SU„2… formulation of the t-J model: Application to underdoped cuprates
Patrick A. Lee
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Naoto Nagaosa
Department of Applied Physics, University of Tokyo, Tokyo 113, Japan
Tai-Kai Ng
Department of Physics, Hong Kong University of Science and Technology, Clearwater Bay, Hong Kong
Xiao-Gang Wen
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
~Received 22 January 1997!
We develop a slave-boson theory for the t-J model at finite doping that respects an SU~2! symmetry: a
symmetry previously known to be important at half filling. The mean-field phase diagram is found to be
consistent with the phases observed in the cuprate superconductors, which contain d-wave superconductor,
spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux
phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral
function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at
high-doping concentrations. The close relation between the SU~2! and the U~1! slave-boson theory is dis-
cussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes
@which were overlooked in the U~1! theory# are identified. @S0163-1829~98!08109-0#
I. INTRODUCTION
It is well established that high-temperature superconduc-
tivity appears in cuprates when holes are doped into the par-
ent compound, which is understood to be Mott-Hubbard an-
tiferromagnetic ~AF! insulators. Since the parent compound
is insulating only by virtue of strong correlation, it stands to
reason that a strongly correlated model is the requisite start-
ing point to describe the cuprates. The simplest such model
is the two-dimensional t-J model and a large effort has been
made to study how the phase diagram evolves from a
Heisenberg antiferromagnet when a concentration x of holes
is introduced. The doping of a Mott-Hubbard insulator is a
relatively new problem in condensed-matter physics and in-
volves issues quite different from the doping of a band insu-
lator. A key question is the evolution of the Fermi surface
with doping. At low doping, the unit cell is doubled in the
AF state and the first holes will form small pockets, not
unlike the doping of band insulators. The pockets are cen-
tered at (
p
/2,
p
/2).
1
On the other hand, when the hole con-
centration is large, it is known that a large Fermi surface is
formed, with an area given by 12 x, in agreement with Lut-
tinger theorem.
2
The point is that the local moments on the
copper are now counted as part of the conduction electron
that makes up the Fermi sea. The key question is how this
evolution takes place as a function of doping. It seems quite
likely that the state for intermediate doping may contain fea-
tures not encountered before. Indeed, concepts such as quan-
tum spin liquid states and spin-charge separation were intro-
duced early on and much work has gone into the
development of a formal theory that exhibits some of these
features.
3
One line of approach is to start from mean-field
decoupling
4–7
and study fluctuations about the mean-field so-
lution, which turns out to be a U~1! gauge theory.
8–10
On the
experimental front, much work has focused on the under-
doped region, defined as the region of hole concentration
between the onset of superconductivity and the maximal T
c
,
because many anomalous properties are found in the metallic
state in this regime. For example, unlike optimally doped
systems where the magnetic susceptibility
x
and the Knight
shift are temperature independent, underdoped cuprates gen-
erally show a reduction in
x
below 400 K or so.
11
At the
same time the specific heat is found to be suppressed relative
to the T linear behavior expected for conventional metals.
12
This behavior suggests the formation of a gap in the excita-
tion spectrum. This gap also shows up in the c-axis
frequency-dependent conductivity,
13
but the conductivity in
the plane is not so strongly affected. The in-plane dc conduc-
tivity shows a suppression below about 200 K relative to the
linear T resistivity observed at higher temperatures. This
suppression can be attributed to a reduction of the width of
the Drude-like peak by a factor of 2 with little effect on the
spectral weight.
14
The reduction of the conductivity is due to
the scattering rate rather than to the carrier concentration.
15
These observations suggest that the gap appears only in the
spin and not the charge degrees of freedom in the two-
dimensional plane and has been loosely referred to as the
spin gap. We should add that the strongest gaplike behavior
has been seen in the Cu NMR relaxation rate and in neutron
scattering, both of which are sensitive to spin excitation at
momentum Q5 (
p
,
p
). This latter phenomenon usually
starts at a lower temperature of order 200 K and it has been
argued that it is observed only in bilayer or trilayer
materials.
