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# Origin of the maximal critical temperature disparities in one - layer
cuprate superconductors.

Baruch Rosenstein  Electrophysics Department, National Yang Ming Chiao Tung
University, Hsinchu 30050, Taiwan, R. O. C<EMAIL_ADDRESS>B. Ya. Shapiro
Physics Department, Bar-Ilan University, 52900 Ramat-Gan, Israel
<EMAIL_ADDRESS>Electrophysics Department, National Yang Ming Chiao Tung
University, Hsinchu 30050, Taiwan, R. O. C  Physics Department, Bar-Ilan
University, 52900 Ramat-Gan, Israel  Electrophysics Department, National Yang
Ming Chiao Tung University, Hsinchu 30050, Taiwan, R. O. C  Physics
Department, Bar-Ilan University, 52900 Ramat-Gan, Israel

###### Abstract

Recently a phonon exchange d - wave pairing mechanism in cuprates was
proposed. The phonons are the lateral apical oxygen atoms vibrations. They
generate the attractive pairing potential peaked at $\Gamma$ point of the
Brillouin zone, $V\left(k\right)\propto\exp\left[-2kd_{a}\right]$, where
$d_{a}$ is distance from the $CuO$ planes. The model explains a rather
paradoxical well known negative correlation of the optimal doping critical
temperature $T_{c}^{\max}$ with $d_{a}$ and increase of $T_{c}^{\max}$ with
pressure. However the large disparities in $T_{c}^{\max}$, especially in one -
layer cuprate superconductors, from $39K$ for $La_{2-x}Sr_{x}CuO_{4}$, to
$95K$ for $HgBa_{2}CuO_{4+\delta}$, cannot be attributed by differencies in
$d_{a}$. Other important material parameters include the hopping amplitudes
$t,t^{\prime}$ and the on site Coulomb repulsion $U$. It is shown (within weak
coupling).that $T_{c}^{\max}$ is highest for materials close to the
topological (Lifshitz) transition from an open to a close Fermi surface. The
transition occurss for $t^{\prime}=-0.19t$ and rather small values of
effective value of $U=2t$ at optimal doping. Analytic expressions for
$T_{c}^{\max}$ are derived both near criticality and away from it.

###### pacs:

PACS: 74.20.Rp, 74.72.-h, 74.25.Dw

##  I Introduction

The physical cause of high temperature superconductivity in cuprates is still
under intensive debate despite enormous body of experimental facts and
numerous theoretical proposals. It is generally accepted that the normal state
including the Mott anti - ferromagnetic (AF) insulator at low doping, the
pseudogap and strange metal phases at intermediate dopings are due to strong
on site electron repulsion $U$ in the $CuO_{2}$ planes. Since the pairing is
d-wave, it was conjectured that an in - plane mechanism like the AF spin
fluctuations play a major role. One of the obvious questions one asks about
the pairing mechanism in cupratesKaminski08  is why structurally identical
members of the cuprate family like the one - layered $La_{2-x}Sr_{x}CuO_{4}$
(”La”) and $HgBa_{2}CuO_{4+\delta}$ (”Hg”) have so different maximal critical
temperatures ($39K$ and $95K$ respectively). A purely $CuO$ mechanism when
applied to whole family of cuprates raises several questions. Moreover it is
believed now that La has larger $U$ then Hg, so that a weaker coupled system
develops much larger pairing then an ”iconic” strongly coupled system La.

Recent first principle calculations provide increasingly accurate estimates of
$U$ that should be used in Hubbard - like modeling on the ”mesoscopic” scale.
Limiting ourselves to simplest one - layer cuprates, one estimatesDFTsmallU
$U=6.5t$ for lower $T_{c}$ parent compounds $La_{2}CuO_{4}$ and
$Bi_{2}Sr_{2}CuO_{6}$, while the highest $T_{c}^{\max}$ compounds, Hg and
$Tl_{2}Ba_{2}CuO_{6}$ have smaller on site Coulomb repulsion $U=4.5t$, see
Table 1. The nearest neighbors hoping is roughly the same ( $t=450-470meV$)
for all the materials. Other recent calculationsDFTlargeU  give slightly
larger values of $U$, but ratios of $U$ for various materials are the same.
The critical temperature is maximal at the universal hole doping,
$p_{opt}\simeq 0.16$, for the four compounds. In the framework of
”unconventional” pairing mechanism (as the spin fluctuations within the $CuO$
layer) the disparity in critical temperature is counter - intuitive: stronger
correlations should lead to larger $T_{c}^{\max}$ at least in the intermediate
coupling (namely $U<W$, where the bandwidth is $W\simeq 5-8t$). At very large
$U$ the dependence of $T_{c}^{\max}$ is not clear. Other presumably important
characteristics of these materials in addition to $t$ and $U$ are the apical
distance and the next to nearest hopping amplitudes determining the dispersion
relation.

The structure of the compounds is quite similar. The lattice spacing is almost
the same, $a=3.8-3.9A$, while the two apical oxygen atoms are above and below
the $Cu$ atoms. These distances are $d_{a}=2.4A$ for the low $T_{c}$ materials
(La, Bi) and$\ d_{a}=2.8A$ for higher $T_{c}$ (Hg,Tl), see Table 1.
Correlation of $T_{c}^{\max}$ with apical distance $d_{a}$ (despite the small
differences between materials) and the next to nearest neighbor hoping,
$t^{\prime}$, was noticed early on in numerous cuprate familiesPavarini01 . On
the basis of ARPES measurement it was notedKaminski08 , that near the optimal
doping the Fermi surfaces have different topology: the diamond shape
(”closed”) for La and Bi, while it is hyperbolically (”open”) for $Hg$ and
$Tl$, see Fig.1. This fact was first to blame for a factor $2.5$ disparity in
critical temperature. To quoteKaminski08 : ”Since the size of the
superconducting gap is largest in the antinodal region, differences in the
band dispersion at the antinode may play a significant role in the pairing and
therefore affect the maximum transition temperature”. This is also
qualitatively in accord with values of $t^{\prime}$ estimatedPavarini01 ,
$\left|t^{\prime}\right|/t=0.17$ in La and $0.33$ in Hg since at certain value
of $t^{\prime}/t$ there is a topological (Lifshitz) transition from an open to
close Fermi surface, see Fig.1, where the FS for $p_{opt}=0.16$ is given for
five values of $t^{\prime}$. It is not clear how close the two materials are
to the topological transition point (center panel in Fig.1), but apparently
La,Bi are far from it (on the left), while Tl,Hg are much closer (the right)
to the transition.

