
framework is that the electron, as a combination of
charge and spin degrees of freedom, will not explicitly
violate the time-reversal (T) and parity (P) syinmetries
due to some fundamental physical reasons. Hence introducing the nonlocal phase effect will be a useful way in
correladealing with the short-range antiferromagnetic
tion in 2D, while the physical properties in the meantime
are not necessarily showing "exotic" behavior in the
sense that no T and P symmetries are really broken in
contrast to the predictions in the ordinary anyon
theory. '
(RVB) type of
So far several resonating-valence-bond
mean-field states have been proposed for the 2D t-J model at small doping. Among them the commensurate-fluxshow
phase'
(CFP) and the d-wave RVB states '
for a reasonably
competitive variational energies '
small J/t and at an appropriate small doping regime. Because of the small energy difference, the variational energy alone would not be enough to decide which one of
them is more stable. The d-wave RVB state usually has a
Recently the
slightly lower variational kinetic energy.
from
kinetic energy in the CFP has been also estimated
an optical conductivity calculation which shows good
The present
agreement with the exact diagonalization.
state has a close connection with the CFP. The main distinction between them is that the uniform lattice flux in
the CFP case will be quantized and bound to the particles
(spinons and holons) in the present state. In the CFP, it
has been argued ' that the binding of a quantized flux
to the particle can occur due to a Chem-Simons term appearing in the gauge fluctuation. Interesting variational
'
have been also given with regard to this
arguments
instability. On the other hand,
statistics-transmutation
for a single-species free-particle system on a lattice exact
do show a finite binding endiagonalization calculations
ergy for the flux-binding state (e.g. , semion state) as compared to the uniform-flux state.
It is noted that in the context of the CFP, the possible
restoration of T and P symmetries as well as the Meissner
in the longeffect have been also discussed
wavelength, low-energy limit. Ho~ever, these predictions actually hold at the half-filling limit since the conclusions crucially rely on the cancellation of the zero-field
Hall conductances of the spinon and holon in the CFP,
which becomes exact only when 5~0 (Ref. 32) (5 is the
doping concentration of holes). It will be pointed out
that both the flux binding as well as the constraint between the charge and spin degrees of freedom are the
essential reasons for the restoration of T and P symin the
metries at general frequency and momentum
"
'
present state. Experimentally, the search for T and P
symmetry-broken
signals in the optical measurements
(see Ref. 33 and the references therein) and muon-spinof high-T, materials have
rotation (@SR) measurements
been not encouraging so far. The absence of chirality or' of the
der is also indicated in the exact diagonalization
t-J model on a small lattice. The present flux-binding
state is consistent with these results.
The rest of the paper will be divided into the following
parts: In Sec. II, we shall express r Jmod-el (1.1) in a
generalized slave-particle formalism which includes the
nonlocal effect as a generalization from 1D. The meanfield decoupling of the charge and spin in this scheme will
lead to a topological state, which coincides with the ~flux state at half-filling. In Sec. III, we shall study the
around this mean-field state
fluctuations
low-energy
through a gauge-field formulation. In Sec. IV, we shall
show that the present state is an intrinsic superconducting ground state and point out that the cancellation of
term in the electron
the T and P symmetry-broken
response is robust. The final section will be devoted to a
discussion and conclusion.
II.
GENERALIZED SLAVE-PARTICLE METHOD
formalism
We shall begin with the slave-boson
h;, where f, is a fermion annihilation operator
c, =
and h, is a boson creation operator satisfying the on-site
hard-core condition t h;, h; ) =1 and (h, ) =0.
As discussed in the Introduction, the no double occupancy constraint (1.2) on c; is replaced by an equality
(1.3) in the slave-boson formalism. Because of such a
constraint, one may rewrite c, as'
f
i a+(4, (), (I) n)
lpga
.
.
e
i aX(%, 8,. ( I )( n(" —i
).
)
for arbitrary a and 8;(l) where n}i=+ f& f& and
h
it is only an equivalent transforman& =h& ht. Although
tion, some nonlocal effect has been explicitly introduced
in Eq. (2. 1). To see it, for example, let us first consider
the 1D case below.
chain, one may choose
For an one-dimensional
8;(l)=~8(i —l) where 8(x) is the step function so that
the phase factors in Eq. (2. 1) become the Jordan-Wignerlike transformation exp(iamb(& (nt '"). If these nonlocal
fields are bound to f, and h, , respectively, their statistics
for a general 0..
will be changed into those of anyons
For instance, if a= 1, then Eq. (2. 1) reads c, h=h, f,
'~r! (I "i . It
'"'Xi (I "(
andh, t =h, t e
where f, =f; e
is straightforward to verify that f, is a hard-core boson
operator while h, becomes a fermion operator. Hence,
one could get the slaue fermion formalis-m in Eq. (2. 1)
starting originally from the slave-boson formulation. The
significance of the formalism (2. 1) lies in the following
fact: It has been known in the 1D case that the charge
and spin parts in (2. 1) could be treated as decoupled for a
particular value o. =+—,' if only the long-wavelength, 1owenergy physics is concerned. However, if Eq. (2. 1) is substituted back into the 1D t- model, these JordanWigner-like fields will be just canceled out in the nearest-
"
'"
J
49
FLUX BINDING AND SUPERCONDUCTIVITY IN THE t-J MODEL
neighboring coupling terms of Eq. (1.1). Since it has no
effec on the Hamiltonian, the long-distance decomposition in 1D has no direct connection with the short-range
process which could have quite different character. In
fact, as mentioned in the Introduction, the long-distance
behavior could be obtained only after a careful treatment
of the short-range process.
In two dimensions, the short-range process and longrange behavior usually would be more connected. The
generalized decomposition (2. 1) will become nontrivial in
this case. We shall define 8;(I) in resemblance to the 2D
generalization of the Jordan-%igner transformation,
which is given by
8, (I ) = Irn ln(z;
character of (2.2) is
gb8; (l)=2m,
(2.3)
where the link variable b, 8; (1) =8, (l) —
8 (l) is defined at
the nearest-neighbor bond (ij ) and the summation is over
loop C of links with the site l enclosed. Notice that the
expression (2.2) is not unique for a given vorticity of (2.3).
But it will be shown later that various choices of 8;(l)
with the same vorticity are equivalent due to a gauge in-
variance.
Using the generalized slave-particle formulation
one may rewrite the t-J model (1.1) in the form
(2. 1),
(2.4)
where
(2.5)
(2.6)
and (2.4) is subject to the no double occupancy constraint
(1.3). The nonlocal gauge phases A (~'" in Eqs. (2.5) and
(2.6) are defined, respectively, as
g [8;(l)—8&(l)]n(,
"
A, "=a
, g [8;(l) —8)(l)](n, —1) .
(2.7)
l/i, j
(2.8)
The t-J model in the usual slave-boson representation
can be recovered if a=O and A, '" disappear from Eqs.
(2.5) and (2.6).
The nonlocal effect introduced in Eq. (2. 1) will show up
once one decouples the spin and charge degrees of freedom in Hamiltonian (2.4) at the mean-field level. The
natural mean fields in Eq. (2.4) are characterized by
go= (y; ) and Go = ( g," ). In this approximation, one
finds two decoupled Hamiltonians for
and h, respec-
f
tively,
J'gy;~+H. c.
Hf = —
&ij
Hq
=
t'
(2.9)
,
&
g C;, + H. c.
(ij)
J
J
f
f
(2.11)
—z& ),
with z;=x,. +iy;. The topological
determined by its vorticity
Af~=a
Here the coupling
constants J' = /2go+ tGO and
t ' = tgo —
in a self/2 GO need to be determined
consistent way, and the constraint (1.3) is replaced by a
global one g;(nf+n, ")=N where N is the total number
of sites.
