# Beyond positivity in Ehrhart Theory

Karim Alexander Adiprasito  Karim Adiprasito  _and_ Vasiliki Petrotou  _and_
Johanna Steinmeyer, Einstein Institute of Mathematics, Hebrew University of
Jerusalem, 91904 Jerusalem, Israel _and_ Department of Mathematical Sciences,
University of Copenhagen, 2100 Copenhagen, Denmark<EMAIL_ADDRESS>_and_<EMAIL_ADDRESS>_and_<EMAIL_ADDRESS>,  Stavros
Argyrios Papadakis  Stavros Papadakis, Department of Mathematics, University
of Ioannina, Ioannina, 45110, Greece<EMAIL_ADDRESS>,  Vasiliki Petrotou
and  Johanna Kristina Steinmeyer

(Date: October 19, 2022)

###### Abstract.

We study semigroup algebras arising from lattice polytopes, compute their
volume polynomials (particularizing work of Hochster), and establish strong
Lefschetz properties (generalizing work of the first three authors). This
resolves several conjectures concerning unimodality properties of the
$h^{\ast}$-polynomial of lattice polytopes arising within Ehrhart theory.

###### Key words and phrases:

Gorenstein polytopes, Lefschetz property, unimodality

##  1\. Lattice polytopes and semigroup algebras

The main player of this paper is a convex polytope $P$ all whose vertices lie
in a lattice $\Lambda$. Ehrhart theory analyzes the generating function

$\mathrm{Ehr}_{P}(t)\ :=\ \sum_{i=0}^{\infty}\\#\\{iP\cap\Lambda\\}t^{i}.$

We can then write

$\mathrm{Ehr}_{P}(t)\ =\ \frac{h^{\ast}_{P}(t)}{(1-t)^{d+1}},$

where $d$ is the dimension of $P$ and $\
h^{\ast}_{P}(t)=h_{0}^{\ast}+h_{1}^{\ast}t+\ldots+h_{d}^{\ast}t^{d}\ $ is a
polynomial of degree at most $d$.

To interpret this algebraically, we recall the following well-known
construction: Embed the polytope in $\mathbb{R}^{d+1}$ at height $1$ and
consider the cone over it,

$\operatorname{cone}(P)\coloneqq\mathbb{R}_{\geq 0}(P\times\\{1\\}).$

As such, it generates a semigroup algebra

$\mathbbm{k}[^{*}P]\coloneqq\mathbbm{k}^{*}[\operatorname{cone}(P)\cap\mathbb{Z}^{d+1}],$

graded by the last coordinate. Here $\mathbbm{k}$ is any field, though we
generally assume the field to be infinite to ensure the existence of an
Artinian reduction. In this case $\mathbbm{k}[P]$ is Cohen-Macaulay [Hoc72]
with Hilbert series $\mathrm{Ehr}_{P}(t)$. For a choice of linear system of
parameters $\theta_{0},\ldots,\theta_{d}\in\mathcal{A}^{1}(P)$, the Artinian
reduction

$\mathcal{A}^{*}(P)\coloneqq\mathbbm{k}^{*}[P]/\left<\theta_{0},\ldots,\theta_{d}\right>$

has $\dim\mathcal{A}^{k}(P)=h^{\ast}_{k}$. It follows that the $h^{\ast}_{k}$
are nonnegative.

It has been a central question in the theory of lattice polytopes to determine
additional properties for these coefficients. In particular, Hibi and Ohsugi
conjectured that under two special conditions, the coefficients form a
unimodal sequence [OH06], see also [Bra16, SVL13].

The first of these conditions is that $P$ has the integer decomposition
property, short IDP: every lattice point of $\operatorname{cone}(P)$ is a
nonnegative integral combination of lattice points in $P\times\\{1\\}$, or
equivalently that $\mathbbm{k}^{*}[P]$ is generated in degree one.

The second property is the reflexive property: there is a lattice point $p$ in
$\mathbb{Z}^{d}\times\\{1\\}$ such that

$\operatorname{cone}^{\circ}(P)\cap\mathbb{Z}^{d+1}=\
p+\operatorname{cone}(P)\cap\mathbb{Z}^{d+1},$

where $\operatorname{cone}^{\circ}(P)$ is the interior of
$\operatorname{cone}(P)$. This is equivalent to $\mathbbm{k}[P]$ being
algebraically Gorenstein (that is, a Poincaré duality algebra after any
Artinian reduction) with socle degree $d$ as well as to
$h_{k}^{\ast}=h_{d-k}^{\ast}$ for all $k\leq\nicefrac{{d}}{{2}}$. Numerically
on the level of the $h^{*}$-vector, the restriction to reflexive polytopes
rather than all Gorenstein polytopes is without loss of generality. Bruns and
Herzog [BR07] showed that for every Gorenstein polytope there is a reflexive
polytope with the same $h^{*}$-vector.

We resolve the following conjecture of Hibi and Ohsugi.

######  Conjecture 1.1 (Hibi-Ohsugi).

For any IDP reflexive lattice polytope $P\subset\mathbb{R}^{d}$, the
coefficients of the $h^{\ast}$-polynomial are unimodal:

$h_{0}^{\ast}\leq h_{1}^{\ast}\leq\ldots\leq
h_{\lfloor\nicefrac{{d}}{{2}}\rfloor}^{\ast}=h_{\lceil\nicefrac{{d}}{{2}}\rceil}^{\ast}\geq\ldots\geq
h_{d}^{\ast}$

This is the updated form of a conjecture of Hibi [Hib92], after Mustaţa and
Payne gave an example showing the necessity of the IDP assumption [MP05].
These conjectures in turn go back to a more general one of Stanley [Sta89],
who proposed that the unimodality may hold for a general Gorenstein standard
graded integral domain.

In fact, we shall prove statements that are more powerful than this, and apply
to more general cases. For instance, in the case of polytopes that have only
the IDP, we still obtain monotone decreasing coefficients in the second half,
i.e.

$h^{\ast}_{\lfloor\nicefrac{{d}}{{2}}\rfloor}\geq\ldots\geq h_{d}^{\ast}.$

Moreover, we refine this further by studying also the _local_
$h^{\ast}$-polynomial of a polytope, and obtain unimodality of this refined
invariant for any IDP lattice polytope.

We prove Conjecture 1.1 by proving a Lefschetz property for a generic Artinian
reduction of the associated semigroup algebra.

######  Theorem 1.2.

If $P$ is an IDP reflexive polytope, and the characteristic of $\mathbbm{k}$
is $2$ or $0$, then a generic Artinian reduction $\mathcal{A}^{*}(P)$ of
$\widetilde{\mathbbm{k}}^{*}[P]$ has the Lefschetz property, i.e. there is a
linear element $\ell\in\mathcal{A}^{1}(P)$ such that for any
$k\leq\nicefrac{{d}}{{2}}$, the map

$\mathcal{A}^{k}(P)\xrightarrow{\ \cdot\ell^{d-2k}\ }\ \mathcal{A}^{d-k}(P)$

is an isomorphism.

Here, _generic_ shall mean that the Artinian reduction is taken by linear
forms

$\theta_{0},\ldots,\theta_{d},\quad\text{where}\quad\theta_{i}=\sum_{p\in
P\cap\Lambda}\theta_{i,p},$

with transcedentally independent coefficients $\theta_{i,p}$, necessitating
passing to a transcendental field extension $\widetilde{\mathbbm{k}}$ of
$\mathbbm{k}$.

This is a rather extreme choice of linear system of parameters, necessitated
by the proof via anisotropy. We want to emphasize the importance of the choice
of l.s.o.p., as it makes a crucial difference for the ring we end up working
with. For a specific, and somewhat canonical, choice of l.s.o.p.,
$\mathcal{A}^{*}(P)$ is isomorphic to the orbifold Chow ring of the associated
toric Deligne-Mumford stack [BCS05]. In contrast to this choice, we make of a
generic Artinian reduction here. As graded vector spaces, the results are
isomorphic and the inequalities on the dimensions of the graded pieces remain
true. Yet, as observed in [Adi18], whether an Artinian reduction admits a
Lefschetz element depends on the Artinian reduction, and not only on
$\mathbbm{k}^{*}[P]$. That such “bad” Artinian reductions exist in the present
context was observed in [BD16], giving an example of an IDP reflexive simplex
and an Artinian reduction of the associated semigroup algebra which does not
even admit a weak Lefschetz element. The linear system of parameters chosen
there is however not the canonical system for the orbifold Chow ring, and
whether that ring has the Lefschetz property remains open.

