A. D. Mirlin, F. Evers, I. V. Gornyi, P. M. Ostrovsky
The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed. We focus on the character of criticality as well as on underlying symmetries and topologies that are crucial for understanding phase diagrams and the critical behavior.
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We present an analytic theory of quantum criticality in the quasi one-dimensional topological Anderson insulators of class AIII and BDI. We describe the systems in terms of two parameters $(g,χ)$ representing localization and topological properties, respectively. Surfaces of half-integer valued $χ$ define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow describing the class A quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entir…
Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $ν$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulat…
The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed. The term ``Anderson transition'' is understood in a broad sense, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states. The emphasis is put on recent developments, which include: multifractality of critical wave functions, criticality in the power-law random banded matrix model, symmetry classification of disordered electronic systems, mechanisms of criticality in quasi-one-dimensional and two-dimensional systems and survey of …
We investigate the emergence of complex dynamics in networks with heavy-tailed connectivity by developing a non-Hermitian random matrix theory. We uncover the existence of an extended critical regime of spatially multifractal fluctuations between the quiescent and active phases. This multifractal critical phase combines features of localization and delocalization and differs from the edge of chaos in classical networks by the appearance of universal hallmarks of Anderson criticality over an extended region in phase space. We show that the rich nonlinear response properties of the extended crit…
We present a large N solution of a microscopic model describing the Mott-Anderson transition on a finite-coordination Bethe lattice. Our results demonstrate that strong spatial fluctuations, due to Anderson localization effects, dramatically modify the quantum critical behavior near disordered Mott transitions. The leading critical behavior of quasiparticle wavefunctions is shown to assume a universal form in the full range from weak to strong disorder, in contrast to disorder-driven non-Fermi liquid ("electronic Griffiths phase") behavior, which is found only in the strongly correlated regime…
In contrast to finite dimensions where disordered systems display multifractal statistics only at criticality, the tree geometry induces multifractal statistics for disordered systems also off criticality. For the Anderson tight-binding localization model defined on a tree of branching ratio K=2 with $N$ generations, we consider the Miller-Derrida scattering geometry [J. Stat. Phys. 75, 357 (1994)], where an incoming wire is attached to the root of the tree, and where $K^{N}$ outcoming wires are attached to the leaves of the tree. In terms of the $K^{N}$ transmission amplitudes $t_j$, the tota…
We present a complete analytical and numerical solution of the Typical Medium Theory (TMT) for the Anderson metal-insulator transition. In this theory, we self-consistently calculate the typical amplitude of the electron wave-We present a complete analytical and numerical solution of the Typical Medium Theory (TMT) for the Anderson metal-insulator transition. This approach self-consistently calculates the typical amplitude of the electronic wave-functions, thus representing the conceptually simplest order-parameter theory for the Anderson transition. We identify all possible universality class…
We review recent numerical studies of two-dimensional (2D) Dirac fermion theories that exhibit an unusual mechanism of topological protection against Anderson localization. These describe surface-state quasiparticles of time-reversal invariant, three-dimensional (3D) topological superconductors (TSCs), subject to the effects of quenched disorder. Numerics reveal a surprising connection between 3D TSCs in classes AIII, CI, and DIII, and 2D quantum Hall effects in classes A, C, and D. Conventional arguments derived from the non-linear $σ$-model picture imply that most TSC surface states should A…
We study the propagation of coherent waves in a nonlinearly-induced random potential, and find regimes of self-organized criticality and other regimes where the nonlinear equivalent of Anderson localization prevails. The regime of self-organized criticality leads to power-law decay of transport [Phys. Rev. Lett. 121, 233901 (2018)], whereas the second regime exhibits exponential decay.
