arXiv subjects

Yi-Jun Chang

Publications and source records attributed to Yi-Jun Chang.

At least 55 records · Page 3Linked to original sources

arXiv2023

The Energy Complexity of Diameter and Minimum Cut Computation in Bounded-Genus Networks

This paper investigates the energy complexity of distributed graph problems in multi-hop radio networks, where the energy cost of an algorithm is measured by the maximum number of awake rounds of a vertex. Recent works revealed that some problems, such as broadcast, breadth-first search, and maximal matching, can be solved with energy-efficient algorithms that consume only $\text{poly} \log n$ energy. However, there exist some problems, such as computing the diameter of the graph, that require $Ω(n)$ energy to solve. To improve energy efficiency for these problems, we focus on a special graph class: bounded-genus graphs. We present algorithms for computing the exact diameter, the exact global minimum cut size, and a $(1 \pmε)$-approximate $s$-$t$ minimum cut size with $\tilde{O}(\sqrt{n})$ energy for bounded-genus graphs. Our approach is based on a generic framework that divides the vertex set into high-degree and low-degree parts and leverages the structural properties of bounded-genus graphs to control the number of certain connected components in the subgraph induced by the low-degree part.

cs.DS↗

arXiv2022

Locally Checkable Problems in Rooted Trees

Consider any locally checkable labeling problem $Π$ in rooted regular trees: there is a finite set of labels $Σ$, and for each label $x \in Σ$ we specify what are permitted label combinations of the children for an internal node of label $x$ (the leaf nodes are unconstrained). This formalism is expressive enough to capture many classic problems studied in distributed computing, including vertex coloring, edge coloring, and maximal independent set. We show that the distributed computational complexity of any such problem $Π$ falls in one of the following classes: it is $O(1)$, $Θ(\log^* n)$, $Θ(\log n)$, or $n^{Θ(1)}$ rounds in trees with $n$ nodes (and all of these classes are nonempty). We show that the complexity of any given problem is the same in all four standard models of distributed graph algorithms: deterministic $\mathsf{LOCAL}$, randomized $\mathsf{LOCAL}$, deterministic $\mathsf{CONGEST}$, and randomized $\mathsf{CONGEST}$ model. In particular, we show that randomness does not help in this setting, and the complexity class $Θ(\log \log n)$ does not exist (while it does exist in the broader setting of general trees). We also show how to systematically determine the complexity class of any such problem $Π$, i.e., whether $Π$ takes $O(1)$, $Θ(\log^* n)$, $Θ(\log n)$, or $n^{Θ(1)}$ rounds. While the algorithm may take exponential time in the size of the description of $Π$, it is nevertheless practical: we provide a freely available implementation of the classifier algorithm, and it is fast enough to classify many problems of interest.

cs.DC↗

arXiv2022

Efficient Classification of Locally Checkable Problems in Regular Trees

We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the $[Θ(\log n), Θ(n)]$ region, in two settings. We present one algorithm for unrooted regular trees and another algorithm for rooted regular trees. The algorithms take the description of a locally checkable labeling problem as input, and the running time is polynomial in the size of the problem description. The algorithms decide if the problem is solvable in $O(\log n)$ rounds. If not, it is known that the complexity has to be $Θ(n^{1/k})$ for some $k = 1, 2, \dotsc$, and in this case the algorithms also output the right value of the exponent $k$. In rooted trees in the $O(\log n)$ case we can then further determine the exact complexity class by using algorithms from prior work; for unrooted trees the more fine-grained classification in the $O(\log n)$ region remains an open question.

cs.DC↗

arXiv2022

Experimental Quantum Simulation of Dynamic Localization on Curved Photonic Lattices

Dynamic localization, which originates from the phenomena of particle evolution suppression under an externally applied AC electric field, has been simulated by suppressed light evolution in periodically-curved photonic arrays. However, experimental studies on their quantitative dynamic transport properties and application for quantum information processing are rare. Here we fabricate one-dimensional and hexagonal two-dimensional arrays, both with sinusoidal curvature. We successfully observe the suppressed single-photon evolution patterns, and for the first time measure the variances to study their transport properties. For one-dimensional arrays, the measured variances match both the analytical electric field calculation and the quantum walk Hamiltonian engineering approach. For hexagonal arrays, as anisotropic effective couplings in four directions are mutually dependent, the analytical approach suffers, while quantum walk conveniently incorporates all anisotropic coupling coefficients in the Hamiltonian and solves its exponential as a whole, yielding consistent variances with our experimental results. Furthermore, we implement a nearly complete localization to show that it can preserve both the initial injection and the wave-packet after some evolution, acting as a memory of a flexible time scale in integrated photonics. We demonstrate a useful quantum simulation of dynamic localization for studying their anisotropic transport properties, and a promising application of dynamic localization as a building block for quantum information processing in integrated photonics.

