# Quantum Transports in Two-Dimensions with Long Range Hopping

Si-Si Wang  School of Physics and Materials Science, Guangzhou University,
510006 Guangzhou, China  School of Mathematics and Information Science,
Guangzhou University, 510006 Guangzhou, China  Huangpu Research and Graduate
School of Guangzhou University, 510700 Guangzhou, China  Kangkang Li
Department of Physics, Zhejiang Normal University, Jinhua 321004, China  Yi-
Ming Dai  School of Physics and Materials Science, Guangzhou University,
510006 Guangzhou, China  Hui-Hui Wang  School of Physics and Materials
Science, Guangzhou University, 510006 Guangzhou, China  Huangpu Research and
Graduate School of Guangzhou University, 510700 Guangzhou, China  Yi-Cai Zhang
School of Physics and Materials Science, Guangzhou University, 510006
Guangzhou, China  Yan-Yang Zhang  School of Physics and Materials Science,
Guangzhou University, 510006 Guangzhou, China  Huangpu Research and Graduate
School of Guangzhou University, 510700 Guangzhou, China

###### Abstract

We investigate the effects of disorder and shielding on quantum transports in
a two dimensional system with all-to-all long range hopping. In the weak
disorder, cooperative shielding manifests itself as perfect conducting
channels identical to those of the short range model, as if the long range
hopping does not exist. With increasing disorder, the average and fluctuation
of conductance are larger than those in the short range model, since the
shielding is effectively broken and therefore long range hopping starts to
take effect. Over several orders of disorder strength (until $\sim 10^{4}$
times of nearest hopping), although the wavefunctions are not fully extended,
they are also robustly prevented from being completely localized into a single
site. Each wavefunction has several localization centers around the whole
sample, thus leading to a fractal dimension remarkably smaller than 2 and also
remarkably larger than 0, exhibiting a hybrid feature of localization and
delocalization. The size scaling shows that for sufficiently large size and
disorder strength, the conductance tends to saturate to a fixed value with the
scaling function $\beta\sim 0$, which is also a marginal phase between the
typical metal ($\beta>0$) and insulating phase ($\beta<0$). The all-to-all
coupling expels one isolated but extended state far out of the band, whose
transport is extremely robust against disorder due to absence of
backscattering. The bond current picture of this isolated state shows a
quantum version of short circuit through long hopping.

## Introduction

Long range orders and behaviors from short range coupling are among the most
important themes of condensed matter physics. Even theoretical models based on
nearest neighbor hopping and interaction can be used to describe vast amounts
of physical phenomena, including the energy band, magnetism, metal-insulator
transition (MIT) and topological states, etc.[1, 2, 3, 4, 5]. On the other
hand, systems with long range coupling have attracted many interests recently.
For example, in the photonic lattice, controllable long range coupling can be
realized by shaping the spectrum of the optical pump[6], or by the optical
gain[7]. Simulating long range hopping is proposed by periodically driven
superconducting qubits[8]. Rydberg atomic arrays are considered as a promising
platform for quantum information application[9, 10], where the ultra-long
coupling between atoms plays a crucial role[11, 12, 13, 10, 14, 15, 16, 17].
Among them, two-dimensional (2D) arrays with long range interactions have been
recently fabricated and studied[9, 10, 17].

The long range coupling between atoms or molecules can also be realized by the
mediation of phonons[18, 19], photons[20, 21] or an optical cavity[22, 23, 24]
in different physical systems, including conventional electronic systems[22,
25, 26]. Another interesting realization of quantum lattice model is the
electric circuit network[27, 28, 29, 30]. Since wires can be connected in
arbitrary ways, in principle this system can simulate models with arbitrary
ranges of hopping in any spatial dimensions.

Long range coupling systems possess many special and useful properties. In a
Rydberg array for quantum computing, the long range coupling can compensate
for low fidelity and therefore enables better algorithmic performances[10]. A
one-dimensional dimerized superconducting circuit lattice with long-range
hopping is proposed to be a phase-robust topological router[31]. Rich
phenomena of localization from some long range coupling models have been
recently noticed[32]. Some of them may be rather counterintuitive. For
example, although long range hopping seems to greatly enhance connectivity
between sites, it does not necessarily result in an enhanced quantum
transport[32, 33].

Even for one-dimensional (1D) systems, theoretical studies have shown many
novel properties, for example, breaking of ergodicity[34], cooperative
shielding in many-body systems[35], weakened topological bulk-boundary
correspondence[36], and subdiffusive phases in an open clean system[37]. The
measurement-induced phase transitions of long range coupling systems are
theoretically investigated[38, 39, 40, 41], with practical applications in
qubits. Based on a model with all-to-all and distance-independent hopping[42],
some interesting phenomena in 1D have been predicted recently[33, 43, 44]. The
first one is a cooperative shielding in the single particle picture, i.e.,
absence of effects from the long range hopping for most of the states in the
clean limit[33]. The second is the disorder-enhanced and disorder-independent
transport, if a large-bias current is considered[43].

In this manuscript, we generalize the above all-to-all model to a 2D version,
and theoretically investigate its quantum transports. By presenting transport
evidences, we find that the cooperative shielding persists in the weak
disorder limit. Although a fixed boundary condition breaks the perfect
shielding, this breaking will be negligible for a sufficiently large sample.
With increasing disorder, the shielding is destroyed and the transports will
be remarkably different from the short range counterpart. We reveal the
microscopic pictures of these transports by showing the real space
distributions of bond currents, so that the roles of bonds with different
ranges can be seen vividly. Furthermore the localization property is discussed
by calculating the fractal dimension of eigenstates, and by performing size
scaling of the conductance. Over several orders of strong disorder, most
states exhibit a hybrid feature from (or a marginal feature between)
localization and delocalization. A unique feature of this model is the
existence of a single isolated state far away from the band states[42, 33],
with a large gap proportional to the size of the sample. We find it is an
extended isolated state with very robust transport. The physical origins are
also discussed.

