# On the derivations of the Equation of Hydrostatic Equilibrium

Kṛṣṇa Dev  Department of Physics and Astronomy, Dickinson College, Carlisle PA

Abstract

We develop and present a geometrical and an analytical derivation of the
equation of hydrostatic equilibrium in spherically symmetric stars with a
generalized stress tensor. The analytical derivation is based on the Navier-
Cauchy equation. We also critically examine the derivation of this equation
found in textbooks on stellar astrophysics and show that there are errors in
many of the derivations presented in textbooks.

##  I Introduction

On page 45 of his 1938 manuscript An Introduction to the Theory of Stellar
Structure [1], Chandrasekhar wrote, “Consider a perfect gas configuration in
gravitational equilibrium. Then
$\frac{dp}{dr}=-\frac{GM(r)}{r^{2}}\rho~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm(55)}^{\rm{\@slowromancap
i@}}$ where $r$ denotes the radius vector with the center of configuration as
origin. Furthermore, $M(r)$ is the mass inclosed inside a spherical surface of
radius $r$ and $p=R\rho T$. The foregoing equation is an elementary
consequence of hydrostatic equilibrium”. A survey of textbooks on stellar
astrophysics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], finds that the derivation of
this “elementary consequence of hydrostatic equilibrium” is not an elementary
task. We found that the derivations presented in textbooks including
Chandrasekhar’s book are incorrect and/or logically inconsistent.

The aim of this paper is to present logically consistent derivations of the
equation of hydrostatic equilibrium (EHE) in spherically symmetric Newtonian
stars with a generalized stress tensor. We will develop both an analytical and
a geometrical derivation of the equation of the EHE using the principles of
Newtonian mechanics and hydrostatic balance. The more widespread studies of
the EHE considers the pressure to be isotropic. In this paper we will take the
pressure to be anisotropic. A spherically symmetric configuration with
anisotropic pressure is defined as a system where the radial pressure is not
equal to the tangential pressure. An isotropic system is one in which the
radial and tangential pressures are equal.

This paper is organized as follows. In the next section we will present our
case that the derivations of the EHE based on spherical volume elements found
in textbooks are wrong. In sections 3 we will describe our geometrical
derivation. In section 4 we will return to the derivations found in textbooks
and consider those that use cylindrical volume elements. In section 5 we will
present an analytical derivation of the EHE based on the Navier-Cauchy
equation. In section 6 we will summarize our work and give some general
comments about derivations of the EHE found in textbooks.

In order to give a clear and detailed description of our reservations about
various derivations of the EHE found in textbooks we will find it necessary to
reproduce short extracts from some of the books. We will rewrite verbatim the
extracts from the books and in order to avoid confusion between the equations
and figures from the textbooks and our equations and figures and we will label
the equations and figures from the textbooks in blue.

We will use $r,\theta$ and $\phi$ for spherical coordinates and $s,\phi$ and
$z$ for cylindrical coordinates.

##  II  Derivation of the EHE using a spherical volume element.

All derivations of the equation of hydrostatic equilibrium in Newtonian stars
relie on the principle of hydrostatic balance and some elementary geometrical
considerations. Hydrostatic balance is condition where there is equilibrium
between the pressure forces and gravitational forces acting on a mass element
in the fluid. There are two different geometrical derivations of the EHE that
are found in textbooks. These derivations differ from each other in the choice
of the geometry of the volume element under consideration. The authors either
study the forces acting on a spherical volume element or a cylindrical volume
element. We will first consider derivations where spherical volume elements
employed.

The derivations of the EHE using a spherical volume element found in Phillips
[2] and Choudhuri [3] are incorrect. Kippenhahn and Weigert [4] and Hansen,
Kawaler and Trimble [5] consider spherical shells in their derivations and
their derivations also suffer from the curvature problem as we will describe
below.

We will quote verbatim short extracts from both Choudhuri’s and Philips’s
books and then we will discuss our reservations about their derivations.

On page 62 of his book Choudhuri writes: Let us now consider a small portion
of the shell between $r$ and $r+dr$. If $dA$ is the transverse area of this
small element, the forces exerted by pressure acting on its inward and outward
surfaces are $PdA$ and $-(P+dP)dA$, where $P$and $P+dP$ are respectively the
pressures at radii $r$and $r+dr$. So the net force arising out of pressure is
$-dPdA$, which should be balanced by gravity under equilibrium conditions. The
gravitational field at r is caused by the mass M(r) inside $r$ and is equal to
$-GM_{r}/r^{2}$. Since the mass of the small element under consideration is
$\rho drdA$, the force balance condition for it is

${\color[rgb]{0,0,1}-dPdA-\frac{GM_{r}}{r^{2}}\rho drdA=0}$

from which

${\color[rgb]{0,0,1}\frac{dP}{dr}=-\frac{GM_{r}}{r^{2}}\rho~{}~{}~{}~{}~{}~{}~{}(3.2)}$

This is the second of the stellar structure equations. This is the end of the
extract from Choudhuri’s book.

Phillips in his book on page 6 writes: We begin by finding an expression for
the acceleration of a mass element located at a distance $r$ from the centre.
The matter enclosed by a spherical shell of radius $r$ has mass

${\color[rgb]{0,0,1}m(r)=\int_{0}^{r}\rho(r^{\prime})4\pi(r^{\prime})^{2}dr^{\prime}}$

and acts as a gravitational mass situated at the centre giving rise to an
inward gravitational acceleration equal to

${\color[rgb]{0,0,1}g(r)=\frac{Gm(r)}{r^{2}}}$

There is also, in general, a force arising form the pressure gradient. To find
this we consider a small volume element located between radii $r$ and $r+dr$
of cross sectional area $\Delta A$ and volume $\Delta r\Delta A$ as
illustrated in Fig 1.1 A net force arises if the pressure on the outer surface
of the volume is not equal to the pressure on the inner surface. Indeed, the
inward force on the volume element due to the pressure gradient is

$\displaystyle{\color[rgb]{0,0,1}\left[P(r)+\frac{dp}{dr}\Delta
r-P(r)\right]~{}\Delta A=\left[\frac{dP}{dr}\right]\Delta r\Delta A}$

Bearing in mind that the mass of the volume element is
${\color[rgb]{0,0,1}\Delta M=\rho\Delta r\Delta A}$ we deduce that the inward
acceleration of any element of mass at a distance r from the center of gravity
and pressure is

$\displaystyle{\color[rgb]{0,0,1}-\frac{d^{2}r}{dt^{2}}=g(r)+\frac{1}{\rho(r)}\frac{dP(r)}{dr}}$

Note that to oppose gravity the pressure must increase towards the centre.
This ends the extract from Phillips’s book.

Figure 1:  This figure illustrates the derivation of the EHE found in [2]. It
is taken directly from page 6 the book.

The derivations of the EHE presented by Phillips and Choudhuri are wrong on
two identical issues. We will refer to these as (i) the curvature problem and
(ii) the tangential force problem.

The Curvature Problem. The authors considers the forces acting on spherical
volume elements. A figure from Phillips’s book [2] is shown in fig 1. They
both claim that the force due to the radial pressure difference between the
inner and outer surface area of the mass element is $dF(P_{r})=-dP_{r}dA$. (We
have highlighted the relevant text from the books in red.) This claim is
incorrect, because it implicitly assumes that the areas of the two
hemispherical caps at $r$ and $r+dr$ are the same. We note that if we take the
area at $r$ to be $dA$ as proposed in the textbooks then the area at $r+dr$
must be of the form be $dA+\delta A$ in order to account for the difference in
curvature of the two hemispherical caps and for $dF_{P_{r}}$ we should write

$\displaystyle dF_{P_{r}}$ $\displaystyle=$ $\displaystyle
P_{r}(r+dr)(dA+\delta A)-P_{r}(r)dA$ $\displaystyle=$
$\displaystyle\left[\frac{dP_{r}}{dr}\right]drdA+P_{r}(r)\delta A.$

Both terms on the right hand side of the above equation are of the same order
of smallness, thus ${\it a\,priori}$ there is no reason to neglect the term
$P_{r}(r)\delta A$. It is standard practice however, in most textbooks to take
the area of the two caps to be equal. This is clearly wrong. We will refer to
this issue as the the curvature problem

The Tangential Force Problem. The tangential force problem is the assumption
is that the forces on the lateral side of the cone sum to zero. This is not
shown, it is considered to be obvious. However, we will show that there is a
nonzero force $F_{t}$, that arises from the pressure acting on the lateral
side of the cone. This force $F_{t}$, has a radial component that opposes the
gravitational force on the volume element. In an extraordinarily fortuitous
scenario in fluids with isotropic pressure the radial component of the force
on the lateral side cancel the $p(r)\delta A$ term that is missing in (II).
Thus the two errors, the area problem and the tangential force problem
compensate each other and the correct EHE is found for spheres with isotropic
pressure.