quant-ph↗

arXiv2022

The Energy Complexity of Las Vegas Leader Election

We consider the time and energy complexities of randomized leader election in a multiple-access channel, where the number of devices $n\geq 2$ is unknown. It is well-known that for polynomial-time randomized leader election algorithms with success probability $1-1/poly(n)$, the optimal energy complexity is $Θ(\log\log^*n)$ if receivers can detect collisions, and $Θ(\log^*n)$ otherwise. Without collision detection, all existing randomized leader election algorithms using $o(\log\log n)$ energy are Monte Carlo in that they may fail with some small probability, and they may consume unbounded energy and never halt when they fail. Though the optimal energy complexity of leader election appears to be settled, it is still an open question to attain the optimal $O(\log^*n)$ energy complexity by an efficient Las Vegas algorithm that never fails. In this paper we address this fundamental question. $\textbf{Separation between Monte Carlo and Las Vegas:}$ Without collision detection, we prove that any Las Vegas leader election algorithm with finite expected time complexity must use $Ω(\log\log n)$ energy, establishing a large separation between Monte Carlo and Las Vegas algorithms. $\textbf{Exponential improvement with sender collision detection:}$ In the setting where senders can detect collisions, we design a new leader election algorithm that finishes in $O(\log^{1+ε}n)$ time and uses $O(ε^{-1}\log\log\log n)$ energy in expectation, showing that sender collision detection helps improve the energy complexity exponentially. $\textbf{Optimal deterministic leader election algorithm:}$ As a side result, via derandomization, we show a new deterministic algorithm that takes $O(n\log(N/n))$ time and $O(\log(N/n))$ energy to elect a leader from $n$ devices, where each device has a unique identifier in $[N]$. This algorithm is time-optimal and energy-optimal.

cs.DS↗

arXiv2021

Distributed graph problems through an automata-theoretic lens

The locality of a graph problem is the smallest distance $T$ such that each node can choose its own part of the solution based on its radius-$T$ neighborhood. In many settings, a graph problem can be solved efficiently with a distributed or parallel algorithm if and only if it has a small locality. In this work we seek to automate the study of solvability and locality: given the description of a graph problem $Π$, we would like to determine if $Π$ is solvable and what is the asymptotic locality of $Π$ as a function of the size of the graph. Put otherwise, we seek to automatically synthesize efficient distributed and parallel algorithms for solving $Π$. We focus on locally checkable graph problems; these are problems in which a solution is globally feasible if it looks feasible in all constant-radius neighborhoods. Prior work on such problems has brought primarily bad news: questions related to locality are undecidable in general, and even if we focus on the case of labeled paths and cycles, determining locality is $\mathsf{PSPACE}$-hard (Balliu et al., PODC 2019). We complement prior negative results with efficient algorithms for the cases of unlabeled paths and cycles and, as an extension, for rooted trees. We introduce a new automata-theoretic perspective for studying locally checkable graph problems. We represent a locally checkable problem $Π$ as a nondeterministic finite automaton $\mathcal{M}$ over a unary alphabet. We identify polynomial-time-computable properties of the automaton $\mathcal{M}$ that near-completely capture the solvability and locality of $Π$ in cycles and paths, with the exception of one specific case that is $\mbox{co-$\mathsf{NP}$}$-complete.