Figure 1:  Fermi surface (in the first quarter of the BZ) of the tight binding
binding model at fixed doping (here the universal optimal doping p=0.16).
Topological transition occurs on the central frame. FL is closed around the
$\Gamma$ point on the left, while open (equivalently centered at the $M$
point) on the right.

An idea that the peak in $T_{c}$ appears on the topological transition line is
not unique to the high $T_{c}$ cuprates. For example it was predictedWSM17
that when a Weyl semimetal undergoes a Lifshitz transition from the type I
(closed FS, elliptic) to the type II (open FS, hyperbolic) such a peak (the
s-wave) should appear. It was observed in $HfTe_{5}$ by applying pressureHfTe
. The main reason is clearly the spectral weight enhancement at points at
which the pairing occurs. Returning to cuprates, one notices that at the open
- close transition FS passes the van Hove point
$\mathbf{X}=\left(\pi,0\right)$ of the BZ, see central frame in Fig.1, The van
Hove singularity in cuprates was observed experimentallyFischer11 .
Theoretically the van Hove singularity’s effect on the s - wave
superconductivity was studied vHove  (in some cases in relation to
cupratesTsuei ), but the result was that the enhancement was found to be
negligible.

The question of the superconductivity dependence on the dispersion relation
(like tuning $t^{\prime}$ to the van Hove singularity) within the spin
fluctuation pairing theory typically based on the Hubbard model was not
addressed for a number of reasons. First it has not been convincingly
established that the model has the d - wave ground state around $p_{opt}$ .
The weak coupling, $U<4t$, just leads to an exponentially small d - wave
instabilityKivelson , while recently it was demonstrated that at strong
coupling, $U>5t$, competing correlated states take overnodwave ). Second, the
strongly coupled theory requires application of very complicated
methodsKatzenelson  that makes a delicate fine tuning of the band parameters
nearly impossible.

Recently a phonon exchange d - wave pairing mechanism in cuprates was
proposedI . It was argued that the only phonon modes capable of generating the
d -wave pairing in the $CuO$ plane are the lateral apical oxygen atoms
vibrations. Better studied the in - plane breathing and buckling oxygen
vibrationsBulut  and the apical oxygen c - direction modesapicz  are not able
to generate a significant d-wave $T_{c}$. The apical lateral oxygen vibrations
in ionic environment generateI  an attractive pairing in - plane potential,
$V\left(k\right)\propto\exp\left[-2kd_{a}\right]$. This phonon exchange model
will be referred to as ALLP. The $d_{a}$ dependence explains a well known
correlation of the optimal doping critical temperature $T_{c}^{\max}$ with
pressure,but, as mentioned above is impossible to explain the large
disparities in $T_{c}^{\max}$.

The model at intermediate $U$ describes wellI  various normal properties in
the doping range not too close to the Mott insulator. In an underdoped cuprate
(with locally anti - ferromagnetic pseudogap) FS have four Fermi pockets,
while the transition to the overdoped samples occurs via the Lifshitz point
(optimal doping, different from the open - close FS Lifshitz transition
discussed above) where the pockets coalesce into a single Fermi surface.
Importantly the phonon generated $T_{c}$ is maximal at the optimal doping,
where the pseudogap disappears. In the overdoped strange metal linear in
temperature resistivity was obtainedpreprint .

In the present paper we apply the ALLP of the d - wave superconductivity at
optimal doping in the case when the system is near the open-close FS
topological transition. The main feature in this case is that of the Fermi
surface approaches the van Hove singularity point $\mathbf{X}$. In particular
it paradoxically results in the largest $T_{c}^{\max}$ for materials with
lowest $U$. The paper is organized as follows. In Section II a model of the
lateral optical phonons in ionic crystal and an effective $t-t^{\prime}$ model
of the correlated electron gas is presented. This $50mev$ phonon mode and its
coupling including the matrix elements are described sufficiently well by the
Born - Meyer approximation that has been applied to cupratesFalter . In
Section III symmetrized Hartree - Fock approximationLi19  (valid for
intermediate strength $U$) is used to investigate the Fermi surface topology
change. We concentrate on the optimal doping and consider the topological
close to open Fermi surface transition. In Section IV we develop a BCS - like
d - wave theory of superconductivity in the framework of dynamic approach. The
effect of the van Hove singularity is taken into account approximately.
Section V contains results for different one - layer cuprates obtained
directly from numerical solution of Gorkov equations. In the last Section
results are summarized and discussed.

##  II The model

Our model consists of the 2DEG interacting with phonons of a polar insulator:

$H=H_{e}+H_{ph}+H_{e-ph}\text{.}$ (1)

We start with the phonon term. The electron part is the Hubbard model, while
the coupling between the electronic and vibrational degrees of freedom,
$H_{e-ph}$, is subject of the last Subsection.

