UNIFORM ENTROPY AND ENERGY BOUNDS FOR FULLY NON-LINEAR EQUATIONS 111Work
supported in part by the National Science Foundation under grant DMS-22-03273.

Bin Guo and Duong H. Phong

###### Abstract

Energy bounds which are uniform in the background metric are obtained from
upper bounds for entropy-like quantities. The argument is based on auxiliary
Monge-Ampère equations involving sublevel sets, and bypasses the Alexandrov-
Bakelman-Pucci maximum principle. In particular, it implies uniform
$L^{\infty}$ bounds for systems coupling a fully non-linear equation to its
linearization, generalizing the cscK equation.

##  1 Introduction

Let $(X,\omega_{X})$ be a compact $n$-dimensional Kähler manifold. If
$\varphi$ is any smooth $\omega_{X}$-plurisubharmonic function, its entropy
${\rm Ent}(\varphi)$ is defined as the entropy of the measure
$\omega_{\varphi}^{n}=(\omega_{X}+i\partial\bar{\partial}\varphi)^{n}$ with
respect to the measure $\omega_{X}^{n}$,

$\displaystyle{\rm Ent}(\varphi)=\int_{X}\,{\rm
log}\,({\omega_{\varphi}^{n}\over\omega_{X}^{n}})\omega_{\varphi}^{n}$ (1.1)

and, if $\varphi$ is normalized so that ${\rm sup}_{X}\varphi=0$, its energy
$E(\varphi)$ is defined as

$\displaystyle
E(\varphi)=\int_{X}(-\varphi)\omega_{\varphi}^{n}=\|\varphi\|_{L^{1}(\omega_{\varphi}^{n})}.$
(1.2)

Both notions are essential for the study of the Monge-Ampère equation and the
problem of constant scalar curvature Kähler metrics. For example, the entropy
is the leading term in the Mabuchi functional, while the energy is closely
related to the well-known Aubin-Yau functionals $I(\varphi)$ and $J(\varphi)$
of Kähler geometry. In a recent major breakthrough, it had actually been shown
by X.X. Chen and J.R. Cheng [5] that, in the constant scalar curvature
equation, an upper bound for the entropy is equivalent to bounds for $\varphi$
to all orders. In another direction, it has been known for some time that
bounds for the entropy imply bounds for the energy [4], and more precise
embeddings of spaces of potentials with finite entropy into
$L^{p}(\omega_{\varphi}^{n})$ have now been established in [10].

Our interest in entropy-like quantities comes from a related but different
source: entropy-like quantities can also be defined for general fully non-
linear equations on Kähler manifolds, and they turn out to be central to the
existence of a priori $L^{\infty}$ estimates [12]. More precisely, let
$\omega$ be a Kähler form on $X$, and consider an equation of the form

$\displaystyle f(\lambda[h_{\varphi}])=c_{\omega}e^{F_{\omega}},\quad{\rm
sup}_{X}\varphi=0,\quad\lambda[h_{\varphi}]\in\Gamma,$ (1.3)

where
$(h_{\varphi})^{j}{}_{k}=\omega_{X}^{j\bar{m}}(\omega_{\varphi})_{\bar{m}k}$
is the relative endomorphism between $\omega_{X}$ and
$\omega_{\varphi}=\omega+i\partial\bar{\partial}\varphi$,
$\lambda[h_{\varphi}]$ the un-ordered vector of eigenvalues of $h_{\varphi}$,
and $f(\lambda)$ a given function defined on a cone $\Gamma\subset{\bf R}^{n}$
satisfying the conditions (1-4) spelled out in §2 below. The function
$F_{\omega}$ is normalized to satisfy
$\int_{X}e^{nF_{\omega}}\omega_{X}^{n}=\int_{X}\omega_{X}^{n}$, and we set
$V_{\omega}=\int_{X}\omega^{n}$ to be the volume of $\omega$. The case
considered in [12] is when $\omega=\chi+t\omega_{X}$, $t\in(0,1]$, where
$\chi$ is a given non-negative closed $(1,1)$-form, and the estimates were
required to be uniform in $t$. This case includes the one corresponding to a
fixed background metric, which can be obtained by setting $\chi=0$ and $t=1$.
It is then proved in Theorem 2, [12] that for such $\omega$ and any $p>n$, we
must have

$\displaystyle{\rm sup}_{X}|\varphi|\leq C$ (1.4)

where $C$ depends only on $\omega_{X},\chi,n,p,\gamma$, and upper bounds for
the following three quantities

$\displaystyle{c_{\omega}^{n}\over V_{\omega}},\quad
E(\omega)={c_{\omega}^{n}\over
V_{\omega}}\int_{X}(-\varphi)f(\lambda[h_{\omega}])^{n}\omega_{X}^{n},\quad{\rm
Ent}_{p}(e^{nF_{\omega}})=\int_{X}e^{nF_{\omega}}|F_{\omega}|^{p}\omega_{X}^{n}.$
(1.5)

In the particular case of the Monge-Ampère equation
$f(\lambda)=(\prod_{j=1}^{n}\lambda_{j})^{1\over n}$,
$\Gamma=\\{\lambda_{j}>0,1\leq j\leq n\\}$, the first two quantities in the
above list can be bounded by elementary arguments, so we obtain uniform bounds
for the complex Monge-Ampère equation depending only on an upper bound for
${\rm Ent}_{p}(\omega)$ for any fixed $p>n$, thus recovering the classic
estimate of Kolodziej [25], as well as the uniform version established by
Demailly and Pali [8] and Eyssidieux, Guedj, and Zeriahi [11].

The proof of the $L^{\infty}$ bound for fully non-linear PDE’s which we just
described was based on a comparison with an auxiliary Monge-Ampère equation
involving integrals $A_{s}$ on the sublevel sets $\Omega_{s}=\\{z\in
X;\varphi<-s\\}$ of the function $\varphi$. This method turns out to be
particularly effective: it has been extended by various authors to stability
estimates [13], nef classes [14], moduli of continuity [15, 19], lower bounds
for the Green’s function [16, 18], as well as Hermitian manifolds [17] and
parabolic equations [6].

However, a natural question concerning the above general $L^{\infty}$ estimate
was whether uniform bounds for the energy $E(\omega)$ can be obtained from
bounds for the entropy-like quantity ${\mathrm{E}nt}_{p}(\omega)$. For fixed
background metric $\omega$, this had been done in Theorem 3, [12], using an
argument inspired by Chen-Cheng [5] which relied on the Alexandrov-Bakelman-
Pucci (ABP) maximum principle. The ABP maximum principle is a powerful method
pioneered by Blocki ([3]; see more applications in e.g. [26, 27]), but its
dependence on the background metric can be delicate, so uniform energy bounds
are still lacking.

The main goal of the present paper is to supply such uniform bounds for the
energy $E(\omega)$. It turns out that that they can be obtained once again by
a modification of the auxiliary Monge-Ampère equation involving sublevel sets
of $\varphi$ used in [12]. This auxiliary Monge-Ampère equation bypasses the
ABP maximum principle, and does yield uniform estimates. As an indirect
consequence, it can be used to simplify parts of the arguments in [5], and
generalize the $C^{0}$ estimates there to uniform estimates as well. We
provide a description of the precise results in the next section.

##  2 Statement of the main results

We begin by stating the precise conditions on the nonlinear operator
$f(\lambda)$. As in [12], we require that $f:\Gamma\to{\bf R}_{+}$ satisfies

(1) $\Gamma\subset{\bf R}^{n}$ is a symmetric cone with

$\Gamma_{n}\subset\Gamma\subset\Gamma_{1};$ (2.1)

Here $\Gamma_{k}$ is the cone of vectors $\lambda$ with
$\sigma_{j}(\lambda)>0$ for $1\leq j\leq k$, where $\sigma_{j}(\lambda)$ is
the $j$-th symmetric polynomial in $\lambda$. In particular, $\Gamma_{1}$ is
the half-space defined by $\lambda_{1}+\cdots+\lambda_{n}>0$, and $\Gamma_{n}$
is the first octant, defined by $\lambda_{j}>0$ for $1\leq j\leq n$.

(2) $f(\lambda)$ is symmetric in
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\Gamma$ and it is homogeneous of
degree one;

(3) $\frac{\partial f}{\partial\lambda_{j}}>0$ for each $j=1,\ldots,n$ and
$\lambda\in\Gamma$;

(4) There is a $\gamma>0$ such that

$\prod_{j=1}^{n}\frac{\partial
f(\lambda)}{\partial\lambda_{j}}\geq\gamma,\quad\forall\lambda\in\Gamma.$
(2.2)

It is well-known that equations such as the Monge-Ampère equation, with
$f(\lambda)=(\prod_{j=1}^{n}\lambda_{j})^{1\over n}$, or the Hessian equation
with $f(\lambda)=\sigma_{k}(\lambda)^{1\over k}$, or the $p$-Monge-Ampère
equation of Harvey and Lawson [20, 21] with

$f(\lambda)=\Big{(}\prod_{I}\lambda_{I}\Big{)}^{\frac{n!}{(n-p)!p!}}$

where $I$ runs over all distinct multi-indices
$1\leq{i_{1}}<\cdots<{i_{p}}\leq n$,
$\lambda_{I}=\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}$, and $\Gamma$ is the cone
defined by $\lambda_{I}>0$ for all $p$-indices $I$, all satisfy the structural
condition (4). In a remarkable recent development, Harvey and Lawson [22]
showed that the condition (4) actually holds for very large classes of non-
linear operators, including all invariant Garding-Dirichlet operators. As
noted in [22], the condition (4) also arose independently in [2] in the study
of $W^{2,p}$ interior regularity.

###### Theorem 1

Let $(X,\omega_{X})$ is a compact $n$-dimensional Kähler manifold. Let
$\omega$ be any Kähler form on $X$ with

$\displaystyle\omega\leq\kappa\,\omega_{X}$ (2.3)

for some constant $\kappa>0$. Consider the equation (1.3) with the operator
$f(\lambda)$ satisfying the conditions (1-4). Then for any $p>0$, any $C^{2}$
solution $\varphi$ of (1.3) satisfies the following

(i) Trudinger-like inequalities

$\int_{X}e^{\alpha(-\varphi)^{q}}\omega_{X}^{n}\leq C_{T},$ (2.4)

(ii) and energy-like inequalities

$\int_{X}(-\varphi)^{pq}e^{nF_{\omega}}\omega_{X}^{n}\leq C_{e}.$ (2.5)

Here the exponent $q$ is given by $q=\frac{n}{n-p}$ if $p<n$, and can be any
strictly positive exponent if $p\geq n$. The constants $C_{T}$ and $C_{e}$ are
computable constants depending only on $n,p,q,\omega_{X},\kappa,\gamma$, and
upper bounds for the following two quantities

$\displaystyle{c_{\omega}^{n}\over
V_{\omega}},\quad{\mathrm{Ent}}_{p}(e^{nF_{\omega}})=\int_{X}e^{nF_{\omega}}|F_{\omega}|^{p}\omega_{X}^{n},$
(2.6)

and the $\alpha>0$ in (2.4) is a constant that depends only on
$n,p,\gamma,{c^{n}_{\omega}\over V_{\omega}}$ and $\kappa$.

We observe that, in the case of a fixed background Kähler metric $\omega$,
this theorem was proved as Theorem 3 in [12]. The point of the new theorem is
to have uniform estimates, even as the background metric $\omega$ may
degenerate to the boundary of the Kähler cone. For this same reason, we
include the case $p>n$ in the statement. When $p>n$ and the background Kähler
form $\omega$ is fixed, it follows from Theorem 1, [12], that the solution
$\varphi$ of the equation is actually bounded, and the above Trudinger-like
and energy-like inequalities follow at once. But here again, the existing
results do not give the inequalities uniform in $\omega$ that we seek.

To obtain estimates which are uniform with respect to the background metric
$\omega$, we have to improve on the proof for fixed $\omega$ in [12], which
was modeled on the arguments of [5] for the constant scalar curvature
equation, and made essential use of the ABP maximum principle. What appears
needed for uniform estimates is rather arguments in the spirit of Theorem 2,
[12], and indeed, it turns out that these arguments can be adapted to the case
at hand.

