# Spatial and temporal structures of four financial markets in Greater China

F. Y. Ouyang  B. Zheng  X. F. Jiang  Department of Physics, Zhejiang
University, Hangzhou 310027, P. R. China

###### Abstract

We investigate the spatial and temporal structures of four financial markets
in Greater China. In particular, we uncover different characteristics of the
four markets by analyzing the sector and subsector structures which are
detected through the random matrix theory. Meanwhile, we observe that the
Taiwan and Hongkong stock markets show a negative return-volatility
correlation, i.e., the so-called leverage effect. The Shanghai and Shenzhen
stock markets are more complicated. Before the year 2000, the two markets
exhibit a strong positive return-volatility correlation, which is called the
anti-leverage effect. After 2000, however, it gradually changes to the
leverage effect. We also find that the recurrence interval distributions of
both the trading volume volatilities and price volatilities follow a power law
behavior, while the exponents vary among different markets.

###### keywords:

Econophysics, Complex system, Random matrix theory, Leverage effect

##  1 Introduction

Financial markets are complex systems with many-body interactions. In recent
years, much attention of physicists has been paid to the financial dynamics,
and physical concepts and methods are applied to analyze the dynamic behavior.
As large amounts of financial data are available now, it allows to explore the
fine structure of the financial dynamics and achieve various empirical results
[1, 2, 3, 4, 5, 6, 7, 8]. With rapid development of the economy, the financial
markets in Greater China attract more attention from the world. Let us now
focus on four stock markets, i.e., the Shanghai stock market, Shenzhen stock
market, Taiwan stock market and Hongkong stock market. Due to different
political and economic systems, the dynamic behavior varies much among the
four markets. The economy style of Taiwan is a typical export-oriented one.
The stock market developed much through several important economic policies,
such as import substitution, export expansion and structural adjustment.
Hongkong is a financial center in Asia, and the economy is prosperous. The
Shanghai and Shenzhen stock markets are both in mainland China, and undergoing
a rapid development in recent years. To the best of our knowledge, there have
not been literatures focusing on the comparative study of the spatial and
temporal structures of the four stock markets, although some relevant works
could be found such as the comparison between the response dynamics in
transition economies and developed countries [9]. In this paper, we intend to
provide a comparative study about the four stock markets, and understand how
political and economic environments may influence the financial dynamics.

In the past years the properties of the cross-correlation matrix of individual
stock prices have been analyzed, e.g., with the random matrix theory (RMT),
and much effort has been made to identify the business sectors by the
components in the eigenvectors of the cross-correlation matrix [10, 11, 12,
13, 14, 15, 16, 17]. In this paper, the analysis of the so-called spatial
structure is just an analysis about the cross-correlations between individual
stocks based on the RMT theory. After taking into account the signs of the
components in an eigenvector, a sector may be further separated into two
subsectors, i.e., the positive and negative subsectors [18]. A purpose of this
paper is to investigate the spatial structures of the four stock markets in
Greater China, and uncover characteristics of the sector and subsector
structures for each market.

The dynamic behavior of the stock prices has been studied for years, and
various results have been obtained. For example, the probability distribution
of the price return usually exhibits a power-law tail, the price volatility is
long-range correlated in time, while the price return itself is short-range
correlated [2, 19, 20, 21]. To better understand the dynamic behavior of the
stock prices, one may consider a higher-order time correlation, i.e., the
return-volatility correlation [5, 6, 14, 22]. A negative return-volatility
correlation, which is called the leverage effect, was first discovered by
Black in 1976 [23, 24]. The leverage effect is observed in most of the stock
markets in the world, while a positive return-volatility correlation, which is
called the anti-leverage effect, was detected in the stock markets of mainland
China [1, 4, 6]. The leverage and anti-leverage effects are crucial for the
understanding of the price dynamics [5, 6, 22]. In this study, we analyze the
return-volatility correlation function of the four corresponding stock-market
indices, i.e., the Shanghai Composite Index, Shenzhen Composite Index, Taiwan
Weighted Index and Hang Seng Index.

The analysis of the recurrence interval may deepen the understanding of the
dynamic behavior in financial markets [21, 25]. Recently, statistical
properties of the recurrence intervals of volume volatilities and price
volatilities have been studied [26, 27, 28, 29]. We present a comparative
study on the recurrence interval distributions of the four stock markets. For
each market, we analyze the recurrence interval distributions for both the
trading volume volatilities and the price volatilities.

The paper is organized as follows. In section 2, we investigate the sector and
subsector structures. In section 3, we analyze the return-volatility
correlation function and the distributions of the recurrence intervals for
both volume volatilities and price volatilities. In section 4, we present the
conclusion.

##  2 Sector and subsector structures

We define the logarithmic price return of the $i$-th stock over a time
interval $\Delta t$ as

$R_{i}(t^{\prime},\triangle t)\equiv ln\,P_{i}(t^{\prime}+\triangle
t)-ln\,P_{i}(t^{\prime}),$ (1)

where $P_{i}(t^{\prime})$ represents the close price at time $t^{\prime}$, and
we set $\Delta t$ to be one day. To ensure that the results are independent of
the fluctuation scales of the stock prices, we introduce the normalized return
of the $i$ -th stock

$r_{i}(t^{\prime},\triangle t)=\frac{R_{i}-\langle R_{i}\rangle}{\sigma_{i}},$
(2)

where $\langle\cdots\rangle$ represents the time average over time
$t^{\prime}$ and the standard deviation of $R_{i}$ is denoted by
$\sigma_{i}=\sqrt{\langle R_{i}^{2}\rangle-\langle R_{i}\rangle^{2}}$ [30].
Then, the elements of the equal-time cross-correlation matrix $C$ are defined
by

$C_{ij}\equiv\langle r_{i}(t^{\prime})r_{j}(t^{\prime})\rangle,$ (3)

which measure the correlations between the returns of individual stocks.
According to the definition, $C$ is a real symmetric matrix with $C_{ii}=1.$
The value of $C_{ij}$ ranges from $-1$ to $1$.

In this paper, we compute the cross-correlation matrix $C$ with the daily
stock prices of the four stock markets in Greater China. The time periods of
259 stocks for each stock market are shown in the first column of Table 1. Why
do we choose 259 stocks for each stock market? On the one hand, we should use
as many stocks as we can. On the other hand, the available data of the stocks
should be as long as possible. Under these conditions we obtain 259 stocks for
the Shanghai stock market. For comparison, we also use this number 259 for the
other three markets. The probability distributions $P(C_{ij})$ of the four
stock markets are displayed in Fig. 1. The average value of $C_{ij}$ is close
to 0.37 for both the Shanghai and Shenzhen markets, larger than 0.26 for the
Taiwan market, and much larger than 0.11 for the Hongkong market. Previous
studies show that the average value of $C_{ij}$ of a mature market is smaller
than that of an emerging market [31, 32]. Our results indicate that the
Hongkong stock market is a mature one, the Shanghai and Shenzhen stock markets
are emerging, and the Taiwan stock market is in between.

We then compute the eigenvalues of the cross-correlation matrix $C$, and
compare it with the so-called Wishart matrix [33, 34]. The Wishart matrix is
derived from non-correlated time series. Assuming $N$ time series with length
$T$, statistical properties of such random matrices are known. In the limit
$N$$\rightarrow\infty$ and $T$$\rightarrow\infty$ with $Q\equiv T/N$$\geqslant
1$, the probability distribution $P_{rm}(\lambda)$ of the eigenvalue $\lambda$
can be given by [33, 34]

$P_{rm}(\lambda)=\frac{Q}{2\pi}\frac{\sqrt{(\lambda_{max}^{ran}-\lambda)(\lambda-\lambda_{min}^{ran})}}{\lambda},$
(4)

the lower and upper bounds are

$\lambda_{min(max)}^{ran}=\left[1\pm(1/\sqrt{Q})\right]^{2},$ (5)

where $\lambda_{min}^{ran}\leq\lambda\leq\lambda_{max}^{ran}$.

For a dynamic system, large eigenvalues deviate from $P_{rm}(\lambda)$,
implying that there exist non-random interactions. In fact, both mature and
emerging stock markets show the same phenomenon that the bulk of the
eigenvalue spectrum $P(\lambda)$ of the cross-correlation matrix is similar to
$P_{rm}(\lambda)$ of the Wishart matrix [12, 14, 16, 35], but some large
eigenvalues deviate greatly from the upper bound $\lambda_{max}^{ran}$. The
probability distributions $P(\lambda)$ of the Taiwan and Hongkong stock
markets are shown in Fig. 2. $P(\lambda)$ of the Shanghai and Shenzhen markets
are similar, therefore are not shown in the figure. The inset displays
$\lambda_{max}$ for each stock market.

