#  Extremely Correlated Superconductors

B Sriram<EMAIL_ADDRESS>
1Physics Department, University of California, Santa Cruz, CA, 95064

(September 2, 2021)

###### Abstract

Superconductivity in the $t$-$J$ model is studied by extending the recently
introduced extremely correlated fermi liquid theory. Exact equations for the
Greens functions are obtained by generalizing Gor’kov’s equations to include
extremely strong local repulsion between electrons of opposite spin. These
equation are expanded in a parameter $\lambda$ representing the fraction of
double occupancy, and the lowest order equations are further simplified near
$T_{c}$, resulting in an approximate integral equation for the superconducting
gap. The condition for $T_{c}$ is studied using a model spectral function
embodying a reduced quasiparticle weight $Z$ near half-filling, yielding an
approximate analytical formula for $T_{c}$. This formula is evaluated using
parameters representative of single layer High-$T_{c}$ systems. In a narrow
range of electron densities that is necessarily separated from the Mott-
Hubbard insulator at half filling, we find a typical $T_{c}$$\sim$$10^{2}$K.

Keywords The tJ Model, cuprates d-wave superconductivity, correlated Gor‘kov
equation

##  1 Introduction

The single band $t$-$J$ model Eq. (1),[1, 2], and the closely related strong
coupling Hubbard model have attracted much attention in recent years. In large
part the interest is due to the potential relevance of these models in
describing the phenomenon of High $T_{c}$ superconductivity, discovered in
cuprate materials in 1987 [3] and later, in other materials. These models lead
to a single sheet of the fermi surface, and are specified by fixing the band
hopping $t$ and the exchange energy $J$ for the $t$-$J$ model, or equivalently
$4t^{2}/U$ for the strong coupling ($U\gg t$) Hubbard model, where the
interaction is given by $V_{Hub.}=U\sum_{i}n_{i\downarrow}n_{i\uparrow}$. The
exotic possibility of superconductivity arising from such inherently repulsive
systems, is surprising from a theoretical perspective, and also challenging.
Significant theoretical work using a variety of tools on the strong coupling
Hubbard model and the extremely strong coupling $t$-$J$ models [4, 5, 6, 7, 8,
9, 11, 10, 12, 13, 14, 15, 16] has given useful insights into the role of
strong correlations in cuprate superconductivity. However given the non-
triviality of the theoretical task of  analytically solving these models,
progress in that direction has been slow.

In this work we extend the extremely correlated fermi liquid theory (ECFL)
[17, 18] recently formulated to overcome the analytical difficulties of the
strong coupling models, to include superconducting type broken symmetry. Upon
cooling the normal metallic state, a superconducting instability is expected
to arise, and our main goal in this study is to determine the conditions for
the occurrence of this state, and to provide its detailed description.

In order to motivate these calculations of the superconducting state, it is
useful to summarize the main features of ECFL theory as applied to the normal
(non-superconducting) state so far. We provide a broad overview next, further
details can be found in Ref. ([17, 18]).

The methodology developed in this theory starts with exact functional
differential equations for the various Greens function, obtained using the
Tomonaga-Schwinger approach of external potentials. These equations
incorporate the modification of the anti-commutation relations between the
fermion operators due to Gutzwiller projection (see Eq. (5)). While providing
a formally exact starting point for us, these equations are not yet amenable
to systematic approximations. The core difficulty is that an additional set of
terms arise from this modified non-canonical anticommutator structure Eq. (5).
These non-canonical terms multiply the most singular term in the equation,
namely the Dirac delta function (originating in the time derivative of the
time ordering $\Theta$ functions in the Greens functions). For an explicit
example, note the $\gamma$ term multiplying the delta function in Eq. (34).

In order to make progress, we therefore need to go beyond the established
framework of Tomonaga-Schwinger. The first development in ECFL is that the
above inconvenient feature of a non-canonical coefficient of the delta
function, is eliminated by factoring the Greens function into two parts, the
auxiliary Greens function ${\bf g}$ and the caparison function
$\widetilde{\mu}$ (see Eq. (46) and the discussion in the text following it).
The auxiliary Greens function ${\bf g}$ now satisfies a canonical equation (as
in Eq. (51) by ignoring the term involving ${\bm{f}}$), while the caparison
function $\widetilde{\mu}$ accounts for the non-canonical nature of the
original equation (as in Eq. (52)). This factorization process and the
resulting equations are exact.

As the next development, we introduce a parameter $\lambda$ in the range
$0\leq\lambda\leq 1$ into these exact equations. Setting $\lambda$$=$$0$ gives
the uncorrelated system, while $\lambda$$=$$1$ gives the exact equations of
the strongly correlated system. The $\lambda$ parameter has a formal
similarity to the expansion parameter $\frac{1}{2S}$ used in the Dyson-Maleev
(or Holstein-Primakoff) formulations [19, 20] of the spin-wave theory of
magnets. The magnetic models involve spin operators satisfying the SU(2)
(angular momentum) Lie algebra. They can be approached using different
strategies. On the one hand we may think of spins as canonical bosons with a
constraint on their occupation number $n^{b}_{i}$ at any site $i$, namely
$n^{b}_{i}=0,1,\ldots 2S$. This constraint can be implemented using a
repulsive interaction between bosons
$Un^{b}_{i}(n^{b}_{i}-1)\ldots(n^{b}_{i}-2S)$, and finally letting
$U$$\to$$\infty$. This bosonic Hubbard model is difficult to solve, since the
large energy scale $U$ makes the use of perturbation theory impractical. On
the other hand we can employ the Dyson-Maleev (or Holstein-Primakoff) non-
linear mappings to bosons, and expand the relevant Heisenberg equations of
motion in a series in $\frac{1}{2S}$. This gives an efficient way of solving
the models to considerable precision at fairly low orders in $\frac{1}{2S}$.
This latter method is parallel to the $\lambda$ expansion employed here, since
the modified anticommutators Eq. (5) also yield a (non-canonical) Lie
algebra.This analogy is discussed further in Ref. ([18]) (Sec. 6). In a
different setting, the parameter $\lambda$ can also be related to the fraction
of doubly occupied states [21] (see Appendix. A).

The parameter $\lambda$ serves two important and related objectives. Firstly
it provides a continuous path between the uncorrelated and the fully
correlated system equations. Since $0\leq\lambda\leq 1$, dialing it up from
$0$ does not involve invoking a large energy scale, unlike for example,
dialing up $U$ in the Hubbard model. This (isothermal) continuity enables the
ECFL method to retain the ideal (i.e. non-interacting) fermi surface volume at
low T. This ideal volume is expected for weakly interacting fermi systems from
the Luttinger-Ward perturbative arguments [22], and importantly, survives the
transition to extremely strongly correlated regions, as argued recently using
non-perturbative arguments [23]. Lastly, the ideal volume is also seen in
photoemission studies of overdoped and optimally doped cuprate superconductors
in the normal state [24], which provide a useful starting point for our study.

The second aspect of $\lambda$ is that it can be used to organize a systematic
power series expansion, analogous in spirit to the skeleton graph expansion of
Dyson [25] in perturbative theories. This $\lambda$ expansion can be carried
out order by order, leading to a set of successive equations that are amenable
to numerical study. A question might arise, whether a low order calculation in
this expansion can capture the strongly correlated limit. For answering this,
it is useful to examine the results for the $d=\infty$ Hubbard model at
$U=\infty$, where numerically exact results are available from the dynamical
mean field theory [26]. The $\lambda$ expansion to ${\cal O}(\lambda^{2})$ is
compared with the exact numerical result from the dynamical mean field theory
[27], in Fig. (6) of Ref. ([28]). This shows that the calculated quasiparticle
weight $Z$ vanishes upon approaching a density of $1$-particle per site, i.e.
half filling. This vanishing is a hallmark of the strong correlation limit,
where the Mott-Hubbard insulating state is realized. In the above $d=\infty$
study, and also in the case of the 2-dimensional $t$-$J$ model [29, 30, 31],
the $\lambda$ expansion describes an extremely correlated Fermi liquid state,
characterized by a small quasiparticle weight that vanishes near the Mott-
Hubbard insulator, accompanied by a rich set of low energy scales located
above the (strongly suppressed) effective fermi temperature. The ${\cal
O}({\lambda}^{2})$ equations for the normal state have been applied to
calculations of the asymmetric photoemission lines[32, 30, 31], and most
recently the calculation of the almost T-linear resistivity in single layer
cuprates[29].

In this paper we extend the above formalism to the case where superconducting
order emerges at low temperatures. This requires a non-trivial generalization
to the superconducting state of the various steps of the ECFL theory
highlighted above. In a similar fashion to the normal state, we first obtain
exact equations for the normal and anomalous Greens functions for the $t$-$J$
model. These equations generalize Gor’kov’s equations for BCS type weak
coupling superconductivity[33] by including the effect of extremely strong
local repulsion between electrons. These equations are studied further using a
specific decomposition of the Greens functions into two pieces (see Eq. (46)).
This step is followed by a systematic expansion in a parameter $\lambda$. This
leads to an set of equations Eq. (51, 52, 54), iterating these in $\lambda$ to
all orders constitutes the exact answer. In the present work, we perform a
leading order calculation.

In order to obtain explicit results, Eq. (51, 52, 54) are further simplified
near $T_{c}$ where the order parameter is small, leading to simplified
versions of these in Eq. (55, 56, 57). These are treated to ${\cal
O}({\lambda}^{2})$, and the lowest order condition for $T_{c}$ is formulated
in Eq. (68). In summary Eq. (68) is the leading order term near $T_{c}$,
within the $\lambda$ expansion, and constitutes an important formal result of
the present work. In principle it should be possible to find further
systematic equations to higher order, and also to extend the results for $T\ll
T_{c}$ following the procedure laid out here. In this work we are content to
study this first set in detail. The transition temperature is given from Eq.
(68), which is expressed in terms of the electronic Greens function,
renormalized by strong correlations. In this renormalization the short ranged
Hubbard-Gutzwiller terms are dominant, and the pairing energy causing the
instability, is provided by the much smaller exchange energy $J$. This
equation exhibits both a tendency towards an insulating state due to a
diminished quasiparticle weight, and a tendency towards superconductivity due
to the exchange term $J$. Their competing tendencies play out in Eq. (68) and
the closely related Eq. (70). These equations determine whether
superconductivity is found at all, and further identifies the model parameters
that promote it. When the superconducting state is found, they also provides
an estimate of the range of densities and temperatures which favor it.

The conditions Eq. (68, 70) are evaluated using a simple phenomenological
electronic spectral function, modeling strong correlations near half filling
in terms of a density dependent quasiparticle weight $Z$ and a wide
background. This model has the advantage of leading to an explicit analytical
formula for $T_{c}$, in terms of the various parameters of the $t$-$J$ model,
thus allowing for a thorough understanding of the role of different parameters
on the result. Evaluating this expression we find that the model supports a
d-wave superconducting phase consistent with data [36, 37], located away from
half filling. The $T_{c}$ is found to be typically $\sim$$10^{2}$K, i.e. an
order of magnitude smaller than that of the model Eq. (2) where the sole
difference from the $t$-$J$ model is that short ranged Hubbard-Gutzwiller type
correlations are ignored, in a range of densities determined by the band
parameters. The temperature-density phase diagram has the form of a tapered
tower Fig. (1). A smooth dome structure reported in cuprates, is replaced here
by a somewhat narrow density range and an exaggerated height near the peak.
The location of the peak can be varied by choosing the hopping parameters, but
always remains well-separated from the insulating limit.