##  Model and Method

Our 2D model is a generalization from the 1D counterpart[33, 43], which is
illustrated as the sample enclosed by the red dashed-line square in Figure 1
(a). It is defined on a square lattice with the spinless Hamiltonian

$\displaystyle H_{\mathrm{LR}}$ $\displaystyle=$ $\displaystyle
H_{\mathrm{NN}}+H_{\mathrm{AA}}$ (1) $\displaystyle\equiv$
$\displaystyle\sum_{\langle ij\rangle}tc^{\dagger}_{i}c_{j}+\sum_{ij}\gamma
c^{\dagger}_{i}c_{j}$

where $c^{\dagger}_{i}$ ($c_{i}$) creates (annihilate) an electron at site
$i$. Here $H_{\mathrm{NN}}$ contains the conventional nearest neighbor hopping
with the magnitude $t$ [black bonds in Figure 1 (a)]. The second term,
$H_{\mathrm{AA}}$ includes all-to-all and distance independent long range
hopping $\gamma$ [red bonds in Figure 1 (a)], which can be realized by a
cavity-assisted technology[43, 22, 23]. It has been argued that this distance
independent long range hopping $\gamma$ grasps the main physics arising from
the coupling of the molecules with the cavity mode, since the coupling to the
cavity mode is the same for all molecules[43]. Throughout this paper, $t=1$
will be used as the energy unit, and $\gamma=1/2$ is identical to previous 1D
counterparts[33, 43]. For simplicity, we adopt the sign convention that
$t,\gamma>0$. In $H_{\mathrm{AA}}$, the diagonal terms $\gamma
c^{\dagger}_{i}c_{i}$ are intentionally included. This global and trivial
energy shift makes the band center of $H_{\mathrm{LR}}$ (referred as the “long
range model” hereafter) identical to that of $H_{\mathrm{NN}}$ (referred as
the “short range model” hereafter), which will be convenient in following
calculations.

Figure 1: (Color online) Illustration of transport setups in our simulation.
(a) Two-terminal conductance setup, where a $4\times 4$ sample (encircled by
the red dashed-line square) with long range hopping is connected to two leads
with short range hopping. Black (red) bonds represent the nearest (long range)
hopping $t$ ($\gamma$). (b) and (c) illustrate the energy band configurations
among the sample (red) and leads (black) used in Sections III and IV,
respectively, where the black dashed line labels the chemical potential.
Created from OriginPro 8 SR0 (URL: http://www.OriginLab.com).

For a finite sample with $N\equiv N_{x}\times N_{y}$ lattice sites, and with
periodic boundary condition for $H_{\mathrm{NN}}$ in both directions (there is
no difference of boundary conditions for $H_{\mathrm{AA}}$), it can be easily
verified that two terms of $H_{\mathrm{LR}}$ are commutable and therefore,

$C_{H}\equiv\big{[}H_{\mathrm{NN}},H_{\mathrm{LR}}\big{]}=0.$ (2)

As a result, $H_{\mathrm{LR}}$ and $H_{\mathrm{NN}}$ have common
eigenfunctions[33]. Furthermore, the operator $H_{\mathrm{AA}}$ has $N-1$
degenerate eigenvalues $0$ and one eigenvalue $N\gamma$. Therefore similar to
the 1D case[33], the eigenvalues of the long range model $H_{\mathrm{LR}}$ are
also identical to those of the short range model $H_{\mathrm{NN}}$, i.e.,

$E^{\mathrm{LR}}_{n}=E^{\mathrm{NN}}_{n},\quad 1\leq n\leq N-1,$ (3)

except the highest one (the lowest one if $\gamma<0$), which is

$E^{\mathrm{LR}}_{N}=4t+N\gamma$ (4)

for $H_{\mathrm{LR}}$. Here and throughout this manuscript, we always index
eigenstates of a Hamiltonian in the ascending order of eigenvalues. Notice
from Equation (4) that $E^{\mathrm{LR}}_{N}$ is size dependent, and this
single state is isolated from the energy band consisting of the rest $N-1$
eigenstates. Thus it will be called the isolated state. Correspondingly, the
rest $N-1$ eigenstates [distributed within $[-4t,4t)$] will be called the
band, which are simply identical to those of the single band of the short
range model [Equation (3)]. We can define the subspace $S_{N}$ (the isolated
state) as the one spanned by this isolated state $|\psi_{N}\rangle$, and the
subspace $S_{N-1}$ (the band) as the one spanned by the rest $N-1$ eigenstates
${|\psi_{i}\rangle}$. Due to the large gap $\Delta\sim N_{x}N_{y}\gamma$,
these two subspaces are barely mixed when the sample size is sufficiently
large and/or the disorder strength is not strong[33]. These novel mathematical
structures lead to interesting consequences. For example, because of the above
mentioned commutability and unmixing between $S_{N}$ and $S_{N-1}$, the
dynamics within $S_{N-1}$ is shielded from long-range hopping, namely it
behaves as if long-range hopping does not exist. This is the the cooperative
shielding, a counterintuitive phenomenon found in the 1D counterpart[33]. In
the following, we will see that this cooperative shielding is also manifested
in 2D quantum transports.

One of our focus is disorder, which is simply included by adding a random
onsite potential as

$V=\sum_{i}W\cdot U_{i}c^{\dagger}_{i}c_{i},$ (5)

where $U_{i}$ are independent random numbers uniformly distributed in
$(-1/2,1/2)$ and $W$ is the single parameter to characterize the disorder
strength. With nonzero $W$, the short range part $H_{\mathrm{NN}}+V$ will not
commute with the all-to-all part $H_{\mathrm{AA}}$ again. The shielding and
corresponding transport phenomena will be one of the primary themes of this
work.