For our datasets, N is equal to 259 for all four stock markets, T is about
2000 days. As shown in Table 1, $\lambda_{max}^{ran}$ takes similar values for
the four markets, according to Eq.(5). $\lambda_{max}$ is rather stable for
sufficiently large T. The dependence of $\lambda_{max}$ on T is already stable
for T around 2000. Therefore, the comparison of $\lambda_{max}$ is meaningful.
$\lambda_{max}$ for the Shanghai and Shenzhen markets are about 98.0, while
the ones for the Taiwan and Hongkong markets are 72.3 and 35.5 respectively.
It is known that $\lambda_{max}$ for the US stock market is about 40.0 [16].
$\lambda_{max}$ for the Hongkong market is close to the one for the US market,
while $\lambda_{max}$ for the Shanghai, Shenzhen and Taiwan stock markets are
much larger. It indicates that the Hongkong stock market is mature, similar to
the western stock markets, while the Shanghai and Shenzhen stock markets are
emerging. The Taiwan stock market is in between.

The eigenvector of a large eigenvalue is dominated by a group of stocks,
usually associated with a business sector [16, 36, 37, 38]. Let
$u_{i}(\lambda_{\alpha})$ denotes the component of the $i$-th stock in the
eigenvector of $\lambda_{\alpha}$. In order to identify the sector structure,
we introduce a threshold $u_{c}$: $|u_{i}(\lambda_{\alpha})|\geq u_{c}$, to
select the dominating components in a particular eigenvector. The threshold
$u_{c}$ should be properly determined. Firstly, for a random matrix,
$\langle|u(\lambda)|\rangle$$\sim 1/\sqrt{N}$ for every eigenmode, where $N$
is the number of stocks. Therefore, the threshold should be larger than this
value. Secondly, $u_{c}$ should not be too large, because for a large
threshold there are not so many stocks in each sector [18]. We choose
different thresholds to test the stability of the results, such as
$u_{c}=0.06,0.08,0.10,0.12$. In fact, the results are stable when changing the
value of the threshold. We just show the typical ones in our paper. After
taking into account the signs of the components in an eigenvector, we may
separate a sector into two subsectors by introducing two thresholds
$u_{c}^{\pm}$$=\pm u_{c}$: $u_{i}$$(\lambda_{\alpha})\geq u_{c}^{+}$ and
$u_{i}$$(\lambda_{\alpha})$$\leq u_{c}^{-}$, which correspond to the positive
and negative subsectors respectively [18].

With the above method, we identify the sector and subsector structures up to
the ninth largest eigenvalue $\lambda_{8}$. The largest eigenvalue
$\lambda_{0}$ represents the market mode, which is driven by interactions
common for stocks in the entire market [18]. Therefore the market mode does
not correspond to a sector. If the financial situations of a company in the
Shanghai market are abnormal, the company will be treated specially, and a
prefix of the acronym “ST” will be added to the stock ticker. The abnormal
financial situations include: the audited profits are negative in two
successive accounting years, the audited net worth per share is less than its
stock’s par value in the recent accounting year. The acronym “ST” will be
removed when the financial situations become normal [16]. The positive
components in the eigenvector of the second largest eigenvalue $\lambda_{1}$
for the Shanghai market are dominated by the ST stocks. Therefore, we define
this subsector as the ST subsector. Since more than half of the stocks in the
negative components in the eigenvector of $\lambda_{1}$ belong to industrial
goods, we call it the Industrial goods subsector. The positive and negative
subsectors of $\lambda_{2}$ are identified to be Real estate and Health care
subsectors. The detected subsectors of other eigenvalues are displayed in
Table 2, while “Null” represents the one we could not identify. The subsectors
for the Shenzhen, Taiwan and Hongkong stock markets are shown in Table 3. From
Table 2 and 3, we can see that a particular subsector may appear several times
for a stock market, and the one which shows up the most often plays a
dominating role in this market. For the Shanghai stock market, there are
fourteen identified subsectors, and the dominating ones are Basic materials
and Industrial goods. For the Shenzhen stock market, the dominating ones are
the Energy and Real estate subsectors.

For the Taiwan stock market, the negative components in the eigenvector of
$\lambda_{1}$ are dominated by the electronic industry stocks which defines
the Electronic industry subsector, while the positive ones remain unknown. The
negative components of $\lambda_{3}$ and the positive ones of $\lambda_{4}$
are identified as the Steel industry subsectors. The dominating subsectors of
this market are the Chemical industry and Daily consumer goods.

For the Hongkong stock market, the finance and real estate stocks are strongly
correlated with each other. The negative components of $\lambda_{3}$ are
dominated by the finance and real estate stocks which defines the Real estate
& Finance subsector. The positive components of $\lambda_{3}$ can be
identified as the Service subsector. The dominating subsector of the Hongkong
stock market is the Real estate & Finance.

To better understand the above results, let us look at the regional economies
of the four stock markets. Shanghai and Shenzhen are both in mainland China
where the business such as basic material, energy and industry plays a vital
role [39, 40, 41]. The real estate attracts much attention in Shenzhen, and it
also plays an important role in the economy of mainland China [42]. The
economy of Taiwan is forefront in Asia. In Taiwan, there are various types of
industries, and the dominating businesses are steel, machinery, computer and
electronics, textile and clothing [43]. Hongkong is changing from a port city
into an industrial one. In Hongkong, the public service is very important, the
businesses such as real estate, finance, commerce, tourism and some other
traditional industries are dominating [44]. Our results of the sector and
subsector structure in Tables 2 and 3 well reflect the features of the
regional economies.

Now we investigate the anti-correlation between the two subsectors for each
eigenmode. The cross-correlation between two stocks can be decomposed into
different eigenmodes,

$C_{ij}=\sum_{\alpha=1}^{N}\lambda_{\alpha}C_{ij}^{\alpha},\quad
C_{ij}^{\alpha}=u_{i}^{\alpha}u_{j}^{\alpha},$ (6)

where $\lambda_{\alpha}$ is the $\alpha$-th eigenvalue, $u_{i}^{\alpha}$ is
the $i$-th component in the eigenvector of $\lambda_{\alpha}$, and
$C_{ij}^{\alpha}$ represents the cross-correlation in the $\alpha$-th
eigenmode. The eigenvalue $\lambda_{\alpha}$ is always positive, and it gives
the weight to the $\alpha$-th eigenmode. According to Eq.(6),
$C_{ij}^{\alpha}$ is positive when the components $u_{i}^{\alpha}$ and
$u_{j}^{\alpha}$ have a same sign. Otherwise, it is negative. When
$C_{ij}^{\alpha}$ is negative, two stocks are referred to be anti-correlated
in this eigenmode, indicating that when the price of the $i$-th stock rises,
the price of the $j$-th stock tends to fall, and vice versa [18]. Therefore,
any two stocks in a particular subsector are positively correlated in this
eigenmode, while those in different subsectors are anti-correlated.

To measure the anti-correlation between the positive and negative subsectors
quantitatively, we construct the combinations of stock price returns in the
two subsectors, $I_{\alpha}^{\pm}(t)=\sum_{i}u_{i}^{\pm}(\alpha)r_{i}(t),$ and
compute the cross-correlation between $I_{\alpha}^{+}(t)$ and
$I_{\alpha}^{-}(t)$ as

$C_{+-}(\alpha)=\langle I_{\alpha}^{+}(t)I_{\alpha}^{-}(t)\rangle,$ (7)

where $u_{i}^{\pm}(\alpha)$ is the $i$-th positive or negative component in
the $\alpha$-th eigenmode selected by a threshold [18]. $u_{c}$ could be
different from this market to another market since different markets may have
different fluctuation scales. We choose $u_{c}^{\pm}=\pm 0.06$ for the Taiwan
stock market, and $u_{c}^{\pm}=\pm 0.08$ for the Shanghai, Shenzhen and
Hongkong stock markets. The cross-correlation $C_{+-}(\alpha)$, in comparison
with that between two random combinations are shown in Fig. 3. For each
market, $C_{+-}(\alpha)$ increases slowly with increasing $\alpha$, and
gradually approaches the curve calculated from two random combinations of
stock price returns.

We denote the cross-correlation $C_{+-}(\alpha)$ computed with
$I_{\alpha}^{+}(t)$ and $I_{\alpha}^{-}(t)$ as $C_{real}$, and the one
computed with two random combinations of stock price returns as $C_{rand}$.
The relative difference $D$ between $C_{real}$ and $C_{rand}$ can be defined
as

$D=(C_{rand}-C_{real})/(C_{rand}+C_{real}).$ (8)

The results for the four stock markets are shown in Fig. 4. A positive value
of $D$ indicates the existence of the anti-correlation between the positive
and negative subsectors. For the Shanghai, Shenzhen and Taiwan markets, $D$
drops to zero for $\alpha\approx 20$. While for the Hongkong market, the value
of $D$ is larger, and it remains non-zero up to $\alpha\approx 40$, which
implies that the anti-correlation between positive and negative subsectors of
the Hongkong stock market is much stronger.