Theorem 1 readily combines with Theorem 2, [12] to give the following
improvement, which we state for easy reference in the future:

###### Theorem 2

Let $(X,\omega_{X})$ be a compact $n$-dimensional Kähler manifold. Let
$\omega$ be a Kähler form satisfying the condition (2.3) for a fixed constant
$\kappa>0$, and consider the equation (1.3) with the operator $f(\lambda)$
satisfying the conditions (1-4). Then for any $p>n$, a $C^{2}$ solution
$\varphi$ of the equation (1.3) must satisfy

$\displaystyle{\rm sup}_{X}|\varphi|\leq C$ (2.7)

where $C$ is a constant depending only on $\omega_{X},n,p,\gamma,\kappa$, and
upper bounds for the following two quantities

$\displaystyle{c_{\omega}^{n}\over V_{\omega}},\quad{\rm
Ent}_{p}(e^{nF_{\omega}}).$ (2.8)

We observe that in [12], Theorem 2 was stated for background Kähler metrics of
the form $\omega=\chi+t\omega_{X}$, $t\in(0,1]$. However, as noted in [18],
the proof applies uniformly for background Kähler forms $\omega$ satisfying
(2.3), as long as we allow a dependence of all relevant constants on the bound
$\kappa$.

We would also like to note that, for the specific case of the Monge-Ampère
equation on strongly pseudoconvex domains in ${\bf C}^{n}$, a proof of
$L^{\infty}$ estimates using the Monge-Amp‘ere energy and corresponding
Sobolev inequalities has been given in [29, 30].

As we have stressed above, the key to the proof of Theorem 1 is an argument
bypassing the use of the ABP maximum principle. As such, it can also apply and
simplify several of those parts in the paper of Chen-Cheng [5] which relied on
the ABP maximum principle. As an illustration, we state here a $C^{0}$
estimate for a coupled system, generalizing the coupled system corresponding
to the constant scalar curvature equation, which is uniform with respect to
the background Kähler form $\omega$.

Let $(X,\omega_{X})$ be a compact $n$-dimensional Kähler manifold as before,
$\omega$ be a Kähler form, and let $\theta$ be a given smooth $(1,1)$-form on
$X$. We consider the coupled system

$\displaystyle
f(\lambda[h_{\varphi}])=c_{\omega}\,e^{F_{\omega}},\quad\sup_{X}\varphi=0$
$\displaystyle\Box_{\omega_{\varphi}}F_{\omega}=-c_{\theta}+G^{i\bar{j}}\theta_{i\bar{j}},$
(2.9)

where $G^{i\bar{j}}=\frac{\partial}{\partial h_{i\bar{j}}}\,{\rm
log}\,f(\lambda[h_{\varphi}])$ is the linearized operator of $\,{\rm
log}\,f(\lambda[h])$ which is positive definite by condition (3) in the
definition of $f(\lambda)$, and
$\Box_{\omega_{\varphi}}F_{\omega}=G^{i\bar{j}}(F_{\omega})_{i\bar{j}}$. The
function $F_{\omega}$ is again normalized (for the purpose of determining
$c_{\omega}$) to satisfy
$\int_{X}e^{nF_{\omega}}\omega_{X}^{n}=\int_{X}\omega_{X}^{n}$, and
$c_{\theta}$ is a constant determined by the equation. We have then

###### Theorem 3

Assume that the function $f(\lambda)$ satisfies the conditions (1-4) spelled
out at the beginning of this section, and $\omega$ satisfies the condition
$\omega\leq\kappa\,\omega_{X}$ for some constant $\kappa$. Fix a number
$p\in(0,n]$. We assume that

${\mathrm{Ent}}_{p}(e^{nF_{\omega}})=\int_{X}|F_{\omega}|^{p}e^{nF_{\omega}}\omega_{X}^{n}\leq
K_{1},\quad{\rm and}\quad\theta\geq-K_{2}\omega$ (2.10)

for some constants $K_{1}>0$ and $K_{2}>0$. Then

$\displaystyle{\rm sup}_{X}|\varphi|\leq C,\,\mbox{and
}\sup\nolimits_{X}F_{\omega}\leq C,$

for a constant $C$ depending only on
$\omega_{X},p,n,\gamma,\kappa,c_{\omega}^{n}/V_{\omega},c_{\theta},K_{1}$ and
$K_{2}$. If we assume further that

$\displaystyle\theta\leq K_{3}\omega$ (2.11)

then the function $F_{\omega}$ is bounded from below by another constant
depending further on $K_{3}$.

We remark that when
$f(\lambda[h_{\varphi}])=(\frac{\omega_{\varphi}^{n}}{\omega_{X}^{n}})^{1/n}$
and $\theta={\rm Ric}(\omega_{X})$ is the Ricci curvature of $\omega_{X}$, the
coupled system (2) is the constant scalar curvature Kähler equation (cscK)
studied in [5]. In this case, the constants $c_{\omega}=V_{\omega}^{1/n}$, and
$c_{\theta}\in{\bf R}$ depend only on the cohomology classes $[\omega]$ and
$c_{1}(X)$. The lower bound for $F_{\omega}$ was established in [23]. One of
the main results of [5] is that, assuming an upper bound for the entropy
${\mathrm{E}nt}_{p=1}(e^{nF})$, one can obtain a priori estimates for
$\varphi$ of all orders. What our result shows is that the $C^{0}$ bounds for
$\varphi$ and $F_{\omega}$ for this particular coupled system still hold even
when $\omega$ degenerates to the boundary of the Kähler cone. We remark that
the condition $-K_{2}\omega\leq\theta\leq K_{3}\omega$ in (2.10) and (2.11) is
not very restrictive for a degenerating family. For example, it holds for
$\omega=\chi+t\omega_{X}$ and $\theta=-\chi$ for some nonnegative $(1,1)$-form
$\chi$.

##  3 Proof of Theorem 1

For notational simplicity, we will omit the subscript $\omega$ in $F_{\omega}$
and simply write $F$ in this section. Suppose $\varphi\in C^{2}(X)$ solves the
equation (1.3) with $\sup_{X}\varphi=0$. Since
$\lambda[h_{\varphi}]\in\Gamma\subset\Gamma_{1}$, we have

${\rm tr}_{\omega_{X}}\omega+\Delta_{\omega_{X}}\varphi>0.$

The assumption (2.3) implies that $\Delta_{\omega_{X}}\varphi\geq-{\rm
tr}_{\omega_{X}}\omega\geq-n\kappa$. An application of the Green’s formula
shows the uniform $L^{1}(X,\omega_{X}^{n})$ estimate of $\varphi$, i.e.

###### Lemma 1

There exists a constant $C_{0}=C_{0}(n,\kappa,\omega_{X})$ such that

$\int_{X}|\varphi|\omega^{n}_{X}\leq C_{0}.$

We will write $K>0$ to be an upper bound of the $p$-th entropy of $e^{nF}$ for
$p\in(0,n]$. As in Theorem 1, let $q=\frac{n}{n-p}$ if $p<n$ and $q>0$ be any
positive number if $p=n$. Let $s>0$ be a positive number and
$\Omega_{s}\subset X$ be the sub-level set

$\Omega_{s}=\\{z\in X|~{}-\varphi(z)-s>0\\}.$ (3.1)

We also define a monotonically decreasing function $\phi(s)$ as in [12]

$\phi(s)=\int_{\Omega_{s}}e^{nF}\omega_{X}^{n}.$ (3.2)

Given these definitions, we have the following lemma about the decay of
$\phi(s)$.

###### Lemma 2

There exists a constant $C_{1}>0$ that depends on $n,p,K,C_{0}>0$ such that
for any $s>1$

$\phi(s)\leq\frac{C_{1}}{(\,{\rm log}\,\,s)^{p}}.$ (3.3)

Proof of Lemma 2. We observe that by the Hölder-Young’s inequality (c.f.
[12]), there is a constant $C_{p}>0$ depending only $p$ such that

$\displaystyle\int_{\Omega_{s}}e^{nF}\omega_{X}^{n}$ $\displaystyle\leq$
$\displaystyle 2^{p}\int_{\Omega_{s}}\Big{(}\frac{2^{-1}\,{\rm
log}\,(-\varphi)}{\,{\rm log}\,s}\Big{)}^{p}e^{nF}\omega_{X}^{n}$
$\displaystyle\leq$ $\displaystyle\frac{2^{p}}{(\,{\rm
log}\,s)^{p}}\int_{\Omega_{s}}\Big{(}e^{nF}(1+n^{p}|F|^{p})+C_{p}e^{\,{\rm
log}\,(-\varphi)}\Big{)}\omega_{X}^{n}$ $\displaystyle\leq$
$\displaystyle\frac{C}{(\,{\rm
log}\,s)^{p}}(1+K+C_{p}C_{0})\leq\frac{C_{1}}{(\,{\rm log}\,s)^{p}}.$

From Lemma 2, we see that $\phi(s)$ can be arbitrarily small if $s>1$ is
sufficiently large. We now prove Theorem 1.

Proof of Theorem 1. We will modify the approach given in [12]. We break the
proof into four steps.

Step 1. Let $\tau_{k}(x):{\bf R}\to{\bf R}_{+}$ be a sequence of positive
smooth functions that converges monotonically decreasingly to the function
$x\cdot\chi_{{\bf R}_{+}}(x)$. Let $a=pq=\frac{np}{n-p}$ ($a$ is any positive
number if $p=n$). We solve the following complex Monge-Ampère equation

$(\omega+i\partial\bar{\partial}\psi_{s,k})^{n}=\frac{\tau_{k}(-\varphi-s)^{a}}{A_{s,k}}c_{\omega}^{n}e^{nF}\omega_{X}^{n},\quad\sup_{X}\psi_{s,k}=0.$
(3.4)

Here the constant $A_{s,k}$ is defined by

$A_{s,k}=\frac{c_{\omega}^{n}}{V_{\omega}}\int_{X}\tau_{k}(-\varphi-s)^{a}e^{nF}\omega_{X}^{n}$
(3.5)

to make the equation (3.4) compatible. By assumption $[\omega]$ is a Kähler
class, so by Yau’s theorem [31], equation (3.4) admits a unique smooth
solution $\psi_{s,k}$. We observe that by dominated convergence theorem that
as $k\to\infty$

$A_{s,k}\to
A_{s}:=\frac{c_{\omega}^{n}}{V_{\omega}}\int_{\Omega_{s}}(-\varphi-s)^{a}e^{nF}\omega_{X}^{n}.$
(3.6)

Step 2. Define a smooth function

$\Phi:=-\varepsilon(-\psi_{s,k}+\Lambda)^{b}-\varphi-s,$

where the constants are given by

$b=\frac{n}{n+a}\in(0,1),\quad\mbox{and
}\varepsilon=\frac{1}{\gamma^{1/(n+a)}(nb)^{n/(n+a)}}A_{s,k}^{\frac{1}{n+a}},$
(3.7)

and $\Lambda$ is chosen so that $\varepsilon b\Lambda^{-(1-b)}=1$, i.e.

$\Lambda=\frac{b^{1/(1-b)}}{(\gamma^{1/(n+a)}(nb)^{n/(n+a)})^{1/(1-b)}}A_{s,k}^{\frac{1}{a}}.$
(3.8)

We claim that $\Phi\leq 0$ on $X$. To see this, we note that $\Phi<0$ on
$X\backslash\Omega_{s}$ by definition. If $\max_{X}\Phi$ is achieved somewhere
on $X\backslash\Omega_{s}$, we are done. So we assume
$\max_{X}\Phi=\Phi(x_{0})$ for some point $x_{0}\in\Omega_{s}$. Let

$G^{i\bar{j}}=\frac{\partial\,{\rm log}\,f(\lambda[h_{\varphi}])}{\partial
h_{i\bar{j}}}=\frac{1}{f}\frac{\partial f(\lambda[h_{\varphi}])}{\partial
h_{i\bar{j}}}$ (3.9)

be the coefficients of the linearized operator of $f(\lambda[h_{\varphi}])$ at
the point $x_{0}$. $G^{i\bar{j}}$ is positive definite by condition (3) of the
function $f$. Hence we have $G^{i\bar{j}}\Phi_{i\bar{j}}\leq 0$ at the point
$x_{0}$. Choosing local holomorphic coordinates at $x_{0}$ such that at this
point $(\omega_{X})_{i\bar{j}}|_{x_{0}}=\delta_{ij}$ and
$(\omega_{\varphi})|_{x_{0}}$ is diagonal with eigenvalues
$\lambda_{1},\ldots,\lambda_{n}$. Then we have

${\rm det}G^{i\bar{j}}=\frac{1}{f^{n}}\prod_{j=1}^{n}\frac{\partial
f}{\partial\lambda_{j}}\geq\frac{\gamma}{f^{n}},$ (3.10)

by the condition (4) in the definition of the nonlinear operator $f(\lambda)$.