Now let us briefly describe the main methods of calculation. At zero
temperature, the two-terminal conductance $G$ of a finite sample is
proportional to the transmission (Landauer formula)[45], and can be expressed
by Green’s functions as[46, 47]

$G=\frac{e^{2}}{h}\mathrm{Tr}\left[\Gamma_{L}G^{r}\Gamma_{R}G^{a}\right],$ (6)

where
$G^{r/a}(E)\equiv\left(E\pm-H-\Sigma^{r/a}_{L}-\Sigma^{r/a}_{R}\right)^{-1}$
is the dressed retarded/advanced Green’s function of the central sample, and
$\Gamma_{L(R)}=i(\Sigma_{L(R)}^{r}-\Sigma_{L(R)}^{a})$ with
$\Sigma_{L(R)}^{r/a}$ being retarded/advanced self energies due to the left
(right) lead, respectively (The spin degeneracy factor 2 is omitted in this
manuscript). The lead self energy is defined as

$\Sigma=\tau g(E)\tau^{\dagger},$ (7)

where $g(E)$ is surface Green’s function of the semi-infinite lead, and $\tau$
is the coupling Hamiltonian from the sample to the lead[46]. Numerically this
self energy can be conveniently calculated from a direct diagonalization
method[48]. In our calculations, as illustrated in Figure 1 (a), we take both
leads to be semi-infinite square lattices with only the nearest hopping
Hamiltonian $H_{\mathrm{NN}}$, since otherwise, it is numerically inaccessible
to calculate the self energy of a semi-infinite lattice with long range
hopping.

The local current from site $i$ to $j$ along the bond is[47, 49]

$J_{i\rightarrow
j}=\frac{2e^{2}}{h}\mathrm{Im}\big{[}H_{ij}G^{n}_{ji}\big{]}(V_{L}-V_{R}),$
(8)

where $H_{ij}$ the matrix element of the bare Hamiltonian, and
$G^{n}=G^{r}\Gamma_{L}G^{a}$ is the correlation function. Since it is defined
in the linear response regime, we simply take the voltage difference
$V_{L}-V_{R}$ between the left (source) and right (drain) leads to be unity.
In this case, the net current through any transverse cross section of the
sample is numerically equal to the conductance calculated from Equation (6).

Notice that all Green’s functions and self energies appearing in Eqs. (6) and
(8) are energy dependent. In the weak disorder regime, the energies in leads
are adopted to vary with that of the sample as illustrated in Figure 1 This is
the case in Figure 2, Figure 3 and Figure 4 (a-e). Such a setup can minimize
the contact resistance due to the mode mismatch between the sample-lead
interface. In the regimes of strong disorder or the isolated state, the
energies in leads are fixed at $E=0$ to provide the maximum number of
conductive channels (which is equal to $N_{y}$). This is applied in Figure 4
(f), Figure 6 and Figure 8. In these cases, the interface scattering is not an
issue because the impurity scattering in the sample is dominating.

## The Band

### A. Shielding at Weak Disorder

Figure 2: (Color online) Results for zero disorder and fixed boundary
condition. (a) Difference of eigenergies $\Delta E_{n}\equiv
E^{\mathrm{LR}}_{n}-E^{\mathrm{NN}}_{n}$ between the long range and short
range models, for a finite sample with sizes $10\times 10$ (black), $20\times
20$ (red) and $30\times 30$ (green) respectively. The latter two curves are
shifted vertically for visual clarity. (b)-(d): the two-terminal conductance
$G$ as a function of Fermi energy $E$, for a short range (blue curve) and long
range model (red curve), with sample size $10\times 10$ (b), $20\times 20$ (c)
and $30\times 30$ (d). Most of the blue curves have been covered by red curves
due to an almost perfect match. The transport setup is shown in Figure 1 (a)
and (b).

Let us first investigate properties in the clean limit, $W=0$. In the
following calculations, all samples are square shaped with size $N=N_{x}\times
N_{x}$ to exhibit the 2D nature. The commutator $C_{H}$ in Equation (2) holds
rigorously for periodic boundary conditions, for which a perfect shielding is
expected. However, such a torus geometry is not physically applicable for a
realistic 2D sheet, especially when it will be connected to conducting leads.
Therefore in the rest of our work, we will employ fixed (hard-wall) boundary
conditions in both directions for the $H_{\mathrm{NN}}$ term (there is no
boundary effects for the $H_{\mathrm{AA}}$ term). In this case, the commutator
matrix $C_{H}$ will not be identically zero. However, we find that nonzero
matrix elements only appear when they are associated with boundary bonds of
$H_{\mathrm{NN}}$, which only constitute an extremely small portion of the
matrix $C_{H}$. With increasing size, this portion will be even smaller due to
the shrinking of boundary-bulk proportion, so the perfect shielding is
expected to recover.

We first check the validity of Equation (3) for isolated samples without being
attached to any leads. In Figure 2 (a), we plot the difference of eigenvalues
from the long range and short range models, without disorder and in the
presence of fixed boundary condition, for three different sample sizes. Notice
the latter two curves are shifted vertically for better visual clarity. One
can see that remarkable differences mostly occur in the high energy region
near the band top (and also near the isolated state), and the difference
shrinks with increasing sample size. Therefore it is reasonable to expect a
perfect shielding effect recovers for sufficiently large sample.

