# A Generalization of the Hausdorff Dimension Theorem for Fractals

(Date: Jun 5, 2021)

###### Abstract.

How many fractals exist in nature or the virtual world? In this note, we
partially answer the second question using Mandelbrot’s fundamental definition
of fractals and their quantities of the Hausdorff dimension and Lebesgue
measure. We prove the existence of beth- two of virtual fractals with a
Hausdorff dimension of a bi-variate function of them and the given Lebesgue
measure. The question remains unanswered for other fractal dimensions.

Mohsen Soltanifar111Biostatistics Division, Dalla Lana School of Public
Health, University of Toronto, Toronto, ON, Canada
e-mail<EMAIL_ADDRESS>
ORCID: https://orcid.org/0000-0002-5989-0082

Keywords Cantor Set, fat Fractals, Hausdorff Dimension, continuum, beth-two

Mathematics Subject Classification (2020). 28A80, 03E10

##  1\. Introduction

Benoit Mandelbrot (1924-2010) coined the term _fractal_ and its dimension in
his 1975 essay on the quest to present a mathematical model for self-similar,
fractured geometric shapes in nature [1]. Both nature and the virtual world
have prominent examples of fractals: Some natural examples include snowflakes,
clouds and mountain ranges, while some virtual instances are the middle third
Cantor set, the Sierpinski triangle and the Vicsek fractal. A key element in
the definition of the term _fractal_ is its dimension usually indexed by the
Hausdorff dimension. For instance, the usual Vicsek has the Hausdorff
dimesnion equal to $\log_{3}(5).$ While each fractal has its Hausdorff
dimension as a unique number in $(0,+\infty)$, there has been little evidence
on the inverse existence statement. Given that constructive existence theorems
play a key role in a wide spectrum of mathematical fields and computer science
[2, 3], one may ask: Is there any fractal for given Hausdorff dimension ? And,
in case of affirmative response, What are the features of the set of such
fractals? To the author’s best knowledge, the only available existence theorem
of fractals in the literature is limited to the case of the fractals with
Lebesgue measure 0 (thin fractals) [4] and its very minor version in
$\mathbb{R}$[5]. However, on one hand, there are fractals in the Euclidean
spaces with a positive Lebesgue measure (fat fractals) applied to model real
physical systems [6, 7]. Examples include the fat Cantor set with Hausdorff
dimension of 1, the fat Vicsek fractal with Hausdorff dimension of 2, and the
fat Menger Sponge with Hausdorff dimension of 3. On the other hand, the
current existential results in the literature[4, 5] lacks their provision for
the highest potential cardinal of beth- two and a given Lebesgue measure. This
issue gets more complex given the fact that fractals with a positive Lebesgue
measure must have a positive integer Hausdorff dimension.

This work presents a parallel existence result for fractals of a given
Hausdorff dimension and a positive Lebesgue measure in n-dimensional Euclidean
spaces. In light of this aim, this work provides a comprehensive deterministic
framework, providing a second proof for the former results and, in particular,
generalizes them in terms of Lebesgue measure and cardinal number.

##  2\. Preliminaries

The reader who has studied fractal geometry is well-equipped with the
following definitions and key properties of the Uniform Cantor sets,
Topological Dimension and the Hausdorff Dimension. Henceforth, in this paper
we consider the n-dimensional Euclidean space $\mathbb{R}^{n}$ with its
conventional Euclidean metric and the Lebesgue measure of
$\lambda_{n}(n\in\mathbb{N})$.

###  2.1. The Uniform Cantor Set

The Uniform Cantor set, or Smith-Volterra-Cantor set, or the fat Cantor set,
was introduced in a series of publications by Henry Smith in 1875, Vito
Volterra in 1881, and George Cantor in 1883, respectively. To construct the
uniform Cantor set, let $\\{\beta_{n}\\}_{n\in\mathbb{N}}$-called the removing
sequence- be a sequence of positive numbers with
$\sum_{n\in\mathbb{N}}\beta_{n}=1-l\in(0,1].$ Take $s\in\mathbb{N},$ and set
$I_{s,0}=[0,1].$ Define $I_{s,n}(n\in\mathbb{N})$ recursively by removing $s$
symmetrically located open intervals with equal length of
$\frac{\beta_{n}}{s(s+1)^{n-1}}$ from middle of each interval in $I_{s,n-1}$.
Then, $I_{s,n}$ is the union of $(s+1)^{n}$ disjoint closed intervals
$I_{s,n}^{(j)}$ of equal length
$\delta_{n}=\frac{1-\sum_{k=1}^{n}\beta_{k}}{(s+1)^{n}}$; furthermore, the
sequence $\\{I_{s,n}\\}_{n\in\mathbb{N}_{0}}$ is decreasing sequence with
finite intersection property in the compact space $[0,1].$

######  Definition 2.1.

The Uniform Cantor set of order $s\geq 1$ and Lebesgue measure $l\geq 0$
associated with the removing sequence $\\{\beta_{n}\\}_{n}$ is defined as:

(1) $C_{\\{\beta_{n}\\}_{n}}(s,l)=\cap_{n=0}^{\infty}I_{s,n}.$

It is well known that the Uniform Cantor set $C_{\\{\beta_{n}\\}_{n}}(s,l)$ is
nowhere dense, totally disconnected, perfect and uncountable [4]. Next, by
definition,
$\lambda_{1}([0,1]-I_{s,n})=\sum_{k=1}^{n}s(s+1)^{k-1}(\frac{\beta_{k}}{s(s+1)^{k-1}})=\sum_{k=1}^{n}\beta_{k}(n\in\mathbb{N}).$
Consequently, the Lebesgue measure of the Uniform Cantor set
$C_{\\{\beta_{n}\\}_{n}}(s,l)$ is given by:

(2) $\displaystyle\lambda_{1}(C_{\\{\beta_{n}\\}_{n}}(s,l))$ $\displaystyle=$
$\displaystyle
1-\lambda_{1}([0,1]-C_{\\{\beta_{n}\\}_{n}}(s,l))=1-\lambda_{1}(\cup_{n\in\mathbb{N}}([0,1]-I_{s,n}))$
$\displaystyle=$ $\displaystyle
1-\lim_{n\rightarrow\infty}\lambda_{1}([0,1]-I_{s,n})=1-\sum_{n\in\mathbb{N}}\beta_{n}=l.$

A prominent family of the Uniform Cantor sets in the literature [4, 8, 9, 10]
is obtained for the case of

(3) $\beta_{n}=\beta_{n}(s,\beta,l)=((s+1)\beta)^{n-1}(1-(s+1)\beta)(1-l)$

where $s\geq 1,0<\beta<\frac{1}{s+1}$ and $0\leq l<1.$ In particular, the
ordinary middle third Cantor set is obtained whenever $s=1,\beta=\frac{1}{3},$
and $l=0.$ It is simultaneously the first term of two distictive sequences of
the Uniform Cantor sets: one including $s\geq 1,\
\beta=\beta(s)=\frac{1}{2s+1}(s\geq 1),\ l=0$ [4]; and, another including
$s=1,\ \beta=\beta(s^{*})=\frac{1}{s^{*}+2}(s^{*}\geq 1),\ l=0,$ [8].

###  2.2. Topological Dimension

The Topological dimension considered here is the Urysohn-Menger small
inductive dimension. It is defined inductively by setting
$dim_{ind}(\phi)=-1.$ We then say that for given $C\subseteq\mathbb{R}^{n},$
we have $dim_{ind}(C)\leq k$ whenever there is a base $(U_{i})_{i\in I}$ of
open sets of $C$ such that $dim_{ind}(U_{i})\leq k-1(i\in I).$ We then say
$dim_{ind}(C)=k$ whenever $dim_{ind}(C)\leq k$ but $dim_{ind}(C)\nleq
k-1(k\in\mathbb{N}_{0}),$ [10].

For the case of a zero-topological dimension, we may consider a more
informative definition based on the idea of clopen (i.e., simultaneously open
and closed) sets as follows [10]:

######  Definition 2.2.

The space $C\subseteq\mathbb{R}^{n}$ has zero topological dimension (i.e.
$dim_{ind}(C)=0$) whenever every finite open cover $\\{U_{i}\\}_{i\in
I}(|I|<\infty)$ of $C$ has a finite refinement $\\{V_{i}\\}_{i\in
J}(|J|\leq|I|<\infty)$ that is a clopen partition of $C.$

Using above definition, it has been shown that for the case of middle third
Cantor set $C=C_{\\{\frac{2^{n}}{2.3^{n}}\\}_{n}}(1,0)$ we have
$dim_{ind}(C)=0,$ [10]. A straightforward generalization of the proof shows
that for the uniform Cantor set $C=C_{\\{\beta_{n}\\}_{n}}(s,l)$ we also have
the same conclusion. The following theorem summarizes some prominent
properties of the small inductive dimension required in this paper[10]:

######  Theorem 2.3.

Let $\\{C_{i}\\}_{i\in I}$ be a countable family of subsets of
$\mathbb{R}^{n}\ (n\geq 1).$ Then:

(i) $dim_{ind}(C_{i})\in\\{-1,0,\cdots,n\\}$ where $i\in I,$
(ii) $dim_{ind}(C_{i})\leq dim_{ind}(C_{j})$ where $C_{i}\subseteq C_{j}(i\neq
j),$
(iii) $dim_{ind}(cC_{i}+d)=dim_{ind}(C_{i})$ where $c,d\in\mathbb{R}^{+}$ and
$i\in I,$
(iv) $dim_{ind}(\cup_{i\in I}C_{i})\leq k$ for some fixed $k\in\mathbb{N}_{0}$
whenever we have $dim_{ind}(C_{i})\leq k(i\in I).$
(v) $dim_{ind}(\prod_{i\in I}C_{i})\leq\sum_{i\in I}dim_{ind}(C_{i})$ whenever
$I$ is finite.