Table 1: Material parameters of one layer cuprates. material | $La_{2-x}Sr_{x}CuO_{4}$ | $Bi_{2}Sr_{2}CuO_{6+\delta}$ | $HgBa_{2}CuO_{4+\delta}$ | $Tl_{2}Ba_{2}CuO_{6+\delta}$
---|---|---|---|---
$T_{c}^{opt}$ | $39$ | $35$ | $95$ | $95$
$U_{parent}/t$ | $6.5$ | $4.06$ | $3.93$ | $3.82$
$\overline{U}_{opt}/t$ | $2.6$ | $2.7$ | $1.9$ | $1.9$
$t^{\prime}/t$ | $-0.13$ | $-0.12$ | $-0.19$ | $-0.20$
$d_{a}\left[A\right]$ | $2.4$ | $2.45$ | $2.79$ | $2.7$

###  II.1 The apical oxygen lateral vibrations contribution to the effective
electron - electron interaction

Although the prevailing hypothesis is that superconductivity in cuprate is
”unconventional”, namely not to be phonon - mediated, the phonon based
mechanism has always been a natural option to explain extraordinary
superconductivity in cuprates. The crystal has very rich spectrum of phonon
modes. However very few have a strong coupling to 2DEG and even fewer can
generate lateral (in plane) forces causing pairing. As mentioned in
Introduction, the most studied phonon ”glue” mode has been the oxygen
vibrations within the $CuO$ planeBulut . It was arguedRosen19  (in the context
of high $T_{c}$ 1UC $FeSe$ on perovskite substrates where this was observed
experimentallyGuo ) that lateral vibrations of the oxygen atoms in the
adjacent ionic perovskite layer can couple strongly to 2DEG residing in the
$CuO$ plane.

The structure of the $La_{2}CuO_{4}$ and $HgBa_{2}CuO_{6}$ unit cell below the
conducting layer is schematically depicted in Fig. 2. Besides the single
$CuO_{2}$ ($Cu$ is drawn in Fig.2 as a brown sphere, $O$ \- small orange
spheres).layer only two insulating oxide layers are assumed to be relevant.
The closest layer at distance $d_{a}$, see Table I, consists of heavy $Ba$
atoms (cyan rings) and light ”apical” oxygen (small red circle). The next
layer is $LaO/HgO$, ($Hg$ \- violet, $O$ \- small dark red circles).
Qualitatively one of the reasons is that the $LaO/BaO$ layer, see Fig. 2,
constitutes a strongly coupled ionic insulator. Unlike the metallic layer
where screening is strong, in an ionic layer screening is practically absent
and a simple microscopic theory of phonons and their coupling existsAbrahamson
. Although phonons in cuprates were extensively studied within the first
principle approach including the oxygen vibration mode the matrix elements are
rarely discussed, so that the simple approach is appropriateFalter . It was
repeatedly noticedGorkov  that vibrations in $c$ directions contribute little
to the in - plane pairing.

Figure 2:  The profile 3D view of three layers comprising relevant part of the
one unit cell :of $La_{2}CuO_{4}$ and $HgBa_{2}CuO_{6}$. The top is the
$CuO_{2}$ ($Cu$ \- brown sphere, $O$ \- orange), ALLP are located inthe
insulating $LaO/BaO$ ($La$ -cyan $Ba$ \- magenta violet) $O$ (dark red). The
third layer: $LaO/HgO$, ($Hg$ \- violet, $O$ \- red). Sizes of atoms are
inversely proportional to the values of the Born - Mayer inter - atomic
potential parameter parameter $b$

Phonons in ionic crystals are described sufficiently well by the Born - Meyer
potential due to electron’s shells repulsionAbrahamson  and electrostatic
interaction of ionic charge,

$V\left(r\right)=\sqrt{A_{1}A_{2}}\exp\left[-\frac{1}{2}\left(b^{1}+b^{2}\right)r\right]+Z_{1}Z_{2}\frac{e^{2}}{r}\text{,}$
(2)

with standard values of coefficients $A$ and $b$ for all the atoms. The ionic
charges $Z$ are estimated from the DFT calculated Milliken chargesaverestov
or determined phenomenologically. In the $LaO/BaO$ layer the charges are
constrained by neutrality. Since oxygen is much lighter than $La/Ba$, the
heavy atoms’ vibrations are negligible. Obviously that way we lose the
acoustic branch, however it is known that the acoustic phonons contribute
little to the pairingGorkov . Atoms in neighboring layers can also be treated
as static. Moreover one can neglect more distant layers. Even the influence of
the yet lower layers (below the last layer shown in Fig. 2) is insignificant
due to the distance. Consequently the dominant lateral displacements,
$u_{\mathbf{m}}^{\alpha}$, $\alpha=x,y$, are of the oxygen atoms directly
beneath the $Cu$ sites at $\mathbf{r}_{\mathbf{m}}=a\left(m_{1},m_{2}\right)$.

The dynamic matrix $D_{\mathbf{q}}^{\alpha\beta}$ is calculated by expansion
of the energy to second order in oxygen displacement (details in Appendix A in
ref.I ), so that the phonon Hamiltonian in harmonic approximation is:

$H_{ph}=\frac{1}{2}\sum\nolimits_{\mathbf{q}}\left\\{M\frac{du_{-\mathbf{q}}^{\alpha}}{dt}\frac{du_{\mathbf{q}}^{\alpha}}{dt}+u_{\mathbf{-q}}^{\alpha}D_{\mathbf{q}}^{\alpha\beta}u_{\mathbf{q}}^{\beta}\right\\}\text{.}$
(3)

Here $M$ is the oxygen mass. Summations over repeated components indices is
implied. Now we turn to derivation of the phonon spectrum. Two eigenvalues,
the transversal optical (TO) and the longitudinal optical (LO) modes are given
in obtained. One observes that there are longitudinal modes are in the range
$\Omega_{\mathbf{q}}\sim$ $45-511mev$ and $25-32mev$ respectively. The energy
of LO modes is larger than that of the corresponding TO, although the sum
$\Omega_{\mathbf{q}}^{LO}+\Omega_{\mathbf{q}}^{TO}$ is nearly dispersionless.
At $\Gamma$ the splitting is small, while due to the long range Coulomb
interaction there is hardening of LO and softening of TO at the BZ edges. The
dispersion of the high frequency modes is small, while for the lower frequency
mode it is more pronounced.