Why would there be the subsector structure with the anti-correlation between
the positive and negative subsectors? For the Shanghai and Shenzhen stock
markets, the positive and negative subsectors of $\lambda_{2}$ are Real estate
and Health care respectively. It is well known that the real estate in
mainland China fluctuates much with the macroeconomy. When the economy is
prosperous, the real estate will also be booming, while the health care will
be less dependent on the economy. For the Taiwan market, we take the two
subsectors of $\lambda_{3}$ as an example. As described in Ref. [18], from the
intrinsic properties, the steel industry is classified as the strongly
cyclical industry, while the daily consumer goods is classified as the weakly
cyclical industry. The weakly and strongly cyclical industries are weakly and
strongly correlated with the macro-economy environment, respectively. Thus,
the strongly cyclical industry is preferred when the macro-economy is booming.
However, the investors would rather choose the weakly cyclical industry when
the macro-economy declines. The result in Ref. [45] also shows that the energy
such as the steel industry fluctuates much with the economy. For the Hongkong
market, the two subsectors of $\lambda_{3}$ are Real estate & Finance and
Service. As we have described above, the real estate & finance fluctuate with
the economy, while the service will remain relatively stable.

##  3 Dynamic properties

We first consider the return-volatility correlation function defined as

$L(t)=[\langle r(t^{\prime})|r(t^{\prime}+t)|^{2}\rangle-L_{0}]/Z,$ (9)

with $Z=\langle|r(t^{\prime})|^{2}\rangle^{2}$ and $L_{0}=\langle
r(t^{\prime})\rangle\langle|r(t^{\prime})|^{2}\rangle$. For $t>0$, $L(t)$
describes how the past returns affect the future volatilities. It was first
observed by Black that the past negative returns increase future volatilities,
i.e., the return-volatility correlation is negative [23, 24], and this
phenomenon is called the leverage effect. The leverage effect is observed in
most of the stock markets in the world. However, a positive return-volatility
correlation, which is now called the anti-leverage effect, is observed in the
stock markets in mainland China, and this is the discovery a few years ago [6,
22, 46].

We compute the return-volatility correlation for the four stock markets in
Greater China, with the daily price returns of each stock-market index. The
results are displayed in Fig. 5 and Fig. 6. $L(t)$ of the Hongkong stock
market shows negative values up to at least 20 days. This is the well-known
leverage effect. Similar to the Hongkong stock market, the Taiwan stock market
also exhibits a leverage effect. Qualitatively, $L(t)$ behaves the same for
the Shanghai and Shenzhen stock markets, therefore we take the average over
the two stock-market indices. The removing of non-tradable shares was proposed
around the year 2000. It is a significant event in the stock markets of
mainland China. Therefore, we divide the whole period into two sub-periods by
before and after the year 2000. We observe that before the year 2000, $L(t)$
shows positive values up to 10 days, i.e., an anti-leverage effect. After
2000, however, $L(t)$ displays negative values up to 20 days, i.e., a leverage
effect. This crossover behavior from the anti-leverage to the leverage effect
is a new observation for the Shanghai and Shenzhen stock markets. It seems
that the two stock markets may be on the way gradually approaching a mature
one.

One important reason for the crossover from the anti-leverage effect to the
leverage effect in the Shanghai and Shenzhen stock markets may be the removing
of non-tradable shares around the year 2000. In early years, the stocks were
divided into tradable shares and non-tradable shares. After that, all shares
are tradable, which makes the two stock markets more regulated and mature.

On the other hand, the trading volume is an important variable which reflects
the liquidity of the financial markets. To analyze the dynamic behavior of the
trading volume, we introduce two basic measures: the volume return $R_{v}$ and
the volume volatility $\nu_{v}$, following the Refs. [7, 21, 25]. The $R_{v}$
is defined as the logarithmic change in the successive daily trading

$R_{v}(t)\equiv ln[V(t)/V(t-1)],$ (10)

where $V(t)$ is the daily trading volume at time $t$. The normalized return is

$r_{v}(t)=\frac{R_{v}(t)-\langle R_{v}\rangle}{\sqrt{\langle
R_{v}^{2}\rangle-\langle R_{v}\rangle^{2}}},$ (11)

the volatility is the absolute value of return: $\nu_{v}=|r_{v}|$.

From a volume volatility time series, we may extract the recurrence intervals
$\tau$ between consecutive volatilities above a threshold $q$, and construct a
series of the recurrence intervals $\left\\{\tau(q)\right\\}$ [26, 27, 28,
29]. The threshold $q$ is measured in unit of the standard deviation of the
volume volatility. It can not be too large, otherwise we could not have
sufficient data points. Let us denote the probability distribution function of
$\left\\{\tau(q)\right\\}$ as $P_{q}(\tau)$. The dependence of $P_{q}(\tau)$
on $q$ for the Shanghai stock market is shown in Fig. 7. After the probability
of the recurrence intervals $P_{q}(\tau)$ being scaled with the mean interval
$\langle\tau(q)\rangle$, all the curves for different thresholds collapse onto
a single curve, and it suggests that $P_{q}(\tau)$ obeys a scaling function

$P_{q}(\tau)=\frac{1}{\langle\tau\rangle}f(\tau/\langle\tau\rangle).$ (12)

As the threshold $q$ increases, the curve tends to be truncated due to the
limited size of the data set [21]. However, the tails of the scaling function
can be approximated by a power law

$f(\tau/\langle\tau\rangle)\sim(\tau/\langle\tau\rangle)^{-\gamma},$ (13)

which is displayed by the dashed line in the figure, and $\gamma$ is the tail
exponent.

We also investigate the statistical properties of the recurrence intervals
constructed from the price volatility time series. The results for the
Shanghai stock market are shown in Fig. 7. The distributions diverge slightly
for small intervals, but exhibit scaling behaviors for large intervals. The
method proposed in Ref. [47] is widely used to test the power-law fit [21, 25,
26, 27, 28, 29]. With this method, the power-law behavior of recurrence
interval distributions for both price volatilities and volume volatilities
pass the Kolmogorov-Smirnov test. In other words, our results support the ones
in Ref. [21] that the recurrence interval distribution of the price
volatilities may follow the power-law behavior. The reason for the difference
between our results and those in some previous papers [25, 26, 27, 28, 29] may
be due to that our results are from the average over 259 stocks.

The values of the power law exponent $\gamma$ are shown in the last two
columns of Table 1. The exponents for the volume volatilities range from $3.7$
to $4.7$. $\gamma=4.7$ for the Taiwan market is the largest one, while,
$\gamma=3.7$ for the Hongkong market is the smallest one. The exponents for
the Shanghai and Shenzhen markets are $4.2$ and $4.3$ respectively. However,
the exponents for the price volatilities of the four stock markets are about
$3.0$. The results indicate that the dynamic behavior of large price
volatilities is rather robust, while that of large volume volatilities is not.

##  4 Conclusion

In the RMT analysis, after taking into account the signs of the components in
an eigenvector of the cross-correlation matrix, one detects that a sector may
split into two subsectors, which are anti-correlated with each other in the
corresponding eigenmode. For the four stock markets in greater China, the
sector and subsector structures exhibit different characteristics. The
Shanghai and Shenzhen markets are dominated by the Basic materials and
Industrial goods subsectors. For the Taiwan market, the dominating subsectors
are Electronic industry and Chemical industry, while those for the Hongkong
market are Real estate & Finance and Service. All these results reflect the
features of the regional economies.

Meanwhile, we analyze the return-volatility correlation function. The Hongkong
and Taiwan markets show a leverage effect. However, the Shanghai and Shenzhen
markets are more complicated. The two markets exhibit a strong anti-leverage
effect before 2000, while it gradually changes to the leverage effect after
2000. This is a new observation for the stock markets in mainland China,
additional to the discovery of the anti-leverage effect in Ref. [22]. We also
study the recurrence interval distributions, and find that the power law
exponents for the volume volatilities range from $3.0$ to $5.0$ for the four
markets, while those for the price volatilities are about $3.0$.

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| Time period | $T$ | $\lambda_{min}^{ran}$ | $\lambda_{max}^{ran}$ | $\lambda_{min}^{real}$ | $\lambda_{max}^{real}$ | $\gamma_{p}$ | $\gamma_{v}$
---|---|---|---|---|---|---|---|---
SH | 2003.1-2011.7 | 2067 | 0.42 | 1.83 | 0.02 | 98.0 | 3.0 | 4.2
SZ | 2003.1-2011.4 | 2000 | 0.41 | 1.85 | 0.12 | 98.0 | 3.1 | 4.3
TW | 2003.1-2011.11 | 2206 | 0.43 | 1.80 | 0.14 | 72.3 | 3.2 | 4.7
HK | 2003.1-2011.9 | 2146 | 0.43 | 1.82 | 0.12 | 35.5 | 3.2 | 3.7
Table 1: The second column shows the time periods of 259 stocks for the
Shanghai (SH), Shenzhen (SZ), Taiwan (TW) and Hongkong (HK) stock markets. $T$
is the total number of the daily data. $\lambda_{min(max)}^{ran}$ represents
the low (up) bound of the eigenvalues of the Wishart matrix, while
$\lambda_{min(max)}^{real}$ is that of the real cross-correlation matrix.
$\gamma_{p}$ is the power law exponent for the price volatility, and
$\gamma_{v}$ is the one for the volume volatility.