In the next section we will present a proper logical derivation of the EHE
using a spherical volume element and then we will examine derivations based on
a cylindrical volume element. We will show that the use of a cylindrical
volume element in the derivation of the EHE is a clever attempt to solve the
two problems the plague derivations based on spherical volume elements.

##  III Geometrical derivation of the equation for hydrostatic equilibrium
with anisotropic pressure.

The equation of hydrostatic equilibrium for the case with anisotropic pressure
is

$\frac{dP}{dr}=-\frac{G\rho(r)M(r)}{r^{2}}+\frac{2}{r}(P_{t}-P_{r})$ (2)

here $G$ is Newton’s gravitational constant. The standard approach to the
derivation of this equation is apply the principle of hydrostatic balance
i.e., to equate the pressure forces with gravitational force acting on a mass
element. We consider a mass element in a spherically symmetric configuration
with anisotropic pressure. In equilibrium the forces acting on a mass element
$dm$ must sum to zero:

${dF}_{g}+{dF}_{P_{r}}+{dF}_{P_{t}}=0~{},$ (3)

where ${dF}_{g}$ is the force due to gravity, ${dF}_{P_{r}}$ is the force due
to the radial pressure, ${dF}_{P_{t}}$ is the force due to the tangential
pressure.

Figure 2:  A volume element $dv$ subjected to stresses from $P_{t}$ and
$P_{r}$

We will now derive expressions for the forces. The gravitational force on a
mass element $dm$ of volume $dv$ located at a distance $r$ from the center of
the star is

$dF_{g}=-\frac{GM\left(r\right){\rm d}m}{r^{2}},$ (4)

here

$M\left(r\right)=\int_{0}^{r}4\pi\rho\left(r\right)r^{2}dr$ (5)

is the total mass contained in a sphere of radius $r$ and

$dm=\rho(r)dv$ (6)

is the mass of the fluid element under consideration. Here we will choose the
volume element to be a conical frustum that lies between $r$ and $r+dr$ with
hemispherical caps. The volume of a conical frustum $dv_{cf}$ is obtained from
the standard spherical volume element $dv=r^{2}\sin\theta drd\theta d\phi$, by
integration over $\phi$ i.e., $dv_{cf}=2\pi r^{2}\sin\theta drd\theta$. Also
since here, $\theta=d\theta$, then $\sin\theta=d\theta$ and $dv_{cf}=2\pi
r^{2}dr(d\theta)^{2}$. The area of the hemispherical cap ar $r$ is $2\pi
r^{2}\sin\theta d\theta\approx 2\pi r^{2}(d\theta)^{2}$. A conical frustum is
shown in fig. 3. This choice of volume element will allow us to readily
compute the forces from the radial and tangential pressures.

Thus the gravitational force on the mass element is

$dF_{g}=-\rho\frac{GM(r)}{r^{2}}2\pi r^{2}dr(d\theta)^{2}.$ (7)

The force on the mass element due to the radial pressure is

$\displaystyle dF_{P_{r}}$ $\displaystyle=$ $\displaystyle
P_{r}(r+dr)dA(r+dr)-P_{r}dA(r)$ $\displaystyle=$
$\displaystyle(P_{r}+\frac{dP_{r}}{dr}dr)2\pi(r+dr)^{2}(d\theta)^{2}-P_{r}2\pi
r^{2}(d\theta)^{2}$ $\displaystyle=$ $\displaystyle 4\pi
rdr(d\theta)^{2}P_{r}+2\pi r^{2}dr(d\theta)^{2}\frac{dP_{r}}{dr}.$

If we consider only the forces due to gravity and the radial pressure on the
mass element then we find that the equation for hydrostatic equilibrium is

$\begin{split}4\pi rdr(d\theta)^{2}P_{r}+2\pi
r^{2}dr(d\theta)^{2}\frac{dP_{r}}{dr}=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
-\rho\frac{GM(r)}{r^{2}}2\pi r^{2}dr(d\theta)^{2}.\end{split}$ (9)

This equation simplifies to

$\frac{dP_{r}}{dr}+\frac{2}{r}P_{r}=-\frac{\rho(r)GM(r)}{r^{2}}.$ (10)

If the second term on the left hand side in eq. (10) was not present then
resulting equation is standard equation for hydrostatic equilibrium in stars
with isotropic pressure. This second term is present because the the area of
hemispherical caps at $r$ and $r+dr$ are not equal. This is a crucial point:
because the curvature of the the two hemispherical caps are different, their
areas are also different. In several textbooks on stellar astrophysics this
point is not noted and the authors derive the equation for hydrostatic
equilibrium by making two erroneous assumptions: (i) the areas of the two
hemispherical caps are equal, (the curvature problem) and (ii) the
contribution to the total force from the tangential pressure acting on the
lateral side of the cone is zero (the tangential force problem). We have shown
above that first assumption is incorrect and below we will demonstrate that
there is a non-zero contribution to the net force from the tangential pressure
acting on the lateral side of the of the cone. This extra term exactly cancels
the second term on the left hand side of (10), when the pressure is isotropic.
If the pressure is anisotropic the two terms do not cancel and we arrive at
the equation for hydrostatic equilibrium with anisotropic pressure (2).

Figure 3: A conical frustum located between $r$ and $r+dr$ Figure 4: The force
due on conical frustum located between $r$ and $r+dr$ due to $P_{t}$

The force due to the tangential pressures $P_{\theta}$ and $P_{\phi}$ is

$\displaystyle dF_{P_{t}}=(P_{\theta}+P_{\phi})dA_{cf}$ (11)

where $dA_{cf}$ is the lateral area of a conical frustum between $r$ and
$r+dr$. From fig 3 we find that

$\begin{split}dA_{cf}=\pi(a+b)\times{\rm
length\,of\,slant\,side}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
~{}~{}~{}~{}~{}=\pi(rd\theta+(r+dr)d\theta)dr=2\pi rd\theta dr\end{split}$
(12)

Since $P_{\theta}=P_{\phi}=P_{t}$ for any spherical symmetric configuration,
the component of $dF_{P_{t}}$ that acts along the radial direction is

$\displaystyle(dF_{P_{t}})_{r}=4\pi rdr(d\theta)^{2}P_{t}~{}~{}~{}~{}~{}$ (13)

Hydrostatic equilibrium requires

$dF_{P_{r}}+(dF_{P_{t}})_{r}=dF_{g}.$ (14)

Substituting for $dF_{P_{r}}$ (III), $(dF_{P_{t}})_{r}$ (13), and $dF_{g}$
(7), we find that

$\begin{split}4\pi rdr(d\theta)^{2}P_{r}+2\pi
r^{2}dr(d\theta)^{2}\frac{dP_{r}}{dr}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
-4\pi rdr(d\theta)^{2}P_{t}=-\rho\frac{GM(r)}{r^{2}}2\pi
r^{2}dr(d\theta)^{2}.\end{split}$ (15)

Simplifying the above expression we find:

$\displaystyle\frac{dP_{r}}{dr}=-\frac{\rho(r)GM(r)}{r^{2}}+\frac{2}{r}(P_{t}-P_{r}).$
(16)

This is the equation of hydrostatic equilibrium for Newtonian stars with
anisotropic pressure. In the next section we return to our critique of the
derivations of the EHE found in textbooks and consider derivations based on
cylindrical volume elements.

##  IV  The derivation of the EHE using a cylindrical volume element.

Cylindrical volume elements are mentioned in the derivations found in the
following textbooks: Chandrasekhar [1], Prialnik [6], Clayton [7], Carroll and
Ostlie [8], Zelik, Geogory and Smith [9], Ryden and Peterson [10] and Maoz
[11].

We will first reproduce extracts from Clayton’s and Maoz’s books and then we
describe our basic reservations about this and other derivations based on
cylindrical volume elements.

Figure 5:  This figure is used in the derivation of the EHE found in Clayton’s
book [7]. It is labelled Fig 2.14 in the book.