We calculate at $x_{0}$ as follows:

$\displaystyle\noindent 0$ $\displaystyle\geq$ $\displaystyle
G^{i\bar{j}}\Phi_{i\bar{j}}$ $\displaystyle=$ $\displaystyle\varepsilon
b(-\psi_{s,k}+\Lambda)^{b-1}G^{i\bar{j}}(\psi_{s,k})_{i\bar{j}}+\varepsilon
b(1-b)(-\psi_{s,k}+\Lambda)^{b-2}G^{i\bar{j}}(\psi_{s,k})_{i}(\psi_{s,k})_{\bar{j}}$
$\displaystyle-G^{i\bar{j}}(\omega_{\varphi}-\omega)_{i\bar{j}}$
$\displaystyle\geq$ $\displaystyle\varepsilon
b(-\psi_{s,k}+\Lambda)^{b-1}G^{i\bar{j}}(\omega_{\psi_{s,k}})_{i\bar{j}}-1+(1-\varepsilon
b(-\psi_{s,k}+\Lambda)^{b-1})G^{i\bar{j}}\omega_{i\bar{j}}$
$\displaystyle\geq$ $\displaystyle n\varepsilon
b(-\psi_{s,k}+\Lambda)^{b-1}({\rm det}G^{i\bar{j}})^{1/n}({\rm
det}(\omega_{\psi_{s,k}})_{i\bar{j}})^{1/n}-1+(1-\varepsilon
b\Lambda^{-(1-b)})G^{i\bar{j}}\omega_{i\bar{j}}$ $\displaystyle\geq$
$\displaystyle n\varepsilon
b(-\psi_{s,k}+\Lambda)^{b-1}\gamma^{\frac{1}{n}}\Big{(}\frac{(-\varphi-s)^{a}}{A_{s,k}}\Big{)}^{\frac{1}{n}}-1.$

Here we have applied the arithmetic-geometric inequality, (3.10), and the
equation (3.4) of $\omega_{\psi_{s,k}}$. It follows easily from the above and
the choice of constants in (3.7) that at $x_{0}$

$(-\varphi-s)\leq\frac{A_{s,k}^{1/a}}{(\gamma^{1/n}n\varepsilon
b)^{n/a}}(-\psi_{s,k}+\Lambda)^{(1-b)\frac{n}{a}}=\varepsilon(-\psi_{s,k}+\Lambda)^{b}$

that is, $\Phi(x_{0})\leq 0$.

Step 3. From the previous step, $\Phi\leq 0$. Thus on $\Omega_{s}$ we have

$\frac{(-\varphi-s)}{A_{s,k}^{1/(n+a)}}\leq
C(-\psi_{s,k}+CA_{s,k}^{1/a})^{\frac{n}{n+a}},$ (3.11)

for some uniform constant $C>0$ that depends on $n,p,\gamma$. Taking
$((n+a)p/n)$-th power on both sides of (3.11), and multiplying $e^{nF}$, we
obtain that on $\Omega_{s}$

$\frac{(-\varphi-s)^{\frac{p(n+a)}{n}}}{A_{s,k}^{p/n}}e^{nF}\leq
C(-\psi_{s,k}+A_{s,k}^{1/a})^{p}e^{nF}\leq
C_{2}[(-\psi_{s,k})^{p}e^{nF}+A_{s,k}^{p/a}e^{nF}],$ (3.12)

for some constant $C_{2}>0$ depending only on $n,p,\gamma$. We note that by
Hölder-Young’s inequality, for any $\beta>0$ there is a constant $C_{p}>0$
depending only on $p$ such that

$(-\frac{\beta}{2}\psi_{s,k})^{p}e^{nF}\leq
e^{nF}(1+|nF|^{p})+C_{p}e^{-\beta\psi_{s,k}}.$ (3.13)

Since $\omega+i\partial\bar{\partial}\psi_{s,k}>0$ and by the assumption
(2.3), that is $\omega\leq\kappa\omega_{X}$, we see that $\psi_{s,k}\in
PSH(X,\kappa\omega_{X})$. Hence there exists a
$\beta=\beta(X,\kappa\omega_{X})>0$ such that ([24, 28])

$\int_{X}e^{-\beta\psi_{s,k}}\omega_{X}^{n}\leq C_{X},$ (3.14)

for some uniform constant $C_{X}=C_{X}(\kappa\omega_{X},n)$. We integrate both
sides of (3.12) against $\omega_{X}^{n}$ over $\Omega_{s}$ and apply (3.13)
and (3.14),

$\int_{\Omega_{s}}\frac{(-\varphi-s)^{\frac{p(n+a)}{n}}}{A_{s,k}^{p/n}}e^{nF}\omega_{X}^{n}\leq
C_{3}+C_{2}A_{s,k}^{p/a}\int_{\Omega_{s}}e^{nF}\omega_{X}^{n}=C_{3}+C_{2}A_{s,k}^{p/a}\phi(s).$
(3.15)

Here $C_{3}>0$ is a uniform constant depending on $n,p,\kappa,C_{X}$ and $K$,
the upper bound of ${\mathrm{Ent}}_{p}(e^{nF})$. Letting $k\to\infty$ in
(3.15), we obtain that

$\int_{\Omega_{s}}(-\varphi-s)^{\frac{p(n+a)}{n}}e^{nF}\omega_{X}^{n}\leq
C_{3}A_{s}^{p/n}+C_{2}A_{s}^{\frac{p}{a}+\frac{p}{n}}\phi(s),$ (3.16)

where $A_{s}$ is given by (3.6). On the other hand, by Hölder inequality we
have

$\displaystyle A_{s}$ $\displaystyle=$
$\displaystyle\frac{c_{\omega}^{n}}{V_{\omega}}\int_{\Omega_{s}}(-\varphi-s)^{a}e^{nF}\omega_{X}^{n}$
(3.17) $\displaystyle\leq$
$\displaystyle\frac{c_{\omega}^{n}}{V_{\omega}}\Big{(}\int_{\Omega_{s}}(-\varphi-s)^{\frac{p(n+a)}{n}}e^{nF}\omega_{X}^{n}\Big{)}^{\frac{na}{p(n+a)}}\Big{(}\int_{\Omega_{s}}e^{nF}\omega_{X}^{n}\Big{)}^{1-\frac{na}{(n+a)p}}$
$\displaystyle\leq$
$\displaystyle\frac{c_{\omega}^{n}}{V_{\omega}}\Big{(}C_{3}A_{s}^{p/n}+C_{2}A_{s}^{\frac{p}{a}+\frac{p}{n}}\phi(s)\Big{)}^{\frac{na}{p(n+a)}}\phi(s)^{1-\frac{na}{(n+a)p}}$
$\displaystyle\leq$ $\displaystyle
C_{4}\frac{c_{\omega}^{n}}{V_{\omega}}A_{s}^{\frac{a}{n+a}}\phi(s)^{1-\frac{na}{(n+a)p}}+C_{5}\frac{c_{\omega}^{n}}{V_{\omega}}A_{s}\phi(s).$

Here the constant $C_{5}>0$ depends only on $n,p,\gamma$ and $C_{4}>0$ depends
additionally on $\kappa,K$. Note that the inequality (3.17) holds for any
$s>0$, and the constants $C_{4},C_{5}$ are independent of $s$. We remark that
by the choice of $q$, $\frac{na}{(n+a)p}=1$ when $p\in(0,n)$, and
$\frac{na}{(n+a)p}<1$ when $p=n$, which justifies the Hölder inequality used
above.

We now apply Lemma 2 to conclude that when

$s\geq\bar{s}=\max(1,\,{\rm
exp}\,[(2C_{1}C_{5}c_{\omega}^{n}/V_{\omega})^{1/p}])$ (3.18)

we have

$\phi(s)\leq\frac{C_{1}}{(\,{\rm
log}\,s)^{p}}\leq\frac{1}{2}\frac{1}{C_{5}c_{\omega}^{n}/V_{\omega}}.$ (3.19)

Combining (3.19) and (3.17), we see that when $s\geq\bar{s}$

$A_{s}\leq
2C_{4}\frac{c_{\omega}^{n}}{V_{\omega}}A_{s}^{\frac{a}{n+a}}\phi(s)^{1-\frac{na}{p(n+a)}}.$

Dividing both sides by $A_{s}^{\frac{a}{n+a}}$, we easily obtain that when
$s\geq\bar{s}$

$A_{s}\leq(2C_{4})^{\frac{n+a}{n}}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{n+a}{n}}\phi(s)^{\frac{n+a}{n}(1-\frac{na}{p(n+a)})}\leq
C_{6}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{n+a}{n}}.$ (3.20)

By the definition of $A_{s}$ in (3.6), we see from (3.20) that

$\int_{\Omega_{\bar{s}}}(-\varphi-\bar{s})^{a}e^{nF}\omega_{X}^{n}\leq
C_{6}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{a}{n}}.$ (3.21)

Since $a>0$, by the calculus inequality $|x-y|^{a}\leq 2^{a}(x^{a}+y^{a})$ for
any $x,y>0$, we easily obtain from (3.21) that

$\int_{\Omega_{\bar{s}}}(-\varphi)^{a}e^{nF}\omega_{X}^{n}\leq
2^{a}C_{6}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{a}{n}}+C_{7}\bar{s}^{a}.$
(3.22)

Finally note that on $X\backslash\Omega_{\bar{s}}$,
$0\leq-\varphi\leq\bar{s}$, hence from (3.22)

$\displaystyle\int_{X}(-\varphi)^{a}e^{nF}\omega_{X}^{n}$ $\displaystyle\leq$
$\displaystyle\int_{\Omega_{\bar{s}}}(-\varphi)^{a}e^{nF}\omega_{X}^{n}+\int_{X\backslash\Omega_{\bar{s}}}\bar{s}^{a}e^{nF}\omega_{X}^{n}$
$\displaystyle\leq$ $\displaystyle
2^{a}C_{6}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{a}{n}}+C_{8}\bar{s}^{a}$
$\displaystyle=$ $\displaystyle
2^{a}C_{6}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{a}{n}}+C_{8}\,{\rm
exp}\,[a(2C_{1}C_{5}c^{n}_{\omega}/V_{\omega})^{1/p}]=:C_{e}.$

Here $C_{e}>0$ is the desired constant in (2.5) with an explicit dependence on
the relative volume $c^{n}_{\omega}/V_{\omega}$.