$\lambda_{i}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{4}$
---|---|---|---|---
Signs | $+$ | $-$ | $+$ | $-$ | $+$ | $-$ | $+$ | $-$
Sector | ST | IG | RE | Heal | Null | IG | BM | Ener
$u_{c}$$=\pm 0.08$ | $\nicefrac{{23}}{{33}}$ | $\nicefrac{{11}}{{18}}$ | $\nicefrac{{19}}{{19}}$ | $\nicefrac{{10}}{{19}}$ | $34$ | $\nicefrac{{11}}{{17}}$ | $\nicefrac{{16}}{{26}}$ | $\nicefrac{{17}}{{28}}$
$u_{c}$$=\pm 0.10$ | $\nicefrac{{16}}{{25}}$ | $\nicefrac{{6}}{{6}}$ | $\nicefrac{{17}}{{17}}$ | $\nicefrac{{4}}{{7}}$ | $18$ | $\nicefrac{{5}}{{7}}$ | $\nicefrac{{7}}{{11}}$ | $\nicefrac{{16}}{{17}}$

$\lambda_{i}$ | $\lambda_{5}$ | $\lambda_{6}$ | $\lambda_{7}$ | $\lambda_{8}$
---|---|---|---|---
Signs | $+$ | $-$ | $+$ | $-$ | $+$ | $-$ | $+$ | $-$
Sector | EI | BM | BM | RE | BM | DG | IG | Null
$u_{c}=\pm 0.08$ | $\nicefrac{{12}}{{23}}$ | $\nicefrac{{11}}{{26}}$ | $\nicefrac{{17}}{{32}}$ | $\nicefrac{{11}}{{23}}$ | $\nicefrac{{11}}{{22}}$ | $\nicefrac{{12}}{{24}}$ | $\nicefrac{{11}}{{22}}$ | $27$
$u_{c}=\pm 0.10$ | $\nicefrac{{9}}{{15}}$ | $\nicefrac{{8}}{{20}}$ | $\nicefrac{{6}}{{11}}$ | $\nicefrac{{5}}{{10}}$ | $\nicefrac{{9}}{{16}}$ | $\nicefrac{{7}}{{9}}$ | $\nicefrac{{7}}{{15}}$ | $12$

Table 2: The subsectors for the SH stock market. The fraction is the number of well-identified stocks over the total number of stocks in the subsector. The abbreviations of the business subsectors are as follows. ST: Specially Treated stocks; RE: Real estate; BM: Basic materials; Ener: Energy; Heal: Health care; DG: Daily consumer goods; IG: Industrial goods; EI: Electronic industry; Null: No obvious category. Those non-fractions are the total counts of stocks in the corresponding subsectors. |  | SZ | TW | HK
---|---|---|---|---
$\lambda_{1}$ | + | Ener | Null | Null
| - | IG | EI | RE&Fin
$\lambda_{2}$ | + | RE | RE | IG
| - | Heal | CI | RE&Fin
$\lambda_{3}$ | + | RE | DG | Serv
| - | Ener | SI | RE&Fin
$\lambda_{4}$ | + | Null | SI | Null
| - | RE | Null | RE&Fin
$\lambda_{5}$ | + | RE | DG | RE&Fin
| - | DG | CI | Null
$\lambda_{6}$ | + | Null | CI | RE&Fin
| - | Ener | EI | Null
$\lambda_{7}$ | + | Ener | DG | Null
| - | Null | CI | RE&Fin
$\lambda_{8}$ | + | EI | CI | Null
| - | DG | DG | RE&Fin
Table 3: The subsectors for the SZ, TW and HK stock markets. CI: Chemical
industry; SI: Steel industry; Serv: Service; RE&Fin: Real estate & Finance.
The other abbreviations can be seen in the caption of Table 2. Figure 1: The
probability distributions of $C_{ij}$ for the SH, SZ, TW and HK stock markets.
The abbreviations of the stock markets are introduced in the caption of Table
1.

Figure 2: The probability distributions of the eigenvalues of the correlation
matrix $C$ for the TW and HK stock markets. The inset shows the largest
eigenvalue for each market. $\lambda_{max}$ for the SH and SZ stock markets
are about 98.0, while those for the TW and HK stock markets are 72.3 and 35.5
respectively. Figure 3: $C_{+-}(\alpha)$ for the SH and TW stock markets
compared with that between two random combinations of stock price returns.
Error bars are given to the random curves. Figure 4: The quantity $D$ for each
stock market.

Figure 5: The return-volatility correlation of the Hang Seng Index (HK) from
the year 1990 to 2011, and that of the Taiwan Weighted Index (TW) from 1997 to
2011. The dashed line shows an exponential fit. Figure 6: $L(t)$ for the
Shanghai Composite Index (SH) and Shenzhen Composite Index (SZ) in two time
periods. Dashed lines show the exponential fits. Figure 7: Probability
distributions of the scaled recurrence intervals of the volume and price
volatilities for different thresholds in the SH stock market. $\gamma$ is the
power law exponent.


#  $\eta$-pairing ground states in the non-Hermitian Hubbard model

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

The introduction of non-Hermiticity has greatly enriched the research field of
traditional condensed matter physics, and eventually led to a series of
discoveries of exotic phenomena. We investigate the effect of non-Hermitian
imaginary hoppings on the attractive Hubbard model. The exact bound-pair
solution shows that the electron-electron correlation suppresses the non-
Hermiticity, resulting in off-diagonal long-range order (ODLRO) ground state.
In a large negative $U$ limit, the ODLRO ground state corresponds to
$\eta$-spin ferromagnetic states. We also study the system with mixed hopping
configuration. The numerical result indicates the existence of the transition
from normal to $\eta$-pairing ground states by increasing the imaginary
hopping strength. Our results provide a promising approach for the non-
Hermitian strongly correlated system.

##  I Introduction

Non-Hermitian systems that can only be described by non-Hermitian Hamiltonians
are ubiquitous in nature. Many open systems, which are not fully isolated from
the rest of world, belong to this class. Comparing to the Hermitian systems,
the probability of the non-Hermitian system effectively becomes nonconserving
due to the exchange of energy, particles, and information with external
degrees of freedom that are out of the Hilbert space. Mainly driven by
experimental progress in atomic physics Bloch et al. (2008); D’Alessio et al.
(2016), the last two decades have witnessed remarkable developments in studies
of out-of-equilibrium dynamics in isolated quantum many-body systems. It has
become possible to study open many-body physics in a highly controlled manner
Syassen et al. (2008); Barontini et al. (2013); Patil et al. (2015); Rauer et
al. (2016); Lüschen et al. (2017); Li et al. (2019); Corman et al. (2019).
Within this burgeoning field, the treasure hunt is sprouting into fascinating
new directions ranging from non-Hermitian extensions of Kondo effect Nakagawa
et al. (2018); Lourenço et al. (2018), many-body localization Hamazaki et al.
(2019), to fermionic superfluidity Yamamoto et al. (2019); Okuma and Sato
(2019).

Recent advances in quantum simulations of the Hubbard model with ultracold
atoms have offered a multifunctional platform to unveil low-temperature
properties of the strongly correlated system Bakr et al. (2009); Parsons et
al. (2015); Cheuk et al. (2015, 2016); Parsons et al. (2016); Esslinger
(2020). A series of cornerstone works have reshaped our understanding of the
dissipative strongly correlated system Syassen et al. (2008); Fausti et al.
(2011); Hu et al. (2014); Zhu et al. (2014); Kaiser et al. (2014); Mitrano et
al. (2016); Tomita et al. (2017); Sponselee et al. (2018); Cantaluppi et al.
(2018); Sato et al. (2019); McIver et al. (2020); Booker et al. (2020);
Tindall et al. (2020); Zhang et al. (2020). One of the most tantalizing
findings is the possible superconductivity in which $\eta$-pairing state plays
a vital role. This stimulates a plethora of non-equilibrium protocols
including photodoping schemes Iwai et al. (2003); Rosch et al. (2008);
Sensarma et al. (2010); Eckstein and Werner (2011); Ichikawa et al. (2011);
Lenarčič and Prelovšek (2013); Stojchevska et al. (2014); Mitrano et al.
(2014); Werner et al. (2019a); Peronaci et al. (2020); Li et al. (2020) and
dissipation-induced schemes Diehl et al. (2008); Kraus et al. (2008);
Coulthard et al. (2017); Werner et al. (2019b); Zhang and Song (2020) to
selectively generate such superconducting-like states. However, few people
discuss the impact of non-Hermiticity on the low-lying energy spectrum and
quantum magnetism of the strongly correlated system from the level of non-
Hermitian quantum mechanics Nakagawa et al. (2020).

It is the aim of this paper to investigate the effect of non-Hermiticity on
the strongly correlated system in the context of the non-Hermitian quantum
mechanics. We show that the non-Hermitian imaginary hopping can indeed induce
a robust $\eta$-pairing ground state for a wide range of parameters $U$
(particle-particle interaction) and $t$ (hopping strength), by considering the
bipartite non-Hermitian Hubbard system. An exact solution of the bound pair is
employed to elucidate the underlying paring mechanism and pave the way to
extend the results to dilute gas. In physics, the particle-particle
interaction suppresses the non-Hermiticity leading to the off-diagonal long-
range order (ODLRO) ground state with real energy, and the non-Hermitian
imaginary hopping, in turn, suppresses the antiferromagnetic correlation, thus
ensuring the system has a $\eta$-spin ferromagnetic ground state. Numerical
results of 1D and 2D systems with corrugation patterns indicate that such
property is insensitive to the disorder and the strength of the interaction
even though the on-site interaction breaks the $SO(4)$ symmetry, which
suggests a promising scheme in a real experiment. We further demonstrate that
there can exist a transition from normal to $\eta$-pairing ground state
associating with the sudden change of the doublon-doublon correlation.