Clayton writes on page 130 of his book: “Consider the small cylindrical volume
element with axis of length $dr$ parallel to the radius vector at the point
$r$ and having cross-sectional area equal $dA$. Let $dP$ be the pressure
increment associated with $dr$ Then the radial force on this volume element
due to the pressure differential is
${\color[rgb]{0,0,1}F_{P}=Pda-(P+dP)dA=-dPdA~{}~{}~{}~{}~{}~{}(\operatorname{2-177})}$
since the pressure will actually decrease in the radial direction, the
differential $dP$ will be negative and the pressure force will then be
positive. If the volume element is not to be accelerated upward by $F_{P}$, it
is necessary that it be exactly cancelled by the central gravitational force
in the volume element. If $dm$ is the mass of the volume element and $M(r)$ is
the mass interior to the spherical surface at radius $r$, the gravitational
force is
${\color[rgb]{0,0,1}F_{g}=-G\frac{M(r)dm}{r^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\operatorname{2-178})}$
where  ${\color[rgb]{0,0,1}M(r)=\int_{0}^{r}\rho(r)4\pi
r^{2}dr~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\operatorname{2-179})}$ The
gravitational constant $G=6.670\times 10^{-8}\,dyne\,cm^{2}\,g^{-2}$. In
writing these equations, we have clearly considered only the spherically
symmetric problem. The important problem of rotation is discussed in Chap. 6.
Because the mass of the small volume element is $\rho dAdr=dm$, we have
${\color[rgb]{0,0,1}0=F_{P}+F_{G}=-\frac{dP}{dr}-G\frac{\rho
M}{r^{2}}~{}~{}~{}~{}~{}(\operatorname{2-180})}$ This force balance together
with the definition of ${\rm{\color[rgb]{0,0,1}Eq.(\operatorname{2-180})}}$,
constitutes one of the most important condition to be satisfied by stellar
structures.”

This ends the extract from Clayton’s book. A shorter but similar discussion is
given in Maoz’s book [11]. Maoz writes on page 23 of his book:

Figure 6:  This figure is used in the derivation of the EHE found in Maoz’s
book [11]. It is labelled Figure 3.1 in the book.

Consider now a small, cylinder-shaped mass element inside a star, with A the
area of the cylinder’s base, and $dr$ its height see Fig. (3.1). If there is a
pressure difference $dP$ between the top of the cylinder and its bottom, this
will lead to a net force $AdP$ on the mass element, in addition to the force
of gravity toward the center. Equilibrium will exist if

${\color[rgb]{0,0,1}-\frac{GM(r)dm}{r^{2}}-AdP=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3.17)}$

But

${\color[rgb]{0,0,1}dm=\rho(r)Adr,~{}~{}~{}~{}~{}~{}~{}(3.18)}$

which leads us to the first equation of stellar structure, the equation of
hydrostatic equilibrium

${\color[rgb]{0,0,1}\frac{dP(r)}{dr}=-\frac{G\rho(r)M(r)}{r^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3.19)}$

This is the end of the extract from Maoz’s book.

We find that derivations of the EHE that use cylindrical volume elements
carefully circumnavigate the problems encountered when using spherical volume
elements, i.e., the curvature problem and the tangental force problem. We can
see this by examining fig 5 taken from Clayton’s book and fig 6 taken from
Maoz’s book. An inspection of the diagrams shows that the surface area of the
caps of cylindrical volume elements are flat discs. Thus, this geometry
immediately eliminates the the curvature problem. Further, since pressure
forces always act directly perpendicular to a given surface, the pressure
forces on the curved surface of the cylinder will sum to zero and this solves
the the tangential force problem. However, we note that in the interior of the
stellar configurations that we are considering both the pressure and the
gravitational field have spherical symmetry. Thus, the use of volume elements
with cylindrical symmetry in a spherical symmetric medium should be
problematic. We have some reservations about the use of cylindrical volume
elements in the derivation of the EHE. We will refer to these as (i) the area
problem and (ii) the anisotropy problem.

(i) The area problem. We found in our studies of the various derivations of
the EHE using cylindrical volume elements that all authors arrive at an
expression of the following form:

${\color[rgb]{0,0,1}-\frac{GM(r)}{r^{2}}drA-dPA=0.}$

This is Equation 3.17 in Maoz’s book. The term $dPA$ represents the force that
arises due to the pressure difference between the caps. An implicit assumption
here is that the radial pressure i.e., a spherically symmetric quantity is
constant over flat surfaces with cylindrical symmetry. Maoz does not
explicitly state this in his derivation but he could not have written his
expression for the pressure force without this assumption. Similarly, Clayton
could only arrive at $F_{P}=-dPdA$ in his Eq (2-177) using this assumption.
However, we disagree with this assumption. Our reasoning is as follows. We are
asked to consider a cylindrical volume element of height $r$ and surface area
$dA$. The change in height $dr$ is necessary for there to exist a radial
pressure gradient between the top and bottom of the cylinder, i.e., when
$r\rightarrow r+dr$, then $P(r+dr)-P(r)=(dP/dr)dr\neq 0$. We do not object to
this preposition.

Figure 7:  This figure illustrates the area problem

However, we have a conceptual difficulty in understanding how is it possible
for the radial pressure, a spherically symmetric quantity, to be constant over
a flat area that is perpendicular to the radial direction (see fig 7). The
pertinent question here is, how small is the area $dA$? Certainly, it must be
at least $\pi(ds)^{2}$, this is because the smallest possible value the
cylindrical radial coordinate $s$ can have is $ds$. Any quantity smaller than
$ds$ is a point. Now we have established that $dA$ must have a radius of
minimum length $ds$, we ask why is there a pressure difference with a change
in height $dr$ but there is no pressure difference along the surface when $s$
changes by at least $ds$, certainly when $s\rightarrow s+ds$, we must have
$P(r(s+ds))-P(r(s))=(dP/dr)(dr/ds)ds\neq 0$. If $ds=0$ then we are not
considering an infinitesimal area, the geometrical object under consideration
is a point and the derivation is moot. We will call this the area problem.

The relevant question now in view of the area problem, is what form does the
EHE assume when using a cylindrical volume element? The expression for the
gravitational force on the mass element is undisputed. We now need to find an
expression for the force due to the pressure difference.

The gravitational force on the cylindrical mass element $dm=\rho dv$ is

$\displaystyle F_{g}$ $\displaystyle=$
$\displaystyle-\frac{GM(r)}{r^{2}}dm=-\frac{GM(r)}{r^{2}}\rho dv$
$\displaystyle=$
$\displaystyle-\frac{GM(r)\rho}{r^{2}}dzA_{c}\approx-\frac{GM(r)\rho}{r^{2}}drA_{c}.$

In the last line in the above equation we have used the approximation $dz=dr$.
The force due to the pressure on the bottom cap is given by an expression of
the form

$\displaystyle F_{P_{b}}$ $\displaystyle=$ $\displaystyle\int P(r(s))dA_{c}$
$\displaystyle=$
$\displaystyle\int\left(P(r)+\frac{dP(r(s))}{ds}ds\right)dA_{c}.$

Here $dA_{c}$ is an area element of the flat cap. The force due to the
pressure on the top cap is given by an expression of the form

$\displaystyle F_{P_{t}}$ $\displaystyle=$ $\displaystyle\int
P(r(s)+dr(s))dA_{c}$ $\displaystyle=$
$\displaystyle\int\left(P(r)+\frac{dP(r(s))}{ds}ds+\frac{dP(r)}{dr}dr\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}\left.+\frac{d^{2}P(r(s))}{dsdr}dsdr\right)dA_{c}.$

The net force due to the pressure difference is

$\displaystyle dF_{P}$ $\displaystyle=$
$\displaystyle\int\left(\frac{dP(r)}{dr}dr+\frac{d^{2}P(r(s))}{dsdr}dsdr\right)dA_{c}$
$\displaystyle=$
$\displaystyle\frac{dP}{dr}drA_{c}+\int\frac{d^{2}P(r(s))}{dsdr}dsdrdA_{c}.$

For equilibrium we need

$F_{g}+F_{P}=0.$ (21)

Substituting into this equation we find

$-\frac{GM(r)\rho}{r^{2}}drA_{c}-\frac{dP}{dr}drA_{c}-\int\frac{d^{2}P(r(s))}{dsdr}dsdrdA_{c}=0.$
(22)

This the EHE for a cylindrical volume element in a spherically symmetric
system. We will obtain the standard EHE in a spherically symmetric medium if
$dP/ds=0$, however, if this condition holds, then the caps have a surface area
equal to zero and the cylinder is reduced to a needle.

(ii). The anisotropy problem. We note that despite our dedicated efforts, we
were unable to derive an EHE for spherically symmetric systems with
anisotropic pressure using a cylindrical volume element. This is probably not
surprising, since a cylindrical volume element has no surface over which the
components of a spherically symmetric pressure tensor are directly
perpendicular or parallel. Thus the pressure forces on each surface is due to
a mixture of both the radial and tangential pressures acting on those
surfaces. We think it is very challenging mathematical problem to obtain the
correct EHE for systems with spherical symmetry and anisotropic pressure by
studying the forces on a cylindrical volume element. This is the anisotropy
problem.