According to Eqs. (2.5) and (2.6), fermion
and boson h in the Hamiltonians (2.9) and (2. 10) always experience their self-generated fictitious gauge fields A,f and
A;", respectively. For example, if we let fermion
go
through a closed loop C on the lattice, it will see a Berry
phase
—
(2. 10)
in terms of (2.7) and (2.3). The left-hand side of Eq. (2. 11)
is equal to the total fictitious Aux enclosed in the loop C
and the right-hand side term tells that each fermion enclosed in the loop contributes a Aux 2m. a. In other words,
each fermion in the Hamiltonian (2.9) carries a 2@a-flux
tube with it. Consistent with the discussion at the beginning of the section, this Aux binding is related to the
statistics transmutation.
At half-filling, each site is occupied by one fermion
(n) =1) so that the flux (2. 11) becomes the lattice flux
with Q~Af~=2ma per plaquette. If a=1/2, one gets m.
flux per plaquette.
Thus the present mean-field state
(with a= 1/2) reduces to the n.-flux state ' at half-filling.
It is well known that the n-flux state (which is equivalent
to the d-wave RVB at this limit ) has the lowest meanfield energy among the various uniform RVB states. At
finite doping, the average flux of (2. 11) per plaquette at
a = 1/2, which is equal to n. ( n f ), is just the same as in
the commensurate Aux phase. ' But in the present state,
the lattice flux in the CFP is quantized into m-flux tubes
and bound to the fermions as described by Eq. (2. 11). A
similar discussion can be done to the boson Hamiltonian
(2. 10). We shall come back to this part later in the paper.
Recently, a study' of a quasi-1D model as an assembly
of infinite number parallel chains with weak interchain
hopping shows that the charge-spin separation found in
1D does persist into (anisotropic) 2D and the nonlocal
string field will turn the holon into semion by attaching it
a half-quantized (n ) flux tube. Later we shall find a similar separation happening in the present 2D flux-binding
state at a= —, Therefore, there could be a profound
connection between the one-dimensional
and twodirnensional cases as emphasized by Anderson.
Since the CFP has a close connection with the meanfield state under the Hamiltonian (2.9) and (2. 10), we shall
briefly discuss this state in the following. Liang and
Trivedi
have calculated the ground-state energy of the
CFP by the variational Monte Carlo (VMC) method and
it shows a competitive ground-state energy at small doping with a proper small parameter
/t [e.g. , for
5-0. 1, J/t could be as low as 0.2 (Ref. 24)]. In their approach, however, no decomposition of the electron operator has been employed. In that case, one has to introduce
the phase associated with the holes to minimize the kinetic energy subject to the well-known flux frustration.
In
the present decomposition scheme, there is no need to in-
'.
J
610
49
Z. Y. WENG, D. N. SHENG, AND C. S. TING
troduce such phases due to the gauge invariance, and we
have checked the variational ground-state energy by the
VMC method. The trial wave function is a product of
two species, i.e., fermion and boson, and each of them is
composed of the wave function of a single particle in a
uniform magnetic field as the average of (2. 11) at a = 1/2.
At the particular doping concentration 5=p/q where p
and q are integers without a common factor, the singleparticle states for both fermion and boson form the
Hofstadter band structure. ' The fermion tria1 wave
function is obtained by filling up the Hofstadter subbands. For the superexchange term, the variational energy is as good as that in Ref. 24 at the same flux. But the
kinetic energy crucially depends on the choice of the boson wave function. We have considered two options.
First, all bosons are condensed at the bottom of the band.
Since there are degenerate states at the bottom, one could
get different variational results for different choices. We
have found that a space-uniform superposition state usually gives the lowest trial energy and it coincides with the
Liang-Trivedi result. Thus properly choosing a lowest
state for a boson to condense plays a similar role as the
phase adjustment in the Liang-Trivedi scheme. Another
choice of the boson trial wave function is to let bosons fill
to the lowest
up a bunch of subbands, corresponding
Landau level in the continuum limit, instead of condensing into the bottom, hoping to further reduce the phase
frustration in the hopping term. Unfortunately, for the
boson many-body wave function, we have no so-called
reduces
which substantially
method
inverse-updating
the VMC computation time as for the fermion case.
Therefore, we have had to use statistics-transmutation
methods, similar to what has been introduced earlier in
this section, to change a boson into a fermion. The cost
is that some additional nonlocal phase similar to 3; but
with +=1 will appear in the hopping term without a
counterpart 3;~ to cancel it. Then one may again use an
average version of the phase A, " to get the trial wave
The variational
function in the fermion representation.
kinetic energy obtained in this way is close but always
slightly higher than that in Ref. 24. But if one uses the
replace the
average phase factor to approximately
boson-fermion transmutation in the Hamiltonian, a variational kinetic energy as low as 5% of the Liang-Trevidi
results is found. Our best guess would be that the direct
use of the boson trial wave function with the lowest Landau level filled may give a better variational kinetic energy than that in Ref. 24 for the same flux, but a further
computation is still needed. Some related VMC calculation in the decomposition scheme can be found in Ref. 44
~here the trial wave functions in the continuum limit
have been used at a special doping concentration.
For a single-species fermion system on the 1attice, the
fiux-binding (semion) state usually has lower energy than
the uniform-flux state as indicated by exact diagonalizations on a small lattice. In the present two-species system, such a type of calculation would be much more
dificult. But we expect that the fIux-binding state would
be still the stable version of the CFP. Some variational
approaches ' have been given towards a similar problem. In the following we shall mainly focus on the fluc-
tuations around the present flux-binding state and study
the stability of this state against the flux-unbinding fluctuation.
III.
GAUGE THEORY DESCRIPTION
The Hamiltonian (2.4) can be reformulated by the functional integral method in which the auxiliary fields will be
introduced to describe the low-lying fluctuations around
the saddle-point state in a gauge-invariant form.
It will be convenient for us to work in the imaginarytime representation. The zero-temperature results can be
In a stanrecovered through an analytic continuation.
function
dard procedure,
we express the partition
Z=Tr(e ~ ) by the functional integral formula in the
coherent states of h and
f,
Z
= [df][df
f
— (Lo+H)dr
f
][dh][dh ][dA, ]exp
(3. 1)
where
L
=gft r) f;
+QhtdQ;++A,
;
haft
f; +h; h; —1
(3.2)
&= —t+G, ;X;, —gX,;X;,
(ij
ij
(3.3)
&
in which X,, and 0; are defined in Eqs. (2.5) and (2.6) (at
a = 1/2) with the replacement of f, and h; operators by
the corresponding fields in the coherent states. In (3.3),
the term JGG ~ J5 has been neglected just for simplicity
of the formulation as usual in the literature. Including
such a term will not change our conclusion. The constraint of no double occupancy (1.3) has been enforced by
introducing the Lagrangian multiplier A, in L0.
As discussed in Sec. II, we are interested in the lowstate with
the mean-field
around
physics
energy
So we introduce the followXo=(X, ) and Go=(G,
ing auxiliary fields to decouple the Hamiltonian (3.3) in a
similar way as in Ref. 8 for the uniform RVB state. Inserting the following identity into Eq. (3. 1):
).
f dX;, dX, —X;, »(X, —X;,
= f dg; de;. dX; dX,'e
1=
',
', &(X;,
)
(,
x —x
]
(3.4)
one finds
f
Z= [df][df ][dh][dht][dk][dX][dq]
Xexp — de(r)
f
(3.5)
0
with
L=LO+
g [ —tX, C;, +iq, ,X;, +c.c. ]
*,
(i j )
(i, j)
IX;, I'
~
—
n;, X,', +in;, X;,
Even though L is not quadratic in the fields h and
(3.6)
f,
one
FLUX BINDING AND SUPERCONDUCTIVITY IN THE t-J MODEL
49
also put to the fermion part in (3.8) through the variable
ange
The Lagrangian (3.8} is invariant under the local gauge
e ', a,"+qzt A,., '
transformation
h, ~h, e ', f;
may still formally integrate out them first in Eq. (3.5) to
define an effective Lagrangian for the auxiliary fields
g;. , g;, and A, ;.. Then the saddle point we are interested in is determined
&=go, irt,
by y,
and A, =A, o with A, o and the
=t(C;~ &+J/2(y;~&
determining
chemical
potential
g, n,."= N5 and
"=(j,
=J'
~f;
g; n f =N(1 —5).