11,16
We shall take the point of view that the behav-
PHYSICAL REVIEW B 1 MARCH 1998-IIVOLUME 57, NUMBER 10
57
0163-1829/98/57~10!/6003~19!/$15.00 6003 © 1998 The American Physical Society
ior at (
p
,
p
) may be a more delicate issue depending on
nesting properties at the Fermi surface, etc., and for the rest
of the paper we shall use the term ‘‘spin gap’’ to refer to
properties mentioned earlier that are characteristic of single-
layer as well as multilayer cuprates.
Very recently, angle-resolved photoemission experiments
have yielded important information concerning the electronic
excitations of underdoped cuprates. It was discovered that a
gap in the spectral functions already existed in the normal
state.
17,18
Furthermore, the size of this gap and its depen-
dence on k
W
space is similar to the d-wave-type gap observed
in the superconducting state. The difference is that in the
normal state, the gap appears to close in a finite segment near
(
p
/2,
p
/2), leaving a ‘‘Fermi-surface segment.’’ If this en-
ergy gap is related to the spin gap, this observation gives an
important boost to the notion of spin-charge separation. This
is because when an electron is removed from the plane, as in
photoemission and in c-axis conductivity, one is forced to
pay the energy cost to break the singlet pairs in the plane,
whereas for in-plane conductivity, charge transport may oc-
cur within the spin singlet sector. Such a behavior is in fact a
natural consequence of the mean-field phase diagram of the
t-J model that has been in existence for some time.
6,7
In this
theory the constraint of no double occupancy is enforced by
writing the electron operator c
a
i
in terms of auxiliary fermi-
ons and boson particles c
a
i
5 f
a
i
b
i
†
and demanding that each
site is occupied by either a fermion or a boson. In a mean-
field treatment, the order parameters
x
ij
5
^
f
a
i
†
f
a
j
&
and D
ij
5
^
f
1i
f
2j
2f
2i
f
1j
&
describe the formation of singlets envi-
sioned in Anderson’s resonating valence-bond ~RVB!
picture.
3
Above the Bose condensation temperature of the
bosons, spin charge separation occurs at the mean-field level.
In particular, in the underdoped regime the fermions are
paired in a d-wave state, leading to a gap in the spin excita-
tion but no gap in the charge excitation. This scenario
has been used as an explanation of the spin-gap
phenomenon.
19,20
While the conventional U~1! mean-field theory has many
attractive features, it suffers from a number of defects. First,
when an attempt was made to improve the theory by includ-
ing gauge fluctuations, it was found that the d-wave state
was unstable.
21
Second, in the underdoped regime, there are
indications that the system is unstable to the spontaneous
generation of gauge fluxes at finite wave vectors.
22
Such in-
stabilities will lead to a breaking of translation symmetry that
is not observed experimentally. We note that it has been
suggested recently that a modified d-wave state with a large
gap at the (0,
p
) point and vanishing gap along a segment
near (
p
/2,
p
/2) may be stable against gauge fluctuations.
23
However, the question about finite wave-vector instabilities
remains. Such considerations motivated us to produce a for-
mulation of the constraint that generalizes the SU~2! theory
for the half-filled t-J model to the t-J model away from half
filling.
24
Our hope is that since SU~2! gauge symmetry is an
exact symmetry at half filling, the mean-field approximation
of our formulation may capture more accurately the low-
energy degrees of freedom and may be a better starting point
for small x. Indeed, we found that in the underdoped region,
the mean-field solution may be understood as a d-wave pair-
ing state, or equivalently as a staggered-flux (s-flux! phase,
where the gauge flux alternates on even or odd sublattices.