The paper is organized as follows. In Section (2) we define the $t$-$J$
Hamiltonian, express it in terms of the correlated fermionic operators, and
outline the method of external potentials employed to generate the exact
dynamical equations for the electron Greens function ${\cal G}$ and the
Gor’kov anomalous Greens function ${\cal F}$. In Section (3) the equation is
expanded in $\lambda$ and further simplified near $T_{c}$. In Section (4) the
condition for $T_{c}$ is evaluated using a model spectral function. This
section contains expressions that involve only the electronic spectral
function, and might be directly accessible to readers who are more interested
in the concrete results. In Section (5) we conclude with a discussion of the
results.

##  2 Theoretical Preliminaries

The $t$-$J$ Hamiltonian [1, 2] is

$\displaystyle H_{\mbox{ tJ}}$ $\displaystyle=$ $\displaystyle
H_{\mbox{t}}+H_{\mbox{J}}$ (1) $\displaystyle H_{\mbox{t}}$ $\displaystyle=$
$\displaystyle-\sum_{ij\sigma}t_{ij}\widetilde{c}^{\dagger}_{i\sigma}\widetilde{c}_{j\sigma}-{\bm{\mu}}\sum_{i}n_{i}$
$\displaystyle H_{\mbox{J}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{ij}J_{ij}(\vec{S}_{i}.\vec{S}_{j}-\frac{n_{i}n_{j}}{4})$

where $t_{ij}$ are the band hopping matrix elements detailed below, $J_{ij}$
the nearest neighbor exchange and ${\bm{\mu}}$ the chemical potential,  with
the density operator
$n_{i}=\sum_{\sigma}\widetilde{c}^{\dagger}_{i\sigma}\widetilde{c}_{i\sigma}$,
and spin density operator
$S_{i}^{\alpha}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}\widetilde{c}^{\dagger}_{i\sigma}\tau^{\alpha}_{\sigma\sigma^{\prime}}\widetilde{c}_{i\sigma^{\prime}}$,
$\tau^{\alpha}$ is a Pauli matrix and the correlated fermi destruction
operator $\widetilde{c}_{i}$ is found from the plain (i.e. canonical or
unprojected) operators $c_{i}$, by sandwiching it between two Gutzwiller
projection operators $\widetilde{c}_{i\sigma}=P_{G}c_{i\sigma}P_{G}$, where
$P_{G}\equiv\prod_{j}(1-n_{j\uparrow}n_{j\downarrow})$ [38]. It acts by
eliminating all states with double occupancy in the state space. The creation
operators follow by taking their hermitean conjugate. The physical meaning of
this sandwiching process is that the fermi operators act within the subspace
where projector $P_{G}$ enforces single occupancy at each site. The $t$-$J$
model may be obtained by taking the large $U$ limit of the Hubbard model [1].
It has also been argued [2] to be the low energy effective Hamiltonian for an
underlying three-band model, describing the copper oxygen lattice of the
cuprate superconductors, where it is found by eliminating high energy states
of the model.

In the following work we will also find it useful to study the model

$\displaystyle H_{\mbox{unc-
tJ}}=-\sum_{ij}t_{ij}{c}^{\dagger}_{i\sigma}{c}_{j\sigma}-{\bm{\mu}}\sum_{i}n_{i}+\frac{1}{2}\sum_{ij}J_{ij}(\vec{S}_{i}.\vec{S}_{j}-\frac{n_{i}n_{j}}{4}).$
(2)

We may view it as an uncorrelated $t$-$J$ model in contrast to the correlated
version Eq. (1), here the ultra strong short ranged Hubbard-Gutzwiller
correlations with $U\gg\mbox{max}\\{|t_{ij}|\\}$ are turned off, while the
relatively weak exchange term $J\ll\mbox{max}\\{|t_{ij}|\\}$ is retained. All
operators that appear in Eq. (2), including the density and spin, are defined
by the same expression as Eq. (1) but with the unprojected fermion operators
$c_{i\sigma},c^{\dagger}_{i\sigma}$’s. In this model the exchange term, which
is usually viewed as the mechanism for antiferromagnetism, doubles up to play
the role of a superconducting pairing potential. This fruitful observation of
Anderson, Baskaran and Zou[5, 6] follows from viewing the interaction in the
crossed or Cooper channel. It is paralleled in our discussion later (see
paragraph below Eq. (30)), where the exchange term, after a rearrangement
amounting to a crossed channel, leads to a mean Cooper pair expectation in Eq.
(31). Its superconducting solution, found by standard BCS-Gor’kov meanfield
theory, is presented below (see Eqs. (74, 75)), and serves as a useful
reference point in the study of the strongly correlated $t$-$J$ model.

It is convenient for our calculations to use the operators invented by Hubbard
Ref. ([39, 40]) to represent this projection process.  Ref. ([41]) (Sec.8)
discusses the origin of difficulties of the early work employing the Hubbard
operators, in reproducing the Luttinger-Ward Fermi surface volume at low
temperatures. In contrast the present ECFL formalism achieves this goal
successfully, using continuity with the Fermi gas and the $\lambda$ expansion
described in [17, 41] and below. We denote

$\displaystyle\widetilde{c}^{\dagger}_{i\sigma}\leftrightarrow X_{i}^{\sigma
0},\;\widetilde{c}_{i\sigma}\leftrightarrow
X_{i}^{0\sigma},\;\widetilde{c}^{\dagger}_{i\sigma}\widetilde{c}_{i\sigma^{\prime}}\leftrightarrow
X_{i}^{\sigma\sigma^{\prime}}.$ (3)

These operators satisfy the following fundamental anti-commutation relations
and their adjoints:

$\displaystyle\\{X_{i}^{0\sigma_{i}},X_{j}^{0\sigma_{j}}\\}$ $\displaystyle=$
$\displaystyle 0$ (4)
$\displaystyle\\{X_{i}^{0\sigma_{i}},X_{j}^{\sigma_{j}0}\\}$ $\displaystyle=$
$\displaystyle\delta_{ij}\left(\delta_{\sigma_{i}\sigma_{j}}-\sigma_{i}\sigma_{j}X_{i}^{\bar{\sigma}_{i}\bar{\sigma}_{j}}\right),\;\;\bar{\sigma}=-\sigma.$
(5)

In physical terms, for a given site index $i$ and with $\\{a,b\\}\in$
$\\{0,\uparrow,\downarrow\\}$ limited to the three allowed initial and final
states of the projected Hilbert space, the symbol $X_{i}^{ab}$ represents an
operator representing all allowed matrix elements. To yield the correct
fermion antisymmetry, the creation operator $X_{i}^{\sigma_{i},0}$ anti-
commutes with creation or destruction operators at different sites with any
spin. In terms of these operators we can rewrite

$\displaystyle H_{t}$ $\displaystyle=$
$\displaystyle-\sum_{ij\,\sigma}t_{ij}X_{i}^{\sigma
0}X_{j}^{0\sigma}-{\bm{\mu}}\sum_{i\,\sigma}X_{i}^{\sigma\sigma}$ (6)
$\displaystyle H_{J}$ $\displaystyle=$
$\displaystyle=-\frac{1}{4}\sum_{ij\,\sigma_{i}\sigma_{j}}J_{ij}\sigma_{i}\sigma_{j}X_{i}^{\sigma_{i}\sigma_{j}}X_{j}^{\bar{\sigma}_{i}\bar{\sigma}_{j}}.$
(7)

In the following we employ a convenient repeated internal spin summation
convention. We shall follow the convention that in an equation defining any
object, often (but not always) indexed by external spin indices, all the
internal and repeated spin indices are to be summed over. As an example, we
could drop the explicit summation over spins in Eq. (6, 7), but not in Eq. (5)
where $\sigma_{i},\sigma_{j}$ are external spin indices that appear on the
left hand side. We also use a repeated internal site index below.

In order to calculate the Greens functions for this model, we add an imaginary
time $\tau$ dependent external potential (or source term) ${\cal A}$ to the
definition of thermal averages. The expectation of an arbitrary observable
$Q(\tau_{1},\ldots)$, composed e.g. of a product of several (imaginary) time
ordered Heisenberg picture operators, is written in the notation

$\displaystyle\langle\langle
Q(\tau_{1},\ldots)\rangle\rangle=Tr\;P_{\beta}\;T_{\tau}\\{e^{-{\cal
A}}Q(\tau_{1},\ldots)\\}.$ (8)

Here $T_{\tau}$ is the time-ordering operator, an external potential term
${\cal A}=\int_{0}^{\beta}d\tau{\cal A}(\tau)$, and $P_{\beta}=e^{-\beta
H}/{Tr\left(e^{-\beta H}T_{\tau}e^{-{\cal A}}\right)}$ is the Boltzmann weight
factor including ${\cal A}$. Here ${\cal A}(\tau)$ is a sum of two terms,
${\cal A}_{\rho}(\tau)$ involving a density-spin dependent external potential
${\cal V}$, and ${\cal A}_{C}(\tau)$ involving ${\cal J}$ (${\cal J}^{*}$)
Cooper pair generating (destroying) external potentials. These are given by

$\displaystyle{\cal A}_{\rho}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{i}{\cal
V}_{i}^{\sigma_{i}\sigma_{j}}(\tau)X_{i}^{\sigma_{i}\sigma_{j}}(\tau)$
$\displaystyle{\cal A}_{C}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{ij}\left({\cal
J}^{*}_{j\sigma_{j}i\sigma_{i}}(\tau)X_{i}^{0\sigma_{i}}(\tau)X_{j}^{0\sigma_{j}}(\tau)+{\cal
J}_{i\sigma_{i}j\sigma_{j}}(\tau)X_{i}^{\sigma_{i}0}(\tau)X_{j}^{\sigma_{j}0}(\tau)\right),$

where the repeated internal spin convention implies summing over
$\sigma_{i},\sigma_{j}$, and where we require the antisymmetry ${\cal
J}_{i\sigma_{i};j\sigma_{j}}=-{\cal J}_{j\sigma_{j};i\sigma_{i}}$ and likewise
for ${\cal J}^{*}$. The external potentials ${\cal J},{\cal J}^{*}$ in Eq.
(LABEL:Sources) couple to operators that add and remove Cooper pairs of
correlated electrons, and are essential to describe the superconducting phase.
At the end of the calculations, the external potentials are switched off, so
that the average in Eq. (8) reduces to the standard thermal average.
Tomonaga[42] in 1946 and Schwinger[43] in 1948 (TS) pioneered the use of such
external potentials [25, 44]. We next illustrate this technique for the
present problem.

###  2.1 Using external potentials

The advantage of introducing these external potential ( or “sources”) is that
we can take the (functional) derivatives of Greens function with respect to
the added external potentials in order to generate higher order Greens
functions. If we abbreviate the external term as ${\cal A}=\sum_{i}{\cal
U}_{j}(\tau)V_{j}(\tau)$, where ${\cal U}_{j}(\tau)$ is one of the above
c-number potential, and $V_{j}(\tau)$ is the corresponding operator in the
imaginary-time Heisenberg picture, and $Q_{i}(\tau)$ an arbitrary observable,
straightforward differentiation leads to the TS identity

$\displaystyle TrP_{\beta}T_{\tau}\\{e^{-{\cal
A}}Q_{i}(\tau^{\prime})V_{j}(\tau)\\}=\langle\langle
Q_{i}(\tau^{\prime})\rangle\rangle\;\langle\langle
V_{j}(\tau)\rangle\rangle-\frac{\delta}{\delta{\cal
U}_{i}(\tau)}\langle\langle Q_{i}(\tau^{\prime})\rangle\rangle$ (10)

This important identity can be found by taking the functional derivative of
Eq. (8) with respect to ${\cal U}_{j}(\tau)$ (see e.g. Ref. ([21]) Eq. (18)),
and is now illustrated with various choices of the external potential.