As the composition of individual multiplications with $\ell$, the Lefschetz
isomorphism gives us an injection in the first half and a surjection in the
second half, and thus the desired inequalities on the dimensions of the graded
pieces:

######  Theorem 1.3.

The $h^{\ast}$-polynomial of a reflexive IDP lattice polytope has a unimodal
sequence coefficients. Stronger, we get that
$(h_{i+1}^{\ast}-h_{i}^{\ast})_{0\leq i\leq\nicefrac{{d}}{{2}}}$ is an
$M$-vector.

Here the $M$-vector property too immediately follows from the Lefschetz
property. The vector of differences $\max\\{h^{\ast}_{i}-h^{\ast}_{i-1},0\\}$
is the Hilbert vector of a standard graded algebra, namely

${\raisebox{3.00003pt}{$\mathcal{A}^{\ast}(P)$}\Big{/}\raisebox{-3.00003pt}{$\ell\mathcal{A}^{\ast}(P)$}}.$

We follow the recent breakthroughs of Adiprasito [Adi18], Papadakis and
Petrotou [PP20, APP21], though our work requires a critical new ingredient.
This is due to the fact that the proofs employed previously for Stanley-
Reisner rings made use of explicitly combinatorial techniques to reach the
desired goal, and the algebra we investigate here is not immediately as
governed by a combinatorial struture as the previous one, being cut out by
binomial ideals rather than monomials. And so while Hochster was the one to
gift us with understanding of the canonical module, he was unable to determine
a natural normalization of the fundamental class. We give one here, compatible
with the case of Stanley-Reisner rings. From this, we then get the key
identity, reminiscent of Parseval’s identity in Fourier analysis.

The paper is organized as follows. We start in Section 2 by generalizing the
setup in order to state our main theorem and deduce the individual numerical
corollaries within Erhart theory. For sake of completeness, we then recall in
Section 3 the necessary parts of the machinery of [Adi18, PP20, APP21] in
order to prove the Lefschetz statements by way of anisotropy. Following this
thorough setup, we give our new contributions to the theory. Section 4
contains the normalization of the fundamental class and an auxiliary
differential identity, while Section 5 contains the key identity of Parseval
type. We finish with a discussion of remaining open questions in Section 6.

##  2\. Poincaré, Gorenstein and the Lefschetz properties

For $P$ reflexive, the semigroup algebra $\mathbbm{k}^{*}[P]$ is Gorenstein of
Krull dimension equal to the dimension of the polytope plus one [BH93]. After
an Artinian reduction using a linear system of parameters of length equal to
the Krull dimension, we arrive at a Poincaré duality algebra of socle degree
$d$.

For general IDP polytopes, the situation is a little more delicate. One can
force Poincaré duality however, using the usual trick: we allow for relative
objects. The $\mathbbm{k}^{*}[P]$-module defined by

$\mathbbm{k}^{*}[P,\partial
P]\coloneqq\mathbbm{k}^{*}[(\operatorname{cone}(P)\setminus\partial\operatorname{cone}(P))\cap\Lambda^{\prime}]$

is the canonical module of the Cohen-Macaulay ring $\mathbbm{k}^{*}[P]$
[Hoc72]. After Artinian reduction, we are left with a perfect bilinear pairing

$\mathcal{A}^{k}(P)\ \times\ \mathcal{A}^{d+1-k}(P,\partial P)\
\longrightarrow\ \mathcal{A}^{d+1}(P,\partial P)\ \cong\ \mathbbm{k}.$

######  Theorem 2.1.

If $P$ is an IDP polytope of dimension $d$, and the characteristic of
$\mathbbm{k}$ is $2$ or $0$, then a generic Artinian reduction
$\mathcal{A}^{*}(P)$ of $\widetilde{\mathbbm{k}}^{*}[P]$ has the relative
Lefschetz property, i.e. there exists a linear element
$\ell\in\mathcal{A}^{1}(P)$ such that for all $k\leq\nicefrac{{d+1}}{{2}}$,

$\mathcal{A}^{k}(P,\partial P)\ \xrightarrow{\ \cdot\ell^{d+1-2k}\ }\
\mathcal{A}^{d+1-k}(P)$

is an isomorphism. Moreover, the system of parameters can be chosen such that
the restriction to any face forms a linear system of parameters when
restricted to that face (meaning that its length does not exceed the depth of
the module on that face.)

The final part here is important for the refined statement about local
$h^{\ast}$-vectors, and we call it the _perverted_ linear system, because it
is the most generic linear system agreeing with the intersection cohomology of
the associated orbifold.

Note that Theorem 2.1 implies Theorem 1.2. For $P$ reflexive, we have an
isomorphism of semigroup algebras
$\mathbbm{k}^{k}[P]\xrightarrow{\cdot\mathbf{x}_{p}}\mathbbm{k}^{k+1}[P,\partial
P]$ which passes to the Artinian reduction. For $k\leq\nicefrac{{d}}{{2}}$, we
then get the Lefschetz isomorphism as the composition

$\mathcal{A}^{k}(P)\xrightarrow{\cdot\mathbf{x}_{p}}\mathcal{A}^{k+1}(P,\partial
P)\xrightarrow{\ \cdot\ell^{d-2k-1}}\mathcal{A}^{d-k}(P).$

For general IDP polytopes, Theorem 2.1 still gives surjections

$\mathcal{A}^{d-k}(P)\xrightarrow{\cdot\ell}\mathcal{A}^{d+1-k}(P)\quad\text{and}\quad\mathcal{A}^{k}(P)\xrightarrow{\cdot\ell^{d+1-2k}}\mathcal{A}^{d+1-k}(P)$

for $k\leq\lfloor\nicefrac{{d}}{{2}}\rfloor$ and thus

######  Corollary 2.2.

The $h^{\ast}$-polynomial of an IDP lattice polytope of dimension $d$ has
monotone decreasing coefficients in the second half, i.e.

$h^{\ast}_{\lfloor\nicefrac{{d}}{{2}}\rfloor}\geq\ldots\geq h_{d}^{\ast}.$

Moreover, for all $k\leq\nicefrac{{d+1}}{{2}}$, we have

$h^{\ast}_{k}\ \geq\ h^{\ast}_{d+1-k}.$

In particular, for any IDP lattice polytope, Stapledon’s $a$-polynomial has
unimodal coefficients [Sta09].

The last part also follows from Theorem 2.4, by observing that the
$a$-polynomial corresponds exactly to the $h^{*}$-polynomial of $\partial P$
as a lattice complex.

In addition, Theorem 2.1 has interesting consequences if we know at what
height the interior of $\operatorname{cone}^{\circ}(P)$ is generated. Suppose
all minimal elements of $\operatorname{cone}^{\circ}(P)$ are of heights at
most $j$. Then, for $k\leq\lceil\frac{d-j}{2}\rceil$, every nontrivial $u$ in
$\mathcal{A}^{k}(P)$ multiplies with one of these elements, say
$\mathbf{x}_{\alpha}$. Hence in the composition

$\mathcal{A}^{k}(P)\ \xhookrightarrow{\ \cdot\mathbf{x}_{\alpha}\ }\
\mathcal{A}^{k+j}(P,\partial P)\xrightarrow{\ \cdot\ell^{d+1-2k-2j}\
}\mathcal{A}^{d+1-k-j}(P),$

the element $u$ is nontrivial. Hence $\ell^{d+1-2k-2j}u$ is nontrivial.

######  Corollary 2.3.

The $h^{\ast}$-polynomial of an IDP lattice polytope of dimension $d$ with
$\operatorname{cone}^{\circ}(P)$ generated at height $\leq j$ has monotone
increasing coefficients in the initial part, i.e.

$h^{\ast}_{0}\leq\ldots\leq h^{\ast}_{\lceil\frac{d-j}{2}\rceil}.$

Moreover, for $k\leq\lceil\frac{d-j}{2}\rceil$ we have

$h_{k}^{*}\leq h^{*}_{d+1-j-k}.$

For the finer statements regarding the local $h^{\ast}$-vector, we need a more
powerful algebraic statement. To this end, notice that given any abstract
polytopal complex $\Sigma$ consisting of lattice polytopes (short lattice
complex), with the property that the lattices coincide in common
intersections, we obtain an analogous Cohen-Macaulay ring
$\mathbbm{k}^{*}[\Sigma]$ [BBR07]. In analogy with the proofs of the g-theorem
of [AHK18, PP20, APP21], we have

######  Theorem 2.4.

If $\Sigma$ is an IDP lattice sphere or ball (that is, a polyhedral sphere or
ball with lattice structure in which every element has the IDP property) of
dimension $d$, and the characteristic of $\mathbbm{k}$ is $2$ or $0$, then a
generic Artinian reduction $\mathcal{A}^{*}(\Sigma)$ of
$\widetilde{\mathbbm{k}}^{*}[\Sigma]$ has the Lefschetz property

$\mathcal{A}^{k}(\Sigma,\partial\Sigma)\ \xrightarrow{\ \cdot\ell^{d-2k}\ }\
\mathcal{A}^{d+1-k}(\Sigma).$

Similar results hold for manifolds and cycles (again in direct analogy to
[APP21]) but seem less immediately relevant here.

