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On PNDP-manifold

A. Pigazzini, C. $\ddot{O}$zel, P. Linker and S. Jafari

Dedicated to Professor Maximilian Ganster on the occasion his retirement

and Renata Albizetti 08/11/1925 - 06/01/2020

Abstract

We provide a possible way of constructing new kinds of manifolds which we will
call Partially Negative Dimensional Product manifold (PNDP-manifold for
short).
In particular a PNDP-manifold is an Einstein warped product manifold of
special kind, where the base-manifold $B$ is a Remannian (or pseudo-
Riemannian) product-manifold
$B=\Pi_{i=1}^{q^{\prime}}B_{i}\times\Pi_{i=(q^{\prime}+1)}^{\widetilde{q}}B_{i}$,
with $\Pi_{i=(q^{\prime}+1)}^{\widetilde{q}}B_{i}$ an Einstein-manifold, and
the fiber-manifold $F$ is a derived-differential-manifold (i.e., $F$ is the
form: smooth manifold ($\mathbb{R}^{d}$)+ obstruction bundle, so it can admit
negative dimension).
Since the dimension of a PNDP-manifold is not related with the usual geometric
concept of dimension, from the speculative and applicative point of view, we
try to define this relation using the concept of desuspension to identify the
PNDP with another kind of ”object”, introducing a new kind of hidden
dimensions.

Keywords: PNDP-manifolds; Einstein warped product manifolds; negative
dimensional manifolds; derived-manifold; desuspension; point-like manifold,
virtual dimension.

2010 Mathematics Subject Classification: 53C25

1\. Introduction and Preliminaries

The concept of negative dimensional space is already used in linguistic
statistics [1]. Also in supersymmetric theories in Quantum Field Theory,
negative dimensional spaces are used [2].

Let $E\cong M\times F$ be a fiber bundle with base space $M$ and its fiber
$F$. We will discuss now a case, where the fiber has negative dimension. Note
that the total dimension of the fiber bundle is given by the relation $\dim
E=\dim M+\dim F$. We will consider the case, where the base manifold has
greater positive dimension than the negative dimension of the fiber is $\dim
M>-\dim F$. In this case, the dimension of the total fiber bundle is still
positive. Since the base manifold is obtained by projection of the fiber
bundle along the fiber by projection operator $\pi_{F}$, we have $\pi_{F}E=M$
i.e. the projection of the lower-dimensional fiber-bundle along the fiber
yields the higher dimensional base manifold space. Therefore, the projection
operator $\pi_{F}$ along the negative-dimensional fiber, is a suspension
operator that raises the dimension of topological spaces.
Let $\dim F=-d,d>0$. Then, the operator $\pi_{F}$ will be a $d$-fold
suspension (a single suspension extends a topological space by dimension one).
By a slight redefinition of the notion of a fiber bundle to the new
mathematical structure, the inverse fiber bundle, we are enable to turn fiber
bundles with negative-dimensional fibers into inverse fiber bundles with
positive-dimensional fibers. More precisely, we define the inverse fiber
bundle as follows: \- An inverse fiber bundle $E^{*}$ is locally isomorphic to
the space
$M^{*}\times^{*}F^{*}$ with some base manifold $M^{*}$ and a fiber $F^{*}$.

\- However, we have a suspension along the fiber $\Sigma_{F^{*}}$ such that
$\Sigma_{F^{*}}E^{*}=M^{*}$ instead of a projection along a fiber in an
inverse fiber bundle. \- Because of the use of suspension instead of
projections, the inverse cartesian product $\times^{*}$ is not the ordinary
cartesian product. It can be defined functorially as follows: in the category
of fiber bundle topology $Fib$, the ordinary cartesian product is a morphism
between some topological spaces (the objects in this category). By defining an
inversion functor $\mathcal{F}$ that maps the category $Fib$ to the category
of inverse fiber bundles $InvFib$ by mapping projection operators (are also
morphisms) to suspensions $\mathcal{F}(\pi_{F})=\Sigma_{F^{*}}$. Objects
remain the same, but on the fiber, the sign of the dimension is changed
(negative fibers in $Fib$ change to positive ones in $InvFib$). Thus, it holds
$\mathcal{F}(\times)=\times^{*}$. We define this product to be compatible with
suspensions instead of projections. The functor also preserves commutative
diagrams that are frequently used in the definition of fiber bundles. It is
also easy to generalize the above discussion to the case of negative fibers
with arbitrary negative dimension. We can treat a fiber bundle with negative-
dimensional fiber as an inverse fiber bundle, where the fibers will be
positive-dimensional. Cartesian products, however, must be changed to inverse
cartesian products. The base manifold in inverse fiber bundles has the highest
dimension, since it is obtained by a suspension. Imposing local isomorphisms
between an inverse fiber bundle to the inverse cartesian product of base
manifold with fiber, we can see that the inverse cartesian product reduces
dimensions by some amount. It holds: $\dim
E^{*}=\dim(E^{*}\times^{*}F^{*})=\dim M^{*}-\dim F^{*}$.
Definition 1: We say that dimension of $X$ is $-1$ if suspension of $X$,
$\Sigma X$ is diffeomorphic to a point. By induction and suspension operation
we can define all negative dimensions. Since if $X$ and $Y$ are difeomorphic
then their suspensions $\Sigma X$ and $\Sigma Y$ are diffeomorphic. Also a
metric on $X$ is compatible with $\Sigma X$. It is also well-known that
differentials and suspensions are compatible on manifolds. This means that the
differential on the suspension space $\Sigma X$ is induced from the
differential on $X$.

The operator $\times^{*}$ acts similar as the topological quotient, but in
such a way that it is compatible with respect to above defined functor. The
inverse cartesian product we can also view as a quotient operator obtained by
using the $\mathcal{F}$-functor. We can alternatively write: ×
$\times^{*}=/_{\mathcal{F}}$.

Definition 2: In general, given an n-dimensional space $X$, the suspension
$\Sigma X$ has dimension $n+1$. Thus, the operation of suspension creates a
way of moving up in dimension. The inverse operation $\Sigma^{-1}$, is called
desuspension. Therefore, given an $n$-dimensional space $X$, the desuspension
$\Sigma^{-1}X$ has dimension $n-1$, (see [14]).

1.1 PNDP-manifolds

Einstein warped product manifolds are a mathematical object that is considered
in current research. We will treat an Einstein warped product manifold in this
section of the paper, where we will introduce also the concept of negative
dimensions. As ”mathematical tool”, for the fiber-manifold, we will use
derived geometry and about this we recall that if $A\rightarrow M$ and
$B\rightarrow M$ are two transversal submanifolds of codimension $a$ and $b$
respectively, then their intersection $C$ is again a submanifold, of
codimension $a+b$. Derived geometry explains how to remove the transversality
condition and make sense out of a nontransversal intersection $C$ as a derived
smooth manifold of codimension $a+b$. In particular, $dimC=dimM-a-b$, and the
latter number can be negative. So the obtained dimensions are not related to
the usual geometry concept of ”dimension”, but they are ”virtual dimensions”,
and therefore from a speculative/applicative point of view we will try to
relate the dimensions obtained from the PNDP-manifolds with desuspensions
(special projections), interpreting this, as the negative dimensions of the
fiber-manifold ”hide” the positive dimensions of the base- manifold, making
the PNDP-manifold appears as if desuspensions had been performed.

Definition 3: A warped product manifold
$(M,\bar{g})=(B,g)\times_{f}(F,\ddot{g})$ (where ($B,g$) is the base-manifold,
($F,\ddot{g}$) is the fiber-manifold), with $\bar{g}=g+f^{2}\ddot{g}$, is
Einstein if only if:

(1)
$\bar{Ric}=\lambda\bar{g}\Longleftrightarrow\begin{cases}Ric-\frac{d}{f}\nabla^{2}f=\lambda
g\\\ \ddot{Ric}=\mu\ddot{g}\\\ f\Delta f+(d-1)|\nabla f|^{2}+\lambda
f^{2}=\mu\end{cases}$

where $\lambda$ and $\mu$ are constants, $d$ is the dimension of $F$,
$\nabla^{2}f$, $\Delta f$ and $\nabla f$ are,
respectively, the Hessian, the Laplacian and the gradient of $f$ for $g$, with
$f:(B)\rightarrow\mathbb{R}^{+}$ a smooth positive function.

Contracting first equation of (1) we get:

(2) $R_{B}f^{2}-f\Delta fd=nf^{2}\lambda$
where $n$ and $R_{B}$ is the dimension and the scalar curvature of $B$
respectively, and from third equation, considering $d\neq 0$ and $d\neq 1$, we
have:

(3) $f\Delta fd+d(d-1)|\nabla f|^{2}+\lambda f^{2}d=\mu d$
Now from (2) and (3) we obtain:

(4) $|\nabla
f|^{2}+[\frac{\lambda(d-n)+R_{B}}{d(d-1)}]f^{2}=\frac{\mu}{(d-1)}$.