We study numerically scattering and transport statistical properties of the one-dimensional Anderson model at the metal-insulator transition described by the Power-law Banded Random Matrix (PBRM) model at criticality. Within a scattering approach to electronic transport, we concentrate on the case of a small number of single-channel attached leads. We observe a smooth transition from localized to delocalized behavior in the average scattering matrix elements, the conductance probability distribution, the variance of the conductance, and the shot noise power by varying $b$ (the effective bandwi…
Topological Anderson transitions, which are direct phase transitions between topologically distinct Anderson localised phases, allow for criticality in 1D disordered systems. We analyse the statistical properties of an emsemble of critical wavefunctions at such transitions. We find that the local moments are strongly inhomogeneous, with significant amplification towards the edges of the system. In particular, we obtain an analytic expression for the spatial profile of the local moments which is valid at all topological Anderson transitions in 1D, as we verify by direct comparison with numerica…
Topological surface-state destruction via trivializing proximity effect: Lattice localization despite continuum criticality
研究揭示了在三维类CI拓扑晶格模型中，表面态易受弱随机性和额外平凡2D带的混合影响而安德森局域化，尽管连续模型表现出临界性。
An extensive numerical analysis of the scattering and transport properties of the power-law banded random matrix model (PBRM) at criticality in the presence of orthogonal, unitary, and symplectic symmetries is presented. Our results show a good agreement with existing analytical expressions in the metallic regime and with heuristic relations widely used in studies of the PBRM model in the presence of orthogonal and unitary symmetries. Moreover, our results confirm that the multifractal behavior of disordered systems at criticality can be probed by measuring scattering and transport properties,…
q-Gaussian Crossover in Overlap Spectra towards 3D Edwards-Anderson Criticality
研究通过谱方法揭示三维Edwards-Anderson自旋玻璃中q-Gaussian分布的临界行为，发现q指数从-1到1的变化，为理解无序系统中的临界现象提供新视角。
The quantum kicked rotor (QKR) driven by $d$ incommensurate frequencies realizes the universality class of $d$-dimensional disordered metals. For $d>3$, the system exhibits an Anderson metal-insulator transition which has been observed within the framework of an atom optics realization. However, the absence of genuine randomness in the QKR reflects in critical phenomena beyond those of the Anderson universality class. Specifically, the system shows strong sensitivity to the algebraic properties of its effective Planck constant $\tilde h \equiv 4π/q$. For integer $q$, the system may be in a glo…
Non-Hermitian Delocalization Realizes Random Dirac Criticality in One Dimension
通过非厄米性，一维系统在周期边界条件下实现随机狄拉克费米子普适类的内在临界性，揭示了谱拓扑驱动的退局域化机制。
We investigate boundary multifractality of critical wave functions at the Anderson metal-insulator transition in two-dimensional disordered non-interacting electron systems with spin-orbit scattering. We show numerically that multifractal exponents at a corner with an opening angle θ=3π/2 are directly related to those near a straight boundary in the way dictated by conformal symmetry. This result extends our previous numerical results on corner multifractality obtained for θ< πto θ> π, and gives further supporting evidence for conformal invariance at criticality. We also propose a refinement o…
Employing scaling analysis of the localization length, we deduce the critical exponent of the metal-topological insulator (TI) transitions induced by disorder. The obtained exponent nu~2.7 shows no conspicuous deviation from the value established for metal-ordinary insulator transitions in systems of the symplectic class. We investigate the topological phase diagram upon carrier doping to reveal the nature of the so-called topological Anderson insulator (TAI) region. The critical exponent of the metal-TAI transition is also first estimated, shown to be undistinguishable from the above value wi…
We study the interplay of superconductivity and a wide spectrum of critical (multifractal) wave functions ("spectrum-wide quantum criticality," SWQC) in the one-dimensional Aubry-André and power-law random-banded matrix models with attractive interactions, using self-consistent BCS theory. We find that SWQC survives the incorporation of attractive interactions at the Anderson localization transition, while the pairing amplitude is maximized near this transition in both models. Our results suggest that SWQC, recently discovered in two-dimensional topological surface-state and nodal superconduct…
We establish the theory of critical transport in amorphous Chern insulators and show that it lies beyond the current paradigm of topological criticality epitomized by the quantum Hall transitions. We consider models of Chern insulators on percolation-type random lattices where the average density determines the statistical properties of geometry. While these systems display a two-parameter scaling behaviour near the critical density, the critical exponents and the critical conductance distributions are strikingly nonuniversal. Our analysis indicates that the amorphous topological criticality r…
Using a cutoff-free formulation of the coherent transport theory, we show that the interference terms at the origin of localization strongly affect the transport anisotropy. In contrast to the common hypothesis, we then find that the anisotropies of incoherent and coherent diffusion are significantly different, in particular at criticality. There, we show that the coherent transport anisotropy is mainly determined by the properties of the disorder-averaged effective scattering medium while the incoherent transport contributions become irrelevant.