Step 4. We now show the Trudinger-like inequality (2.4). We take
$(\frac{n+a}{n})$-th power on both sides of (3.11) and multiply the resulted
inequality by a small constant $\alpha>0$ to be determined. Then it follows
that on $\Omega_{s}$

$\alpha(-\varphi-s)^{\frac{n+a}{n}}\leq C_{9}\alpha
A_{s,k}^{1/n}(-\psi_{s,k}+A_{s,k}^{1/a}).$ (3.23)

Taking exponential on both sides of (3.23) and integrating it over
$\Omega_{s}$, we then obtain

$\int_{\Omega_{s}}e^{\alpha(-\varphi-s)^{\frac{n+a}{n}}}\omega_{X}^{n}\leq
e^{C_{9}\alpha A_{s,k}^{\frac{n+a}{na}}}\int_{\Omega_{s}}e^{-C_{9}\alpha
A_{s,k}^{1/n}\psi_{s,k}}\omega_{X}^{n}.$ (3.24)

Note that by (3.20), $A_{\bar{s}}\leq
C_{6}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{n+a}{n}}$ and
$A_{\bar{s},k}\to A_{\bar{s}}$, so when $k$ is large enough we have

$A_{\bar{s},k}\leq 2C_{6}(\frac{c^{n}_{\omega}}{V_{\omega}})^{\frac{n+a}{n}}.$

If we choose $\alpha>0$ small enough such that

$C_{9}A_{\bar{s},k}^{1/n}\alpha\leq(2C_{6})^{1/n}C_{9}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{n+a}{n^{2}}}\alpha<\alpha(X,\kappa\omega_{X})$

where $\alpha(X,\kappa\omega_{X})>0$ is the $\alpha$-invariant of the Kähler
manifold $(X,\kappa\omega_{X})$. Then from (3.24) we get

$\int_{\Omega_{\bar{s}}}e^{\alpha(-\varphi-\bar{s})^{\frac{n+a}{n}}}\omega_{X}^{n}\leq\,{\rm
exp}\,\Big{(}{C_{10}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{(n+a)}{na}}}\Big{)}.$
(3.25)

It is then elementary to see that (3.25) implies

$\int_{\Omega_{\bar{s}}}e^{\alpha(-\varphi)^{\frac{n+a}{n}}}\omega_{X}^{n}\leq\,{\rm
exp}\,\Big{(}{C_{10}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{(n+a)}{na}}}+2\alpha\bar{s}^{\frac{n+a}{n}}\Big{)}.$
(3.26)

Again observing that $-\varphi\leq\bar{s}$ on $X\backslash\Omega_{\bar{s}}$,
we conclude from (3.26) that

$\int_{X}e^{\alpha(-\varphi)^{\frac{n+a}{n}}}\omega_{X}^{n}\leq
V_{\omega_{X}}e^{\alpha\bar{s}^{\frac{n+a}{n}}}+\,{\rm
exp}\,\Big{(}{C_{10}(\frac{c_{\omega}^{n}}{V_{\omega}})^{\frac{(n+a)}{na}}}+2\alpha\bar{s}^{\frac{n+a}{n}}\Big{)}=:C_{T}.$
(3.27)

Since $\bar{s}$ is explicitly given in (3.18), the constant $C_{T}$ has an
explicit dependence on $c^{n}_{\omega}/V_{\omega}$. Finally note that
$\frac{n+a}{n}=1+\frac{p}{n-p}=\frac{n}{n-p}=q$. This completes the proof of
the inequality (2.4). Q.E.D.

##  4 Proof of Theorem 3

Again, we drop the subindex $\omega$ from $F_{\omega}$ for notational
simplicity. Let $(\varphi,F)$ solve the coupled system (2) stated in §2. Fix
the number $p\in(0,n]$.

Let $\delta=\frac{1}{10}K_{2}$. We solve the auxiliary complex Monge-Ampère
equation

$(\omega+i\partial\bar{\partial}\psi_{k})^{n}=\frac{\tau_{k}(-\varphi+\delta
F)^{p}}{A_{k}}c_{\omega}^{n}e^{nF}\omega_{X}^{n},\quad\sup_{X}\psi_{k}=0,$
(4.1)

where

$A_{k}=\frac{c_{\omega}^{n}}{V_{\omega}}\int_{X}\tau_{k}(-\varphi+\delta
F)^{p}e^{nF}\omega_{X}^{n}\to\frac{c_{\omega}^{n}}{V_{\omega}}\int_{\Omega}(-\varphi+\delta
F)^{p}e^{nF}\omega_{X}^{n}=:A_{\infty},$ (4.2)

as $k\to\infty$, where $\Omega=\\{-\varphi+\delta F>0\\}$. Note that the
constant $q>1$ in Theorem 1, so by Hölder inequality and (2.5) in Theorem 1,
we have $\int_{X}(-\varphi)^{p}e^{nF}\omega_{X}^{n}\leq C$ for some constant
$C>0$ depending additionally on $K_{1}$ and $c_{\omega}^{n}/V_{\omega}$.
Moreover, by the assumption (2.10),
$\int_{\Omega}|F|^{p}e^{nF}\omega_{X}^{n}\leq K_{1}$, so we have
$A_{\infty}\leq C$ for some constant depending on $K_{1}$ and
${c_{\omega}^{n}/}{V_{\omega}}$. Thus $A_{k}\leq C$ for $k$ sufficiently
large.

We consider the function

$\Psi=-\varepsilon(-\psi_{k}+\Lambda)^{\frac{n}{n+p}}-\varphi+\delta F.$

with

$\varepsilon=\Big{(}\frac{(n+p)(1+\delta
c_{\theta})}{n^{2}}\Big{)}^{\frac{n}{n+p}}A_{k}^{\frac{1}{n+p}},$

and $\Lambda$ is chosen so that $\Lambda^{p/(n+p)}=\frac{2n}{n+p}\varepsilon$,
i.e.

$\Lambda=(\frac{2n}{n+p})^{\frac{n+p}{p}}\Big{(}\frac{(n+p)(1+\delta
c_{\theta})}{n^{2}}\Big{)}^{{n/p}}A_{k}^{1/p}.$

We claim that $\Psi\leq 0$. By the definition of $\Omega$, it suffices to show
the case when $\max_{X}\Psi=\Psi(x_{0})$ for some point $x_{0}\in\Omega$. We
calculate at $x_{0}$ as in the previous section.

$\displaystyle 0$ $\displaystyle\geq$ $\displaystyle
G^{i\bar{j}}\Psi_{i\bar{j}}$ $\displaystyle\geq$
$\displaystyle\frac{\varepsilon
n}{n+p}(-\psi_{k}+\Lambda)^{-\frac{p}{n+p}}G^{i\bar{j}}(\omega_{\psi_{k}})_{i\bar{j}}-\frac{\varepsilon
n}{n+p}(-\psi_{k}+\Lambda)^{-\frac{p}{n+p}}G^{i\bar{j}}\omega_{i\bar{j}}$
$\displaystyle-G^{i\bar{j}}(\omega_{\varphi})_{i\bar{j}}+G^{i\bar{j}}\omega_{i\bar{j}}-\delta
c_{\theta}+\delta G^{i\bar{j}}\theta_{i\bar{j}}$ $\displaystyle\geq$
$\displaystyle\frac{\varepsilon
n^{2}}{n+p}(-\psi_{k}+\Lambda)^{-\frac{p}{n+p}}\Big{(}\frac{(-\varphi+\delta
F)^{p}}{A_{k}}\Big{)}^{1/n}-1-\delta c_{\theta}+(1-\frac{\varepsilon
n\Lambda^{-\frac{p}{n+p}}}{n+p}-\frac{1}{10})G^{i\bar{j}}\omega_{i\bar{j}}$
$\displaystyle\geq$ $\displaystyle\frac{\varepsilon
n^{2}}{n+p}(-\psi_{k}+\Lambda)^{-\frac{p}{n+p}}\Big{(}\frac{(-\varphi+\delta
F)^{p}}{A_{k}}\Big{)}^{1/n}-1-\delta c_{\theta}.$

By the choice of $\varepsilon$, it follows that at $x_{0}$,

$(-\varphi+\delta F)^{p}\leq\Big{(}\frac{(n+p)(1+\delta
c_{\theta})}{n^{2}\varepsilon}\Big{)}^{n}A_{k}(-\psi_{k}+\Lambda)^{\frac{np}{n+p}}=\varepsilon^{p}(-\psi_{k}+\Lambda)^{\frac{np}{n+p}},$

that is $\Psi(x_{0})\leq 0$. This proves the claim that $\Psi\leq 0$ on $X$.
Since $A_{k}\leq C$ for $k$ large enough, we derive from $\Psi\leq 0$ that in
$X$

$\delta F\leq-\varphi+\delta F\leq
C(-\psi_{k}+1)^{\frac{n}{n+p}}\leq-\epsilon\psi_{k}+C_{\epsilon},$

where the last inequality follows from the elementary inequality
$(x+1)^{n/(n+p)}\leq\epsilon x+C_{\epsilon}$ for any $\epsilon>0$. In
particular, this shows that for any $r\geq 1$

$\int_{X}e^{r\delta F}\omega_{X}^{n}\leq
C\int_{X}e^{-r\epsilon\psi_{k}}\omega_{X}^{n}\leq C_{r},$ (4.3)

where we choose $\epsilon>0$ small so that $\epsilon r<$ the alpha-invariant
of $(X,\kappa\omega_{X})$. In particular, this implies that $e^{nF}$ is
bounded in $L^{p^{\prime}}(X,\omega_{X}^{n})$ for any $p^{\prime}>1$. An
application of [12] then implies that the $L^{\infty}$ norm of $\varphi$ is
bounded by a constant depending only
$\|e^{nF}\|_{L^{{}^{p^{\prime}}}(X,\omega_{X}^{n})}$ for some $p^{\prime}>1$
and $c_{\omega}^{n}/V_{\omega}$ (c.f. Theorem 2 in §2), and subsequently by
(4.3), $\|\varphi\|_{L^{\infty}}$ depends only on
$\kappa,K_{1},K_{2},c_{\theta},\gamma$ and $c_{\omega}^{n}/V_{\omega}$.

Moreover, under the assumption (2.10), we will show $\sup_{X}F$ is bounded
above by a uniform constant. To see this, we begin with a mean-value type
inequality which was proved in [16] for complex Monge-Ampère equations and the
arguments there can be easily adapted to the current situation. But for
convenience of the readers, we include a proof of this inequality in Lemma 3
below.

###### Lemma 3

Let $u\in C^{2}(X)$ be a $C^{2}$ function on $X$ that satisfies the
differential inequality

$\Box_{\omega_{\varphi}}u\geq-a,$ (4.4)

then there is a constant $C>0$ depending on
$n,\kappa,\gamma,\kappa,p,K_{1},c_{\omega}^{n}/V_{\omega}$ and $a\geq 0$ such
that

$\sup\nolimits_{X}u\leq
C(1+\frac{c_{\omega}^{n}}{V_{\omega}}\int_{X}|u|e^{nF}\omega_{X}^{n}).$

Proof of Lemma 3. As in [16], we may assume that
$N:=\frac{c_{\omega}^{n}}{V_{\omega}}\int_{X}|u|e^{nF}\omega_{X}^{n}\leq 1$,
otherwise, we can consider the rescaled function $\tilde{u}=u/N$, which still
satisfies the differential inequality (4.4) with the same $a$. We also assume
$\\{u>0\\}\neq\emptyset$, otherwise this lemma is trivial.

Let $s>0$ be a positive number such that the super-level set
$U_{s}:=\\{u>s\\}$ is non-empty. We consider the auxiliary Monge-Ampère
equation

$(\omega+i\partial\bar{\partial}\psi_{s,k})^{n}=\frac{\tau_{k}(u-s)}{A_{s,k}}c_{\omega}^{n}e^{nF}\omega_{X}^{n},\quad\sup_{X}\psi_{s,k}=0,$
(4.5)

where as $k\to\infty$

$A_{s,k}=\frac{c_{\omega}^{n}}{V_{\omega}}\int_{X}\tau_{k}(u-s)e^{nF}\omega_{X}^{n}\to
A_{s}=\frac{c_{\omega}^{n}}{V_{\omega}}\int_{U_{s}}(u-s)e^{nF}\omega_{X}^{n}.$

The condition that $N\leq 1$ implies that $A_{s}\leq 1$, so if $k>1$ is
sufficiently large, we have $A_{s,k}\leq 2$.