Definition 4: We called PNDP-manifold a warped product manifold
$(M,\bar{g})=(B,g)\times_{f}(F,\ddot{g})$ that satisfies (1), where the base-
manifold $(B,g)$ is a Riemannian (or pseudo-Riemannian) product-manifold
$B=\Pi_{i=1}^{q}B_{i}=B_{1}\times B_{2}\times...$ with $g=\Sigma g_{i}$, which
we can write as $B=B^{\prime}\times\widetilde{B}$ where $\widetilde{B}$ is an
Einstein manifold $\Pi_{i=(q^{\prime}+1)}^{\widetilde{q}}B_{i}$ (i.e.,
$\widetilde{R}ic=\lambda\widetilde{g}$ where $\lambda$ is the same for (1) and
$\widetilde{g}$ is the metric for $\widetilde{B}$), with
$dim\widetilde{B}=\widetilde{n}$ and $B^{\prime}=\Pi_{i=1}^{q^{\prime}}B_{i}$
with $dimB^{\prime}=n^{\prime}$, so $dimB=n=n^{\prime}+\widetilde{n}$. The
warping function $f:B\rightarrow\mathbb{R}^{+}$ is
$f(x,y)=f^{\prime}(x)+\widetilde{f}(y)$ (where each is a function on its
individual manifold, i.e., $f^{\prime}:B^{\prime}\rightarrow\mathbb{R}^{+}$
and $\widetilde{f}:\widetilde{B}\rightarrow\mathbb{R}^{+}$) and can also be a
constant function. The fiber-manifold $(F,\ddot{g})$ is a derived differential
Riemann-flat manifold with negative integer dimensions $m$, where with derived
differential manifold we consider the derived manifolds as smooth Riemannian
flat manifolds by adding a vector bundle of obstructions. In particular we
consider, for $F$, only the spaceforms $\mathbb{R}^{d}$, with orthogonal
Cartesian coordinates such that $g_{ij}=-\delta_{ij}$, by adding a vector
bundle of obstructions, $E\rightarrow\mathbb{R}^{d}$, (so
$F:=\mathbb{R}^{d}+E$), with dimension $m=d-rank(E)$, where $rank(E)>d$ (for
issues related to derived-geometry or obstruction bundle see [9]), and more
specifically we consider $F$ such that its dimension is $m=-d$. Since
$F=\mathbb{R}^{d}+E$, and on $E$ (obstruction bundle) the Riemannian geometry
does not work, we consider and define that every operation of Riemannian
geometry done and defined on the underlying $\mathbb{R}^{d}$, is considered
done and defined on $F$ (i.e., for example, if we calculate the Ricci
curvature of $\mathbb{R}^{d}$, which is obviously zero, then we will say that
the Ricci curvature of $F$ is zero, this because it will be the definition of
Ricci curvature for $F$). Moreover if we want the quantity $n-d>0$ then we
must set $n^{\prime}=d=-m$, and in the special case where $n-d>0$ with also
$B^{\prime}$ an Einstein-manifold with the same Einstein-$\lambda$, then
$B^{\prime}_{i}=\widetilde{B}_{i}$, or $B^{\prime}=\widetilde{B}$.

Important Note 1: Since the usual Riemannian geometry works for the underlying
Riemannian manifold as for ”ordinary” manifolds, we can, therefore, consider
ourselves to be able to work with the (pseudo-)Riemannian geometry on our
derived fibers-manifold $F$, defining all Riemannian geometry operations on
$\mathbb{R}^{d}$ as Riemannian geometry operations made on $F$, paying
attention only to the dimension; In particular, as mentioned above, we work on
$F$ by actually working on the underlying $\mathbb{R}^{d}$, (in the sense
that, since $F:=\mathbb{R}^{d}+E$, the calculation results obtained on
$\mathbb{R}^{d}$ are considered to be obtained and defined on $F$), but taking
into account that $F$ has negative dimensions, in fact on the obstruction
bundle we cannot put the metric, then we do not work on the obstruction bundle
with the Riemannian geometry, but it gives negative dimensions on $F$.
Obviously for what has been said, the tangent space and the vector fields are
those of $\mathbb{R}^{d}$. The scalar product with two arbitrary vector fields
${{}^{F}}g\langle V,W\rangle$ is define on $F$ as:
$g_{ij}v^{i}w^{j}=-\delta_{ij}v^{i}w^{j}=-(v^{i}w^{i})$.

We want to underline that the analysis does not differ from the usual Einstein
warped product manifold analysis, and that the Riemannian curvature tensor and
the Ricci curvature tensor of the product Riemannian manifold can be written
respectively as the sum of the Riemannian curvature tensor and the Ricci
curvature tensor of each Riemannian manifold (see [15]). This could be also a
special kind of Einstein Sequential warped product manifold
$(M_{1}\times_{h}M_{2})\times_{\bar{h}}M_{3}$, (see [16]), where $h=1$,
$M_{2}$ is an Einstein-manifold and $M_{3}$ is a derived-smooth-manifold with
negative dimensions, that said:

Proposition: If we write the B-product as $B=B^{\prime}\times\widetilde{B}$,
where:
i) $Ric^{\prime}$ is the Ricci tensor of $B^{\prime}$,
ii) $\widetilde{R}ic$ is the Ricci tensor of $\widetilde{B}$,
iii) $g^{\prime}$ is the metric tensor referred to $B^{\prime}$,
iv) $\widetilde{g}$ is the metric tensor referred to $\widetilde{B}$,
v) $f=f^{\prime}+\widetilde{f}$, is the smooth warping function
$f:B\rightarrow\mathbb{R}^{+}$, where
$f^{\prime}:B^{\prime}\rightarrow\mathbb{R}^{+}$ and
$\widetilde{f}:\widetilde{B}\rightarrow\mathbb{R}^{+}$,
vi) $\nabla^{2}f=\tau^{\prime*}\nabla^{\prime
2}f^{\prime}+\widetilde{\tau}^{*}\widetilde{\nabla}^{2}\widetilde{f}$ is the
Hessian referred on its individual metric, but since $\widetilde{B}$ is
Einstein, we have $\widetilde{\tau}^{*}\widetilde{\nabla}^{2}\widetilde{f}=0$,
(where $\tau^{\prime*}$ is the pullback),
vii) $\nabla f$ is the gradient (then $|\nabla
f|^{2}=|\nabla^{\prime}f^{\prime}|^{2}+|\widetilde{\nabla}\widetilde{f}|^{2}$),
and
viii) $\Delta f=\Delta^{\prime}f^{\prime}+\widetilde{\Delta}\widetilde{f}$ is
the Laplacian, (since
$\widetilde{\tau}^{*}\widetilde{\nabla}^{2}\widetilde{f}=0$, then
$\widetilde{\Delta}\widetilde{f}=0$), so the Ricci curvature tensor will be:

(1****)
$\begin{cases}\bar{Ric}(\Sigma_{i=1}^{q^{\prime}}X_{i},\Sigma_{i=1}^{q^{\prime}}Y_{i})=Ric^{\prime}(\Sigma_{i=1}^{q^{\prime}}X_{i},\Sigma_{i=1}^{q^{\prime}}Y_{i})-\frac{d}{f}\nabla^{\prime
2}f^{\prime}(\Sigma_{i=1}^{q^{\prime}}X_{i},\Sigma_{i=1}^{q^{\prime}}Y_{i})\\\
\bar{Ric}(\Sigma_{i=(q^{\prime}+1)}^{\widetilde{q}}X_{i},\Sigma_{i=(q^{\prime}+1)}^{\widetilde{q}}Y_{i})=\widetilde{R}ic(\Sigma_{i=(q^{\prime}+1)}^{\widetilde{q}}X_{i},\Sigma_{i=(q^{\prime}+1)}^{\widetilde{q}}Y_{i})\\\
\bar{Ric}(U,V)=\ddot{Ric}(U,V)-f^{2}\ddot{g}(U,V)f^{*}\\\
\bar{Ric}(\Sigma_{i=1}^{q^{\prime}}X_{i},\Sigma_{i=(q^{\prime}+1)}^{\widetilde{q}}X_{i})=0\\\
\bar{Ric}(\Sigma_{i=1}^{q}X_{i},U)=0,\end{cases}$
where $f^{*}=\frac{\Delta^{\prime}f^{\prime}}{f}+(d-1)\frac{|\nabla
f|^{2}}{f^{2}}$, $\Sigma_{i=1}^{q^{\prime}}X_{i}=X^{\prime}$,
$\Sigma_{i=1}^{q^{\prime}}Y_{i}=Y^{\prime}$,
$\Sigma_{i=(q^{\prime}+1)}^{\widetilde{q}}X_{i}=\widetilde{X}$ and
$\Sigma_{i=(q^{\prime}+1)}^{\widetilde{q}}Y_{i}=\widetilde{Y}$.

Theorem: A warped product manifold with derived differential fiber-manifold
$F:=\mathbb{R}^{d}+E$, and $dimF$ a negative integer, is a PNDP-manifold, as
defined in
Definition 4, if and only if:

(1***)
$\bar{Ric}=\lambda\bar{g}\Longleftrightarrow\begin{cases}Ric^{\prime}-\frac{d}{f}\tau^{\prime*}\nabla^{\prime
2}f^{\prime}=\lambda g^{\prime}\\\
\widetilde{\tau}^{*}\widetilde{\nabla}^{2}\widetilde{f}=0\\\
\widetilde{R}ic=\lambda\widetilde{g}\\\ \ddot{Ric}=0\\\
f\Delta^{\prime}f^{\prime}+(d-1)|\nabla f|^{2}+\lambda f^{2}=0,\end{cases}$
(since $Ric$ is the Ricci curvature of $B$, then
$Ric=Ric^{\prime}+\widetilde{R}ic=\lambda(g^{\prime}+\widetilde{g})+\frac{d}{f}\tau^{\prime*}\nabla^{\prime
2}f^{\prime}$).