The paper is organized as follows. Sec. II discusses the non-Hermitian Hubbard
model, the non-Hermiticity of which originates from the imaginary hopping.
Sec. III introduces the exactly two-particle solution providing the mechanism
of the formation of the $\eta$-pairing ground state. Sec. IV gives the
effective magnetic Hamiltonian under the large negative $U$ limit. Sec. V
shows the numerical results and the analytical understanding of
superconductive $\eta$-pairing ground state. Sec. VI demonstrates the
transition from a normal to superconductive ground state. Sec. VII concludes
this paper. Some details of our calculations are placed in the Appendixes.

##  II Model

We consider a non-Hermitian Hubbard model on a bipartite lattice

$\displaystyle H$ $\displaystyle=$ $\displaystyle
i\sum_{j,l}\sum_{\sigma=\uparrow,\downarrow}t_{jl}(c_{j,\sigma}^{\dagger}c_{l,\sigma}+c_{l,\sigma}^{\dagger}c_{j,\sigma})$
(1) $\displaystyle+U\sum_{j}n_{j,\uparrow}n_{j,\downarrow},$

with the following notation: the operator $c_{j,\sigma}$
($c_{j,\sigma}^{\dagger}$) is the usual annihilation (creation) operator of a
fermion with spin $\sigma\in\left\\{\uparrow,\downarrow\right\\}$ at site $j$,
and $n_{j,\sigma}=c_{j,\sigma}^{\dagger}c_{j,\sigma}$ is the number operator
for a particle of spin $\sigma$ on site $j$; the symbol $i=\sqrt{-1}$
represents an imaginary number; $U$ and $t_{jl}$ are required to be real and
play the role of interaction and kinetic energy scales, respectively; the
system can be divided into two sublattices $A$ and $B$ such that $t_{jl}=0$
whenever $j\in\left\\{A\right\\}$ and $l\in\left\\{A\right\\}$ or
$j\in\left\\{B\right\\}$ and $l\in\left\\{B\right\\}$. The non-Hermiticity of
$H$ stems from the imaginary hopping $it_{jl}$ that can be realized by the
judicious design of the loss and the magnetic flux Li et al. (2015, 2017)
which are within the reach of cold atom experiments Lee and Chan (2014);
Schreiber et al. (2015). Notice that such non-Hermiticity is distinct from the
complex particle-particle interaction adopted to describe the inelastic
collision of two particles Zhang et al. (2017); Nakagawa et al. (2020). When
the uniform Hermitian hopping is taken, the Hamiltonian can feature a Mott
insulating ground state with a strong antiferromagnetic correlation that is
generically nonsuperconducting. Evidently, the imaginary hopping inevitably
competes with the interaction leading to the unique properties of the
considered system. It can be expected that the introducing of such non-
Hermiticity will significantly alter the magnetic correlation of the system.

In this paper, we focus on whether the system can favor the ground state with
superconductivity in this non-Hermitian setting. To gain physical insight into
this system, we first investigate the symmetry of the considered model. It has
two sets of commuting $SU(2)$ symmetries. The first is the spin symmetry
characterized by the generators

$\displaystyle s^{+}$ $\displaystyle=$
$\displaystyle\left(s^{-}\right)^{\dagger}=\sum_{j}s_{j}^{+},$ (2)
$\displaystyle s^{z}$ $\displaystyle=$ $\displaystyle\sum_{j}s_{j}^{z},$ (3)

where the local operators $s_{j}^{+}=c_{j,\uparrow}^{\dagger}c_{j,\downarrow}$
and $s_{j}^{z}=\left(n_{j,\uparrow}-n_{j,\downarrow}\right)/2$ obey the Lie
algebra, i.e., $[s_{j}^{+},$ $s_{j}^{-}]=2s_{j}^{z}$, and $[s_{j}^{z},$
$s_{j}^{\pm}]=\pm s_{j}^{\pm}$. Large values of the spin quantum number $s$
corresponds to ferromagnetism. The second often referred to as $\eta$ symmetry
has the generators

$\displaystyle\eta^{+}$ $\displaystyle=$
$\displaystyle\left(\eta^{-}\right)^{\dagger}=\sum_{j}\eta_{j}^{+},$ (4)
$\displaystyle\eta^{z}$ $\displaystyle=$ $\displaystyle\sum_{j}\eta_{j}^{z},$
(5)

with $\eta_{j}^{+}=\lambda c_{j,\uparrow}^{\dagger}c_{j,\downarrow}$ and
$\eta_{j}^{z}=\left(n_{j,\uparrow}+n_{j,\downarrow}-1\right)/2$ satisfying
commutation relation, i.e., $[\eta_{j}^{+},$ $\eta_{j}^{-}]=2\eta_{j}^{z}$,
and $[\eta_{j}^{z},$ $\eta_{j}^{\pm}]=\pm\eta_{j}^{\pm}$. Here we assume a
bipartite lattice and $\lambda=1$ for $j\in\left\\{A\right\\}$ and $-1$ for
$j\in\left\\{B\right\\}$. Notice that under a particle-hole transformation,
$c_{j,\downarrow}\rightarrow\lambda c_{j,\downarrow}^{\dagger}$, which maps
the attractive Hubbard model to a repulsive one in the parent Hermitian
Hamiltonian (1), the role of the two sets of $SU(2)$ generators is
interchanged. Straightforward algebra shows that

$\displaystyle\left[H,\eta^{\pm}\right]$ $\displaystyle=$ $\displaystyle\pm
U\eta^{\pm},$ (6) $\displaystyle\left[H,\eta^{z}\right]$ $\displaystyle=$
$\displaystyle 0,$ (7)

which indicates that one can construct many exact eigenstates
$H\left(\eta^{+}\right)^{N}\left|\mathrm{Vac}\right\rangle=NU\left(\eta^{+}\right)^{N}\left|\mathrm{Vac}\right\rangle$
with $\left|\mathrm{Vac}\right\rangle$ being the vacuum state of fermion
$c_{j,\sigma}$. Correspondingly, the large values of the $\eta$ quantum number
are related to a staggered ODLRO and superconductivity Yang (1962); Singh and
Scalettar (1991).

Figure 1: Comparison of the two-particle spectrum within the subspace $(0,0)$
between the non-Hermitian setting and its parent Hermitian system for (a)
$U=-0.8t$, (b) $U=-2t$, and (c) $U=-4.5t$, respectively. The upper and lower
panels present the spectrum of the Hermitian and non-Hermitian system,
respectively. The red circle and gray shading denote the bound pair and
scattering state. The parent Hermitian system can be obtained by assuming
$it\rightarrow t$. For the Hermitian system, the bound pair with the lowest
energy lies in the $K=0$ subspace while the ground state of two-particle non-
Hermitian setting locates on the subspace indexed by $K=\pi$. It is shown that
the presence of the imaginary hopping not only makes all scattering energy
bands imaginary but also reverses the whole bound band. In the condition of
small $U$, there can exist an EP characterized by the divergence of
$\partial\epsilon_{K}/\partial K$. Such non-Hermiticity alters significantly
the paring mechanism and hence favors superconductivity. Figure 2: Plots of
the overlap $\mathcal{F}$ and correlator $C_{j}$ as a function of the strength
of interaction disorder $b$ for (a) $t=1$, $a=0.1t$, $U=-0.5t$ (b) $t=1$,
$a=0.3t$, $U=-1.5t$ (c) $t=1$, $a=0$, $U=-4t$, and (d) $t=1$, $a=0.2t$,
$U=-4t$. The numerical simulation is performed for the $6$ site 1D Hubbard
model at half filling and $s^{z}=0$. Here the strength of the hopping disorder
$a$ is set to be constant for each subfigure and the correlator $C_{j}$ is
averaged over all sites separated by a distance $j$. When $b=0$, no matter
what value $a$ takes, as long as $U$ is non-zero, one can always get a perfect
$\eta$-pairing ground state. The variation of $C_{j}$ indicates that the
increase of $b$ will not result in the significant change of the
$\eta$-pairing ground state; the presence of the hopping disorder can suppress
the fluctuation of $C_{j}$ compared to the disorder free case, which can be
seen from (c)-(d). Therefore, the value of the correlator is the consequence
of the interplay between two such disorders, which provides a scheme to
prepare $\eta$-pairing ground state in the experiment.