###  II.2 The $t-t^{\prime}$ Hubbard model of the 2DEG in $CuO$ layers.

The electron gas is described by an effective single band $t-t^{\prime}$
model. Hamiltonian in momentum space is:

$H_{e}=\sum\nolimits_{\mathbf{k}}c_{\mathbf{k}}^{\sigma\dagger}\left(\epsilon_{\mathbf{k}}-\mu\right)c_{\mathbf{k}}^{\sigma}+U\sum\nolimits_{\mathbf{i}}n_{\mathbf{i}}^{\uparrow}n_{\mathbf{i}}^{\downarrow}\text{,}$
(4)

Here $c_{\mathbf{k}}^{\sigma\dagger}$ is the electron creation operator with
spin projection $\sigma=\uparrow,\downarrow$. Only nearest and next to nearest
neighbors hopping terms are included:

$\epsilon_{\mathbf{k}}=-2t\left(\cos\left[ak_{x}\right]+\cos\left[ak_{y}\right]\right)-4t^{\prime}\cos\left[ak_{x}\right]\cos\left[ak_{y}\right]\text{.}$
(5)

The dispersion relation thus is simplified with respect to a ”realistic” one
in which more distant hops are included. The on site repulsion is described by
the on site Hubbard repulsion term with
$n_{\mathbf{i}}^{\sigma}=c_{\mathbf{i}}^{\sigma\dagger}c_{\mathbf{i}}^{\sigma}$
being the spin $\sigma$ occupation on the site $\left\\{i_{x},i_{y}\right\\}$
and $\mu$ is the chemical potential. Due to the repulsion, even the model
without phonons is nontrivial and will be treated approximately in the next
Section. Now we turn to the electron - phonon coupling.

###  II.3 Electron - phonon coupling

The lateral apical oxygen phonon’s interaction with the 2DEG on the adjacent
$CuO$ layer $d_{a}$ above the $LaO/BaO$ plane is determined by the electric
potential created the charged apical oxygen vibration mode
$\mathbf{u}_{\mathbf{m}}$ at arbitrary point $\mathbf{r}$ is:

$\Phi\left(\mathbf{r}\right)=\sum\nolimits_{\mathbf{m}}\frac{Ze}{\sqrt{\left(\mathbf{r}-\mathbf{r}_{\mathbf{m}}-\mathbf{u}_{\mathbf{m}}\right)^{2}+d_{a}^{2}}}\text{,}$
(6)

The apical oxygen charge was taken as $Z=1.3$. The electron-phonon interaction
(EPI) Hamiltonian that accounts for the hole charge distribution in the $CuO$
plane is derived in Appendix A of ref.I . The result in momentum space has a
density - displacement form

$H_{eph}=Ze^{2}\sum\nolimits_{\mathbf{q}}n_{-\mathbf{q}}g_{\mathbf{q}}^{\alpha}u_{\mathbf{q}}^{A\alpha}\text{,}$
(7)

with EPI matrix element,

$\mathbf{g}_{\mathbf{q}}=2\pi\widehat{\mathbf{q}}e^{-\left|q\right|d_{a}}\text{.}$
(8)

Here $\left|\mathbf{q}\right|^{2}\equiv
4\left(\sin^{2}\left[q_{x}/2\right]+\sin^{2}\left[q_{y}/2\right]\right)$. It
is well known that only longitudinal phonons contribute to the effective
electron - electron interaction, as is clear from the scalar product form of
the Eq.(7). The precision of the last equality is 2%, see figure 9,10 in
Appendix A of ref.I .

To conclude Eqs.(4,3,7) define our microscopic model. Now we turn to
description of the normal state

##  III Tuning to the open - close Fermi surface transition

###  III.1  Mean field description of the Hubbard model at intermediate
coupling

The normal state of cuprates exhibits a host of phenomena including pseudogap
in underdoped regime resulting in fracture of the Fermi surface. In the
intermediate to weak coupling range, $U/t=1-2.5$, one can use a much simpler
approximation scheme: the Hartree - Fock approximation with its extension
including the RPA type coupling $U$ renormalization Maier07 ; I  and
symmetrized HFLi19 . It provides a good agreement with the more sophisticated
methods. The coupling $U$ that enters the Hubbard description generally
depends on doping. It is largest for the parent material (Mott insulator in
all holed doped cuprates). The first principle values were given in
Introduction. However in the present paper we focus on the optimal doping
$p_{opt}=0.16$ for which the value $\overline{U}$ is significantly reduced. In
some first principle calculations doping is simulated by periodic
substitution. For example the band structure of the $p=0.25$ La was simulated
recentlyMarkiewcz18 . As discussed above host of normal and superconducting
properties in overdoped, optimal and slightly underdoped materials can be
described in the intermediate coupling regime.

Generally the HF approximation takes into account the ground state
reconstruction due to the electron - hole pairs generated by the Coulomb
repulsion. In translation invariant case it results in renormalization of the
chemical potential

$\mu_{r}=\mu-Un/2\text{.}$ (9)

Within the HF approximation the density at temperature $T$ at optimal (and
above) doping is related to the dispersion relation and chemical potential
$\mu$ by

$n=\frac{2}{\left(2\pi\right)^{2}}\int_{BZ}\frac{1}{1+\exp\left[\left(\epsilon_{\mathbf{k}}-\mu_{r}\right)/T\right]}\text{.}$
(10)

From now on we chose units in which $a=t=\hbar=1$.

Let us find the value of parameters at which the renormalized FS ”reaches” the
van Hove singularity point $\mathbf{X}=\left(\pi,0\right)$,
$\epsilon_{\mathbf{X}}-\mu_{r}=0$. As can be seen from Fig.3c and Fig.1,
precisely at this value the electronic system undergoes a Lifshitz transition
from closed FS (like in La) to an open one (like in Hg). For our dispersion
relation, Eq.(5), this becomes

$\mu_{r}=\epsilon_{\mathbf{X}}=4t^{\prime}\text{.}$ (11)

At this point Eq.(10) depends on $t^{\prime}$ only:

$n=\frac{2}{\left(2\pi\right)^{2}}\int_{BZ}\frac{1}{1+\exp\left[\left(\epsilon_{\mathbf{k}}-\epsilon_{\mathbf{X}}\right)/T\right]}.$
(12)

The doping $p=1-n$ is given for $T=0.02$ $t$ in Fig.3. It is almost
independent of temperature for $T<0.1$ $t\approx 500K$ for $t=450$ $meV$ and
allows a linear fit, $p=0.84$ $t^{\prime}/t$. In particular for the optimal
doping $p=0.16$ one should have $t^{\prime}=-0.19$ $t$. If the Tl and Hg
cuprates are very close to the transition due their high $T_{c}^{\max}$, they
should have this value of the next to nearest neighbor hopping.