Therefore (2) and (3), for $n-d=0$ and $n-d<0$, become:

(1**)
$\bar{R}=\lambda\bar{n}\Longleftrightarrow\begin{cases}R^{\prime}f-\Delta^{\prime}f^{\prime}d=n^{\prime}f\lambda\\\
\widetilde{\Delta}\widetilde{f}=0\\\ \widetilde{R}=\lambda\widetilde{n}\\\
\ddot{Ric}=0\\\ f\Delta^{\prime}f^{\prime}+(d-1)|\nabla f|^{2}+\lambda
f^{2}=0.\end{cases}$

where $n^{\prime}$ and $R^{\prime}$ are the dimension and the scalar curvature
of $B^{\prime}$ respectively, while for $n-d>0$, we must set $d=n^{\prime}$.
We have
(1*)
$\bar{R}=\lambda\bar{n}\Longleftrightarrow\begin{cases}R^{\prime}f-\Delta^{\prime}f^{\prime}n^{\prime}=n^{\prime}f\lambda\\\
\widetilde{\Delta}\widetilde{f}=0\\\ \widetilde{R}=\lambda\widetilde{n}\\\
\ddot{Ric}=0\\\ f\Delta^{\prime}f^{\prime}+(n^{\prime}-1)|\nabla
f|^{2}+\lambda f^{2}=0.\end{cases}$

Proof. We applied the condition that the warped product manifold of system
(1****) is Einstein. $\square$

Recapitulating, we used the Derived-geometry to define the fiber-manifold,
which explains how to remove the transversality condition and make sense of a
non-transversal intersection and therefore admit the presence of negative
dimensions. Since we consider the fiber-manifold $F$ as $\mathbb{R}^{d}$
adding a vector bundle of obstructions (then we work with the Riemannian
geometry on $F$ working only on the underlying $\mathbb{R}^{d}$, such that all
the Riemannian geometry operations made on $\mathbb{R}^{d}$ are considered as
defined on $F$), then the classical construction for the warped product
manifold (see [10]) is the same; we consider the vertical vector fields $U$,
$V$, as lift of vector fields of $F$ the development of the formulas remains
the same; $X,Y$ are lift from $B$, they are horizontal and so constant on
fibers, then for example $V[X,Y]=0$, and we continue to consider the inner
product between a vector field on $B$ with one on $F$ as zero (i.e., $\langle
X,V\rangle=0$). We obtain an Einstein warped product manifold
$(M,\bar{g})=(B,g)\times_{f}(\mathbb{R}^{d},\ddot{g})$, because from the
geometric point of view the fiber-manifold is $\mathbb{R}^{d}$, but taking
into account that the manifold obtained has the fiber-manifold with negative
dimensions, then the outcome manifold is $(n+(d-rank(E)))$-dimensional
manifold.
Another important observation is that we consider $\mathbb{R}^{d}$ because,
from a speculative point of view, negative dimensional space is used only to
make the PNDP-manifold dimension smaller than the base-manifold dimension.
Therefore for this purpose we consider it in the simplest possible form.

PNDP-metric: Referring to a PNDP-manifolds, with negative dimensional fiber,
and for not confusing its metric with the metrics of a ”classic” Einstein
warped product manifold, we denote the Riemannian or pseudo-Riemannian metric
of the fiber-manifold with the following notation to indicate that $F$ has
negative dimensions:
$\bar{g}=g-f^{2}(\Sigma^{n}_{i=1}(d\psi^{i})^{2})_{(m)}$, where $g$ is the
metric of the base-manifold $B$, and $m$ is the negative dimension of $F$.
Then the general metric form of a PNDP-manifold is:
$\bar{g}=g-f^{2}(\Sigma^{n}_{i=1}(d\psi^{i})^{2})_{(m)}=(g^{\prime}+\widetilde{g})+(f^{\prime}+\widetilde{f})^{2}(\Sigma^{n}_{i=1}(d\psi^{i})^{2})_{(m)}$.
Example 1 - A type of flat PNDP-manifold:
The manifold
$(\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R})\times[(\mathbb{R}^{2}+E)]$
(with $rank(E)=4$) is a $(4-2)$-PNDP-manifold Ricci-flat. In fact it satisfies
(1*) for constant $f=1$ ($f^{\prime}$ and $\widetilde{f}$ both constants), we
have $n+m=n-d=4-2=2>0$. Thus we have to consider $dimB^{\prime}=dimF$. In this
special case the B-product is composed of only Einstein-manifolds where
$B^{\prime}=\widetilde{B}$, that is $\mathbb{R}^{2}$, and its metric is:
$ds^{2}=dt^{2}+dx^{2}+dy^{2}+dz^{2}-(du^{2}+dv^{2})_{(-2)}$.

Example 2 - A type of PNDP-point-like manifold: For this purpose, we consider
fiber-manifold $F=(\mathbb{R}^{3}+E)$ (with $rank(E)=6$), 1-dimensional
$B^{\prime}$-manifold and $\lambda=0$. Like in the previous case, if for
example we choose $\widetilde{B}$ as $\mathbb{R}^{2}$, then our PNDP-manifold
will be:
$M=(\mathbb{R}^{2}\times\mathbb{R})_{f}(\mathbb{R}+E)$, and with the following
metric:
$ds^{2}=dx^{2}+dy^{2}+dz^{2}+h(y)^{2}(du^{2}+dv^{2}+dw^{2})_{(-3)}$.

Example 3 - A type of PNDP-manifold with negative ”virtual” dimension:
Following what was done in the previous examples, keeping the same setting,
and considering, for example, $dim(F)=-4$, we obtain that space ”will emerge”
as a desuspension of a point, i.e., negative dimension.

2\. Possible speculative interpretation of the types of PNDP manifolds

We assume that negative quantities exist in nature, here we consider the
possible existence of negative spatial dimensions, but the fact remains that
the PNDP-manifolds could be used to describe anything in nature that has to do
with negative quantities.

From the speculative point of view, in this session we consider 2 types of
PNDP-manifolds and try to relate their ”virtual dimensions” to the usual
concept of dimensions in Riemannian geometry, and this interpreting that,
through the product, the negative virtual dimensions of the fiber-manifold
”hide” the positive dimensions of the base-manifold, making the PNDP-manifold
appears as if desuspensions had been performed.
In practice the derived-manifold $F$ has negative dimensions, which are far
from the usual geometric concept of ”dimension”. Being that with the ”warped
product” we must consider both the dimension of $F$ (virtual) and the
dimension of B, (which is a Riemannian or pseudo-Riemannian manifold and
therefore its dimension is in perfect relationship with the usual concept of
”dimension”), the obtained virtual dimension of our PNDP-manifold is
equivalent to the dimension of the base-manifold $B$ with the dimension of the
underlying manifolds $\mathbb{R}^{d}$ subtracted, and we get that the result
will be smaller than the dimension of $B$. From the speculative point of view,
we want to consider the dimensions obtained for the PNDP-manifold interpreting
that when the dimensions of $B$ come into ”contact” with the dimensions of
$F$, these latter interact by ”hiding” the dimensions of $B$. This means that
the PNPD-manifold, will ”visually appears” as if a kind of desuspensions had
been performed on $B$, then with extra hidden dimensions which we can not see.
The same thing can be assumed for $F$; it ”will appear” only as desuspensions
of the point, with extra hidden dimensions which we can not see. Below we have
consider two kinds of PNDP-manifolds:

Type I) the $(n,-n)$-PNDP manifold that has overall, zero-dimension
($dimM=dimB+dimF=n+(-n)=0$). Then the result may be interpreted as an
”invisible” manifold but made up of two manifolds with $n$ and $-n$
dimensions, respectively. Then we try to consider it as a kind of ”point-like
manifold” (zero-dimension), but with ”hidden” dimensions, and

Type II) the $(n,-d)$-PNDP manifold, where $n$ (the base-manifold dimension)
is different from $d$ such that $dim=n-d>0$. The particular feature of this
manifold is that it appears as another Einstein-manifold with $(n-d)$
dimension.

In (Type I) we try to consider $dim=0$ referring to a point-like manifold. It
would be thought that the points of $B$ and of the underlying $\mathbb{R}^{d}$
of $F$ appear as if they were ”projected” on the PNDP-manifold, which in this
case degenerates into a point (dim = 0), and we observe the manifold as a
point-like; in effect this perception is given by our interpretation that
negative dimensions ”hide” positive dimensions and show us the manifold as if
it were a point, i.e. the PNDP-manifold is ”hidden” except for a point.
Also in (Type II) case, since the dimension of the PNDP-manifold is calculated
by the dimension of the base-manifold with the dimension of the fiber-manifold
subtracted, we try to interpret it as if the negative dimensions ”hide” the
positive dimensions of the base-manifold. Since the total dimension will be
smaller than the dimension of B, but still positive,we consider that the PNDP-
manifold appears as a projection of $B$ with $dim=n-d$.
Therefore our projection interpret this operation by considering that each
negative dimension can act equally on each positive dimension, so that the
final dimension is the difference between positive and negative dimensions.
In our speculative interpretation we consider $(n,-d)$-PNDP manifold like an
Einstein-manifold projected from the product-manifold $B$, which has hidden
dimensions, but we will see this part better in the next section.
Concluding this section, we can say that our interpretation makes Type I and
II ”appear” as point or classic Einstein manifold, different from what they
actually are.

3\. Desuspension as Projection

This section helps to better define how the desuspensions work.
As we have mentioned in the previous paragraph, we consider the relation
between PNDP-manifold dimension and the usual geometric concept of
”dimension”, as desuspensions interpreted as projections, and for this, since
each negative dimension can act equally on each positive dimension, in order
not to be faced with the choice on which projection of B the result can be, we
must ”orient” the result of the projections to what reflects certain
characteristics and to do this we must consider $B$ as a Riemannian base-
manifold expressible as a Riemannian product-manifold
$B=B^{\prime}\times\widetilde{B}$ with
$g_{B}=g_{B^{\prime}}+g_{\widetilde{B}}$, such that $\widetilde{B}$ is an
Einstein manifold with the same dimension and the same constant $\lambda$ of
the PNDP-manifold, the latter will be the result of the projection we are
going to consider, so:

$B=\Pi_{i=1}^{q^{\prime}}B_{i}\times\Pi_{i=(q^{\prime}+1)}^{\widetilde{q}}B_{i}$
and $F=\mathbb{R}^{d}+E$, then PNDP$=B\times_{f}F=[\Pi_{i\in
I}B_{i}\times_{f}F]$, with $dimB-dim\mathbb{R}^{d}=dim$PNDP$=dim(\pi_{(n-d)}:$
PNDP$)=(n-d)$.
\- If $(n-d)>0$ (i.e. system solutions (1*)) we have the projection:
$\pi_{(n-d)}:$PNDP$\rightarrow(\Pi_{i=(q^{\prime}+1)}^{\widetilde{q}}B_{i})=\widetilde{B}$,
\- if $(n-d)=0$, (i.e., system solutions (1**)), we have the projection:
$\pi_{0}:$PNDP$\rightarrow$ point-like manifold, and
\- if $(n-d)<0$, (i.e., system solutions (1**)), we have the projection:
$\pi_{(n-d)}:$PNDP$\rightarrow\Sigma^{n-d}(p)$, with $\Sigma^{n-d}(p)$, we
mean the $(n-d)$-th desuspension of point.