cs.DC↗

arXiv2021

Near-Optimal Time-Energy Trade-Offs for Deterministic Leader Election

We consider the energy complexity of the leader election problem in the single-hop radio network model, where each device has a unique identifier in $\{1, 2, \ldots, N\}$. Energy is a scarce resource for small battery-powered devices. For such devices, most of the energy is often spent on communication, not on computation. To approximate the actual energy cost, the energy complexity of an algorithm is defined as the maximum over all devices of the number of time slots where the device transmits or listens. Much progress has been made in understanding the energy complexity of leader election in radio networks, but very little is known about the trade-off between time and energy. $\textbf{Time-energy trade-off:}$ For any $k \geq \log \log N$, we show that a leader among at most $n$ devices can be elected deterministically in $O(k \cdot n^{1+ε}) + O(k \cdot N^{1/k})$ time and $O(k)$ energy if each device can simultaneously transmit and listen, where $ε> 0$ is any small constant. This improves upon the previous $O(N)$-time $O(\log \log N)$-energy algorithm by Chang et al. [STOC 2017]. We provide lower bounds to show that the time-energy trade-off of our algorithm is near-optimal. $\textbf{Dense instances:}$ For the dense instances where the number of devices is $n = Θ(N)$, we design a deterministic leader election algorithm using only $O(1)$ energy. This improves upon the $O(\log^* N)$-energy algorithm by Jurdziński et al. [PODC 2002] and the $O(α(N))$-energy algorithm by Chang et al. [STOC 2017]. More specifically, we show that the optimal deterministic energy complexity of leader election is $Θ\left(\max\left\{1, \log \frac{N}{n}\right\}\right)$ if the devices cannot simultaneously transmit and listen, and it is $Θ\left(\max\left\{1, \log \log \frac{N}{n}\right\}\right)$ if they can.

cs.DS↗

arXiv2021

Strong-Diameter Network Decomposition

Network decomposition is a central concept in the study of distributed graph algorithms. We present the first polylogarithmic-round deterministic distributed algorithm with small messages that constructs a strong-diameter network decomposition with polylogarithmic parameters. Concretely, a ($C$, $D$) strong-diameter network decomposition is a partitioning of the nodes of the graph into disjoint clusters, colored with $C$ colors, such that neighboring clusters have different colors and the subgraph induced by each cluster has a diameter at most $D$. In the weak-diameter variant, the requirement is relaxed by measuring the diameter of each cluster in the original graph, instead of the subgraph induced by the cluster. A recent breakthrough of Rozhoň and Ghaffari [STOC 2020] presented the first $\text{poly}(\log n)$-round deterministic algorithm for constructing a weak-diameter network decomposition where $C$ and $D$ are both in $\text{poly}(\log n)$. Their algorithm uses small $O(\log n)$-bit messages. One can transform their algorithm to a strong-diameter network decomposition algorithm with similar parameters. However, that comes at the expense of requiring unbounded messages. The key remaining qualitative question in the study of network decompositions was whether one can achieve a similar result for strong-diameter network decompositions using small messages. We resolve this question by presenting a novel technique that can transform any black-box weak-diameter network decomposition algorithm to a strong-diameter one, using small messages and with only moderate loss in the parameters.

cs.DS↗

arXiv2021

Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

We study connections between distributed local algorithms, finitary factors of iid processes, and descriptive combinatorics in the context of regular trees. We extend the Borel determinacy technique of Marks coming from descriptive combinatorics and adapt it to the area of distributed computing. Using this technique, we prove deterministic distributed $Ω(\log n)$-round lower bounds for problems from a natural class of homomorphism problems. Interestingly, these lower bounds seem beyond the current reach of the powerful round elimination technique responsible for all substantial locality lower bounds of the last years. Our key technical ingredient is a novel ID graph technique that we expect to be of independent interest. We prove that a local problem admits a Baire measurable coloring if and only if it admits a local algorithm with local complexity $O(\log n)$, extending the classification of Baire measurable colorings of Bernshteyn. A key ingredient of the proof is a new and simple characterization of local problems that can be solved in $O(\log n)$ rounds. We complement this result by showing separations between complexity classes from distributed computing, finitary factors, and descriptive combinatorics. Most notably, the class of problems that allow a distributed algorithm with sublogarithmic randomized local complexity is incomparable with the class of problems with a Borel solution. We hope that our treatment will help to view all three perspectives as part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn -- arXiv 2004.04905].