These states are related by local SU~2! gauge transformations
and do not break translational symmetry. Furthermore, these
states are connected smoothly to the
p
-flux phase at half
filling that has large excitation energy at the (0,
p
) point,
comparable to that at (0,0). This is in agreement with pho-
toemission experiments on the insulating cuprates, suggest-
ing that the AF state may resemble the
p
-flux phase at short
distances.
25,26
Furthermore, in the experiment the state at
(0,
p
) moves towards the Fermi surface with doping, which
can be understood in the mean-field theory as a gradual clos-
ing of the spin gap. In this work
24
we also introduced a
residual attraction between the boson and fermions and show
that this gives to Fermi-surface segments near the (
p
/2,
p
/2)
point that grows with doping. Thus the SU~2! mean-field
theory allows us to answer the fundamental question of how
the Fermi surface evolves from hole pockets near the
(
p
/2,
p
/2) point near half filling to a large Fermi surface for
large-doping concentration.
In this paper we give a more detailed description of the
SU~2! theory and we also offer an alternative formulation
that has some advantage over the original SU~2! mean-field
theory, particularly in the approach to large doping. More
specifically, in Sec. II we show that the SU~2! theory is in-
timately related to the original U~1! theory. This leads us to
a formulation in terms of a
s
model of slowly varying boson
fields. This is discussed in Secs. III and IV. In Sec. V we
present detailed calculations of the electron spectral function,
comparing the original SU~2! mean-field approach and the
present
s
-model formulation. We also made some modifica-
tions of the interaction potential between fermions and
bosons, which lead to considerable improvement of the spec-
tral function when compared with experiments. In Sec. VI
we discuss the collective excitations of the theory, which are
SU~2! gauge fields, and we point out the important massless
gauge fields in different parts of the phase diagram. In par-
ticular, the existence of a massless mode in the staggered
flux phase is an important feature of the SU~2! theory com-
pared to the U~1! formulation. We also briefly discuss the
response to an electromagnetic field of the normal and super-
conducting states.
II. RELATION OF THE SU„2… FORMULATION TO U„1…
THEORY
Affleck et al.
27
pointed out that the t-J model at half fill-
ing obeys an exact SU~2! symmetry. They introduced the
SU~2! doublets
c
1i
5
S
f
1i
f
2i
†
D
,
c
2i
5
S
f
2i
2 f
1i
†
D
~1!
to represent the destruction of a spin up and spin down on
site i, respectively. This expresses the physical idea that a
physical up spin can be represented by an up-spin fermion or
the absence of a down-spin fermion once the constraint is
imposed. The theory is invariant under the local transforma-
tion
c
a
i
→g
i
c
a
i
, where g isa232 matrix representation of
the SU~2! group. In the original formulation, which we shall
refer to as the U~1! theory, this symmetry is broken upon the
introduction of holes.
6004 57LEE. NAGAOSA, NG, AND WEN
In Ref. 24 a formulation of the constraint of no double
occupation in the t-J model was introduced that preserves
the SU~2! symmetry even away from half filling. The key
step is the introduction of a doublet of bosons
h
i
5
S
b
1i
b
2i
D
~2!
on each site, so that the physical electron operator can be
written as an SU~2! singlet, i.e.,
c
1i
5
1
A
2
h
i
†
c
1i
5
1
A
2
~
b
1i
†
f
1i
1 b
2i
†
f
2i
†
!
,
c
2i
5
1
A
2
h
i
†
c
2i
5
1
A
2
~
b
1i
†
f
2i
2 b
2i
†
f
1i
†
!
. ~3!
The t-J Hamiltonian
H5
(
^
i,j
&
@
J
~
S
W
i
•S
W
j
2
1
4
n
i
n
j
!
2 t
~
c
a
i
†
c
a
j
1 H.c.
!