For the case that $\Sigma$ is a sphere, we of course have
$\mathcal{A}^{*}(\Sigma,\partial)=\mathcal{A}^{*}(\Sigma)$, letting us
consider the standard graded algebra
${\raisebox{3.00003pt}{$\mathcal{A}^{\ast}(\Sigma)$}\Big{/}\raisebox{-3.00003pt}{$\ell\mathcal{A}^{\ast}(\Sigma)$}}$.
We denote the dimensions of the graded pieces by $g_{k}^{*}$.

For any polytope $P$ with the integer decomposition property, we can now
combine the relative property for $P$ with the Lefschetz property for the
boundary sphere $\partial P$.

Consider to that effect the exact sequence

$\widetilde{\mathbbm{k}}^{*}[P,\partial P]\ \longrightarrow\
\widetilde{\mathbbm{k}}^{*}[P]\ \longrightarrow\
\widetilde{\mathbbm{k}}^{*}[\partial P]\ \longrightarrow\ 0.$

In order to pass to a statement on the Artinian reductions, we take a
perverted linear system of parameters for $P$, ensuring that the restriction
to $\partial P$ is again a proper linear system of parameters. With the last
element $\theta_{d}$ acting as Lefschetz element for $\mathcal{A}^{*}(\partial
P)$, we get the short exact sequence

$0\ \longrightarrow\ \mathrm{im}\Big{(}\mathcal{A}^{*}(P,\partial
P)\to\mathcal{A}^{*}(P)\Big{)}\ \longrightarrow\ \mathcal{A}^{*}(P)\
\longrightarrow\ {\raisebox{3.00003pt}{$\mathcal{A}^{\ast}(\partial
P)$}\Big{/}\raisebox{-3.00003pt}{$\theta_{d}\mathcal{A}^{\ast}(\partial P)$}}\
\longrightarrow\ 0.$

We want to use this observation to analyze the local $h^{*}$-polynomial
$\ell^{*}(t)$ (also called the Box polynomial or $\tilde{S}$-polynomial)
considered by Betke-McMullen, Stanley, and others [MB85, Sta92, BM03, KS16].
From Stanley [Sta92][Theorem 7.8, Example 7.13], we get for any Cohen-Macaulay
lattice complex $\Sigma$ that

$h_{\Sigma}^{*}(t)\ =\sum_{\varnothing\leq F\in\Sigma}\ell_{F}^{*}(t)\cdot
h_{[F,\Sigma]}(t)\ =\sum_{\varnothing\leq F\in\Sigma}\ell_{F}^{*}(t)\cdot
g_{[F,\Sigma)}(t),$

where the sum goes over all faces of $\Sigma$ and
$h_{[F,\Sigma]}(t),g_{[F,\Sigma)}(t)$ are the toric $h$\- and $g$-polynomial
respectively, of the posets $[F,\Sigma]$ and $[F,\Sigma)$, given as an
interval in the face lattice of $\Sigma$.

From this we deduce

$\displaystyle\ell^{*}_{P}(t)\ $ $\displaystyle=\ h_{P}^{*}(t)\
-\sum_{\varnothing\leq F\in\partial P}\ell_{F}^{*}(t)\cdot h_{[F,P]}(t)$
$\displaystyle=\ h_{P}^{*}(t)\ -\sum_{\varnothing\leq F\in\partial
P}\ell_{F}^{*}(t)\cdot g_{[F,\partial P]}(t)$ $\displaystyle=\
h^{*}_{P}(t)-g^{*}_{\partial P}(t),$

implying that the dimensions of the graded pieces of
$\mathrm{im}\Big{(}\mathcal{A}^{*}(P,\partial P)\to\mathcal{A}^{*}(P)\Big{)}$
are exactly $\ell^{*}_{k}$.

The relative Lefschetz map now restricts to
$\mathrm{im}\Big{(}\mathcal{A}^{*}(P,\partial P)\to\mathcal{A}^{*}(P)\Big{)}$
and thus

######  Theorem 2.5.

The local $h^{\ast}$-vector $\ell^{\ast}$ of any IDP polytope is unimodal.

##  3\. Anisotropy

To deduce the Lefschetz property, we employ a reduction found in [Adi18], see
also [APP21]: It is enough to demonstrate a nondegeneracy property of the
Poincaré pairing at certain ideals. The following theorem then finally implies
the Lefschetz theorems.

######  Theorem 3.1.

If $P$ is an IDP $d$-polytope, and the characteristic of $\mathbbm{k}$ is $2$,
then a generic Artinian reduction $\mathcal{A}^{*}(P,\partial P)$ of
$\widetilde{\mathbbm{k}}^{*}[P,\partial P]$ has the anisotropy property, that
is, for every nontrivial $u\in\mathcal{A}^{k}(P,\partial P)$ of degree
$k\leq\nicefrac{{d}}{{2}}$, we have $u^{2}\ \neq\ 0$.

The analogous result applies to lattice discs and spheres.

Most of the remainder of this paper is devoted to proving this theorem. Before
we do that, however, let us note that from here, the Lefschetz property is a
cakewalk. For sake of completeness, we remind the reader of the derivation of
the Lefschetz property from anisotropy, based on [Adi18].

Consider an IDP lattice sphere of dimension $d-1$. We say that
$\mathcal{A}^{\ast}(\Sigma)$, with socle degree $d$, satisfies the Hall-Laman
relations in degree $k\leq\frac{d}{2}$ and for ideal
$\mathcal{I}^{\ast}\subset\mathcal{A}^{\ast}(\Sigma)$ if there exists an
$\ell$ in $\mathcal{A}^{1}(\Sigma)$, such that the pairing

$\begin{array}[]{ccccc}\mathcal{I}^{k}&\times&\mathcal{I}^{k}&\longrightarrow&\
\mathcal{I}^{d}\cong\mathbbm{k}\\\ a&&b&{{\xmapsto{\ \ \ \ }}}&\
\mathrm{deg}(ab\ell^{d-2k})\end{array}$ (1)

is nondegenerate.

Note that that the Hall-Laman relations for an ideal
$\mathcal{A}^{\ast}(\Delta,\partial\Delta)$ corresponding to a disk $\Delta$
of dimension $d$ in $\Sigma$ are precisely the relative Lefschetz property of
that disk; hence the Hall-Laman relations give us a way to discuss relative
Lefschetz properties properly.

To in turn prove the Hall-Laman relations, consider the suspension
$\operatorname{susp}\Sigma$ of the lattice disc or sphere $\Sigma$, which is
again IDP. Note that $\Sigma$ itself is a face of $\operatorname{susp}\Sigma$,
so two points with one in each of the cones over $\Sigma$ do not lie in a
common face. On the level of semigroup algebras, a suspension corresponds to
the introduction of two variables which multiply to zero. Label the two
vertices (corresponding to the newly introduced indeterminates) of the
suspension $\mathbf{n}$ and $\mathbf{s}$ (for north and south). Let $\uppi$
denote the projection along $\mathbf{n}$, and let $\vartheta$ denote the
height over that projection.

######  Lemma 3.2 ([Adi18, Lemma 7.5]).

Considering $\operatorname{susp}\Sigma$ realized in $\mathbbm{k}^{d+1}$, and
$k<\frac{d}{2}$, the following two are equivalent:

  1. (1)

The Hall-Laman relations in degree $k+1$ for
$\mathcal{I}\cdot\mathbf{x}_{\mathbf{n}}$ in
$\mathcal{A}^{k+1}(\operatorname{susp}\Sigma)$ with respect to
$\mathbf{x}_{\mathbf{n}}$.

  2. (2)

The Hall-Laman relations in degree $k$ for $\mathcal{I}$ in
$\mathcal{A}^{k}(\Sigma)$ with respect to $\vartheta$.