Example 4: Let $(B^{\prime}_{1}\times B^{\prime}_{2}\times
B^{\prime}_{3}\times\widetilde{B}_{4}\times\widetilde{B}_{5}\times\widetilde{B}_{6})\times_{f}F$
be a $(6-3)$-PNDP-manifold with $f$ non-constant, and since $n+m=n-d=6-3=3>0$,
then $dimB^{\prime}=dimF$.
So our PNDP-manifold is such that:
$\widetilde{B}_{4}\times\widetilde{B}_{5}\times\widetilde{B}_{6}$ will be an
Einstein-manifold, i.e.,
$Ric_{(\widetilde{B}_{4}\times\widetilde{B}_{5}\times\widetilde{B}_{6})}=\lambda(g_{\widetilde{B}_{4}}+g_{\widetilde{B}_{5}}+g_{\widetilde{B}_{6}})$,
and since $n-d=3$ we have the desuspension/projection:
$\pi_{3}:$$(6-3)$-PNDP$\rightarrow(\Pi_{i=4}^{6}B_{i})$, by identifying the
$(6-3)$-PNDP manifold with the Einstein-manifold
$\widetilde{B}_{4}\times\widetilde{B}_{5}\times\widetilde{B}_{6}$.

Example 5: Let $(B^{\prime}_{1}\times
B^{\prime}_{2}\times\widetilde{B}_{3}\times\widetilde{B}_{4})\times_{f}F$ be a
$(8-4)$-PNDP-manifold with $f$ non-constant, and also in this case, since
$n+m=n-d=6-4=4>0$, $dimB^{\prime}=dimF$.
Then our PNDP-manifold is such that:
$\widetilde{B}_{3}\times\widetilde{B}_{4}$ will be Einstein, i.e.,
$Ric_{(\widetilde{B}_{3}\times\widetilde{B}_{4})}=\lambda(g_{\widetilde{B}_{3}}+g_{\widetilde{B}_{4}})$,
then we will have: $\pi_{4}:(8-4)$-PNDP
$\rightarrow(\widetilde{B}_{3}\times\widetilde{B}_{4})$. Hence we identify the
$(8-4)$-PNDP manifold with the Einstein-manifold
$\widetilde{B}_{3}\times\widetilde{B}_{4}$ .

Example 6: If we consider the manifold of the Example 1, where we have the
special case $B^{\prime}=\widetilde{B}$, that is $\mathbb{R}^{2}$, the
desuspension/projection will be:
$\pi_{2}:(\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R})\times(\mathbb{R}^{2}+E)\rightarrow\mathbb{R}^{2}$,
i.e., we identify the $(4-2)$-PNDP manifold with $\mathbb{R}^{2}$.

Example 7: If we consider the manifold of the Example 2, the
desuspension/projection will be:
$\pi_{0}:(\mathbb{R}^{2}\times\mathbb{R})\times_{f}(\mathbb{R}^{3}+E)\rightarrow$
point-like manifold (zero-dimension).

Example 8: If we consider the manifold of the Example 3, the
desuspension/projection will be:
$\pi_{-1}:(\mathbb{R}^{2}\times\mathbb{R})\times_{f}(\mathbb{R}^{4}+E)\rightarrow\Sigma^{-1}(p)$,
i.e., a desuspension of a point.

Important Note 2: In summary, from a speculative point of view, we hypothesize
that in nature there are hidden dimensions that can be interpreted as the
interaction of negative dimensions, (which we have introduced with manifolds
generated by non-transversal intersections, derived-geometry), with positive
dimensions, which by closing together, are not perceived, therefore hidden. To
show this effect, we used special projections such as desuspensions, which
show only a part of the manifold, its virtual dimension, while the remaining
part is hidden. In this speculative context the PNDP is identified by
$(\pi_{(n-d)},\lambda,(n,m),g)$.

4\. Possible speculative application about the $(n,-n)$-PNDP-manifolds

and $(n,-q)$-PNDP-manifolds

In this section we present possible applications in light of the
interpretations made above.
Since $m=d-rankE$, where $rankE>d$ (such that $m=-d$), here we consider $d=n$,
then $m=-n$, where $n$ is the dimension of the base-manifold. Referring to a
manifold with $n$ negative dimensions, we denote the pseudo-Riemannian metric
of the fiber-manifold with the following notation to indicate that $F$ has
negative dimensions: $\ddot{g}=-(\Sigma^{n}_{i=1}(d\psi^{i})^{2})_{(-n)}$,
where (-n) is the dimension of the fiber-manifold $F$. In this way we don’t
confuse it with the metrics of a ”classic” product manifold.
Then the $(n,-n)$-PNDP metric has the form:
$\bar{g}=g-f^{2}(\Sigma^{n}_{i=1}(d\psi^{i})^{2})_{(-n)}$, where $g$ is the
metric of the base-manifold $B$.
First of all we recall and highlight that the purpose of the PNDP-manifolds is
precisely to present the point-like manifolds from a mathematical point of
view, and introduce a type of manifold with a new kind of hidden dimensions.
In [12], Capozziello et al. introduced the concept of the ”point-like
manifold” building superconductors with graphene. In particular they argued
that superconductor graphene can be produced by molecules organized in point-
like structures where sheets are constituted by $(N+1)$-dimensional manifold.
Particles like electrons, photons and “effective gravitons” are string modes
moving on this manifold. In fact, according to string theory, bosonic and
fermionic fields like electrons, photons and gravitons are particular “states”
or “modes” of strings. In their important work, they show that at the
beginning, there are point-like polygonal manifolds (with zero spatial
dimension) in space which strings are attached to them, where all interactions
between strings on one manifold are the same and are concentrated on one point
which the manifold is located on it. They also show that by joining these
manifolds, 1-dimensional polygonal manifolds are emerged on which gauge fields
and gravitons live and so, these manifolds glued to each other build higher
dimensional polygonal manifolds with various orders of gauge fields and
curvatures.
As in [12], the authors in [13], propose a version of Moffat’s Modified
Gravity (MOG) without anomaly, where they showed the same ”configuration” for
space-time where at first, there are only point-like manifolds with scalars
attached to them.
In these contexts, the $(n,-n)$-PNDP manifolds can play an important role.
In fact $(n,-n)$-PNDP appears as a point (point-like), because in general from
our interpretation, it is a point (positive and negative dimensions hide each
other out and and the total dimension equals zero), but in special it is
composed by two manifolds, $B$ and $F$ with nonzero dimensions.

Let’s begin with the MOG without anomaly case, and try to consider a tPNDP-
manifold to describe the point-like version of the spacetime where to induce a
Morris-Thorne wormhole or a Schwarzchild black hole.
Our aim is to try to demonstrate that with this manifold and the Time, we are
able to present a new concept of flat spacetime, that it is point-like
version: a PNDP-manifold with $dim=n+m=0$ plus Time.
From 5th equation of (1*), i.e.,
$f\Delta^{\prime}f^{\prime}+(n^{\prime}-1)|\nabla f|^{2}+\lambda f^{2}=0$,
since $R^{\prime}=0$ and $\lambda=0$, we obtain $|\nabla f|^{2}=0$, then
$f=$constant.
For this situation, we consider a negative real line $I=-e^{2\Phi(r)}t$ (where
$t$ is the Time and $\Phi(r)$ is an adjustable function of $r$) and choose
$n=3$, so a (3,-3)-PNDP manifold $T=B\times F$ where
$B=\mathbb{R}\times\mathbb{R}\times\mathbb{R}$, but for simplicity we will
write as $B=\mathbb{R}^{3}$ and the same for $F$ that we will write as
$F_{(-3)}=\mathbb{R}^{3}+E$ (bundle of obstruction), such that $dimF=-3$.
Then:

(5) $L=\mathbb{R}^{3}\times F_{(-3)}\times I$, with
$ds^{2}=-e^{2\Phi(r)}dt+dx^{2}+dy^{2}+dz^{2}-(d\psi^{2}+d\varphi^{2}+d\sigma^{2})_{(-3)}$.
In [11], the authors, show that a $(2+1)$-dimensional wormhole can be induced
from $(3+1)$-dimensional flat space-time.
They considered the $(3+1)$-dimensional Minkowski space-time in the
cylindrical coordinates. Then, following the authors’ analysis, we can write
the metric of $B$ in cylindrical coordinates:
(6)
$ds^{2}=-e^{2\Phi(r)}dt^{2}+dr^{2}+dz^{2}+r^{2}d\vartheta^{2}-(d\psi^{2}+d\varphi^{2}+d\sigma^{2})_{(-3)}$.
If from now we work only temporarily on $B$ plus Time and set $z=\xi(r)$, we
find:

(7)
$ds^{2}=-e^{2\Phi(r)}dt^{2}+(1+\xi^{\prime}(r)^{2})dr^{2}+r^{2}d\vartheta^{2}$,
(and if we consider $\Phi(r)=0$ and $\xi^{\prime}(r)=0$, we have
$(2+1)$-Minkowski spacetime), so the form of energy density becomes:
$\varrho=\frac{\xi^{\prime}\xi^{\prime\prime}}{r(1+\xi^{\prime 2})^{2}}$.
The relation (7) is comparable with the Morris-Thorne static wormhole:
(8)
$ds^{2}=-e^{2\Phi(r)}dt^{2}+(\frac{1}{1-\frac{b(r)}{r}})dr^{2}+r^{2}d\vartheta^{2}$,
in our specific case we have:
$b(r)=r(\frac{\xi^{\prime}(r)^{2}}{1+\xi^{\prime}(r)^{2}})$.