From Eq. (10) we note the frequently used result

$\displaystyle\langle\langle\sigma_{i}\sigma_{j}X_{i}^{\bar{\sigma}_{i}\bar{\sigma}_{j}}(\tau)Q(\tau^{\prime})\rangle\rangle=\left(\gamma_{\sigma_{i}\sigma_{j}}(i\tau)-{\cal
D}_{\sigma_{i}\sigma_{j}}(i\tau)\right)\langle\langle
Q(\tau^{\prime})\rangle\rangle$ (11)

where

$\displaystyle\gamma_{\sigma_{i}\sigma_{j}}(i,\tau)$ $\displaystyle=$
$\displaystyle\sigma_{i}\sigma_{j}\langle\langle
X_{i}^{\bar{\sigma}_{i}\bar{\sigma}_{j}}(\tau)\rangle\rangle$
$\displaystyle{\cal D}_{\sigma_{i}\sigma_{j}}(i,\tau))$ $\displaystyle=$
$\displaystyle\sigma_{i}\sigma_{j}\frac{\delta}{\delta{\cal
V}_{i}^{\bar{\sigma}_{i}\bar{\sigma}_{j}}(\tau)},$ (12)

The singlet Cooper pair operator is

$\displaystyle\left(X_{i}^{0\uparrow}X_{j}^{0\downarrow}-X_{i}^{0\downarrow}X_{j}^{0\uparrow}\right)=\sigma
X_{i}^{0\sigma}X_{j}^{0\bar{\sigma}},$ (13)

where summation over $\sigma$ is implied on the right hand side, and its
Hermitean conjugate

$\displaystyle-\left(X_{i}^{\uparrow 0}X_{j}^{\downarrow 0}-X_{i}^{\downarrow
0}X_{j}^{\uparrow 0}\right)=\bar{\sigma}X_{i}^{\sigma
0}X_{j}^{\bar{\sigma}0}.$ (14)

We define the (singlet) Cooper pair correlation functions at time $\tau$ as

$\displaystyle C_{ij}(\tau)=\langle\langle\sigma
X_{i}^{0\sigma}(\tau)X_{j}^{0\bar{\sigma}}(\tau)\rangle\rangle$ (15)
$\displaystyle C^{*}_{ij}(\tau)=\langle\langle\bar{\sigma}X_{i}^{\sigma
0}(\tau)X_{j}^{\bar{\sigma}0}(\tau)\rangle\rangle,$ (16)

where $\sigma$ is summed over. We note that $C^{*}_{ij}$ equals the complex
conjugate of $C_{ij}$ only after the external potentials are finally turned
off, but not so in the intermediate steps.

The basic equation Eq. (10) for the Cooper pair operators for an arbitrary
operator $Q$ are

$\displaystyle\frac{\delta}{\delta{\cal
J}^{*}_{i\sigma_{i}j\sigma_{j}}(\tau)}\langle\langle
Q\rangle\rangle=\langle\langle
X_{j}^{0\sigma_{j}}(\tau)X_{i}^{0\sigma_{i}}(\tau)\rangle\rangle\langle\langle
Q\rangle\rangle-\langle\langle
X_{j}^{0\sigma_{j}}(\tau)X_{i}^{0\sigma_{i}}(\tau)Q\rangle\rangle$ (17)
$\displaystyle\frac{\delta}{\delta{\cal
J}_{i\sigma_{i}j\sigma_{j}}(\tau)}\langle\langle
Q\rangle\rangle=\langle\langle
X_{i}^{0\sigma_{i}}(\tau)X_{j}^{0\sigma_{j}}(\tau)\rangle\rangle\langle\langle
Q\rangle\rangle-\langle\langle
X_{i}^{0\sigma_{i}}(\tau)X_{j}^{0\sigma_{j}}(\tau)Q\rangle\rangle$ (18)

From these relations the Cooper-pair correlations can be found by summing over
the spins

$\displaystyle\langle\langle\sigma
X_{i}^{0\sigma}(\tau)X_{j}^{0\bar{\sigma}}(\tau)Q\rangle\rangle=\left[C_{ij}(\tau)-{\cal
K}_{ij}(\tau)\right]\langle\langle Q\rangle\rangle$ (19)
$\displaystyle\langle\langle\bar{\sigma}X_{i}^{\sigma
0}(\tau)X_{j}^{\bar{\sigma}0}(\tau)Q\rangle\rangle=\left[C^{*}_{ij}(\tau)-{\cal
K}^{*}_{ij}(\tau)\right]\langle\langle Q\rangle\rangle$ (20)

where

$\displaystyle{\mathcal{K}}_{ij}(\tau)=\bar{\sigma}\frac{\delta}{\delta{\cal
J}^{*}_{i\sigma;j\bar{\sigma}}(\tau)}$ (21) $\displaystyle{\cal
K}^{*}_{ij}=\bar{\sigma}\frac{\delta}{\delta{\cal
J}_{i\sigma;j\bar{\sigma}}(\tau)},$ (22)

where $\sigma$ is summed over.

###  2.2 Greens functions and their dynamical equations

We are interested in the electron Greens function  (see e.g. Ref. ([21]) Eq.
(17)) expressed compactly by

$\displaystyle{\cal
G}_{i\sigma_{i}j\sigma_{f}}(\tau,\tau^{\prime})=-\langle\langle
X_{i}^{0\sigma_{i}}(\tau)X_{j}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle,$
(23)

where the Dyson time ordering $T_{\tau}$ and the external potential factor
$e^{-{\cal A}}$ are included in the definition of the brackets Eq. (8). To
describe the superconductor, following Gor’kov [33] we define the anomalous
Greens function :

$\displaystyle{\cal
F}_{i\sigma_{i}j\sigma_{f}}(\tau,\tau^{\prime})=\bar{\sigma}_{i}\langle\langle
X_{i}^{\bar{\sigma}_{i}0}(\tau)X_{j}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle$
(24)

where $\bar{\sigma}\equiv-\sigma$, and as in Eq. (23), the Dyson time ordering
$T_{\tau}$ and the external potential factor $e^{-{\cal A}}$ are included in
the definition of the brackets Eq. (8)

We note that the Cooper pair correlation functions Eq. (16), which plays a
crucial role in defining the order parameter of the superconductor, can be
expressed in terms of the anomalous Greens function using

$\displaystyle C^{*}_{ij}(\tau)$ $\displaystyle=$ $\displaystyle{\cal
F}_{i\sigma j\sigma}(\tau,\tau),$ (25)

where $\sigma$ is to be summed over, as per the convention used. We will also
need the equal time correlation of creation operators ${C}_{ij}(\tau)$ Eq.
(15). It is straightforward to show that when the external potentials ${\cal
A}$ are switched off, this object is independent of $\tau$ and can be obtained
by complex conjugation of $C_{ij}^{*}$. It is possible to add another
anomalous Greens function with two destruction operators as in Eq. (24),
corresponding to Nambu’s generalization of Gor’kov’s work. In the present
context it adds little to the calculation and is avoided by taking the complex
conjugate of ${C}^{*}_{ij}$ to evaluate ${C}_{ij}$.

####  2.2.1 Greens function ${\cal G}$

The equations for the Greens functions follow quite easily from the Heisenberg
equations, followed by the use of the identity Eq. (10), and has been
discussed extensively by us earlier. There is one new feature, concerning an
alternate treatment of the $H_{J}$ (exchange) term, necessary for describing
superconductivity described below. In this section we make use of the internal
repeated site index summation convention quite extensively.

Taking the $\tau$ derivative of ${\cal G}$ we obtain

$\displaystyle\partial_{\tau}\langle\langle
X_{i}^{0\sigma_{i}}(\tau)X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle=\delta(\tau-\tau^{\prime})\delta_{if}(\delta_{\sigma_{i}\sigma_{f}}-\gamma_{\sigma_{i}\sigma_{f}}(i\tau))$
$\displaystyle+\langle\langle[H_{t}+H_{J}+{\cal
A}(\tau),X_{i}^{0\sigma_{i}}(\tau)]\;\;X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle$
(26)

We work on the terms on the right hand side. At time $\tau$ we note

$\displaystyle[H_{t}+{\cal
A}_{\rho},X_{i}^{0\sigma_{i}}]={\bm{\mu}}X_{i}^{0\sigma_{i}}-{\cal
V}_{i}^{\sigma_{i}\sigma_{j}}X_{i}^{0\sigma_{j}}+t_{ij}(\delta_{\sigma_{i}\sigma_{j}}-\sigma_{i}\sigma_{j}X_{i}^{\bar{\sigma}_{i}\bar{\sigma}_{j}})X_{j}^{0\sigma_{j}},$
(27)

where the repeated internal indices $\sigma_{j}$ and $j$ are summed over. From
this basic commutator, using Eq. (10), Eq. (11) and the definitions Eq. (12)
we obtain

$\displaystyle\langle\langle[H_{t}+{\cal
A}_{\rho}(\tau),X_{i}^{0\sigma_{i}}(\tau)]\;\;X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle=\left({\bm{\mu}}\delta_{\sigma_{i}\sigma}-{\cal
V}_{i}^{\sigma_{i}\sigma}\right)\langle\langle
X_{i}^{0\sigma}(\tau)X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle$
$\displaystyle+t_{ij}\langle\langle
X_{j}^{0\sigma_{i}}(\tau)X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle-
t_{ij}(\gamma_{\sigma_{i}\sigma}(i,\tau)-{\cal
D}_{\sigma_{i}\sigma}(i,\tau))\langle\langle
X_{j}^{0\sigma}(\tau)X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle,$ (28)

where the repeated spin index $\sigma$, and the site index $j$ are summed
over, while $\sigma_{i},\sigma_{f}$ and site indices $i,f$ are held fixed.

For the exchange term

$\displaystyle[H_{J},X_{i}^{0\sigma_{i}}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}J_{ij}\sigma_{i}\sigma\;X_{i}^{0\sigma}X_{j}^{\bar{\sigma}_{i}\bar{\sigma}}$
(29) $\displaystyle=$
$\displaystyle-\frac{1}{2}J_{ij}\sigma_{i}X_{j}^{\bar{\sigma}_{i}0}\left(X_{i}^{0\uparrow}X_{j}^{0\downarrow}-X_{i}^{0\downarrow}X_{j}^{0\uparrow}\right),$
(30)

where the repeated internal indices $\sigma$ and $j$ are summed over. In order
to obtain Eq. (30) from Eq. (29), we used
$X_{j}^{\bar{\sigma}_{i}\bar{\sigma}}=X_{j}^{\bar{\sigma}_{i}0}X_{j}^{0\bar{\sigma}}$
and anticommuted the equal time operators
$X_{i}^{0\sigma}X_{j}^{\bar{\sigma}_{i}0}$ into
$-X_{j}^{\bar{\sigma}_{i}0}X_{i}^{0\sigma}$, followed by an explicit sum over
$\sigma$. This subtle step is essential for obtaining the superconducting
phase, as discussed (para following Eq. (2)) in the Introduction, since the
role of exchange in promoting Cooper pairs manifests itself here. Using Eq.
(19) we find

$\displaystyle\langle\langle[H_{J},X_{i}^{0\sigma_{i}}(\tau)]X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle=-\frac{1}{2}J_{ij}\sigma_{i}\left(C_{ij}(\tau^{+})-{\cal
K}_{ij}(\tau^{+})\right)\langle\langle
X_{j}^{\bar{\sigma}_{i}0}(\tau)X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle,$
(31)

where the repeated internal index $j$ is summed over, and with $\eta$ is a
positive infinitesimal we indicate here and elsewhere
$\tau^{+}\equiv\tau+\eta$ and $\tau^{-}\equiv\tau-\eta$ .