Topological quantum criticality of the disordered Chern insulator
该研究针对二维无序拓扑陈绝缘体，提出了弱无序下识别临界态的几何判据并解析推广至强无序区，绘制出能量、拓扑参数与无序强度相空间中的临界曲面，辅以数值分析验证了临界态位置与多重分形性质。
In this paper, we explore quantum criticality in the disordered Aubry-André (AA) model. For the pure AA model, it is well-known that it hosts a critical point separating an extended phase and a localized insulator phase by tuning the strength of the quasiperiodic potential. Here we unearth that the disorder strength $Δ$ contributes an independent relevant direction near the critical point of the AA model. Our scaling analyses show that the localization length $ξ$ scales with $Δ$ as $ξ\proptoΔ^{-ν_Δ}$ with $ν_Δ$ a new critical exponent, which is estimated to be $ν_Δ\approx0.46$. This value is r…
The scaling theory of Anderson localization is based on a global conductance $g_L$ that remains a random variable of order O(1) at criticality. One realization of such a conductance is the Landauer transmission for many transverse channels. On the other hand, the statistics of the one-channel Landauer transmission between two local probes is described by a multifractal spectrum that can be related to the singularity spectrum of individual eigenstates. To better understand the relations between these two types of results, we consider various scattering geometries that interpolate between these …
The article reviews the physics of Anderson localization on random regular graphs (RRG) and its connections to many-body localization (MBL) in disordered interacting systems. Properties of eigenstate and energy level correlations in delocalized and localized phases, as well at criticality, are discussed. In the many-body part, models with short-range and power-law interactions are considered, as well as the quantum-dot model representing the limit of the "most long-range" interaction. Central themes -- which are common to the RRG and MBL problems -- include ergodicity of the delocalized phase,…
The probability density function (PDF) for critical wavefunction amplitudes is studied in the three-dimensional Anderson model. We present a formal expression between the PDF and the multifractal spectrum f(alpha) in which the role of finite-size corrections is properly analyzed. We show the non-gaussian nature and the existence of a symmetry relation in the PDF. From the PDF, we extract information about f(alpha) at criticality such as the presence of negative fractal dimensions and we comment on the possible existence of termination points. A PDF-based multifractal analysis is hence shown to…
We analyze the scattering properties of a periodic one-dimensional system at criticality represented by the so-called power-law banded random matrix model at the metal insulator transition. We focus on the scaling of Wigner delay times $τ$ and resonance widths $Γ$. We found that the typical values of $τ$ and $Γ$ (calculated as the geometric mean) scale with the system size $L$ as $τ^{\tiny typ}\propto L^{D_1}$ and $Γ^{\tiny typ} \propto L^{-(2-D_2)}$, where $D_1$ is the information dimension and $D_2$ is the correlation dimension of eigenfunctions of the corresponding closed system.