#
~4!
can now be written in terms of our fermion-boson ~FB!
fields. The Hilbert space of the FB system is larger than that
of the t-J model. However, the local SU~2! singlets satisfy-
ing (
1
2
c
a
i
†
t
W
c
a
i
1 b
i
†
t
W
b
i
)
u
phys
&
5 0 form a subspace that is
identical to the Hilbert space of the t-J model. On a given
site, there are only three states that satisfy the above con-
straint. They are f
1
†
u
0
&
, f
2
†
u
0
&
, and 1/
A
2(b
1
†
1b
2
†
f
2
†
f
1
†
)
u
0
&
corresponding to a spin-up electron, and a spin-down elec-
tron, and a vacancy, respectively. Furthermore, the FB
Hamiltonian, as a SU~2! singlet operator, acts within the sub-
space and has the same matrix elements as the t-J Hamil-
tonian. The projection to the physical subspace is accom-
plished by introducing a set of three auxiliary fields
a
0i
l
, l 5 1,2,3, on each site i. The partition function is
written after a standard Hubbard-Stratonovich transformation
as
Z5
E
Dh Dh
†
D
c
D
c
†
Da
W
0
DU exp
S
2
E
0
b
L
;
D
, ~5!
where the Lagrangian L
;
is given by
L
;
5
J
˜
2
(
^
ij
&
Tr
@
U
ij
†
U
ij
#
1
1
2
(
i,j,
a
c
a
i
†
~
]
t
d
ij
1J
˜
U
ij
!
c
a
j
1
(
i,l
a
0i
l
S
1
2
c
a
i
†
t
l
c
a
i
1h
i
†
t
l
h
i
D
1
(
i,j
h
i
†
@~
]
t
2
m
!
d
ij
1 t
˜
U
ij
#
h
j
. ~6!
The matrix
U
ij
5
F
2
x
ij
*
D
ij
D
ij
*
x
ij
G
, ~7!
where
x
ij
represents fermion hopping and D
ij
represents fer-
mion pairing, respectively, and J
˜
5 3J/8, t
˜
5 t/2.
28
The
density of physical holes equals the total density of bosons
^
12 c
a
i
†
c
a
i
&
5
^
h
i
†
h
i
&
5
^
b
1
†
b
1
1 b
2
†
b
2
&
5 x ~8!
and is enforced by the chemical potential
m
.
The a
0i
l
enforces the local constraint
^
1
2
c
a
i
†
t
l
c
a
i
1 h
i
†
t
l
h
i
&
5 0. ~9!
In particular, for
l 5 3 we have
^
f
a
i
†
f
a
i
1 b
1i
†
b
1i
2 b
2i
†
b
2i
&
5 1. ~10!
The Lagrangian is invariant under the local SU~2! transfor-
mation
c
a
i
→g
i
†
c
a
i
, h
i
→g
i
†
h
i
, U
ij
→g
i
†
U
ij
g
j
,
a
0i
l
t
l
→g
i
†
a
0i
l
t
l
g
i
2g
i
]
t
g
i
†
, ~11!
where g
i
(
t
)isa232 matrix that represents an SU~2! group
element.
Equations ~5! and ~6! are a faithful representation of the
t-J model, just as the more standard U~1! representation is.
The two representations must be equivalent, as long as we
include all the fluctuations. To understand the relation be-
tween the SU~2! and the U~1! theory, we will rewrite the
SU~2! theory to make it as similar to the U~1! theory as
possible. In Appendix A we will do the reverse, i.e., we will
start with the U~1! theory and write it in the form of the
SU~2! theory; we will also discuss some subtleties of the
relation.
The key ingredient is that the two-component boson field
in the SU~2! representation is nothing but an SU~2! rotation
of the standard slave boson b
i
, i.e.,
h
i
5 g
i
S
b
i
0
D
. ~12!