The assumption that the doping is optimal means that it obeys an additional
equation for the AF-paramagnet criticality (assuming the second order
transition), see ref. I . This determines $U$ as the following integral over
the magnetic BZ:

$U^{-1}=\frac{1}{4\left(2\pi\right)^{2}}\int_{k_{1}=0}^{\pi}\int_{k_{y}=-\pi}^{\pi}\frac{1}{\left|h_{\mathbf{k}}\right|}\left(\tanh\left[\frac{\varepsilon_{\mathbf{k}}^{\prime}-\mu_{r}+\left|h_{\mathbf{k}}\right|}{2T}\right]-\tanh\left[\frac{\varepsilon_{\mathbf{k}}^{\prime}-\mu_{r}-\left|h_{\mathbf{k}}\right|}{2T}\right]\right)$
(13)

Here $k_{1}=k_{x}-k_{y}$ and,

$\displaystyle h_{\mathbf{k}}$ $\displaystyle=$ $\displaystyle
1+e^{i\left(2k_{1}\right)}+e^{ik_{y}}+e^{i\left(2k_{1}-k_{y}\right)};$ (14)
$\displaystyle\varepsilon_{\mathbf{k}}^{\prime}$ $\displaystyle=$
$\displaystyle-4t^{\prime}\cos\left[k_{1}\right]\cos\left[k_{1}-k_{y}\right]\text{.}$

At van Hove chemical potential it takes a form:

$U^{-1}=\frac{1}{4\left(2\pi\right)^{2}}\int_{k_{1}=0}^{\pi}\int_{k_{y}=-\pi}^{\pi}\frac{1}{\left|h_{\mathbf{k}}\right|}\left(\tanh\left[\frac{\varepsilon_{\mathbf{k}}^{\prime}-4t^{\prime}+\left|h_{\mathbf{k}}\right|}{2T}\right]-\tanh\left[\frac{\varepsilon_{\mathbf{k}}^{\prime}-4t^{\prime}-\left|h_{\mathbf{k}}\right|}{2T}\right]\right)$
(15)

This gives the coupling at optimal doping for the van Hove material when
$t^{\prime}=-0.19$, $U_{vH}=1.74$ $t$, see Fig.3b. Away from the van Hove
value of $t^{\prime}$ one obtains couplings shown in Fig.3b. For smaller
values of $t^{\prime}$ corresponding to closed FS one obtains larger $U$ but
still within the range of perturbatively. improved HF theorypreprint .

The range of acceptable values of $t^{\prime}/t$ is rather limited. If one
chooses $\left|t^{\prime}\right|/t<0.1$, the Mott state at very low doping
does not appearIrkhin16 . At values larger than
$\left|t^{\prime}\right|/t>0.25$ the shape of the Fermi surface in the
underdoped regime is qualitatively different from the one observed by ARPES.
The value of $t^{\prime}=-0.19t$ for Hg and Tl is chosen to tune the Lifshitz
(topological) transition from the full Fermi surface to the fractured one
(four arcs) occurs at experimentally observed doping $x^{opt}=0.16$.

.

Figure 3: a. The optimal doping as function of the ration of hoping amplitudes
$t^{\prime}/t$. b. An effective on site Coulomb repulsion. c. Energy
difference between the Femi level and the van Hove energy. .

##  IV The d-wave phonon mediated superconductivity.

It is well established that phonons cause s - wave pairing in low $T_{c}$
materials, however d-wave pairing is possible when forward scattering peak of
Eq.(8) is present. Early work in this direction was summarized in ref.
Kulicrev . The transition temperature $T_{c}$ as function of the hole doping
within the present model was studied in ref. I  demonstrates the maximum at
$p_{opt}$. We start from the derivation of the phonon exchange d wave
”potential” (appearing mainly near the $\Gamma$ point of BZ).

Figure 4: a. Effective electron - electron attraction potential. Maximum at
$\mathbf{q=0}$ is the result of electrostatics due to the distance $d_{a}$.
point. b. The effective potential for the d - wave pairing corresponding to
the ALLP potential. The peak now moved to points
$\mathbf{X}=\left(\pi/a,0\right)$ and $\mathbf{Y}=\left(0,\pi/a\right)$

###  IV.1 Effective phonon generated electron - electron interactions in spin
singlet channel

In order to describe superconductivity, one should ”integrate out” the phonon
and the spin fluctuations degrees of freedom to calculate the effective
electron - electron interaction. We start with the phonons. The Matsubara
action for EPI, Eq.(7), and phonons, Eq.(3), are,

$\frac{1}{T}\sum\nolimits_{m\mathbf{,q}}\left(Ze^{2}n_{-m,-\mathbf{q}}g_{\mathbf{q}}^{\alpha}u_{m,\mathbf{q}}^{\alpha}+\frac{M}{2}u_{-m,-\mathbf{q}}^{\alpha}\Pi_{m,\mathbf{q}}^{\alpha\beta}u_{m\mathbf{,q}}^{\beta}\right)\text{,}$
(16)

where
$n_{-n,-\mathbf{q}}=\sum\nolimits_{\mathbf{k,}m}\psi_{\mathbf{k}-\mathbf{q},m-n}^{\ast\sigma}\psi_{\mathbf{k},m}^{\sigma}$
and $\mathbf{g}$ was defined in Eq.(8). The polarization matrix is defined via
the dynamic matrix of Eq.(3):
$\Pi_{n,\mathbf{q}}^{\alpha\beta}=\left(\omega_{n}^{b}\right)^{2}\delta_{\alpha\beta}+M^{-1}D_{\mathbf{q}}^{\alpha\beta}$,
$\alpha,\beta=x,y$. Since the action is quadratic in the phonon field
$\mathbf{u}$, the partition function is gaussian and can be integrated out
exactly, see details in ref.Rosen19 . As a result one obtains the effective
density - density interaction term for of electrons