In treating this term we could have proceeded differently by sticking to Eq.
(29), using Eq. (10) with a different external potential term as in Eq. (11)
to write

$\displaystyle\langle\langle[H_{J},X_{i}^{0\sigma_{i}}(\tau)]X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle=\frac{1}{2}J_{ij}\sigma_{i}\sigma\;\langle\langle
X_{i}^{0\sigma}(\tau)X_{j}^{\bar{\sigma}_{i}\bar{\sigma}}(\tau)X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{2}J_{ij}\;\left(\gamma_{\bar{\sigma}_{i}\bar{\sigma}}(j,\tau)-{\cal
D}_{\bar{\sigma}_{i}\bar{\sigma}}(j,\tau)\right)\langle\langle
X_{i}^{0\sigma}(\tau)X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle,$

where the repeated spin index $\sigma$, and the site index $j$ are summed
over, while $\sigma_{i},\sigma_{f}$ and site indices $i,f$ are held fixed.
These two expressions Eq. (31) and Eq. (LABEL:exchange-4) are alternate ways
of writing the higher order Greens functions [45]. In order to describe a
broken symmetry solution with superconductivity, we are required to use Eq.
(31), since using the other alternative disconnects the normal and anomalous
Greens functions altogether, thereby precluding a superconducting solution.

The term $\langle\langle[{\cal
A}_{C}(\tau),X_{i}^{0\sigma_{i}}(\tau)]X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle$
generates a term that is linear in ${\cal J}$ which is treated similarly and
the final result quoted in Eq. (34).

We summarize these equations compactly by defining

$\displaystyle G^{-1}_{0i\sigma_{i}j\sigma_{j}}$ $\displaystyle=$
$\displaystyle\delta_{ij}\delta_{\sigma_{i}\sigma_{j}}\left({\bm{\mu}}-\partial_{\tau}\right)+t_{ij}\delta_{\sigma_{i}\sigma_{j}}-\delta_{ij}{\cal
V}_{i}^{\sigma_{i}\sigma_{j}}$ $\displaystyle Y_{i\sigma_{i}j\sigma_{j}}$
$\displaystyle=$ $\displaystyle t_{ij}\gamma_{\sigma_{i}\sigma_{j}}(i,\tau)$
$\displaystyle X_{i\sigma_{i}j\sigma_{j}}$ $\displaystyle=$ $\displaystyle-
t_{ij}{\cal D}_{\sigma_{i}\sigma_{j}}(i,\tau),$ (33)

and write the exact equation

$\displaystyle(G^{-1}_{0i\sigma_{i}j\sigma_{j}}-Y_{i\sigma_{i}j\sigma_{j}}-X_{i\sigma_{i}j\sigma_{j}}){\cal
G}_{j\sigma_{j}f\sigma_{f}}(\tau,\tau^{\prime})=\delta(\tau-\tau^{\prime})\delta_{if}(\delta_{\sigma_{i}\sigma_{f}}-\gamma_{\sigma_{i}\sigma_{f}}(i,\tau))$
$\displaystyle+\frac{1}{2}J_{ij}\left(C_{ij}(\tau)-{\mathcal{K}}_{ij}(\tau)\right)\;{\cal
F}_{j\sigma_{i}f\sigma_{f}}(\tau,\tau^{\prime})$ $\displaystyle+{\cal
J}_{j\sigma_{j};i\sigma_{k}}\left(\delta_{\sigma_{i},\sigma_{k}}-\gamma_{\sigma_{i}\sigma_{k}}(i,\tau)+{\cal
D}_{\sigma_{i}\sigma_{k}}(i,\tau)\right)\sigma_{j}{\cal
F}_{j\bar{\sigma}_{j}f\sigma_{f}}(\tau,\tau^{\prime}),$ (34)

where the spins $\sigma_{j},\sigma_{k}$ and the site index $j$ are summed
over, while $\sigma_{i},\sigma_{f}$ and site indices $i,f$ are held fixed. The
final term drops off when we switch off the external potential ${\cal J}$.
Viewing the spin and site indices as joint matrix indices, these equations and
their counterparts Eq. (40), are transformed into matrix equations below.

####  2.2.2 Greens function ${\cal F}$

The Gor’kov Greens function ${\cal F}$ in Eq. (24) satisfies an exact equation
that can be found as follows. First we note

$\displaystyle\partial_{\tau}\langle\langle
X_{i}^{\bar{\sigma}_{i}0}(\tau)X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle=\langle\langle[H_{t}+H_{J}+{\cal
A}(\tau),X_{i}^{\bar{\sigma}_{i}0}(\tau)]\;\;X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle$
(35)

A part of the right hand side satisfies

$\displaystyle\langle\langle[H_{t}+{\cal
A}_{\rho}(\tau),X_{i}^{\bar{\sigma}_{i}0}(\tau)]\;\;X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle=-\left({\bm{\mu}}\delta_{\sigma_{i}\sigma}-{\cal
V}_{i}^{\bar{\sigma}_{i}\bar{\sigma}}\right)\langle\langle
X_{i}^{\bar{\sigma}0}(\tau)\;\;X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle$
$\displaystyle-t_{ij}\langle\langle
X_{j}^{\bar{\sigma}_{i}0}(\tau)\;\;X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle+t_{ij}(\gamma_{\bar{\sigma}\bar{\sigma}_{i}}(i\tau)-{\cal
D}_{\bar{\sigma}\bar{\sigma}_{i}}(i\tau))\langle\langle X_{j}^{\sigma
0}(\tau);\;X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle,$ (36)

where the repeated spin index $\sigma$, and the site index $j$ are summed
over, while $\sigma_{i},\sigma_{f}$ and site indices $i,f$ are held fixed. The
exchange term is treated similarly to Eq. (29)

$\displaystyle[H_{J},X_{i}^{\bar{\sigma}_{i}0}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}J_{ij}\left(X_{i}^{\uparrow 0}X_{j}^{\downarrow
0}-X_{i}^{\downarrow 0}X_{j}^{\uparrow 0}\right)\sigma_{i}X_{j}^{0\sigma_{i}}$
(37)

so that using Eq. (20) we get

$\displaystyle\langle\langle[H_{J},X_{i}^{\bar{\sigma}_{i}0}]\;X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle=-\frac{1}{2}J_{ij}\sigma_{i}\left({C}^{*}_{ij}(\tau^{-})-{\cal
K}^{*}_{ij}(\tau^{-})\right)\langle\langle
X_{j}^{0\sigma_{i}}(\tau)X_{f}^{\sigma_{f}0}(\tau^{\prime})\rangle\rangle,$

where the repeated internal index $j$ is summed over

We gather and summarize these equation in terms of the variables that are
“time-reversed” partners of Eq. (34) and hence denoted with hats:

$\displaystyle\widehat{G}^{-1}_{0i\sigma_{i}j\sigma_{j}}=\delta_{ij}\delta_{\sigma_{i}\sigma_{j}}\left({\bm{\mu}}+\partial_{\tau}\right)+t_{ij}\delta_{\sigma_{i}\sigma_{j}}-\delta_{ij}{\cal
V}_{i}^{\bar{\sigma}_{i}\bar{\sigma}_{j}}$
$\displaystyle\widehat{Y}_{i\sigma_{i}j\sigma_{j}}=t_{ij}\gamma_{\bar{\sigma}_{j}\bar{\sigma}_{i}}(i,\tau)$
$\displaystyle\widehat{X}_{i\sigma_{i}j\sigma_{j}}=-t_{ij}{\cal
D}_{\bar{\sigma}_{j}\bar{\sigma}_{i}}(i,\tau)$ (39)

So that

$\displaystyle\left(\widehat{G}^{-1}_{0i\sigma_{i}j\sigma_{j}}-\widehat{Y}_{i\sigma_{i}j\sigma_{j}}-\widehat{X}_{i\sigma_{i}j\sigma_{j}}\right){\cal
F}_{j\sigma_{j}f\sigma_{f}}(\tau,\tau^{\prime})=-\frac{1}{2}J_{ij}\left({C}^{*}_{ij}-{\cal
K}^{*}_{ij}\right){\cal G}_{j\sigma_{i}f\sigma_{f}}(\tau,\tau^{\prime})$
$\displaystyle+\sigma_{i}\sum_{m}{\cal
J}^{*}_{i\bar{\sigma}_{n}m\sigma_{m}}(\delta_{\sigma_{i},\sigma_{n}}-\gamma_{\bar{\sigma}_{n}\bar{\sigma}_{i}}(i,\tau)+{\cal
D}_{\bar{\sigma}_{n}\bar{\sigma}_{i}}(i,\tau)){\cal
G}_{m\sigma_{m}f\sigma_{f}}(\tau,\tau^{\prime})$ (40)

where the repeated spin indices $\sigma_{j},\sigma_{n},\sigma_{m}$ and site
index $j$ are summed over, while $\sigma_{i},\sigma_{f}$ and $i,f$ are held
fixed. The final term arising from $\langle\langle[{\cal
A}_{C},X_{i}^{\bar{\sigma}_{i}0}]X_{f}^{\bar{\sigma}_{f}0}(\tau^{\prime})\rangle\rangle$
drops off when we switch off the external potential ${\cal J}^{*}$.

####  2.2.3 Summary of Equation in Symbolic Notation

The equations Eq. (34) and Eq. (40) are exact in the strong correlation limit.
Noting that all terms containing $\gamma$ and ${\cal D}$ in Eq. (34) and Eq.
(40) arise from Gutzwiller projection, we obtain the corresponding equations
for the uncorrelated $t$-$J$ model in Eq. (2) by dropping these terms. Recall
also that the external potentials ${\cal J,J}^{*}$ represent the imposed
symmetry-breaking terms that force superconductivity, and are meant to be
dropped at the end. In this uncorrelated case, let us understand the role of
the terms with the Cooper pair derivatives ${\cal K,K}^{*}$. If we ignore
these terms and also set ${\cal J,J}^{*}\to 0$ right away, the equations Eq.
(34) and Eq. (40) reduce to the Gor’kov mean-field equations for the
uncorrelated model [33], with the equation Eq. (25) providing a self
consistent determination of $C_{ij}^{*}$ in terms of ${\cal F}$. Thus by
neglecting the terms with ${\cal K,K}^{*}$, the role of the exchange $J$ is
confined to providing the lowest order electron-electron attraction in the
Cooper channel. This amounts to neglecting the ${\cal O}(J^{2})$ dressings of
the electron self energies and irreducible interaction i.e. the pairing kernel
in Eq. (64). When retained, the normal state studies (see Ref. ([31]) Figs.
(22,23,24-(a))) show that the self energy terms arising from $J$ change the
spectral functions of the model only slightly. Regarding the irreducible
interaction in the superconducting channel, the ${\cal O}(J)$ term is already
attractive. Since we are in the regime of $J\ll\mbox{max}\\{|t_{ij}|\\}$ the
retained term is expected to dominate the neglected higher order term.  In
summary, strong Hubbard-Gutzwiller type short ranged interactions renormalize
the Greens function to ${\cal G}$ from $G_{0}$, and the self energy terms due
to $J$ are minor[17, 31]. The role of $J$ is significant  only insofar as it
provides a mechanism for superconducting pairing,  and potentially magnetic
instabilities close to half filling. Keeping these considerations in mind, we
drop the terms involving ${\cal K,K}^{*},{\cal J,J}^{*}$ in Eq. (34) and Eq.
(40). This suffices for our initial goal, of generalizing a Gor’kov type[33]
mean-field treatment of Eq. (2) to the strongly correlated problem Eq. (1).