This reduces the Lefschetz property iteratively to a pairing property for
$d+1=2k$ and such that

$\begin{array}[]{ccccc}\mathcal{J}^{k}&\times&\mathcal{J}^{k}&\longrightarrow&\
\mathcal{J}^{d}\cong\mathbbm{k}\\\ a&&b&{{\xmapsto{\ \ \ \ }}}&\
\mathrm{deg}(ab)\end{array}$

is nondegenerate on some $d$-dimensional lattice disc or sphere. But this is
guaranteed for any $\mathcal{J}$ by the anisotropy property of Theorem 3.1.
The result for spheres and discs follows simply by linearity, as each monomial
lies in a polytopal face.

We are left with the task of proving Theorem 3.1 in the case when $2k$ equals
the dimension $d$ of the polytope plus one, so that

$\mathcal{A}^{2k}(P,\partial P)\ \cong\ \widetilde{\mathbbm{k}}.$

Given the Lefschetz property over _some_ field of characteristic $2$, note
that the Lefschetz property over _any_ field of characteristic $2$ or $0$
immediately follows: The injectivity of the multiplication map is a condition
on the determinant of a linear map. If it does not vanish in characteristic
$2$, it cannot vanish in characteristic $0$.

##  4\. Normalization of the integration map

We now want to prove anisotropy for a general IDP lattice polytope $P$
relative to its boundary, so that in a generic Artinian reduction,
$\mathcal{A}^{d+1}(P,\partial P)\cong\tilde{\mathbbm{k}}$ is the fundamental
class. As this is a statement about elements of the fundamental class not
vanishing, it serves us well to find a way to make this last isomorphism
explicit. We denote the resulting map by
$\mathrm{deg}:\mathcal{A}^{d+1}(P,\partial P)\to\tilde{\mathbbm{k}}$. It is
usually called the degree map, but to avoid confusion with the degree of a
monomial, we shall instead call this identification the _integration map_
(alluding to the fact that what we are aiming to understand is actually the
volume polynomial).

One curiosity of lattice polytopes is that even though we know the canonical
module thanks to Hochster, we do not actually know of an identification of the
top degree with the base field, that is, we do not have a canonical choice of
integration map. The proof of Theorem 3.1 is reliant on understanding this
integration. In the situation of classical algebraic geometry, there is a
canonical such identification, which leads to a classical combinatorial
formula in toric geometry [Bri97]. In our case, no such canonical
identification seems to have been explored. We provide it here.

Stopping short of a full combinatorial formula, we contend ourselves with
determining it uniquely. As this is an isomorphism between to copies of the
same field, it suffices to give one nontrivial affine condition and prove
consistency in order to determine the function.

If $P$ has a unimodular boundary facet $\tau$ in $\partial P$, then we obtain
the desired normalization by matching the face ring picture and setting

$1\ =\
\sum_{p\in(P\setminus\tau)\cap\Lambda}\mathrm{deg}(\mathbf{x}_{p}\mathbf{x}_{\tau})\det(\Theta_{|\tau,p})$
(2)

where $\Theta=(\theta_{i,j})$ is the matrix of coefficients in the linear
system of parameters. In general, consider a flag $(\tau_{i})$ of faces of $P$
such that $\tau_{d}=P$ and such that $\tau_{i}$ is a facet of $\tau_{i+1}$. We
say a set
$\sigma=\\{\sigma_{0},\ldots,\sigma_{d}\\}\subset(P\cap\Lambda)^{d+1}$ without
repetitions is coherent with $(\tau_{i})$ if it intersects $\tau_{i}$ in a set
of cardinality $i+1$. We then normalize by setting

$1\ =\ \sum_{\sigma\text{ coherent with
}(\tau_{i})}\mathrm{deg}(\mathbf{x}_{\sigma})\det(\Theta_{|\sigma})$ (3)

######  Lemma 4.1.

The normalization is independent of the flag $(\tau_{i})$ chosen.

###### Proof.

A direct computational proof is rather uninformative, but we can use an
indirect argument without getting our fingers dirty that informs what is
really happening. Let us assume first that $P$ has a boundary consisting of
unimodular simplices. Consider then the cone over $P$ as a polytope; it
follows that

$(v\ast\partial P)\ \cup\ P$

is a lattice complex in the sense of [BBR07], and that it defines a Poincaré
duality algebra, meaning that the integration map on $P$ extends uniquely to a
integration map on $(v\ast\partial P)\cup P$. For the simplices of
$v\ast\partial P$ the integration map is uniquely determined and given by
$\det^{-1}(\Theta_{|X})$ for a simplex $X$. The consistency follows.

For an arbitrary polytope $P$, we argue in a similar fashion. However, we can
use the fact that the faces of $v\ast\partial P$ are decidedly more simplicial
than $P$: they are cones over polytopes of codimension one. We can repeat that
trick and obtain double cones over arbitary polytopes as new faces, and so on;
finally the facets are simplicial and we once again understand the integration
map in terms of smooth toric varieties. This implies consistency of the
integration map conditions. ∎

In other words, the normalization is compatible to the natural integration map
in toric geometry, and is consistent with the same if we use gluing to smooth
affine varieties. This means that the normalization is somewhat canonical; in
fact, it also coincides with the normalization obtained by Kustin-Miller
unprojection [Pap07], a connection which we shall explore in an upcoming work.

This definition of the integration map as well as the anisotropy we wish to
prove is dependent of our choice of $\theta_{i}$, so we will consider
$\mathrm{deg}$ as a rational function in the variables $\theta_{i,j}$. This
allows us to formulate the auxiliary lemma on the way to anisotropy.

######  Lemma 4.2.

For $P$ a $(2k-1)$-dimensional polytope, the integration normalized, and
$F=\\{f_{1},\ldots,f_{k}\\}\subset(P\cap\Lambda)^{k}$ is without repetitions,
in $\mathcal{A}(P)$ over any infinite field of characteristic $2$, we have

$\partial_{F}\mathrm{deg}(u^{2})\ =\ \mathrm{deg}(u\cdot\mathbf{x}_{F})^{2}.$

Here, $\partial_{F}$ is the differential operator obtained as the composition
of the differential in directions
$\theta_{1,f_{1}},\theta_{2,f_{1}},\theta_{3,f_{2}},\theta_{4,f_{2}},\theta_{5,f_{3}},\dots$,
applied to the integration map as a rational function.

We postpone the proof of this lemma to the end of the next section, where we
will derive it from the key identity of Parseval type.

From this differential identity, anisotropy follows at once: every nonzero
element $u$ as in the lemma multiplies with _some_ $\mathbf{x}_{F}$ due to
Poincaré duality. Hence $\partial_{F}\mathrm{deg}(u^{2})$ is not identically
zero. Thus for a choice of transcendentally independent coefficients,
$\mathrm{deg}(u^{2})\neq 0$, proving Theorem 3.1.

##  5\. Parseval identities

While the differential identity is similar to identities proven in the case of
simplicial cycles in [APP21, PP20], the case of lattice polytopes is
comparatively harder: we understand the integration map using the following
nonhomogeneous equation that takes the form of an identity of the Parseval
type. We consider polytopes of dimension $d$.

Assume $\beta=(\beta_{0},\dots,\beta_{d})\in(P\cap\Lambda)^{d+1}$. We set as
usual

$\mathbf{x}_{\beta}\ =\ \prod_{0\leq i\leq d}\mathbf{x}_{\beta_{i}}.$

######  Lemma 5.1 (The Parseval identity (disguised)).

For an IDP lattice $d$-polytope $P$, we have in $\mathcal{A}^{*}(P,\partial
P)$ over characteristic $2$

$\mathrm{deg}(\mathbf{x}_{\alpha})\ =\
\sum_{\beta\in(P\cap\Lambda)^{d+1}}\mathrm{deg}(\mathbf{x}_{\frac{\alpha+\beta}{2}})^{2}\cdot\prod_{0\leq
i\leq d}\theta_{i,\beta_{i}}$ (4)

Here, $\mathrm{deg}(\mathbf{x}_{\frac{\alpha+\beta}{2}})$ is defined to be
$\mathrm{deg}(\mathbf{x}_{\gamma})$ if there is an
$\mathbf{x}_{\gamma}\in\mathbbm{k}[P]$ such that
$\mathbf{x}_{\alpha}\mathbf{x}_{\beta}=\mathbf{x}_{\gamma}^{2}$, and 0
otherwise.