With the relaxation of the local constraint of no double
occupancy, the saddle-point energy of (3.6) is usually substantially higher than the variational energy obtained by
using the saddle-point state as the trial wave function
with the double occupancy being projected out in the
VMC. Thus we expect that the fluctuations around the
saddle point would be important to restore the constraint
effect. It has been shown by Nagaosa and Lee in the uniform RVB state that the local no double occupancy constraint could be enforced by the fluctuation of the Lagrangian multiplier field A, and the phase fluctuations of
and g,". The latter actually leads to the conservation
law of currents,
J~+ J"=0
(3.7)
for spinon and holon. That is, a spinon hopping forward
should be always accompanied by a backward hopping of
either a holon or a spinon with opposite spin. Obviously
the constraints (1.3) and (3.7) are connected to each other. This abundance in fact reflects an extra freedom of
the system related to the gauge invariance as will be
shown below.
Thus we may write down a model Lagrangian to replace (3.6) as follows:
X=gft
(t},+A, +5k, , )f; +baht(B, +A, +5k, ;)h;
gauge invariance is a direct consequence of the decomposition of c,. in Eq. (2. 1), and presumably the gauge fields
and h
M, , and a; will restore the constraint between
fields as discussed earlier. Because of this gauge invariance, one finds that different choices of 8;(1) in A f~'" satisfying the same vorticity (2.3) are all equivalent by the
gauge transformations which do not change (2.3). Thus
the uncertainty in the usual statistics transmutation has
disappeared in the present gauge-invariance formulation.
Furthermore, one may eliminate the additional gauge degree of freedom by using a gauge transformation such
that i B,8; =5k, ;. Then the "temporal component" of the
iB,O, =0. This is consistent with
gauge field 5A, , ~5k, ; —
the fact that the current conservation law (3.7) plus the
global constraint is equivalent to the local no double occupancy constraint (1.3).
The gauge field a," is different from the electromagnetic
field which by itself can propagate. The dynamics of a;
in the Lagrangian (3.8) is generated from its coupling
and boson h fields. The effective acwith the fermion
tion of a; is defined by integrating out the fermion and
boson fields,
f
f
~
—gM, ;+const —t'ge
I
&ij
i(a" +q
i(gCXt+g
"
i
"+q " g/l"h;h~
X
)
A~)
—t'g(e
(3.8)
&ij
=+1 and
mean-field
Lagrangian
at
0
ij
ij
the
is
aext
(3.9)
2 = g f t (8,+ ko)f; + gh;~(B, + ho)h, + const
)
&ij)
= —l.
"" and
is obtained by
Equation (3.
—ia,8)"
"
=
J'e
into the Lai
=yoe
g;
y;
(3.6). Here we have ignored the amplitude fluctuations of y; and g;. as well as the out-of-phase mode
between y, z and g; as usual. As pointed out above, the
quadratic fluctuations of the gauge field a," in (3.8) will
lead to the current conservation law in (3.7) (see Appendix A). The external electromagnetic field a,.'"' has also
been introduced in (3.8) with the gauge in which the temporal part is set equal to zero. Notice that a -"' can be
with qz
inserting
grangian
f [dh][dh'][df][df']e
f [dh][dht][df][dft]e
—S(a)
where
I
LCT
611
q&
—ia,
" "h,
+c.c. ) —J'g(e
h
"f; f& +c.c. ).
&)
)
(3.10}
X
Notice that
and
2
are not simply quadratic
in the
f and h. In general one could not perform the integration appeared on the right-hand side of Eq. (3.9).
fields
But if the gauge fluctuation is small so that the quadratic
fluctuation dominates, one may expand the exponential in
the right-hand nutnerator in terms of a," (and a,'"'), and
then formally carry out the integrations and reexponentiate it. To the quadratic order (in the gauge M, ;=0),
one gets
I
1
p
[a] = — drdr'
g
[[a; (r)+a ."'(r)]II,
".
~
(r, r')[a& (r')+ f"'a(r')]+
(ar)~II;
&
(r, r')a&
(r')];
(3.11)
&ij)&im)
)&'
here
II,",'~
(r, r )= (J,",' (1)J," ( r'—
5( } (;.}5(r r') ( K;"'
and the average
( .-
&
I
Lagrangian
&
(3.12}
is defined under the saddle-point
(3.10) as
=Zo
'
f [dh][dh'][dfl[df']
. .
(3. 13)
".'~ and K;"-'~ in
J,
Eq. (3.12) are the current and local
Z. Y. WENG, D. N. SHENG, AND C. S. TING
612
kinetic operators, respectively, defined by
" 'h,
J,", =it'(e
JI=iJ'
e
iq
a;. (~)=(r,. —r
—H. c. ),
h
t ~~'gf, f; .
)
49
r,
a
(3. 14b)
7
2
(3. 14a)
H—
. c.
+r-
(3. 15)
[(r, +r )/2] is defined on a new square lattice
at the center of the link (ij )
otice that for a given link
component a of a has no
(ij ), the perpendicular
definition in Eq. (3. 15). So a and a are actually defined
on two different lattices composed of the centers of the
links along x and y directions, respectively. ] Then one
may expand a as follows:
in which a
[N.
K
= —t'(e
K,/=
—J'
" 'h;
+H. c. ),
h
(3.14c)
I "gf;"f/ +H. c.
e
(3. 14d)
We note that in the usual gauge theory, the small gauge
fiuctuation is realized to 1/N order in the large Nex-pansion of the t-J model with the generalization of SU(2)
spins to SU(N) fiavors. In the present case the gauge
itself will be suppressed
in the longfluctuation
wavelength regime as shown later which would justify the
present approximation of keeping the quadratic fluctua-
tion.
To introduce the Fourier transformation,
1
SIa]= —
g[Ia
we write
(k, co„)+a'"'(k, co„)]II"&(—k,
;~ =(PNa
aa
)
'
ga
(k, co„)e
k, co
(3. 16)
where a is the lattice constant and co„=2itn/P such that
a (0)=a (P). In this momentum-frequency
representation, the action S [a ] in Eq. (3. 11) can be rewritten in the
form
co„)ta—
&(k, co„)+a&"'(k,co„)]+a (k, co„)II tt( —k, —
co„)a&(k, co„)],
kco
(3. 17)
where a (k, co„)=a
II"tI(k, co„)=
(
—k, —co„) and
6 &(K
I dre
—
0
here
J 'I(k)
'I)—
"
(J
'
(k, ~)Jtt' (k, O));
is the Fourier transformation
(3. 18)
of the current
J ' [(r;+rj )/2] defined through the link current
—
J;~ =(r, r, ) J [(r, +r, )/2] and (K" I) =(K,"/).
,
'
action given by (3.17), it is straightD t3(k, r)
the propagator
to determine
= —(T,a (k, r)att( —k, O)) for the gauge field a . In
momentum and frequency space, it has the form [setting
a'"' =0 in the action (3. 17)]
With
forward
the
D p(k, co„)= —[II"(k,co„)+III(k, co„)] p',
(3. 19)
here the symmetry relation II tI(k, co„)=IItt' ( —
k, —
co„)
has been used. The action (3.17) also determines the
response of the system to an external electromagnetic
field a'"'. Since S [a } is quadratic in
we can easily into get '
tegrate out the internal
gauge field a
a,
(3.20)
= 6S'/6a '"'
response function defined by
=K &a&. Equation (3.18) is in the same form of the
Ioffe-Larkin formula, which has been pointed out to be
a general combination rule due to the current conservation (3.7).