As an example of physical observable, the two-terminal conductance $G$ of a
sample connected with leads (short range model) will be calculated by using
Equation (6). Notice that all Green’s functions and self energies appearing in
Equation (6) are energy dependent. Here, as illustrated in Figure 1 (b), we
adopt a uniform Fermi energy $E$ among the central sample and leads, i.e.,
$G(E)=\frac{e^{2}}{h}\mathrm{Tr}\left[\Gamma_{L}(E)G^{r}(E)\Gamma_{R}(E)G^{a}(E)\right]$,
so that wavefunctions in these regions can have a best mode match, which is
expected from the perfect cooperative shielding.

The resulting $G$ of the long range model sample as a function of Fermi energy
$E$ is presented as red curves in Figure 2 (b)-(d), for different sample
sizes. For comparison, the result for a sample replaced by a short range model
($\gamma=0$) with the same size is also plotted as blue curves in each panel.
For the latter case, since the sample and leads are completely identical and
perfectly matched, the transmission is therefore perfectly quantized as
typical integer steps[46]. Then, it is interesting to see that the result from
the former case also matches these quantization steps nicely. Distinguishable
differences only occur occasionally in the high energy region in Figure 2 (b),
i.e., for the smallest sample.

To understand these results, we should remember that our simulation of quantum
transports is based on a fully coherent picture, which makes the results
highly sensitive to any imperfection or mismatch in the structure[47]. Even
without any disorder in the sample, the mismatch between the sample and leads
is sufficient to introduce remarkable scattering at interfaces and therefore
destroy the conductance quantization remarkably. For example, a recent study
on a low-dimensional structure with long range hopping shows that such a
mismatch can even cause a subdiffusive transport through the whole clean
sample, instead of a ballistic one[37]. That is to say, in the coherent limit,
the scattering at the interface alone can be strong enough to alter the
transport property qualitatively. Therefore here, the perfect quantization of
coherent transmission through a _heterostructure_ [as shown in Figure 1 (a)]
is a nontrivial phenomenon, which implies the absence of scattering in the
sample _and_ at the lead-sample interfaces, or a perfect match of wavefunction
modes between the sample (long rang model) and leads (short range model),
despite the apparent lattice mismatch. In other words, the electron travels as
if the long range hopping did not exist, which is a manifestation of the
cooperative shielding[33] in 2D quantum transport, when disorder is absent.
This shielding is a direct result of Eqs. (2) and (3). Small deviations from
perfect shielding in transports [Figure 2 (b)] are a consequence of the
boundary effect (which is not periodic), and will practically vanish for
larger sample sizes owing to weaker boundary effects.

Figure 3: (Color online) The disorder averaged conductance $\langle G\rangle$
(first row) and its standard deviation $\Delta G$ (second row) as functions of
Fermi energy $E$, for different disorder strengths $W$ as shown. The blue
(red) curve is for the short (long) range model. The sample size is $40\times
40$ and each data point is an average over 1000 disorder samples. All energies
are in units of $t$.

We have seen that in the clean limit, the quantum transport through the 2D
long range model is identical to that in the short range model, which is a
manifestation of the perfect shielding in 2D. Now we will investigate the
effect from disorder, $W\neq 0$. Disorder breaks Equation (2) and thus may
also break the perfect shielding. In Figure 3, we present the disorder
averaged conductance $\langle G\rangle$ (first row) and its standard deviation
$\Delta G$ (second row) as functions of the Fermi energy $E$, for different
disorder strengths $W$. When $W=3$ [panels (a) and (e)], similar to the clean
limit, the results for the short range (blue curves) and the long range (red
curves) models are identical. This suggests that the perfect shielding
practically survives through weak disorder, which was also observed in 1D
[33].

With increasing disorder, for example, $W=5$ [Figure 3 (b) and (f)], the
results for short and long range models start to deviate gradually. In the
case of stronger disorder, $W=7,\,11$, both the conductance and its
fluctuation of the long range model are remarkably larger than those of the
short range counterpart. The existence of long range hopping gives rise to
better transport (i.e., larger conductance) in the strong disorder limit.

Figure 4: (Color online) Distribution of bond currents for a certain disorder
configuration on a $20\times 20$ lattice, with the source (drain) lead
connected to the left (right) boundary. (a)-(e): Fermi energy $E=1$ in the
band, with disorder strength $W=0,\,3,\,5,\,7,\,11$, respectively. (f): Sample
Fermi energy $E=203.71$ at the EIS, with $W=7$, discussed in Section IV. Each
arrow connecting two sites represents the corresponding bond (thus its length
indicating the bond length), with the size of the _arrowhead_ alone to
indicate the magnitude of its current. For visual clarity, only currents with
$J\geq 0.4J_{\mathrm{max}}$ are displayed, and long (short) bonds are plotted
in red (blue) color, while intermediately long bonds in purple color.

In order to have a microscopic understanding of these quantum transports, for
example, contributions from short and long range hoppings, we present
distributions of bond currents associated with some typical regimes in Figure
4. Due to the all-to-all nature of the hopping, there are too many
($\sim(N_{x}N_{y})^{2}$) bonds with varying lengths intersecting each other,
and densely distributed on the lattice. This makes it difficult to present a
full and visually distinguishable picture of _all_ bond currents, as in
conventional short range models[49]. Therefore for the purpose of displaying
the dominating physics clearly, we adopt some technical tactics in plotting
Figure 4. Firstly, a small sample with $20\times 20$ sites is used. This is
smaller than those used in Figure 3, but we have checked (but not shown here)
that the physics is identical. Secondly, only currents with magnitudes $J\geq
0.4J_{\mathrm{max}}$ are displayed, where $J_{\mathrm{max}}$ is the maximum
magnitude of bond current in this sample. Therefore the displayed currents do
not obey the current conservation. Thirdly, the magnitude of a bond current is
represented by the size of the _arrowhead_ only, while the size of the arrow
shaft still represents the real size of the bond connecting two sites. For a
better visual clarity, longer (shorter) bonds are plotted in red (blue) color.
We do so because, besides the current flowing on it, the actual position and
length of each bond are also important information.