This specializes to the following identity for $\alpha=2\alpha^{\prime}$,
which explains the naming of this identity:

######  Lemma 5.2 (The Parseval identity (revealed)).

For an IDP lattice $d$-polytope $P$, in $\mathcal{A}^{*}(P,\partial P)$ over
characteristic $2$ and for $d+1$ even, we have

$\mathrm{deg}(u^{2})\ =\
\sum_{\beta\in(P\cap\Lambda)^{d+1}}\mathrm{deg}(u\cdot\mathbf{x}_{\frac{\beta}{2}})^{2}\cdot\prod_{0\leq
i\leq d}\theta_{i,\beta_{i}}$ (5)

Here for a non-monomial $u^{2}$ we get the result by linearity over
characteristic $2$:

$\displaystyle\mathrm{deg}(u^{2})\ $ $\displaystyle=\
\mathrm{deg}\bigg{(}\big{(}\sum_{a}\lambda_{a}\mathbf{x}_{a}\big{)}^{2}\bigg{)}\
=\ \sum_{a}\lambda_{a}^{2}\ \mathrm{deg}(\mathbf{x}_{a}^{2})$ $\displaystyle=\
\sum_{a}\lambda_{a}^{2}\sum_{\beta\in(P\cap\Lambda)^{d+1}}\mathrm{deg}(\mathbf{x}_{a}\cdot\mathbf{x}_{\frac{\beta}{2}})^{2}\cdot\prod_{0\leq
i\leq d}\theta_{i,\beta_{i}}$ $\displaystyle=\
\sum_{\beta\in(P\cap\Lambda)^{d+1}}\mathrm{deg}(u\cdot\mathbf{x}_{\frac{\beta}{2}})^{2}\cdot\prod_{0\leq
i\leq d}\theta_{i,\beta_{i}}$

In order to prove the Parseval identity, we recall first a well-known identity
for the integration map:

######  Lemma 5.3 (The balancing identity).

Consider any two elements $I,J\in(P\cap\Lambda)^{d}$, where at least one point
of $I$ lies in the interior of $P$. Then we have

$R_{I,J}\ :=\ \sum_{p\in
P\cap\Lambda}\det(\Theta_{|J,p})\mathrm{deg}(\mathbf{x}_{I}\mathbf{x}_{p})\ =\
0.$

This is a consequence (see [PP20][11.1]) of the following, simpler identity:

######  Lemma 5.4 (The linear identity).

Consider any $0\leq s\leq d$, and $x_{I}$ a monomial of degree $d$ of
$\widetilde{\mathbbm{k}}[P,\partial P]$. Then in $\mathcal{A}(P,\partial P)$
we have $\sum_{p\in
P\cap\Lambda}\theta_{s,p}\mathrm{deg}(\mathbf{x}_{I}\mathbf{x}_{p})\ =\ 0$.

This last identity arises simply because we quotient by the linear elements

$\theta_{j}=\sum_{p\in P\cap\Lambda}\theta_{j,p}\mathbf{x}_{p}.$

###### Proof of Lemma 5.2.

We establish a special case first, and argue by induction on dimension.
Consider the case $d=1$.

Instead of proving the Lemma for every lattice line segment separately, we
instead prove it for the halfline $P=\mathbb{Z}_{\geq 0}$, and restrict to
segments afterwards; this is possible because normalization requires the
existence of a flag of boundary faces only.

To simplify our work we index the monomials by the sum of their lattice
points, as the integration map of a monomial only depends on that invariant by
the arithmetic relations imposed by the lattice structure.

We then start by considering the normalization Identity (2), and multiply both
sides by $\mathrm{deg}(\mathbf{x}_{\alpha})$ to obtain

$\mathrm{deg}(\mathbf{x}_{\alpha})\ =\ \sum_{\sigma\text{ coherent with
}\tau}\mathrm{deg}(\mathbf{x}_{\alpha})\mathrm{deg}(\mathbf{x}_{|\sigma|})\det(\Theta_{|\sigma})$

where $|\sigma|$ denotes the sum of lattice points of $\sigma$. We denote the
right side by $N_{\alpha}$.

There exists a natural action

$S_{d+1}\times(P\cap\Lambda)^{d+1}\ \longrightarrow\ (P\cap\Lambda)^{d+1}$

of the symmetric group $S_{d+1}$ on $d+1$ elements on $(P\cap\Lambda)^{d+1}$
and we set

$\omega_{b}\ =\ \sum_{\beta\in\mathrm{Orb}(b)}\prod_{0\leq i\leq
d}\theta_{i,\beta_{i}}.$

We can then rewrite Identity (4) as

$\mathrm{deg}(\mathbf{x}_{\alpha})\ =\
\sum_{b\in\nicefrac{{(P\cap\Lambda)^{d+1}}}{{S_{d+1}}}}\omega_{b}\cdot\mathrm{deg}(\mathbf{x}_{\frac{\alpha+\beta}{2}})^{2}.$

Now, given $i,j\in\mathbb{Z}_{\geq 0}$, we set $\lambda_{i,j}$ to be
$\delta_{i,j}\cdot\mathrm{deg}(\mathbf{x}_{\alpha+j-i})$. Here $\delta_{i,j}$
is $1$ if $\nicefrac{{\alpha}}{{2}}<i<j$ or
$\nicefrac{{\alpha}}{{2}}-j<i<\nicefrac{{\alpha}}{{2}}$, and $0$ otherwise.
Consider then

$\displaystyle 0\ =\ $ $\displaystyle\sum_{i,j}\lambda_{i,j}R_{i,j}$
$\displaystyle=\ $
$\displaystyle\sum_{i,j}\lambda_{i,j}\sum_{t\in\mathbb{Z}_{\geq
0}}\det(\Theta_{|j,t})\mathrm{deg}(\mathbf{x}_{i+t})$ $\displaystyle=\ $
$\displaystyle N_{\alpha}\ +\ \sum_{t<j\in\mathbb{Z}_{\geq
0}}\omega_{t,j}\cdot\left(\mathrm{deg}(\mathbf{x}_{\frac{\alpha+t+j}{2}})^{2}\
+\
\mathrm{deg}(\mathbf{x}_{\frac{\alpha}{2}+t})\mathrm{deg}(\mathbf{x}_{\frac{\alpha}{2}+j})\right).$

Applying the linear identities we obtain that

$0\ =\ N_{\alpha}\ +\ \sum_{t\leq j\in\mathbb{Z}_{\geq
0}}\omega_{t,j}\cdot\mathrm{deg}(\mathbf{x}_{\frac{\alpha+t+j}{2}})^{2}.$

This proves the Parseval identity for $d=1$ for the halfline. Now we can
restrict to any line segment $P^{\prime}=\\{t\in\mathbb{Z}_{\geq 0}:p\leq
o\\}$ to obtain the general statement: here, we note that the coefficients
$\lambda_{i,j}$ are chosen providently to vanish whenever $i$ is not in the
interior of the given line segment.

To prove the Parseval identity for $d>1$, we consider again first the
polyhedral cone $P=\mathbb{Z}_{\geq 0}^{[d]}$, $[d]=\\{1,\dots,d\\}$, where
the initial flag is chosen to be.

$\mathbb{Z}_{\geq 0}^{[0]}=\\{0\\}\ \subset\ \mathbb{Z}_{\geq 0}^{[1]}\
\subset\ \mathbb{Z}_{\geq 0}^{[2]}\cdots.$

Define the lexicographic order so that the this flag is minimal.