The matrix g
i
can be parametrized as
g
i
5
S
z
i1
2 z
i2
*
z
i2
z
i1
*
D
, ~13!
with the constraint
(
a
z
i
a
*
z
i
a
5 1, which is satisfied by the
parametrization
z
i1
5 e
2 i
~
a
/2
!
e
2 i
~
f
/2
!
cos
u
2
, z
i2
5 e
2 i
~
a
/2
!
e
i
~
f
/2
!
sin
u
2
.
~14!
It is natural to introduce the isospin vector I
W
I
W
5 z
a
*
t
W
ab
z
b
5
~
sin
u
cos
f
,sin
u
sin
f
,cos
u
!
. ~15!
Furthermore, it is easy to check that
g
i
t
3
g
i
†
5
t
W
•I
W
. ~16!
Thus I
W
has the meaning of the local quantization axis param-
etrized by the polar coordinates
u
and
f
. The angle
a
in z
i
and g
i
is redundant and can be absorbed into the phase of b
i
in Eq. ~12!. Using Eq. ~12! we can write Eqs. ~5! and ~6! as
Z5
E
Dg Db Db
†
D
c
D
c
†
Da
W
0
DU exp
S
2
E
0
b
L
8
D
,
~17!
57 6005
SU~2! FORMULATION OF THE t-J MODEL: . . .
where
L
8
5
J
˜
2
(
^
ij
&
Tr
~
U
ij
†
U
ij
!
1
1
2
(
i,j,
a
c
a
i
†
@~
]
t
1a
W
0
•
t
W
!
d
ij
1J
˜
U
ij
#
c
a
j
1
(
i,j
b
i
*
S
]
t
2
m
1
1
2
Tr
$
t
3
@
g
i
†
a
W
0
•
t
W
g
i
2
~
]
t
g
i
†
!
g
i
#
%
d
ij
1
t
˜
2
Tr
@~
11
t
3
!
g
i
†
U
ij
g
j
#
D
b
j
. ~18!
We see that the path integral of the SU~2! theory is very
similar to that of the U~1! theory. Here note that the first line
of Eq. ~18! is invariant against the local gauge transforma-
tion ~11!. To see this we transform of the integral variables
c
a
i
5 g
i
†
c
˜
a
i
, g
i
†
U
ij
g
j
5U
˜
ij
, and g
i
†
a
W
0
•
t
W
g
i
2 (
]
t
g
i
†
)g
i
5 a
˜
W
0
•
t
W
. Then Eqs. ~17! and ~18! become
Z5
E
Dg Db Db
†
D
c
˜
D
c
˜
†
Da
˜
W
0
DU
˜
exp
S
2
E
0
b
L
˜
8
D
~19!
and
L
˜
8
5
J
˜
2
(
^
ij
&
Tr
@
U
˜
ij
†
U
˜
ij
#
1
1
2
(
i,j,
a
c
˜
i
a
†
F
S
]
t
1
(
a51
3
a
˜
0
a
t
a
D
d
ij
1J
˜
U
˜
ij
G
c
˜
j
a
1
(
i
b
i
†
~
]
t
2
m
1a
˜
0
3
!
b
i
2 t
˜
(
i,j
x
ij
b
j
†
b
i
. ~20!
Note that L
˜
8
no longer depends on g so that the g integral
can be dropped. If we drop the a
0
1,2
integral, Eqs. ~19! and
~20! have the same form as the U~1! formulation with an
exception that there t
˜
is replaced by t5 2 t
˜
. It is not our
purpose to derive the exact equivalence between the U~1!
and SU~2! path integrals, but rather we want to point out
how low-lying fluctuations in the SU~2! formulation may be
reproduced in the U~1! picture.
The U~1! mean-field theory corresponds to fixing g to be
unity ~so that I
W
5 z
ˆ
) and finding U
ij
0
and a
W
0
(0)
, which mini-
mizes the action after summing over
c
and b. In the under-
doped region, it was found that U
ij
(0)
corresponds to d-wave
pairing of fermions. Thus the SU~2! symmetry at half filling
is broken by the boson term for finite x. At the same time, it
is clear that for x! 1, there is a host of U~1! mean-field states
U
ij
5g
i
†
U
ij
(0)
g
j
that are close in energy to the d-wave state.