It is well-known that the topological structure of a wormhole includes a
throat that connects two asymptotically flat spaces and in order to have and
maintain this structure, the geometric flare-out condition is necessary, i.e.
the minimum dimension to the throat. In the case of Morris-Thorne type
wormhole, this condition is given by the huge surface tension compared to the
energy density times the square of the speed of light and this implies that,
if $r=r_{0}$ is the location of the throat, then (*) $b(r_{0})=r_{0}$ and (**)
for $r>r_{0}$, $b^{\prime}(r)<\frac{b(r)}{r}$. In the new setting, (*) implies
that at the throat $\xi^{\prime}=\pm\infty$ and (**) states that
$\xi^{\prime}\xi^{\prime\prime}<0$ for $r>r_{0}$. Besides these conditions at
the throat, we have $z=\xi(r_{0})=0$.
By the same token, we can construct a Schwarzchild Black Hole, by setting:
$\xi(r)=\frac{4M}{(\frac{2M}{r-2M})^{1/2}}$, and
$\Phi(r)=(1/2)ln(\frac{2M}{r}-1)$.
Then we can conclude that from our (5) we can describe a point-like spacetime
in which to induce: $((-1)+1)-$spatial dimension Morris-Thorne wormhole or
$((-1)+1)-$spatial dimension Schwarzchild black hole.
Our $(n,-n)$-PNDP manifold model consists of two manifolds with nonzero
dimensions (one with $n$-dimension and one with $-n$-dimension, where these
two manifolds can be thought as a result of intersection between other
manifolds). Then we can consider these two manifolds as contained in a
”$p$-dimensional BULK”, but their product (which generates the $(n,-n)$-PNDP)
will create the point-like manifolds.
Returning to the Graphene wormhole case, we can carry out the same analysis
done above considering $R^{\prime}=\widetilde{R}=0$ and $f=const.$, and
setting a $(n,-q)$-PNDP-manifolds with $n=4$ and $m=-q=-3$. In this way we
obtain the following metric:
(9)
$ds^{2}=-e^{2\Phi(r)}dt+dx^{2}+dy^{2}+dz^{2}+dw^{2}-(d\psi^{2}+d\varphi^{2}+d\sigma^{2})_{(-3)}$,
that as we saw at the beginning of the section, it is a 1-dimensional manifold
plus Time, which in this case we consider to appear as $\mathbb{R}$.
Therefore proceeding with the analysis made for the previous case, but with an
extra dimension to the base-manifold $B$, we can obtain:
(10)
$ds^{2}=-e^{2\Phi(r)}dt^{2}+(\frac{1}{1-\frac{b(r)}{r}})dr^{2}+r^{2}(d\vartheta^{2}+sin^{2}\vartheta
d\zeta^{2})-(d\psi^{2}+d\varphi^{2}+d\sigma^{2})_{(-3)}$,
which is $[(3+(-3))+1]$-wormhole, i.e., the point-like spatial version plus
Time of the graphene wormhole.
So we can end this ”application-session” by saying that the $(n,-n)$-PNDP
manifold could be considered as possible mathematical interpretation of point-
like manifolds.

The PNDP-manifold is a new type of Einstein warped-product manifold that uses
the derived-geometry to introduce a concept of virtual dimension, which, from
the speculative point of view, is the dimension with which we perceive our
PNDP-manifold, the rest is hidden, invisible.

5\. Conclusions and Remarks

We have introduced this new kind of Einstein warped product manifold:
($\Pi_{i=1}^{q^{\prime}}B_{i}\times\Pi_{i=(q^{\prime}+1)}^{\widetilde{q}}B_{i})\times_{f}F$,
where the base-manifold contains an Einstein-manifold
$\Pi_{i=(q^{\prime}+1)}^{\widetilde{q}}B_{i}$ (or $\widetilde{B}$), and where
$\widetilde{B}$ has the same dimension and the same
Einstein-$\lambda$-constant of the PNDP-manifold.
For this new kind of manifold we have shown a Ricci-flat example with $f$
constant, and we have shown the existence of a non-Ricci-flat case with non-
constant warping function $f$. While from the speculative point of view we
have relating its dimension to the usual geometric concept of dimension, by
means of desuspensions (special projections) $\pi_{(n-d)}:$ PNDP, where in the
case $n-d>0$ it is identified with another Einstein-manifold
($\widetilde{\Pi}_{(i=q^{\prime}+1)}^{\widetilde{q}}B_{i}$) with equal
dimension and where the respective scalar curvatures have the same constant
$\lambda$, in the case $n-d=0$ it is identified with a point, and in the case
$n-d<0$ it is identified with $(n-d)$-th desuspensions of a point.

To conclude, in addition to the possible application seen as point-like
versions of the graphene-wormhole and spacetime, the PNDP-manifold could
describe a new nature of time. In fact the negative dimension of the fiber-
manifold could represent the ”past” time, the positive dimension of the base-
manifold could represent the time ”future” (for example 1-dimensional time
”future”), while the ”present” time which is instantaneous, could be
represented by a point (dimension zero dim(M)=0). It is well-known that
Einstein considered the five dimensional Kaluza-Klein theory in his
investigation for a unified field theory, i.e., a unification of gravity and
electromagnetism, during the years 1938-1943. But by finding a non-singular
particle solution, he discarded this theory as an impossible model for a
unified field theory. Since then some other theories have been formulated
which are based on Kalazu-Klein’s theory [[3],[4], [5]]. In this paper we gave
some ideas concerning a supersymmetric Kaluza-Klein theory in which tPNDP-
manifold can be considered as hidden extra 1-time-like dimensions in a theory
with more than 10 dimensions. To get rid of massless ghosts we have a
4-dimensional spacetime, a spontaneous compactification of ground state is
necessary by assuming that the cosmological constant is zero. Indeed, in
Einstein theory the vierbein field of the 4-dimensional spacetime has 10
degrees of freedom. But 5-dimensional Kaluza-Klein theory the metric field has
5 more degrees of freedom.
Suppose that ground state is $\Sigma_{5}\times S_{1}$ with a radius of Planck
length. Moreover, let us assume that the metric field is independent of the
fifth coordinate. The provided action creates a theory of $\Sigma_{4}$ which
together with the Brans-Dicke scalar field has 15 degrees of freedom. It is
well-known that 11-dimensional supergravity is an important theory among other
supergravity theories since eleven is the maximal space-time dimension which
does not allow particles with helicity greater than two (see: [6]). Eleven
dimensional supergravity has been discarded as it does not provide a realistic
model in 4 dimensions. But it is considered as description of low-energy
dynamics of $M$-theory (see: [5]). It is true that the eleven-dimensional
super gravity constructed by Cremmer et al. [7], did not consider the chiral
phenomenology since it is eleven-dimensional and eleven is odd. But as Wang
Mian [8] proposed, one can construct a consistent supergravity theory with an
extra hidden time-like dimensions. We believe that it is possible to construct
a consistent theory of supergravity with hidden extra time-like dimensions,
and these hidden extra time-like dimensions are identified with our PNDP-
manifold with metric
$ds^{2}=dt_{1}^{2}+dt_{2}^{2}-(dt_{3}^{2}+dt_{4}^{2})_{-2}$, which we denote
by $M_{PNDP}$.
Our $M_{PNDP}$ is a hidden extra time-like manifold. In fact the ”role” of the
negative dimension of the fiber-manifold is to hide the dimension of the base-
manifold making our $M_{PNDP}$ as a point-manifold (recall, in general, that
$dim((n,-n)-PNDP)$ is $dimB+dimF=n-n=0$).
Now it is possible to have the topology of vaccum as
$\Sigma_{12}=\Sigma_{4}\times S_{2}\times S_{2}\times S_{1}\times
H^{3}\backslash\Gamma$, where $\Sigma_{4}$ is the 4-spacetime with the
condition that the cosmological constant is zero. Moreover
$H^{3}\backslash\Gamma$ is a hyperbolic manifold and $\Gamma$ is a discrete
isometry which is not acting freely. This implies that there is no Killing
vectors in this quotient space.
This suggests that we have no gauge field which is associated with the hidden
extra time-like dimensions, and our gravitational model can be written as:
$\Sigma_{12}\times M_{PNDP}$. In this way the Time has 5 dimensions, the usual
temporal dimension of space-time and 4 extral ”hidden” dimensions. We believe
that the theory can successfully describe a super-Yang-Mills field theory
coupled with gravitation after spontaneous compactification.

## References

  * [1] V.P. Maslov Negative dimension in general and asymptotic topology, ArXiv (2006).
  * [2] P. Cvitanović Negative dimensions and E7 symmetry, Nucl. Phys. B, vol. 188 (2) (1981),
pp. 373–396,

  * [3] M. J. Duff, hep-th$\backslash$, 9410046 (1994)
  * [4] M. J. Duff, An Introduction to Kaluza-Klein Theories, Ed. by H. C. Lee World Scientific (1983).
  * [5] M. J. Duff, The World in Eleven Dimensions: Supergravity, Supermembranes and M-theory,
Ed. M. J. Duff IOP Publishing (1999).