As a corollary of this theorem, for the case of a finite family of uniform
Cantor sets $\\{C_{i}\\}_{i\in I},$ it follows that $dim_{ind}(\cup_{i\in
I}C_{i})=0$ and $dim_{ind}(\prod_{i\in I}C_{i})=0.$

###  2.3. Hausdorff Dimension

The Hausdorff dimension, or Hausdorff-Besicowitch dimension, is considered as
the one of the most prominent dimensions for fractals. It was first introduced
by Felix Hausdorff in 1918 and was later improved in terms of computational
techniques by Abram S. Besicovitch. It is definable for any subset of the real
line as follows[9]:

######  Definition 2.4.

Let $C\subseteq\mathbb{R}$ and given $s\geq 0.$ Then, given s-dimensional
Hausdorff measure of $C$ by

(4) $H^{s}(C)=\lim_{\delta\rightarrow
0}\Big{(}\inf\\{\sum_{i\in\mathbb{N}}|U_{i}|^{s}|\ \
C\subseteq\underset{i\in\mathbb{N}}{\cup}U_{i}:0<|U_{i}|\leq\delta\\}\Big{)},$

the Hausdorff dimension of $C$ is defined as:

(5) $dim_{H}(C)=\inf\\{s\geq 0|H^{s}(C)=0\\}.$

Using this definition, Hausdorff showed that the middle third Cantor set
$C=C_{\\{\frac{2^{n}}{2.3^{n}}\\}_{n}}(1,0)$ has Hausdorff dimension equal to
$\frac{log(2)}{log(3)}$. More generally[10]:

######  Theorem 2.5.

The Hausdorff dimension of the Uniform Cantor set
$C=C_{\\{\beta_{n}\\}_{n}}(s,l)$ is given by:

(6)
$dim_{H}(C_{\\{\beta_{n}\\}_{n}}(s,l))=\begin{cases}\frac{log(s+1)}{log(s+1)+\underset{n\rightarrow\infty}{\limsup}(-\frac{log(\beta_{n})}{n-1})}&\textnormal{if
$s\geq 1,\ \ l=0,\
\underset{n\in\mathbb{N}}{\inf}(\frac{\beta_{n}}{1-\sum_{k=1}^{n-1}\beta_{k}})>0,$}\\\
1&\textnormal{if $s\geq 1,\ \ 0<l<1.$}\end{cases}$

As a Corollary, the Hausdorff dimension of the above prominent family of
Cantor sets in equation (3) is given by:

(7)
$dim_{H}(C_{\\{\beta_{n}(s,\beta,l)\\}_{n}}(s,l))=\begin{cases}\frac{log(s+1)}{-log(\beta)}&\textnormal{if\
\ $s\geq 1,\ 0<\beta<\frac{1}{s+1},\ l=0$}\\\ 1&\textnormal{if\ \ $s\geq 1,\
0<\beta<\frac{1}{s+1},\ 0<l<1.$}\end{cases}$

Next, we consider useful notation for the Uniform Cantor sets with Hausdorff
dimension $r>0$ as follows:

######  Notation 2.6.

The Linear Transform of the Uniform Cantor set of order $s\geq 1$, Hausdorff
dimension $r>0$ and Lebesgue measure $l\geq 0$ associated with the removing
sequence $\\{\beta_{n}\\}_{n}$ is denoted as:

(8) $F_{r,l,s}=c\times C_{\\{\beta_{n}\\}_{n}}(s,l)+d\ \text{for some}\
c>0,d\geq 0.$

The following theorem summarizes some key properties of the Hausdorff
Dimension required in this paper[9]:

######  Theorem 2.7.

Let $\\{C_{i}\\}_{i\in I}$ be a countable family of subsets of
$\mathbb{R}^{n}\ (n\geq 1).$ Then:

(i) $0\leq dim_{H}(C_{i})\leq n$ where $i\in I,$
(ii) $dim_{H}(C_{i})\leq dim_{H}(C_{j})$ where $C_{i}\subseteq C_{j}(i\neq
j),$
(iii) $dim_{H}(cC_{i}+d)=dim_{H}(C_{i})$ where $c,d\in\mathbb{R}^{+}$ and
$i\in I,$
(iv) $dim_{H}(\cup_{i\in I}C_{i})=\sup_{i\in I}(dim_{H}C_{i}),$
(v) $dim_{H}(\prod_{i\in I}C_{i})=\sum_{i\in I}dim_{H}(C_{i})$ whenever $I$ is
finite and one of $C_{i}$s is a uniform Cantor set.
(vi) $dim_{H}(C_{i})=n(i\in I)$ whenever $\lambda_{n}(C_{i})>0(i\in I).$

Finally, throughout this paper we refer to the _fractal_ in terms of
Mandelbrot’s definition [1]:

######  Definition 2.8.

A subset $C\subseteq\mathbb{R}^{n}$ with Hausdorff dimension $dim_{H}(C)$ and
the topological inductive dimension $dim_{ind}(C)$ is a fractal whenever:

(9) $dim_{H}(C)>dim_{ind}(C).$

###  2.4. General Cartesian Product Distribution over Unions

We finish this section with a review of the relationship between union and
Cartesian products. As the Cartesian product is distributive over unions [11],
we have the following general result easily proved by induction on the
dimension of the product $n$:

######  Theorem 2.9.

Let $I_{j}(1\leq j\leq n)$ be indexing sets and $\\{C_{i_{j}}\\}_{i_{j}\in
I_{j}}(1\leq j\leq n)$ be families of subsets of $\mathbb{R}$ indexed by them,
respectively. Then:

(10) $\prod_{j=1}^{n}(\bigcup_{i_{j}\in
I_{j}}C_{i_{j}})=\bigcup_{(i_{1},\cdots,i_{n})\in\prod_{j=1}^{n}I_{j}}(\prod_{j=1}^{n}C_{i_{j}}),$

where $\bigcup$ denotes the union and $\prod$ denotes the Cartesian product.

##  3\. Main Results

We generalize the existence Hausdorff Dimension Theorem from fractals with a
Lebesgue measure 0 (thin fractals) to those with a non-negative Lebesgue
measure (fat fractals) with existence of the higher cardinal of beth- two. The
construction process is accomplished in four stages: (i) Showing the existence
of fractals with a plausible Hausdorff dimension and a Lebesgue measure in
$\mathbb{R}$; (ii) Expanding the cardinality of fractals in the first stage to
the continuum; (iii) Exending the result in the second stage to the higher
dimensional Euclidean spaces $\mathbb{R}^{n}(n>1),$ and, (iv) Generalizing the
result in the third stage to the cardinal of beth- two. We begin with the
following existence result whose very special case has been stated in [5]:

######  Lemma 3.1.

For any real $0<r\leq 1$ and $l\geq 0$ there is a fractal with the Hausdorff
dimension $1_{[0]}(l).r+1_{(0,\infty)}(l)$ and the Lebesgue measure $l$ in
$\mathbb{R}.$

###### Proof.

We consider three scenarios:

(i) $0<r<1,l=0:$
Fix $s_{0}\in\mathbb{N},$ and consider the family of Uniform Cantor sets
$C_{\\{\beta_{n}(s_{0},\beta,0)\\}}(s_{0},0),$ $(0<\beta<\frac{1}{s_{0}+1}).$
Then, their Hausdorff dimension are given by
$f_{s_{0}}(\beta)=\frac{log(s_{0}+1)}{-log(\beta)},$
$(0<\beta<\frac{1}{s_{0}+1}).$ Since $f_{s_{0}}$ is a continuous increasing
function from $(0,\frac{1}{s_{0}+1})$ onto $(0,1),$ such that $f(0^{+})=0$ and
$f(\frac{1}{s_{0}+1}^{-})=1,$ by an application of the Intermediate Value
Theorem, there exists $\beta_{0}\in(0,\frac{1}{s_{0}+1})$ such that
$f_{s_{0}}(\beta_{0})=r.$ Now, it is enough to consider the following fractal
in the interval $[0,1]:$

(11) $F_{r,0,s_{0}}=C_{\\{\beta_{n}(s_{0},\beta_{0},0)\\}}(s_{0},0)$

(ii) $r=1,l=0:$
Again, fix $s_{0}\in\mathbb{N}$ and consider the sequence
$r_{n}=\frac{n}{n+1}\ (n\in\mathbb{N}).$ By part (i), there is a corresponding
sequence of fractals $F_{r_{n},0,s_{0}}\ (n\in\mathbb{N})$ such that
$dim_{H}(F_{r_{n},0,s_{0}})=r_{n},dim_{ind}(F_{r_{n},0,s_{0}})=0,$ and
$\lambda_{1}(F_{r_{n},0,s_{0}})=0,\ (n\in\mathbb{N}).$ Take:

(12) $F_{1,0,s_{0}}^{*}=\underset{n\in\mathbb{N}}{\cup}F_{r_{n},0,s_{0}}$

Accordingly, two consecutive applications of Theorem 2.3(iv) and
Theorem2.7(iv) proves the desired result.

(iii) $r=1,l>0:$
Finally, for fixed $s_{0}\in\mathbb{N}$ consider
$F_{1,\frac{l}{[l]+1},s_{0}}=([l]+1)\times
C_{\\{\beta_{n}(s_{0},\beta_{0},\frac{l}{[l]+1})\\}}(s_{0},\frac{l}{[l]+1})$
with $\beta_{n}$ given by equation 3. Hence, two consecutive applications of
Theorem 2.3(iii) and Theorem2.7(iii) yields the desired result. ∎

A closer look at the proof of the Lemma 3.1 and changing values
$s_{0}\in\mathbb{N}$ shows that indeed for any real $0<r\leq 1$ and $l\geq 0$
there are countably infinite fractals with the Hausdorff dimension
$1_{[0]}(l).r+1_{(0,\infty)}(l)$ and the Lebesgue measure $l$ in $\mathbb{R}$.
The following Lemma expands this result to the cardinality of the Continuum.

######  Lemma 3.2.

For any real $0<r\leq 1$ and $l\geq 0$ there is a continuum of distinctive
fractals with the Hausdorff dimension $1_{[0]}(l).r+1_{(0,\infty)}(l)$ and the
Lebesgue measure $l$ in $\mathbb{R}.$

###### Proof.

We consider three scenarios similar to the proof of Lemma 3.1. In each
scenario, we partition a given interval (e.g. $[0,1]$) to an infinite union of
its shrinking sub-intervals and, using linear transformations, put into them
Cantor Fractals.

(i) $0<r<1,l=0:$
By construction, in the part(i) of the proof of Lemma3.1, there is sequence of
fractals $\\{F_{r,0,s}\\}_{s\in\mathbb{N}}$ with
$F_{r,0,s}=\frac{1}{2^{s}}C_{\beta_{n}(s,\beta_{s},0)}+\frac{1}{2^{s}}$ in the
interval $[\frac{1}{2^{s}},\frac{1}{2^{s-1}}]$ such that:
$dim_{H}(F_{r,0,s})=r,dim_{ind}(F_{r,0,s})=0,$ and $\lambda_{1}(F_{r,0,s})=0\
(s\in\mathbb{N}).$ Let $I\in\mathcal{P}(\mathbb{N})$ be infinite and define:

(13) $F_{r,0}^{I}=\underset{s\in I}{\cup}F_{r,0,s}$

Then, again an application of Theorem 2.3(iii,iv) and Theorem 2.7(iii,iv) and
uncountability of the set of such $I^{\prime}s$ proves the assertion.