Since these states are degenerate at x5 0, we may expect an
energy cost of order xJ per hole or x
2
J per unit cell. An
example of special interest is the staggered flux phase that
has a Dirac spectrum E
k
5
A
j
k
2
1 D
k
2
at (
p
/2,
p
/2). Since the
density of states of the Dirac spectrum is linear in energy, the
energy cost is ;
m
F
3
/DJ for a given fermion chemical poten-
tial. To satisfy the fermion number constraint,
m
F
'
A
xDJ so
that in this case the energy cost is expected to be
A
DJx
3/2
per unit cell. At finite temperatures, we expect that these
low-energy configurations should be included in the partition
function sum. This additional degree of freedom is just rep-
resented by the functional integral over g in Eq. ~17! and this
is the motivation for adopting the SU~2! formulation.
In Ref. 24 a mean-field theory was introduced for the
SU~2! action ~5! and ~6!. The mean field is a saddle point of
the action with respect to U
ij
and a
W
0
, after integrating over
c
,
c
†
and h, h
†
, which is possible because the action is qua-
dratic in these variables. We find that the mean-field phase
diagram is only slightly modified from the U~1! case and
consists of six different phases. ~i! In the staggered flux
(s-flux! phase
U
i,i1 x
ˆ
52
t
3
x
2i
~
2
!
i
x
1i
y
D,
U
i,i1y
ˆ
52
t
3
x
1i
~
2
!
i
x
1i
y
D, ~21!
and a
0i
l
5 0. In the U~1! slave-boson theory, the staggered
flux phase breaks translational symmetry. Here the breaking
of translational invariance is a gauge artifact. In
fact, a site-dependent SU~2! transformation W
i
5 exp
@
i(2 1)
i
x
1i
y
(
p
/4)
t
1
#
maps the s-flux phase to the
d-wave pairing phase of the fermions: U
i,i1 x
ˆ
,y
ˆ
52
x
t
3
6D
t
1
, which is explicitly translationally invariant. In the
s-flux phase the fermion and boson dispersions are given by
6 E
f
and 6 E
b
, where E
f
5
A
(
e
f
2 a
0
3
)
2
1
h
f
2
,
e
f
5
2 2 J
˜
(cos k
x
1cos k
y
)
x
,
h
f
522J
˜
(cos k
x
2cos k
y
)D, with a
similar result for E
b
with J
˜
replaced by t
˜
. Since ia
0
3
50we
have
^
f
a
i
†
f
a
i
&
5 1 and
^
b
1
†
b
1
&
5
^
b
2
†
b
2
&
5 x/2. ~ii! The
p
-flux
phase is the same as the s-flux phase, except here
x
5 D. ~iii!
The uniform RVB ~URVB! phase is described by Eq. ~21!
with a
0i
l
5 D5 0. ~iv! A localized spin phase has U
ij
50 and
a
0i
l
5 0, where the fermions cannot hop. ~v! The d-wave su-
perconducting ~SC! phase is described by U
i,i1 x
ˆ
,y
ˆ
52
x
t
3
6D
t
1
and a
0
3
Þ0, a
0
1,2
5 0,
^
b
1
&
Þ0, and
^
b
2
&
5 0. ~vi! The
Fermi-liquid ~FL! phase is similar to the SC phase, except
that there is no fermion pairing (D5 0).
The connection with the U~1! mean-field theory is now
clear by using Eq. ~18!. The SU~2! mean-field consists of
fixing U
ij
5U
ij
(0)
and a
W
0
5 a
W
0
(0)
. For each
$
g
i
%
the integral
over
c
,
c
†
, b,b
†
gives the free energy of a U~1! mean-field
theory with
U
ij
~
g
!