math.CO↗

arXiv2020

Constructing higher-order topological states in higher dimension

Higher-order topological phase as a generalization of Berry phase attracts an enormous amount of research. The current theoretical models supporting higher-order topological phases, however, cannot give the connection between lower and higher-order topological phases when extending the lattice from lower to higher dimensions. Here, we theoretically propose and experimentally demonstrate a topological corner state constructed from the edge states in one dimensional lattice. The two-dimensional square lattice owns independent spatial modulation of coupling in each direction, and the combination of edge states in each direction come up to the higher-order topological corner state in two-dimensional lattice, revealing the connection of topological phase in lower and higher dimensional lattices. Moreover, the topological corner states in two-dimensional lattice can also be viewed as the dimension-reduction from a four-dimensional topological phase characterized by vector Chern number, considering two modulation phases as synthetic dimensions in Aubry-Andre-Harper model discussed as example here. Our work deeps the understanding to topological phases breaking through the lattice dimension, and provides a promising tool constructing higher topological phases in higher dimensional structures.

quant-ph↗

arXiv2020

Tight Distributed Listing of Cliques

Much progress has recently been made in understanding the complexity landscape of subgraph finding problems in the CONGEST model of distributed computing. However, so far, very few tight bounds are known in this area. For triangle (i.e., 3-clique) listing, an optimal $\tilde{O}(n^{1/3})$-round distributed algorithm has been constructed by Chang et al.~[SODA 2019, PODC 2019]. Recent works of Eden et al.~[DISC 2019] and of Censor-Hillel et al.~[PODC 2020] have shown sublinear algorithms for $K_p$-listing, for each $p \geq 4$, but still leaving a significant gap between the upper bounds and the known lower bounds of the problem. In this paper, we completely close this gap. We show that for each $p \geq 4$, there is an $\tilde{O}(n^{1 - 2/p})$-round distributed algorithm that lists all $p$-cliques $K_p$ in the communication network. Our algorithm is \emph{optimal} up to a polylogarithmic factor, due to the $\tildeΩ(n^{1 - 2/p})$-round lower bound of Fischer et al.~[SPAA 2018], which holds even in the CONGESTED CLIQUE model. Together with the triangle-listing algorithm by Chang et al.~[SODA 2019, PODC 2019], our result thus shows that the round complexity of $K_p$-listing, for all $p$, is the same in both the CONGEST and CONGESTED CLIQUE models, at $\tildeΘ(n^{1 - 2/p})$ rounds. For $p=4$, our result additionally matches the $\tildeΩ(n^{1/2})$ lower bound for $K_4$-\emph{detection} by Czumaj and Konrad [DISC 2018], implying that the round complexities for detection and listing of $K_4$ are equivalent in the CONGEST model.

cs.DC↗

arXiv2020

Deterministic Distributed Expander Decomposition and Routing with Applications in Distributed Derandomization

There is a recent exciting line of work in distributed graph algorithms in the $\mathsf{CONGEST}$ model that exploit expanders. All these algorithms so far are based on two tools: expander decomposition and expander routing. An $(ε,ϕ)$-expander decomposition removes $ε$-fraction of the edges so that the remaining connected components have conductance at least $ϕ$, i.e., they are $ϕ$-expanders, and expander routing allows each vertex $v$ in a $ϕ$-expander to very quickly exchange $\text{deg}(v)$ messages with any other vertices, not just its local neighbors. In this paper, we give the first efficient deterministic distributed algorithms for both tools. We show that an $(ε,ϕ)$-expander decomposition can be deterministically computed in $\text{poly}(ε^{-1}) n^{o(1)}$ rounds for $ϕ= \text{poly}(ε) n^{-o(1)}$, and that expander routing can be performed deterministically in $\text{poly}(ϕ^{-1})n^{o(1)}$ rounds. Both results match previous bounds of randomized algorithms by [Chang and Saranurak, PODC 2019] and [Ghaffari, Kuhn, and Su, PODC 2017] up to subpolynomial factors. Consequently, we derandomize existing distributed algorithms that exploit expanders. We show that a minimum spanning tree on $n^{o(1)}$-expanders can be constructed deterministically in $n^{o(1)}$ rounds, and triangle detection and enumeration on general graphs can be solved deterministically in $O(n^{0.58})$ and $n^{2/3 + o(1)}$ rounds, respectively. We also give the first polylogarithmic-round randomized algorithm for constructing an $(ε,ϕ)$-expander decomposition in $\text{poly}(ε^{-1}, \log n)$ rounds for $ϕ= 1 / \text{poly}(ε^{-1}, \log n)$. The previous algorithm by [Chang and Saranurak, PODC 2019] needs $n^{Ω(1)}$ rounds for any $ϕ\ge 1/\text{poly}\log n$.