For $I,J\in\mathbb{Z}_{\geq 0}^{[d]}$, we set $\lambda_{I,J}$ to be
$\delta_{I,J}\cdot\mathrm{deg}(\mathbf{x}_{\alpha+J-I})$, where we choose
$\delta_{I,J}$ inductively on dimension: given $j$ the lexicographically
minimal element of $J$, set $\delta_{I,J}:=\delta_{I,J\setminus\\{j\\}}$ with
respect to the pullback to the ideal of $\mathbf{x}_{j}$. This finishes the
proof. ∎

###### Proof of Lemma 4.2 using Lemma 5.2.

For $P$ a $(2k-1)$-dimensional polytope, the integration map normalized and
$F$ a $k$-set of lattice points in $P$ without repetitions, let
$u\in\mathcal{A}^{2k}(K_{P})$. Consider $\mathrm{deg}(u^{2})$ using (4). We
have the revealed Parseval identity

$\mathrm{deg}(u^{2})\ =\
\sum_{\beta\in(P\cap\Lambda)^{d+1}}\mathrm{deg}(u\cdot\mathbf{x}_{\frac{\beta}{2}})^{2}\cdot\prod_{0\leq
i\leq d}\theta_{i,\beta_{i}}$

Now differentiation by $F$ yields

$\partial_{F}\mathrm{deg}(u^{2})\ =\
\sum_{\beta\in(P\cap\Lambda)^{d+1}}\mathrm{deg}(u\cdot\mathbf{x}_{\frac{\beta}{2}})^{2}\cdot\partial_{F}\left(\prod_{i=0}^{d}\theta_{i,\beta_{i}}\right),$

in characteristic two, as differentiating
$\mathrm{deg}(u\cdot\mathbf{x}_{\frac{\beta}{2}})^{2}$ introduces a factor of
2. For all nonzero summands there exists a $b$ such that $2b=\beta$. Now note
that
$\partial_{F}\big{(}\prod_{i=0}^{d}\theta_{i,\beta_{i}}\big{)}=\delta_{F,b}$,
so we get the desired identity

$\partial_{F}\mathrm{deg}(u^{2})\ =\
\mathrm{deg}(u\cdot\mathbf{x}_{F})^{2}.\qed$

##  6\. Open questions

The non-lattice cases of Stanley’s conjecture remain. Even for IDP lattice
polytopes or stronger yet, for lattice polytopes with a regular unimodular
triangulation, we are left with a gap in the inequalities restricting
$h^{\ast}$ if the interior of the cone is generated in higher degree. We
conjecture that the Lefschetz property, and in fact the unimodality of
$h^{\ast}$ fails in general. The intuition here is that lattice polytopes
behave like triangulated disks, which can have non-unimodal $h$-vectors. The
idea here could rely on constructing appropriate connected sums: as we saw
above, Gorenstein polytopes have $h^{\ast}$-polynomials that peak at half of
their socle degree (which is $d+1-s$, $s$ being the minimal dilation constant
so that the polytope has an interior vertex). By connected sum of polytopes
with different socle degree, one could hope to turn a dromedary into a camel
(though a mythical beast with more humps is not beyond our imagination, alas
such a creature has to be high-dimensional).

A word of caution, however, lies in an inequality for the
$h^{\ast}$-polynomial arising from work of Eisenbud and Harris [Sta91]: we
have that for any nonnegative $k$, and $s$ the degree of the
$h^{\ast}$-polynomial, we have

$h^{\ast}_{0}\ +\ \dots\ +\ h^{\ast}_{k}\ \leq\ h^{\ast}_{s}\ +\ \dots\ +\
h^{\ast}_{s-k}.$

This inequality is special to domains, and prevents us from introducing a hump
below half the socle degree easily; it remains to understand the impact of
this inequality in general. The most promising approach is then to look among
polytopes whose interior is generated in high degree, and look for non-
unimodality between half the degree of the $h^{\ast}$-polynomial and half the
dimension of the polytope.

Acknowledgements. We thank Benjamin Nill for asking us about the local
$h^{\ast}$ polynomial, and Paco Santos for insightful discussions. K. A., V.
P. and J. S. are supported by Horizon Europe ERC Grant number: 101045750 /
Project acronym: HodgeGeoComb.

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# Quantum Anomalous Hall Phase Stabilized via Realistic Interactions on a
Kagome Lattice

Yafei Ren  ICQD, Hefei National Laboratory for Physical Sciences at
Microscale, and Synergetic Innovation Center of Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, Anhui
230026, China  Department of Physics and Astronomy, California State
University, Northridge, California 91330, USA  CAS Key Laboratory of Strongly-
Coupled Quantum Matter Physics, and Department of Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China  T.-S. Zeng
Department of Physics, The University of Texas at Dallas, Richardson, Texas
75080, USA  W. Zhu  Theoretical Division, T-4 and CNLS, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  D. N. Sheng  Department of
Physics and Astronomy, California State University, Northridge, California
91330, USA

###### Abstract

We study the quantum phases of spinless fermion at one-third filling on a
Kagome lattice featuring a quadratic band touching Fermi point. In the
presence of weak first and second nearest-neighbor repulsive interactions
($V_{1}$ and $V_{2}$), we demonstrate an interaction driven quantum anomalous
Hall effect by employing exact diagonalization and density-matrix
renormalization group methods. The time-reversal symmetry is broken
spontaneously by forming loop currents that exhibit long-range correlation.
Quantized Hall conductance corresponding to Chern number of $\pm 1$ is
obtained by measuring the pumped charge through inserting flux in a cylinder
geometry. We find that the energy gap, which topologically protects the
emerging ground states, can be enhanced remarkably by a moderate $V_{2}<V_{1}$
via calculating the spectrum and charge excitation gaps, which highlights the
experimentally feasible scheme of realizing interaction driven topological
phase by spatially decaying interactions on topologically trivial lattice
models.

Introduction.— As an analogy of quantum Hall effect discovered in the presence
of strong perpendicular magnetic fields, the quantum anomalous Hall effect
(QAHE) was first proposed by Haldane for honeycomb lattice with staggered
magnetic flux breaking time-reversal symmetry QAHE_Haldane_88 . Such QAHEs
have been widely explored in non-interacting systems where nontrivial band
topology arise from magnetization and spin-orbit coupling kane ; Liu2016 ;
QAHE_TIthinfilm_10 ; QAHE_Graphene_Qiao_10 . Meanwhile, in strongly
interacting systems, the searching of exotic ground states has stimulated the
interests of interaction-driven QAHE from topologically trivial bands where
spontaneous time-reversal symmetry breaking can be realized by interaction
driven loop currents QAHE_DiracMott_Raghu_08 ; Rachel2018 . The first example
was proposed in Dirac band with Hubbard interaction based on mean-field
analysis and functional renormalization group theory QAHE_DiracMott_Raghu_08 .
Further theoretical studies report controversial results depending on boundary
conditions from exact diagonalization (ED) QAHE_DiracMott_Herbut_OPC_14 ;
QAHE_DiracMott_ED_13 ; QAHE_DiracMott_ED_14 ; QAHE_DiracMott_PRB_15_ED ,
whereas more reliable density-matrix renormalization group (DMRG) algorithm
suggests that the spontaneous QAHE is predominated by other competing phases
Motruk2015 .

A key ingredient to realize the interaction-driven QAHE is the presence of
Fermi touching point, which also emerges in bands with quadratic crossings
protected by both time-reversal symmetry and point group symmetry Chong2008 .
This stimulates a broad research interest in various lattice models
QAHE_QBC_Sun_09 ; QAHE_QBC_Sun_ColdAtom_12 ; QAHE_KagomeDecGra_MF_10 ;
QAHE_QBC_Kagome_MF_10 ; QAHE_Kagome_NN_ColdAtom_18 ; QAHE_Star_ED_MF_18 ;
QAHE_BLGMott_SU4_10 ; QAHE_BLGMott_FanZhang_11 ; QAHE_BLGMott_Science_11 ;
QAHE_Lieb_Tsai_15 ; QAHE_QBC_root3_16 ; QAHE_QBC_CkB_WuHQ_16 ; QAHE_16_PRL_Zhu
; MIT_Kagome_Nishimoto_10 , where mean-field calculations suggest that the
QAHE emerges as long as weak repulsive interactions are introduced. Although
some ED results indicate the presence of interaction driven QAHE
QAHE_QBC_CkB_WuHQ_16 ; QAHE_Star_ED_MF_18 , those evidences, i.e., double
degeneracy of ground state and finite loop currents for small systems, are
subjected to the finite size effect. Solid numerical evidences are demanded to
confirm whether these phases are stable against quantum fluctuation in the
thermodynamic limit.

Recently, several numerical evidences of interaction-driven QAHE are reported
on both Kagome QAHE_16_PRL_Zhu  and checkerboard Zeng2018 ; Sur2018  lattices
based on the state of the art DMRG studies. The story for the Kagome systems
turns out to be interesting. Up to third nearest-neighbor interactions with
comparable strengths are required in the lattice model to realize strong and
robust QAHE with a larger excitation gap, which makes it difficult to be
realized experimentally. Moreover, in contrast to the mean-field results where
infinitesimal interaction can induce QAHE instability, finite interactions may
be required to stabilize the QAHE QAHE_16_PRL_Zhu , leaving the physics in the
weak interaction regime unsettled. With only first nearest-neighbor hopping
and interaction on Kagome lattice, earlier DMRG results suggest that the
system remains a metal for weak interaction MIT_Kagome_Nishimoto_10 , which is
in contrast to mean-field results QAHE_KagomeDecGra_MF_10 ;
QAHE_QBC_Kagome_MF_10 . The main goals of this work are to address the fate of
the system in the presence of weak interactions, and to explore more realistic
conditions for realizing QAHE for potential experimental systems.