##  III $\eta$-pairing state in two-particle subspace

Based on the symmetry of the system, we first elucidate the paring mechanism
through the exact solution within the two-particle subspace. Supposing that
the Hamiltonian (1) describes a 1D homogeneous ring system in which
$it_{jl}=it$. Owing to the translation symmetry, the basis of such invariant
subspace can be constructed as follow

$\displaystyle\left|\phi_{0}^{-}\left(K\right)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{j}e^{iKj}c_{j,\uparrow}^{\dagger}c_{j,\downarrow}^{\dagger}\left|\mathrm{Vac}\right\rangle,$
(8) $\displaystyle\left|\phi_{r}^{\pm}\left(K\right)\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2N}}e^{iKr/2}\sum_{j}e^{iKj}(c_{j,\uparrow}^{\dagger}c_{j+r,\downarrow}^{\dagger}$
(10) $\displaystyle\pm
c_{j,\downarrow}^{\dagger}c_{j+r,\uparrow}^{\dagger})\left|\mathrm{Vac}\right\rangle,$

and

$\frac{s^{\pm}}{\sqrt{2}}\left|\phi_{r}^{+}\left(K\right)\right\rangle=\frac{1}{\sqrt{N}}e^{iKr/2}\sum_{j}e^{iKj}c_{j,\pm\uparrow}^{\dagger}c_{j+r,\pm\uparrow}^{\dagger}\left|\mathrm{Vac}\right\rangle,$
(11)

where $N$ is an even number and $K=2n\pi/N$ is the momentum vector indexing
the subspace. $r$ represents the relative distance between the two particles.
These bases are eigenvectors of the operators $s^{2}$ and $s^{z}$, which
satisfies

$\displaystyle s^{2}\left|\phi_{r}^{-}\left(K\right)\right\rangle$
$\displaystyle=$ $\displaystyle 0,\text{ }$ (12) $\displaystyle
s^{z}\left|\phi_{r}^{-}\left(K\right)\right\rangle$ $\displaystyle=$
$\displaystyle 0,$ (13) $\displaystyle
s^{2}\left|\phi_{r}^{+}\left(K\right)\right\rangle$ $\displaystyle=$
$\displaystyle 2\left|\phi_{r}^{+}\left(K\right)\right\rangle,\text{ }$ (14)
$\displaystyle s^{z}\left|\phi_{r}^{+}\left(K\right)\right\rangle$
$\displaystyle=$ $\displaystyle\left|\phi_{r}^{+}\left(K\right)\right\rangle.$
(15)

Evidently, each subspace labeled by $K$ can be further decomposed into four
subspaces with $(s,$ $s^{z})=(0,$ $0)$, $(1,$ $0)$ and $(1,$ $\pm 1)$ in term
of spin symmetry. Aiding by the detailed calculation in the Appendix, the
bound pair emerges in the $(0,$ $0)$ subspace with eigen energy being
$\epsilon_{K}=\mathrm{sgn}\left(U\right)\sqrt{U^{2}+4\lambda_{K}^{2}}$ in
which $\lambda_{K}=2it\cos\left(K/2\right)$. The bound pair state is
$\left|\varphi_{K}^{\mathrm{b}}\right\rangle=\sum_{r}f_{K}^{-}\left(r\right)\left|\phi_{r}^{-}\left(K\right)\right\rangle$
with

$f_{K}^{-}\left(j\right)=\left\\{\begin{array}[]{c}1/\sqrt{2}\text{, }j=0\\\
e^{-\beta j}\text{, }j\neq 0\end{array}\right.,$ (16)

where $\beta=\ln[(-U\pm\sqrt{U^{2}+4\lambda_{K}^{2}})/2\lambda_{K}]$. Here
$\pm$ denotes negative and positive $U$, respectively. We concentrate on the
negative $U$ in the following unless stated otherwise. In the absence of on-
site interaction $U$, only the scattering eigenstate with imaginary
eigenenergy presents and the system does not accommodate the bound pair state.
The nonzero interaction $U$ leads to the emergence of the bound pair. When
$\left|U\right|>\left|4t\right|$, the system possesses the full real bound
pair spectrum. However, a small $U$ results in the appearance of the imaginary
bound pair energy. The corresponding eigenstate is in the form of an
oscillation damping wave rather than a monotonic damping wave of the Hermitian
parent system. Notice that if $\left|U\right|$ $\leqslant$ $\left|4t\right|$,
then an exceptional point (EP) $\left|U\right|=\left|2\lambda_{K_{c}}\right|$
presents, at which the coalescent eigenstate approaches to a unidirectional
plane wave with $\beta=0$ or $\pi$ corresponding to $K=0$ or $2\pi$. In this
sense, the non-Hermiticity of the system is suppressed through the pairing
mechanism. The emergence of real energy is the consequence of the competition
between the on-site interaction and imaginary hopping. Furthermore, the lowest
real eigenenergy appears in the $K=\pi$ subspace no matter whether the system
possesses the full real spectrum. The corresponding ground state is
$\eta$-pairing state with the form

$\left|\phi_{0}^{-}\left(K\right)\right\rangle=\left(\eta^{+}\right)/\sqrt{N}|\mathrm{Vac}\rangle,$
(17)

and thus it favors superconductivity. This is in stark difference from the
Hermitian system, i.e., $it\rightarrow t$. In that case, the ground state of
the two-particle system locates on the $K=0$ rather than $K=\pi$ subspace such
that the $\eta$-pairing state has the highest eigenenergy than the other bound
pair state. Fig. 1 shows the typical energy spectrum of subspace $(0,0)$. It
demonstrates that the imaginary hopping flips the bound pair spectrum of the
parent Hermitian spectrum so that $\eta$-pairing state becomes the ground
state of the system. It is worthy pointing out that if we consider the dilute
Fermi gas formed by many bound pairs in which the pair-pair interaction is
neglected, then the mechanism for a single bound pair can be extended to this
type of dilute gas.

Figure 3: Numerical simulation of 2D corrugation pattern for (a1) $8$ site
Hubbard model with $4$ filled particles, (b1) $9$ site Hubbard model with $6$
filled particles, and (c1) $13$ site Hubbard model with $2$ filled particles.
The different sizes of solid circles and different lengths of red edges denote
the disorder of the interaction strength $U_{j}$ and hopping strength $t_{j}$
of Eq. (23). (a2)-(c2) Plots of $\mathcal{F}$ and $C_{1}$ as function of
disorder strength $b$. The system parameters are (a) $t=1$, $a=0$, $U=-0.8t$
(b) $t=1$, $a=0.2t$, $U=-0.8t$ and (c) $t=1$, $a=0.2t$, $U=-0.5t$. Similar
with 1D system, the induced fluctuation of $C_{1}$ is minor in comparison with
disorder free case ($b=0$) indicating that the existing result of 1D can be
extended to 2D or higher dimensional system.

##  IV $\eta$-pairing state in the large $U$ limit

Now we turn to investigate the situation with arbitrary filling but in the
large $U$ limit. system. Following the standard step of quantum mechanics, the
system can be divided into the kinetic part $H^{\prime}$ and interaction part
$H_{0}$, where

$\displaystyle H^{\prime}$ $\displaystyle=$ $\displaystyle
i\sum_{j,l}\sum_{\sigma=\uparrow,\downarrow}t_{jl}(c_{j,\sigma}^{\dagger}c_{l,\sigma}+c_{l,\sigma}^{\dagger}c_{j,\sigma}),$
(18) $\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle
U\sum_{j}n_{j,\uparrow}n_{j,\downarrow}.$ (19)

Here the imaginary hopping is assumed to be homogeneous $it_{jl}=it$. In the
strongly correlated regime $\left|U\right|\gg t$, the kinetic term
$H^{\prime}$ can be treated as a perturbation and one can derive an effective
Hamiltonian for the degenerate space. To second order in perturbation theory,
the effective Hamiltonian is given by

$H_{\mathrm{eff}}=P_{0}H_{0}P_{0}+P_{0}H^{\prime}P_{1}\frac{1}{E_{0}-H_{0}}P_{1}H^{\prime}P_{0}+O\left(\frac{t^{3}}{U^{2}}\right),$
(20)

where $P_{0}$ is a projector onto the Hilbert subspace in which there are $M$
lattice sites occupied by two particles with opposite spin orientation, and
$P_{1}=1-P_{0}$ is the complementary projection. Here the energy $E_{0}$ of
the unperturbed state is set to $E_{0}=MU$ where $M$ denotes the number of
doublons. Since $H^{\prime}$ acting on states in $P_{0}$ annihilates only one
double occupied site, all states in $P_{1}H^{\prime}P_{0}$ have exactly $N-1$
doubly occupied sites. Therefore, the effective Hamiltonian regarding doublon-
hole creation and recombination process is given as

$H_{\mathrm{eff}}=MU+\frac{4t^{2}}{U}\sum_{j}\left(\bm{\eta}_{j}\cdot\bm{\eta}_{j+1}-\frac{1}{4}\right),$
(21)

where $\bm{\eta}_{j}=(\eta_{j}^{x},$ $\eta_{j}^{y},$ $\eta_{j}^{z})$. In the
Appendix, a simple two-site case is provided to elucidate this mechanism. This
indicates that the non-Hermitian virtual exchange mediates an interaction
between the pseudo spins. It is similar to the Heisenberg interaction in the
Hermitian Hubbard model in the way that the doubly occupied and vacuum sites
correspond to spin up and spin down states, respectively. Owing to the fact
that $4t^{2}/U<0$, the effective Hamiltonian is ferromagnetic Heisenberg model
of pseudo spins. The eigenstate of the lowest eigenenergy within each doublon
subspace is the $\eta$-pairing state with different pair number. As such the
ground state of $H_{\mathrm{eff}}$ is
$\left(\eta^{+}\right)^{N}\left|\mathrm{Vac}\right\rangle$ dubbed as
$\eta$-spin ferromagnetic state. Note in passing that for the case of
repulsive Hubbard model ($U>0$) at half filling, the extra minus sign induced
by the non-Hermitian virtual exchange leads to an effective ferromagnetic
rather than an antiferromagnetic Heisenberg Hamiltonian

$H_{\mathrm{eff}}^{\mathrm{l}}=-\frac{4t^{2}}{U}\sum_{j}\left(\bm{s}_{j}\cdot\bm{s}_{j+1}-\frac{1}{4}\right)$
(22)

describing the behavior of the ground state and low energy excitations. The
eigenenergy of the ground state is zero. Notably, the interplay between the
imaginary hopping and particle-particle interaction fundamentally alters the
magnetism of the Hubbard model leading to sign reversal of magnetic
correlation.