(ii) $r=1,l=0:$
Again, consider the sequence $r_{n}=\frac{n}{n+1}(n\in\mathbb{N}).$ By part
(i), there is a corresponding sequence of fractals
$F_{r_{n},0}^{I}(n\in\mathbb{N})$ and an infinite set
$I\in\mathcal{P}(\mathbb{N})$ such that
$dim_{H}(F_{r_{n},0}^{I})=r_{n},dim_{ind}(F_{r_{n},0}^{I})=0,$ and
$\lambda_{1}(F_{r_{n},0}^{I})=0(n\in\mathbb{N}).$ Put :

(14)
${F^{*}}_{1,0}^{I}=\underset{n\in\mathbb{N}}{\cup}(\frac{1}{2^{n}}F_{r_{n},0}^{I}+\frac{1}{2^{n}})$

Now, by two applications of Theorem 2.3(iii,iv) and Theorem2.7(iii,iv) and
uncountability of the set of such $I^{\prime}s$, the assertion follows.

(iii) $r=1,l>0:$
By construction, in the part(iii) of the proof of Lemma3.1, there is sequence
of fractals $\\{F_{1,\frac{l}{[l]+1},s}\\}_{s\in\mathbb{N}}$ in the interval
$[0,[l]+1]$ such that:
$dim_{H}(F_{1,\frac{l}{[l]+1},s})=1,dim_{ind}(F_{1,\frac{l}{[l]+1},s})=0,$ and
$\lambda_{1}(F_{1,\frac{l}{[l]+1},s})=l\ (s\in\mathbb{N}).$ Let
$F_{1,\frac{l}{[l]+1},s}^{*}=\frac{[l]+1}{2^{s}}F_{1,\frac{l}{[l]+1},s}+\frac{[l]+1}{2^{s}}(s\in\mathbb{N}),$
and $I\in\mathcal{P}(\mathbb{N})$ be infinite and define:

(15) ${F^{*}}^{I}_{1,l}=\frac{l}{\lambda_{1}(\underset{s\in
I}{\cup}F_{1,\frac{l}{[l]+1},s}^{*})}\times\underset{s\in
I}{\cup}F_{1,\frac{l}{[l]+1},s}^{*}$

Finally, by two applications of Theorem 2.3(iii,iv) and Theorem2.7(iii,iv) and
uncountability of the set of such $I^{\prime}s$ the assertion follows. ∎

An investigation into Lemma 3.2 reveals its limitation to providing fractals
with Hausdorff dimension on the range $0<r\leq 1.$ To obtain fractals with
Hausdorff dimension $r>1$ we need to consider the higher dimensional Euclidean
spaces. Using the idea of the countable unions of n-dimensional Cantor Fractal
dusts (as the Cartesian product of the Uniform Cantor sets defined above), our
next result addresses this situation.

######  Lemma 3.3.

For any real $r>0$ and $l\geq 0,$ there is a continuum of distinctive fractals
with the Hausdorff dimension $1_{[0]}(l).r+1_{(0,\infty)}(l).n$ and Lebesgue
measure $l$ in $\mathbb{R}^{n}$ where $(-[-r]\leq n)$.

###### Proof.

Let $r>0,l\geq 0$ and for given $-[-r]\leq n$ define
$r^{\prime}=\frac{r}{n}\in(0,1].$Then, using equations (13-15), we consider
the following scenarios with the Hausdorff dimension $r^{\prime}$ and Lebesgue
measure $l\geq 0$ given different values of $r^{\prime},l:$

Table 1. Construction of fractals with the given Hausdorff dimension $r^{\prime}$ and Lebesgue measure $l$ in $\mathbb{R}$ $(r^{\prime},l)$ | $l=0$ | $l>0$
---|---|---
$0<r^{\prime}<1$ | $F_{r^{\prime},l}^{I}(Eq.13)$ | DNE
$r^{\prime}=1$ | ${F^{*}}_{r^{\prime},l}^{I}(Eq.14)$ | ${F^{*}}_{r^{\prime},l}^{I}(Eq.15)$

where $I\in\mathcal{P}(\mathbb{N})$ is infinite. Now, we consider n copies
from each cell in Table 1 and consider the following Cartesian products:

Table 2. Construction of fractals with given a Hausdorff dimension $r$ and Lebesgue measure $l$ in $\mathbb{R}^{n}$ $(r,l)$ | $l=0$ | $l>0$
---|---|---
$r\notin\mathbb{N}$ | $\prod_{j=1}^{n}F_{r^{\prime},l}^{I_{j}}$ | DNE
$r\in\mathbb{N}$ | $\prod_{j=1}^{n}{F^{*}}_{r^{\prime},l}^{I_{j}}$ | $(\frac{1}{l})^{n-1}\times\prod_{j=1}^{n}{F^{*}}_{r^{\prime},l}^{I_{j}}$

where $I_{j}\in\mathcal{P}(\mathbb{N})(1\leq j\leq n)$ are infinite. Finally,
considering Table 2, the assertion follows by an application of Theorem
2.3(v), Theorem2.7(iii),(v) and Theorem 2.9; and that there are uncountably
infinite sets of $I_{j}\in\mathcal{P}(\mathbb{N})(1\leq j\leq n).$ ∎

The constructed fractals in the Lemma 3.3 have two key features as follows:
First, they are centrally asymmetric with respect to the point symmetric of
$\overrightarrow{c}_{n}=(\frac{1}{2},\cdots,\frac{1}{2})$ where $-[-r]\leq n.$
This is a direct result of the fact that being centrally asymmetric is
invariant under Cartesian products and unions; and, the building block of the
fractals in the Lemma are centrally asymmetric fat Cantor sets with point
symmetric of $\overrightarrow{c}_{1}=\frac{1}{2}.$ However, the Lemma can be
expanded to the symmetric fractals with consideration of transformed symmetric
fat Cantor sets. The details of the proof are minor modifications of our proof
for the asymmetric case presented here with replacement of fractals $F$ in
equations (13-15) with $\frac{1}{2}((F\cup(-F))+1)$. Second, there is a
continuum of them with the same Hausdorff dimension and the same Lebesgue
measure in the Euclidean spaces $\mathbb{R}^{n+1}-\mathbb{R}^{n}\ \ (-[-r]\leq
n)$ as those in $\mathbb{R}^{n}\ \ (-[-r]\leq n).$ Here, $\mathbb{R}^{n}$ is
considered isomorphic to the subspace of $\mathbb{R}^{n}\times
0\subset\mathbb{R}^{n+1}.$ This result is the direct consequence of the
construction process in the proof of the Lemma. Finally, while the Lemma 3.3
guarantees at least a continuum of fractals in n-dimensional Euclidean space
with given properties, it does not guarantee the existence of the beth- two
fractals given the generalized continuum hypothesis [12]. This lack of
precision of the cardinal number (between either the continuum or beth- two)
creates ground for further investigation. Our main result provides the answer.

######  Theorem 3.4.

(The General Hausdorff Dimension Theorem) For any real $r>0$ and $l\geq 0,$
there is a set of beth- two distinctive fractals with the Hausdorff dimension
$1_{[0]}(l).r+1_{(0,\infty)}(l).n$ and Lebesgue measure $l$ in
$\mathbb{R}^{n}$ where $(-[-r]\leq n)$.

###### Proof.

We accomplish the proof in four steps as follows:

First, let $r>0,l\geq 0,$ and fix $n\geq-[-r].$ Then, for the function
$h_{n}(\beta)=\frac{n.log(2)}{-log(\beta)},$ $\ (0<\beta<\frac{1}{2}),$ given
$h_{n}(0^{+})=0$ there is $\beta_{00}\in(0,\frac{1}{2})$ such that
$h_{n}(\beta_{00})<\frac{r}{2}.$
Second, take a fractal $F_{00}=(C+[l]+1)^{n}$ in $\mathbb{R}^{n}$ where
$C=C_{\beta_{n}}$ with $\beta_{n}=\beta_{n}(1,\beta_{00},0)$ $\ (n\geq 1)$ as
in equation (3). Then, $\lambda(F_{00})=0,\ \ \dim_{ind}(F_{00})=0,$ and
$dim_{H}(F_{00})<\frac{r}{2}.$ Define the power set of $F_{00}$ with cardinal
of beth- two as follows:

(16) $\mathcal{F}_{1}(r,l)=\\{F\subseteq\mathbb{R}^{n}|F\subseteq
F_{00},dim_{H}(F)<\frac{r}{2},\lambda(F)=0,dim_{ind}(F)=0\\}.$

Third, consider the family of fractals given by Lemma 3.3:

(17)
$\mathcal{F}_{2}(r,l)=\\{F\subseteq\mathbb{R}^{n}|dim_{H}(F)=1_{[0]}(l).r+1_{(0,\infty)}(l).n,\
\lambda(F)=l,dim_{ind}(F)=0\\}.$

Fourth, given two families in equations (16),(17) define the following family
of fractals with cardinal of beth- two:

(18) $\mathcal{F}(r,l)=\\{F_{1}\cup
F_{2}|F_{1}\in\mathcal{F}_{1}(r,l),F_{2}\in\mathcal{F}_{2}(r,l)\\}.$

Finally, let $F\in\mathcal{F}(r,l).$ Then, by construction, there are
$F_{i}\in\mathcal{F}_{i}(i=1,2)$ such that: $F=F_{1}\cup F_{2}.$ Accordingly,
by another application of Theorem 2.3(iv), and Theorem 2.7(iv):

(19) $\displaystyle dim_{H}(F)$ $\displaystyle=$
$\displaystyle\max(dim_{H}(F_{1}),dim_{H}(F_{2}))$ $\displaystyle=$
$\displaystyle dim_{H}(F_{2})=1_{[0]}(l).r+1_{(0,\infty)}(l).n,$ (20)
$\displaystyle\lambda(F)$ $\displaystyle=$ $\displaystyle\lambda(F_{2})=l,$
(21) $\displaystyle dim_{ind}(F)$ $\displaystyle=$ $\displaystyle 0.$

This completes the proof. ∎

The assertion of Theorem 3.4 and its proof methodology have two immediate
consequences: First, in terms of having the cardinality of beth-two, the
theorem’s assertion and an application of the Cantor-Schroder-Bernstein
theorem [13] put the set of virtual fractals of $\mathbb{R}^{n}$ the set of
virtual fractals of $\mathbb{R}^{n}$ in the same category of mathematical sets
such as the sigma-algebra of Lebesgue measurable sets in $\mathbb{R}^{n},$ the
power set of $\mathbb{R}^{n},$ the set of functions from $\mathbb{R}^{n}$ to
$\mathbb{R}^{n},$ and, the Stone-Cech compactification of $\mathbb{R}^{n}.$
Second, the theorem’s proof can be modified to show that the cardinality of
the set of non-fractals in $\mathbb{R}^{n}$ is beth-two as well. To see this,
let $n\geq 1,C$ be the standard middle third Cantor set, and $I^{n}$ be the
unit cube in $\mathbb{R}^{n}$. Define the family $\mathcal{F}(n)=\\{I^{n}\cup
P_{C,n}|P_{C,n}\subseteq(C+1)^{n}\\}$. Then, for any $F\in\mathcal{F}(n)$ we
have $dim_{ind}(F)=n=dim_{H}(F).$ Hence, by Mandelbrot’s definition 2.8,$F$ is
a non-fractal. Consequently, $\mathcal{F}(n)$ is a subset of the set of non-
fractals in $\mathbb{R}^{n}$. Accordingly, the assertion follows from
$\mathcal{F}(n)$ having the cardinality of beth-two and applying the Cantor-
Schroder-Bernstein theorem [13].