Figure 4 (a) is the result for zero disorder, corresponding to the case shown
in Figure 2 (c). It can be seen that now the currents are almost uniformly
carried by nearest bonds (blue arrows), and contributions from most long range
bonds are small. This is a clear picture of the perfect shielding: long range
coupling hardly plays a role as a result of a delicate quantum coherent
effect. Small currents along a few long range bonds (red arrows with small
arrowheads) can be attributed to resonant states between hard wall boundaries.

In the presence of very weak disorder, Figure 4 (b), the current distribution
is also disordered, and contributions from long range bonds start to increase.
We notice that this corresponds to the transport shown in in Figure 3 (a) and
(e), where the conductances from the short range and long range models still
match well. In other words, although the transports are carried by short range
and long range bonds, the total current is still very close to that of the
short range model. This is another subtle manifestation of the word, the
_cooperative_ shielding[33]. With larger disorder, as shown in Figure 4 (c)
and (d), the currents distribute in a more chaotic pattern, with long range
bonds trying to connect localization centers of the wavefunction.

With these pictures, now we can understand more about Figure 3. When disorder
is nonzero but still weak ($W>3$), although these two subspaces are still
shielded from each other, long range hops start to manifest themselves by
contributing currents and therefore enhancing the conductance. We call it a
regime of unperfect shielding, compared to the perfect one (zero effect from
long range hopping) for $W=0$ as shown in Figure 2 (c,d) and Figure 4 (a). In
short range models of localization, it was recently found that
microscopically, the dominating transport path can be pinned within a certain
range of model parameters (e.g., Fermi energy and disorder potential) [50],
since endeavors must be made to find another continuous path composed of
sequential short range bonds. However here, thanks to the all-to-all
connectivity, an entirely new path can be found more easily upon parameter
changes, so that the pinning effect is reduced. Therefore, the transport will
be more mutable and sensitive with the change of model parameters, e.g., the
disorder potential. This leads to larger fluctuations of the conductance at
stronger disorder displayed in Figure 3 (g) and (h).

### B. Localization Properties at Strong Disorder

Figure 5: (Color online) Eigenstates of a $80\times 80$ sample for the long
range model $\gamma=0.5$ (upper row) and the short range model $\gamma=0$
(lower row). Left column: the fractal dimension $\Gamma_{n}$ versus
eigenenergy $E_{n}$, with the disorder strength $W=256$. Middle column: the
spatial distribution of the squared magnitude of an eigenstate
$|\psi_{n}|^{2}$ with $\Gamma_{n}\sim\Gamma_{\mathrm{typ}}$. Right column:
$\Gamma_{\mathrm{typ}}$ as a function of the disorder strength $W$ (logarithm
scale).

The above discussions were focused on the shielding effect, with rather weak
disorder strength $W\lesssim 10$. In the 1D case, the long range model
exhibits surprisingly rich behaviors of transports over several orders of the
disorder strength $W$[43], where all states of short range models should have
been completely localized. To have a first observation of our 2D long model
with stronger disorder, we now turn to investigate eigenstates of an finite
sample detached from any conducting leads. The localization property of the
$n$-th eigenstate $\psi_{n}$ of the sample can be characterized by its fractal
dimension defined as

$\Gamma_{n}=\frac{\ln(\mathrm{IPR})_{n}}{\ln N_{x}},$ (9)

where $(\mathrm{IPR})_{n}$ is the inverse participation ratio
$(\mathrm{IPR})_{n}\equiv\sum_{i}|\psi_{n,i}|^{4}$ [3, 68, 69]. In a 2D
system, an extended state corresponds to $\Gamma_{n}\sim 2$ while a localized
state corresponds to $\Gamma_{n}\sim 0$.

In the left column of Figure 5, the fractal dimensions $\Gamma_{n}$ and
eigenenergies $E_{n}$ of band states of a $80\times 80$ sample are plotted,
for a certain disorder configuration at $W=256$, with panels (a) and (d)
corresponding to the long range and short range models respectively. For the
short range model shown in panel (b), as expected, most eigenstates are
localized. In fact, the typical value of the fractal dimension

$\Gamma_{\mathrm{typ}}=\exp(\frac{1}{N_{x}N_{y}}\sum_{n=1}^{N_{x}N_{y}}\ln\Gamma_{n}),$
(10)

for data displayed in Figure 5 (d) is $0.00209\sim 0$. This vanishing
dimension suggests that most eigenstates have been localized to a single site,
which is verified from the spatial distribution of a typical wavefunction
shown in Figure 5 (e). Figure 5 (f) shows $\Gamma_{\mathrm{typ}}$ as a
function of $W$, from which we can see that such a single-site localization
has been realized when $W\gtrsim 100$.

On the other hand, the behaviors of the long range model are remarkably
different. In Figure 5 (a), most eigenstates possess a significantly nonzero
fractal dimension $\Gamma_{n}$, with a typical value
$\Gamma_{\mathrm{typ}}=0.18588$. The spatial distribution of a typical
eigenstate in Figure 5 (b) exhibits several localization centers [instead of
one single peak in the short range model shown in panel (e)] which can be
connected by long range hoppings. This is similar to that in the case of 1D
long range model where the wavefunction has an extended tail[43]. In other
words, although the disorder breaks the fully extended nature of the
wavefunction, the long range hoppings prevent it from being localized into a
single site, even at very strong disorder. From the dependence of
$\Gamma_{\mathrm{typ}}$ on $W$ in Figure 5 (c), one can see that such a
robustness against a single-site localization (and even the value of
$\Gamma_{\mathrm{typ}}$ itself) persists over several orders of disorder
strength $W$ (Notice the logarithmic scale of $W$).