The following argument is similar to that in Step 2 in the proof of Theorem 1.
We have known that $\sup_{X}|\varphi|$ is bounded. So take
$\Lambda_{0}=\sup_{X}|\varphi|+1$. We consider the function

$\Phi_{u}:=-\varepsilon(-\psi_{s,k}+\varphi+\Lambda_{0})^{\frac{n}{n+1}}+u-s,$

and we claim that for $\varepsilon=\varepsilon(s,k,a)>0$ satisfying the
equation

$\varepsilon^{n+1}=A_{s,k}(a+\frac{n\varepsilon}{n+1})^{n},$ (4.6)

we have $\sup_{X}\Phi_{u}\leq 0$. First we observe that from equation (4.6)
and $A_{s,k}\leq 2$ it holds that $\varepsilon\leq CA_{s,k}^{1/(n+1)}$ for
some $C=C(n,a)>0$. If the maximum of $\Phi_{u}$ is achieved at some point
$x_{0}\in U_{s}$, then at this point by maximum principle

$\displaystyle 0$ $\displaystyle\geq$
$\displaystyle\Box_{\omega_{\varphi}}\Phi_{u}(x_{0})$ $\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{s,k}+\varphi+\Lambda_{0})^{-\frac{1}{n+1}}\Big{(}G^{i\bar{j}}(\omega_{\psi_{s,k}})_{i\bar{j}}-G^{i\bar{j}}(\omega_{\varphi})_{i\bar{j}}\Big{)}-a$
$\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{s,k}+\varphi+\Lambda_{0})^{-\frac{1}{n+1}}\Big{(}n(\frac{u-s}{A_{s,k}})^{1/n}-1\Big{)}-a$
$\displaystyle\geq$
$\displaystyle\frac{n^{2}\varepsilon}{n+1}(-\psi_{s,k}+\varphi+\Lambda_{0})^{-\frac{1}{n+1}}\Big{(}\frac{u-s}{A_{s,k}}\Big{)}^{1/n}-\frac{n\varepsilon}{n+1}-a.$

It is then elementary to see that $\Phi_{u}\leq 0$ by the choice of
$\varepsilon$ in (4.6). This immediately implies

$\int_{U_{s}}\,{\rm
exp}\,\Big{(}\alpha\frac{(u-s)^{\frac{n+1}{n}}}{A_{s,k}^{1/n}}\Big{)}\omega_{X}^{n}\leq
C(n,a)\int_{U_{s}}e^{-\alpha C(n,a)\psi_{s,k}}\omega_{X}^{n}\leq C,$

if we choose $\alpha>0$ small enough so that $\alpha C(n,a)$ is less than the
alpha invariant of $(X,\kappa\omega_{X})$. Letting $k\to\infty$ we obtain

$\int_{U_{s}}\,{\rm
exp}\,\Big{(}\alpha\frac{(u-s)^{\frac{n+1}{n}}}{A_{s}^{1/n}}\Big{)}\omega_{X}^{n}\leq
C.$ (4.7)

This equation together with Hölder-Young’s inequality yield that for any $r>n$

$\int_{U_{s}}(u-s)^{\frac{(n+1)r}{n}}e^{nF}\omega_{X}^{n}\leq CA_{s}^{r/n}.$
(4.8)

From now on we fix an $r>n$. Then we can apply Hölder inequality to obtain

$\displaystyle A_{s}$ $\displaystyle=$
$\displaystyle\frac{c^{n}_{\omega}}{V_{\omega}}\int_{U_{s}}(u-s)e^{nF}\omega_{X}^{n}$
$\displaystyle\leq$
$\displaystyle\frac{c_{\omega}^{n}}{V_{\omega}}\Big{(}\int_{U_{s}}(u-s)^{\frac{(n+1)r}{n}}e^{nF}\omega_{X}^{n}\Big{)}^{\frac{n}{r(n+1)}}\Big{(}\int_{U_{s}}e^{nF}\omega_{X}^{n}\Big{)}^{1-\frac{n}{r(n+1)}}$
$\displaystyle\leq$
$\displaystyle\frac{c_{\omega}^{n}}{V_{\omega}}CA_{s}^{1/(n+1)}\phi_{u}(s)^{1-\frac{n}{r(n+1)}},$

where $\phi_{u}(s)=\int_{U_{s}}e^{nF}\omega_{X}^{n}$. Then we have

$A_{s}\leq
C(\frac{c_{\omega}^{n}}{V_{\omega}})^{1+\frac{1}{n}}\phi_{u}(s)^{1+\frac{r-n}{rn}}.$

This combined with the definition of $A_{s}$ implies that for any $t>0$, there
exists a constant $\bar{C}>0$ depending on $n,\gamma,\kappa,a,K_{1},K_{2},$
such that

$t\phi_{u}(s+t)\leq\bar{C}(\frac{c_{\omega}^{n}}{V_{\omega}})^{1/n}\phi_{u}(s)^{1+\frac{r-n}{rn}}.$

Let $s_{0}>0$ be a number such that
$\bar{C}(\frac{c_{\omega}^{n}}{V_{\omega}})^{1/n}\phi_{u}(s_{0})^{\frac{r-n}{rn}}\leq
1/2$. This $s_{0}$ can be chosen as
$(2\bar{C})^{\frac{rn}{r-n}}(\frac{c_{\omega}^{n}}{V_{\omega}})^{n/(r-n)}$,
since by the assumption $N\leq 1$

$\phi_{u}(s_{0})\leq\frac{N}{s_{0}}\frac{1}{c_{\omega}^{n}/V_{\omega}}\leq\frac{1}{s_{0}}\frac{1}{c_{\omega}^{n}/V_{\omega}}.$

Then a De Giorgi type iteration argument of Kolodziej [25] (see also [12] and
[9]) implies that $\phi_{u}(s)=0$ for $s\geq S_{\infty}$ for some uniform
constant

$S_{\infty}=s_{0}+\frac{1}{1-2^{-{(r-n)/}{rn}}},$

and this gives $u\leq S_{\infty}$ as desired. Q.E.D.

We now apply Lemma 3 to $u:=F-K_{2}\varphi$, which satisfies

$\Box_{\omega_{\varphi}}u=-c_{\theta}+G^{i\bar{j}}\theta_{i\bar{j}}-K_{2}+K_{2}G^{i\bar{j}}\omega_{i\bar{j}}\geq-
c_{\theta}-K_{2}=:-a.$

Lemma 3 yield the upper bound of $u$ (hence that of $F$) in terms of its
$L^{1}$ integral, while the latter is bounded since $e^{nF}$ is bounded in
$L^{r}(\omega_{X}^{n})$ for any $r>1$ and $\varphi$ is bounded in $L^{\infty}$
by [12].

If in addition we assume (2.11), that is

$\theta\leq K_{3}\omega,$

then apply Lemma 3 to $u:=-F-K_{3}\varphi$, we can also get a uniform lower
bound of $F$ depending on $K_{3}$. Q.E.D.

## References

  * [1]
  * [2] S. Abja and G. Olive, “Local regularity for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation”, arXiv:2102.07553v1.
  * [3] Z. Blocki, “On the uniform estimate in the Calabi-Yau theorem II”, Science China Math. 54 (2011) 1375-1377.
  * [4] R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, “Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties”. J. Reine Angew. Math. 751 (2019), 27 - 89.
  * [5] X.X. Chen and J.R. Cheng, “On the constant scalar curvature Kähler metrics I - a priori estimates”, J. Amer. Math. Soc. (2021) DOI: https://doi.org/10.1090/jams/967, arXiv: 1712.06697.
  * [6] X.X. Chen and J.R. Cheng, “The $L^{\infty}$ estimates for parabolic complex Monge-Ampère and Hessian equations”, arXiv:2201.13339.
  * [7] E. De Giorgi, “Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari”. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 1957 25 - 43.
  * [8] J.P. Demailly and N. Pali, “Degenerate complex Monge-Ampère equations over compact Kähler manifolds”, Intern. J. Math. 21 (2010) no. 3, 357-405.
  * [9] S. Dinew, “Lectures on pluripotential theory on compact Hermitian manifolds”. Complex non-Kähler geometry, 1 - 56, Lecture Notes in Math., 2246, Fond. CIME/CIME Found. Subser., Springer, Cham, 2019.
  * [10] E. Di Nezza, V. Guedj, and C.H. Lu, “Finite energy vs finite entropy”, arXiv: 2006.07061.
  * [11] P. Eyssidieux, V. Guedj, and A. Zeriahi, “Singular Kähler-Einstein metrics”, J. Amer. Math. Soc. 22 (2009) 607-639.
  * [12] B. Guo, D.H. Phong, and F. Tong, “On $L^{\infty}$ estimates for complex Monge-Ampère equations”, arXiv:2106.02224
  * [13] B. Guo, D.H. Phong, and F. Tong, “Stability estimates for the complex Monge-Ampr̀e and Hessian equations”, arXiv:2106.03913
  * [14] B. Guo, D.H. Phong, F. Tong, and C. Wang, “On $L^{\infty}$ estimates for Monge-Ampère and Hessian equations on nef classes”, arXiv:2111.14186
  * [15] B. Guo, D.H. Phong, F. Tong, and C. Wang, “On the modulus of continuity of solutions to complex Monge-Ampère equations”, arXiv:2112.02354
  * [16] B. Guo, D.H. Phong, and J. Sturm, “Green’s functions and complex Monge-Ampère equations”, arXiv:2202.04715.
  * [17] B. Guo and D.H. Phong, “On $L^{\infty}$ estimates for fully nonlinear partial differential equations on Hermitian manifolds”, arXiv:2204.12549
  * [18] B. Guo, D.H. Phong, J. Song, and J. Sturm, to appear.
  * [19] B. Guo and J. Song, “Local noncollapsing for complex Monge-Ampère equations”, arXiv:2201.02930
  * [20] F. R. Harvey and H. B. Lawson, “Dirichlet duality and the nonlinear Dirichlet problem”. Comm. Pure Appl. Math. 62 (2009), no. 3, 396 - 443.
  * [21] F. R. Harvey and H. B. Lawson, “Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds”, J. Differential Geom. 88 (2011), no. 3, 395 - 482.
  * [22] F.R. Harvey and H.B. Lawson, “Determinant majorization and the work of Guo-Phong-Tong and Abja-Olive”, arXiv: 2207.01729.
  * [23] W.Y. He. “On the regularity of the complex Monge-Ampère equations”. Proc. Amer. Math. Soc. 140 (2012), no. 5, 1719-1727.
  * [24] L. Hörmander, “An introduction to complex analysis in several variables”, Van Nostrand, Princeton, NJ, 1973
  * [25] S. Kolodziej, “The complex Monge-Ampère equation”, Acta Math. 180 (1998) 69-117.
  * [26] D.H. Phong and D.T. To, “Fully non-linear parabolic equations on Hermitian manifolds”, Ann. Scient. Ecole Normale Sup. 54 (2021), 793-829.
  * [27] G. Szekelyhidi, “Fully non-linear elliptic equations on compact Hermitian manifolds”, J. Differential Geometry 109 (2018) no. 2, 337-378.
  * [28] G. Tian, “On Kähler-Einstein metrics on certain Kähler manifolds with $C_{1}(M)>0$”, Invent. Math. 89 (1987), no. 2, 225–246
  * [29] J.X. Wang, X.J. Wang, and B. Zhou, “A priori estimates for the complex Monge-Ampère equation”, arXiv:2003.06059.
  * [30] J.X. Wang, X.J. Wang, and B. Zhou, “Moser-Trudinger inequality for the complex Monge-Ampère equation”, arXiv:2003.06056
  * [31] S.T. Yau, “On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I”, Comm. Pure Appl. Math. 31 (1978) 339-411.
  * [32]

Department of Mathematics & Computer Science, Rutgers University, Newark, NJ
07102 USA

<EMAIL_ADDRESS>
Department of Mathematics, Columbia University, New York, NY 10027 USA

<EMAIL_ADDRESS>


# Phase coherence of pairs of Cooper pairs as quasi-long-range order of half-
vortex pairs in a two-dimensional bilayer system

Feng-Feng Song1 and Guang-Ming Zhang1,2<EMAIL_ADDRESS>1State Key
Laboratory of Low-Dimensional Quantum Physics and Department of Physics,
Tsinghua University, Beijing 100084, China.
2Frontier Science Center for Quantum Information, Beijing 100084, China.

###### Abstract

It is known that the loss of phase coherence of Cooper pairs in two-
dimensional (2D) superconductivity corresponds to the unbinding of vortex-
antivortex pairs with the quasi-long-range order (quasi-LRO) in the order-
parameter phase field, described by the Berezinskii-Kosterlizt-Thouless (BKT)
transition of a 2D XY model. Here we show that the second-order Josephson
coupling can induce an exotic superconducting phase in a bilayer system. By
using tensor-network methods, the partition function of the 2D classical model
is expressed as a product of 1D quantum transfer operator, whose eigen-
equation can be solved by an algorithm of matrix product states rigorously.
From the singularity shown by the entanglement entropy of the 1D quantum
analogue, various phase transitions can be accurately determined. Below the
BKT phase transition, an inter-layer Ising long-range order is established at
$T_{Ising}$, and the phase coherence of both intra-layers and inter-layers is
locked together. For two identical layers, the Ising transition coincides with
the BKT transition at a multi-critical point. For two inequivalent layers,
however, there emerges an intermediate quasi-LRO phase
($T_{Ising}<T<T_{BKT}$), where the vortex-antivortex bindings occur in the
layer with the larger intra-layer coupling, but only half-vortex pairs with
topological strings exist in the other layer, corresponding to the phase
coherence of pairs of Cooper pairs. So our study provides a promising way to
realize the charge-4e superconductivity in a bilayer system.