##  4\. Discussion

This work presented an existence theorem for fractals of a given Hausdorff
dimension and a Lebesgue measure by considering finite Cartesian products of
the countable unions of Uniform Cantor sets with plausible Lebesgue measure.
In addition, it generalized the former existence theorem in terms of the
Lebesgue measure and cardinal number.

This work’s contributions to the fractal geometry literature covers two
perspectives: First, it highlights the advantage of the Cantor sets to other
well-known classical fractals in showing existence of fractals with any
Hausdorff dimension and Lebesgue measure. Other prominent fractals lack this
feature given not being defined in the one dimensional Euclidean space
$\mathbb{R}$ and having Hausdorff dimension larger than 1. Examples include
the Sierpinski triangle, Takagi curve, Julia set, Triflake, Koch curve, and
Apollonian gasket. Second, the existence theorem is equipped with constructive
proof (versus pure existence proof) presenting real fractals for given
Hausdorff dimension and the Lebesgue measure [14]. This key feature helps us
to explore other properties of the constructed fractals yielding to more
comprehensive information on them.

There are some limitations in this work that create four new lines of research
for the interested reader. First, we considered only Mandelbrot’s strict
mathematical definition to present the existence result. However, there are
other agreed-upon key descriptive features in the definition of fractal that
need to be considered. Examples of these characteristics are self-similarity
type(exact, quasi, statistical), fine structure, irregularity(local, global),
and the recursive definition[9]. Second, while we considered only Hausdorff
dimension as the index of fractal dimension in the Mandelbrot’s definition,
the existence case for the other indices of fractal dimension remains to be
investigated. For instance, we can consider the Renyi dimension (with special
cases, such as the Minkowski dimension [9], Information dimension [15],
Correlation dimension [16]), the Higuchi dimension [17], the Lyapunov
dimension [18], Packing dimension[19], the Assouad dimension[20] and the
generalization of the Hausdorff dimension[21]. Third, a more rigorous and
comprehensive method is to investigate the existence problem of fractals for
the generalized fractal space equipped with a fractal structure and the
generalized fractal dimension [22]. Finally, the existence result in this work
is limited to deterministic fractals constructed by their associated
deterministic recursive processes. However, its validity for the more general
random fractals remains an open question. We summarize the above points as the
following set of open problems:

###  Open Problems

Given the n-dimensional Euclidean space $\mathbb{R}^{n}(n\geq 1).$

(1) Does the precise cardinality of the set of all distinctive fractals vary
by the applied fractal dimension ?

(2) Does the precise cardinality of the set of all distinctive fractals depend
on the fractal structure ?

(3) Does the precise cardinality of the set of all distinctive fractals depend
on the deterministic or random nature of the fractal?

##  5\. Conclusion

There has been a growing interest in fractal dimension and its calculations
from different methods since the early 1900s without retrospective exploration
of the existence of fractals for a given dimension. This work addressed the
problem in part by presenting a constructive existence result from
deterministic viewpoint, paving the way for exploring a more generalized
random perspective.

## References

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  * [13] Bajnok, B. An Invitation to Abstract Mathematics (2ed); Springer: Cham, Switzerland, 2020; p.325
  * [14] Bridges, D; Palmgren, E. Constructive Mathematics; in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy: Metaphysics Research Lab, Stanford University; San Francisco, USA, 2018: retrieved 2020-06-17
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  * [16] Peter, G. Generalized Dimensions of Strange Attractors. Physics Letters A1983, 97(6): 227‒230.
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  * [18] Frederickson, F.; Kaplan, J.; Yorke, E.; Yorke, J. The Liapunov dimension of strange attractors . Journal of Differential Equations1983, 49(2): 185–207.
  * [19] Tricot, Jr., C. Two definitions of fractional dimension. Mathematical Proceedings of the Cambridge Philosophical Society 1982, 91(1): 57–74.
  * [20] Robinson, J.C. Dimensions, Embeddings, and Attractors: Cambridge University Press: Cambridge, UK, 2010; p.5.
  * [21] Fernández-Martínez, M.; Sánchez-Granero, M.A. Fractal dimension for fractal structures: a Hausdorff approach revisited.Journal of Mathematical Analysis and Applications 2014, 409 (1), 321-330.
  * [22] Fernández-Martínez, M.; Sánchez-Granero, M.A. Fractal dimension for fractal structures. Topology and its Applications 2014, 163, 93-111.


# Transition from insulating to conducting states in the bilayer Hubbard model
by perpendicular quench field

X. Z. Zhang  College of Physics and Materials Science, Tianjin Normal
University, Tianjin 300387, China  Z. Song<EMAIL_ADDRESS>School of
Physics, Nankai University, Tianjin 300071, China

###### Abstract

A many-body quantum system with different parameters may support two distinct
quantum states in the same energy shell. It allows the dynamical transition
from the ground state of pre-quench Hamiltonian to a steady state of post-
quench one. We study the dynamical response of the ground states of a two-
layer half-filled Hubbard model to a perpendicular electric field. We show
that the steady state is conducting state when the field is resonant to the
on-site repulsion, while the initial state is a Mott-insulating state. In
addition, two layers exhibit the same conducting behavior due to the formation
of the long-lived dopings, which are manifested by the performance of the
charge fluctuation. The key to achieving this dynamic transition is the
cooperation between on-site interaction and the resonant field rather than the
role of either. Our finding provides an alternative mechanism for field-
induced conductivity in a strongly correlated system.

##  I Introduction

Understanding the static and dynamical properties of strongly correlated
quantum systems poses a major challenge in both theory and experiment Dagotto
(1994); Georges et al. (1996); Yeshurun et al. (1996); Dukelsky et al. (2004);
Basov and Timusk (2005); Bloch et al. (2008); Aoki et al. (2014). Although the
presence of strong quantum correlations often precludes intuitive models, the
interplay between particle-particle interaction and kinetic energy gives rise
to novel effects including the magnetism and superconductivity Auerbach
(1994); Sierra and Martín-Delgado (1997); Giamarchi et al. (2016). The Hubbard
model provides a minimal paradigm for studying these properties and, as such,
is attracting the attention of theoreticians from many different corners of
condensed matter Hubbard and Flowers (1964); Fisher et al. (1989); Yang
(1989); Georges et al. (1996); Lee et al. (2006); Sachdev (2011); Keimer et
al. (2015); Zhang and Song (2021); Xie et al. (2023). Recent years have seen a
significant progress in simulating quantum many-body physics with ultracold
atoms that are well isolated from the environment and therefore evolve under
their intrinsic quantum dynamics Giorgini et al. (2008); Bloch et al. (2008);
Randeria et al. (2012); Christian and Immanuel (2017). In particular,
ultracold fermionic atoms in an optical lattice can be used to realize the
Hubbard model Köhl et al. (2005); Jördens et al. (2008); Schneider et al.
(2008); Esslinger (2010); Taie et al. (2012); Hart et al. (2015); Duarte et
al. (2015); Cocchi et al. (2016); Hartke et al. (2020). However, most real
materials possess rather complex lattice structures, which can be approximated
as a system of coupled layers. Fortunately, a bilayer Fermi–Hubbard model can
be realized through a highly stable vertical superlattice González-Tudela and
Cirac (2019); Koepsell et al. (2020); Gall et al. (2021); Meng et al. (2021),
which provides a platform to study the basic property and enlarge our
understanding of the multi-layer interacting system.

Driven by the advances of high-resolution microscopy, the theory describing
the various quantum phase transitions has advanced remarkably in the context
of interacting many-body quantum systems at equilibrium. However, the behavior
of such systems is far less understood when it comes to the non-equilibrium
regime, whose relevance has rapidly grown triggered by the significant
experimental progress. Examples of systems that offer a large degree of
control such as ultracold atoms in optical lattices Greiner et al. (2002);
Bloch et al. (2008); Estève et al. (2008); Schneider et al. (2008); Bakr W. et
al. (2010); Abanin et al. (2019), trapped ions Albiez et al. (2005); Schumm et
al. (2005); Hofferberth et al. (2007); Bakr W. et al. (2010); Blatt and Roos
(2012), superconducting qubits Roushan et al. (2017); Xu et al. (2018), as
well as nuclear and electron spins associated with impurity atoms in diamond
Doherty et al. (2013); Schirhagl et al. (2013). The tunability and long
coherence times of these systems, along with the ability to prepare highly
non-equilibrium states, enable one to probe quantum dynamics and
thermalization in closed systems Abanin et al. (2019). Hence, it seems very
timely to investigate the non-equilibrium behavior of the bilayer Hubbard
system and unravel the potential exotic behavior.

The bilayer Hubbard model served as a paradigm to study band to Mott insulator
crossover Kancharla and Okamoto (2007); Gall et al. (2021). In the large $U$
limit, whether the ground state is in a planar antiferromagnetically ordered
Mott insulating phase or a band insulating phase is fully determined by the
ratio of the inter-and intra-layer coupling. But no matter what phase the
system is in, its ground state is non-conductive. In this work, we propose a
way to realize the conductivity of the two-layer system through non-
equilibrium dynamics. To this end, we first prepare the system into a Mott
insulator phase with a strong antiferromagnetic correlation. Then we switch on
a resonant perpendicular electric field such that the single-occupied Mott
insulating states and non-ferromagnetic state doped with doublons and holes
hybridize in the same energy shell. This guarantees the significant charge
fluctuation of the low-energy eigenstates of the post-quench Hamiltonian since
the conducting proportion overwhelms the insulating ratio. Furthermore, these
states reside in the low-lying sector protected by the energy gap. This keeps
the two layers with dopings for a long time providing a platform to facilitate
the mobility of the particles in each layer under the non-ferromagnetic
background. Accordingly, the evolved state driven by the post-quench
Hamiltonian can hardly return to the initial insulating state with zero charge
fluctuation. Correspondingly, the system enters into a non-equilibrium
conducting phase featured by the large charge fluctuation. The key ingredient
of our proposal is the formation of the long-lived dopings in each layer
arising from the cooperation between the on-site interaction and the resonant
electric field. It is hoped that these results can inspire further studies on
the non-equilibrium bilayer even multi-layer interacting system.