Notice that the gauge field a, appears in the same posias the
J
tion as A, J.' in the Lagrangian (3.8). There is a simple
picture for the role of a, here. Let us consider, for example, the fermion degree of freedom. A fermion that adiabatica11y goes through a closed loop C will gain a Berry
phase
g(a;, +q&AIJ)=pa;J+q&ir
C
c
g nj
(3.21)
leC
according to the Lagrangian (3.8). Thus it is different
form (2. 11), which is obtained in the saddle-point state,
by an extra flux. This additional flux is contributed by
the transverse part of the gauge field a; [the longitudinal
part has the form a,' =P, —
P which does not contribute
the fiux in Eq. (3.21)]. So the transverse a, provides an
extra degree of freedom for the flux to deviate from the
exact fiux-binding condition (2. 11). If the present fiuxbinding state is stable, then one expects that the transverse gauge field a," would be suppressed. Otherwise a
strong fluctuation of the transverse a,-, which represents
the effect of flux unbinding from the fermions and bosons,
will be found in the gauge-field propagator D & (3.19). In
the following, we shall discuss the properties of the boson
and fermion response functions II"' which decide the
gauge-field
propagator D & and the electromagnetic
response function E &.
IV. CHARGE-SPIN SEPARATION,
SUPERCONDUCTIVITY, AND THE RESTORATION
OF T AND P SYMMETRIES
According to the previous gauge-field formulation, the
electronic properties of the flux-binding state
low-energy
FLUX BINDING AND SUPERCONDUCTIVITY IN THE t-J MODEL
are determined
tions from the
First of all, we
the fermion and
through a combination of the contribuboson and fermion degrees of freedom.
calculate the response functions II & for
boson parts at their saddle-point states.
A. Calculation of II
&
According to the scheme outlined in Sec. III, Hf~ as
the fermion response function defined in Eq. (3.18} is
determined by the ground state of the Hamiltonian
Hf= —J'
g
e
(ij )o.
f "f; f~ +H. c.
(4. 1)
,
the total number of particles is given by
As discussed in Sec. II, A,f in
(4. 1) represents the phase contributed by the ir-flux tubes
carried by all the other fermions. The Hamiltonian (4. 1}
is thus similar to a lattice semion model in the fermion
representation with an additional spin index. The exact
for a single-species semion
diagonalization calculation
gas of a small system has shown that the lattice model exhibits a similar superfluid behavior as that found in the
continuum model. '
But so far no analytical attempt
has been given to go beyond the mean-field solution of
(4. 1) because of the complexity in the Hofstadter structure of the mean-field state. ' As an approximation, we
shall replace the Hamiltonian (4. 1) by its continuum version
in which
g;~f; f; =N(1 —5).
'
Hf'"=
1
2mf
d r
—i —qfA
r
r
(4.2)
cr
gauge field (5A, and ao"' } has been always set as zero due
to the gauge invariance, a and P in (4.5) only refer to the
space indexes x and y. e & in (4.5} is defined as
0 and e~y = —ay~ = 1 . Here the first term on
Ezz
6yy
the right-hand side represents the transverse component
which relates the transverse part of the induced current
to the transverse field. This term in (4.5) shows a
Meissner-like effect since it is finite at ~=0 and
The second term is the longitudinal part in which there is
a gapless pole at m=ufk. This longitudinal part is also
connected to the temporal part through the gaugeinvariance condition. The third term (e & term) in Eq.
(4.4) couples the transverse and longitudinal components.
It leads to the zero-field Hall effect
k~0.
Uf k
qf
4~
co
Afj= J „'dr A through the relation
—
—Im 1n(z~ —z& ) = 1r zX(r rI)
Im ln(z; —
z& )
2
r—
rI
''
I
B. Calculation of II "&
Ht,
(4.4)
continuum
counterpart
of (4. 1), one has
' and
(r) in the Hamiltonian
mf =(2J'a )
f;
(4.2).
At zero temperature the random-phase-approximation
(RPA) calculation of Iif& under the Hamiltonian (4.2) is
similar to that of a single-species semion gas. '
The
procedure of derivation is outlined in the Appendix B.
For small k and ~, we find
the
~af
=
t'
ge
k kp
k2
2
k2
k2
~
8~p
Q)q
~
~
2
2
lqf
U2k2
~~
f2k
2
f
Since in the present approach
(4.6)
(4.7)
is
related
—
g~P;, = m. for
to the lattice flux which satisfies
a plaquette. Similar to the fermion case,
given by
(4.8)
U2k2
(4.5)
and p=(1 —
5)a
the temporal part of the
W=np/mf, vf =&mplmf,
H—
The nonlocal phase A,." can be separated into two parts at
a = 1/2 in terms of Eq. (2.8),
Cis
Uf k
2
+ . c.
A,.~ in (4.7) describes the contribution of the m-flux tubes
carried by the holons. That is, the Aux enclosed in a loop
U2k2
CO
4~
2
" "h; h
(ij )
'
~f
diagrams
The response function II"& defined in Eq. (3.18) is
determined under the boson Hamiltonian
which satisfies
where
fk
which is a manifestation of the T and P broken symmetry'
in the fermion degree of freedom.
The validity of the RPA calculation for a free semion
gas has been explored recently by including certain
At zero temperature, it has
higher-order diagrams.
been found that the fractional statistics fluid is described
fairly well by the RPA. On the other hand, the prediction of the RPA at finite temperature may not be correct
due to the failure account of the vortex-antivortex effects
in the thermal excitations. In the present approach, we
are mainly concerned with the ground-state properties
(4.3)
As
U
high-order
and thus the effects involving
beyond the RPA have not been considered.
where
613
If there is no lattice-flux phase P, , H& in Eq. (4.6) wou"ld
simply describe the holon as a free semion in the boson
representation.
But P;. is certainly important here. At
first sight, one may choose a treat A;" as a perturbation at
Z. Y. WENG, D. N. SHENG, AND C. S. TING
614
small doping and expand Eq. (4.6) in the representation
of the m. -fiux state decided by P, Unfortunately, A,". alwith short-wavelength
ways mixes the long-wavelength
processes and one could not justify the effective mass expansion at the band bottom of the ~-flux state. A related
piece of evidence is the Hofstadter spectrum for particles
in the uniform lattice flux which could show quite a
different subband structure even when the flux is very
close to m flux per plaquette. '
On the other hand, since h; is a hard-core boson satisrelation, one may infying the on-site anticommutation
—h, . Then A; reads
troduce an "eon" operator e, =
H,
".
1
—
g
A;" = —
j
[0,. (l) —8
——A, .
(l)]e& e& =
(4.9)
1&i,
the Hamiltonian
(4.6) becomes a lattice
Apparently,
semion model in the eon picture. Between h and e there
is no usual hole-particle symmetry as in the Fermi-liquid
system.
Since we are interested in the small doping regime, the
hard-core condition for a holon may be negligible, while
it becomes certainly crucial in the eon representation.
This dilemma is very much like the one we have met in
the original electron representation of the t-J mode1. In
the same logic as the treatment of electron operator, we
may introduce a decomposition scheme for h, ,
h;=h, e,
(4. 10)
In Eq. (4. 10), h; and e; are defined as the conventional boson operators in which e; will absorb the phase (4.9) to
become a semion, while h; is introduced to keep the track
of the hole characteristic of (4. 10). The hard-core condition for h; will be enforced by the constraint
h;h;+e;e;=1 .
(4. 11)
The total particle numbers for h and e are N6 and
N(1 —5), respectively. Thus the behavior of h, is described as a combination of those of h; and e, . The latter
can be handled in the more controllable approximations.
Using Eq. (4. 11), one still finds Eq. (4.9). But now the
eon is described as a conventional boson subject to the
constraint (4. 11). Such a constraint could be impleinented by the current conservation law J,&+ J,-, =0 in a gauge
theory description, as discussed in Sec. III. Following a
similar procedure outlined there, the Hamiltonian (4.6) is
then replaced by two separated ones [see Eqs. (4. 13) and
(4. 14) below] describing h and e, respectively. These two
degrees of freedom will be recombined through a gauge
coupling by a model Lagrangian similar to (3.8). Correspondingly, the response function II p under the Hamiltonian (4.6), after integrating out the internal gauge field
is expressed as
—1 —1
11h
(4. 12)
[ lib 1+ 11e ]
like K & in Eq. (3.20). And II and II' in Eq. (4. 12) are
the electromagnetic response functions determined by the
following Hamiltonians
H1,
= —t'Bog h; h +H. c.