5g
i
†
U
ij
~
0
!
g
j
~22!
and
a
W
0
•
t
W
5 g
i
†
a
0
~
0
!
•
t
W
g
i
1 g
i
†
]
t
g
i
. ~23!
Upon integration over
$
g
i
%
, we see that the SU~2! mean-field
theory includes the U~1! mean-field state
$
U
ij
(0)
,a
W
0
(0)
%
and all
the configurations
$
U
ij
,a
W
0
%
connected to it by SU~2! rota-
tions. Thus, for x! 1 all the low-energy excitations are in-
cluded in the partition sum. This is the reason why we be-
lieve the SU~2! mean-field theory is a better starting point for
underdoped cuprates.
We note that with the exception of the superconducting
and Fermi liquid phases, a
W
0
(0)
50 in the SU~2! mean-field
solution. This means that
^
f
a
i
†
f
a
i
&
5 1 and the constraint ~10!
is satisfied by
^
b
1
†
b
1
&
2
^
b
2
†
b
2
&
5 0. Unlike the U~1! case, the
density of fermions is not necessarily 12 x. It is this feature
6006 57LEE. NAGAOSA, NG, AND WEN
that allows the staggered-flux and d-wave states to be gauge
equivalent descriptions in the s-flux phase, for instance. One
consequence is that the node in the gap function of the fer-
mion excitation is pinned at (
p
/2,
p
/2). In Ref. 24 it was
found that by including an attraction between the boson and
fermion due to the exchange of a
0
fluctuations, Fermi-
surfacelike features can be recovered in the physical electron
spectral weight that is shifted away from (
p
/2,
p
/2).
A similar situation appears in the URVB phase. The fer-
mion Fermi surface encloses area 1 and one must go beyond
mean-field theory to produce electron Fermi-surfacelike fea-
tures that obey the Luttinger theorem. The problem is even
more serious in the FL phase. Even though a
0
3
is now not
equal to zero, the fermion Fermi surface area approaches 1
2 x only very slowly with increasing x and decreasing tem-
perature. The FL state exists only for x>J/t, so the motiva-
tions behind the SU~2! mean-field theory are no longer ap-
plicable. Nevertheless, this observation means that the SU~2!
mean-field theory does not evolve towards the U~1! mean-
field theory in a way that is acceptable.
We believe the origin of these difficulties lies in fixing
a
W
0
(0)
as a mean-field parameter from the beginning. For a
W
0
(0)
50, the constraint is satisfied on the average by
^
h
†
t
W
h
&
5 0.
For example, this implies
^
b
1i
†
b
2i
&
5 0. Using Eqs. ~13! and
~15!, this suggests that the isospin vector I
W
is randomized so
that
^
I
W
i
&
5 0. On the other hand, as we approach the super-
conducting phase boundary T
c
from above or the Fermi-
liquid boundary from the URVB side, the boson field h
i
becomes phase coherent and we expect that it should be
slowly varying in space and time. In these regions, the short-
range correlation of the boson field is not captured by the
SU~2! mean-field theory. This motivates us to formulate an
alternative effective theory for the SU~2! partition function,
which we shall refer to as the
s
-model description.
Our strategy is to pick a mean-field configuration U
ij
(0)
and consider a slowly varying configuration h
i
in Eq. ~6! or,
equivalently, a slowly varying g
i
and b in Eq. ~18!. For each
configuration, a
W
0
is solved to satisfy the constraint locally,
after performing the integral over
c
,
c
†
. Thus, in principle a
W
0
is a functional of
$
h
W
i
%
. Our final goal is to produce an effec-
tive Lagrangian for
$
h
W
i
%
that will take the form of some
nonlinear
s
model to describe the low-energy physics of the
problem. This is the opposite limit to the SU~2! mean-field
theory: The assumption of a uniform a
W
0
is valid when the h
i
configurations are rapidly varying on the scale of the fermion
correlation length, which is of order
j
0
5
e
F
/D in the s-flux
phase. This picture is valid at high temperatures, whereas the
s
-model approach is expected to be applicable near the su-
perconducting transition and the crossover to the Fermi-
liquid state. The truth most likely lies in between the two
extreme limits in most parts of the phase diagram and it will
be of interest to explore the consequences of both limits.