  * [6] W. Nahm, Supersymmetries and their representations, Nucl. Phys. B, 135 (1978), 149.
  * [7] E. Cremmer, B. Julia and J. Scherk, Supergravity theory in 11 dimensions, Phys. Lett. B, 76 (1978), 409.
  * [8] W. Mian, Proc. of the Fourth Marcel Grossmann Meeting on General Relativity, Ed. R. Ruffini (1986), p. 1459.
  * [9] D. Joyce, Kuranishis and Symplectic Geometry, Volume II, (2017)
  * [10] B. O’Neill, Semi-Riemannian Geometry with applications to Relativity, Academic Press, (1983).
  * [11] S. H. Mazharimousavi, M. Halilsoy - 2+1-dimensional traversable wormholes supported by positive energy \- European Physical Journal C, 75(2) september (2014).
  * [12] S. Capozziello, R. Pincak, E. N. Saridakis, Constructing superconductors by graphene Chern–Simons wormholes, Annals of Physics (Elsevier), 390 (2018), pp. 303-333.
  * [13] Alireza Sepehri, Tooraj Ghaffary, Yaghoob Naimi - MOG without anomaly \- Physics of the Dark Universe, Elsevier, 19 (2018) 91–103.
  * [14] L. Wolcott, E. McTernan - Imagining Negative-Dimensional Space \- Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture (2012) p. 637-642.
  * [15] M. Atceken and S. Keles - On the product Riemannian manifolds \- Differential Geometry - Dynamical Systems (2003) p. 1-8.
  * [16] U. Chand Dea, S. Shenawyb, B. Unal - Sequential Warped Products: Curvature and Conformal Vector Fields \- Filomat 33:13 (2019), 4071–4083.

Alexander Pigazzini: Mathematical and Physical Science Foundation, Sidevej 5,
4200 Slagelse, Denmark, E-Mail<EMAIL_ADDRESS>
Cenap Ozel: King Abdulaziz University, Department of Mathematics, 21589 Jeddah
KSA, E-Mail<EMAIL_ADDRESS>
Patrick Linker: Department of Materials Testing, University of Stuttgart,
70569 Stuttgart, Germany, E-Mail<EMAIL_ADDRESS>
Saeid Jafari: Mathematical and Physical Science Foundation, Sidevej 5, 4200
Slagelse, Denmark, E-mail<EMAIL_ADDRESS>


# Frustration induced incommensurate solids in the extended Bose-Hubbard model

Kwai-Kong Ng<EMAIL_ADDRESS>Department of Applied Physics, Tunghai
University, Taichung 40704, Taiwan (R.O.C.)  Min-Fong Yang<EMAIL_ADDRESS>Department of Applied Physics, Tunghai University, Taichung 40704, Taiwan
(R.O.C.)

###### Abstract

The extended Bose-Hubbard model with nearest-neighbor and next-nearest-
neighbor repulsive interactions on a square lattice is investigated by using
the quantum Monte Carlo method. We find that, for the cases of weak next-
nearest-neighbor interactions and small hoppings, incommensurate solids of
fractional densities varying from 1/4 to 1/2 can be stabilized in the
thermodynamic limit. We further show that the continuous changes of ordering
wave vectors from $(\pi,\pi/2)$ [or $(\pi/2,\pi)$] to $(\pi,\pi)$ within the
incommensurate solid phase can be understood by the mechanism of domain wall
formation. The related ground-state phase diagram and thermal phase
transitions are also discussed.

###### pacs:

##  I introduction

Polar molecules trapped in optical lattices provide an unique opportunity to
study the dipole-dipole interactions in real experimental setups [1, 2, 3, 4].
One primary investigation is to identify possible quantum phases that may
uniquely arise from these long-range interactions but cannot be observed in
systems with only short-range interactions. Recent numerical studies [5, 6, 7,
8] of hard-core bosons with infinite-range interactions on two-dimensional
(2D) square lattices have presented evidence of Mott insulators with
checkerboard, stripe, and star ordering at densities $\rho=$ 1/2, 1/3, and
1/4, respectively. Furthermore, supersolids around the Mott lobes with $\rho=$
1/2 and 1/4 are also found by doping the solids with particles or vacancies.
Besides, devil’s staircase (that is, a sequence of commensurate phases
separated by a series of jumps) is observed for finite-size systems, which
signals the presence of the incommensurate phases in the thermodynamic limit.
A question is: Are all these interesting phases uniquely stabilized by long-
range interactions?

Actually, the Mott insulators and the supersolids can be realized in short-
range models as well. In previous numerical investigations on the extended
Bose-Hubbard model that includes only the nearest-neighbor (nn) and the next-
nearest-neighbor (nnn) interactions, Mott insulators at 1/2 and 1/4 fillings
and the associated supersolids have indeed been observed [9, 10, 11, 12, 13,
14]. The model Hamiltonian for the hard-core bosons on a 2D square lattice is,

$\displaystyle H=$ $\displaystyle-t\sum_{\langle
i,j\rangle}(b_{i}^{\dagger}b_{j}+h.c.)+V\sum_{\langle
i,j\rangle}n_{i}n_{j}+V^{\prime}\sum_{\langle i,j\rangle^{\prime}}n_{i}n_{j}$
$\displaystyle-\mu\sum_{i}n_{i}\;.$ (1)

Here $b_{i}$ ($b^{\dagger}_{i}$) is the annihilation (creation) bosonic
operator on site $i$, $t$ the hopping integral, $n_{i}$ the particle number on
site $i$, and $V$ ($V^{\prime}$) the nn (nnn) repulsive interaction. $\mu$
denotes the chemical potential which controls the particle density in the
grand canonical ensemble. For the cases of strong $V^{\prime}$
($V^{\prime}>V/2$), Mott insulators of striped and star orders are observed at
1/2 and 1/4 fillings respectively. Both of them are associated with
supersolids of the same orderings. Therefore, the long-range nature of the
dipole-dipole interactions is not indispensable to the stability of these
phases.

Moreover, the discovery of incommensurate supersolid phases for hard-core
bosons with nn interactions on anisotropic triangular lattices [15, 16] shows
that long-range interactions are not crucial for the appearance of
incommensurate phases. This motivates us to explore the possibility of
incommensurate phases in short-range models. In addition, the nature of
incommensurate phases and their stabilities is of great interest on its own,
the understanding of which should shed light on the other systems with
frustrated interactions. However, these fundamental issues are hard to address
for infinite-range models. The realization of incommensurate phases in short-
range models will thus help us to analyze the mechanism of their formation and
then to improve our understanding of these phases.

In this work, we employ the quantum Monte Carlo (QMC) method with a stochastic
series expansion algorithm [17, 18] to study the short-range model in Eq. (1).
We find that, at small hopping $t$ and weak nnn interactions $V^{\prime}$
($V^{\prime}<V/2$), incommensurate solids of densities ranging from 1/4 to 1/2
can be stabilized in the thermodynamic limit. The ground-state phase diagram,
as one of our main results, is presented in Fig. 1, where incommensurate
solids emerge between the half-filled checkerboard and quarter-filled solids.
In contrast to the strong $V^{\prime}$ cases [9, 10, 12, 13, 11], there exists
no supersolid phases in the present case of weak $V^{\prime}$. The
incommensurate solids are found to be characterized by a continuous change in
ordering wave vectors as model parameters are varied. The formation of such a
phase can be explained by the proliferation of domain walls from half-filled
or quarter-filled solids. Following the analysis in Ref. [16] in the context
of the anisotropic triangular lattice, analytical calculations on the domain-
wall excitation energies are made for the present system on a square lattice.
Our analytical predictions on the phase boundaries agree very well with the
QMC numerical results. Furthermore, the quantum phase transitions from the
incommensurate solids to the half-filled or quarter-filled solids are found to
be continuous in the thermodynamic limit, while the transition to the
superfluid phase is first order. Upon increasing temperatures, the
incommensurate solids will melt into normal fluids via a first-order thermal
phase transition, which is demonstrated by hysteresis of the structure factors
and double peaks in the histogram.

This paper is organized as follows. The ground-state phase diagram at fixed
$V$ and $V^{\prime}$ is identified in Sec. II. The general features of the
incommensurate solids are explained in Sec. III by the domain wall formation.
The energetic perspective of the domain wall is elucidated in Sec. IV.
Analytic expressions of phase boundaries of the incommensurate solids and of
the domain wall density are then derived. More quantum and thermal phase
transitions out of the incommensurate solids are discussed in Sec. V. We
summarize our work in Sec. VI.

##  II Quantum phase diagram

In this work we utilized the well established stochastic series expansion
algorithm [17, 18] in the QMC calculations as the Hamiltonian $H$ in Eq. (1)
is sign-problem free. Unless mentioned otherwise, we set $V^{\prime}=1$ as the
energy scale. The inverse temperature is set to be $\beta=1/(2L)$, where $L$
is the linear size of lattice ($L=12$, 24, 36, and 48 are used). To identify
various quantum phases, besides the particle density $\rho$, two more
quantities are measured in our simulations., The superfluidity $\rho_{s}$ is
calculated to signal off-diagonal U(1) symmetry breaking, which is defined by
the fluctuations of winding numbers $W_{x}$ and $W_{y}$,

$\rho_{s}=\frac{1}{2\beta}(\langle W_{x}^{2}\rangle+\langle
W_{y}^{2}\rangle).$ (2)

On the other hand, the translational broken symmetry is characterized by the
structure factor

$S(\textbf{Q})=\frac{1}{L^{2}}\sum_{i,j}\langle n_{i}n_{j}\rangle
e^{i\textbf{Q}\cdot\textbf{r}_{ij}},$ (3)

where Q denotes the wave vector and rij the displacement between two sites $i$
and $j$.

Figure 1:  Ground-state phase diagram with $V=4$ obtained from QMC
calculations after extrapolating to thermodynamic limit. The dotted lines are
the phase boundaries predicted by the domain wall analysis. The insets show
the CBS and the QFS lattice structures. Here $V^{\prime}=1$ as the energy
unit.