,
(4. 13)
= —t'Hope
"
'e, te, +H. c.
(4. 14)
,
)
&ij
respectively. In Eqs. (4. 13) and (4.14). 80 = ( e " "e, e~. )
and Ho = (h, h ) need to be determined self-consistently.
In the Hamiltonian
(4. 13), the boson h experiences
Bose condensation at T =0. At a suSciently low temperinteraction should sustain a
ature, the short-range
superfluid phase for h. Usually the gauge fluctuation is
believed to suppress the superfluid phase of bosons similar to the situation in an external magnetic field. h actually interacts with two types of gauge fields: One is the
internal gauge interaction between h and e. Another one
is the gauge field determined by the propagator (3. 19).
As will be shown later, these gauge fluctuations are
suppressed so that II calculated according to the unperturbed Hamiltonian (4. 13) with Bose condensation becomes meaningful. At T =0, we find
-
rr"=
ap
s-
k kp
k2
ap
pl,
+
m
k kp
k2
pl,
CO
m
CO
2
k
2mb
(4. 15)
=5a and mz =(2t'Boa )
Hamiltonian (4. 14) is equivalent to a lattice
eon
The
semion model. In the same spirit as in the previous treatment of the fermion, we shall derive II'p under the continuum version of the Hamiltonian (4. 14) which is given
as
in which pz
""=
H,COI1-
A'—
q—
)
z
(4. 16)
—r')
1
—
r' zX(r
d 2,
—r'/' e (r')e(r'),
(4. 17)
iV
d re (r)(
2m~
e(r);
here
A'(r)=
/r
=
=
' (2t'5a
with m, (2t 'H11a )
)
The continuum Hamiltonian (4. 16) as a free-semion gas
model in the boson representation has been already studand the randomied by the hydrodynamic
approach
In the RPA calculation
phase approximation (RPA).
of Mori, it has been found at T =0 that
II;(k, ~) =
~c
CO
co
II;(k, cv) =
II;~(k, co) =
at
a
small
Uq
(4. 18a)
k
1—
(4. 18b)
8 1Tp
iq„—
vqk
4~
co, k
co
limit
U~
(4. 18c)
k
with
co',
= (~/m,
)p
and
=&1rplm„where II„'„(k,=k, kz =0;~)=111(k,co),
II' (k„=k, k =0;co)=II;(k,co),
II„' (k„=k,
and
v,
k =0;co)=111z(k,co). Generally,
ten as (see Appendix B)
II'&(k, co) can be writ-
FLUX BINDING AND SUPERCONDUCTIVITY IN THE t-J MODEL
49
k kp
II'p(k, co) = 5 p—
+
11;(k,~)
k
k kp
IIi(k,
k
)+E pII;2(k, co) .
(4. 19)
Finally, II& can be obtained from Eqs. (4. 15) and (4. 19)
by using the combination rule presented in (4.12).
C. Charge-spin separation and superconductivity
The gauge-field fluctuation a is characterized by its
propagator D &(k, co) in Eq. (3.19). We have pointed out
in Sec. III that the transverse component of a actually
describes the freedom for the quantized-flux tubes to be
unbound from the fermions and bosons. According to
Hfp calculated in Secs. IVA and IVB, the transverse
propagator D p has the form
T
k kp
D,p(k, co)= — 5 p—
-
c
co
2
—
2k2
3
2
22+
—ufk
c
co
3
2
(4.20}
8'
'+p(mf '+m, '} and
with p =pzmh
~2
p
k
U2k2
—u,22
k
((or))1,
there is no pole
Since v-const when co/Uf, k
appearing in D p if co and k are small. When m-uf, k,
one could find a pole in (4.20) but near such a pole
(co —
co&, D & behaves like
col, ) '. The density of
states for such a mode vanishes fast at small k or co. Thus
the transverse gauge fluctuation described by D p is
and low-energy resuppressed in the long-wavelength
gime. A finite density of states for the transverse gauge
fluctuation may come from some higher-energy (gapped)
mode. To know the exact character of the high-energy
mode in D p, one needs to include the higher orders in
the co and k expansion, which is beyond our interest here.
The fact that there is no softened transverse gauge fluc-
-k
615
tuation proves the stability of the flux-binding state described by the phases A,f'" in Eqs. (2.7) and (2.8) at
a= 1/2. The topological properties (2.11) and (4.8) are
thus exact at a large-length scale. On the other hand, the
transverse gauge fluctuation could be still important at a
short distance or higher energy, representing the chirality
fluctuation around the saddle point. '
Recall that in the
fluxless (i.e., A fi'" = 0) state
the
high-temperature
transverse gauge fluctuation becomes so softened in the
long-wavelength that the charge and spin degrees of freeThe
dom are confined through this gauge force.
suppression of the gauge fluctuation in the present fluxbinding state implies a charge-spin separation in the
long-distance, low-energy limit. But the observable effect
of the charge-spin separation in 2D still needs to be further explored.
One may similarly discuss the longitudinal propagator
D p and find a sound mode in its pole. The longitudinal
fluctuation of a; is related to the temporal component
through a gauge transformation as discussed in Sec. III.
It is presumably important at short range in order to enforce the local constraint (1.3). But it has been argued
that the long-wavelength effect of the longitudinal gauge
fluctuation is not important due to the screening effect of
the density fluctuation.
Physically it may be undereffect mainly comes
stood as that the long-wavelength
from the antiferromagnetic
correlation, which is described either by the singular transverse gauge field in the
fluxless state or by the fluxes carried
high-temperature
by particles in the present state.
As with the suppression of the transverse gauge field,
the external electromagnetic field, which appears at the
same place as the gauge field a in the Lagrangian (3.8),
will be reduced and expelled out of the system in a similar
fashion. The electromagnetic response function K is a
combination of IIf and II" as shown in Eq. (3.20). Instead of II and II", the response function K is an experimentally measurable quantity. II and II" are 2X2 matrices whose inverse matrices are easily obtained in terms
of Eqs. (4.5), (4. 15), (4. 19), and (4. 12),
I
(Ilf
(II"
').,= 5.,—k k'kp
)
p=
k
mf
+ k'kp
p
k kp
mh
5p—
+
Ph
m,
P
mf
Q7
p
+ kakp
uf k
+ E'gp &qf
~
k2
4p
mh
+ me
~
(4.21)
N
ue
k
P
Ph
Wh
k2
4p
co
(4.22)
I
where only the leading terms at small k and co have been
kept. Notice that the e & terms in Eqs. (4.21) and (4.22)
without
only differ by a sign since qf = —1 and qh
the mass dependence. Such anomalous
e
terms
exactly
p —
—1
+II" 1 . The eleccancel each other in the sum of H
tromagnetic response function K is finally found to be
with
=+1,
L
U
K p(k, co)= 5
+
—k
p
kp
1
kz
k kp
k2 4~A.
~2 —
U2k2
mh
a2
5
a
(4.23)
mh5
+
+mf
m
1
—5
(mP+m, ')
'+(mf +m, )(1 —5)
(4.24)
(4.25)
part of K &, the first term, on the righthand side of Eq. (4.23), shows a finite value in the zerolimit k~0, which
frequency co=0 and long-wavelength
will give rise to the Meissner effect.
In fact, if only the
The transverse
4m', L
L2
4~
Z. Y. WENG, D. N. SHENG, AND C. S. TING
616
transverse part of a'"' is considered, K & will lead to the
London equation
and A, L in Eq. (4.24) corresponds to
the London penetration
depth. In the limit 6 && 1,
m&=m*, m, =m*/5 with m*=(2ta ) ' as the band
mass of the electron and one finds A, L ~ 6/2m * which is
in agreement with the experiment. The longitudinal part
of K & in Eq. (4.23) shows a gapless mode at the pole
of this sound mode is
The velocity
co = Uk.