The remainder of the paper is structured as follows: In Sec. II, we introduce
the bilayer Hubbard model subjected to a perpendicular electric field. Aiding
by the symmetries of the considered system, the role of the external field is
elucidated in the absence of the on-site interaction. Sec. III is devoted to
investigating the formation of the possible energy shell that is determined by
the cooperation between the resonant field and on-site interaction. In Sec.
IV, the mechanism of the field-induced conducting state is sketched in which
the long-lived dopings play the pivotal role. Then a non-equilibrium scheme
for generating a steady conducting state is proposed. Finally, we summarize
our results in In Sec. VI.

##  II Two-layer Hubbard Model

Figure 1: Schematic illustration of the bilayer-Hubbard system subjected to a
perpendicular electric field. Experimentally, the bilayer system is realized
by a bichromatic superlattice in the $z$ direction trapping atoms in several
bilayer sheets Gall et al. (2021). The intra-layer hoppings $t_{\mathrm{A}}$
and $t_{\mathrm{B}}$ between lattice sites are controlled by the depth of the
optical lattices; the inter-layer tunnelling $t_{\mathrm{AB}}$, on the other
hand, is independently induced by microwaves Meng et al. (2021). Using
tomographic imaging, (spin-) densities in a single layer are detected.

We consider a two-layer Hubbard model system, described by the Hamiltonian

$H=H_{\mathrm{A}}+H_{\mathrm{B}}+H_{\mathrm{AB}},$ (1)

with sub-Hamiltonian on each layer

$\displaystyle H_{\mathrm{A}}$ $\displaystyle=$ $\displaystyle
t_{\mathrm{A}}\sum_{\sigma=\uparrow,\downarrow}\sum_{\left\langle
i,j\right\rangle}(a_{i,\sigma}^{{\dagger}}a_{j,\sigma}+\mathrm{H.c.})+U\sum_{i=1}^{N}a_{i,\uparrow}^{{\dagger}}a_{i,\downarrow}^{{\dagger}}a_{i,\downarrow}a_{i,\uparrow}$
$\displaystyle+V_{\mathrm{A}}\sum_{i=1}^{N}\left(a_{i,\uparrow}^{{\dagger}}a_{i,\uparrow}+a_{i,\downarrow}^{{\dagger}}a_{i,\downarrow}\right),$
$\displaystyle H_{\mathrm{B}}$ $\displaystyle=$ $\displaystyle
t_{\mathrm{B}}\sum_{\sigma=\uparrow,\downarrow}\sum_{\left\langle
i,j\right\rangle}(b_{i,\sigma}^{{\dagger}}b_{j,\sigma}+\mathrm{H.c.})+U\sum_{i=1}^{N}b_{i,\uparrow}^{{\dagger}}b_{i,\downarrow}^{{\dagger}}b_{i,\downarrow}b_{i,\uparrow}$
(2)
$\displaystyle+V_{\mathrm{B}}\sum_{i=1}^{N}\left(b_{i,\uparrow}^{{\dagger}}b_{i,\uparrow}+b_{i,\downarrow}^{{\dagger}}b_{i,\downarrow}\right),$

and inter-layer hopping term

$H_{\mathrm{AB}}=t_{\mathrm{AB}}\sum_{\sigma=\uparrow,\downarrow}\sum_{j=1}^{L}(b_{j,\sigma}^{{\dagger}}a_{j,\sigma}+\mathrm{H.c.}),$
(3)

where $a_{i,\sigma}$ and $b_{j,\sigma}$ are fermion operators with
spin-$\frac{1}{2}$ polarization $\sigma=\uparrow,\downarrow$ in layers $A$ and
$B$, respectively. The parameters $t_{\mathrm{A}}(t_{\mathrm{B}})$ and
$t_{\mathrm{AB}}$ are intra- and inter-layer hopping strengths, and taken to
be real in this paper. The on-site interaction $U$ tends to localize the
fermions by establishing a Mott insulator at half band filling. Here
$H_{\mathrm{A}}$ and $H_{\mathrm{B}}$ are identical, describing a standard
Hubbard model, which is restricted as simple bipartite lattice with no
restriction on their geometries. In particular, the key features of the setup
are: (i) $H_{\mathrm{AB}}$ is the rung of the whole system representing the
tunneling between two subsystems $H_{\mathrm{A}}$ and $H_{\mathrm{B}}$. (ii)
The on-site potentials $V_{\mathrm{A}}$ and $V_{\mathrm{B}}$. The gradient of
such two uniform potentials acts as the electric field that can bring the
migration of particles. Such two ingredients play a pivotal role in the
subsequent quench dynamics. The schematic of the system is presented in Fig.
1. Driven by the advances of high-resolution microscopy, such a bilayer
Fermi–Hubbard model can be realized experimentally by using ultracold atoms in
an optical lattice. In particular, the bichromatic optical superlattice in the
vertical direction can be employed to control the bilayer systems and inter-
layer tunneling $t_{\mathrm{AB}}$ Koepsell et al. (2020); Hartke et al.
(2020); Gall et al. (2021). The other Hamiltonian parameters $t_{\mathrm{A}}$,
$t_{\mathrm{B}}$, $V_{\mathrm{A}}$, $V_{\mathrm{B}}$, $U$, the geometry
Soltan-Panahi et al. (2011); Wirth et al. (2011); Tarruell et al. (2012); Jo
et al. (2012); Shintaro et al. (2015), and the dimensionality are
experimentally tunable, thus providing access to a large parameter range.

The Hamiltonian in Eq. (1) has a rich structure due to the symmetry it
possesses. To gain further insight into this model, we define pseudo-spin
operators

$\displaystyle s^{+}$ $\displaystyle=$
$\displaystyle\left(s^{-}\right)^{{\dagger}}=\sum_{i=1}^{N}\left(a_{i,\uparrow}^{{\dagger}}a_{i,\downarrow}+b_{i,\uparrow}^{{\dagger}}b_{i,\downarrow}\right),$
(4) $\displaystyle s^{z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{N}\left(a_{i,\uparrow}^{{\dagger}}a_{i,\uparrow}+b_{i,\uparrow}^{{\dagger}}b_{i,\uparrow}-a_{i,\downarrow}^{{\dagger}}a_{i,\downarrow}-b_{i,\downarrow}^{{\dagger}}b_{i,\downarrow}\right),$
(5)

which satisfy the Lie algebra commutation relations
$\left[s^{\mu},s^{\nu}\right]=\sum_{\lambda=x,y,z}2i\epsilon^{\mu\nu\lambda}s^{\lambda}$,
and are employed to characterize the symmetry of the system. It can be shown
that the Hamiltonian has SU(2) symmetry, obeying

$\left[H,s^{z}\right]=\left[H,s^{\pm}\right]=0.$ (6)

It has been shown that the ground state of $H$ with zero field at half-filling
is singlet with $s=0$ Lieb (1989), or anti-ferromagnetic insulating state. The
presence of the on-site potentials $V_{\mathrm{A}}$ and $V_{\mathrm{B}}$ does
not break the SU(2) symmetry. It has two implications: (i) The appearance of
the potential brings the possibility to change the property of the original
system, such as improving the mobility of particle. (ii) The retained spin
symmetry maintains the invariant subspaces, which allows us to investigate the
problem in each sector. And every sector can be selected by adding an electric
field in the experiment.

Figure 2: Sketch of the possible movement of the particles driven by a
resonant electric field in a ladder system. Here we present several typical
configurations to show the underlying mechanism of the transition. (a) The
system is initialized in a single-occupied state such that the cooperation
between the resonant electric field $V_{\mathrm{A}}=-V_{\mathrm{B}}=U/2$ and
interaction $U$ only allows the migration of the particles from the
$\mathrm{A}$ layer to the $\mathrm{B}$ layer. As a consequence, a hole is left
in the $\mathrm{A}$ layer while a doublon is formed in the $\mathrm{B}$ layer.
But the hopping of particles within each layer is strictly prohibited. (b)-(d)
denote the multi-doublon and hole cases. With the constraint of energy
conservation, the electron and hole can move in the non-ferromagnetic
background. In particular, the spin-up particle can move freely in the upper
layer while the spin-down particle (doublon) can effectively move as a single
particle with the aid of the background spin configuration. The principle of
the particle migration is that doublon cannot appear in the $\mathrm{A}$ layer
and holes cannot appear in the $\mathrm{B}$ layer. The hopping processes
denoted by red arrows are allowed, while green arrows are forbidden.

Based on the aforementioned statement, we examine the response of the system
to the quenched perpendicular electric field when the interaction is free for
simplicity. To this end, we introduce the Fourier transformation in two sub-
systems,

$\displaystyle a_{j,\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{k}e^{ikj}a_{k,\sigma},$ (7)
$\displaystyle b_{j,\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{k}e^{ikj}b_{k,\sigma},$ (8)

where the wave vector $k=2n\pi/N$, $\left(n\in\left[-L/2,\text{
}L/2-1\right]\right)$. This transformation block diagonalizes the Hamiltonian
due to its translational symmetry, i.e.,

$H=\sum_{k,\sigma}H_{k,\sigma}=\sum_{k,\sigma}\psi_{k,\sigma}^{\dagger}h_{k,\sigma}\psi_{k,\sigma},$
(9)

satisfying $\left[H_{k,\sigma},\text{
}H_{k^{\prime},\sigma^{\prime}}\right]=0$. Here $H$ is rewritten in the Nambu
representation with the basis

$\psi_{k,\sigma}=\left(\begin{array}[]{c}a_{k,\sigma}\\\
b_{k,\sigma}\end{array}\right),$ (10)

and $h_{k,\sigma}$ is a core matrix with the form

$h_{k,\sigma}=\left(\begin{array}[]{cc}\lambda_{\mathrm{A}}&t_{\mathrm{AB}}\\\
t_{\mathrm{AB}}&\lambda_{\mathrm{B}}\end{array}\right),$ (11)

where $\lambda_{\mathrm{\sigma}}=-2t_{\parallel}\cos k+V_{\mathrm{\sigma}}$
($\mathrm{\sigma=A,}$ $\mathrm{B}$). Note that
$t_{\mathrm{A}}=t_{\mathrm{B}}=t_{\parallel}$ is assumed for simplicity. The
Hamiltonian can be diagonalized by taking the linear transformation on
$h_{k,\sigma}$. Straightforward algebra shows that