$\displaystyle\mathcal{A}_{eff}^{ph}$ $\displaystyle=$
$\displaystyle\frac{1}{2T}\sum\nolimits_{\mathbf{q},n}n_{n,\mathbf{q}}V_{n\mathbf{q}}n_{-n,-\mathbf{q}};\text{
\
}V_{n,\mathbf{q}}=-\frac{\gamma}{\omega_{n}^{b2}+\Omega^{2}}e^{-2d_{a}\left|\mathbf{q}\right|}+U;$
(17) $\displaystyle\text{\ \ \ \ \ }\gamma$ $\displaystyle=$
$\displaystyle\frac{\left(2\pi Ze^{2}\right)^{2}}{M}\text{.}$

The central peak is clearly seem in Fig. 4a.

The standard superconducting gap equation is,

$\Delta_{m\mathbf{k}}=-T\sum\nolimits_{n\mathbf{p}}\frac{V_{m-n,\mathbf{k-p}}\Delta_{n\mathbf{p}}}{\omega_{n}^{2}+\left(\epsilon_{\mathbf{p}}-\mu_{r}\right)^{2}+\left|\Delta_{n\mathbf{p}}\right|^{2}}\text{.}$
(18)

Here the (Matsubara) gap function is related to the anomalous GF,
$\left\langle\psi_{m\mathbf{k}}^{\sigma}\psi_{n\mathbf{p}}^{\rho}\right\rangle=\delta_{n+m}\delta_{\mathbf{k+p}}\varepsilon^{\sigma\rho}F_{m\mathbf{k}}$
($\varepsilon^{\sigma\rho}$ \- the antisymmetric tensor), by

$\Delta_{m\mathbf{k}}=T\sum\nolimits_{n\mathbf{p}}V_{m-n,\mathbf{k-p}}F_{n\mathbf{p}}\text{.}$
(19)

The gap equation was solved numerically by iteration by discretization the BZ
with $N=256$ and $64$ frequencies. It converges to the d - wave solution. The
absolute value of the Matsubara gap function has a maximum near the
crystallographic $\mathbf{X}$ point, $\left(0,\pi\right)$. At critical
temperature the gap disappears. The values for various hoping amplitudes at
optimal doping are presented in Fig.5 (as blue points). It can be fitted well
by a Lorenzian (blue dashed curve):

$T_{c}^{\max}=\frac{T_{L}}{1+\alpha\left(t^{\prime}-t_{L}^{\prime}\right)^{2}}\text{,}$
(TL)

with maximal value of temperature at the open to close Lifshitz point,
$T_{L}=92K$ achieved next to nearest hoping amplitude
$t_{L}^{\prime}/t=-0.19$. Other parameters characterizing the electron -
phonon interactions, coupling $U$ and the electron gas were given in Section
II. The Coefficient $\alpha$ is different for
$\left|t^{\prime}\right|<\left|t_{L}^{\prime}\right|$, $\alpha_{<}=180,$ and
$\left|t^{\prime}\right|>\left|t_{L}^{\prime}\right|$, with $\alpha_{>}=240$.

To understand qualitatively the results, we derive next approximate formulas
for $T_{c}$ in terms of the ”effective” d - wave pairing strength
$\lambda_{d}$ analogous to the well known s - wave coupling $\lambda_{s}$. The
case of the Lifshitz transition requires a special treatment near the van Hove
point.

Figure 5: Critical temperature at optimal doping as function of $t;/t$.
Numerical solution is very close to the BCS formula (green dashed line) away
from topological criticality. The BCS expression is ”cut off” by the van Hove
dominated critical value (light blue line). The numerical results are fitted
by Lorentzian (blue line) given in Eq.().

###  IV.2 The d -wave analog of the asymptotic expression for $T_{c}$

Near $T_{c}$ one neglects the second term in denominator and obtains

$\Delta_{m\mathbf{k}}=-T_{c}\sum\nolimits_{n\mathbf{p}}V_{m-n,\mathbf{k-p}}\frac{\
\Delta_{n\mathbf{p}}}{\omega_{n}^{2}+\left(\epsilon_{\mathbf{p}}-\mu_{r}\right)^{2}}\text{.}$
(20)

Numerical solution of the gap equation shows that the gap function decreases
slowly like $1/\omega$ at large $\omega$ and is often used at intermediate
couplingMaier  and at relevant relatively small frequencies can be
approximated by its $n=0$ component:

$\Delta_{n\mathbf{p}}=\Delta_{\mathbf{p}}=f_{d}\left(\mathbf{p}\right)\Delta,$
(21)

The quasi - momentum dependence turns out to be quite universal:

$f_{d}\left(\mathbf{p}\right)=\cos\left[p_{x}\right]-\cos\left[p_{y}\right].$
(22)

This is an analog of the BSC assumption $\Delta_{n\mathbf{p}}=\Delta$ for the
s -wave. The normalization,
$\sum\nolimits_{\mathbf{k}}f\left(\mathbf{k}\right)^{2}=1$, is convenient for
the following derivation.

After multiplication of the $T_{c}$ equation, Eq.(20), by
$f\left(\mathbf{p}\right)$ and summation over both $\mathbf{p}$ and $n$, it
becomes:

$1=-\sum\nolimits_{\mathbf{p}}\frac{\rho_{\mathbf{k-p}}^{d}}{\Omega^{2}}\
\frac{\tanh\left[\left(\epsilon_{\mathbf{p}}-\mu_{r}\right)/2T_{c}\right]}{2\left(\epsilon_{\mathbf{p}}-\mu_{r}\right)}\text{.}$
(23)

Here

$\rho_{\mathbf{p}}^{d}=f_{d}\left(\mathbf{p}\right)\sum\nolimits_{\mathbf{k}}\rho_{\mathbf{k-p}}\
f_{d}\left(\mathbf{k}\right),$ (24)

is an effective form factor” of the phonon mediated interaction, see Fig.4b.
For d-wave the nodal region is irrelevant, while the anti-nodal region is
strongly emphasized. For the s-wave it is simpler: $\rho_{\mathbf{p}}^{s}=$
$\rho_{\mathbf{p}}$.