In the parameter regime we studied in our model, the off-diagonal and the
diagonal orderings do not coexist, which means no supersolid is found. In Fig.
1, the ground-state phase diagram $\mu$ vs $t$ with $V=4$ contains a
superfluid (SF) and two solid states of definite ordering wave vectors Q
signaling the peak in the structure factor. Their phase boundaries are
determined by the abrupt changes of $\rho_{s}$ or $S(\textbf{Q})$ (see Fig. 2)
and then by extrapolating to the thermodynamic limit with
$L\rightarrow\infty$. The half-filled phase is a checkerboard solid (CBS) with
$\textbf{Q}=(\pi,\pi)$, instead of a striped solid, because the nn repulsion
is dominant ($V>2V^{\prime}$) so that nn occupation is avoided. Reducing the
chemical potential such that $\rho=1/4$, the ground state is then a quarter-
filled solid (QFS) of star ordering [12, 13] with $\textbf{Q}=(\pi/2,\pi)$ or
$(\pi,\pi/2)$. While there are two types of star orders that are doubly
degenerate at $t=0$, the degeneracy is lifted for finite $t$. The preferred
QFS ordering is shown in the inset of Fig. 1 as it prevents nn occupation up
to second order of hopping $t$ to reduce the cost of the nn potential energy.
Interestingly, for finite but small $t$, the phase in between the CBS and the
QFS is neither a superfluid nor supersolid, but an incommensurate solid (ICS)
with ordering wave vectors Q continuously varying from $(\pi/2,\pi)$ [or
$(\pi,\pi/2)$], to $(\pi,\pi)$ in the thermodynamic limit. As explained in the
next section, we found that ICS can be understood by inserting linear domain
walls into the solid phases. By increasing hopping $t$, all solid phases
eventually melt into a superfluid phase directly without passing an
intermediate supersolid phase. Further discussions on these phase transitions
will be provided in Sec. V.

##  III incommensurate solids

Figure 2:  Order parameters as functions of $\mu$ with $t=0.4$ and $V=4$.
Upper panel: the particle density $\rho$ as a function of $\mu$. The
fractional numbers are the corresponding densities of the plateaus. Solid
(Open) circles represent the data for $L=36$ ($L=24$). Lower panel:
superfluidity $\rho_{s}$ (filled squares) and structure factors
$S(\textbf{Q})$ (open circles) as functions of $\mu$. The corresponding
ordering wave vectors of the data are, from left to right, Q=($2m\pi/L$,
$\pi$) with $m=L/4$, $(L/4)+1$, …, $L/2$. For simplicity, the structure
factors for the ordering wave vectors Q=($\pi$, $2m\pi/L$) are not shown. Note
that the data of $S(\pi,\pi)$ is reduced by five times.

For a given hopping $t=0.4$, the particle density $\rho$ is shown in Fig. 2
for $L=36$. A series of density plateaus is observed from $\rho=1/4$ to 1/2,
in between of the QFS and the CBS. We note that the plateaus occur at
densities of $m/L$, with $m$ being an integer from $L/4$ to $L/2$. This is
very similar to the results found in the model with infinite-range
interactions [3, 5]. In that case, although a smaller lattice size ($L=12$) is
used, many more plateaus of rational fillings are found between $\rho$ =1/4
and 1/2. The reason for this difference is that the infinite-range nature of
the interactions allows more possible configurations of particle ordering to
be stabilized, which, unfortunately, also makes the physics more difficult to
be analyzed. In contrast, for the present system with short-range
interactions, we can understand the origin of the density plateaus by a simple
picture of linear fluctuating domain walls. While both domain walls along the
$x$ and $y$ directions are observed in our simulations, for simplicity, only
the domain walls along the $y$ direction will be discussed hereafter.

Under doping the half-filled CBS with holes, the potential energy gain is
$4V^{\prime}$ for each added isolated hole. Nevertheless, holes aligned
together to form a domain wall [Fig. 3 (b)] can further gain kinetic energy
through particle hoppings, which will be explained in details in the next
section when we discuss the domain wall dynamics. For convenience, we define
$L_{x}$ ($L_{y}$) to be the lattice size in the $x$ ($y$) direction. We note
that even numbers of domain walls are required to maintain the periodic
boundary condition imposed in our system. Because $L_{y}/2$ particles must be
removed in order to create a domain wall along the $y$ direction out of the
CBS, the total number of removed particle becomes $nL_{y}$ when $2n$ domain
walls appear in the lattice for $n=0$, 1, 2, …, $L_{x}/4$. Consequently, when
there are $L_{x}/4$ pairs of domain walls, the particle density is reduced by
$1/4$ and the state becomes the QFS with $\rho=1/2-1/4=1/4$ [Fig. 3 (d)]. For
the states containing $2n$ domain walls, the particle densities become
$\rho=(L_{x}-2n)/(2L_{x})$, which exactly correspond to those of the plateaus
found in Fig. 2 for both $L=24$ and 36 (note that $L=L_{x}=L_{y}$ in all our
simulations). Our analysis clearly shows that the series of plateaus is formed
by adding pairs of linear domain walls from the CBS.

Furthermore, the observed shift in the ordering wave vector Q in Fig. 2
implies the presence of linear domain walls as well. We note that the
insertion of domain walls at equal distance will not immediately destroy the
order completely. Instead, due to producing long-period superstructures in the
$x$ direction, the ordering wave vector will be shifted by ($2\pi/L_{x},0$)
for each added pair of linear domain walls along the $y$ direction. Hence the
plateau states in ICS with $2n$ domain walls along the $y$ direction can be
characterized by the ordering wave vectors $\textbf{Q}=(\pi-2n\pi/L_{x},\pi)$
or $\textbf{Q}=(2m\pi/L_{x},\pi)$ with $m$ being an integer from $L_{x}/4$ to
$L_{x}/2$. (The same discussions apply as well for the states with $2n$ domain
walls along the $x$ direction.) This conclusion is consistent with the
numerical results shown in Fig 2.

Another implication from the above analysis is that as $L\rightarrow\infty$,
the number of plateaus will become infinity such that $\rho$ changes
continuously from 1/4 to 1/2. The density jump at the phase boundary between
QFS (CBS) and ICS is then expected to be reduced to zero and the corresponding
commensurate-incommensurate transition should thus be continuous, rather than
of first order. The location of these phase boundaries in the ground-state
phase diagram for small $t$ can be determined analytically by considering the
domain wall dynamics, as discussed in the next section.

##  IV domain wall dynamics

Figure 3:  Illustrations of domain wall motion. (a) Removing a column of
bosons. (b) Shifting the right half of the lattice costs no extra potential
energy but gains kinetic energy due to the fluctuation of the domain wall (red
line). (c) Multiple domain walls. (d) QFS state with a domain wall density
$\rho_{D}=1/2$.

Here we follow the analysis in Ref. [16] for incommensurate supersolids on
anisotropic triangular lattices to discuss the domain wall dynamics in the
present model. Started from the half-filled CBS state, one can split the CBS
into half by removing a column of bosons (i.e., removing $L_{y}/2$ bosons), as
depicted in Fig. 3(a). This gives a cost in potential energy per unit length
$\Delta E_{p}/L_{y}=(\mu-4V^{\prime})/2$. After shifting half of the lattice
upwards, a domain wall is created without further cost in potential energy. To
be specific, we define the domain walls by connecting the center of unoccupied
bonds, shown as the red solid zigzag line in Fig. 3(b). Note that the domain
wall can fluctuate to gain the kinetic energy, because the bosons around the
shifted boundary are now free to move sideways. As illustrated in Fig. 3(b),
after the hopping of bosons, parts of the domain wall fluctuate in the
opposite direction of the particle movement (shown as the red dotted lines).
To estimate the gain in kinetic energy, one can map the one-dimensional (1D)
domain wall onto a 1D spin-1/2 XY chain [16, 19, 20]. That is, by tracing the
zigzag chain along the $y$ direction, a up (down) spin is assigned if the
$x$-coordinate of a given unoccupied bond is increased (decreased) by one
unit. The kinetic energy of a domain wall thus corresponds to the ground-state
energy of the 1D spin chain and has the value (per unit length) $\Delta
E_{k}/L_{y}=-2t/\pi$ [19]. Therefore, the total excitation energy (per unit
length) of a domain wall becomes $\Delta E/L_{y}=\Delta E_{p}/L_{y}+\Delta
E_{k}/L_{y}=(\mu-4V^{\prime})/2-2t/\pi$. When its excitation energy is reduced
to zero by decreasing $\mu$, the phase transition out of the CBS induced by
proliferation of domain walls will occur. This gives the critical $\mu_{c1}$
for the CBS-ICS phase transition, i.e.

$\mu_{c1}=4V^{\prime}+\frac{4t}{\pi}.$ (4)

This quantum phase boundary is plotted in Fig. 1 (the upper dotted line),
which is in good agreement with the QMC results.

For the transition from QFS to ICS, the same argument applies. Doping the QFS
with a column of bosons, it actually generates two domain walls and the total
cost in potential energy per unit length equals to $(4V^{\prime}-\mu)/2$. Here
we define the domain wall as the line connecting nn bosons. Again, by mapping
each of the two domain walls onto a 1D spin-1/2 chain, the gain in kinetic
energy per unit length of the two domain walls is deduced to be $2(-2t/\pi)$.
Therefore, the critical $\mu_{c2}$ for the QFS-ICS phase transition is given
by

$\mu_{c2}=4V^{\prime}-\frac{8t}{\pi},$ (5)

which is again in good agreement with the numerical results. Deviations in
analytical predictions of $\mu_{c1}$ and $\mu_{c2}$ are expected for large
$t$, since higher order corrections and SF fluctuations become important then.

Following the analysis in Ref. [16], we also determine the domain wall density
$\rho_{D}$ as a function of $\mu$, which is defined as:

$\rho_{D}=\frac{1}{L_{x}L_{y}}\sum_{x,y}\bar{n}_{x,y}\bar{n}_{x+1,y}\;,$ (6)

where $\bar{n}_{x,y}=1-n_{x,y}$ is the hole number at site $(x,y)$. Here the
summation of $\bar{n}_{x,y}\bar{n}_{x+1,y}$ over $x$ coordinates counts the
number of hole-hole bonds in the $x$ direction, and then gives the number of
linear domain walls along the $y$ direction. The QMC results of $\rho_{D}$ are
shown in Fig. 4.