It is clear that any U
˜
ij
(0)
related to U
ij
(0)
byaSU~2!gauge
transformation will give an equivalent description. Thus we
can start with any U~1! mean-field configuration. In prin-
ciple, we should optimize the parameters
x
and D at the end
of the calculation, but in practice we expect these parameters
to be not so different from that given by the U~1! mean-field
theory. We also find that a judicial choice of U
ij
(0)
that ex-
hibits the symmetry of a given phase yields a
s
model that
exhibits the proper symmetry. As a first example we discuss
the URVB state.
III.
s
MODEL OF THE FERMI LIQUID
AND THE URVB PHASES
In U~1! mean-field theory the matrix U
ij
(0)
in the URVB
state is given by U
ij
(0)
5(
0
2
x
ij
*
x
ij
0
). Here we make the choice
x
ij
5i
x
0
, so that U
ij
(0)
5i
x
0
I is proportional to the identity
element. Thus U
ij
(0)
itself is invariant under a global SU~2!
transformation.
For a
0
1
5 a
0
2
5 0 the bosons b
1
and b
2
are diagonalized by
the energy dispersion
E
b
1,2
522t
x
0
~
sink
x
1 sink
y
!
6 a
0
3
2
m
. ~24!
In the Fermi-liquid phase, the boson condenses to the bottom
of the band, located for this choice of gauge at Q
0
5 (
p
/2,
p
/2).
As explained in Ref. 24 the SU~2! mean-field theory so-
lution for the Fermi liquid is given by a
0
3
, 0 and b
1
contains
a Bose-condensed part so that
^
b
1
&
5 b
0
e
iQ
W
i
•r
W
. Note that at
finite T, thermal excitations make
^
b
2
†
b
2
&
Þ0. From Eqs. ~8!
and ~10! we see that the fermion density
K
(
a
f
i
a
†
f
i
a
L
5 12 x1 2
^
b
2
†
b
2
&
~25!
is not equal to 12 x, so that Luttinger theorem is not obeyed.
As discussed in the Introduction, this motivates us to try the
s
-model approach, where we write
h
i
5 h
˜
i
e
iQ
W
0
•r
W
~26!
and look for h
˜
i
that is slowly varying in space and
t
.Wecan
further parametrize h
˜
i
5 g
i
(
0
b
). Locally we can consider g
i
5 g
0
as constant. By introducing
c
˜
5 g
i
†
c
we see that L
8
in
Eq. ~18! takes the U~1! form
L
8
5
1
2
(
i,j,
a
c
˜
i
a
†
@~
]
t
1a
W
0
8
•
t
W
!
d
ij
1J
˜
U
ij
~
0
!
#
c
˜
i
a
1
(
i,j
b
i
*
3
S
]
t
2
m
1
1
2
Tr
@
t
3
~
a
W
0
8
•
t
W
!
#
d
ij
1
t
˜
2
Tr
@~
11
t
3
!
U
ij
~
0
!
#
D
b
j
, ~27!
where
a
W
0
8
•
t
W
5 g
i
†
a
W
0
•
t
W
g
i
2
~
]
t
g
†
!
g. ~28!
The local U~1! mean-field solution of Eq. ~26! is given by
a
W
0
8
5 a
00
z
ˆ
and a
00
is the fermion chemical potential chosen in
a way that ensures that the
c
˜
fermion density is 12 x. From
Eqs. ~26! and ~16! we find that
a
W
0i
5 a
00
I
W
~
g
i
!
. ~29!
57 6007
SU~2! FORMULATION OF THE t-J MODEL: . . .