The anomalous Floquet Anderson insulator (AFAI) is a two dimensional periodically driven system in which static disorder stabilizes two topologically distinct phases in the thermodynamic limit. The presence of a unit-conducting chiral edge mode and the essential role of disorder induced localization are reminiscent of the integer quantum Hall (IQH) effect. At the same time, chirality in the AFAI is introduced via an orchestrated driving protocol, there is no magnetic field, no energy conservation, and no (Landau level) band structure. In this paper we show that in spite of these differences th…
For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green functions of the remaining sites [H. Aoki, J. Phys. C13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions $d=2,3$, we study numerically the statistical properties of the renormalized on-site energies $ε$ and of the renormalized hoppings $V$ as a function of the linear size $L$. We find that the renormalized on-site energies $ε$ remain finite in the localized phase in $d=2,3$ and at criticality ($d=3$), with a finite density …
Exploring quantum criticality in a 4D quantum disordered system
本研究利用超冷原子量子模拟器首次观测并表征了四维 Anderson 转变，通过测量临界指数验证其服从 Wegner 标度律，首次实验证明该转变在四维下仍具非平均场特性。
We consider a simple model of quantum disorder in two dimensions, characterized by a long-range site-to-site hopping. The system undergoes a metal-insulator transition -- its eigenfunctions change from being extended to being localized. We demonstrate that at the point of the transition the eigenfunctions do not become fractal. Their density moments do not scale as a power of the system size. Instead, in one of the considered limits our result suggests a power of the logarithm of the system size. In this regard, the transition differs from a similar one in the one-dimensional version of the sa…
In the absence of spin-orbit coupling, the conventional dogma of Anderson localization asserts that all states localize in two dimensions, with a glaring exception: the quantum Hall plateau transition (QHPT). In that case, the localization length diverges and interference-induced quantum-critical spatial fluctuations appear at all length scales. Normally QHPT states occur only at isolated energies; accessing them therefore requires fine-tuning of the electron density or magnetic field. In this paper we show that QHPT states can be realized throughout an energy continuum, i.e. as an "energy sta…
For Anderson localization on the Cayley tree, we study the statistics of various observables as a function of the disorder strength $W$ and the number $N$ of generations. We first consider the Landauer transmission $T_N$. In the localized phase, its logarithm follows the traveling wave form $\ln T_N \simeq \bar{\ln T_N} + \ln t^*$ where (i) the disorder-averaged value moves linearly $\bar{\ln (T_N)} \simeq - \frac{N}{ξ_{loc}}$ and the localization length diverges as $ξ_{loc} \sim (W-W_c)^{-ν_{loc}}$ with $ν_{loc}=1$ (ii) the variable $t^*$ is a fixed random variable with a power-law tail $P^*(…
Critical phenomena and quantum phase transitions are paradigmatic concepts in modern condensed matter physics. A central example in the field of mesoscopic physics is the localization-delocalization (metal-insulator) quantum phase transition driven by disorder -- the Anderson transition. Although the notion of localization has appeared half a century ago, this field is still full of surprising new developments. The most recent arenas where novel peculiar localization phenomena have been studied are graphene and topological insulators, i.e., bulk insulators with delocalized (topologically prote…
We investigate the possibility of an Anderson type transition in the quantum kicked rotor with a smooth potential due to dynamical localization of the wavefunctions. Our results show the typical characteristics of a critical behavior i.e multifractal eigenfunctions and a scale-invariant level-statistics at a critical kicking strength which classically corresponds to a mixed regime. This indicates the existence of a localization to delocalization transition in the quantum kicked rotor. Our study also reveals the possibility of other type of transitions in the quantum kicked rotor, with a kickin…