Introduction. -Superconductivity arises from electron pairing and its phase
coherence. In conventional Bardeen-Cooper-Schrieffer superconductors, the
electron pairing and condensation of Cooper pairs always happen
simultaneously, and the superconducting transition is determined by the
pairing temperature. In two dimensions (2D), however, the true transition can
be substantially below the pairing temperature and is controlled primarily by
thermal fluctuations in the phase field of the order parameterBeasley et al.
(1979); Emery and Kivelson (1995); Carlson et al. (1999); Li et al. (2007);
Ding et al. (2008); Kang et al. (2020); Faeth et al. (2021). In the Ginzburg-
Landau theory, when the magnitude fluctuation of the order parameter is
frozen, the phase field fluctuation can be characterized by the 2D XY spin
model, and the loss of phase coherence among the Cooper pairs corresponds to
the unbinding of vortex-antivortex pairs with the quasi-long-range order
(quasi-LRO), characterized by the Berezinskii-Kosterlizt-Thouless (BKT) phase
transitionBerezinsky (1971); Kosterlitz and Thouless (1973); Kosterlitz
(1974).

In recent years there has been the increasing interest in a bilayer structure
of coupled 2D superconducting systemsBojesen et al. (2013, 2014); Kobayashi et
al. (2019); Bighin et al. (2019); Zeng et al. (2021). When a direct Josephson
coupling is present, the relative phase of the order parameters is pinned to a
fixed value, so both phase locking and phase coherence of the Cooper pairs are
characterized by a single BKT transitionParga and Van Himbergen (1980).
However, when the direct Josephson coupling is suppressedLi et al. (2007);
Ding et al. (2008), the second-order Josephson coupling is dominant, and an
Ising-like transition for the phase locking occurs at $T_{Ising}$, which is
usually lower than the BKT transition temperature $T_{BKT}$. For the
inequivalent coupled layers, it was argued the existence of an intermediate
phase ($T_{Ising}<T<T_{BKT}$) with partial order: one layer is in disordered
phase and the other layer have vortex-antivortex pairs with quasi-LROGranato
and Kosterlitz (1986). Due to the lack of sharp thermodynamic signatures for
the BKT transition, it cannot unambiguously determine whether the transition
for the identical coupled layers is a single or double transitions with an
intervening unlocked phaseGranato and Kosterlitz (1986). Actually, the nature
of the intermediate phase with partial order has not been fully explored, so
it is a great challenge to determine the global phase diagram and calculate
the properties of the intermediate phase accurately.

Recently, tensor network methods have become a powerful tool to characterize
correlated quantum many-body phases and their phase transitions in the
thermodynamic limitVerstraete et al. (2008); Orús (2014). Since the partition
function of a 2D statistical model can be represented as a tensor product of
1D quantum transfer operatorHaegeman and Verstraete (2017), the correspondence
eigen-equation can be efficiently solved by the algorithm of variational
uniform matrix product statesZauner-Stauber et al. (2018a); Fishman et al.
(2018); Vanderstraeten et al. (2019a, b). In this Letter, we apply this method
to the bilayer system. According to the singularity displayed by the
entanglement entropy of the 1D quantum analogue, various phase transitions can
be precisely determinedLi et al. (2020); Song and Zhang (2021), and various
correlation functions of local observables are calculated rigorously.

Figure 1:  (a) The finite-temperature phase diagram of the bilayer system.
Here we choose $K=0.5J_{1}$. In the low temperature phase, there emerges an
inter-layer Ising-like long-range order. The BKT and Ising transitions merge
together at the point $P$. (b) The schematic picture of the quasi-LRO phase-2,
while the quasi-LRO phase-1 is obtained by switching the upper and lower
layers. (c) The schematic picture of the low-temperature ordered phase.

The derived global phase diagram is displayed in Fig.1(a). As the temperature
decreases, the BKT transition first occurs before a local inter-layer Ising
long-range order is established. The Ising transition accompanies with the
vortex-antivortex bindings in both intra-layers and inter-layers, as shown in
Fig.1(c). For two identical layers, the Ising transition coincides with the
BKT transition at the multi-critical point $P$. However, for two inequivalent
layers, we find that the intermediate phase has a quasi-LRO: vortex-antivortex
bindings occur only in the layer with the larger intra-layer coupling while
half-vortex pairs emerge in the other layer, schematically shown in Fig.1(b).
Since the half-vortices are point singularities around which spin directions
rotate through an angle $\pi$ on circumnavigation, each pair of half-vortices
is connected by a topological stringLee and Grinstein (1985); Carpenter and
Chalker (1989); Shi et al. (2011); Serna et al. (2017). More importantly, as
the quasi-LRO of the phase fields can be viewed as the condensation of the
Cooper pairs of 2D superconductivity, the half-vortex pairs with a quasi-LRO
imply the condensate of pairs of the Cooper pairs in the absence of phase
coherence among the Cooper pairs, corresponding to the charge-4e
superconductivityDouçot and Vidal (2002); Babaev (2004); Berg et al. (2009);
Jiang et al. (2017); Fernandes and Fu (2021).

Model and tensor-network methods. -The Hamiltonian of a two-coupled XY spin
model on a square lattice is defined by

$\displaystyle H$ $\displaystyle=$ $\displaystyle-J_{1}\sum_{\langle
i,j\rangle}\cos(\theta_{i}-\theta_{j})-J_{2}\sum_{\langle
i,j\rangle}\cos(\varphi_{i}-\varphi_{j})$ (1)
$\displaystyle+K\sum_{i}\cos(2\theta_{i}-2\varphi_{i}),$

where $\theta_{i}$ and $\varphi_{i}\in[0,2\pi]$ are two $U(1)$ phase fields
describing the pairing order-parameters on the upper and lower layers,
respectively, $J_{1}$ and $J_{2}$ are their respective nearest-neighbour
intra-layer couplings, and $K$ denotes the second-order Josephson inter-layer
coupling. Due to the nature of the low-temperature phase, the inter-layer
coupling is always relevant for any finite value of $K$, and the phase fields
$\theta$ and $\varphi$ are no longer two independent $U(1)$ variables. At low
temperatures, the relative phase $\sigma_{i}\equiv\theta_{i}-\varphi_{i}$ is
reduced to a $\mathbb{Z}_{2}$ variable, which can be explicitly displayed in
the limit of $K\rightarrow\infty$, $\varphi_{i}=\theta_{i}+\pi s_{i}/2$ with
$s_{i}=\pm 1$. The reduced Ising-XY coupled model was intensively studied by
various numerical methodsChoi and Doniach (1985); Granato et al. (1991); Lee
et al. (1991); Li and Cieplak (1994); Nightingale et al. (1995).

Figure 2:  (a) Tensor network representation of the partition function. (b)
The construction of the local tensor $O$ in the partition function. (c) Eigen-
equation for the fixed-point uMPS $|\Psi(A)\rangle$ of the 1D quantum transfer
operator $T$. (d) Two-point correlation function represented by contracting a
sequence of channel operators.

In the tensor network framework, the partition function is expressed as a
contraction of local tensors defined on the original square lattice, given by

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\prod_{i}\iint\frac{\mathrm{d}\theta_{i}\mathrm{d}\varphi_{i}}{\left(2\pi\right)^{2}}\prod_{\langle
i,j\rangle}\mathrm{e}^{\beta J_{1}\cos(\theta_{i}-\theta_{j})}$ (2)
$\displaystyle\times\mathrm{e}^{\beta
J_{2}\cos(\varphi_{i}-\varphi_{j})}\mathrm{e}^{-\beta
K\cos(2\theta_{i}-2\varphi_{i})},$

where $\beta=1/T$ is the inverse temperature. To obtain its tensor network
representation, we apply a duality transformation that maps the phase
variables on each lattice site to the number indices on the nearest-neighbor
links. Such a map is achieved by the character decomposition
$\mathrm{e}^{x\cos\theta}=\sum_{n=-\infty}^{\infty}I_{n}(x)\mathrm{e}^{in\theta}$
for each Boltzmann factor, where $I_{n}(x)$ is the modified Bessel function of
the first kind. Then the partition function is represented as

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\prod_{i}\iint\frac{\mathrm{d}\theta_{i}\mathrm{d}\varphi_{i}}{\left(2\pi\right)^{2}}\prod_{l\in\mathcal{L}}\sum_{n_{l},m_{l},k_{l}}I_{n_{l}}(\beta
J_{1})I_{m_{l}}(\beta J_{2})$ (3) $\displaystyle\times I_{k_{l}}(-\beta
K)\mathrm{e}^{in_{l}(\theta_{i}-\theta_{j})}\mathrm{e}^{im_{l}(\varphi_{i}-\varphi_{j})}\mathrm{e}^{i2k_{l}(\theta_{i}-\varphi_{i})},$

where $n_{l}$ ($m_{l}$) runs over every link on the upper (lower) layer, and
$k_{l}$ corresponds to every vertical link between $\theta_{i}$ and
$\varphi_{i}$. By integrating out all the phase degrees of freedom, the
partition function is transformed into a double tensor network as shown in
Fig. 2(a)

$Z=\mathrm{tTr}\prod_{i}O_{n_{1}m_{1},n_{2}m_{2}}^{n_{3}m_{3},n_{4}m_{4}}(i),$
(4)

where “tTr” denotes the tensor contraction over all auxiliary links. The
details are given in Supplemental Materials. As displayed in Fig. 2(b), each
local tensor $O$ is defined by

$\displaystyle O_{n_{1}m_{1},n_{2}m_{2}}^{n_{3}m_{3},n_{4}m_{4}}$
$\displaystyle=$ $\displaystyle\sum_{k}\left(\prod_{l=1}^{4}I_{n_{l}}(\beta
J_{1})I_{m_{l}}(\beta J_{2})\right)^{1/2}$ (5) $\displaystyle\times
I_{k}(\beta
K)\delta_{n_{1}+n_{2}+2k}^{n_{3}+n_{4}}\delta_{m_{1}+m_{2}}^{m_{3}+m_{4}+2k},$

where the inter-layer $k$ indices are summed over and the corresponding intra-
layer $m_{l}$ and $n_{l}$ indices are grouped together. The global $U(1)$
invariance of the bilayer model is encoded in each local tensor:
$O_{n_{1}m_{1},n_{2}m_{2}}^{n_{3}m_{3},n_{4}m_{4}}\neq 0$ only if
$n_{1}+m_{1}+n_{2}+m_{2}=n_{3}+m_{3}+n_{4}+m_{4}$. Since the expansion
coefficients in the Bessel function $I_{n}(x)$ decrease exponentially as
increasing $n$, an accurate truncation can be performed on the virtual indices
of the local tensors.

In the tensor-network approach, the row-to-row transfer matrix composed of an
infinite row of $O$ tensors is a 1D quantum transfer operator, whose
logarithmic form gives rise to a 1D quantum model with complex spin-spin
interactions. Under such a correspondence, the finite-temperature properties
of the 2D statistical problem are exactly mapped into a 1D quantum model at
zero temperature. In the thermodynamic limit, the value of the partition
function is determined by the dominant eigenvalues of the transfer operator,
whose eigen-equation sketched in Fig. 2(c) is

$T|\Psi(A)\rangle=\Lambda_{\max}|\Psi(A)\rangle,$ (6)

where $|\Psi(A)\rangle$ is the leading eigenvector represented by uniform
matrix product states (uMPS) consisting of local $A$ tensorsZauner-Stauber et
al. (2018b). This eigen-equation can be accurately solved by the algorithm of
variational uniform matrix product statesZauner-Stauber et al. (2018a);
Fishman et al. (2018); Vanderstraeten et al. (2019a, b), and the largest
eigenvector $|\Psi(A)\rangle$ corresponds to the fixed-point solution. The
precision of this approximation is controlled by the auxiliary bond dimension
$D$ of the local $A$ tensors.