Multiplying the $\gamma$ and ${\cal D}$ terms, or equivalently the $X$ and $Y$
terms with $\lambda$ and expanding the resulting equations systematically in
this parameter constitutes the $\lambda$-expansion that we discuss below.

With these remarks in mind we make the following changes to the equations Eq.
(34) and Eq. (40):

  1. (i)

We drop the terms proportional to ${\cal J,J}^{*}$ and the corresponding
derivative terms ${\cal K,K}^{*}$.

  2. (ii)

Defining the gap functions:

$\displaystyle\Delta_{ij}=\frac{1}{2}J_{ij}C_{ij}\;\;\mbox{and}\;\;\Delta^{*}_{ij}=\frac{1}{2}J_{ij}{C}^{*}_{ij}$
(41)

  3. (iii)

We scale the each occurrence of $\gamma,X,Y$,$\widehat{X},\widehat{Y}$ by
$\lambda$.

With these changes we write the modified Eq. (34) and Eq. (40):

$\displaystyle(G^{-1}_{0i\sigma_{i}j\sigma_{j}}-\lambda
Y_{i\sigma_{i}j\sigma_{j}}-\lambda X_{i\sigma_{i}j\sigma_{j}}){\cal
G}_{j\sigma_{j}f\sigma_{f}}$
$\displaystyle=\delta(\tau-\tau^{\prime})\delta_{if}(\delta_{\sigma_{i}\sigma_{f}}-\lambda\gamma_{\sigma_{i}\sigma_{f}}(i,\tau))+\Delta_{ij}\;{\cal
F}_{j\sigma_{i}f\sigma_{f}}$ (42)
$\displaystyle\left(\widehat{G}^{-1}_{0i\sigma_{i}j\sigma_{j}}-\lambda\widehat{Y}_{i\sigma_{i}j\sigma_{j}}-\lambda\widehat{X}_{i\sigma_{i}j\sigma_{j}}\right){\cal
F}_{j\sigma_{j}f\sigma_{f}}=-\Delta^{*}_{ij}{\cal
G}_{j\sigma_{i}f\sigma_{f}},$ (43)

where $\sigma_{j}$ is summed over in both Eq. (42) and Eq. (43). Note that the
self consistency condition Eq. (16) and Eq. (25) fix the correlation functions
$C$’s in terms of ${\cal F}$. As $\lambda\to 0$ we get back the meanfield
equations of Gor’kov for the uncorrelated-J model. The $\lambda$ parameter
governs the density of doubly occupied states, and hence a series expansion in
this parameter builds in Gutzwiller type correlations systematically. We
expand the Greens functions to required order in $\lambda$ and finally set
$\lambda=1$.

We write Eq. (42) and Eq. (43) symbolically as

$\displaystyle({\bf g}_{0}^{-1}-\lambda Y-\lambda X).{\cal G}$
$\displaystyle=$ $\displaystyle(\bm{1}-\lambda\gamma)+\Delta.{\cal F}$ (44)
$\displaystyle(\widehat{{\bf
g}}_{0}^{-1}-\lambda\widehat{Y}-\lambda\widehat{X}).{\cal F}$ $\displaystyle=$
$\displaystyle-\Delta^{*}.{\cal G}$ (45)

where the symbols ${\cal G},{\cal F}$ etc are regarded as matrices in the
space, spin and time variables, $\bm{1}$ is the Dirac delta function in time
and a Kronecker delta in space and spin, with the dot indicating matrix
multiplication or time convolution. In the case of $X,\widehat{X}$ it also
indicates taking the necessary functional derivatives.

##  3 Expansion of the Equations in $\lambda$

We decompose of both Greens functions in Eq. (44) and Eq. (45) as

$\displaystyle{\cal G}={\bf g}.\widetilde{\mu},\;\;{\cal
F}={\bm{f}}.\widetilde{\mu}$ (46)

where $\widetilde{\mu}$ is a function of spin, space and time that is common
to both Greens function. As an example of the notation, the equation ${\cal
G}={\bf g}.\widetilde{\mu}$ stands for ${\cal
G}_{i\sigma_{i}j\sigma_{j}}(\tau_{i},\tau_{j})=\sum_{k\sigma_{k}}\int_{0}^{\beta}d\tau_{k}\;{\bf
g}_{i\sigma_{i}k\sigma_{k}}(\tau_{i},\tau_{k})\;\widetilde{\mu}_{k\sigma_{k}j\sigma_{j}}(\tau_{k},\tau_{j})$.
Here $\widetilde{\mu}$ is called the caparison (i.e. a further dressing)
function, in a similar treatment of the normal state Greens function. The
terms ${\bf g}$ and ${\bm{f}}$ are called the auxiliary Greens function. The
basic idea is that this type of factorization can reduce Eq. (44), to a
canonical type equation fo ${\bf g}$, where the terms $\bm{1}-\lambda\gamma$
is replaced by $\bm{1}$. We remark that this is a technically important step
since the term $\bm{1}-\lambda\gamma$ modifies the coefficient of the delta
function in time, and encodes the distinction between canonical and non-
canonical fermions.

To simplify further, we note that $X$ contains a functional derivative with
respect to ${\cal V}$, acting on objects to its right. When acting on a pair
of objects, e.g. $X.{\cal G}=X.{\bf g}.\widetilde{\mu}$, we generate two
terms. One term is $(X.g).\widetilde{\mu}$, where the bracket, temporarily
provided here, indicates that the operation of $X$ is confined to it. The
second term has the derivative acting on $\widetilde{\mu}$ only, but the
matrix product sequence is unchanged from the first term. We write the two
terms together as

$\displaystyle X.{\bf g}.\widetilde{\mu}$ $\displaystyle=$
$\displaystyle\wick{\c{X}.\c{\GH}.\widetilde{\mu}}+\wick{\c{X}.{\bf
g}.\c{\wmu}},$ (47)

so that the ‘contraction’ symbol refers to the differentiation by $X$, and the
‘.’ symbol refers to the matrix structure. We may view this as the Leibnitz
product rule.

Let us now operate with $X$ on the identity ${\bf g}.{\bf g}^{-1}={\bf 1}$,
where ${\bf g}^{-1}$ is the matrix inverse of ${\bf g}$. Using the Leibnitz
product rule, we find

$\displaystyle\wick{\c{X}.\c{\GH}}=-\left(\wick{\c{X}.\bf{g}.\c{\bf
g}}^{-1}\right).{\bf g}$ (48)

and hence we can rewrite Eq. (47) in the useful form

$\displaystyle X.{\bf g}.\widetilde{\mu}$ $\displaystyle=$
$\displaystyle-\left(\wick{\c{X}.\bf{g}.\c{\bf g}}^{-1}\right).{\bf
g}+\wick{\c{X}.{\bf g}.\c{\wmu}}.$ (49)

With this preparation we rewrite Eq. (44) the equation for ${\cal G}$ as

$\displaystyle({\bf g}_{0}^{-1}-\lambda
Y+\lambda\left(\wick{\c{X}.\bf{g}.\c{\bf g}}^{-1}\right)).{\bf
g}.\widetilde{\mu}=(\bm{1}-\lambda\gamma)+\Delta.{\bm{f}}.\widetilde{\mu}+\lambda\wick{\c{X}.{\bf
g}.\c{\wmu}}$ (50)

We now choose ${\bf g},{\bm{f}}$ such that

$\displaystyle({\bf g}_{0}^{-1}-\lambda
Y+\lambda\left(\wick{\c{X}.\bf{g}.\c{\bf g}}^{-1}\right)).{\bf g}={\bf
1}+\Delta.{\bm{f}}.$ (51)

Substituting Eq. (51) into Eq. (50), we find that $\widetilde{\mu}$ satisfies
the equation

$\displaystyle\widetilde{\mu}=(\bm{1}-\lambda\gamma)+\lambda\wick{\c{X}.{\bf
g}.\c{\wmu}}.$ (52)

Note that Eq. (51) has the structure of a canonical equation since we replaced
the $\bm{1}-\lambda\gamma$ term by $\bm{1}$ in Eq. (50). Thus the non-
canonical Eq. (44) for ${\cal G},{\cal F}$ is replaced by a pair of canonical
equations for ${\bf g},\widetilde{\mu}$. In Eq. (51) we note that the action
of $X$ is confined to the bracket $\lambda\left(\wick{\c{X}.\bf{g}.\c{\bf
g}}^{-1}\right)$, unlike the term $\lambda X.{\cal G}$ in the initial Eq. (44)
. We may thus view the term in bracket in Eq. (51) as a proper self energy for
${\bf g}$.

For treating the equation for ${\cal F}$ Eq. (45) we use the same scheme Eq.
(46) and find

$\displaystyle\widehat{X}.{\cal
F}=\widehat{X}.{\bm{f}}.\widetilde{\mu}=-\left(\wick{\c{\wh{X}}.{\bm{f}}.\c{\f}^{-1}}\right).{\bm{f}}.\widetilde{\mu}+\wick{\c{\wh{X}}.{\bm{f}}.\c{\wmu}}$
(53)

With this we rewrite Eq. (45) after cancelling an overall right multiplying
factor $\widetilde{\mu}$

$\displaystyle(\widehat{{\bf
g}}_{0}^{-1}-\lambda\widehat{Y}+\lambda\wick{\c{\wh{X}}.{\bm{f}}.\c{\f}^{-1}}).{\bm{f}}$
$\displaystyle=$ $\displaystyle-\Delta^{*}.{\bf
g}+\lambda\wick{\c{\wh{X}}.{\bm{f}}.\c{\wmu}}.\widetilde{\mu}^{-1}$ (54)

Summarizing we need to solve for ${\bm{f}},{\bf g},\widetilde{\mu},\Delta^{*}$
from Eqs. (51,52,54) by iteration in powers of $\lambda$.

###  3.1 Simplified Equations near $T_{c}$

For the present work, we note that the equation Eq. (54) simplifies
considerably, if we work close to $T_{c}$. In this regime ${\bm{f}}$ may be
assumed to be very small, enabling us to throw away all terms of ${\cal
O}(f^{2})$ and also to discard terms of ${\cal O}(\lambda f)$. This truncation
scheme is sufficient to determine $T_{c}$ for low orders in $\lambda$.