In this work, we numerically map out the quantum phase diagram driven by first
and second nearest-neighbor repulsive interactions $(V_{1},V_{2})$ at one-
third filling of spinless fermion on Kagome lattice through ED and DMRG
simulations. Our extensive ED calculations demonstrate the emergence of QAHE
by doubly degenerate ground states and finite loop currents apart from the
$V_{1}$-only case where the energy gap is found to be vanishingly small. The
presence of $V_{2}$ interaction enhances the energy gap remarkably, signaling
the robustness of the topological phase. With a finite $V_{2}<V_{1}$, the
nontrivial topology is confirmed by large-scale DMRG calculations, which give
rise to a uniform circulating loop current spontaneously. The quantized Hall
conductance corresponding to a quantized Chern number $C=\pm 1$ is also
identified by pumping a unit charge from one side of a cylinder to the other
side through inserting $U(1)$ charge flux into the cylinder adiabatically.
When the strength of interactions increase, we reveal a continuous quantum
phase transition from the QAHE to a charge density wave (CDW) without any
intermediate phase. Our results provide unbiased numerical evidences of
spontaneous QAHE on Kagome lattice stabilized by weak interactions making this
model experimentally feasible.

Model and methods.— We consider a spinless fermion-Hubbard model on Kagome
lattice with first and second nearest-neighbor interactions. The Hamiltonian
is written as

$\displaystyle H=t\sum_{\langle
ij\rangle}c_{i}^{\dagger}c_{j}+V_{1}\sum_{\langle
ij\rangle}n_{i}n_{j}+V_{2}\sum_{\langle\langle ij\rangle\rangle}n_{i}n_{j},$
(1)

where $c_{i}^{\dagger}$ ($c_{i}$) is the creation (annihilation) operator of a
fermion at site $i$ and $n_{i}=c_{i}^{\dagger}c_{i}$ is the particle number
operator. $V_{1}$ and $V_{2}$ are the strengths of repulsive interactions
between the first ($\langle\ldots\rangle$) and second
($\langle\langle\ldots\rangle\rangle$) nearest neighbors. We focus on the one-
third filling case in a finite system of $N_{x}\times N_{y}$ unit cells with
total number of sites $N_{s}=3\times N_{x}\times N_{y}$ and the number of
fermions $N_{e}=N_{s}/3$. Here, we take $t=1$, in which the lower energy band
is flat having a quadratical crossing with the middle one.

To characterize the topological property of the ground states driven by
interactions, we employ the DMRG algorithm combined with ED method. In ED
calculations, we study systems up to 36 sites. In a periodic torus geometry,
the energy eigenstates can be labeled by the total momentum
$\bm{k}=(k_{x},k_{y})$. To explore larger systems, we exploit finite or
infinite DMRG on either periodic torus or cylinder geometry where the boundary
is open (periodic) along $x$ ($y$)-direction. In DMRG calculations, we set
$N_{y}$ up to 5 unit cells (15 lattice sites) and keep the DMRG states up to
$M=4800$ to guarantee a good convergence (with the truncation error around
$10^{-5}$).

Figure 1: Phase diagram from ED calculation on $N_{s}=3\times 4\times 3$ tours
based on color maps of (a) the loop current amplitude and (b) energy
difference between lowest two states. The phase boundary is indicated by a
white dashed line. The thick black line on $V_{1}$-axis indicates that the
quantum phase of $V_{2}=0$ is undetermined due to the vanishingly small energy
gap. Contour plots of static density structure factors $S({\bm{k}})$ are shown
for (c) QAHE and (d) $q=(0,0)$ CDW phase. White dashed lines indicate the
first Brillouin zone.

Phase diagram.— In the presence of interactions, we map out the quantum phase
diagram in the parameter space spanned by $V_{1}$-$V_{2}$, based on both bond
current and charge density orders. Two topologically distinct phases, i.e.,
QAHE and CDW phases, are displayed in Fig. 1(a). The QAHE phase is
characterized by doubly near degenerate ground states $|\psi_{\pm}\rangle$. We
make complex superposition of these lowest two states, which possess opposite
chiralities and are related to each other by time-reversal operation. Such a
near degeneracy is reflected by a small energy difference between the lowest
two energy levels as shown in Fig. 1(b). For a complex superposition state
from these lowest energy states, the time-reversal symmetry is broken by
forming loop currents measured by the expectation of current operator
$\langle\hat{j}_{ij}\rangle=i\langle
c_{i}^{\dagger}c_{j}-c_{j}^{\dagger}c_{i}\rangle$. The magnitude of bond
current is mapped to color in Fig. 1(a) where a sizable current is found in
the whole QAHE region.

We point out that a thick black line is plotted for $V_{1}$ only case, i.e.,
$V_{2}=0$, to distinguish it from the QAHE phase. In this case, although
finite loop currents appear, the energy gap protecting the QAHE is found to be
vanishingly small, which is consistent with a gapless state. However, the QAHE
emerges with the turn on of a weak $V_{2}$ interaction. Furthermore, when
$V_{1}$ or $V_{2}$ increases to cross the phase boundary, the bond current
decreases rapidly suggesting the transition to a trivial phase. The CDW phase
exhibits a charge distribution imbalance among different sublattice sites
characterized by the density structure factor $S({\bm{k}})$. As shown in Figs.
1(c) and 1(d) for QAHE, $S({\bm{k}})$ defined for the density correlations of
the same sublattice is featureless in the whole Brillouin zone suggesting a
uniform density distribution, whereas for the CDW phase, $S({\bm{k}})$ shows a
strong Bragg peak at the center of the first Brillouin zone.

Figure 2: Enhanced energy gap by $V_{2}$. (a) Color map of spectrum gap
between the third and second lowest energy levels calculated by ED method in
$N_{s}=36$ site system. The spectrum gap is vanishingly small for $V_{1}$ only
case and increases significantly in the presence of $V_{2}$. (b) Charge-hole
gap $\Delta(N_{s})$ vs $V_{1}$ for $V_{2}=0$ and $V_{2}=V_{1}/2$ calculated by
ED in systems of different sites and by DMRG in system of $N_{s}=72$ sites.
Gray dashed line indicates the critical point between QAHE and CDW phase for
$V_{2}=V_{1}/2$.

Enhanced gap by second nearest-neighbor interaction.— To demonstrate the
dependence of energy gap on $V_{1}$-$V_{2}$, we show the color map of energy
difference between the third and the second lowest energy levels as the
spectrum gap in Fig. 2(a). Inside the CDW phase, the energy difference is
extremely small, manifesting the three-fold degeneracy of the ground state. In
the QAHE phase, however, the spectrum gap exhibits strong dependence on
$V_{2}$. Extremely small spectrum gap appears near the line of $V_{2}=0$ even
for $V_{1}=2$ in agreement with Ref. MIT_Kagome_Nishimoto_10 , which reported
a gapless metallic phase for $V_{1}$ only model. Nevertheless, the presence of
$V_{2}$ can enhance the spectrum gap remarkably. As shown in Fig. 2(a), the
spectrum gap increases linearly with $V_{2}$ and shows weak dependence on
$V_{1}$. A robust gap appears when $V_{2}$ is moderately large comparable to
$t=1$.

Such an enhancement of the gap by $V_{2}$ is further confirmed by the charge
excitation gap
$\Delta(N_{s})=E_{0}(N_{s},N_{e}+1)+E_{0}(N_{s},N_{e}-1)-2E_{0}(N_{s},N_{e})$
where $E_{0}(N_{s},N)$ is the ground state energy of the system with $N_{s}$
sites and $N$ particles. Figure 2(b) shows the dependence of $\Delta$ on
$V_{1}$ for different second neighbor interactions $V_{2}=0$ and
$V_{2}=V_{1}/2$ for different values of $N_{s}$. For $V_{2}=0$, the charge gap
$\Delta$ is vanishingly small for different $N_{s}$ in good agreement with the
spectrum gap. In the presence of finite $V_{2}=V_{1}/2$, $\Delta$ grows up
gradually as $V_{1}$ increases, signaling the emergence of an incompressible
gapped QAHE phase. The linear dependence of energy gap $\Delta$ on $V_{2}$ is
extraordinary comparing to the exponentially small gap for weak interaction
predicted by mean-field calculations QAHE_KagomeDecGra_MF_10 ;
QAHE_QBC_Kagome_MF_10 ; QAHE_QBC_Sun_09 ; QAHE_Kagome_NN_ColdAtom_18 .