From the fixed-point uMPS for the 1D quantum transfer operator, various
physical quantities can be estimated accurately. As far as the phase
transitions are concerned, the quantum entanglement entropy is the most
efficient measureVidal et al. (2003); Pollmann et al. (2009), which can be
directly determined via the Schmidt decomposition of $|\Psi(A)\rangle$:
$S_{E}=-\sum_{\alpha=1}^{D}s_{\alpha}^{2}\ln s_{\alpha}^{2}$, where
$s_{\alpha}$ are the singular values. And the two-point correlation function
of the local observable $h_{i}$ defined by $G(r)=\langle h_{j}h_{j+r}\rangle$
can be evaluated by the trace of an infinite sequence of channel operators
containing two local impurity tensors $M_{j}$ and $M_{j+r}$, as shown in Fig.
2(d). The details can be found in Supplementary Materials.

Phase Diagram. -Since the inter-layer coupling is always relevant, the
structure of the complete phase diagram is independent of its value, so we
simply choose a practical value $K/J_{1}=0.5$. Importantly we have noticed
that the entanglement entropy $S_{E}$ of the fixed-point uMPS for the 1D
quantum transfer operator exhibits singularity, which provides an accurate
criterion to determine the transition points. To obtain the phase diagram, we
have to numerically calculate the entanglement entropy under a wide range of
intra-layer coupling ratios $J_{2}/J_{1}$. In Fig. 3(a), the entanglement
entropy along $J_{2}/J_{1}=1.5$ develops two sharp peaks at $T_{c1}\simeq
1.21J_{1}$ and $T_{c2}\simeq 1.44J_{1}$, respectively. When $J_{2}$ approaches
$J_{1}$, these two peaks merge together, leading to a single peak at
$T_{\ast}\simeq 1.095J_{1}$ as shown in Fig. 3 (b). These peak positions are
nearly unchanged under the bond dimensions $D=90,100,110$. So the phase
boundaries can be determined with high precision and the complete phase
diagram is displayed in Fig. 1(a).

Figure 3:  (a) and (b) The entanglement entropy as a function of temperature
for $J_{2}/J_{1}=1.5$ and $J_{1}=J_{2}$ with $K=0.5J_{1}$. (c) and (d) The
specific heat and the local Ising order parameter along $J_{2}/J_{1}=1.5$ and
$J_{1}=J_{2}$, respectively.

In order to gain insight into the essential physics of different phases, we
calculate the specific heat. Within the tensor-network framework, the internal
energy per site is calculated as

$u=-2J_{1}\langle\mathrm{e}^{i(\theta_{j}-\theta_{j+1})}\rangle-2J_{2}\langle\mathrm{e}^{i(\varphi_{j}-\varphi_{j+1})}\rangle+K\langle\mathrm{e}^{i(2\theta_{j}-2\varphi_{j})}\rangle,$

and the specific heat is obtained by $C_{V}=\partial u/\partial T$. As shown
in Fig. 3(c), along the line $J_{2}/J_{1}=1.5$, the specific heat exhibits a
logarithmic divergence at $T_{c1}$ but a small bump around $T_{c2}$. However,
for $J_{2}/J_{1}=1$, a single logarithmic singularity is observed at
$T_{\ast}$ as displayed in Fig. 3(d). The logarithmic specific heat at the
lower temperature reminds us of a 2D Ising phase transition with a
$\mathbb{Z}_{2}$ symmetry breaking, while the small bump at the higher
temperature indicates the nearby BKT transition.

At low temperatures, since the relative phase field
$\sigma_{i}\equiv\theta_{i}-\varphi_{i}$ is reduced to a $\mathbb{Z}_{2}$
variable, a local inter-layer Ising order parameter can be defined by
$\tau=\langle\sin\sigma_{i}\rangle$. As shown in Fig. 3(c) and (d), $\tau$ is
finite below $T_{c1}$, indicating that the phase lock occurs between the upper
and lower layers. When $J_{2}/J_{1}=1$, the Ising transition coincides with
the BKT transition exactly at the multi-critical point $P$, where there is an
interplay between the Ising and BKT degrees of freedom at the microscopic
level and exhibits a new universality class of critical properties with
emerged supersymmetryHuijse et al. (2015).

Correlation functions and spin stiffness. -To further explore the nature of
the intermediate temperature phase, we calculate the two-point correlation
functions of the XY spins and nematic spins, which represent the integer
vortices and half-integer vortices variables in the bilayer system,
respectively. The results are summarized in Table.1.

Table 1: Properties of correlation functions in the different phases of the phase diagram in Fig.1. | disordered | quasi-LRO-1 | quasi-LRO-2 | ordered
---|---|---|---|---
$\\!\langle\mathrm{e}^{i(\varphi_{i}-\varphi_{j})}\rangle\,$ | $\sim\mathrm{e}^{-r/\xi_{\varphi}}$ | $\sim\mathrm{e}^{-r/\xi_{\varphi}}$ | $\sim r^{-\eta_{\varphi}}$ | $\sim r^{-\eta_{\varphi}}$
$\\!\langle\mathrm{e}^{i2(\varphi_{i}-\varphi_{j})}\rangle\,$ | $\sim\mathrm{e}^{-r/\xi_{2\varphi}}$ | $\sim r^{-\eta_{2\varphi}}$ | $\sim r^{-\eta_{2\varphi}}$ | $\sim r^{-\eta_{2\varphi}}$
$\\!\langle\mathrm{e}^{i(\theta_{i}-\theta_{j})}\rangle\,$ | $\sim\mathrm{e}^{-r/\xi_{\theta}}$ | $\sim r^{-\eta_{\theta}}$ | $\sim\mathrm{e}^{-r/\xi_{\theta}}$ | $\sim r^{-\eta_{\theta}}$
$\\!\langle\mathrm{e}^{i2(\theta_{i}-\theta_{j})}\rangle\,$ | $\sim\mathrm{e}^{-r/\xi_{2\theta}}$ | $\sim r^{-\eta_{2\theta}}$ | $\sim r^{-\eta_{2\theta}}$ | $\sim r^{-\eta_{2\theta}}$
$\\!\langle\mathrm{e}^{i(\theta_{i}-\varphi_{j})}\rangle\,$ | $\sim\mathrm{e}^{-r/\xi_{\theta\varphi}}$ | $\sim\mathrm{e}^{-r/\xi_{\theta\varphi}}$ | $\sim\mathrm{e}^{-r/\xi_{\theta\varphi}}$ | $\sim r^{-\eta_{\theta\varphi}}$
$\\!\langle\mathrm{e}^{i(\sigma_{i}-\sigma_{j})}\rangle\,$ | $\sim\mathrm{e}^{-r/\xi_{\sigma}}$ | $\sim\mathrm{e}^{-r/\xi_{\sigma}}$ | $\sim\mathrm{e}^{-r/\xi_{\sigma}}$ |  $\sim$ const.

For $J_{2}/J_{1}>1$, the spin-spin correlation function of the lower layer
$G_{\varphi}(r)$ starts to decay algebraically at $T_{c2}$ as the temperature
decreases. When approaching $T_{c2}$ from above, the spin correlation length
$\xi_{\varphi}$ is well-fitted by an exponentially divergent form

$\xi(T)\propto\exp(\frac{b}{\sqrt{T-T_{C}}}),\quad T\rightarrow T_{C}^{+},$
(7)

where $b$ is a non-universal positive constant. This is the characteristic
feature of the BKT transition. Below $T_{c1}$, the spin-spin correlation
functions of both the intra-layer $G_{\theta}(r)$ and the inter-layer
$G_{\theta\varphi}(r)$ exhibit the algebraic behavior, implying the vortex-
antivortex bindings in both intra-layers and inter-layers, a fully phase-
coherent state of the Cooper pairs in the bilayer system.

When we focus on the quasi-LRO-2 phase, the spin-spin correlation function
$G_{\varphi}(r)$ in the lower layer decays algebraically, while in the upper
layer it is the correlation function of the nematic spins $G_{2\theta}(r)$
that exhibits an algebraic behavior, instead of the correlation function of
the XY spins $G_{\theta}(r)$

$\displaystyle G_{\theta}(r)=\langle
e^{i\left(\theta_{j}-\theta_{j+r}\right)}\rangle\sim\mathrm{e}^{-r/\xi_{\theta}},$
$\displaystyle G_{2\theta}(r)=\langle
e^{i\left(2\theta_{j}-2\theta_{j+r}\right)}\rangle\sim r^{-\eta_{2\theta}}.$
(8)

For a given value of $J_{2}/J_{1}=1.5$ and $T/J_{1}=1.3$, the comparison
between the spin-spin correlation function and nematic correlation function is
displayed in Fig. 4(a) and (b). Such a behavior indicates that the integer
vortices in the upper layer are fractionalized into half-integer vortex pairs
due to the presence of the inter-layer squared cosine interaction. Since the
half-integer vortices are point-like topological defects about which the phase
angles of spins wind by $\pi$, each pair of half-vortices should be connected
by a topological string across which spins are antiparallel. Because the
integer vortex-antivortex pairs with quasi-LRO are regarded as the phase
condensation of the Cooper pairs in 2D, the half-integer vortex pairs with
quasi-LRO can be regarded as the condensation of pairs of the Cooper pairs in
the absence of the phase coherence among the Cooper pairsDouçot and Vidal
(2002); Babaev (2004). Such a phenomenon is just the characteristics of the
charge-4e superconductivityBerg et al. (2009); Jiang et al. (2017); Fernandes
and Fu (2021).

Figure 4:  The properties of the quasi-LRO phase-2 when $J_{2}/J_{1}=1.5$,
$K/J_{1}=0.5$, and $T/J_{1}=1.3$. (a) The correlation function of the XY spins
shows an exponential decay. (b) The correlation function of the nematic spins
exhibits a power law decay.

To access the superfluid response of the bilayer system, we calculate the spin
stiffness or the helicity modulus defined by the second derivative of the free
energy density with respect to a twist $v$ along a given directionFisher et
al. (1973); Nelson and Kosterlitz (1977),
$\rho_{s}=\frac{\partial^{2}f}{\partial^{2}v}|_{v=0}$. The twist needs to be
taken in a way that respects the joint $U(1)$ symmetry of the coupled bilayer,
and the spin stiffness is expressed in terms of two-point functions within the
framework of tensor network methodsVanderstraeten et al. (2015, 2016). Since
the process is more technical, the details are given in the Supplementary
Materials. The jump of spin stiffness should be altered from the BKT
predictions $\rho_{s}/T_{BKT}=2/\pi$ due to the emergence of half
vorticesHübscher and Wessel (2013). In Fig. 5, the numerical spin stiffness as
a function of temperature is shown for $J_{2}/J_{1}=1.0\sim 1.8$ with the
inter-layer coupling $K/J_{1}=0.5$. It can be seen that the spin stiffness
starts to dramatically increase from zero around the BKT transition
temperature $T_{c2}$. When the temperature decreases, a cusp point forms in
the further increase of the spin stiffness, corresponding to the Ising phase
transition $T_{c1}$ precisely. Surprisingly, the cusp points for given values
of $J_{2}/J_{1}$ sit on a straight line, which is a key experimental feature
of the presence of the Ising phase transition within the superconducting
phase.

Figure 5: The spin stiffness as a function of temperature for given values of
$J_{2}/J_{1}$. The inter-layer coupling is chosen as $K=0.5J_{1}$ and the bond
dimension of the local tensor is $D=110$. The red straight line indicates the
temperature of the Ising transition.

Conclusion. -We have used the tensor-network methods to study the bilayer
system of two-coupled 2D XY spin models. The global finite temperature phase
diagram has been accurately determined. It has been found that, as the
temperature decreases, the BKT transition always happens above the phase
locking of the bilayer system, which corresponds to an inter-layer Ising long-
range order. More importantly, for two inequivalent coupled bilayers, there
exists an intervening unlocked phase, where the half-integer vortex pairs form
in one layer with the smaller intra-layer coupling, coexisting with the
integer vortex-antivortex pairs in the other layer. When a weak direct
Josephson coupling is also present, we have further proved that the Ising
phase transition below the BKT transition survives and the main results of
this work are still valid, because two local minima always exist to lock the
phase fields of the upper and lower layers.