Figure 6: (Color online) Disorder averaged $\langle\ln G_{\mathrm{I}}\rangle$
for the long range model (upper row) and the short range model (lower row).
Left three columns: $\langle\ln G_{\mathrm{I}}\rangle$ versus Fermi energy $E$
for different disorder strengths $W$ . Rightmost column: $\langle\ln
G_{\mathrm{I}}\rangle$ versus disorder strength $W$ (logarithmic scale) at
Fermi energy $E=0$. Curves in different colors correspond to different sample
sizes: $10\times 10$ (black), $20\times 20$ (red), $40\times 40$ (green),
$80\times 80$ (blue), and $160\times 160$ (magenta). The number of disorder
configurations for averaging is 1000 for $80\times 80$ and $160\times 160$,
and 5000 for other sizes.

Now we discuss this question: in the presence of disorder, will the long range
hopping lead to a really metallic (delocalized) state in the thermodynamic
limit? To answer this, one needs to perform size scaling on some transport
quantities. A commonly used scaling quantity is the localization length
normalized by the sample size, which can be extracted from the transfer matrix
method[51, 52]. However, a transfer matrix can only be applicable to a model
with a very finite hopping range. Here instead, we use the numerical scaling
on the intrinsic conductance $G_{\mathrm{I}}$ which is defined as[53]

$\frac{1}{G_{\mathrm{I}}}=\frac{1}{G}-\frac{1}{M_{c}},$ (11)

with $M_{c}$ the number of active channels at Fermi energy in the lead and $G$
is the above used Landauer conductance. The second term $\frac{1}{M_{c}}$ is
used to deduct the effect of contact resistance at the sample-lead interfaces,
so that $G_{\mathrm{I}}$ can manifest the intrinsic transport property of the
bulk sample, which is found to be closely related to the conductance derived
from the Kubo formula or Thouless formula [53]. When $G$ is small (strong
disorder) and/or $M_{c}$ is large (large size), $G_{\mathrm{I}}\thickapprox
G$. This intrinsic conductance ($N_{x}\times N_{x}$) has been widely employed
to investigate the occurrence, scaling and critical properties of the MIT in
2D and 3D systems[53, 54, 55, 56, 57]. For example, it can be used to evaluate
the standard scaling function $\beta=\frac{d\langle\ln
G_{\mathrm{I}}\rangle}{d\ln N_{x}}$ [67, 2, 3] of MIT, where
$\langle\cdots\rangle$ still stands for averaging over the disorder ensemble.
An increase (decrease) of $\ln G_{\mathrm{I}}$ with increasing $\ln N_{x}$
indicates a metal with $\beta>0$ (insulator with $\beta<0$) phase[54, 58]. In
fact experimentally, scaling of the conductance is also the standard method to
distinguish metal and insulator phases of materials[59, 60, 61, 62, 63, 64].

Again let us first take a look at the short range model that has been well
understood, results of which are displayed in the lower rows of Figure 6. In
panels (e), (f) and (g), $\langle\ln G_{\mathrm{I}}\rangle$ as a function of
the energy $E$ is plotted for different disorder strengths, and panel (h) is
$\langle\ln G_{\mathrm{I}}\rangle$ versus $W$ at a fixed energy $E=0$. In each
panel, curves in different colors correspond to results from different sample
sizes. Here, the conductance is monotonically decreasing with the sample size
and disorder strength for all states, which means that they are trivially
localized. Moreover, this decrease is more rapid for a larger sample size or a
larger disorder strength.

On the other hand, in the upper row of Figure 6, the results for the long
range model are quite different. The first obvious feature is that for a
definite size and disorder strength, the conductance of the long range model
is several orders larger than that of the short range model. For $W=64$ shown
in Figure 6 (a), $\ln G_{\mathrm{I}}$ is decreasing with increasing the sample
size, in the whole energy region. Such decreasing can also be seen for weaker
disorder strengths shown in the Supplementary Material. These seem to indicate
that all states of the long range model are localized in the thermodynamic
limit. However, for a larger disorder $W=256$ as shown in Figure 6 (b), the
conductance has a significant increase when the size is increased from
$N_{x}=10$ (black) to $N_{x}=20$ (red) before a decrease again with larger
sizes. For an even larger disorder $W=1024$ presented in Figure 6 (c), the
conductance increases up to size $N_{x}=40$ and then saturate there when the
size is doubled as $N_{x}=80$.

In Figure 6 (d), we fix the fermi energy at $E=0$ and display the development
of the size scaling with increasing the disorder strength $W$. From the first
glance, it seems to be a scaling pattern for the MIT, where the conductance is
increasing ($\beta>0$) or decreasing ($\beta<0$) with $N_{x}$ on two ends of
the $W$ axis, respectively[53, 54, 56, 57]. However a careful scrutinize can
reveal several distinct differences from the scaling pattern of the standard
MIT. First, here curves associated with different sizes do _not_ cross at a
single critical value $W_{c}$. Instead, curves for larger sizes cross at
larger $W$. Second, on the larger $W$ side, although the conductance is
clearly increasing with the scaling at small sizes, this increase eventually
tends to saturate at large sizes. For example, at $W=4096$, although the
conductance is still increasing when the size is doubled from $40\times 40$
(green) to $80\times 80$ (blue), it saturates when the size is doubled again
to $160\times 160$ (magenta). Therefore at $W=8192$, although the conductance
of size $160\times 160$ is slightly larger than that of $80\times 80$, it is
reasonable to imagine it will also saturate at, say, size $320\times 320$
(which is beyond our calculation capability). Third, with increasing size, on
the side of increasing or saturating conductance (large $W$), the magnitude of
the conductance is smaller than that on the side of decreasing conductance
(small $W$). This is contrary to the case of the standard disorder induced
MIT[42], where the conductance of the metal phase (increasing conductance) is
surely larger than that of the insulator phase (decreasing conductance). In
one word, at strong disorder, the transport property of this 2D long range
model is neither a typical insulating phase ($\beta>0$) nor a typical metal
phase ($\beta>0$), but seems to be marginal phase ($\beta\simeq 0$) which is a
mixture of both. This is consistent with the physics shown in the upper row of
Figure 5, where the disorder destroys the fully extended nature of the
wavefunction ($\Gamma\ll 2$), but meanwhile the long range hopping prevents
the wavefunction from a complete localization ($\Gamma\simeq 0$). Such a
marginal state persists over several orders of $W$, typically with a constant
fractal dimension $\Gamma$ [Figure 5 (c)], or a slowly decreasing conductance
$G_{\mathrm{I}}$ [Figure 6 (d)].