We consider the use of a cylindrical volume element in the derivation of the
EHE to be a “geometrical ruse” designed to circumnavigate the problems
encountered when using spherical volume elements, i.e., the curvature problem
and the tangental force problem, and not a serious logically consistent
mathematical derivation of the EHE.

##  V An analytical derivation of the equation for hydrostatic equilibrium
with anisotropic pressure.

A pertinent question to ask is since the EHE is such a well known and
extensively studied equation, does there exist proper analytical derivations
of this equation that does not rely on geometrical subtleties and “ruses” ?
The answer is yes, such derivations do exist. Examples of analytical
derivations can be found in Collins [12] and Thorne and Blandford [13].

In his derivation Collins studies the flow of fluids. He starts with the
Boltzmann transport equation and develops the Euler-Lagrange equations of
hydrodynamic flow:

$\frac{\partial\vec{u}}{\partial
t}+(\vec{u}\cdot\vec{\nabla})\vec{u}=-\vec{\nabla}\Phi-\frac{\vec{\nabla}\cdot{\overleftrightarrow{P}}}{\rho}$
(23)

Here $\vec{u}$ is the flow velocity of a particle, $\Phi$ is the potential
associated with an external central force acting on the particles and
$\overleftrightarrow{P}$ is the pressure tensor. When the flow velocity is
constant and isotropic the entire left-hand side of (23) is zero and we obtain

$\vec{\nabla}\cdot{\overleftrightarrow{P}}=-\rho\vec{\nabla}\Phi$ (24)

This is the EHE and here it is an expression for the conservation of momentum.

Thorne and Blandford in their approach to derivation of the EHE, applies the
ideas and concepts developed for studying elastostatics in solids to the study
of hydrostatics in fluids. The fundamental equation of elastodynamics is the
Navier-Cauchy equation. The Navier-Cauchy equation describes the displacements
in an elastic solid under the influence of external forces or internal
stresses. The Navier-Cauchy equation can be written as

$\overrightarrow{\nabla}\cdot{{{\overleftrightarrow{\sigma}}}}+\vec{f}=\rho\frac{d^{2}\vec{\xi}}{dt^{2}}.$
(25)

Here ${\overleftrightarrow{\sigma}}$ is the Cauchy stress tensor, $\vec{f}$ is
external force density and $\vec{\xi}$ is the displacement vector.
Elastostatic equilibrium is achieved when the time derivatives are set to
zero. Thus the fundamental equation of elastostatics is

$\overrightarrow{\nabla}\cdot{{{\overleftrightarrow{\sigma}}}}=-\vec{f}$ (26)

This equation expresses elastostatic balance between the internal stresses in
the solid and the external applied force(s). In hydrostatics a similar balance
is achieved between the pressure forces and the applied external forces. The
fundamental equation of hydrostatics has the same form as (26) except now the
Cauchy stress tensor is replaced by the fluid’s stress tensor,
$\overleftrightarrow{T}$. Thus the fundamental equation of hydrostatics is

$\overrightarrow{\nabla}\cdot{{{\overleftrightarrow{T}}}}=-\vec{f}$ (27)

We are considering fluids with anisotropic pressure that have no shear or
vorticity thus here the stress tensor has a diagonal form :

${\overleftrightarrow{T}}=\begin{bmatrix}P_{r}(r)&0&0\\\ 0&P_{t}(r)&0\\\
0&0&P_{t}(r)\end{bmatrix}$ (28)

In the interior of stars the only body force acting is gravity and it has only
a radial component. In evaluating (27) we find that only the $r$ component of
$\overrightarrow{\nabla}\cdot{{{\overleftrightarrow{T}}}}$ gives a non-trivial
expression. The $\theta$ and $\phi$ equations gives zero on both sides of the
equations. Writing out the $r$ component of (27) we find

$\begin{split}\frac{\partial T_{rr}}{\partial r}+\frac{1}{r}\frac{\partial
T_{r\theta}}{\partial\theta}+\frac{1}{r\sin\theta}\frac{\partial
T_{r\phi}}{\partial\phi}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
+\frac{1}{r}(2T_{rr}-T_{\theta\theta}-T_{\phi\phi}+T_{r\theta}\cot\theta)=-f.\end{split}$
(29)

Substituting $f_{r}=\rho g$ ($g$ is the gravitational acceleration), and
taking into consideration the form of ${\overleftrightarrow{T}}$ we find that
(29) reduces to

$\frac{\partial P_{r}(r)}{\partial r}+\frac{1}{r}(2P_{r}(r)-2P_{t}(r))=-\rho
g$ (30)

In the interior of a spherically symmetric star at a distance $r$ from the
center

$g=\frac{GM(r)}{r^{2}}$ (31)

Substituting this expression for $g$ in (30) and rearranging terms we find

$\displaystyle\frac{dP_{r}}{dr}=-\frac{\rho(r)GM(r)}{r^{2}}+\frac{2}{r}(P_{t}-P_{r})$
(32)

This is the equation of hydrostatic equilibrium for Newtonian stars with
anisotropic pressure. Thus we have derived the EHE analytically without the
need to consider geometrical subtleties.

##  VI Conclusion

We have given two derivations of the equation of hydrostatic equilibrium. The
first derivation applied the principles of Newtonian mechanics and simple
geometrical arguments to obtain the EHE.

Our second derivation is purely analytical and it based on applying the
principles of elastostatics in solids to hydrostatics in fluids. We have also
shown that the derivation of this equation as it is found in almost all
textbooks on stellar astrophysics is flawed.

Finally, our advice for authors who find it necessary to include a derivation
of the EHE in their manuscripts is to either adopt the correct derivations
presented in this paper and elsewhere [13], or follow the path taken by Bhom-
Vitense [14] or Winn [15]. They both present an elementary derivation of the
equation of hydrostatic equilibrium in a constant gravitational field that is
found in introductory physics textbooks and then generalize the EHE to
situations with spherical symmetry. The EHE is well known, there is no need to
present incorrect derivations or resort to “geometrical ruses”.

##  VII Acknowledgements

We thank Lars English for reading and editing the manuscript.

## References

  * [1] S. Chandrasekhar, An Introduction to the Theory of Stellar Structure, (Univ. of Chicago, Chicago, 1938).
  * [2] A. C. Phillips, The Physics of Stars, (Wiley, New York, 1993).
  * [3] A. R. Choudhuri, Astrophysics for Physicists, (Cambridge, New York, 2010).
  * [4] R. Kippenhahn and A. Weigert, Stellar Structure and Evolution, (Springer-Verlag, Berlin, 1991).
  * [5] C. J. Hansen, S. D. Kawaler and V. Trimble Stellar Interiors: Physical Principles, Structure and Evolution, (Springer-Verlag, Berlin, 2004).
  * [6] D. Prialnik, An Introduction to the Theory of Stellar Structure and Evolution, (Cambridge, New York 2000).
  * [7] D. D. Clayton, Principles of Stellar Evolution and Nucleosynthesis, (The University of Chicago Press, Chicago, 1983).
  * [8] B. Carroll and D. Ostlie An Introduction to Modern Astrophysics, (Cambridge, New York 2017).
  * [9] M. Zeilik, S. Gregory and E. Smith, Introductory to Astronomy and Astrophysics (Saunders, New York 1992).
  * [10] B. Ryden and B. Peterson, Foundations of Astrophysics, (Addison Wesley, New York 2010).
  * [11] D. Maoz. Astrophysics in a Nutshell, (Princeton, Princeton 2016).
  * [12] G. W. Collins, The Fundamentals of Stellar Astrophysics, (W. H. Freeman, San Francisco 1989).
  * [13] K. Thorne and R. Blandford, Modern Classical Physics, (Princeton, Princeton 2017).
  * [14] E. Bhom-Vitense Introduction to Stellar Astrophysics, Volume 3, Stellar Structure and Evolution, (Cambridge, New York 1992).
  * [15] J. N. Winn Introduction to Astrophysics (The Great Courses, Chantilly 2018).