Figure 4: Transition of the ground state driven by the imaginary hopping.
(a)-(b) depict the variation of the energy spectrum with respect to hopping
strength $t$, and (c) presents the switch of the correlator $C_{3}$. The
numerical simulation is performed for the $6$ site Hubbard model with $4$
filled particles. The other system parameters are $t_{0}=1$, and
$U=-1.5t_{0}$. The hopping strength $t$ is in units of $t_{0}$. The red line
denotes the $\eta$\- pairing state. $C_{3}$ experiences an evident jump at the
critical point indicating a dramatic change of the ground state. Such a first-
order transition may associate with quantum phase transition to some extent.

##  V $\eta$-pairing state in system with mixed hoppings

In the aforementioned sections, we have demonstrated that the $\eta$-pairing
state can be either the ground state of the system under the large $U$ limit,
or the ground state of the two-particle subspace with non-zero $U$. Then a
natural question arises: (i) For any nonzero $U$, is the $\eta$-pairing state
still the ground state of the system in the subspace of other particle
numbers? (ii) If yes, can the existing 1D results be extended to 2D or higher
dimensional system? (iii) If the disorder is introduced, does the property of
ground state be changed? To answer these questions, we first investigate the
1D non-Hermitian system with disordered imaginary hoppings and interaction
since the system parameter does not hold the uniform in real experiments. The
corresponding disordered Hamiltonian can be obtained by taking two sets of
random numbers $\left\\{t_{j}\right\\}$ and $\left\\{U_{j}\right\\}$ around
$t$ and $U$ in Eq. (1). The random number parameter can be taken as

$t_{j}=t+\mathrm{rand}(-a,a)\mathrm{\text{, }}U_{j}=U+\mathrm{rand}(-b,b),$
(23)

where $\mathrm{rand}(-a,a)$ denotes a uniform random number within $(-a,a)$.
It is too cumbersome to obtain an analytical result. Hence, we perform the
numerical simulation to check the fidelity between the ground state and target
$\eta$-pairing state, which can be given as

$\mathcal{F}=\left|\langle\psi_{g}\left(m\right)\right|\psi_{c}\left(m\right)\rangle|,$
(24)

with $m$ being an even number representing filled particle number of the
system. Here $|\psi_{g}\left(m\right)\rangle$ is the ground state of such
subspace and the target $\eta$-pairing state is

$|\psi_{c}\left(m\right)\rangle=\Omega^{-1}\left(\eta^{+}\right)^{m/2}|\mathrm{Vac}\rangle,$
(25)

with renormalization coefficient $\Omega=\sqrt{C_{N}^{m/2}}$. Fig. 2 shows
that if $U$ is homogeneous ($b=0$) then the ground state is the $\eta$-pairing
state in the subspace with particle numbers $m=2$, $4$, and $6$. The
corresponding energies are $U$, $2U$, and $3U$ respectively. Such a result
indicates that the formation of the $\eta$-pairing ground state does not
depend on the values of $U$ and is immune to the hopping disorder. However,
the introduction of disordered $U$ will cause the ground state of the system
to deviate from the $\eta$-pairing state. The underlying mechanism is clear,
that is, the system fulfills the $\eta$ symmetry even in the presence of the
hopping disorder, but if one introduce disorder into the interaction, this
symmetry will be destroyed leading to such deviation. One may think that the
disorder of interaction scrambles the background spin configuration and
disturb the spin correlation. Then a question arises: to what extent does the
ground state maintain the superconductivity? To capture superconductivity, the
doublon-doublon correlator

$C_{j}=\sum_{i}\langle\eta_{i}^{+}\eta_{i+j}^{-}\rangle/N$ (26)

is introduced. It is averaged over all sites separated by a distance $j$. The
nonzero value of such quantity implies both the Meissner effect and flux
quantization and hence provides a possible definition of superconductivity.
Yang (1962, 1989) For the target state $|\psi_{c}\left(m\right)\rangle$, the
expectation value can be given as

$\langle\psi_{c}\left(m\right)|\eta_{i}^{+}\eta_{i+j}^{-}|\psi_{c}\left(m\right)\rangle=\left\\{\begin{array}[]{c}\frac{M(N-M)}{N(N-1)}\text{,
for }j\neq 0\\\ \frac{M}{N}\text{, for }j=0\end{array}\right.,$ (27)

where $M=m/2$. Notice that it is irrelative to the distance $j$ and hence the
correlator $C_{j}$ obeys the same law such that $C_{j}=M(N-M)/[N(N-1)]$ for
$j\neq 0$ or $C_{j}=M/N$ for $j=0$. Fig. 2 shows that the value of correlator
and the overlap between the ground state and target state. It indicates that
$\mathcal{F}$ is around $0.9$ and the correlator $C_{j}$ stays at a non-zero
value ensuring the ground state of the system possesses the superconductivity
even though the strong inhomogeneity of interaction presents.

Now we switch gears to the cases of the 2D system. In Fig. 3, the disordered
2D system is sketched. For simplicity, we fix the strength of the hopping
disorder $a$ and examine two quantities $\mathcal{F}$ and $C_{2}$. It is shown
that the system still possesses the $\eta$-pairing state even though the small
homogeneous $U$ and the disordered imaginary hopping present, which is similar
to that of the 1D system. Although the disorder $U$ affects the correlation of
ground state, the correlator $C_{2}$ has a small fluctuation around the value
of uniform case supporting the superconductivity of the ground state.
Therefore, one can conclude that all the results of 1D can be extended to 2D
lattice system. It can be expected that this conclusion is still valid for the
higher dimensional bipartite system.

##  VI Transition from normal to $\eta$-pairing ground states

In this section, we focus on how does the ground state transits from normal to
superconductive state. To observe such a transition, we consider a 1D Hubbard
system with only two nearest neighbour (NN) sites coupled through Hermitian
hopping $t$. The corresponding Hamiltonian can be given as

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\sum_{j,\sigma=\uparrow,\downarrow}t_{j}(c_{j,\sigma}^{\dagger}c_{j+1,\sigma}+c_{j+1,\sigma}^{\dagger}c_{j,\sigma})$
(28) $\displaystyle+U\sum_{j}n_{j,\uparrow}n_{j,\downarrow},$

where homogeneous $U$ is supposed and $t_{1}=t_{0}$ otherwise $t_{j}=0$. The
Hamiltonian still possesses the $\eta$ symmetry and hence supports the
$\eta$-pairing eigenstate. However, such a state is not the ground state of
the system. Now we switch on the other coupling of the NN sites, which are the
imaginary hoppings rather than Hermitian hoppings, that is $t_{j}=it$ for
$j\neq 1$. Fig. 4 shows the variation of the low-lying energy spectrum with
respect to $it$. The $\eta$-pairing state is denoted by the red line, which is
suppressed to the ground state by the increase of imaginary hopping. There
exists a transition window in which the ground state is transformed from a
normal state to a superconducting state. We perform the numerical simulation
to demonstrate this process through the correlator $C_{3}$. Evidently, $C_{3}$
undergoes a leap around the critical point which leads to the divergence of
the first order of derivative $\partial C_{3}/\partial t$. It witnesses the
formation of the $\eta$-pairing ground state. Notice that all the conclusions
can be extended to a higher dimension. Before ending this section, we want to
point out that the imaginary hopping plays the key to achieve the
superconducting ground state, however, it does not mean that the system must
have the $\eta$-pairing ground state as long as the imaginary hopping is
applied. The transition of the ground state always requires a process such
that there is a threshold beyond which the system favors the
superconductivity. Such property is reminiscent of quantum phase transition,
that is, the ground state will experience a dramatic change when the system
crosses the quantum phase transition point. This findings paves the way to
understand the $\eta$-spin ferromagnetic state of the non-Hermitian strongly
correlated system.