'Qtt5/2m *m& at 5 l. As in conventional BCS
v =a
theory, including the long-range Coulomb interaction
will lift such a sound mode to a plasma mode without
changing the Meissner effect.
Finally, we consider the momentum distribution n (k)
of the electrons. n(k) is determined through the singleelectron Green's function
—ik. r,. —r. )
'
' (c, c, ) .
n(k)= iG(k—, 0 )= ge
(4.26)
«
—
(,
IJ
The electron operator c; in (4.26) can be replaced by the
decomposition (2. 1) and the equivalency is guaranteed by
the constraint (1.3). But what we have shown in the
present approach is that at a= 1/2, the charge and spin
are decoupled in the long-wavelength limit such that
(c,~c; )=(h
"h;~)(f,
e
"f;
e
)
iA'
(4.27)
In obtaining the second line above, the decomposition of
(4. 10) has been used where the boson h is condensed
without an effect on the correlation in (4.27). Each average in (4.27) is evidently gauge invariant due to the nonlocal phases A'. . In the continuum limit, A", can be
written as the line integral
h in
A',
/=
e,
f
dr. A'/(r),
(4.28)
I,
A'/ are defined in Eqs. (4. 17) and (4.3), and
&
In the
are the arbitrary paths connecting the sites i and
1D case, the correlation functions in (4.27) can be evaluBut it becomes
ated through the bosonization method.
much more difficult in the present 2D case. Some discussion about this type of correlation function in the context
of the anyon theory can be found in Ref. 50. In the following we shall calculate them in the mean-field approximation, where flux tubes will be smeared uniformly in
space and A' are then replaced by the average phases
A' as defined in (B2). Under this mean-field fiux, the
bosons and fermions will fill the lowest Landau level.
taken as a
The paths
in (4.28) are approximately
&
Thus one has
straight line connecting the sites i and
where
j.
I,
(e
e
iA
(ft e
"e, ) =(e
j.
i
e, )oe
f
'f; ) =(f f;
f
dr-A'
"i,
—if
)ve
I'
(4.29a)
.
dr A
(4.29b)
( e e, ) 0 and ( f, ) o are evaluated on the ground
states with the lowest Landau level being filled, i.e. ,
where
(4.30)
Noticing that
n ( k)
A'= A,
= Cexp[ —k
finally one finds
/k/
],
(4.31)
where k&=a 'V'2ir(1 —
5) is the Fermi momentum for
the free-electron case. Therefore, k& is still the characteristic scale for the electron distribution even in the
present strong-correlated case. The absence of a finite
jump at k = k/ in n (k) is similar to the conventional BCS
ground state. One expects that beyond the mean-field approximation the nonlocal fields A'/ in (4.28) would be
crucial to the electron's spectral function like in the 1D
the
case, where the exact forms of A' determine
correct exponents of n (k) near the singular points of k&
and 3k~.
D. Restoration of time-reversal and parity symmetries
As shown above, the response of the present fluxbinding state to an electromagnetic field is similar to that
of a BCS superconductor and there are no T and P
symmetry-broken signals shown in E & which are related
to the anomalous e & term.
A question may naturally arise: How robust is the
present cancellation of the T and P symmetry-broken
term in the electromagnetic response? In fact, if one goes
beyond the RPA calculation for II" and II or includes
higher-order fluctuations of the gauge field
it would
be hard to imagine the exactness of the cancellation unless there is a fundamental
reason. Furthermore, since
K & has been obtained only in the long-wavelength and
low-energy limit, one would wonder if such a cancellation
and high-frequency
is still true in the short-wavelength
regime. Here we point out that the restoration of the T
and P symmetries in the electronic properties is a general
result of the present flux-binding state. The local current
constraint Jh(r, t)+J/(r, t)=0 and the fiux binding are
the very reasons behind it. In the following we shall give
a heuristic discussion.
Suppose that there is a local electron current flowing
Ji", (r, t)%0 or
along the x direction, J'(r, t) = J/(r, t) = —
a,
u/(
r, t )p/( r, t ) =
ui',
(
r, t )p—i, ( r, t ),
(4.32)
where vugh(r, t) are the velocities and p/h(r, t) are the local densities for the fermion and boson. Then the local
transverse gauge fields A (r, t) and A"(r, t) produced by
the flux tubes carried by particles will generate Lorentz
forces to act on these local fermion and boson currents.
These Lorentz forces are actually responsible for the
zero-field Hall effect in the anyon system'
which
characterizes the T and P violations there. Notice
it-flux part [qP," in Eq. (4.7)]
A (r, t) including a lattice —
m. fiux
which does not contribute to the Lorentz force ( —
is equivalent to m flux by inserting a 2m. flux into each plaquette, which does not change the physics). Then
Lorentz forces on the fermion and boson currents
are found, respectively,
by F&~(r, t ) = q/u/( r, t)B/(r, t )
= —ut (r, t)itp/(r, t) and
Ff(r, t) =q„vi", (r, t)B„(r,t)
and
where
t)
(r,
Bh are the fictitious magt)irph(r,
B&
= vh
netic fields which have commensurate relations with the
local densities of particles through Eq. (2. 11) at
1/2
''
a=
49
FLUX BINDING AND SUPERCONDUCTIVITY IN THE t-J MODEL
and Eq. (4.8). Thus, according to Eq. (4.32), one finds
(4.33)
In other words, the Lorentz forces will bend the boson
and fermion currents into the same y direction. This is
forbidden by the current constraint J&~(r, t)= J—
f(r, t}
which requires the charge current and spin current going
in opposite directions. In gauge theory, the current constraint is implemented through the gauge field a . In the
present case, to enforce the constraint the fictitious electric fields eh and e~& will be produced through the gauge
field a to balance the Lorentz forces in Eq. (4.33). In the
gauge ao =0, the electric field is defined through
e~&(r, t)=eh(r,
t)=
(r,
d, a —
t),
(4.34)
in terms of the Lagrangian (3.8}. From Eqs. (4.33) and
(4.34), the only solution for Jf~ = —
Jf has to be
Ff Q(r, t}+ef $(r, t)=0,
(4.35)
such that
J&~(r, t)
= Jf(r, —
t) =0 .
(4.36)
Thus a local electron current will never experience a
transverse Lorentz-like force in the present ground state.
In other words, there is no local zero-field Hall effect apcorrelation which, in
pearing in the current-current
momentum-frequency
space, corresponds to the e & term
in K & for general k and co. As pointed out in Ref. 8, the
procedure
(4.34) —(4.35) to enforce the constraint
= f is e—
quivalent to the use of the Ioffe-Larkin combination rule in (3.20).
Therefore, we conclude that the exactness of the cancellation of T and P symmetry-broken effects relies on the
exactness of the current constraint (3.7). This result is independent of the detailed approximation.
Thus one expects that there would be no T and P symmetry-broken
signals appearing even for finite k and ~. It is consistent
with the optical measurement
as well as the pSR measurements.
In the above discussion, we have used the flux-binding
condition Bf s =npf „(r,t) where each fermion or boson
carries a a-flux tube. To make a comparison, we note
that in the CFP, 8I, will be replaced by the uniform
fictitious magnetic fields Bf I, =n. (pf h ) and locally one
no longer has the condition (4.33}. In that case, a T and
P symmetry-broken signal should be exhibited again in
the response function. In Sec. IV C, we pointed out that
in the short-distance or higher-energy
scale, the fluxbinding condition in the present state is not necessarily
exact in the sense that the transverse gauge fluctuation of
a (r, t) could become strong. This transverse gauge fluctuation represents the spin-chirality fluctuation ' which
does not break the T and P symmetries. This chirality
fluctuation may possibly be measurable through Raman
scattering ' in a higher-energy scale.