#  Emergence of Chern Supermetal and Pair-Density Wave through Higher-Order
Van Hove Singularities in the Haldane-Hubbard Model

Pedro Castro  Department of Physics, Emory University, 400 Dowman Drive,
Atlanta, GA 30322, USA  Daniel Shaffer  Department of Physics, Emory
University, 400 Dowman Drive, Atlanta, GA 30322, USA  Yi-Ming Wu  Stanford
Institute for Theoretical Physics, Stanford University, Stanford, California
94305, USA  Luiz H. Santos  Department of Physics, Emory University, 400
Dowman Drive, Atlanta, GA 30322, USA

###### Abstract

While advances in electronic band theory have brought to light new topological
systems, understanding the interplay of band topology and electronic
interactions remains a frontier question. In this work, we predict new
interacting electronic orders emerging near higher-order Van Hove
singularities present in the Chern bands of the Haldane model. We classify the
nature of such singularities and employ unbiased renormalization group methods
that unveil a complex landscape of electronic orders, which include
ferromagnetism, density-waves and superconductivity. Importantly, we show that
repulsive interactions can stabilize long-sought pair-density-wave state and
an exotic Chern supermetal, which is a new class of non-Fermi liquid with
anomalous quantum Hall response. This framework opens a new path to explore
unconventional electronic phases in two-dimensional chiral bands through the
interplay of band topology and higher-order Van Hove singularities.

Introduction– The Haldane model Haldane (1988) provides a minimal description
of Chern bands displaying quantized Hall conductance albeit in zero magnetic
field, which are realized when Dirac fermions in graphene bands acquire a
time-reversal broken, inversion symmetric mass at the two valleys
$\pm{\bm{K}}$ of the Brillouin zone (BZ). Besides the realization of Haldane
Chern bands in ultracold fermions Jotzu  _et al._ (2014), new correlation-
driven Chern bands recently identified Spanton  _et al._ (2018); Sharpe  _et
al._ (2019); Serlin  _et al._ (2020); Saito  _et al._ (2021); Xie  _et al._
(2021); Nuckolls  _et al._ (2020); Wu  _et al._ (2021a); Das  _et al._ (2021);
Choi  _et al._ (2021); Park  _et al._ (2021); Stepanov  _et al._ (2021);
Pierce  _et al._ (2021); Yu  _et al._ (2021) in moiré materials open new paths
to explore interaction effects in topological bands. The role of interactions
in Haldane Chern bands has received substantial attention. In particular, in
the strong coupling regime obtained when the interaction scale $U$ is much
stronger than the bandwidth $W$ of a Chern band that is separated from other
bands by an energy gap $\Delta$ such that $\Delta\gg U\gg W$, electronic
correlations can stabilize rich fractionalized phases Neupert  _et al._
(2011); Sheng  _et al._ (2011); Tang  _et al._ (2011); Sun  _et al._ (2011).
Furthermore, correlated phases of the repulsive Haldane-Hubbard model at
commensurate $1/2$ and $1/4$ fillings have been investigated using numerical
methods and mean-field studies Mai  _et al._ (2023); He  _et al._ (2011);
Zheng  _et al._ (2015); Wu  _et al._ (2016); Arun  _et al._ (2016); Vanhala
_et al._ (2016); Imriška  _et al._ (2016); Tupitsyn and Prokof’ev (2019); Yi
_et al._ (2021).

On the other hand, defying expectations that the weak coupling regime $(U\ll
W)$ in Chern bands leads to a stable Fermi liquid (FL) fixed point, recent
analysis Shaffer  _et al._ (2022); Shaffer and Santos (2022) has uncovered new
FL instabilities in partially filled Hofstadter bands Hofstadter (1976)
(generally supporting non-zero Chern number Thouless  _et al._ (1982)) due to
the interplay of repulsive interactions and logarithmic Van Hove singularities
(VHS). In such fractal bands, the magnet flux per unit cell acts a control
knob changing the landscape of VHSs and opening new interaction channels. A
pressing question then arises: Can VHS catalyze FL instabilities in Haldane
Chern bands in the absence of a magnetic field? Notably, while the relation
between VHS and electronic correlations has been greatly emphasized in modern
2D materials, from moiré Li  _et al._ (2010); Kerelsky  _et al._ (2019); Wu
_et al._ (2021b) to kagomé metals Kang  _et al._ (2022); Neupert  _et al._
(2022), little is known about their influence in Chern bands.

In this Letter, we unveil a new scheme to investigate correlation effects in
Chern bands, focusing on the Haldane-Hubbard model as a paradigmatic system to
address the confluence of band topology and electronic correlations. Departing
from previous studies Mai  _et al._ (2023); He  _et al._ (2011); Zheng  _et
al._ (2015); Wu  _et al._ (2016); Arun  _et al._ (2016); Vanhala  _et al._
(2016); Imriška  _et al._ (2016); Tupitsyn and Prokof’ev (2019); Yi  _et al._
(2021), we analyze a new regime characterized by incommensurate fillings
reached when the Fermi energy lies near VHS in the Haldane Chern bands. The
diverging density of states (DOS) near localized pockets in the BZ allows a
treatment of interactions at weak coupling regime using unbiased RG methods.
The main results of this work are:
(1) A novel classification of VHS in Haldane Chern bands: We analytically
identify new logarithmic Van Hove (1953) and higher-order VHS (HOVHS) Shtyk
_et al._ (2017); Yuan  _et al._ (2019) of Haldane Chern bands, beyond the
conventional VHSs in graphene Neto  _et al._ (2009). These new saddle points
are controlled by the second neighbor hopping amplitude $t_{2}$ and the phase
$\phi$ (see Fig. 1.), which break time reversal symmetry while preserving
spatial inversion. Notably, we identify a pair of HOVHS at $\pm{\bm{K}}$
occurring throughout the boundary of regions II and III of Fig. 1. Such HOVHSs
yield diverging DOS $D(\varepsilon)\sim|\varepsilon|^{-1/3}$, promoting
strongly enhanced low temperature susceptibilities in all particle-particle
and particle-hole channels, which, consequently, open a new path to explore
competing electronic orders in Chern bands via these HOVHS.
(2) Novel FL instabilities through competing orders near HOVHS: We employ
perturbative RG Shankar (1994); Maiti and Chubukov (2013) to study FL
instabilities in the vicinity of such pair of HOVHS related by inversion
symmetry. While this HOVHS 2-patch RG was analyzed in bilayer graphene Shtyk
_et al._ (2017) and moiré systems Hsu  _et al._ (2021); Wu  _et al._ (2023),
the situation in Haldane Chern bands is distinct in that band topology non-
trivially constraints the RG flows. This occurs because the topological
winding ($\pm 1$) of Haldane Chern bands is associated with electronic
wavefunctions on opposite $\pm{\bm{K}}$ valleys which have support on opposite
sublattices. This entails a form of sublattice interference (SI), which
suppresses certain interaction channels, profoundly modifying the RG flows and
the resulting orders. While this SI was first noticed in the kagome lattice
Kiesel and Thomale (2012); Wu  _et al._ (2022), to our knowledge, the
existence of this important effect and its connection to competing orders in
Haldane Chern bands has not received earlier consideration, and is one of the
central results of this work.

Incorporating this novel SI into RG analysis brings forth a rich phase
diagram, shown in Fig.3, containing a host of ordered phases including
ferromagnetism (FM), density-wave (DW), superconductivity (SC) and pair-
density-wave (PDW). Remarkably, we also identify a mechanism whereby repulsive
interactions can stabilize an exotic interacting supermetal fixed point Isobe
and Fu (2019), which arises here as a non-Fermi liquid with quantum anomalous
Hall response Nagaosa  _et al._ (2010). We coin this new phase a Chern
supermetal. While the supermetal in Isobe and Fu (2019) relies on an isolated
HOVHS (see also Aksoy  _et al._ (2023)), we demonstrate that SI and
spontaneous generation of a staggered mass under the RG flow in the Haldane
Chern bands is pivotal in suppressing interaction channels between the two
HOVHS that would normally drive the system towards an ordered phase Shtyk  _et
al._ (2017); Classen  _et al._ (2020). The same mechanism also allows for a
pair-density-wave (PDW) state Agterberg  _et al._ (2020) with momentum
$\pm{\bm{K}}$ to emerge from purely repulsive interactions. Our results thus
constitute a new paradigm to explore electronic in correlations topological
Chern bands via HOVHS.

Figure 1: (a) Lattice model. (b) Landscape of VHSs. The boundary between II
and III highlighted in red is where two higher order VHSs located at
$\pm\bm{K}$ emerge. (c) Energy contours show the evolution of VHSs for a fixed
$\phi=\pi/2$ but various $t$, which are marked as the black dots in (b). The
yellow dashed curves are the Fermi surfaces at Van Hove filling.