##  VII Summary

In summary, we have systematically studied the effect of the non-Hermitian
imaginary hopping on the low-lying energy spectrum of the Hubbard model. The
analytical solution within the two-particle subspace shows that the
introduction of the imaginary hopping results in a full imaginary scattering
spectrum and a flip of the bound pair spectrum comparing to its Hermitian
parent model. It indicates that the particle-particle correlation suppresses
the non-Hermiticity making the ground state to be $\eta$-pairing state with
ODLRO. The $\eta$ symmetry plays the vital role in this mechanism. In the
large negative $U$ limit, the magnetism of the Hubbard model is altered
fundamentally due to the interplay between the particle-particle interaction
and non-Hermitian imaginary hopping. The ground state experiences a transition
from normal to $\eta$-spin ferromagnetic states. Such a transition holds for
any pair filled, that is, the ground state in each invariant subspace is
$\left(\eta^{+}\right)^{M}|\mathrm{Vac}\rangle$ with $M$ being the pairs of
particles. Through numerical simulation of 1D and 2D non-Hermitian Hubbard
system, we demonstrate that the $\eta$-pairing ground state can still survival
albeit a small negative $U$ presents. This evidence is robust against disorder
even if the system does not fulfill the $SO(4)$ symmetry. Our results open a
new avenue toward populating a $\eta$-pairing ground state and suppressing
antiferromagnetic correlation of $\eta$ spins in the attractive Hubbard model.

###### Acknowledgements.

We acknowledge the support of the National Natural Science Foundation of China
(Grants No. 11975166, and No. 11874225). X.Z.Z. is also supported by the
Program for Innovative Research in University of Tianjin (Grant No.
TD13-5077).

##  Appendix A Two-particle solutions

In this section, we show the detailed caculation for the two-particle solution
in each invariant subspace. For the simplicity, we only focus on the solutions
in subspaces $\left(0,\text{ }0\right)$ and $\left(1,\text{ }0\right)$, since
the solution in subspace $\left(1,\text{ }\pm 1\right)$ can be obtained
directly from that in subspace $\left(1,\text{ }0\right)$ by operator
$s^{\pm}$. A two-particle state can be given as

$\left|\varphi_{K}^{\pm}\right\rangle=\sum_{r}f_{K,k}^{\pm}\left(r\right)\left|\phi_{r}^{\pm}\left(K\right)\right\rangle\text{,
}\left(f_{K}^{+}\left(0\right)=f_{K,k}^{-}\left(-1\right)=0\right),$ (29)

where $r$ denotes the relative distance between the two particles and the wave
function $f_{K,k}^{\pm}\left(r\right)$ obeys the Schrödinger equations

$\displaystyle
Q_{r}^{K}f_{K,k}^{+}\left(r+1\right)+Q_{r-1}^{K}f_{K,k}^{+}\left(r-1\right)+$
$\displaystyle[\left(-1\right)^{n}Q_{r}^{K}\delta_{r,N_{0}}-\varepsilon_{K}]f_{K,k}^{+}\left(r\right)=0,$
(30)

and

$\displaystyle
Q_{r}^{K}f_{K,k}^{-}\left(r+1\right)+Q_{r-1}^{K}f_{K,k}^{-}\left(r-1\right)+$
$\displaystyle[U\delta_{r,0}+\left(-1\right)^{n}Q_{r}^{K}\delta_{r,N_{0}}-\varepsilon_{K}]f_{K,k}^{-}\left(r\right)=0,$
(31)

with $N_{0}=\left(N-1\right)/2$ and the eigen energy $\varepsilon_{K}$ in the
invariant subspace indexed by $K$. Here factor
$Q_{r}^{K}=-2\sqrt{2}it\cos\left(K/2\right)$ for $r=0$ and
$-2it\cos\left(K/2\right)$ for $r\neq 0$, respectively. $U$ appears in the
$(0,0)$ subspace and therefore admits the bound pair solution. In the large
$N$ limit, we can neglect the effect of on-site potential
$\left(-1\right)^{n+1}2it\cos\left(K/2\right)$ at $N_{0}$th site. The solution
of (31) is equivalent to that of the single-particle semi-infinite tight-
binding chain system with nearest-neighbour (NN) hopping amplitude
$Q_{j}^{K}$, and on-site potentials $U$ at $0$th site, respectively. Moreover
the solution of (30) corresponds to the same chain with infinite $U$. In this
scenario, the bound state solution
$\left|\varphi_{K}^{\mathrm{b}}\right\rangle=\sum_{r}f_{K}^{-}\left(r\right)\left|\phi_{r}^{-}\left(K\right)\right\rangle$
can be determined by substituting the ansatz

$f_{K}^{-}\left(j\right)=\left\\{\begin{array}[]{c}1/\sqrt{2}\text{, }j=0\\\
e^{-\beta j}\text{, }j\neq 0\end{array}\right.$ (32)

into the following equivalent Hamiltonian

$H_{\text{eq}}^{K}=U\left|0\right\rangle\left\langle
0\right|+\sum_{i=0}^{\infty}\left(Q_{i}^{K}\left|i\right\rangle\left\langle
i+1\right|+\text{H.c.}\right).$ (33)

Straightforward algebra shows that
$\beta=\ln[(-U\pm\sqrt{U^{2}+4\lambda_{K}^{2}})/2\lambda_{K}]$ where
$\lambda_{K}=2it\cos\left(K/2\right)$ and $\pm$ denotes negative and positive
$U$, respectively. Correspondingly, the energy of the bound pair is

$\epsilon_{K}=\mathrm{sgn}\left(U\right)\sqrt{U^{2}-16t^{2}\cos^{2}\left(K/2\right)}.$
(34)

For the case of negative $U$, the lowest energy of bound pair is
$\epsilon_{\pi}=-U$ locating on the subspace with $K=\pi$. As such the
corresponding eigenstate is $\left|\phi_{0}^{-}\left(K\right)\right\rangle$
that represents a $\eta$-pairing state in the coordinate space with the form
of $\left(\eta^{+}\right)/\sqrt{N}|\mathrm{Vac}\rangle$.

##  Appendix B Simple example of two-site case for the effective Hamiltonian
$H_{\mathrm{eff}}$

In this subsection, we provide a detailed calculation of the two-site case for
the effective Hamiltonian $H_{\mathrm{eff}}$ which may shed light to obtain
the effective Hamiltonian (21). In the simplest two-site case,
$P_{0}=\sum_{\alpha\in\text{{d.o.}}}|\alpha\rangle\langle\alpha|$ is the
projection operator to the doublon subspace spanned by the configuration
$\left\\{|\text{{x}}0\rangle,|0\text{{x}}\rangle\right\\}$, and
$P_{1}=1-P_{0}=\sum_{a\notin\text{{d.o.}}}|a\rangle\langle a|$ is the
complementary projection. Here the abbreviation d.o. means the doubly occupied
subspace and
$|$x$0\rangle=c_{1,\uparrow}^{\dagger}c_{1,\downarrow}^{\dagger}|$Vac$\rangle$,
$|0$x$\rangle=c_{2,\uparrow}^{\dagger}c_{2,\downarrow}^{\dagger}|$Vac$\rangle$.
The first term of Eq. (21) clear gives $P_{0}H_{0}P_{0}=U$. The second term
can be simplified by noting: (i) the unperturbed energy $E_{0}$ is $U$; (ii)
$P_{1}H^{\prime}P_{0}$ annihilates the doubly occupied site. Then
$H_{\mathrm{eff}}$ can be written as

$\displaystyle H_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle
U+\sum_{\alpha,\beta\in\text{d.o.}}\sum_{a,b\notin\text{d.o.}}|\alpha\rangle\langle\alpha|H^{\prime}|a\rangle\langle
a|$ (35) $\displaystyle\times\frac{1}{U-H_{0}}|b\rangle\langle
b|H^{\prime}|\beta\rangle\langle\beta|$ $\displaystyle=$ $\displaystyle
U+\frac{1}{U}\sum_{\alpha,\beta\in\text{d.o.}}\langle\alpha|\left(H^{\prime}\right)^{2}|\beta\rangle|\alpha\rangle\langle\beta|.$

The second term describes the virtual exchange of the fermions. The non-
Hermitian imaginary hopping brings about an additional sign to this process
yielding that

$H_{\mathrm{eff}}=U-\frac{2t^{2}}{U}\left(|\text{{x}}0\rangle\langle
0\text{{x}}|+|0\text{{x}}\rangle\langle\text{{x}}0|+|\text{{x}}0\rangle\langle\text{{x}}0|+|0\text{{x}}\rangle\langle
0\text{{x}}|\right).$ (36)

Combining the cases in the subspaces of $|\mathrm{xx}\rangle$ and
$|$Vac$\rangle$, the pseudo spin Hamiltonian can be given by the non-Hermitian
Heisenberg-like model

$H_{\mathrm{eff}}=MU+\frac{4t^{2}}{U}\left(\bm{\eta}_{1}\cdot\bm{\eta}_{2}-\frac{1}{4}\right),$
(37)

where $M$ can be $0$, $1$, and $2$ denoting the number of pairs of the doublon
subspace. Evidently, the ground state of $H_{\mathrm{eff}}$ is the $\eta$-spin
ferromagnetic state with the form of $\left(\eta^{+}\right)^{2}|$Vac$\rangle$.

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