cs.DS↗

arXiv2020

The Energy Complexity of BFS in Radio Networks

We consider a model of energy complexity in Radio Networks in which transmitting or listening on the channel costs one unit of energy and computation is free. This simplified model captures key aspects of battery-powered sensors: that battery life is most influenced by transceiver usage, and that at low transmission powers, the actual cost of transmitting and listening are very similar. The energy complexity of tasks in single-hop networks is well understood. Recent work of Chang et al. considered energy complexity in multi-hop networks and showed that $\mathsf{Broadcast}$ admits an energy-efficient protocol, by which we mean each of the $n$ nodes in the network spends $O(\text{polylog}(n))$ energy. This work left open the strange possibility that all natural problems in multi-hop networks might admit such an energy-efficient solution. In this paper we prove that the landscape of energy complexity is rich enough to support a multitude of problem complexities. Whereas $\mathsf{Broadcast}$ can be solved by an energy-efficient protocol, exact computation of $\mathsf{Diameter}$ cannot, requiring $Ω(n)$ energy. Our main result is that $\mathsf{Breadth First Search}$ has sub-polynomial energy complexity at most $2^{O(\sqrt{\log n\log\log n})}=n^{o(1)}$; whether it admits an efficient $O(\text{polylog}(n))$-energy protocol is an open problem. Our main algorithm involves recursively solving a generalized BFS problem on a cluster graph introduced by Miller, Peng, and Xu. In this application, we make crucial use of a close relationship between distances in this cluster graph, and distances in the original network. This relationship is new and may be of independent interest.

cs.DS↗

arXiv2020

Protecting Quantum Superposition and Entanglement with Photonic Higher-Order Topological Crystalline Insulator

Higher-order topological insulator, as a newly found non-trivial material and structure, possesses a topological phase beyond the bulk-boundary correspondence. Here, we present an experimental observation of photonic higher-order topological crystalline insulator and its topological protection to quantum superposition and entanglement in a two-dimensional lattice. By freely writing the insulator structure with femtosecond laser and directly measuring evolution dynamics with single-photon imaging techniques, we are able to observe the distinct features of the topological corner states in C_4 and C_2 photonic lattice symmetry. Especially, we propose and experimentally identify the topological corner states by exciting the photonic lattice with single-photon superposition state, and we examine the protection impact of topology on quantum entanglement for entangled photon states. The single-photon dynamics and the protected entanglement reveal an intrinsic topological protection mechanism isolating multi-partite quantum states from diffusion-induced decoherence. The higher-order topological crystalline insulator, built-in superposition state generation, heralded single-photon imaging and quantum entanglement demonstrated here link topology, material, and quantum physics, opening the door to wide investigations of higher-order topology and applications of topological enhancement in genuine quantum regime.

quant-ph↗

arXiv2020

Observing Movement of Dirac Cones from Single-Photon Dynamics

Graphene with honeycomb structure, being critically important in understanding physics of matter, exhibits exceptionally unusual half-integer quantum Hall effect and unconventional electronic spectrum with quantum relativistic phenomena. Particularly, graphene-like structure can be used for realizing topological insulator which inspires an intrinsic topological protection mechanism with strong immunity for maintaining coherence of quantum information. These various peculiar physics arise from the unique properties of Dirac cones which show high hole degeneracy, massless charge carriers and linear intersection of bands. Experimental observation of Dirac cones conventionally focuses on the energy-momentum space with bulk measurement. Recently, the wave function and band structure have been mapped into the real-space in photonic system, and made flexible control possible. Here, we demonstrate a direct observation of the movement of Dirac cones from single-photon dynamics in photonic graphene under different biaxial strains. Sharing the same spirit of wave-particle nature in quantum mechanics, we identify the movement of Dirac cones by dynamically detecting the edge modes and extracting the diffusing distance of the packets with accumulation and statistics on individual single-particle registrations. Our results of observing movement of Dirac cones from single-photon dynamics, together with the method of direct observation in real space by mapping the band structure defined in momentum space, pave the way to understand a variety of artificial structures in quantum regime.