Figure 4: Domain wall density $\rho_{D}$ as a function of $\mu$ for $t=0.4$
(upper panel) and $t=0.2$ (lower panel). The red solid lines are analytical
predictions given by Eq. (10). The insets show the fittings of the interaction
energy $f(\rho_{D})$ to a power-law function $A\rho_{D}^{\alpha}$ for each
case. Here $V=4$ and $L=24$ are taken.

Clearly, the domain walls will interact with each other. In combination with
the discussions in the first paragraph of this section for isolated domain
walls, the total energy for interacting domain walls at a finite density
$\rho_{D}$ will take the following form:

$E(\rho_{D})=L_{x}L_{y}V^{\prime}\rho_{D}\left[-2+\frac{\mu}{2V^{\prime}}-\frac{2t}{\pi
V^{\prime}}+f(\rho_{D})\right].$ (7)

Here the last term accounts for an effective repulsive interaction energy
between two neighboring domain walls. This yet unknown function $f(\rho_{D})$
can be determined by fitting to the numerical results.

We note that the transition from the state with $2M-2$ domain walls to that
with $2M$ domain walls occurs when $E[(2M-2)/L_{x}]=E(2M/L_{x})$. This
condition leads to a recursive relation of $f(2M/L_{x})$:

$\mu_{M}=\mu_{c1}+2V^{\prime}\left[(M-1)f(\frac{2M-2}{L_{x}})-Mf(\frac{2M}{L_{x}})\right]\;,$
(8)

which can be solved to have

$f(\frac{2M}{L_{x}})=\frac{1}{V^{\prime}}\left[\frac{\mu_{c1}}{2}-\sum^{M}_{i=1}\frac{\mu_{i}}{2M}\right].$
(9)

As a result, the simulated values of $\mu_{M}$ can be used to evaluate the
discrete values of $f(2M/L_{x})$. As shown in the inset of Fig. 4, a power-law
function $A\rho_{D}^{\alpha}$ can be fitted to $f(\rho_{D})$ with the exponent
$\alpha=3.4(2)$ for $t=0.4$ (upper panel). With this result, the domain wall
energy of Eq. 7 is then minimized to obtain the domain wall density

$\rho_{D}=\left[\frac{\mu_{c1}-\mu}{2V^{\prime}A(\alpha+1)}\right]^{1/\alpha}\;,$
(10)

which is found to be in good agreement with the simulated data in Fig. 4. Our
results thus justify the application of the domain wall dynamics in the
present system. Interestingly, we repeated the same calculations on domain
wall density for $t=0.2$ and obtained nearly the same exponent
$\alpha=3.6(3)$. This suggests a possible universal exponent for different
hopping integrals.

##  V Incommensurate-superfluid and thermal phase transitions

Figure 5: Superfluid density $\rho_{s}$ (open circles) and structure factor
$S(\textbf{Q})$ (open squares) as functions of $t$ for (a) $\mu=4.0$ and (b)
$\mu=3.2$. Here $V=4$ and $L=24$. Both quantities show the signatures of
first-order phase transition from the ICS to the SF states.

In our parameter regime, the quantum phase transitions from all solid states
to the SF phase are observed to be discontinuous and there appears no
immediate supersolid phase. These discontinuous transitions are demonstrated
by the abrupt changes of the superfluid density $\rho_{s}$ and the structure
factor $S(\textbf{Q})$ at the phase boundaries. Here we focus on the
transitions out of the ICS states.

In Fig. 5, we present our data for $\mu=4.0$ and 3.2, which give the ICS
states at small hopping $t$ with $\rho=1/3$ and 7/24 respectively. Upon
increasing $t$, we find that both $\rho_{s}$ and $S(\textbf{Q})$ change
abruptly across the phase boundaries. We note that the solid ordering is
destroyed at the same critical $t$ where the superfluidity emerges. Hence no
signs of coexistence of both order parameters is found. The absence of
supersolid phase could be accounted for by the same argument for the CBS and
the QFS that phase separation is more favorable than the supersolid phase [9].

Figure 6: Boson density $\rho$ (circles) and superfluid density $\rho_{s}$
(squares) as functions of $V^{\prime}$ with $V=4$ for (a) $\mu=4.0$ and (b)
$\mu=4.6$. Here $L=36$ and $V=4$.

In the previous discussions, we fix the nnn interaction $V^{\prime}$ to be
unity. We now turn to the effect from varying $V^{\prime}$. When $V^{\prime}$
is increased, the enhanced nnn frustrations will reduce the cost in potential
energy for the domain wall formation (see Sec. IV). Therefore, the boson
density of the system is expected to follow a series of density drops as more
domain walls of holes can be generated for larger $V^{\prime}$. Our numerical
results for $V=4$ at $\mu=4.0$ and 4.6 plotted in Fig. 6 demonstrate this
prediction. At $\mu=4.0$, the system moves from the 1/3-filling state all the
way to the quarter-filling state as $V^{\prime}$ increases. However, at
$\mu=4.6$, when $V^{\prime}>1.24$, the frustration becomes too strong such
that the diagonal long-range order cannot be sustained and the system melts
into a superfluid phase instead. An abrupt jump in superfluid density is
observed that signals a direct first-order phase transition from the ICS to
the SF and there exists no intermediate supersolid phase.

Figure 7: Hysteresis of the structure factors S(Q) for the cases of (a)
$\mu=4.0$ and (b) $\mu=2.0$, which correspond to the 1/3-filling plateau state
and the 1/4-filling QFS state respectively. Here $V=4$ and $L=24$. The insets
show the double-peak feature of S(Q) around the transitions.

We now consider the thermal phase transitions out of the ICS and the QFS
phases. It is expected that the broken translation symmetry in these two
phases will be restored by thermal fluctuations at high temperatures. This
symmetry restoration can be detected by the disappearance of the corresponding
structure factors. Careful studies of the order-disorder transitions indicate
that these transitions are of first order. In Fig. 7, we measure the structure
factors $S(2\pi/3,\pi)$ and $S(\pi/2,\pi)$ of the $\rho=1/3$ ICS state and the
$\rho=1/4$ QFS state respectively. At the first glance, the thermal
transitions seem to be continuous, but a detailed analysis shows the typical
hysteresis behaviors of discontinuous phase transitions around the transition
temperatures. Starting from the order (disorder) states, we increase
(decrease) the temperature slowly in the QMC simulations by using the operator
lists taken from previous temperatures as initial conditions. From Fig. 7, it
is clearly that results from increasing and decreasing temperatures follow
different paths around the phase transitions. Double peaks are also observed
in the histograms (insets in Fig. 7) which again confirm the nature of first-
order transitions. We note that the previous study for another type of
quarter-filled solid suggested a continuous thermal phase transition [14].
This difference can be explained by the distinct broken symmetries of these
two types of quarter-filled solids. Furthermore, it is noticeable that the
hysteresis loop becomes much smaller and the double peaks feature less
significantly in the ICS than what happen in the QFS. This indicates that, as
the chemical potential $\mu$ increases from the QFS to the ICS, the thermal
transition becomes a more weakly first-order one. This is consistent with the
fact that the thermal transition of half-filled CBS is known to be continuous,
so that approaching the CBS from the QFS via the ICS, the thermal transitions
are expected to evolve from strongly first-order to weakly first-order and
then to second-order. Since the ordering wave vector $\mathbf{Q}$ of the
maximal structure factor varies continuously in the ICS phase into the CBS, it
is natural to expect a smooth evolution of transition order instead of an
abrupt change as $\mathbf{Q}$ approaches to ($\pi$, $\pi$).

##  VI Conclusion

We have shown that the ICS states can emerge in a bosonic model on a square
lattice under short-range frustrated interactions. This observation has been
overlooked in previous studies of the extended Bose-Hubbard model. The ICS
phase appears in between the half-filled and the quarter-filled solids and is
characterized by a series of density plateaus with fractional values of
densities. The short-range character of the interactions in our model
simplifies the analysis and allows us to address fundamental questions about
the nature of the plateau states. By following the analysis in Ref. [16], we
show that the fractional particle densities and the ordering wave vectors of
the plateau states can be explained by the domain wall formation. This
indicates that the incommensurate phase originates from proliferation of
domain walls. Furthermore, the measured domain wall densities $\rho_{D}$ agree
well with the predicted values from the theory of the interacting domain
walls. We find that the interaction term behaves as a power-law function of
$\rho_{D}$ with an exponent likely to be independent of the system
parapmeters. In the thermodynamic limit, we expect the ordering wave vector
$\mathbf{Q}$ characterizing different broken symmetries changes continuously
from $\mathbf{Q}=(\pi/2,\pi)$ for the quarter-filled star order to
$\mathbf{Q}=(\pi,\pi)$ for the checkerboard order. In addition, the widths of
the intermediate plateau states become diminished. There is no superfluidity
found in the ICS phase, which differs from the case of anisotropic triangular
lattice where an incommensurate supersolid is observed [16]. It is not
surprising since, in contrast to the case of the square lattice, bosons have
more degrees of freedom to hop around on the triangular lattice. Based on this
observation, incommensurate supersolids may come out in our model if the nnn
hopping is turned on. It is also interesting to consider the effect of strong
frustration (large $V^{\prime}$) on the ICS phase. We show that, by increasing
$V^{\prime}$, either the ICS state is melted into a SF or the system is
transformed to be a QFS. Nevertheless, it is known that for even stronger
$V^{\prime}$ another type of star order becomes favorable at quarter filling.
Whether similar ICS phases can still appear around such quarter-filled solids
is not clear without extensive studies on a wide range of parameter regime.
Furthermore, one may wonder if the present mechanism of domain wall formation
may be applied to explain the results observed in the infinite-range models.
This is worth investigating in a further study.

###### Acknowledgements.

The numerical computations are performed in the Center for High Performance
Computing of the THU. This work is financial supported by the Ministry of
Science and Technology of Taiwan under Grant No. MOST 108-2112-M-029 -005 and
MOST 109-2112-M-029-006. M.F.Y. is also supported by the Ministry of Science
and Technology of Taiwan under Grant No. MOST 109-2112-M-029-005.

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