Model. We investigate the spin-degenerate single particle Hamiltonian Haldane
(1988)

$H_{0}=\sum_{\sigma=\uparrow,\downarrow}\sum_{{\bm{k}}\in\textrm{BZ}}c^{\dagger}_{{\bm{k}}\sigma}\mathcal{H}_{{\bm{k}}}c_{{\bm{k}}\sigma}\,,\quad\mathcal{H}_{{\bm{k}}}=B_{0,{\bm{k}}}\tau_{0}+{\bm{B}}_{{\bm{k}}}\cdot{\bm{\tau}}\,,$
(1)

where BZ is the first Brillouin zone,
$c_{{\bm{k}}\sigma}=(c_{A,{\bm{k}},\sigma},c_{B,{\bm{k}},\sigma})^{T}$ is the
fermionic operator on sublattices $A$ and $B$; $\tau_{\mu}$ ($\mu=0,1,2,3$)
represents the $2\times 2$ identity and three Pauli matrices acting on the
sublattice degrees of freedom, and

$\displaystyle
B_{0,{\bm{k}}}=2t_{2}\cos\phi\sum_{i=1}^{3}\cos{\bm{k}}\cdot\bm{b}_{i}\,,$
$\displaystyle\,{\bm{B}}_{{\bm{k}}}=\sum_{i=1}^{3}\begin{pmatrix}t_{1}\cos{\bm{k}}\cdot\bm{a}_{i}\\\
-t_{1}\sin{\bm{k}}\cdot\bm{a}_{i}\\\
2t_{2}\sin\phi\sin{\bm{k}}\cdot\bm{b}_{i}\end{pmatrix}\,,$ (2)

where $\bm{a}_{1}=a(1,0),\ \bm{a}_{2}=\tfrac{a}{2}(-1,\sqrt{3}),\
\bm{a}_{3}=\tfrac{a}{2}(-1,-\sqrt{3})$ are vectors connecting A to nearest
neighbor B sites and
$\bm{b}_{1}={\bm{a}}_{2}-{\bm{a}}_{3},\bm{b}_{2}={\bm{a}}_{3}-{\bm{a}}_{1},\bm{b}_{3}={\bm{a}}_{1}-{\bm{a}}_{2}$.
$a$ is the lattice constant that we henceforth set to one and we define
$t=t_{2}/t_{1}$. The single particle Hamiltonian 1 is diagonalized
$H_{0}=\sum_{\sigma}\sum_{n=\pm}\,\varepsilon_{n,{\bm{k}}}\psi^{\dagger}_{n,\sigma,{\bm{k}}}\psi_{n,\sigma,{\bm{k}}}$
by the unitary transformation
$c_{s,{\bm{k}},\sigma}=\sum_{n=\pm}u_{sn}({\bm{k}})\psi_{n,{\bm{k}},\sigma}$,
($s=A,B$) leading to spin degenerate energy bands
$\varepsilon_{\pm,{\bm{k}}}=B_{0,{\bm{k}}}\pm|{\bm{B}}_{{\bm{k}}}|\,.$ In
order to classify the critical points
${\bm{\nabla}}_{{\bm{k}}}\varepsilon_{\pm,{\bm{k}}}=0$, we analyse the
determinant of the Hessian
$\mathbb{H}_{n,{\bm{k}}}=\textrm{det}\Big{(}\frac{\partial^{2}\varepsilon_{n,{\bm{k}}}}{\partial
k_{i}\partial k_{j}}\Big{)}\,,$ which yields the local minima/maxima
($\mathbb{H}_{n,{\bm{k}}}>0$), conventional saddle points
($\mathbb{H}_{n,{\bm{k}}}<0$) and higher order saddles
($\mathbb{H}_{n,{\bm{k}}}=0$).

Henceforth we focus on the upper band $\varepsilon_{+}$ with analogous
considerations holding for $\varepsilon_{-}$. The model at $t=0$ reduces to
graphene Neto  _et al._ (2009) whose critical points are $\bm{\Gamma}=(0,0)$,
$\pm\bm{K}=\pm\left(\frac{2\pi}{3},\frac{2\pi}{3\sqrt{3}}\right)$,
$\bm{M}_{1}=\left(\frac{2\pi}{3},0\right)$,
$\bm{M}_{2}=\left(\frac{\pi}{3},\frac{\pi}{\sqrt{3}}\right)$ and
$\bm{M}_{3}=\left(-\frac{\pi}{3},\frac{\pi}{\sqrt{3}}\right)$. $\bm{\Gamma}$
is the maximum point, the valleys $\pm{\bm{K}}$ corresponds to minima and
$\bm{M}_{i}$ are saddle points. By tracking the behavior of the Hessian at
these points, we arrive at the diagram displayed in Fig. 1-(b) (only the
region $\phi=[0,\pi]$ is displayed since the Hessian is invariant under
$\phi\rightarrow 2\pi-\phi$). Analysis of critical points reveals:
Regions I, I’ and I”: In these, ${\bm{M}}_{i}$ are saddle points. In I,
$\Gamma$ is a maximum and $\pm{\bm{K}}$ are minima; this pattern reverses in
I’ and I”. The boundary between regions I and II is defined by
$t=\tau_{12}(\phi)=\frac{1}{32\sin^{2}\phi}\left[\cos\phi+\sqrt{\cos^{2}\phi+32\sin^{2}\phi}\right],$
along which the Hessian vanishes, signaling a HOVHS as shown in the first
panel of Fig. 1-(c) In particular, at $\phi=\pi/2$,
$\varepsilon_{+}({\bm{M}}_{1}+{\bm{p}})\approx\varepsilon_{+}({\bm{M}}_{1})+(9/4)p^{2}_{x}-(27/16)p^{2}_{x}p^{2}_{y}$
describes a HOVHS with $D(\varepsilon)\sim|\varepsilon|^{-1/4}$.
Region II: Crossing the boundary into region II, where $t>\tau_{12}(\phi)$,
each HOVHS splits into two conventional VHS equidistant from $\bm{M}_{i}$ on
the BZ boundary. As we increase $t$ and get closer to the boundary with region
III, the VHS tend towards the $\pm{\bm{K}}$ (second panel of Fig. 1-(c)).
Region III: The boundary between II and III is defined by
$t=\tau_{23}(\phi)=3^{1/4}{[18\sqrt{3}\sin^{2}\phi-18|\sin\phi|\cos\phi}]^{-1/2}$
where the Hessian vanishes when evaluated at the BZ zone corners $\pm\bm{K}$.
We note that this curve has an asymptote at $\phi=\pi/6$. Along the curve
$\tau_{23}(\phi)$, $\pm\bm{K}$ are HOVHSs with
$D(\varepsilon)\sim|\varepsilon|^{-1/3}$ (third panel of Fig.1-(c)), which
separate region II and III where $\pm\bm{K}$ are local minima or maxima,
respectively. As we cross the boundary into region (III), the VHSs split from
the $\pm\bm{K}$ points and move towards the center of the BZ (last panel in
Fig. 1-(c)). We note that regions IV and V are similar to III and II, except
for the existence of 2 groups of six quasidegenerate VHS, but they are not the
focus of this work.

Figure 2: (a) Band structure with HOVHS at $\phi=\pi/2$. (b) Two patch model
of the HOVHS and the sublattice weight for the upper band.

Two HOVHS patches: Henceforth, we focus on the boundary between II and III,
where a pair of HOVHS are located at $\pm{\bm{K}}$ for each band. A typical
band structure for $\phi=\pi/2$ is presented in Fig.2(a). Without loss of
generality, we focus on the upper band (orange band in the figure). Close to
HOVHS filling, we can effectively expand the dispersion within small patches
centered at $\pm\bm{K}$ up to fourth order:

$\begin{split}&\,\varepsilon_{\pm\bm{K}}(\bm{p})=\varepsilon_{0}\pm\kappa_{1}(p_{y}^{3}-3p_{x}^{2}p_{y})-\kappa_{2}(p_{x}^{2}+p_{y}^{2})^{2}\,,\end{split}$
(3)

where $\varepsilon_{0}$ is the energy at the HOVHS,
$\kappa_{1}=\frac{9t\cos\phi}{8}+\frac{\sqrt{3}|\sin\phi|(6t^{2}+\csc^{2}\phi)}{16t}$
and
$\kappa_{2}=\frac{27}{64}t\cos\phi-\frac{|\sin\phi|(162t^{4}+27t^{2}\csc^{2}\phi-\csc^{4}\phi)}{128\sqrt{3}t^{3}}$
are two coefficients that vary continuously along the boundary between II and
III, determined by the curve $t=\tau_{23}(\phi)$. $\kappa_{2}\neq 0$
quantifies the deviation from perfect nesting, which occurs at
$(\phi,t)=(\pi/6,\tau_{23}(\pi/6))$.