JI
J
f
617
tion. The similarity between the present 2D ground state
and the 1D ground state has been demonstrated.
For
both cases, the decoupling of the charge and spin degrees
of freedom are found. While a nonlocal phase shift
causes the Luttinger-liquid behavior in 1D, its generalization to 2D leads to superconductivity.
The profound
connection between 1D and 2D in the t-J model and the
Hubbard model needs to be further investigated through,
e.g. , the quasi-1D system. '
Since the nonlocal effect in 2D corresponds to the attachment of fictitious quantized-flux tubes to the particles
(spinons and holons), one has found a close relation of the
present flux-binding state with the anyon (semion) system. ' But we have shown that the T and P symmetrybroken signals, the characteristics of an anyon system, '
do not show up in the electron's electromagnetic response
function. This is due to the exact cancellation of such
anomalous effects between the charge and spin degrees of
freedom. The robustness of this conclusion is found to
rest on the current constraint between charge and spin as
well as the flux-binding condition. In the present theory,
the flux-binding effect has been shown to be exact in the
long-length scale due to the suppression of the transverse
gauge fluctuation. Such a topological property does not
necessarily hold at short distances or high energies where
the gauge fluctuation, representing the flux-unbinding
effect, is presumably strong. This chirality fluctuation
does not break the T and P symmetries and could be still
detectable through, e.g. , the Raman-scattering measurement.
The present flux-binding state is presumably
stable at small doping
in the t-J model.
The
commensurate-flux
phase could be regarded as its
"mean-field" version after the flux tubes are averagely
smeared out. Without the topological effect of flux binding, the T and P symmetry-broken signal would generally
show up again in the CFP at the finite doping.
The long-wavelength,
low-energy
physics of the
present flux-binding state is described by three species,
i.e., holon, spinon, and eon, which are connected by the
gauge fields to decide the electron properties. The constraints enforced by the gauge fields will be essential in
the elementary
determining
excitations in the fluxbinding state. The problem of whether there exists a
pairing order parameter is also related to the constraint
problem which, together with other superconducting
state properties, is still under investigation. On the other
hand, the charge excitation could be constructed in a
straightforward way which can lead to a simple Lagrangian
describing the charge component in the normal
state of the flux-binding phase. This Lagrangian provides
a simple and consistent explanation
of all the main
anomalous transport properties (the resistivity, the Hall
effect, and thermopower) in the normal state of the highT, copper-oxide materials, which gives strong experimental support.
'
ACKNOWLEDGMENTS
V. CONCLUSIONS
We have studied an exotic ground state of the t-J model in two dimensions based on a slave-particle decomposi-
The authors would like to thank B. Douqot, S. P.
Feng, R. B. Laughlin, P. A. Lee, T. K. Lee, G. Levine,
O. Starykh, Z. B. Su, J. Yang, and Lu Yu for helpful dis-
Z. Y. WENG, D. N. SHENG, AND C. S. TING
618
cussions. The present work is supported by a grant from
the Robert A. Welch Foundation and by the Texas
Center for Superconductivity at the University of Houston.
APPENDIX A: CONSERVATION LAW
OF LOCAL CURRENTS
X=X —g(J;1+J;J)a;J —g
(ij )
.
(K;"+K—
; )a;, (Al)
(ij )
=gf;
t'
out.
+A,
~
), the gauge field a, will be inSo one can make the variable change
(A6) such that
in'
L„=/+a'g
(K;—
"+K()A,' .
1
(A10)
1
According to Eqs. (A9a) and (A9b), we have (J, ) =0
and ( J; J~~) =0 if
or aWP. All the high-order correlations also vanish in a similar way. Since the current
correlations are always zero for different sites or different
time, one concludes that
i'
(8, +AD)f; +baht(B, +AD)h, +const
(—
e "
g
(ij )
"h, h,
—J'g(e f
(ij)
+ c.c. )
"ft f,
+c.c. ),
(A2}
=(r; —r
a
)
r;+r
(A 1 1)
APPENDIX B: THE RANDOM-PHASE
APPROXIMATION
and J;J', K;~'f are defined in Eq. (3. 14).
One may redefine the link variables on the new lattices
as introduced in, e.g. , Eq. (3. 15), that is
a;
etc. , where J;=—J;+J,
In the average (
s,"+sf=o .
where
X
(A9b)
tegrated
a, ~a,
We shall show in the following that the quadratic fluctuation of a,~ in the Lagrangian (3.8) will lead to the
current constraint (3.7).
Choosing the gauge M, ; =0 in (3.8) and expanding X to
the quadratic order of a;. (setting a,~"'=0 for simplicity),
one finds
49
The electromagnetic response function for the Hamiltonian (4.2) could be calculated by the RPA method used
Here we have an exin the single-species semion gas. '
tra spin degree of freedom.
Following Refs. 14 and 15, the vector potential Af(r)
in (4.2) could be separated into two parts
'
Af(r)= Af(r)+ —
f d r'
(A3)
2
(A4)
X
r —r'
2
haft(r')f (r') —p
(Bl)
where
1
—
BzXr
(A5)
A f (r}=
represents
&=go —a g(J, +Jf) a, —a
g
(K;
+Kf )a;
.
—
(A6)
I
I
The currents for the boson and fermion on a lattice are
defined by, respectively,
J
=J+Ka
1
(A7a)
I
sf=sf+a, fa,
-
(A7b)
.
We have put a tilde on characters to distinguish them
from those operators defined in (A4). To determine the
correlations of the currents, we may introduce an external vector potential term in X,
X „=X+a
g( J,"+Jf).A;,
such that the partition
one has
function Z becomes
(A8)
Z~.
Then
average
fictitious
magnetic
field
f (r)f (r)) =(1 —5)a ] after the fiux
smeared out uniformly in space. Since a fer-
tubes are
mion now sees the average fluxes carried by other fermions with both spins, one finds that all the fermions will
just fill up the lowest Landau level instead of the lowest
two Landau levels as in the single-species semion case'
(the degeneracy for each level and each spin is p/2).
Correspondingly the Hamiltonian (4.2) can be written
as two terms
'
H""
f =Hf +H f
(B3)
'
where Hf has a similar form as Hf" but with Af replaced by Af. H~ is a little bit complicated. Its singular
part has the form
Hff = — d rd
f
where
p=0, x,
operator
(A9a)
the
[p=(g
B=irp
The Lagrangian (Al) is then rewritten as
(82)
r'j„(r)V„(r,r')j (r'),
and y, and jo(r)=p(r) —
p.
j is defined by
—iV —qf Af
j(r)=gft)
mf
f
(r) .
(B4)
The current
FLUX BINDING AND SUPERCONDUCTIVITY IN THE t-J MODEL
49
The rest of the interaction
involves the three-body couis
pling and is negligible if the density fluctuation
small. '
The singularity in the matrix V is shown in its Fourier
transform as
'
p~2
0
mfk'
0
filled as mentioned earlier) and after some tedious algebra, finally one gets
llf. (k„=k, k =0;~)=
0
k
(2))),
which is related to the noninteracting
2)0 according to
2) = $0+2)o V2) .
1
(B9)
—Vf2I 2
—
CO
8~p
k
2Vfk
~2 —
V2
f
(B7)
Vfk
iqf—
4~
6)
(Bl 1)
Vfk
at small co, k limit with co, =(m lmf )p and Uf =&mp/mf
The main difference between Eqs. (B9)—(Bl 1) and those
obtained in a single-species semion system'
is that the
velocity vf here is reduced by &2 factor.