Figure 3: (a) Current-current correlation vs distance for cylinder geometry of
$N_{y}=4$ and $N_{y}=5$ from infinite DMRG. Finite constant correlation as
distance increases indicates the presence of long-range order. Inset shows the
current pattern in a $N_{s}=3\times 4\times 4$ torus from finite DMRG
algorithm. Red arrows indicate the current directions. Color maps the current
of each triangle. (b) Net charge transfer $\Delta Q=Q(\theta)-Q(0)$ pumped by
threading a $U(1)$ charge flux $\theta$ adiabatically through the hole of a
cylinder as illustrated in the inset.

Current and charge pumping.— In relation to the phase diagram, we now perform
a numerical DMRG exploration of the QAHE under weak interactions in large
systems. In DMRG, we obtain two different ground states $|\psi_{\pm}\rangle$
with near identical energies and opposite loop currents. As these two states
are related by time-reversal operation, we focus on $|\psi_{+}\rangle$
hereinafter. The loop current is schematically shown in the inset of Fig.
3(a), which distributes almost uniformly and circulates clockwise (anti-
clockwise) around each triangle (hexagon) leading to a vanishing total flux
similar to Haldane-honeycomb model QAHE_Haldane_88 . In Fig. 3(a), we plot the
current-current correlation $\langle\hat{j}_{i,j}\hat{j}_{i_{0},j_{0}}\rangle$
as a function of bond distance $|{\bf r}_{i,j}-{\bf r}_{i_{0},j_{0}}|$ in an
infinite cylinder geometry with different widths, where $\hat{j}_{i,j}$ is the
current operator between nearest-neighbor sites $i,j$. For different system
sizes, the correlation functions tend to converge to finite constants for
large distance limit, indicating the behavior of time-reversal symmetry
breaking in the thermodynamic limit.

The quantized topological nature of the QAHE is characterized by Chern number
$C=1$ obtained by calculating the topological Laughlin pumping in the
$x$-direction by adiabatically inserting a $U(1)$ charge flux $\theta$ into
the cylinder hole (as a twist boundary phase in the $y$-direction) based on
the recently developed adiabatic DMRG Gong2014 ; Flux_Zhu_15 . Here, we
partition the infinite cylinder along the $x$-direction into two halves. The
transverse transfer of total particle number from the right cylinder edge to
the left edge is encoded by the variation of the total charge in the left part
$Q(\theta)={\rm tr}[\widehat{\rho}_{L}(\theta)\widehat{N}_{L}]$, where
$\widehat{N}_{L}$ and $\widehat{\rho}_{L}$ are particle number operator and
reduced density matrix of left part, respectively. The change $\Delta
Q=Q(\theta)-Q(0)$ indicates the transverse transfer of particle as shown in
Fig. 3(b). In one cycle, a unit of particle $C_{+}=Q(2\pi)-Q(0)\simeq 1$ is
pumped, visualizing a quantized transverse Hall conductance
$\sigma_{xy}=C_{+}e^{2}/h$ for $|\psi_{+}\rangle$ state.

Figure 4: (a) Lowest energy levels vs interaction $V_{1}=V_{2}$ for at
momentum sector $(\pi,0)/(0,0)$ for $N_{s}=36/27$ site system. Inset of
$36$-site panel shows the continuous variation of ground state energy as
$V_{1}=V_{2}$ increases. (b) Fidelity susceptibility $\chi$ vs $V_{1}=V_{2}$
for the lowest (red solid square) and second lowest (green solid triangle)
energy levels of $N_{s}=36$ system as well as lowest energy level (blue solid
circle) of $N_{s}=27$ system. (c) DMRG results for bond current magnitude and
particle density imbalance between different sublattices in a cylinder of
$N_{x}\times N_{y}$. Gray dashed line indicates the critical point. (d) DMRG
results for fidelity susceptibility of the ground state in a cylinder of
$N_{x}\times N_{y}=16\times 4$.

Phase transition.— In this part, we turn to analyze the phase transition
between QAHE and CDW as interaction strength increases. Without loss of
generality, we focus on the case with $V_{2}=V_{1}$. Figure 4(a) shows the
evolution of the lowest several energy levels at the momentum sector where
ground states live. As the interaction strength increases, the lowest energy
level is varying smoothly, and does not show level crossing with excited
levels. We further calculate the fidelity $f(V)=\langle\psi(V-\delta
V)|\psi(V)\rangle$ between two wavefunctions with slightly different
interaction strengths. For two states belong to the same phase, $f(V)$ is
close to 1 and the phase transition can be reflected by the peak of fidelity
susceptibility $\chi=2(1-f(V))/\delta V^{2}$ Gu2010 . As plotted in Fig. 4(b),
a smooth function $\chi$ with a single peak structure indicates a continuous
quantum phase transition from QAHE to CDW, without any evidence of an
intermediate phase.

To further verify the continuous transition nature, we also exploit an
unbiased DMRG approach for larger systems. The bond current and maximal
density imbalance between different sublattices are presented in Fig. 4(c).
Similar to our ED analysis, both order parameters exhibit continuous evolution
consistent with a continuous phase transition. Meanwhile, similar single-peak
behavior of $\chi$ is also observed from DMRG calculations in Fig. 4(d). Thus,
our ED and DMRG studies consistently support the direct continuous phase
transition between QAHE and CDW.

Experimental realization with cold atom systems.— Our results based on
$V_{1}$-$V_{2}$ model indicate QAHE can be stabilized by the extended
repulsive interactions, which is feasible for experimental implementations.
With fermionic polar molecules ${}^{40}\text{K}^{87}\text{Rb}$ Moses2017  and
${}^{23}\text{Na}^{40}\text{K}$ Wu2012  loaded into the Kagome optical lattice
ColdAtomKagome , the effective interaction potential between fermionic
particles is expected to be in the form $V({\bf r}-{\bf
r}^{\prime})=d^{2}/|{\bf r}-{\bf r}^{\prime}|^{3}$ versus distance when the
dipole moment $d$ is aligned in the $z$-direction by a strong external field.
By including up to the third nearest-neighbor interactions and truncating off
the tiny terms for longer distance couplings, we confirm that QAHE survives,
and verify the robustness of the QAHE from measurements of bond current and
energy gap in finite system sizes $N_{s}=36,27$ for moderate dipolar
interaction strength $d^{2}/a^{3}\sim t$. Thus our identification of the key
role played by weak tails of repulsion interaction suggests that the cold atom
trapped polar molecules can naturally realize a QAHE phase on the Kagome
lattice.

Summary and discussion.— We have demonstrated a remarkably stable QAHE by
neighboring $V_{1}$-$V_{2}$ interactions on a Kagome lattice, evidenced by
doubly degenerate ground states, spontaneous bond currents with long-range
correlation, and quantized Hall conductance. Without second nearest-neighbor
interaction $V_{2}$, we found vanishingly small spectrum and charge excitation
gaps in agreement with previous works. In the presence of $V_{2}$, we found
the energy gap that protects the ground state exhibits linear dependence on
$V_{2}$, which is strongly enhanced comparing to the exponentially small gap
induced by weak interaction $V_{1}$ predicted by mean-field studies. By tuning
the interactions $V_{1}$ and $V_{2}$, the QAHE undergoes a continuous quantum
phase transition into a CDW phase. Finally, these results imply that the QAHE
can be realized by spatially decaying dipolar interactions on a Kagome
lattice, making the emergence of the topological phase promising within
current experimental technologies. Furthermore, our numerical methods of
identifying such a topological phase can find wide applications for studying
interaction driven topological phases including quantum spin Hall effect, and
QAHE in strongly correlated Mott systems.

Acknowledgments.— Y.F.R. acknowledges the financial support from National Key
Research and Development Program (Grant Nos. 2016YFA0301700, 2017YFB0405703),
the China Government Youth 1000-Plan Talent Program, and the NNSFC(Grant No.
11474265). T.S.Z is supported by Air Force Office of Scientific Research
(FA9550-16-1-0387), National Science Foundation (PHY-1505496), and Army
Research Office (W911NF-17-1-0128). W.Z. is supported by DOE National Nuclear
Security Administration through Los Alamos National Laboratory LDRD Program.
D.N.S. is supported by the DOE, through the Office of Basic Energy Sciences
under the grant No. DE-FG02-06ER46305.

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