cond-mat.mes-hall↗

arXiv2020

Quantum Slide and NAND Tree on a Photonic Chip

In the age of post-Moore era, the next-generation computing model would be a hybrid architecture consisting of different physical components such as photonic chips. In 2008, it has been proposed that the solving of NAND-tree problem can be sped up by quantum walk. Such scheme is groundbreaking due to the universality of NAND gate. However, experimental demonstration has never been achieved so far, mostly due to the challenge in preparing the propagating initial state. Here we propose an alternative solution by including a structure called "quantum slide", where a propagating Gaussian wave-packet can be generated deterministically along a properly-engineered chain. In this way, the optical computation can be achieved with ordinary laser light instead of single photon, and the output can be obtained by single-shot measurements instead of repeated quantum measurements. In our experimental demonstration, the optical NAND-tree is capable of solving computational problems with a total of four input bits, based on the femtosecond laser 3D direct-writing technique on a photonic chip. These results remove one main roadblock to photonic NAND-tree computation, and the construction of quantum slide may find other interesting applications in quantum information and quantum optics.

quant-ph↗

arXiv2020

Streaming Complexity of Spanning Tree Computation

The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an $n$-node input graph to be read sequentially in $p$ passes using $\tilde{O}(n)$ space. In this model, some graph problems, such as spanning trees and $k$-connectivity, can be exactly solved in a single pass; while other graph problems, such as triangle detection and unweighted all-pairs shortest paths, are known to require $\tildeΩ(n)$ passes to compute. For many fundamental graph problems, the tractability in these models is open. In this paper, we study the tractability of computing some standard spanning trees. Our results are: (1) Maximum-Leaf Spanning Trees. This problem is known to be APX-complete with inapproximability constant $ρ\in[245/244,2)$. By constructing an $\varepsilon$-MLST sparsifier, we show that for every constant $\varepsilon > 0$, MLST can be approximated in a single pass to within a factor of $1+\varepsilon$ w.h.p. (albeit in super-polynomial time for $\varepsilon \le ρ-1$ assuming $\mathrm{P} \ne \mathrm{NP}$). (2) BFS Trees. It is known that BFS trees require $ω(1)$ passes to compute, but the naïve approach needs $O(n)$ passes. We devise a new randomized algorithm that reduces the pass complexity to $O(\sqrt{n})$, and it offers a smooth tradeoff between pass complexity and space usage. (3) DFS Trees. The current best algorithm by Khan and Mehta {[}STACS 2019{]} takes $\tilde{O}(h)$ passes, where $h$ is the height of computed DFS trees. Our contribution is twofold. First, we provide a simple alternative proof of this result, via a new connection to sparse certificates for $k$-node-connectivity. Second, we present a randomized algorithm that reduces the pass complexity to $O(\sqrt{n})$, and it also offers a smooth tradeoff between pass complexity and space usage.

cs.DS↗

arXiv2019

Observation of Magic Angle and Wall State in Twisted Bilayer Photonic Graphene

Graphene, a one-layer honeycomb lattice of carbon atoms, exhibits unconventional phenomena and attracts much interest since its discovery. Recently, an unexpected Mott-like insulator state induced by moiré pattern and a superconducting state are observed in magic-angle-twisted bilayer graphene, especially, without correlations between electrons, which gives more hints for the understanding and investigation of strongly correlated phenomena. The photon as boson, behaving differently with fermion, can also retrieve the unconventional phenomena of graphene, such as the bearded edge state which is even never been observed in graphene due to the unstability. Here, we present a direct observation of magic angle and wall state in twisted bilayer photonic graphene. We successfully observe the strong localization and rapid diffusion of photon at the regions with AA and AB stacking order around the magic angle, respectively. Most importantly, we find a wall state showing the photon distribution distinctly separate at the regions with AA and AB/BA stacking order in the lowest-energy band. The mechanism underlying the wall states may help to understand the existence of both Mott-like insulating state and superconducting state in magic-angle twisted bilayer graphene. The accessibility of magic angle in twisted bilayer photonic graphene adds the boson behavior into graphene superlattice and the observation of wall state will also deep the understanding of matter.

cond-mat.mes-hall↗