For a general k, we may redefine the k direction as the
x axis due to the rotational symmetry. Then
'
response function
aII
CX
CKP
aP = a
k
k
—
a—
H
k
ZZ
(B8)
zXk
Note that the response function IIf is determined
correlation in Eq. (3.18).
through the current-current
The current 3 in the continuum version is given by
—iV —qf Af
mf
f (r)
f
zXk
yy
k
zXk
f
(B9)
and (Ef) in (3.18) will be replaced by —
p/mf. The
— will contribute additional terms and
difference
only the leading ones are kept. '
Using Eq. (B8) and
the expression' of 2)o under the unperturbed Hamiltonian IIf (in the present case only the first Landau level is
'
P. W. Anderson, Science 235, 1196 (1987).
2C. Gros, R. Joynt, and T. M. Rice, Phys. Rev. B 36, 8910
(1987).
Z. Y. Weng, D. N. Sheng, C. S. Ting, and Z. B. Su, Phys. Rev.
Lett. 67, 3318 (1991);Phys. Rev. B 45, 7850 (1992).
P. W. Anderson and Y. Ren (unpublished); P. W. Anderson
and Y. Ren, in High Temperature Superconductivity,
edited
by K. S. Bedell et al. (Addison-Wesley, Reading, MA, 1990).
5S. E. Barnes, J. Phys. F 6, 1375 (1976); P. Coleman, Phys. Rev.
B 29, 3035 (1984); N. Read and D. Newns, J. Phys. C 16, 3237
(1983).
P. W. Anderson, Int. J. Mod. Phys. B 4, 181 (1990).
L. B. Ioffe and A. I. Larkin, Phys. Rev. B 39, 8988 (1989).
8N. Nagaosa and P. A. Lee, Phys. Rev. Lett. 64, 2450 (1990);
Phys. Rev. B 46, 5621 (1992).
L. B. Ioffe and P. B. Wiegmann, Phys. Rev. Lett. 65, 653
(1990); L. B. Ioffe and G. Kotliar, Phys. Rev. B 42, 10348
~
CO~
IIf„(k„=k,k =0;co) =
0
T( j„( 1)j
2)„„(1,2)= i ( —
j j
2
(B 1 0)
One notices that a Coulomb-type interaction appears in
the matrix (B6). The summation over the bubble diagrams in the RPA is similar to that in the electron gas.
Define the correlation function
(r)
N
cO
0
gf 7T
Jf(r)=gf
CO
7T
11f,(k„=k, k, =0;~) =
—i gf K
619
(1990).
Z. Y. Weng, D. N. Sheng, and C. S. Ting, Mod. Phys. Lett. B
7, 565 (1993).
H. Mori, Phys. Rev. B 46, 10952 (1992).
Z. Y. Weng, D. N. Sheng, and C. S. Ting, Phys. Lett. A 175,
455 (1993).
R. B. Laughlin, Science 242, 525 (1988); Phys. Rev. Lett. 60,
zXk
f
yX
k
(B12)
are given in Eqs.
II„ IIyy and II y
IIy
—
terms
and
In
of
(B12)
Eqs. (B9)—(Bl1),
(B9) (Bl1).
Eq.
we find Eq. (4.4).
where
2677 (1988).
A. L. Fetter, C. B. Hanna, and R. B. Laughlin, Phys. Rev. B
39, 9679 (1989).
Y. H. Chen, F. Wilczek, E. Witten, and B. I. Halperin, Int. J.
Mod. Phys. B 3, 1001 (1989).
J. March-Russell and F. Wilczek, Phys. Rev. Lett. 61, 2066
(1988); B. I. Halperin, J. March-Russell, and F. Wilczek,
Phys. Rev. B 40, 8726 (1989); X. G. Wen and A. Zee, Phys.
Rev. Lett. 62, 2873 (1989)
~7V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, 2095
(1987).
P. W. Anderson, Phys. Scr. T 27, 60 (1989); P. W. Anderson,
B. S. Shastry, and D. Hristopulos, Phys. Rev. B 40, 8939
(1989).
P. Lederer, D. Poilblanc, and T. M. Rice, Phys. Rev. Lett. 63,
1519 (1989); Y. Hasegawa, P. Lederer, T. M. Rice, and P. B.
Wiegmann, ibid. 63, 907 (1989).
X. G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B 39, 11413
(1989).
C. Gros, R. Joynt, and T. M. Rice, Z. Phys. B 68, 425 (1987).
F. C. Zhang, C. Gros, T. M. Rice, and H. Shiba, Supercond.
Sci. Technol. 1, 36 (1988).
C. Gros, Phys. Rev. B 38, 931 (1988).
24S. Liang and N. Trivedi, Phys. Rev. Lett. 64, 232 (1990).
~
25We thank
49
Z. Y. WENG, D. N. SHENG, AND C. S. TING
620
T. K. Lee for providing
us with their numerical
re-
sults.
A. M. Tikofsky, R. B. Laughlin, and Z. Zou, Phys. Rev. Lett.
69, 3670 (1992).
Y. Hasegawa, O. Narikiyo, K. Kuboki, and H. Fukuyama, J.
Phys. Soc. Jpn. 59, 822 (1990).
Z. Zou, J. L. Levy, and R. B. Laughlin, Phys. Rev. B 45, 993
(1992).
R. B. Laughlin and Z. Zou, Phys. Rev. B 41, 664 (1990); Z.
Zou and R. B. Laughlin, ibid. 42, 4073 (1990)
G. S. Canright, S. M. Girvin, and A. Brass, Phys. Rev. Lett.
63, 2291 (1989); 36, 2295 (1989).
P. B. Wiegmann, Phys. Rev. Lett. 65, 2070 (1990).
J. P. Rodriguez and B. Doucot, Phys. Rev. B 42, 8724 (1990);
45, 971 (1992); J P. Rodriguez (unpublished).
33T. W. Lawrence, A. Szoke, and R. B. Laughlin, Phys. Rev.
Lett 69, 1439 (1992).
34R. E. Kiefl et al. , Phys. Rev. Lett. 64, 2082 (1990); N. Nishida
and H. Miyatake, Hyperfine Interact. 63, 183 (1990).
35E. Dagotto, A. Moreo, F. Ortolani, D. Poilblanc, and J. Riera,
Phys. Rev. B 45, 10741 (1992).
~
~
Usually in the slave-particle decomposition, the on-site commutation relations for h; and f; are not determined, as long
as the constraint (1.3) holds. Choosing the hard-core condition for h will be important for the decomposition in the following approach.
G. Baskaran, in Two-dimensional Strong Correlation Systems,
B. Su et al. (Gordon and Breach, New York,
1988).
38E. Fradkin, Phys. Rev. Lett. 63, 322 (1989).
Y. R. Wang, Phys. Rev. B 43, 3786 (1991)
F. Wilczek, Fractional Statistics and Anyon Superconductivity
(World Scientific, Singapore, 1990).
'I. AfBeck and J. B. Marston, Phys. Rev. B 37, 3774 (1988); G.
Kotiliar, ibid. 37, 3664 (1988).
I. Aleck, Z. Zou, T. Hsu, and P. W. Anderson, Phys. Rev. B
38, 745 (1988).
D. Ceperley, G. V. Chester, and M. H. Kalos, Phys. Rev. B
16, 3081 (1976).
"4M. Ogata, B. Dougot, and T. M. Rice, Phys. Rev. B 43, 5582
edited by Z.
~
(1991).
4sQ. Dai, J. L. Levy, A. L. Fetter, C. B. Hanna, and R. B.
Laughlin, Phys. Rev. B 46, 5642 (1992).
X. G. Wen and A. Zee, Phys. Rev. B 41, 240 (1990).
47H. Mori, Phys. Rev. B 42, 184 (1990).
4sJ. R. Schrieffer, Theory of Superconductivity (Benjamin/Cummings, New York, 1964)
J. Yang, W. P. Su, and C. S. Ting, Phys. Rev. B 43, 1249
~
(1991).
5oE. Fradkin, Field Theory of Condensed Matter Systems
(Addison-Wesley, Reading, MA, 1991).
~'B. S. Shastry and B. Shrainman, Phys. Rev. Lett. 65, 1068
(1990).
~2Z.
Y. Weng, D. N. Sheng, and C. S. Ting (unpublished).