Before closing this section, let us add some remarks on $W$ dependencies in
Figure 5 (c) and Figure 6 (d). In the former figure, the typical fractal
dimension of wavefunctions $\Gamma_{\mathrm{typ}}$ is almost constant when the
disorder strength $W$ increases from 128 to 4096, which means the extent of
the wavefunction’s localization is _independent_ of the $W$. However in the
latter figure, the conductance $G_{\mathrm{I}}$ is clearly _decreasing_ with
$W$. There seems to be a discrepancy: is the transport independent of or
decreasing with the disorder strength? To answer this, we need to remember
that the conductance is a contribution from _all_ states around the Fermi
energy[46, 47]. As a result, the conductance is determined both by the
“diffusion coefficient” (which can be characterized by the localization extent
in real space or the level curvature in momentum space) of each wavefunction,
and also by the number of wavefunctions (i.e., the density of states) round
the Fermi energy[53]. Although Figure 5 (c) states that increasing $W$ does
not change the localization extent of each state, it reduces the density of
states by broadening the energy band. This leads to a reduction of the
conductance with increasing $W$ displayed in Figure 6 (d).

## The Isolated State

One should remember that there is a single state (4) isolated far away ($\sim
N^{2}_{x}\gamma$) from the band investigated above. In 1D with periodic
boundary condition, it was found that the wavefunction of this isolated state
is fully extended with a uniform magnitude and phase among all lattice
sites[33]. In 2D and in the fixed boundary condition here, we have checked
(but do not show here) that this uniformity is only slightly changed, and the
energy value of this state is also slightly different from that given in
Equation (4).

Figure 7: (Color online) The fractal dimension $\Gamma_{n}$ of the $n$-th
eigenstate as a function of disorder strength $W$, for a certain sample with
size $40\times 40$ (a), and size $80\times 80$ (b). Blue (red) color is for
the short (long) range model. Notice horizontal scales in two panels are
different.

As a first step, we also study the fractal dimension $\Gamma_{n}$ of
eigenstates defined in equation 9. In Figure 7, red symbol-curves are
$\Gamma_{n}$ of four representative states as functions of disorder strength
$W$: the band bottom (square), the band center (circle), the band top (up
triangle), and the isolated state (down triangle). Panels (a) and (b) are for
a $40\times 40$ and a $80\times 80$ sample respectively. For comparison,
results from a short range model are also shown as blue symbol curves. We can
see that for eigenstates in the band, $\Gamma_{n}$ decays away from 2 rapidly
with increasing disorder.

On the contrary, the fractal dimension for the isolated state is robustly
quantized as 2 until $W$ approaches $\sim N^{2}_{x}\gamma$, the gap from the
band. We remind that it is a _single_ state instead of a flat band consisting
of a continuum of _states_. Thus for an electron in this state, there is _no_
other state for it to be scattered onto, and therefore any back-scattering or
skew-scattering is forbidden. The robustness of this state’s transport is
protected by the large gap from the band, unless the impurity strength is
strong enough $\sim N^{2}_{x}\gamma$ to overcome this large gap. This is the
physical origin of the robust transport of this state, which we call it the
extended isolated state (EIS) hereafter.

Figure 8: (Color online) Two-terminal conductance $G$ as a function of the
Fermi energy in the sample, for the extended isolated state. (a) For two
disorder configurations, C1 (black curve) and C2 (red curve). The
corresponding DOS $\rho$ (referenced to the right axis) are also plotted as
dots. (b) For different disorder strengths $W$. (c) For different sample
sizes. (d) With different energies $E_{L}$ in leads. If not otherwise stated
in the panel, parameters are as follows: $W=16$, $E_{L}=0$ and size $40\times
40$. In panels (b), (c) and (d), all curves have been horizontally shifted to
the same peak center.

The above results are from a sample decoupling from the environment. Now we
still simulate the quantum transports of the EIS by attaching leads to both
ends of the sample. Here, since the condition Equation (2) for perfect
shielding is partially destroyed by the fixed boundary condition, we have
checked that this EIS is not an eigenstate of the short range model (played as
leads). In other words, no state in the lead can actually match the EIS in the
central sample. Therefore in this section, when calculating transports by
using Eqs. (6) and (8), as illustrated in Figure 1 (c), we simply treat the
leads as electron reservoirs with a fixed Fermi energy $E_{L}$ at the band
center and vary the Fermi energy of the sample $E$ around the EIS, e.g.,
$G(E)=\frac{e^{2}}{h}\mathrm{Tr}\left[\Gamma_{L}(E_{L})G^{r}(E)\Gamma_{R}(E_{L})G^{a}(E)\right]$.