Function $\tanh\left[\varepsilon/2T_{c}\right]/\varepsilon$ is peaked at
$\varepsilon=0$ with width $T_{c}$. Using steepest descent approximation (the
function $\rho_{\mathbf{k}}^{d}$, drawn in Fig. 4b varies much slower), one
arrives at:

$1\simeq-\frac{1}{2\left(2\pi\right)^{2}\Omega^{2}}\int_{BZ}d^{2}\mathbf{p}\text{
}\rho_{\mathbf{p}}^{d}|_{\epsilon_{\mathbf{p}}=\mu_{r}}\frac{\tanh\left[\left(\epsilon_{\mathbf{p}}-\mu_{r}\right)/2T_{c}\right]}{\epsilon_{\mathbf{p}}-\mu_{r}}\text{.}$
(25)

Making change of variables $\left\\{p_{x},p_{y}\right\\}$ to geodesic
coordinates $\left\\{p_{n},p_{t}\right\\}$ (the normal and the tangential
momenta), and then from $p_{n}$ to
$\varepsilon=\epsilon\left(p_{x},p_{y}\right)-\mu_{r}$, one obtains the BCS
formula:

$1=\lambda^{d}\int_{\varepsilon=-\Omega}^{\Omega}d\varepsilon\frac{\tanh\left[\varepsilon/2T_{c}\right]}{2\varepsilon}\simeq\lambda^{d}\log\frac{2\Omega\gamma_{E}}{\pi
T_{c}}\text{.}$ (26)

Here the d - wave coupling constant is defined by,

$\lambda^{d}=-\frac{1}{\Omega^{2}\left(2\pi\right)^{2}}\oint_{\varepsilon}dp_{t}\frac{1}{v\left(p_{t}\right)}\rho_{p_{t},\varepsilon}^{d}=\frac{D\left(\mu_{r}\right)}{2\Omega^{2}}\rho_{p_{t},\varepsilon}^{d}\text{,}$
(27)

where the Fermi velocity is $\mathbf{v=\bigtriangledown}\epsilon$ and the
element of length along the FS is calculated numerically as
$dp_{t}=\sqrt{\left(v_{y}/v_{x}\right)^{2}+1}dp_{y}$. Finally one obtains the
usual BCS \- like expression for $T_{c}$:

$T_{c}=\frac{2\gamma_{E}}{\pi}\Omega\exp\left[-\frac{1}{\lambda^{d}}\right]\text{.}$
(28)

The values of $T_{c}^{\max}$ at optimal doping are given as green lines in
Fig.5. This agrees well with numerical solution of the gap equation except in
the vicinity of the Lifshitz transition point at $t_{L}^{\prime}=-0.19$ where
the naive BCS $T_{c}$ erroneously enters the strong coupling regime. The
reason of the breakdown is that the steepest descent approximation used is
invalidated near the $\mathbf{X}$ point due to the fact that the DOS changes
faster than $\tanh\left[\varepsilon\right]/\varepsilon$ . In particular this
leads to the logarithmically divergent density of states. Here a more
sophisticated approximate calculation is required.

Figure 6: The doping - hopping ($t^{\prime}/t$) phase diagram of the single
later cuprates. Materials with very high $T_{c}$ (Hg and Tl based cuprates)
are described by tp slightly above the critical open - close topological
transition hoping value of $t\prime/t=-0.19$. Lower $T_{c}$ La and Bi based
cuprates have much lower values around $t^{\prime}=0.12$ significantly below
the transition line (magenta). Near the optimal doping $p=0.16$ all the
materials with different hoping ratios undergo the second topological
transition (green) at which the open or closed Fermi surface fractures into
four small pockets.

###  IV.3 The van Hove singularity dominated critical system

In a particular case of the close to open transition at $t^{\prime}=-0.19$,
the FS approaches the van Hove singularity at $\mathbf{k}=(0,\pi)$ and the
steepest descent approximation is invalid. However precisely in this case one
can simplify the gap equation by expanding the dispersion relation around the
dominant $\mathbf{X}$ and $\mathbf{Y}$ points. Replacing the sums by integrals
in Eq.(20) and changing the variables to
$\left\\{\varepsilon=\epsilon\left(p_{x},p_{y}\right),p_{t}\right\\}$, one
obtains:

$1=-\frac{1}{\left(2\pi\right)^{2}\Omega^{2}}\int_{\varepsilon=-\Omega}^{\Omega}d\varepsilon\
\oint_{\varepsilon}dp_{t}\frac{1}{v\left(p_{t}\right)}\rho_{\mathbf{p}}^{d}\frac{\tanh\left[\varepsilon/2T_{c}\right]}{2\varepsilon}\text{.}$
(29)

This can be written via DOS, approximating
$\rho_{\mathbf{p}}^{d}=\rho_{\mathbf{X}}^{d}=0.16\cdot\gamma$, see Fig. 4b, as

$1=-\frac{\rho_{\mathbf{X}}^{d}}{4\Omega^{2}}\int_{\varepsilon=-\Omega}^{\Omega}d\varepsilon\frac{\tanh\left[\varepsilon/2T_{c}\right]}{\varepsilon}D\left(\varepsilon\right)\text{.}$

Near the $\mathbf{X}$ point the DOS is:

$\displaystyle D\left(\varepsilon\right)$ $\displaystyle=$
$\displaystyle\frac{4}{\left(2\pi\right)^{2}\left(1+2t^{\prime}/t\right)^{3/2}}\ln\frac{b}{\varepsilon};$
(30) $\displaystyle b$ $\displaystyle=$
$\displaystyle\left(2\pi\right)^{2}\left(1-2t^{\prime}/t\right)^{2}/\left(1+\sqrt{2}\right)^{2}\text{.}$