$H=\sum_{k,\sigma}\left(\zeta_{k}+\varepsilon\right)\alpha_{k,\sigma}^{\dagger}\alpha_{k,\sigma}+\left(\zeta_{k}-\varepsilon\right)\beta_{k,\sigma}^{\dagger}\beta_{k,\sigma},$
(12)

where

$\displaystyle\zeta_{k}$ $\displaystyle=$ $\displaystyle-2t_{\parallel}\cos
k+\frac{V_{\mathrm{A}}+V_{\mathrm{B}}}{2},$ (13) $\displaystyle\varepsilon$
$\displaystyle=$
$\displaystyle\sqrt{t_{\mathrm{AB}}^{2}+\left(\frac{V_{\mathrm{A}}-V_{\mathrm{B}}}{2}\right)^{2}},$
(14)

and

$\displaystyle\alpha_{k,\sigma}^{\dagger}$ $\displaystyle=$
$\displaystyle\cos\frac{\theta}{2}a_{k,\sigma}^{\dagger}+\sin\frac{\theta}{2}b_{k,\sigma}^{\dagger},$
(15) $\displaystyle\beta_{k,\sigma}^{\dagger}$ $\displaystyle=$
$\displaystyle\sin\frac{\theta}{2}a_{k,\sigma}^{\dagger}-\cos\frac{\theta}{2}b_{k,\sigma}^{\dagger},$
(16)

with $\cos\theta=\left(V_{\mathrm{A}}-V_{\mathrm{B}}\right)/2\varepsilon$.
Evidently, the role of the tilted potential and tunneling $t_{\mathrm{AB}}$ is
to separate two identical single-particle energy bands. Hence, the ground
state of the system is the direct product of spin-singlets and triplets
belonging to the different $k$ subspace, given that the influence of the
electric field is not considered and the system is at half-filling. The
corresponding ratio between such two components is determined by
$t_{\mathrm{AB}}$. In particular, the ground state is a band insulator of
spin-singlets along the bonds between the layers when
$t_{\mathrm{AB}}/t_{\parallel}>2$. Without loss of generality, we take the
ground state of $H$ with $V_{\mathrm{A}}=V_{\mathrm{B}}=0$ at half-filling as
the initial state

$\left|\psi\left(0\right)\right\rangle=\prod_{\left|k^{\prime}\right|<k_{c}^{\prime},\sigma}a_{k^{\prime},\sigma}^{\dagger}b_{k^{\prime},\sigma}^{\dagger}\prod_{k_{c}^{\prime}<\left|k\right|<k_{c},\sigma}\frac{a_{k,\sigma}^{\dagger}-b_{k,\sigma}^{\dagger}}{\sqrt{2}}\left|\mathrm{Vac}\right\rangle,$
(17)

where $\left|\mathrm{Vac}\right\rangle$ is the vacuum state of fermion. Here
$k_{c}^{\prime}=$min$\\{$arccos$(t_{\mathrm{AB}}/2t_{\parallel})$,
arccos$(-t_{\mathrm{AB}}/2t_{\parallel})\\}$, and $k_{c}=\pi-k_{c}^{\prime}$.
After a quench, the on-site potentials $V_{\mathrm{A}}$ and $V_{\mathrm{B}}$
are applied, then the evolve state driven by $H$ can be given as

$\displaystyle\left|\psi\left(t\right)\right\rangle$ $\displaystyle=$
$\displaystyle\prod_{\left|k^{\prime}\right|<k_{c}^{\prime},\sigma}e^{-2i\zeta_{k^{\prime}}t}a_{k^{\prime},\sigma}^{\dagger}b_{k^{\prime},\sigma}^{\dagger}\times$
(18)
$\displaystyle\prod_{k_{c}^{\prime}<\left|k\right|<k_{c},\sigma}e^{-i\zeta_{k}t}\frac{P_{-}a_{k,\sigma}^{\dagger}-P_{+}b_{k,\sigma}^{\dagger}}{\sqrt{2}}\left|\mathrm{Vac}\right\rangle,$

where $P_{\pm}=\cos\varepsilon t+i\sin\varepsilon
t\left(\sin\theta\pm\cos\theta\right)$. To measure the migration of particles,
the ratio of the particle density between two layers

$r(t)=\frac{\langle\sum_{\sigma,i=1}^{N}a_{i,\sigma}^{{\dagger}}a_{i,\sigma}\rangle}{\langle\sum_{\sigma,i=1}^{N}b_{i,\sigma}^{{\dagger}}b_{i,\sigma}\rangle},$
(19)

is introduced. For the evolved state $\left|\psi\left(t\right)\right\rangle$,
its transfer rate can be given as

$r(t)=\frac{\sum_{\left|k^{\prime}\right|<k_{c}^{\prime},\sigma}1+\frac{1}{2}\sum_{k_{c}^{\prime}<\left|k\right|<k_{c},\sigma}\left[1-\sin
2\theta\sin^{2}\varepsilon
t\right]}{\sum_{\left|k^{\prime}\right|<k_{c}^{\prime},\sigma}1+\frac{1}{2}\sum_{k_{c}^{\prime}<\left|k\right|<k_{c},\sigma}\left[1+\sin
2\theta\sin^{2}\varepsilon t\right]}.$ (20)

Evidently, $r(t)$ exhibits a periodical behavior with period
$\tau=\pi/\varepsilon$. Further, the averaged transfer ratio over a long time
interval $T$ is defined as

$\overline{r}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}r(t)\mathrm{d}t.$
(21)

Direct derivation shows that

$\overline{r}=\frac{\sum_{\left|k^{\prime}\right|<k_{c}^{\prime},\sigma}1+\frac{1}{2}\sum_{k_{c}^{\prime}<\left|k\right|<k_{c},\sigma}\left[1-\sin\theta\cos\theta\right]}{\sum_{\left|k^{\prime}\right|<k_{c}^{\prime},\sigma}1+\frac{1}{2}\sum_{k_{c}^{\prime}<\left|k\right|<k_{c},\sigma}\left[1+\sin\theta\cos\theta\right]}.$
(22)

It indicates that the cooperation of the perpendicular field and
$t_{\mathrm{AB}}$ leads to the transfer of particles between the different
layers, which serves as the building block to realize the conducting phase
from a Mott insulator as $U$ switches on. Note, in passing, that the direction
of particle transfer depends on the choice of the initial state and is not
necessarily related to the direction of gradient of the chemical potential. An
obvious example is the sign reversal of $\theta$ in Eq. (22) for the highest
excited state, where particles tend to go from layer $\mathrm{B}$ to layer A.
It is in stark difference from the classical system in which the flow of
particles is along the direction of the gradient of the potential. On the
other hand, in the absence of $U$, the transfer rate between two layers
oscillates such that the system acquires the information of the initial state
once the time $t=n\tau$ ($n$ is a positive integer). In other words, the
presence of potential only induces the particle migration and does not result
in the imbalance of particle density between two layers over a long time
interval.

In the following, we study the dynamical response of the ground state of $H$
(large $U$) to a quenched electric field with the resonant value.

Figure 3: Plots of the local charge fluctuation $\Delta n_{a,1}$ for the low
energy excitation of the Hubbard model at half-filling. The geometry of the
considered two-layer system is illustrated in Fig. 3(a). The other system
parameters are given as (b) $t_{\mathrm{B}}/t_{\mathrm{A}}=1$, $V=0$, and
$U/t_{\mathrm{A}}=20$ (c) $t_{\mathrm{B}}/t_{\mathrm{A}}=1$,
$V/t_{\mathrm{A}}=5$, and $U/t_{\mathrm{A}}=20$. In Fig. 3(b), the system
favors the Mott-insulating phase and hence $\Delta n_{a,1}$ of low-energy
eigenstates are around $0$. However, $\Delta n_{a,1}$ is around $0.5$ except
for very few eigenstates when the resonant field $V$ presents. The cooperation
between $U$ and $V$ results in a dramatic change in the system conductivity.

##  III Resonant field

Meanfiled theory Sachdev (2011) has shown that the phase diagram of a Hubbard
model is described by the Mott lobe, which is a function of $U$ and chemical
potential. The formation of the anti-ferromagnetic insulating ground state
requires large $U$ and half-filling. For the bilayer system, we note that the
total particle number of the two-layer is conservative, while the one of each
layer is not. In this work we consider the case with fixed particle number,
which equals to $2N$. Thus when taking $V_{\mathrm{A}}=V_{\mathrm{B}}$, the
ground state is an anti-ferromagnetic insulating state. However, when taking
$V_{\mathrm{A}}\neq V_{\mathrm{B}}$, the chemical potentials can adjust the
particle density on each layer, since one layer can act as a source of
particles for the other. It happens by the difference of the chemical
potential, or the perpendicular electric field. In the following, we will show
that an optimal field can provide a resonant channel for the directional
migration of particles from one layer to the other.

We start with the simplest case with
$t_{\mathrm{A}}=t_{\mathrm{B}}=t_{\mathrm{AB}}=V_{\mathrm{A}}=V_{\mathrm{B}}=0$.
The Hamiltonian only contains the repulsive term, i.e.,
$U\sum_{i=1}^{N}\left(a_{i,\uparrow}^{{\dagger}}a_{i,\downarrow}^{{\dagger}}a_{i,\downarrow}a_{i,\uparrow}+b_{i,\uparrow}^{{\dagger}}b_{i,\downarrow}^{{\dagger}}b_{i,\downarrow}b_{i,\uparrow}\right)$,
and then the ground states in the subspace with zero $s^{z}$ are multi-
degenerate. The energy is zero and the degenerate degree is
$\mathcal{N}_{\mathrm{I}}=C_{2N}^{N}$, the explicit expression of which is
$C_{2N}^{N}=(2N)!/[(2N-N)!(N)!]$. These states construct singlet ground state
when parameters $t_{\mathrm{A}}=t_{\mathrm{B}}=t_{\mathrm{AB}}$ are switched
on in large $U$ limit, and are referred to as the insulating set (I-set) of
states in this paper.