This two patch model admits three momentum conserving interactions
$g_{1},g_{2}$ and $g_{3}$ (see Fig.2(b)) defined as

$H_{I}=g_{1}\psi_{\alpha}^{\dagger}\psi^{\dagger}_{\alpha}\psi_{\alpha}\psi_{\alpha}+\sum_{\alpha\neq\beta}g_{2}\psi_{\alpha}^{\dagger}\psi_{\beta}^{\dagger}\psi_{\beta}\psi_{\alpha}+g_{3}\psi_{\alpha}^{\dagger}\psi_{\beta}^{\dagger}\psi_{\alpha}\psi_{\beta}$
(4)

where we leave the spin structure and spin summation implicit, which should be
$\sigma,\sigma^{\prime},\sigma^{\prime},\sigma$ for the four fermions
operators in each term, and denote the patches by $\alpha,\beta=\pm$ with
$\psi_{\pm}=\psi_{\pm{K}+\bm{p}}$. These interactions are obtained from
projecting the lattice fermion interactions onto band fermions. Quite
interestingly, as long as we focus on patches around $\pm\bm{K}$, the SI
effect always associates each valley with a distinct sublattice index. We show
this effect by plotting the sublattice weights (for the upper band)
$|u_{A+}(\bm{k})|^{2}$ (red) and $|u_{B+}(\bm{k})|^{2}$ (blue) in Fig.2(b).
Crucially, fermions near $+\bm{K}$ ($-\bm{K}$) are solely from A (B)
sublattice, so there is a one-to-one correspondence between $c_{i\in A/B}$ and
$\psi_{\alpha}$. An important consequence is that the onsite Hubbard
interaction
$H_{U}=U\sum_{i}c_{i\uparrow}^{\dagger}c_{i\downarrow}^{\dagger}c_{i\downarrow}c_{i\uparrow}$
only contributes to $g_{1}$ and the nearest neighbor interaction
$H_{V}=V\sum_{\langle
ij\rangle}c^{\dagger}_{i\sigma}c^{\dagger}_{j\sigma^{\prime}}c_{j\sigma^{\prime}}c_{i\sigma}$
only contributes to $g_{2}$. Note that if there were no SI, $H_{U}$ would
contribute to all $g_{i}$. Moreover, the SI also implies that $g_{3}$ only
arises from the more exotic bond-bond interactions $H_{W}=W\sum_{\langle
ij\rangle}c^{\dagger}_{i\sigma}c^{\dagger}_{j\sigma^{\prime}}c_{i\sigma^{\prime}}c_{j\sigma}$
Kivelson  _et al._ (1987, 1988); Nakamura and Itoh (2001). In the extended-
Hubbard model with additional bond-bond interactions ($H_{U}+H_{V}+H_{W}$) the
bare interactions before renormalization are therefore

$g_{1}(0)=U,~{}~{}g_{2}(0)=V,~{}~{}g_{3}(0)=W.$ (5)

Note that this initial condition is robust against imperfect nesting, as long
as the two HOVHS are located at $\pm\bm{K}$. The bond-bond interactions are
typically orders of magnitude smaller than other interactions Hubbard and
Flowers (1963); Gammel and Campbell (1988), and so we will neglect them below.

RG equations: To identify the potential electronic instabilities resulting
from these interactions in Eq.(5), we note that in the presence of HOVHS the
bare particle-hole and particle-particle bubbles at zero momentum or
$\pm\bm{K}$ all diverge in the same power-law manner. These bubbles constitute
the renormalization of $g_{i}$’s at one-loop level, resulting in the following
RG flow equations Shtyk  _et al._ (2017); Wu  _et al._ (2023):

$\begin{split}\dot{g_{1}}&=\epsilon
g_{1}+(d_{2}-d_{3})g_{1}^{2}-2d_{2}g_{2}^{2}\,,\\\ \dot{g_{2}}&=\epsilon
g_{2}+(d_{1}-1)g_{2}^{2}-2d_{2}g_{1}g_{2}.\end{split}$ (6)

Here we defined the running parameter as $y=\Pi_{pp}(\bm{q}=0)$, and the
nesting parameters are defined as $d_{1}\approx\Pi_{ph}(\bm{K})/y$,
$d_{2}\approx\Pi_{ph}(0)/y$ and $d_{3}\approx\Pi_{pp}(\bm{K})/y$. Within our
patch model, these bubbles are calculated via

$\displaystyle\Pi_{pp}(0)\,,\Pi_{pp}(\bm{K})=\int_{\bm{p}}\frac{1-f[\epsilon_{\bm{K}}(\bm{p})]-f[\epsilon_{\mp\bm{K}}(-\bm{p})]}{\epsilon_{\bm{K}}(\bm{p})+\epsilon_{\mp\bm{K}}(-\bm{p})},$
(7)
$\displaystyle\Pi_{ph}(\bm{K})=\int_{\bm{p}}\frac{f[\epsilon_{-\bm{K}}(\bm{p})]-f[\epsilon_{\bm{K}}(\bm{p})]}{\epsilon_{\bm{K}}(\bm{p})-\epsilon_{-\bm{K}}(\bm{p})},~{}\Pi_{ph}(0)=\int_{\bm{p}}\frac{-\partial
f[\epsilon]}{\partial\epsilon}$

where $\int_{\bm{p}}=\int_{|\bm{p}|<\Lambda}\frac{d^{2}\bm{p}}{4\pi^{2}}$ with
$\Lambda$ being the high energy cutoff, and in the first line ‘+’ is for
$\Pi_{pp}(\bm{K})$ while ‘-’ is for $\Pi_{pp}(0)$. In the perfect nesting case
when $\kappa_{2}$ in Eq.(3) vanishes, Eq.(7) can be evaluated analytically,
which leads to $d_{1}=1$, $d_{2}=d_{3}\approx\frac{1}{3}$. But in general
$d_{i}$’s are determined by both the energy scale and the parameter $\phi$ and
can strongly deviate from their perfect nesting values. The tree-level term
$\epsilon$ results from rescaling the field operators, with $\epsilon=0$ for
the Gaussian fixed point and $\epsilon=1/3$ for the supermetal fixed point,
which occurs at $g_{2}=0$, $g_{1}=\frac{1}{3(d_{3}-d_{2})}$. Note that the
strong coupling fixed points are the same for both choices.

To probe possible symmetry breaking orders, we introduce the following order
parameters in both particle-particle and particle-hole channels:

$\displaystyle\Delta^{s}_{\text{SC}}=\langle\psi^{\dagger}_{+}\psi^{\dagger}_{-}+\psi^{\dagger}_{-}\psi^{\dagger}_{+}\rangle,~{}~{}\Delta^{t}_{\text{SC}}=\langle\psi^{\dagger}_{+}\psi^{\dagger}_{-}-\psi^{\dagger}_{-}\psi^{\dagger}_{+}\rangle$
(8)
$\displaystyle\Delta_{\text{PDW}}=\langle\psi_{\alpha}\psi_{\alpha}\rangle,~{}~{}\Delta_{\text{CDW}}=\langle\psi^{\dagger}_{+}\psi_{-}\rangle,~{}~{}\Delta_{\text{SDW}}=\langle\psi^{\dagger}_{+}\hat{s}\psi_{-}\rangle$
$\displaystyle\Delta_{\text{FM1}}=\sum_{\alpha}\langle\psi_{\alpha}^{\dagger}s_{z}\psi_{\alpha}\rangle,~{}~{}\Delta_{\text{FM2}}=\sum_{\alpha}\alpha\langle\psi_{\alpha}^{\dagger}s_{z}\psi_{\alpha}\rangle$
$\displaystyle\Delta_{\text{PI1}}=\langle\psi_{+}^{\dagger}\psi_{+}+\psi_{-}^{\dagger}\psi_{-}\rangle,~{}~{}\Delta_{\text{PI2}}=\langle\psi_{+}^{\dagger}\psi_{+}-\psi_{-}^{\dagger}\psi_{-}\rangle\,.$