We investigate the effect of nonmagnetic disorder on the stability of the algebraic spin liquid ($ASL$) by deriving an effective field theory, nonlinear $σ$ model ($NLσM$). We find that the anomalous critical exponent characterizing the criticality of the $ASL$ causes an anomalous gradient in the $NLσM$. We show that the sign of the anomalous gradient exponent or the critical exponent of the $ASL$ determines the stability of the "dirty" $ASL$. A positive exponent results in an unstable fixed point separating delocalized and localized phases, which is consistent with our previous study [Phys. R…
Numerical and theoretical studies of the Anderson transition in the chiral symmetry classes with weak topology
研究具有一维弱拓扑的手征对称类中安德森转变，结合晶格模型模拟与场论研究。模拟扩展转移矩阵分析，场论扩展二维重整化群分析。发现弱拓扑诱导准局域相，数值观测的准局域相或为各向同性假设产物，识别需适应各向异性缩放的方法。
Separate surface and bulk topological Anderson localization transitions in disordered axion insulators
本研究通过具有量化Hall电导表面的三维axion insulator，揭示了无序条件下体相与表面迁移率能隙的解耦演化，从而在GUE中实现了二维未量化anomalous Hall metal。
Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two---a metal-insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of th…
We consider a two-dimensional strongly localized system defined in a half-space and whose transfer integral in the edge can be different than in the bulk. We predict an unbinding transition, as the edge transfer integral is varied, from a phase where conduction paths are distributed across the bulk to a bound phase where propagation is mainly along the edge. At criticality the logarithm of the conductance follows the $F_1$ Tracy-Widom distribution. We verify numerically these predictions for both the Anderson and the Nguyen, Spivak and Shklovskii models. We also check that for a half-space, i.…
We investigate numerically the inverse participation ratio, $P_2$, of the 3D Anderson model and of the power-law random banded matrix (PRBM) model at criticality. We found that the variance of $\ln P_2$ scales with system size $L$ as $σ^2(L)=σ^2(\infty)-A L^{-D_2/2d}$, being $D_2$ the correlation dimension and $d$ the system dimension. Therefore the concept of a correlation dimension is well defined in the two models considered. The 3D Anderson transition and the PRBM transition for $b=0.3$ (see the text for the definition of $b$) are fairly similar with respect to all critical magnitudes stud…
Testing the Localization Landscape Theory on the Bethe Lattice
本文在贝叶斯晶格上验证了定位景观理论，发现其与安德森局域化转变的临界行为存在根本差异，揭示了理论在捕捉量子临界性质上的局限性。
Berezinskii-Kosterlitz-Thouless localization-localization transitions in disordered two-dimensional quantized quadrupole insulators
本文研究了二维量子化四极绝缘体中从拓扑角态到正常Anderson局域态的局域-局域转变，通过有限尺寸标度分析揭示了其Hermitian与非Hermitian系统均具有Berezinskii-Kosterlitz-Thouless类型的临界特征。
Two-dimensional Disordered Projected Branes: Stability and Quantum Criticality via Dimensional Reduction
本研究通过Schur分解从三维立方晶格构建二维无序投影膜，证实其可重现三维体系的量子相图与相近的临界指数，为传统二维晶格无法实现的稳定金属、半金属相提供了新的量子全息体系，并指出光子晶格可用于实验验证。
Short-range interactions are irrelevant at the quasiperiodic-driven Luttinger Liquid to Anderson Glass transition
该研究通过多体推广的局域化相图计算方法证明，准周期驱动的 Luttinger 液体—Anderson 玻璃转变中短程相互作用无关，临界指数与非相互作用情形一致。
It has been proposed that adaptation in complex systems is optimized at the critical boundary between ordered and disordered dynamical regimes. Here, we review models of evolving dynamical networks that lead to self-organization of network topology based on a local coupling between a dynamical order parameter and rewiring of network connectivity, with convergence towards criticality in the limit of large network size $N$. In particular, two adaptive schemes are discussed and compared in the context of Boolean Networks and Threshold Networks: 1) Active nodes loose links, frozen nodes aquire new…
Classical versus quantum Anderson localization in disordered systems