When $T\sim T_{c}$, throwing away terms of ${\cal O}(f^{2})$ and ${\cal
O}(\lambda f)$, we obtain the simplified version of Eq. (54)

$\displaystyle{\bm{f}}=-\widehat{{\bf g}}_{0}.\Delta^{*}.{\bf
g}+o(\lambda{\bm{f}}),$ (55)

so that Eq. (51) can be written as

$\displaystyle{\bf g}^{-1}={\bf g}_{0}^{-1}-\lambda
Y+\lambda\left(\wick{\c{X}.\bf{g}.\c{\bf g}}^{-1}\right)+\Delta.\widehat{{\bf
g}}_{0}.\Delta^{*}$ (56)

In this limit the above two are the ${\cal O}(\lambda^{2})$ equations required
to be solved, together with Eq. (52) and the self consistency condition Eq.
(41), Eq. (25). The latter can be combined with Eq. (24) as

$\displaystyle\Delta_{ij}^{*}$ $\displaystyle=$
$\displaystyle\frac{1}{2}J_{ij}{C}^{*}_{ij}=-\frac{1}{2}J_{ij}\sum_{\sigma}{\cal
F}_{i\sigma,j\sigma}(\tau^{+},\tau)$ (57)

and further reduced using Eq. (46). On turning off the external potentials we
regain time translation invariance. We next perform a fourier transform to
fermionic Matsubara frequencies $\omega_{n}=\frac{\pi}{\beta}(2n+1)$ using the
definition ${\cal F}(\tau)=\frac{1}{\beta}\sum_{n}e^{-i\omega_{n}\tau}{\cal
F}(i\omega_{n})$, and write Eq. (46) in the frequency domain as

$\displaystyle{\cal
F}_{p\sigma}(i\omega_{n})={\bm{f}}_{p\sigma}(i\omega_{n})\widetilde{\mu}_{p\sigma}(i\omega_{n}).$
(58)

Thus taking spatial fourier transforms with the definition

$\displaystyle J(q)=2J\left(\cos q_{x}+\cos q_{y}\right),$ (59)

so that the self consistency condition Eq. (57) finally reduces to

$\displaystyle\Delta^{*}(k)$ $\displaystyle=$
$\displaystyle-\frac{1}{2\beta}\sum_{p\sigma\omega_{n}}J(k-p){\bm{f}}_{p\sigma}(i\omega_{n})\widetilde{\mu}_{p\sigma}(i\omega_{n})$
(60)

We may write Eq. (55) as

$\displaystyle{\bm{f}}_{p\sigma}(i\omega_{n})=-\widehat{{\bf
g}}_{0\sigma}(p,i\omega_{n})\Delta^{*}(p){\bf g}_{\sigma}(p,i\omega_{n})$ (61)

where the time reversed free Greens function

$\displaystyle\widehat{{\bf
g}}_{0}(p,i\omega_{n})=\frac{1}{-i\omega_{n}+{\bm{\mu}}_{0}-\varepsilon_{-p}}=\frac{1}{-i\omega_{n}-\xi_{p}}$
(62)

with $\xi=\varepsilon_{p}-{\bm{\mu}}_{0}$ and by using
$\varepsilon_{p}=\varepsilon_{-p}$, and ${\bm{\mu}}_{0}$ is taken as the non-
interacting system chemical potential, discarding the corrections of $\mu$ due
to $\lambda$. Therefore Eq. (60) becomes

$\displaystyle\Delta^{*}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2\beta}\sum_{p\sigma\omega_{n}}J(k-p)\widehat{{\bf
g}}_{0\sigma}(p,i\omega_{n})\Delta^{*}(p){\bf
g}_{\sigma}(p,i\omega_{n})\widetilde{\mu}_{p\sigma}(i\omega_{n})$

Here ${\bf g}$ is taken from Eq. (56), i.e. the ${\cal O}(\lambda^{2})$ Greens
function with a small correction (for $T\sim T_{c}$) from the gap $\Delta$.
Performing the spin summation and recombining ${\bf g}.\widetilde{\mu}={\cal
G}$, we get the equation in terms of the physical electron Greens function

$\displaystyle\Delta^{*}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\sum_{p\omega_{n}}J(k-p)\Delta^{*}(p)\widehat{{\bf
g}}_{0}(p,i\omega_{n}){\cal G}(p,i\omega_{n}).$ (64)

This is an important result of our formalism, it represents the leading order
Gor‘kov equation for the $t$-$J$ model. It is analogous to a refinement of
Gor‘kov’s equation [33], usually called the Eliashberg equation [34], valid
for strong electron-phonon coupling superconductivity. Our $\lambda$ expansion
plays the role of the Migdal theorem [35] in that problem. The analogy with
Migdal [35] and Eliashberg’s [34] work is only superficial, since the strongly
correlated problem does not share the physics of the separation of the
electronic and phonon time scales, underlying those results.

In Eq. (64) the physical electron Greens function ${\cal G}$ is taken from the
${\cal O}(\lambda^{2})$ theory if we neglect the corrections from the gap,
which vanishes above $T_{c}$ anyway. We express the physical Greens function
in terms of its spectral function $A(p,\nu)$

$\displaystyle{\cal G}(p,i\omega_{n})=\int
d\nu\;\frac{A(p,\nu)}{i\omega_{n}-\nu}$ (65)

The frequency integral in Eq. (57) can be performed as

$\displaystyle\frac{1}{\beta}\sum_{\omega_{n}}\widehat{{\bf
g}}_{0}(p,i\omega_{n}){\cal G}(p,i\omega_{n})=\int
d\nu\;A(p,\nu)\frac{1-f(\nu)-f(\xi_{p})}{\nu+\xi_{p}}.$ (66)

where $f$ is the fermi distribution $f(\nu)=1/(1+\exp{\beta\nu})$. Hence

$\displaystyle\Delta^{*}(k)=\sum_{p}J(k-p)\Delta^{*}(p)\;\int
d\nu\;A(p,\nu)\frac{1-f(\nu)-f(\xi_{p})}{\nu+\xi_{p}}.$ (67)

In summary this eigenvalue type equation for $\Delta^{*}(k)$, together with
the spectral function $A(p,\nu)$ determined from the ${\cal O}(\lambda^{2})$
Greens function in Eq. (56), gives the self-consistent gap near $T_{c}$. At
sufficiently high temperatures, i.e. in the normal state $T>T_{c}$
$\Delta^{*}$ vanishes, so that $A$ is independent of $\Delta^{*}$. In this
case Eq. (67) reduces to a linear integral equation for $\Delta^{*}$. We may
then determine $T_{c}$ from the condition that the largest eigenvalue crosses
1. For this purpose we only need the normal state electron spectral function
of the strongly correlated metal.

##  4 Estimate of $T_{c}$

###  4.1 Equation for determining $T_{c}$

The condition for obtaining a d-wave superconducting state is given by setting
$T=T_{c}^{+}$ in Eq. (67) writing $\Delta^{*}(k)=\Delta_{0}(\cos k_{x}-\cos
k_{y})$, using the normal state spectral function for $A$ and canceling an
overall factor $\Delta_{0}(\cos k_{x}-\cos k_{y})$. Following these steps we
get

$\displaystyle 1=J\sum_{p}\left\\{\cos(p_{x})-\cos(p_{y})\right\\}^{2}\;\int
d\nu\;\frac{1-f(\nu)-f(\varepsilon_{p}-\mu_{0})}{\nu+\varepsilon_{p}-\mu_{0}}A(p,\nu)\bigg{|}_{T_{c}}.$
(68)

Instead of working with Eq. (68), it is convenient to make a useful
simplification for the average over angles. Since Eq. (68) is largest when
$\vec{p}$ is on the fermi surface, we factorize the two terms and write

$\displaystyle 1$ $\displaystyle=$ $\displaystyle J\Psi(\mu_{0})\;\Gamma$ (69)
$\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle\sum_{p}\int
d\nu\;\frac{1-f(\nu)-f(\varepsilon_{p}-\mu_{0})}{\nu+\varepsilon_{p}-\mu_{0}}A(p,\nu)\bigg{|}_{T_{c}}$
(70)

where $\Gamma$ is a particle-particle type susceptibility. Here
$\Psi(\mu_{0})$ is more correctly the weighted average of
$\left\\{\cos(p_{x})-\cos(p_{y})\right\\}^{2}$ with a weight function that is
the integrand in Eq. (70). We simplify it to the fermi surface averaged
momentum space d-wavefunction

$\displaystyle\Psi(\mu_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{n(\mu_{0})}\sum_{p}\left\\{\cos(p_{x})-\cos(p_{y})\right\\}^{2})\delta(\varepsilon_{p}-\mu_{0})$
(71)

where $n(\epsilon)$ is the band density of states (DOS) per spin and per site,
at energy $\epsilon$,

$\displaystyle n(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{1}{N_{s}}\sum_{p}\delta(\varepsilon_{p}-\epsilon).$ (72)

Using this simplification and performing the angular averaging over the energy
surface $\varepsilon_{\vec{p}}=\epsilon$ we write the (particle-particle)
susceptibility $\Gamma$ (Eq. (70)) as

$\displaystyle\Gamma=\int\;d\epsilon\int
d\nu\;n(\epsilon)\;A(\epsilon,\nu)\;\frac{1-f(\nu)-f(\epsilon-\mu_{0})}{\nu+\epsilon-\mu_{0}}\bigg{|}_{T_{c}}.$
(73)

where $A(\epsilon,\nu)$ is the angle-averaged version of the spectral function
$A(p,\nu)$. We estimate this expression below for the extremely correlated
fermi liquid, by using a simple model for the spectral function $A$.

In Eq. (73) if we replace the spectral function $A$ by the (fermi gas) non-
interacting result $A_{0}(\epsilon,\mu_{0})=\delta(\nu-\epsilon+\mu_{0})$, we
obtain the Gorkov-BCS mean-field theory, where the susceptibility $\Gamma$
reduces to $\int
d\epsilon\;n(\epsilon)\;\frac{\tanh\frac{1}{2}\beta_{c}(\epsilon-\mu_{0})}{2(\epsilon-\mu_{0})}$.
This is evaluated by expanding around the fermi energy, and utilizing the low
T formula
$\int_{0}^{W_{0}}\frac{d\epsilon}{\epsilon}\,{\tanh\frac{1}{2}\beta_{c}\epsilon}\sim\log\left[\frac{\zeta_{0}W_{0}}{k_{B}T_{c}}\right]$,
where $W_{0}$ is the half-bandwidth and $\zeta_{0}=1.13387\ldots$. Equating
$\Gamma$ to $1/J\Psi(\mu_{0})$ gives the d-wave superconducting transition
temperature for the uncorrelated $t$-$J$ model

$\displaystyle k_{B}T_{c}^{(un)}$ $\displaystyle\sim$ $\displaystyle
1.134\;W_{0}\;e^{-\frac{1}{g}},$ (74)

with the superconducting coupling constant

$\displaystyle g$ $\displaystyle=$ $\displaystyle J\Psi(\mu_{0})n(\mu_{0}).$
(75)

###  4.2 Model spectral function

We next use a simple model spectral function to estimate these integrals. It
has the great advantage that we can carry out most integrations analytically
and get approximate but closed form analytical expressions for $T_{c}$, which
provide useful insights. The model spectral function contains the following
essential features of strong correlations namely:

  * •

A quasiparticle part with fermi liquid type parameters, where the
quasiparticle weight $Z$ goes to 0 at half filling $n=1$, and

  * •

A wide background.