These are singlet and triplet uniform superconductivity
$\Delta_{\text{SC}}^{s/t}$, finite momentum superconductivity (pair density
wave) $\Delta_{\text{PDW}}$, charge- and spin-density wave
$\Delta_{\text{C/SDW}}$, ferromagnetism with different parities
$\Delta_{\text{FM1/2}}$ and Pomeranchuk instabilities with different parities
$\Delta_{\text{PI1/2}}$. Note that $\Delta_{\text{PI1}}$ amounts to shift the
chemical potential by an overall constant, while $\Delta_{\text{PI2}}$
corresponds to shifting the chemical potentials at $\pm\bm{K}$ oppositely. In
other words, in the presence of $\Delta_{\text{PI1/2}}$, the system can still
remain metallic but is shifted away from the Van Hove filling. When the energy
scale is reduced, the order parameters flow as

$\displaystyle\dot{\Delta}_{\text{SC}}^{s/t}=-g_{2}\Delta_{\text{SC}}^{s/t},~{}~{}\dot{\Delta}_{\text{PDW}}=-d_{3}g_{1}\Delta_{\text{PDW}},$
(9)
$\displaystyle\dot{\Delta}_{\text{C/SDW}}=d_{1}g_{2}\Delta_{\text{C/SDW}},~{}~{}\dot{\Delta}_{\text{FM1/2}}=d_{2}g_{1}\Delta_{\text{FM1/2}},$
$\displaystyle\dot{\Delta}_{\text{PI1}}=d_{2}(-g_{1}-2g_{2})\Delta_{\text{PI1}},~{}~{}\dot{\Delta}_{\text{PI2}}=d_{2}(-g_{1}+2g_{2})\Delta_{\text{PI2}}.$

Lastly, we will use the renormalized susceptibilities to identify the leading
order, which obey $\dot{\chi_{i}}=d_{i}|\Delta_{i}|^{2}$, where $d_{i}$ is the
corresponding nesting parameters while for the uniform SC we have $d_{i}=1$.
Close to the onset of the instability, $\chi_{i}\sim(y_{c}-y)^{\gamma_{i}}$,
thus the most negative $\gamma_{i}$ corresponds to the leading instability.

Figure 3: (a) $g_{1}(0)=U,\,g_{2}(0)=V$ phase diagram and RG flow
trajectories. Fixed points are shown by red dots, fixed trajectories are shown
by red lines that also form the phase boundaries. All trajectories in the
green region flow to the supermetal fixed point, while all trajectories in the
blue region A with $g_{2}>0$ (red region B with $g_{2}<0$) flow to some
symmetry-broken strong coupling fixed point. The corresponding phases depend
on temperature $T$ and $\phi$, as shown in (b): the PI2 instability is always
leading in A with a subleading PDW (purple) or C/SDW (yellow, labeled DW)
instability; either a leading PI1 with subleading $s/t$SC or a leading $s/t$SC
occur in B. The fixed trajectory along $g_{2}=0$ is unstable but can become
stabilized at $y=y^{*}$ due to a staggered mass generated by the PI2
instability becoming comparable to the RG cutoff $\Lambda$, as shown in (c),
which also shows the RG flows of $g_{1}$ and $g_{2}$ starting in region A with
repulsive bare interactions. Depending on whether $g_{1}$ changes sign or not
at $y^{*}$ ($y^{*}_{2}$ and $y^{*}_{1}$ labeled in (a)), either an instability
in region C (PDW or FM) or a supermetal phase is realized respectively.

Phase diagram and two-step RG: In Fig.3 (a) we present the phase diagram for
the bare couplings $g_{j}(0)$ with tree level contribution for $\phi=\pi/2$
and $T=0.001t$. The RG flows are also shown by the stream lines. There are
three regions in the phase diagram: two regions with strong coupling
instabilities (blue for $g_{2}>0$ and red for $g_{2}<0$) in which orders
develop, and a region (green) within which no instability of the HOVHS
develops and no symmetry is broken, resulting in a supermetal phase. Note that
the slopes of the boundaries between the regions are functions of $\phi$ and
$T$, but all phase diagrams are qualitatively similar. In particular, the
boundaries are given by
$g_{2}=\left(1-d_{1}\pm\sqrt{(1-d_{1})^{2}+8(3d_{2}-d_{3})}\right)g_{1}/(4d_{2})$
for $g_{1}>0$.

The symmetry broken phases within each region for different values of $\phi$
and temperature are shown in Fig.3(b). For $g_{2}<0$ (region B), singlet and
triplet superconductivities are the leading symmetry-breaking instabilities
(PI1, corresponding to a chemical potential, is generally the leading vertex
correction but does not break any symmetry and does not gap out the Fermi
surface). The degeneracy is lifted by the neglected bond-bond interaction
giving a non-zero $W=g_{3}$, with positive (negative) values favoring triplet
(singlet) SC. For $g_{2}>0$ (region A), the leading instability is the
inversion-symmetry breaking valley-order PI2, with subleading PDW or C/SDW as
shown in Fig.3 (b); observe that C/SDW are Kekulé-type bond orders due to the
sublattice-valley polarization. These are consistent with the phases found in
Wu  _et al._ (2023).

We note, however, that the PI2 instability does not gap out the HOVHS, but
yields an interaction-driven staggered mass in the Haldane model. Once the
generated staggered mass $M$ is sufficiently large, the RG equations Eq. 6 are
no longer valid, as only a single HOVHS at a single valley can be tuned to at
a time, resulting in the situation considered in Aksoy  _et al._ (2023). In
that case the $g_{2}$ and $g_{3}$ processes become forbidden by momentum
conservation. This results in a particularly interesting scenario for the most
relevant case of completely repulsive interactions (positive $g_{j}$). The
instabilities in RG generally occur at some finite critical value $y=y_{c}$,
but it is possible that at some point the running $M(y)$ grows larger than the
cutoff $\Lambda=\Lambda_{0}e^{-y}$ ($\Lambda_{0}$ being the bare cut-off prior
to starting the RG flow) for some $y^{*}<y_{c}$, in which case the RG should
be done in two steps as illustrated in Fig.3(c). When $y^{*}$ is reached, Eq.
6 has to be modified with $g_{2}\rightarrow 0$, with the new RG flow
continuing along the $g_{2}=0$ line in the phase diagram (region C), which is
otherwise an unstable fixed trajectory of the RG flow, until $y_{c}$ is
finally reached.

Depending on whether $g_{1}(y^{*})$ is positive or negative, this mechanism
can realize a Chern supermetal or a broken symmetry phase, respectively. The
possible symmetry broken phases are shown in the third panel of Fig.3(b), and
include either a FM or PDW instabilities. We emphasize that these phases would
be unstable in the absence of the interaction-generated staggered mass which
projects out the $g_{2}$ and the previously neglected $g_{3}$ interactions.
For the supermetal, $g_{3}$ in principle leads to a further instability Shtyk
_et al._ (2017); Wu  _et al._ (2023) due to multiple HOVHS Classen  _et al._
(2020). As the bond-bond interactions $W=g_{3}(0)$ are presumed very small,
however, the resulting instability may only occur at exceedingly low
temperatures making the supermetal stable at finite temperature even in the
absence of a generated staggered mass, thanks to the SI mechanism. The PDW
phase is similarly stabilized once $g_{2}$ and $g_{3}$ are projected out, but
importantly $g_{1}$ changes sign before the projection, with attraction in the
PDW channel thus generated by purely repulsive bare interactions. This
scenario contrasts with models with a single HOVHS Wu  _et al._ (2023); Aksoy
_et al._ (2023), where the PDW instability requires bare attractive
interactions.

In summary, we have classified the manifold of VHSs in inversion symmetric
Haldane bands, uncovering a new framework to investigate electronic
correlations in Chern bands. The identification of a pair of HOVHS at BZ
valleys provides a route to study the interplay of interactions and band
topology once the Fermi energy lies in the vicinity of these higher-order
saddles. Beyond a host of exotic ordered phases including long-sought pair-
density waves, we identified a novel Chern supermetal displaying non-Fermi
liquid behavior and quantum anomalous Hall response, which are salient
features observable in transport and tunneling experiments. Expanding this
framework to other Chern systems realized either in ultracold fermions or in
2D Van der Waals materials is a promising future direction.

-Note: During the completion of this work, Ref. Aksoy  _et al._ (2023) appeared analyzing a single HOVHS in the Haldane model with a staggered mass that explicitly breaks inversion symmetry. Our approach, conversely, starts with an inversion symmetric model with two HOVHSs and leads to different Fermi liquid instabilities.

## Acknowledgments

We thank to Claudio Chamon, Dmitry Chichinadze and Ajit Srivastava for useful
discussions. This research was supported by the U.S. Department of Energy,
Office of Science, Basic Energy Sciences, under Award DE-SC0023327 (D.S. and
L.H.S.) and by the Gordon and Betty Moore Foundation’s EPiQS Initiative
through GBMF8686 (Y.-M.W).

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