Recently a new family of superconductors ACa2Fe4As4F2 (A=K,Rb,Cs) has
synthesizedWang et al. (2016), and these compounds can be viewed as an
intergrowth of AFe2As2 and CaFeAsF layers. The transport and magnetic
measurements on single crystals of CsCa2Fe4As4F2 showed a large resistivity
anisotropy that tends to increase with decreasing temperature, and the 2D
superconducting fluctuations have been observedWang et al. (2019). The
evolution of the in-plane penetration depth shows an inflection point around
$10$ K, indicating that a potentially ”magnetic” phase appears but does not
compete with superconductivityKirschner et al. (2018). These features may be
related to the formation of the inter-layer Ising long-range order and the
manifestation of the phase coherence of pairs of Cooper pairs revealing a cusp
point in the spin stiffness. Therefore, these compounds are good candidate
systems to explore the charge-4e superconductivity.

Acknowledgments. The authors are indebted to Qi Zhang for his stimulating
discussions. The research is supported by the National Key Research and
Development Program of MOST of China (2017YFA0302902).

## References

  * Beasley et al. (1979) M. R. Beasley, J. E. Mooij, and T. P. Orlando, Phys. Rev. Lett. 42, 1165 (1979), URL https://link.aps.org/doi/10.1103/PhysRevLett.42.1165.
  * Emery and Kivelson (1995) V. J. Emery and S. A. Kivelson, Nature 374, 434 (1995).
  * Carlson et al. (1999) E. W. Carlson, S. A. Kivelson, V. J. Emery, and E. Manousakis, Phys. Rev. Lett. 83, 612 (1999), URL https://link.aps.org/doi/10.1103/PhysRevLett.83.612.
  * Li et al. (2007) Q. Li, M. Hücker, G. D. Gu, A. M. Tsvelik, and J. M. Tranquada, Phys. Rev. Lett. 99, 067001 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.99.067001.
  * Ding et al. (2008) J. F. Ding, X. Q. Xiang, Y. Q. Zhang, H. Liu, and X. G. Li, Phys. Rev. B 77, 214524 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.77.214524.
  * Kang et al. (2020) B. L. Kang, M. Z. Shi, S. J. Li, H. H. Wang, Q. Zhang, D. Zhao, J. Li, D. W. Song, L. X. Zheng, L. P. Nie, et al., Phys. Rev. Lett. 125, 097003 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.125.097003.
  * Faeth et al. (2021) B. D. Faeth, S.-L. Yang, J. K. Kawasaki, J. N. Nelson, P. Mishra, C. T. Parzyck, C. Li, D. G. Schlom, and K. M. Shen, Phys. Rev. X 11, 021054 (2021), URL https://link.aps.org/doi/10.1103/PhysRevX.11.021054.
  * Berezinsky (1971) V. Berezinsky, Sov. Phys. JETP 32, 493 (1971).
  * Kosterlitz and Thouless (1973) J. M. Kosterlitz and D. J. Thouless, Journal of Physics C: Solid State Physics 6, 1181 (1973), URL https://doi.org/10.1088.
  * Kosterlitz (1974) J. M. Kosterlitz, Journal of Physics C: Solid State Physics 7, 1046 (1974), URL https://doi.org/10.1088.
  * Bojesen et al. (2013) T. A. Bojesen, E. Babaev, and A. Sudbø, Phys. Rev. B 88, 220511 (2013), URL https://link.aps.org/doi/10.1103/PhysRevB.88.220511.
  * Bojesen et al. (2014) T. A. Bojesen, E. Babaev, and A. Sudbø, Phys. Rev. B 89, 104509 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.89.104509.
  * Kobayashi et al. (2019) M. Kobayashi, M. Eto, and M. Nitta, Phys. Rev. Lett. 123, 075303 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.075303.
  * Bighin et al. (2019) G. Bighin, N. Defenu, I. Nándori, L. Salasnich, and A. Trombettoni, Phys. Rev. Lett. 123, 100601 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.100601.
  * Zeng et al. (2021) M. Zeng, L.-H. Hu, H.-Y. Hu, Y.-Z. You, and C. Wu, arXiv e-prints arXiv:2102.06158 (2021), eprint 2102.06158.
  * Parga and Van Himbergen (1980) N. Parga and J. Van Himbergen, Solid State Communications 35, 607 (1980), ISSN 0038-1098, URL https://www.sciencedirect.com/science/article/pii/003810988090592X.
  * Granato and Kosterlitz (1986) E. Granato and J. M. Kosterlitz, Phys. Rev. B 33, 4767 (1986), URL https://link.aps.org/doi/10.1103/PhysRevB.33.4767.
  * Verstraete et al. (2008) F. Verstraete, V. Murg, and J. Cirac, Advances in Physics 57, 143 (2008), eprint https://doi.org/10.1080/14789940801912366, URL https://doi.org/10.1080/14789940801912366.
  * Orús (2014) R. Orús, Annals of Physics 349, 117 (2014), ISSN 0003-4916, URL http://www.sciencedirect.com/science/article/pii/S0003491614001596.
  * Haegeman and Verstraete (2017) J. Haegeman and F. Verstraete, Annual Review of Condensed Matter Physics 8, 355 (2017), eprint https://doi.org/10.1146/annurev-conmatphys-031016-025507, URL https://doi.org/10.1146/annurev-conmatphys-031016-025507.
  * Zauner-Stauber et al. (2018a) V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete, and J. Haegeman, Phys. Rev. B 97, 045145 (2018a), URL https://link.aps.org/doi/10.1103/PhysRevB.97.045145.
  * Fishman et al. (2018) M. T. Fishman, L. Vanderstraeten, V. Zauner-Stauber, J. Haegeman, and F. Verstraete, Phys. Rev. B 98, 235148 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.235148.
  * Vanderstraeten et al. (2019a) L. Vanderstraeten, J. Haegeman, and F. Verstraete, SciPost Phys. Lect. Notes p. 7 (2019a), URL https://scipost.org/10.21468/SciPostPhysLectNotes.7.
  * Vanderstraeten et al. (2019b) L. Vanderstraeten, B. Vanhecke, A. M. Läuchli, and F. Verstraete, Phys. Rev. E 100, 062136 (2019b), URL https://link.aps.org/doi/10.1103/PhysRevE.100.062136.
  * Li et al. (2020) Z.-Q. Li, L.-P. Yang, Z. Y. Xie, H.-H. Tu, H.-J. Liao, and T. Xiang, Phys. Rev. E 101, 060105 (2020), URL https://link.aps.org/doi/10.1103/PhysRevE.101.060105.
  * Song and Zhang (2021) F.-F. Song and G.-M. Zhang, Phys. Rev. B 103, 024518 (2021), URL https://link.aps.org/doi/10.1103/PhysRevB.103.024518.
  * Lee and Grinstein (1985) D. H. Lee and G. Grinstein, Phys. Rev. Lett. 55, 541 (1985), URL https://link.aps.org/doi/10.1103/PhysRevLett.55.541.
  * Carpenter and Chalker (1989) D. B. Carpenter and J. T. Chalker, Journal of Physics: Condensed Matter 1, 4907 (1989), URL https://doi.org/10.1088.
  * Shi et al. (2011) Y. Shi, A. Lamacraft, and P. Fendley, Phys. Rev. Lett. 107, 240601 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.240601.
  * Serna et al. (2017) P. Serna, J. T. Chalker, and P. Fendley, Journal of Physics A: Mathematical and Theoretical 50, 424003 (2017), URL https://doi.org/10.1088.
  * Douçot and Vidal (2002) B. Douçot and J. Vidal, Phys. Rev. Lett. 88, 227005 (2002), URL https://link.aps.org/doi/10.1103/PhysRevLett.88.227005.
  * Babaev (2004) E. Babaev, Nuclear Physics B 686, 397 (2004), eprint cond-mat/0201547, URL https://www.sciencedirect.com/science/article/pii/S0550321304001294.
  * Berg et al. (2009) E. Berg, E. Fradkin, and S. A. Kivelson, Nature Physics 5, 830 (2009).
  * Jiang et al. (2017) Y.-F. Jiang, Z.-X. Li, S. A. Kivelson, and H. Yao, Phys. Rev. B 95, 241103 (2017), URL https://link.aps.org/doi/10.1103/PhysRevB.95.241103.
  * Fernandes and Fu (2021) R. M. Fernandes and L. Fu, arXiv e-prints arXiv:2101.07943 (2021), eprint 2101.07943.
  * Choi and Doniach (1985) M. Y. Choi and S. Doniach, Phys. Rev. B 31, 4516 (1985), URL https://link.aps.org/doi/10.1103/PhysRevB.31.4516.
  * Granato et al. (1991) E. Granato, J. M. Kosterlitz, J. Lee, and M. P. Nightingale, Phys. Rev. Lett. 66, 1090 (1991), URL https://link.aps.org/doi/10.1103/PhysRevLett.66.1090.
  * Lee et al. (1991) J. Lee, E. Granato, and J. M. Kosterlitz, Phys. Rev. B 44, 4819 (1991), URL https://link.aps.org/doi/10.1103/PhysRevB.44.4819.
  * Li and Cieplak (1994) M. S. Li and M. Cieplak, Phys. Rev. B 50, 955 (1994), URL https://link.aps.org/doi/10.1103/PhysRevB.50.955.
  * Nightingale et al. (1995) M. P. Nightingale, E. Granato, and J. M. Kosterlitz, Phys. Rev. B 52, 7402 (1995), URL https://link.aps.org/doi/10.1103/PhysRevB.52.7402.
  * Zauner-Stauber et al. (2018b) V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete, and J. Haegeman, Phys. Rev. B 97, 045145 (2018b), URL https://link.aps.org/doi/10.1103/PhysRevB.97.045145.
  * Vidal et al. (2003) G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003), URL https://link.aps.org/doi/10.1103/PhysRevLett.90.227902.
  * Pollmann et al. (2009) F. Pollmann, S. Mukerjee, A. M. Turner, and J. E. Moore, Phys. Rev. Lett. 102, 255701 (2009), URL https://link.aps.org/doi/10.1103/PhysRevLett.102.255701.
  * Huijse et al. (2015) L. Huijse, B. Bauer, and E. Berg, Phys. Rev. Lett. 114, 090404 (2015), URL https://link.aps.org/doi/10.1103/PhysRevLett.114.090404.
  * Fisher et al. (1973) M. E. Fisher, M. N. Barber, and D. Jasnow, Phys. Rev. A 8, 1111 (1973), URL https://link.aps.org/doi/10.1103/PhysRevA.8.1111.
  * Nelson and Kosterlitz (1977) D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977), URL https://link.aps.org/doi/10.1103/PhysRevLett.39.1201.
  * Vanderstraeten et al. (2015) L. Vanderstraeten, M. Mariën, F. Verstraete, and J. Haegeman, Phys. Rev. B 92, 201111 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.201111.
  * Vanderstraeten et al. (2016) L. Vanderstraeten, J. Haegeman, P. Corboz, and F. Verstraete, Phys. Rev. B 94, 155123 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.94.155123.
  * Hübscher and Wessel (2013) D. M. Hübscher and S. Wessel, Phys. Rev. E 87, 062112 (2013), URL https://link.aps.org/doi/10.1103/PhysRevE.87.062112.
  * Wang et al. (2016) Z.-C. Wang, C.-Y. He, S.-Q. Wu, Z.-T. Tang, Y. Liu, A. Ablimit, C.-M. Feng, and G.-H. Cao, Journal of the American Chemical Society 138, 7856 (2016), ISSN 1520-5126, URL http://dx.doi.org/10.1021/jacs.6b04538.
  * Wang et al. (2019) Z.-C. Wang, Y. Liu, S.-Q. Wu, Y.-T. Shao, Z. Ren, and G.-H. Cao, Phys. Rev. B 99, 144501 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.99.144501.
  * Kirschner et al. (2018) F. K. K. Kirschner, D. T. Adroja, Z.-C. Wang, F. Lang, M. Smidman, P. J. Baker, G.-H. Cao, and S. J. Blundell, Phys. Rev. B 97, 060506 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.97.060506.