Now we consider the case with $t_{\mathrm{A}}=t_{\mathrm{B}}=t_{\mathrm{AB}}$,
and $V_{\mathrm{A}}=-V_{\mathrm{B}}=U/2$ (hereafter
$V_{\mathrm{A}}=-V_{\mathrm{B}}=V$), which is referred to as the resonant
field. The Hamiltonian can be rewritten in the from

$\displaystyle H_{\mathrm{A}}$ $\displaystyle=$ $\displaystyle
U/2\sum_{i=1}^{N}\left(a_{i,\uparrow}^{{\dagger}}a_{i,\uparrow}+a_{i,\downarrow}^{{\dagger}}a_{i,\downarrow}\right)^{2},$
(23) $\displaystyle H_{\mathrm{B}}$ $\displaystyle=$
$\displaystyle-U/2\sum_{i=1}^{N}\left(b_{i,\uparrow}^{{\dagger}}b_{i,\uparrow}-b_{i,\downarrow}^{{\dagger}}b_{i,\downarrow}\right)^{2},$
(24)

which is convenient for the following analysis. The eigen energy of
$H_{\mathrm{A}}+H_{\mathrm{B}}$ is

$E=U/2\sum_{i=1}^{N}\left[\left(n_{\mathrm{A,}i,\uparrow}+n_{\mathrm{A,}i,\downarrow}\right)^{2}-\left(n_{\mathrm{B,}i,\uparrow}-n_{\mathrm{B,}i,\downarrow}\right)^{2}\right],$
(25)

where $n_{a\mathrm{,}i,\sigma}$ and $n_{b\mathrm{,}i,\sigma}$ are eigen values
of operators $a_{i,\sigma}^{{\dagger}}a_{i,\sigma}$ and
$b_{i,\sigma}^{{\dagger}}b_{i,\sigma}$, respectively. For the half-filled case
in the subspace with zero spin component in $z$ direction, we have the
following constraint conditions

$\displaystyle\sum_{\sigma=\uparrow,\downarrow}\left(N_{a\mathrm{,}\sigma}+N_{b\mathrm{,}\sigma}\right)$
$\displaystyle=$ $\displaystyle 2N,$ (26) $\displaystyle
N_{a\mathrm{,}\uparrow}+N_{b\mathrm{,}\uparrow}-N_{a\mathrm{,}\downarrow}-N_{b\mathrm{,}\downarrow}$
$\displaystyle=$ $\displaystyle 0,$ (27)

where $N_{a\mathrm{,}\sigma}$ and $N_{b\mathrm{,}\sigma}$ denote the eigen
values of operators $\sum_{i=1}^{N}a_{i,\sigma}^{{\dagger}}a_{i,\sigma}$ and
$\sum_{i=1}^{N}b_{i,\sigma}^{{\dagger}}b_{i,\sigma}$, respectively. Notice
that when every site is occupied by a single particle, we have
$\left(n_{a\mathrm{,}i,\uparrow}+n_{a\mathrm{,}i,\downarrow}\right)^{2}=\left(n_{b\mathrm{,}i,\uparrow}-n_{b\mathrm{,}i,\downarrow}\right)^{2}=1$
for all $i\in\left[1,N\right]$ such that the ground state energy is $E=0$. In
the following, we analyse all the possible eigenstates with zero energy, since
the number and configuration of which are important for our results. As the
mentioned above, the number of single-occupied states is
$\mathcal{N}_{\mathrm{I}}$ and the superposition of these states constructs
the anti-ferromagnetic groundstate of the original Hamiltonian with large $U$
and zero $V$. Based on such states, losing a particle in layer $\mathrm{A}$
results in energy lose of $U/2$, while gaining a fermion in layer B results in
energy lose of $-U/2$, generating a new zero energy state. Accordingly, a set
of zero energy states can be constructed by losing $n$
($n\in\left[1,N\right]$) particles in layer $\mathrm{A}$ and gaining $n$
particle in layer B. The number of such kind of states is
$\left(C_{N}^{n}\right)^{2}C_{2N-2n}^{N-n}$, in which term $C_{N}^{n}$ comes
from the configuration of hole in layer $\mathrm{A}$ or the doublon in layer
B, and $C_{2N-2n}^{N-n}$ comes from the spin configuration of the rest single-
occupied particles. Then the total number of zero-energy (ground) states is
$\mathcal{N}_{\mathrm{C}}=\sum_{j=1}^{N}(C_{N}^{j})^{2}C_{2N-2j}^{N-j}$. All
these states are hybridized by the inter-chain and intra-chain hoppings. In
other words, the hopping terms can drive one of these states to all the rest
states.

##  IV the mobility of the free doublons and holes

In this section, we discuss the observable results based on the analysis in
the last section. We can classify all states into two sets: (i) I-set
(insulating set), as mentioned above, which consists of complete single-
occupied states; (ii) C-set (conducting set), which consists of states
involving holes and doublons. The reason for such a classification is that the
superposition of states in I-set describes an antiferromagnetic Mott
insulating state, while the superposition of states in C-set describes a
conducting state. We explain the reason for the classification via the
following examples. For simplicity, we consider the simplest case involving
states with single-hole and single-doublon. A set representative of states in
C-set can be expressed as the form
$a_{i,\uparrow}b_{i,\uparrow}^{{\dagger}}\left|\Uparrow\right\rangle_{\mathrm{A}}\left|\Downarrow\right\rangle_{\mathrm{B}}$,
where states
$\left|\Uparrow\right\rangle_{\mathrm{A}}=\prod_{l=1}^{N}a_{l,\uparrow}^{{\dagger}}\left|\mathrm{Vac}\right\rangle$
and
$\left|\Downarrow\right\rangle_{\mathrm{B}}=\prod_{l=1}^{N}b_{l,\downarrow}^{{\dagger}}\left|\mathrm{Vac}\right\rangle$
denote ferromagnetic states of A and B layers, respectively, and
$\left|\mathrm{Vac}\right\rangle$ is the empty state of the whole system.
Obviously, the dynamics of hole and electron on each layers can act as free
carriers on the ferromagnetic background. Likewise, similar situation occurs
for multi-hole and doublon cases, and for other types of non-ferromagnetic
background. For clarity, we sketch some typical cases in Fig. 2.

Figure 4: Numerical simulation of the quenched bilayer Hubbard system for
different repulsive interactions $U/t_{\mathrm{A}}=5$, $10$, and $20$. The
system consists of two coupled ladders and is initialized in the ground state
of $H_{\mathrm{pre}}$. The small charge fluctuation of the initial ground
state arises from the finite interaction which allows the existence of a
double-occupied state with $s=0$. Such non-ferromagnetic kind of states is
suppressed when $U$ increases. After a quench, the doublons and holes appear
in the evolved state due to the particle flow from the upper layer to the
lower layer as shown in Fig. 3(a). This is protected by the energy shell
residing in the low-energy excited subspace. In the non-ferromagnetic
background, the non-equilibrium system enters into the conducting phase
manifested by the increase of the charge fluctuation. Note that the layers A
and B share the same charge fluctuation $\Delta n_{a,j}=\Delta n_{b,j}=\Delta
n_{j}$ since both free electrons and holes play the same role in the particle
migration.

In order to predict the dynamical behavior of the quench, we estimate the
following two quantities. The first one is the ratio of the dimensions of
insulating and conducting classes. Straightforward calculation gives

$r_{1}=\frac{\mathcal{N}_{\mathrm{I}}}{\mathcal{N}_{\mathrm{C}}}\approx
e^{-\gamma N},$ (28)

where factor $\gamma=0.325$ is obtained from the numerical fitting. The second
one is the ratio of the average particle numbers between insulating and
conducting classes. Considering a mixed state, in which all the zero-energy
states have the same probability, we have the average numbers on each layers

$\displaystyle\overline{N}_{\mathrm{A,}\uparrow}$ $\displaystyle=$
$\displaystyle\overline{N}_{\mathrm{A,}\downarrow}=\frac{\sum_{j=1}^{N}j(C_{N}^{j})^{2}C_{2N-2j}^{N-j}}{\mathcal{N}_{\mathrm{I}}+\mathcal{N}_{\mathrm{C}}}=\frac{N}{3},$
(29) $\displaystyle\overline{N}_{\mathrm{B,}\uparrow}$ $\displaystyle=$
$\displaystyle\overline{N}_{\mathrm{B,}\downarrow}=\frac{\sum_{j=1}^{N}j(C_{N}^{j})^{2}C_{2j}^{j}}{\mathcal{N}_{\mathrm{I}}+\mathcal{N}_{\mathrm{C}}}=\frac{2N}{3},$
(30)

which result in

$r_{2}=\frac{\overline{N}_{\mathrm{A,}\uparrow}+\overline{N}_{\mathrm{A,}\downarrow}}{\overline{N}_{\mathrm{B,}\uparrow}+\overline{N}_{\mathrm{B,}\downarrow}}=1/2.$
(31)

These two quantities imply the following predictions. (i) The initial state
has a very small portion ($r_{1}$) comparing to the possible evolved states,
which will never go back to the initial one. It is similar to the case with a
site initial state in a square lattice. The probability will spread out,
approaching to uniform distribution after a long time. Then the evolved state
probably relaxes to a conducting state. (ii) The value of $r_{2}$ indicates
that the ratio of numbers on two layers turns to be $0.5$ after a long time.
It predicts that there is a particle flow in the quench dynamical process.