The model spectral function used is in the spirit of Landau’s fermi liquid
theory[46, 47, 48] with suitable modifications due to strong correlation
effects[17]. We take the spectral function as

$\displaystyle
A(\epsilon,\nu)=Z\delta(\nu-\frac{m}{m^{*}}\epsilon)+(1-Z)\frac{1}{2W_{0}}\Theta(W_{0}-|\nu|).$
(76)

Here $\Theta(x)=\frac{1}{2}(1+\frac{x}{|x|})$, $W_{0}$ the half-bandwidth
$\frac{m}{m^{*}}$ is the renormalized effective mass of the fermions, and $Z$
is the fermi liquid renormalization factor. The first term is the
quasiparticle part with weight $Z$, and second part represents the background
modeled as an inverted square-well. Integration over $\nu$ gives unity at each
energy $\epsilon$. $Z$ is chosen to reflect the fact that we are dealing with
a doped Mott-Hubbard insulator so it must vanish at $n=1$. For providing a
simple estimate we use Gutzwiller’s result [38, 49]

$\displaystyle Z=1-n.$ (77)

The effective mass is related to $Z$ and the k-dependent Dyson self energy
$\Sigma$ through the standard fermi liquid theory[46, 47, 48] formula

$\displaystyle\frac{m}{m^{*}}=Z\times(1+\frac{\partial\Sigma(\vec{k},\mu)}{\partial\varepsilon_{k}}\bigg{|}_{k_{F}}).$
(78)

The Landau fermi liquid renormalization factor $\frac{m}{m^{*}}$ can be
inferred from heat capacity experiments provided the bare density of states is
assumed known.

Using Eq. (76) in Eq. (73) and decomposing the susceptibility $\Gamma$ into a
quasiparticle and background part, the equation determining $T_{c}$ is:

$\displaystyle\left(\Gamma_{QP}+\Gamma_{B}\right)\bigg{|}_{T\to T_{c}}$
$\displaystyle=$ $\displaystyle\frac{1}{J\Psi(\mu_{0})}$ (79)
$\displaystyle\Gamma_{QP}$ $\displaystyle=$ $\displaystyle Z\int
d\epsilon\,n(\epsilon)\frac{1-f(\epsilon-\mu_{0})-f(\frac{m}{m^{*}}(\epsilon-\mu_{0}))}{(\epsilon-\mu_{0})(1+\frac{m}{m^{*}})}$
(80) $\displaystyle\Gamma_{B}$ $\displaystyle=$
$\displaystyle\frac{(1-Z)}{2W_{0}}\int
d\epsilon\,n(\epsilon)\int_{-W_{0}}^{W_{0}}d\nu\frac{1-f(\epsilon-\mu_{0})-f(\nu)}{(\epsilon-\mu_{0})+\nu}.$

Using the same approximations that lead to Eq. (74) the $\Gamma_{QP}$ can be
evaluated as

$\displaystyle\Gamma_{QP}$ $\displaystyle=$
$\displaystyle\frac{Zn(\mu_{0})}{1+\frac{m}{m^{*}}}\int_{0}^{W_{0}}\frac{d\epsilon}{\epsilon}\
\left(\tanh\frac{\epsilon}{2k_{B}T}-\tanh\frac{\epsilon\,m/m^{*}}{2k_{B}T}\right),$
(82)

and hence at low enough $T$ the estimate

$\displaystyle\Gamma_{QP}\sim
n(\mu_{0})\frac{2Z}{1+\frac{m}{m^{*}}}\log\left[\frac{\zeta_{0}W_{0}\sqrt{\frac{m}{m^{*}}}}{k_{B}T}\right].$
(83)

Unlike the quasiparticle part with this $\log T$ behavior at low T, the
background part is nonsingular as $T\to 0$, since a double integral over the
region of small $\epsilon-\mu_{0}$ and $\nu$ is involved. It can be estimated
by setting $T=0$, $\epsilon-\mu_{0}\sim\epsilon$ and replacing
$n(\epsilon)\sim n(\mu_{0})$. With

$\displaystyle\Gamma_{B}$ $\displaystyle\equiv$ $\displaystyle
n(\mu_{0})\gamma_{B},$ (84) $\displaystyle\gamma_{B}$ $\displaystyle=$
$\displaystyle\frac{(1-Z)}{2W_{0}}\int_{-W_{0}}^{W_{0}}\int_{-W_{0}}^{W_{0}}\frac{1}{2}\frac{\mbox{sign}(\epsilon)+\mbox{sign}(\nu)}{\epsilon+\nu}\,d\epsilon\,d\nu.$
(85)

Integrating this expression we obtain

$\displaystyle\gamma_{B}=(1-Z)\log 4.$ (86)

Combining Eqs. (79,83,86) we find

$\displaystyle k_{B}T_{c}$ $\displaystyle\sim$ $\displaystyle
1.134\;W_{0}\times\sqrt{\frac{m}{m^{*}}}\times e^{-\frac{1}{g_{eff}}}$ (87)

where the effective superconducting coupling:

$\displaystyle g_{eff}$ $\displaystyle=$
$\displaystyle\frac{2Z}{\left(1+\frac{m}{m^{*}}\right)}\left\\{J_{eff}\Psi(\mu_{0})n(\mu_{0})\right\\}$
(88)

and an effective exchange

$\displaystyle J_{eff}=\frac{J}{1-\gamma_{B}J\Psi(\mu_{0})n(\mu_{0})},$ (89)

where the denominator represents an enhancement due to the background spectral
weight. In comparing Eq. (87) with the uncorrelated result Eq. (74) several
changes are visible. The bandwidth prefactor is reduced by correlations due to
the factor of $\sqrt{\frac{m}{m^{*}}}\ll 1$. This factor vanishes as $n\to 1$
thereby diminishing superconducting $T_{c}$ in the close proximity of the
insulator. A similar but even more drastic effect arises from multiplying
factor $\frac{2Z}{(1+\frac{m}{m^{*}})}$ in the coupling $g_{eff}$ Eq. (88).
This term reflects the quasiparticle weight in the pairing process, and
vanishes near the insulating state. Being situated in the exponential, it
kills superconductivity even more effectively than the bandwidth prefactor.
Away from the close proximity of the insulator other terms in $g_{eff}$ become
prominent, allowing for the possibility of superconductivity. Amongst them is
the replacement of the exchange energy by $J_{eff}$. In a density range where
$\Psi(\mu_{0})n(\mu_{0})$ is appreciable, this enhances $J_{eff}$ over $J$ due
to the feedback nature of Eq. (89), and has an important impact on determining
the phase region with superconductivity.

###  4.3 Numerical Estimates of $T_{c}$

We turn to the task of estimating the order of magnitude of the Tc in this
model. When we take typical values for cuprate systems: $W_{0}\sim 10^{4}$K
(i.e $\sim$1 eV) and $J\sim 10^{3}$K (i.e. $\sim$0.1 eV), the transition
temperature of the uncorrelated model $T_{c}^{(un)}$ Eq. (74) is a few
thousand K, at most densities. It remains robustly non-zero at half filling,
since in this formula correlation effects are yet to be built in and the Mott-
Hubbard insulator is missing. For the correlated system, we estimate $T_{c}$
from Eq. (87) using similar values of model parameters. The terms arising from
correlations in Eq. (87) are guaranteed to suppress superconductivity near the
insulating state, since $Z\to 0$ and the quasiparticle is lost. A more refined
question is whether an intermediate density regime ($\delta>0$) can support
superconductivity. And if so, whether the temperature scales are robust enough
to be observable. Within the context and confines of the simplified model
spectral function considered, we answer both questions positively here.

####  4.3.1 Choice of model parameters

In order to estimate the order of magnitude of the Tc, its dependence on $J$
and band parameters, we choose parameters similar to those used in
contemporary studies for the single layer High $T_{c}$ compound
$La_{2-x}Sr_{x}CuO_{4}$. The hopping Hamiltonian
$-\sum_{ij}t_{ij}\widetilde{C}_{i\sigma}^{\dagger}\widetilde{C}_{j\sigma}$,
gives rise to band energy dispersion $\varepsilon(\vec{k})=-2t(\cos k_{x}+\cos
k_{y})-4t^{\prime}\cos k_{x}\cos k_{y}-2t^{\prime\prime}(\cos 2k_{x}+\cos
2k_{y})$ on a square lattice. Thus the hopping amplitudes $t_{ij}$ are equal
to $t$ when $i,j$ are nearest neighbors, $t^{\prime}$ when $i,j$ are second-
nearest neighbors, and $t^{\prime\prime}$ when $i,j$ are third-nearest
neighbors. For this system we will use the values [50, 14]

$\displaystyle t=0.45\,eV,\;\;t^{\prime}/t=-0.16\pm
0.02,\;\;t^{\prime\prime}/t=.01.$ (90)

This parameter set is roughly consistent with the experimentally determined
fermi surface of $La_{2-x}Sr_{x}CuO_{4}$ [50], we comment below on
considerations leading to a more precise choice. The tight binding band
extends from $-W_{0}\leq\epsilon\leq W_{0}$, where $W_{0}=4t$, neglecting a
small shift due to $t^{\prime}$. The exchange energy is chosen to be

$\displaystyle J/t=0.3,\;\;\mbox{or}\;\;J/k_{B}\sim 1550K,$ (91)

as determined from two magnon Raman experiments[51] on the parent insulating
$La_{2}CuO_{4}$. Note that the $t$-$J$ model is obtainable from the Hubbard
model by performing a large $U/t$ super-exchange expansion, giving
$J=\frac{4t^{2}}{U}$. Thus our choice of $J$ corresponds to a strong coupling
type magnitude of $U/t\sim 13.3$ in the Hubbard model, placing it in a
perturbatively inaccessible regime of that model.

We now discuss the enhancement of effective mass $\frac{m}{m^{*}}$ [52]. In
the proximity of the Mott-Hubbard insulating state $n\to 1$, an enhancement in
$\frac{m^{*}}{m}$ is expected on general grounds, reflecting a diminished
thermal excitation energy scale due to band narrowing. For illustrating the
role of this parameter we use two complementary estimates

$\displaystyle\frac{m}{m^{*}}$ $\displaystyle\sim$ $\displaystyle
3.4\;(1-n),\;\;\mbox{(a)}$ (92) $\displaystyle\frac{m}{m^{*}}$
$\displaystyle=$ $\displaystyle Z,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{(b)}$
(93)

where estimate (a) gives a two-fold enhancement of $\frac{m^{\star}}{m}$ at
$\delta=.15$, while estimate (b), obtained by neglecting the $k$ dependence of
the self energy in Eq. (78), gives a seven-fold enhancement. The formulas used
are simple enough so that the effect other estimates for $\frac{m}{m^{*}}$
should be easy enough for the reader to gauge.

####  4.3.2 Results

Figure 1: The superconducting transition temperature for the correlated model
$T_{c}$ (Eq. (87))
($t^{\prime}/t=-0.159,\;t^{\prime\prime}/t=0.01,\;\frac{m}{m^{*}}=Z$). The
scale of the maximum transition temperature is smaller by an order of
magnitude from the uncorrelated model. As the insulator is approached
$\delta\to 0$, and $T_{c}$ decreases drastically. This is easy to understand
since the quasiparticle weight $Z$ shrinks on approaching the insulating
state, killing the coupling $g_{eff}$ Eq. (88). When $\delta$ goes beyond the
peak (optimum) value, the effective superconducting coupling $g_{eff}$ agains
falls off as seen in Fig. (2) and in Fig. (5) due to the other factors in Eq.
(88). When $g_{eff}$ drops below $\sim 0.12$, the resulting $T_{c}$ is
negligible.

In Fig. (1) the superconducting transition temperature for d-wave symmetry is
shown as a function of the hole density $\delta=1-n$ where the band parameters
are indicated in the caption. It shows that $T_{c}$ is maximum at $\delta\sim
0.15$ and falls off rapidly as one moves away from that density in either
direction. The scale of $T_{c}$ is a few hundred K, which is an order of
magnitude lower than that of the uncorrelated system. The small kink-like
features to the right of the peak reflect structure in the DOS shown as inset
in Fig. (2).