In Figure 8 (a), conductances of two disorder configurations [C1 (black curve)
and C2 (red curve)] as functions of $E$ are presented as solid curves, with
size $40\times 40$ and disorder strength $W=16$. The corresponding density of
states (DOS)

$\rho(E)=-\frac{1}{\pi N_{x}^{2}}\mathrm{Im}G^{r}(E),$ (12)

are also plotted as dots. Due to the coupling with leads, both the conductance
and DOS profiles are broadened as smooth peaks. The first obvious observation
is the perfect coincidence between profiles of $G(E)$ and $\rho(E)$, after
appropriate scaling in the vertical direction. This can be understood as
follows. The zero-temperature conductance of a 2D crystal can also be
expressed in the Thouless form [70, 53],

$G=\pi\rho(E)\frac{\partial^{2}E}{\partial k_{x}^{2}}\Big{|}_{k_{x}=0},$ (13)

where $k_{x}$ is the wavevector along the transport direction. Notice the last
factor $\frac{\partial^{2}E}{\partial k_{x}^{2}}$ is a curvature of the band
$E(\bm{k})$. However here, there is only a single state for which no curvature
can be defined, so this factor plays no role and thus has no energy
dependence. As a result for a concrete sample, the energy dependence of the
conductance is simply proportional to that of the DOS, both with the same
spectrum width determined by the imaginary part of the $N$-th eigenvalue of
the dressed (non-Hermitian) Hamiltonian $H+\Sigma_{L}+\Sigma_{R}$.

The second observation from Figure 8 (a) is more important. The peak value of
the conductance is always quantized as unity, which means the EIS carries a
perfectly conducting channel. This reflects the absence of backscattering for
the EIS, as predicted from Figure 7 above. Conductance peaks with other $W$
are shown in Figure 8 (b). The robust transport of EIS persists at least to
$W=4^{4}=256$ for the sample size $40\times 40$, consistent with Figure 7 (a).
To reveal the microscopic origin of this robust transport, we again turn to
the real space distribution of bond currents at the conductance peak, for a
smaller sample as shown in Figure 4 (f). The picture is simple and clear: the
dominating currents are flowing through very long bonds connecting the left
and right boundaries _directly_. This configuration helps the electron to
circumvent any impurities in the sample bulk, leading to a remarkably robust
transport independent of disorder.

The size dependence of the conductance peak is presented in Figure 8 (c),
where larger samples result in sharper conductance peaks. This is not
surprising because a larger size leads to a finer resolution of energy and
thus a smaller broadening. Figure 8 (d) are conductance peaks with different
lead energies $E_{L}$ (respect to its band center). It is interesting to
notice that sharpest peaks correspond to injecting electrons from the band
edge ($E=3.95$) or the band center ($E\sim 0$) of the leads.

## Summary

We numerically investigate the quantum transports of a 2D system with long
range hopping. In the band, the transport is almost identical to that in the
corresponding short range system for a large sample size and weak disorder, as
a manifestation of the cooperative shielding. This shielding is broken at
strong disorder, and the average and fluctuation of the conductance are larger
than those in the short range mode. These can be understood as a better
connectivity and the destruction of path pinning from long range coupling.
Over several orders of strong disorder, the band states exhibit a marginal
feature between metallic and insulating states, which is neither fully
extended nor completely localized. As for the isolated state, its transport of
a unit conducting channel is highly robust against disorder, which can be
depicted as an extended state protected by the large gap from scattering.

## Data availability

On reasonable request, the corresponding author will provide all relevant data
in this paper.

## Code availability

On reasonable request, the corresponding author will provide all numerical
codes in this paper.

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## Acknowledgements

This work was supported by Guangdong Basic and Applied Basic Research
Foundation under No. 2023A1515010698, National Natural Science Foundation of
China under Grant Nos. 12104108 and 11874127, the Joint Fund with Guangzhou
Municipality under Nos. 202201020198 and 202201020137, and the Starting
Research Fund from Guangzhou University under Grant Nos. RQ2020082, RQ 2020083
and 62104360.

## Author contributions statement

S.-S. Wang, Y.-M. Dai and Y.-Y.Z. carried out the theoretical calculations and
wrote the manuscript with the assistance of K. Li, H.-H. Wang and Y.-C. Zhang.
Y.-Y. Zhang and Y.-C. Zhang guided the overall project. All authors reviewed
the manuscript.

## Additional information

Competing financial interests: The authors declare no competing financial
interests.

## Supplementary Material on Quantum Transports in Two-Dimensions with Long
Range Hopping

This supplementary material shows the scaling of conductance for the 1D long
range model.

Figure S1: (Color online) Disorder averaged $\langle\ln G_{\mathrm{I}}\rangle$
as a function of Fermi energy $E$ for different sample sizes: $10\times 10$
(black), $20\times 20$ (red), $40\times 40$ (green), $80\times 80$ (blue), at
different disorder strengths: (a) $W=3$, (b) $W=5$, (c) $W=7$, (d) $W=11$. The
average is over 1000 samples for the largest size and 5000 samples for other
sizes.

In this Supplementary Material, we display the scaling results for weak
disorder. In the extremely weak disorder regime ($W=3$) shown in Figure S1
(a), there are significant finite-size fluctuations for the smallest size
($10\times 10$, black curve). After they are smoothed out in large sizes, it
is clear that $\ln G_{\mathrm{I}}$ is decreasing with increasing sample size,
in the whole energy region. This decreasing is more significant at larger
disorder strengths, as shown in the rest panels of Figure S1, indicating a
strong trend of localization. There seem to be some curve crossings around two
ends of the energy interval. We have checked that this only occurs for the
smallest size $10\times 10$, and that for larger sizes, $\ln G_{\mathrm{I}}$
is still monotonically decreasing. Therefore, within the best numerical
capability we can achieve, all band states with $W\geqslant 3$ is shown to be
localized.