通过比较经典标量波与电子紧束缚模型，发现经典波系统构成独特的约束无序类，声学求和规则阻止其与标准电子无序模型的直接对应，并揭示了局域化相图的定性差异。
Spreading processes on networks are ubiquitous in both human-made and natural systems. Understanding their behavior is of broad interest; from the control of epidemics to understanding brain dynamics. While in some cases there exists a clear separation of time scales between the propagation of a single spreading cascade and the initiation of the next -- such that spreading can be modelled as directed percolation or a branching process -- there are also processes for which this is not the case, such as zoonotic diseases or spiking cascades in neural networks. For a large class of relevant netwo…
Large scale neural quantum states reveal the interplay between superconductivity and quantum criticality in the Hofstadter-Hubbard model
研究采用含432个格点的神经量子态，揭示三角晶格Hofstadter-Hubbard模型中超导性与量子临界性的相互作用，发现掺杂后相变两侧均存在拓扑超导体，超导体能量尺度由相变邻近性决定。
Unveiling quantum criticality of disordered Aubry-André-Harper models via typical fidelity susceptibility
本研究通过典型保真度磁化率揭示无序AAH模型的局域化转变与量子临界性，发现不同势的复杂性影响局域化层级，并提取出不同于无无序AAH和Anderson模型的临界指数，为理解准周期系统提供统一框架。
In contrast to the Landau paradigm, a metal-insulator transition (MIT), driven purely by competition between itinerance and localization and unaccompanied by any conventional (e.g, magnetic) order-disorder instabilities, admits no obvious local order parameter. Here, we present a detailed analysis of the quantum criticality in magneto-transport data on the alloy Nb$_{1-x}$Ti$_{x}$N across a Ti-doping-driven a MIT. We demonstrate, for the first time, clear and novel quantum criticality reflected in the full conductivity tensor across the MIT. Wide ranging, comprehensive accord with recent theor…
We provide a review of recently-develop dynamical mean-field theory (DMFT) approaches to the general problem of strongly correlated electronic systems with disorder. We first describe the standard DMFT approach, which is exact in the limit of large coordination, and explain why in its simplest form it cannot capture either Anderson localization or the glassy behavior of electrons. Various extensions of DMFT are then described, including statistical DMFT, typical medium theory, and extended DMFT, methods specifically designed to overcome the limitations of the original formulation. We provide a…
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybr…
Scale-invariant critical dynamics at eigenstate transitions
本文探讨了本征态临界转变（从量子混沌到局域化）中的标度不变动力学，发现临界点处的存活概率和谱形式因子不仅在晚期，在中期也表现出标度不变性，该结论适用于多种二次与相互作用模型。
Universal Relation between Spectral and Wavefunction Properties at Criticality
研究揭示临界点处谱压缩性与波函数非局域化程度之间的普遍关系，推导出适用于广泛临界系统的普遍函数D₁(r)。
Higher Nishimori Criticality and Exact Results at the Learning Transition of Deformed Toric Codes
本文研究了变形扭码波函数在弱测量下的学习诱导三临界点，揭示其位于更高Nishimori线上，并通过数值模拟验证了Edwards-Anderson相关函数的幂律指数与Ising临界点相同，同时探讨了更高Nishimori临界性在高维空间中的普遍性。
Wegner model in high dimension: U(1) symmetry breaking and a non-standard phase of disordered electronic matter, I. One-replica theory
本研究针对高维安德森转变单参数标度假说的争议，基于U(1)自发对称破缺发展强耦合场论，提出双参数标度机制，为无序电子物质第三相的存在提供了理论基础。
We present a numerical study of the Blume-Capel model with quenched disorder in 3D. The phase diagram is characterized by spin-glass/paramagnet phase transitions of both first and second order in the thermodynamic sense. Numerical simulations are performed using the Exchange-Monte Carlo algorithm, providing clear evidence for inverse freezing. The main features at criticality and in the phase coexistence region are investigated. The whole inverse freezing transition appears to be first order. The second order transition appears to be in the same universality class of the Edwards-Anderson model…
Learning Potts Models and $Z_3$ Toric Codes: Higher and Ordinary Nishimori Criticality
该研究在二维q态Potts模型学习相图中识别出高阶Nishimori临界点，分析其相图结构、普适量及RG流特性，并通过数值验证，还将结果推广至变形$\boldsymbol{\text{Z}_q}$环面码的信息临界性研究。
Disordered monitored free fermions
本文研究了连续监控下随机势场中的自由费米子，发现临界相在有限无序度下保持稳定且符合BKT普适类，同时监控会破坏局域化，为在电子系统中探测相变和量子控制开辟了新途径。
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