In order to capture the main consequence of the particle flow arising from the
cooperation of $U$ and resonant $V$, we further introduce the local charge
fluctuation $\Delta n_{\lambda,j}$ to compare the conductivity of the
eigenstates before and after quench. The definition of $\Delta n_{\lambda,j}$
is given by

$\Delta
n_{\lambda,j}=\sqrt{\Delta_{1,\lambda,j}-\Delta_{2,\lambda,j}^{2}}\quad(\lambda=a,b),$
(32)

where

$\displaystyle\Delta_{1,\lambda,j}$ $\displaystyle=$
$\displaystyle\langle(\sum_{\sigma}\lambda_{j,\sigma}^{{\dagger}}\lambda_{j,\sigma})^{2}\rangle,$
(33) $\displaystyle\Delta_{2,\lambda,j}$ $\displaystyle=$
$\displaystyle\langle\sum_{\sigma}\lambda_{j,\sigma}^{{\dagger}}\lambda_{j,\sigma}\rangle.$
(34)

Specifically, if the system is in the Mott insulating phase, such fluctuation
would become energetically unfavorable, forcing the system into a number state
and exhibiting a vanishing particle number fluctuation $\Delta n_{\lambda,j}$.
Beyond the Mott insulator regime, the fermions are delocalized with the non-
vanishing charge fluctuation. In Fig. 3, we figure out the conductivity of the
eigenstates in the cases of $H\left(V=0\right)$ and $H\left(V=U/2\right)$,
respectively. The system is at symmetric half filling, i.e., $s^{z}=0$, and
its geometry is illustrated in Fig. 3(a). We focus on the low-lying
eigenstates whose energy is around $0$. When $V=0$ and large $U$ limit, the
low energy excitation of the considered half-filled Hubbard model can be
effectively described by the Heisenberg model with two layers. The
corresponding eigenstates are the superposition of bases belonging to the
I-set. As such, the charge fluctuation $\Delta n_{a,1}$ is about $0$ as shown
in Fig. 3(b). Conversely, the low-lying eigenstates of $H\left(V=U/2\right)$
possess the non-zero charge fluctuation since C-set dominates especially in
the large $N$ limit. This can be seen in Fig. 3(c). The local charge
fluctuation for most eigenstates is $0.5$. To gain further insight into the
dynamics of the two-layer system, we consider an initial state expanded in
such energy shell,

$\left|\psi\left(0\right)\right\rangle=\sum_{\alpha}c_{\alpha}\left|E_{\alpha}\right\rangle\text{,}$
(35)

where $\left|E_{\alpha}\right\rangle$ is the eigenstate of $H$ with eigenvalue
$E_{\alpha}$ in the low-lying energy shell. The expectation value of a local
operator $\widehat{O}$ as a function of time thus reads

$\displaystyle O\left(t\right)$ $\displaystyle\equiv$
$\displaystyle\left\langle\psi\left(t\right)\right|\widehat{O}\left|\psi\left(t\right)\right\rangle$
(36) $\displaystyle=$
$\displaystyle\sum_{\alpha}\left|c_{\alpha}\right|^{2}O_{\alpha\alpha}+\sum_{\alpha\neq\beta}O_{\alpha\beta}e^{i\left(E_{\alpha}-E_{\beta}\right)t},$

where $O_{\alpha\alpha}=\left\langle
E_{\alpha}\right|\widehat{O}\left|E_{\alpha}\right\rangle$ and $\widehat{O}$
stands for either the operator
$(\sum_{\sigma}\lambda_{j,\sigma}^{{\dagger}}\lambda_{j,\sigma})^{2}$ or
$\sum_{\sigma}\lambda_{j,\sigma}^{{\dagger}}\lambda_{j,\sigma}$ in the
considered two-layer system. Due to phase cancellation of the off-diagonal
terms in Eq. (36), the long-time average is determined only by the average in
the diagonal element such that

$\overline{O}\left(t\right)=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}O\left(t\right)\mathrm{d}t=\sum_{\alpha}\left|c_{\alpha}\right|^{2}O_{\alpha\alpha}.$
(37)

For the considered two types of the systems $H\left(V=0\right)$ and
$H\left(V=U/2\right)$, $O_{\alpha\alpha}$ approaches a constant number for the
most eigenstates of the low-lying energy sector. Hence,
$\overline{O}\left(t\right)=O_{\alpha\alpha}$ indicating that the eigenstate
and arbitrary evolved state share the same average value of $\Delta
n_{\lambda,j}$. It can be expected that the initial state containing the
insulating component will tend to the conducting state after a long-time
evolution. In the subsequent section, we will demonstrate this point through a
quench process.

Before ending this section, we want to point out that the conducting mechanism
is different from that of the single-layer system wherein the particles are
not half-filling. For the latter system, the maximum of holes is contingent on
the filled particle number. As for the two-layer system, layer A acts as a
source of particles for layer B, and the corresponding particle number of
layer A is not conserved. As a consequence, layer A involves all the possible
configurations of holes in the low-energy sector, which provides more channels
for the transfer of particles between the layers. The cooperation between $U$
and the resonant $V$ can stabilize the system in this phase for a long time.

##  V Dynamical density transfer

The above analysis provides a prediction about the dynamical transition from
an insulating state to a conducting state in a composite system. In a slightly
different language, the main point of the above consideration can be viewed as
follows. A Mott insulating state is essentially the result of Pauli exclusion
principle and Coulomb repulsion at half-filling. It is not a stable
equilibrium since the chemical potential can change the particle density for
an open system. For a half-filled bilayer Hubbard system with zero fields, the
ground state lies at the equilibrium point for each layer. When the resonant
external field is switched on, the low-energy eigenstates changes dramatically
with positive or negative doping in each layer. This exactly compensates for
repulsive interactions presents in the Hubbard model such that the transfer of
particles between the two layers cost free. As a consequence, many nearly
degenerate conducting states lie at the low-energy sector of quenched
Hamiltonian which allows an easier transfer of particles between layers.

We perform numerical simulations on a finite system for the following quench
process. The pre-quench Hamiltonian is assumed as $H_{\mathrm{pre}}=H$ with
$t_{\mathrm{A}}=t_{\mathrm{B}}=t_{\mathrm{AB}}=1$, and $V=0$. The post-quench
Hamiltonian is then $H_{\mathrm{pos}}=H$ with $V=U/2$. Initially, the system
is half-filled and at the ground state, being an insulating state with a
strong anti-ferromagnetic correlation. According to our analysis, the quench
dynamics should result in a steady conducting state, realizing the dynamical
transition from an insulating state to a conducting state. The evolved states
$\left|\Psi\left(t\right)\right\rangle$ for initial anti-ferromagnetic ground
state $\left|\Psi\left(0\right)\right\rangle$ is calculated by exact
diagonalization method. We focus on the following quantities: (i) The ratio of
particle density between two layers $r(t)$ which is a key signature of the
dynamical transition; (ii) The number fluctuation $\Delta n_{\lambda,j}$ of
the evolved state $\left|\Psi\left(t\right)\right\rangle$.

Figure 5: Average $\overline{r}$ and $\overline{\Delta n_{\lambda,j}}$ as
functions of $V$ for $12$-site Hubbard model at half filling and $s_{z}=0$.
The initial state is prepared as the ground state of $H_{\mathrm{pre}}$. The
system parameters are $U/t_{\mathrm{A}}=10$ and cutoff quench time
$T=50/t_{\mathrm{A}}$. It indicates that, as $V$ approaches $U/2$, a dip will
appear verifying our prediction in the text.

The lattice geometry and numerical results are plotted in Fig. 4, where a
$12$-site Hubbard model at half filling is considered. It can be shown that
the resonant electric field can induce the particle transfer from the upper
layer to the lower layer forming the doublons and holes in the evolved state.
This can be demonstrated by the increase of total double occupancy
$D=\sum_{j}\langle D_{j}\rangle$, where the local double-occupation operator
is given by
$D_{\lambda,j}=\lambda_{j,\uparrow}^{\dagger}\lambda_{j,\uparrow}\lambda_{j,\downarrow}^{\dagger}\lambda_{j,\downarrow}$
($\lambda=a,$ $b$). The movable electrons and holes significantly increase the
number fluctuations of both layers and hence enhances the conductivity. Such a
mechanism still holds even in the small $U$ condition which can be shown by
comparing Figs. 4 (b)-(d). The weak interaction means that the ground state
does not fully consist of the single-occupied Mott insulating state such that
the initial double occupancy and charge fluctuation is not zero, which can be
shown in Figs. 4 (b) and (d). Instead, the portion of the other state $s=0$,
i.e., doublon state, increases leading to the non-zero number fluctuation of
the initial state and total transfer rate between the two layers. It indicates
that the effectiveness of the scheme is suppressed due to the tunneling
between the considered energy shell (I- and C-set) and other energy levels. To
further check the validity of the earlier analysis, we introduce the average
number fluctuation

$\overline{\Delta
n}_{\lambda,j}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\Delta
n_{\lambda,j}\mathrm{d}t,$ (38)

to compare the results of the resonant and non-resonant cases. Average
$\overline{r}$ and $\overline{\Delta n}_{\lambda,j}$ as functions of parameter
$V$ for different $j$ are plotted in Fig. 5. It can be shown that the dips
appear in both $\overline{r}$ and $\overline{\Delta n}_{\lambda,j}$ when the
resonant electric field is taken. Another interesting phenomenon is that both
layers share the same conducting behavior manifested by the performance of
$\overline{\Delta n}_{a,j}$ and $\overline{\Delta n}_{b,j}$. This can be
understood as follow: Assume that the initial state is prepared as the direct
product of ferromagnetic states within each layer, i.e.,
$\left|\Uparrow\right\rangle_{\mathrm{A}}\left|\Downarrow\right\rangle_{\mathrm{B}}$.
The resonant electric field makes the spin-up particle migrates from the layer
$\mathrm{A}$ to $\mathrm{B}$ such that a hole and a doublon are formed in two
layers individually. The spin-up particle can move freely based on the spin-
down ferromagnetic background while the formation of the hole enable the
hopping of the spin-down particles. Although the free particles are formed in
different ways in two layers, both facilitate the movement of particles within
each layer and thus increase the conductivity. Note that such a steady state
is the consequence of the non-equilibrium many-body dynamics that is usually
absent in the ground-state phase. Before ending this part, it is worthy
pointing out that the lattice structure of real materials, such as bilayer
graphene, is composed of coupled layers and is therefore not strictly two-
dimensional. Another advantage of our scheme is that it is not only applicable
to the bilayer system. Instead, it still makes sense even in the multi-layer
system. The core is that the interplay between the resonant electric field and
interaction can hybridize a small number of insulating states and a large
number of conducting states such that the particles tend to move in each layer
rather than localize at each site.

##  VI Summary

In summary, we have systematically investigated the quench dynamics of the
bilayer Hubbard model subjected to a perpendicular electric field. In the
presence of the resonance electric field, the system can enter into the steady
conducting phase via a non-equilibrium scheme, which can be demonstrated by
the increase of the charge fluctuation. The underlying mechanism is that the
cooperation between the resonant electric field and on-site interaction $U$
provides an energy shell hybridizing the single-and double-occupied states
with $s=0$. Due to the energy gap determined by the large $U$ and resonant
field, the inter- and intra- layer hopping does not induce the tunneling
between the considered shell and other shells so that the dynamics in the
quasi subspace is protected by the so-called ”energy conservation”. The stable
hybridization of such two kinds of states allow the directional particle flow
between the two layers, which forms the holes in the A layer and doublon in
the B layer. This facilitates the movement of the particles within each layer
and hence enhances the conductivity. The particles exhibit the same conducting
behavior in both layers since the formation of the free particles shares a
similar mechanism in both layers. Due to the large proportion of double-
occupied states in the resonant energy shell, a non-equilibrium system can
accommodate such a conducting state for a long time. Our findings establish a
strategy for enhancing the conductivity of the multi-layer strongly correlated
system from an out of equilibrium method.

###### Acknowledgements.

We acknowledge the support of the National Natural Science Foundation of China
(Grants No. 12275193, No. 11975166, and No. 11874225).

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