The critical temperature at the Lifshitz point ($t^{\prime}=t_{L}^{\prime}$,
$\mu_{r}=\epsilon\left(\mathbf{X}\right)$, using methods described in
ref.Tsuei , is

$T_{c}^{L}=t\exp\left[-\sqrt{\log^{2}\frac{2t}{\Omega}+\frac{2}{\lambda_{L}}}\right],$
(31)

where the d-wave electron - phonon coupling strength is

$\lambda_{L}=\frac{\left|\rho_{\mathbf{X}}^{d}\right|}{\left(2\pi\right)^{2}\left(1+2t_{L}^{\prime}/t\right)^{3/2}\Omega^{2}}\text{.}$

In the present case, $\Omega=50$ $meV$ and $t=460$ $meV$, one obtains
$T_{c}^{\max}=82K$ (dashed light blue line in Fig. 5) compared to the
numerical result $T_{c}^{\max}=92K$. It might be thought of as an upper cutoff
on the naive d - wave BCS estimate in the topological transition region. Just
slightly away from it, $\left|t^{\prime}-t_{L}^{\prime}\right|/t>0.1$, the BCS
formula works well with values of $\lambda^{d}=0.4-0.55$.

##  V Discussion and conclusions

To conclude, large disparity of critical temperatures of one layered cuprates
is explained in the framework of the apical lateral longitudinal phonon (ALLP)
exchange d - wave superconductivity theory. To demonstrate the basic
principles we limited ourselves in this paper to a simple sufficiently generic
model of the electron gas in the $CuO$ planes: the fourfold symmetric
$t-t^{\prime}$ single band Hubbard model with on site repulsion energy $U$ of
moderate strength. It is shown that the highest $T_{c}^{\max}$ materials
$HgBa_{2}CuO_{4+\delta}$ and $Tl_{2}Ba_{2}CuO_{6+\delta}$ ($>90K$) with a
larger next to nearest neighbors hopping amplitude $t^{\prime}/i\simeq 0.20$
are just above the topological transition from an open to a close Fermi
surface, while $La_{2-x}Sr_{x}CuO_{4+\delta}$ and ($<40K$) are well below it
at $t^{\prime}/t\simeq 0.12$ consistent with some first principle
calculations.

As was argued in previous papersI ; preprint  the values of the effective
Coulomb repulsion $U/t$ away from the Mott insulator phase should be in the
intermediate coupling range, $1.5<U/t<3.5$, smaller than the values for the
parent materials. Only in this case the pseudogap and the strange metal
$d\rho/dT$ have experimentally observed order of magnitude. Here we find that
$U$ decreases monotonically with $t^{\prime}$, see Fig. 3b. The correlation
between $t^{\prime}$ and $T_{c}^{\max}$ has been noticedPavarini01  on basis
of early first principle calculations. Of course other relevant for the APPL
theory parameters of the one layer cuprates, the lattice spacing $a$ and the
distance of the apical oxygen $d_{A}$ from the CuO plane and the phonon
frequency $\Omega$ and the apical oxygen charge $Z$ and nearest neighbor
hoping amplitude $t$ are different. Differences in $a$, $d_{A}$ and $t$ are
known to be very small experimentally. Less is known about the APPL electron -
phonon coupling In ref.I  we have estimated generally the range of $\Omega$
and $Z$ (determining the matrix element of the APPL and the CuO electron
states) on the basis of a simple microscopic Born-Meyer model described in
Section II. Looking at Fig.2, one notices that the Born - Meyer parameters for
the apical oxygen later of the two materials are almost identical. The
parameter $A$ in Eq.(2) for La and Ba atoms areAbrahamson  $A_{La}=37.283eV$,
$b_{La}=3.512$ $A^{-1}$ and $A_{La}=36.363$ $eV$, $b_{La}=3.514A^{-1}$
respectively. As a result both $\Omega$ and $Z$ are practically the same.

Fig. 6 summarizes the APPL theory critical temperature dependence on both
doping and $t^{\prime}$. The topological phase diagram contains two different
Lifshitz transitions: at the critical doping $p_{opt}$ (the green line) the FS
fractures into four ”pockets”, while at critical next to nearest neighbor
hoping $t^{\prime}=t_{L}^{\prime}$ the center of FS moves from $\Gamma$ to the
$M$ point of Brillouin zone. Let us discuss some of the limitations of the
present model.

One is the role of the ”unconventional” spin fluctuation d - wave pairingKozik
which at intermediate coupling might still contribute and ”boost”I  $T_{c}$.
In addition the restriction of the description of the dispersion relation to
the one band Hubbard model with just two parameters $t,t^{\prime}$ for nearest
neighbor and next to nearest neighbor hopping obviously makes the model less
realistic to quantitatively describe real materials. If one considers adding
more distant hopping terms like $t^{\prime\prime}$, the condition for the
topological open to close Fermi surface transitions, Eq.(11), is modified to
$\mu_{r}=\epsilon\left(\left\\{0,\pi\right\\}\right)=4t^{\prime}-4t^{\prime\prime}$.
It turns out that the $t^{\prime}\rightarrow t^{\prime}-t^{\prime\prime}$
substitution effectively describes an effect of this more complicated
dispersion relation (generally $t^{\prime}$ is negative, while
$t^{\prime\prime}$ is positive and smaller). Of course the situation in multi
layered cuprates is more involved due to inter - layer tunneling
effectsLifshitz .

The general conclusion is the following: critical temperature of transition to
superconductivity is always peaked at a topologically nontrivial (Lifshitz)
restructuring of the Fermi surface. When two such transitions lines cross, see
Fig. 6, the global maximum is achieved.

Acknowledgements.

We are grateful Prof. D. Li, H.C. Kao, L. L.Wang, C. Q. Jin, J.Y. Lin and Y.
Yeshurun for helpful discussions. Work of B.R. was supported by NSC of R.O.C.
Grants No. 101-2112-M-009-014-MY3.

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