Figure 2: The figure and inset use $t^{\prime}/t=-0.159$,
$t^{\prime\prime}/t=0.01$, and $\frac{m}{m^{*}}=Z$. The effective
superconducting coupling $g_{eff}$ (Eq. (88)) for three Cooper pair
symmetries: (i)(blue) d-wavefunction
$\langle\left\\{\cos(k_{x})-\cos(k_{y})\right\\}^{2}\rangle_{FS}$, (ii)(brown)
extended s-wavefunction
$\langle\left\\{\cos(k_{x})+\cos(k_{y})\right\\}^{2}\rangle_{FS}$, and
(iii)(magenta) $s+id$ wavefunction
$\langle\left\\{\cos^{2}(k_{x})+\cos^{2}(k_{y})\right\\}\rangle_{FS}$. For the
d-wavefunction, the drastic decrease of $T_{c}$ on both sides of the peak
values in Fig. (1) can be understood by referring to the the second y-axis,
giving the temperature scale $T_{c}^{appx}=10^{4}\times
e^{-\frac{1}{g_{eff}}}$ K. This scale provides an order of magnitude of
$T_{c}$ at a given $g_{eff}$ by assuming a prefactor $10^{4}$K. It illustrates
the rapid reduction of $T_{c}$ when $g_{eff}\
{\raise-1.72221pt\hbox{$\buildrel<\over{\sim}$}}\ 0.12$. The other two
symmetries lead to much smaller couplings and are therefore ineffective.
Inset: The band DOS at the fermi energy shows an enhancement around the hole
density $\delta=0.15$.

In Fig. (2) the effective superconducting coupling $g_{eff}$ is shown for
three different symmetries of the Cooper pairs: $d$-wave, extended $s$-wave,
and $s+id$-wave. It is clear that within this theory, only d-wave symmetry
leads to robust superconductivity, the other two symmetries lead to effects
too small to be observable.

Figure 3: The superconducting transition temperature for the correlated model
$T_{c}$ (Eq. (87)) for three parameter sets- (i) (red)
$t^{\prime}/t$$=-0.159$, $t^{\prime\prime}/t$$=0.01$ with
$\delta_{peak}=0.15$, (ii) (blue) $t^{\prime}/t$$=-0.137$,
$t^{\prime\prime}/t=.01$ with $\delta_{peak}=0.13$ and (iii)(purple)
$t^{\prime}/t$$=-0.181$,$t^{\prime\prime}/t$$=0.01$ with $\delta_{peak}=0.17$.
The solid lines use $\frac{m}{m^{*}}=Z$ for and the dashed lines
$\frac{m}{m^{*}}=3.4\delta$. Inset: $\Psi(\mu_{0})$ the fermi surface averaged
d-wavefunction
$\langle\left\\{\cos(k_{x})-\cos(k_{y})\right\\}^{2}\rangle_{FS}$ is shown for
the three sets of band parameters. The peak values occur at the densities
where $T_{c}$ is highest. Their peak magnitude $\sim 3.2$ indicates a strong
constructive interference effect from
$\vec{k}\sim\\{\pm\pi,0\\},\\{0,\pm\pi\\}$, where
$|\cos(k_{x})-\cos(k_{y})|\sim 2$.

From Fig. (3) we see that the peak density is shifted by varying the band
hopping parameters. As the peak density moves towards small $\delta$, its
height falls rapidly. This is understandable as the effect of the
quasiparticle weight $Z$ in the formulas Eqs. (87,88). We also note that the
use of different expressions for the effective mass in Eqs. (92,93) change the
width of the allowed regions somewhat, but are quite comparable.

The inset in Fig. (3) displays the d-wavefunction averages corresponding to
the same sets of parameters. It is interesting to note that the height of the
peaks $\Psi_{max}$$\sim$$3.2$ are close to their upper bound $4$, from a type
of constructive interference that requires comment. Note first that the DOS
can be expressed as a line integral in the octant of the Brillouin zone
$n(\mu_{0})=\frac{2}{\pi^{2}}\int_{0}^{k_{max}}\frac{dk_{x}}{|v^{y}{(\vec{k})}|}$,
where the velocity $v^{y}{(\vec{k})}=2\sin(k_{y})(t+2t^{\prime}\cos
k_{x}+4t^{\prime\prime}\cos k_{y})$ is evaluated with $k_{y}\to
k_{y}(k_{x},\mu_{0})$ on the fermi surface. Thus the region of small $|v^{y}|$
dominates the integral. If $v^{y}$ vanishes on the fermi surface, we get a
(logarithmic van Hove) peak in the DOS. Now the average of $\Psi(\mu_{0})$ is
largest, when $\vec{k}$ is close to $\\{\pm\pi,0\\}$ and $\\{0,\pm\pi\\}$.
Therefore if the fermi surface passes through $\\{\pm\pi,0\\}$ and
$\\{0,\pm\pi\\}$ for an “ideal density”, then we simultaneously maximize the
average of $\Psi$, and obtain a large $DOS$. The condition for this is found
by equating the band energy at $\\{\pm\pi,0\\}$ to the chemical potential
$\mu_{0}=4t^{\prime}-4t^{\prime\prime}$, thereby fixing the corresponding
density $\delta$. It follows that a given $\delta$ can be found from several
different sets of the parameters $t^{\prime},t^{\prime\prime}$. The inset of
Fig. (3) shows the average $\Psi(\mu_{0})$ displays peaks, the middle one
(red) coincides in location with the peak in the DOS in the inset of Fig. (2).

Figure 4: The effective exchange $J_{eff}$ from Eq. (89) for the three
parameter sets- (i) (red) $t^{\prime}/t$$=\\!-0.159$,
$t^{\prime\prime}/t$$=0.01$ with $\delta_{peak}=0.15$, (ii) (blue)
$t^{\prime}/t$$=\\!-0.137$, $t^{\prime\prime}/t=.01$ with
$\delta_{peak}$$=$$0.13$ and (iii)(purple) $t^{\prime}/t$$=\\!-0.181$,
$t^{\prime\prime}/t$$=0.01$ with $\delta_{peak}$$=$$0.17$. Since we assumed
$J/t\sim 0.3$ (Eq. (89)), $J_{eff}/t$ is considerably enhanced in the range of
densities exhibiting high $T_{c}$. This enhancement in turn boosts up
$g_{eff}$, via Eq. (88), and hence plays an important role in giving an
observable magnitude of $T_{c}$ in Fig. (1) and Fig. (3).

In Fig. (4) we illustrate the role of the feedback enhancement of the exchange
$J$ due to the background spectral function discussed in Eq. (89).

Figure 5: The effective superconducting coupling $g_{eff}$ (Eq. (88)) for the
three curves in Fig. (3), with parameter sets- (i) (red)
$t^{\prime}/t$$=-0.159$, $t^{\prime\prime}/t$$=0.01$ with
$\delta_{peak}=0.15$, (ii) (blue) $t^{\prime}/t$$=-0.137$,
$t^{\prime\prime}/t=.01$ with $\delta_{peak}=0.13$ and (iii)(purple)
$t^{\prime}/t$$=-0.181$, $t^{\prime\prime}/t$$=0.01$ with
$\delta_{peak}=0.17$. The drastic decrease of $T_{c}$ on both sides of the
peak values in Fig. (3) can be understood by referring to the the second
y-axis, giving the approximate temperature scale $T_{c}^{appx}=10^{4}\times
e^{-\frac{1}{g_{eff}}}$ K.

For each set of parameters, there is a density region where both the DOS at
the fermi energy and the averaged d-wavefunction are enhanced, and the
confluence directly enhances $J_{eff}$. In turn this is reflected in the
superconducting coupling $g_{eff}$. In Fig. (5) we see how the confluence of
enhancements in the DOS and in the d-wavefunction $\Psi(\mu_{0})$, further
boosts the superconducting coupling $g_{eff}$ and offsets to some extent the
suppression due to a small magnitude of $Z$, as seen in Eq. (88). As a result
of this competition $T_{c}$ turns out to be in the observable range. The
additional y-axis in Fig. (5) translates the superconducting coupling
$g_{eff}$ to an order of magnitude type transition temperature
$T^{appx}_{c}=e^{-1/g_{eff}}\times 10^{4}\;K$. This scale helps us to
understand why $T_{c}$ falls off so rapidly when $\delta$ increases beyond the
peak value where the coupling $g_{eff}$ falls below $\sim 0.12$, thereby
rapidly suppressing $T_{c}$.

##  5 Conclusions

This work presents a new methodology for treating extremely correlated
superconductors. The exact equations of the normal and anomalous Greens
functions in the superconductor are derived. These are further expanded in
powers of a control parameter $\lambda$ related to the density of double
occupancy, and the second order equations are given in Eqs. (51,52,54),
together with the self consistency conditions Eqs. (25,41). A further
simplification is possible for $T$$\sim$$T_{c}$ where the anomalous terms are
small. This leads to a tractable condition for $T_{c}$ given in Eq. (69),
expressed in terms of the electron spectral function. Further analysis uses a
model spectral function Eq. (76), which is simple enough to yield an explicit
expression for $T_{c}$ in Eq. (87). More elaborate calculations should be
feasible upon the availability of reliable spectral functions, when one may
directly solve Eq. (68).

Our calculation delineates the regime of parameters where superconductivity is
possible in the $t$-$J$ model within the ECFL theory. This regime turns out to
be quite constrained. The calculation highlights the requirement of a
substantial magnitude of the d-wavefunction average and the DOS at the fermi
energy. It shows that $T_{c}$ is maximal at a density where $n(\mu_{0})$, the
bare DOS is peaked, and is co-located with the peak of the fermi surface
average of the d-wavefunction Eq. (71) (inset Fig. (3)). The latter aspect is
understandable, since the passing of the Fermi level energy dispersion through
the zone boundary points $\\{\pm\pi,0\\},\\{0,\pm\pi\\}$, promotes a peak in
the DOS, and also leads to the maximization of $(\cos p_{x}-\cos p_{y})^{2}$.
The prediction of a correlation between the peak in $T_{c}$ with a peak in the
d-wavefunction average is testable, since the latter is amenable to
measurement using angle resolved photoemission.

In the approximation used here, the maximum $T_{c}$ is nominally unbounded in
a narrow density range here due to the logarithmic singularity of the DOS. It
is expected to be cutoff to a finite value of ${\cal O}(10^{2}\mbox{K})$ due
to a more exact integration over energies, when using a reliable spectral
function, in the place of the model used here. Such an integration would also
supersede the Gor’kov-type approximation of expanding around the fermi surface
($\int d\epsilon\;n(\epsilon)\sim n(\mu_{0})\int d\epsilon$) employed here,
thereby flattening out the sharp peak into a smoother shape. Finally this
mean-field description of the superconductor is expected to be corrected by
fluctuations of the phase, in a strictly two dimensional case, and by
interlayer coupling, in the physically realistic case of a three dimensional
system of weakly coupled layers.

In conclusion this work contains the essential outline of a new formalism to
treat superconducting states of models with extremely strong correlations,
such as the $t$-$J$ model. A transparent calculation within a low order
approximation is presented here. It demonstrates that the exchange energy $J$
can indeed provide the fundamental binding force between electrons forming
Cooper pairs. It leads to superconductivity with $T_{c}$’s of ${\cal
O}(10^{2}\mbox{K})$, in a finite range of densities located away from the
insulator, as also experimentally found in cuprate superconductors.

##  6 Acknowledgements:

The work at UCSC was supported by the US Department of Energy (DOE), Office of
Science, Basic Energy Sciences (BES), under Award No. DE-FG02-06ER46319.

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