Repeated local measurements of quantum many body systems can induce a phase transition in their entanglement structure. These measurement-induced phase transitions (MIPTs) have been studied for various types of dynamics, yet most cases yield quantitatively similar values of the critical exponents, making it unclear if there is only one underlying universality class. Here, we directly probe the properties of the conformal field theories governing these MIPTs using a numerical transfer-matrix method, which allows us to extract the effective central charge, as well as the first few low-lying scaling dimensions of operators at these critical points. Our results provide convincing evidence that the generic and Clifford MIPTs for qubits lie in different universality classes and that both are distinct from the percolation transition for qudits in the limit of large onsite Hilbert space dimension. For the generic case, we find strong evidence of multifractal scaling of correlation functions at the critical point, reflected in a continuous spectrum of scaling dimensions.
We investigate the structure of many-body wave functions of 1D quantum circuits with local measurements employing the participation entropies. The leading term in system size dependence of participation entropy indicates a model dependent multifractal scaling of the wave-functions at any non-zero measurement rate. The sub-leading term contains universal information about measurement-induced phase transitions and plays the role of an order parameter, being constant non-zero in the error correcting phase and vanishing in the quantum Zeno phase. We provide robust numerical evidence investigating …
This work extends the analysis of the generalized multifractality of critical eigenstates at the spin quantum Hall transition in two-dimensional disordered superconductors [J. F. Karcher et al, Annals of Physics, 435, 168584 (2021)]. A mapping to classical percolation is developed for a certain set of generalized-multifractality observables. In this way, exact analytical results for the corresponding exponents are obtained. Furthermore, a general construction of positive pure-scaling eigenfunction observables is presented, which permits a very efficient numerical determination of scaling expon…
Generalized multifractality characterizes scaling of eigenstate observables at Anderson-localization critical points. We explore generalized multifractality in 2D systems, with the main focus on the spin quantum Hall (SQH) transition in superconductors of symmetry class C. Relations and differences with the conventional integer quantum Hall (IQH) transition are also studied. Using the field-theoretical formalism of non-linear sigma-model, we derive the pure-scaling operators representing generalizing multifractality and then "translate" them to the language of eigenstate observables. Performin…
Using renormalization group methods we study multifractality in percolation at the instance of noisy random resistor networks. We introduce the concept of master operators. The multifractal moments of the current distribution (which are proportional to the noise cumulants $C_R^{(l)} (x, x^\prime)$ of the resistance between two sites x and $x^\prime$ located on the same cluster) are related to such master operators. The scaling behavior of the multifractal moments is governed exclusively by the master operators, even though a myriad of servant operators is involved in the renormalization proced…
We implement a recursive Green function method to extract the Fock space (FS) propagator and associated self-energy across the many-body localization (MBL) transition, for one-dimensional interacting fermions in a random onsite potential. We show that the typical value of the imaginary part of the local FS self-energy, Δ_t, related to the decay rate of an initially localized state, acts as a probabilistic order parameter for the thermal to MBL phase transition; and can be used to characterize critical properties of the transition as well as the multifractal nature of MBL states as a function o…
The disorder induced metal--insulator transition is investigated in a three-dimensional simple cubic lattice and compared for the presence and absence of time-reversal and spin-rotational symmetry, i.e. in the three conventional symmetry classes. Large scale numerical simulations have been performed on systems with linear sizes up to $L=100$ in order to obtain eigenstates at the band center, $E=0$. The multifractal dimensions, exponents $D_q$ and $α_q$, have been determined in the range of $-1\leq q\leq 2$. The finite-size scaling of the generalized multifractal exponents provide the critical …
The phase diagram of the metal-insulator transition in a three dimensional quantum percolation problem is investigated numerically based on the multifractal analysis of the eigenstates. The large scale numerical simulation has been performed on systems with linear sizes up to $L=140$. The multifractal dimensions, exponents $D_q$ and $α_q$, have been determined in the range of $0\leq q\leq 1$. Our results confirm that this problem belongs to the same universality class as the three dimensional Anderson model, the critical exponent of the localization length was found to be $ν=1.622\pm 0.035$. T…
Barkhausen effect in ferromagnetic materials provides an excellent area for investigating scaling phenomena found in disordered systems exhibiting crackling noise. The critical dynamics is characterized by random pulses or avalanches with scale-invariant properties, power-law distributions, and universal features. However, the traditional Barkhausen avalanches statistics may not be sufficient to fully characterize the complex temporal correlation of the magnetic domain walls dynamics. Here we go beyond power laws and focus on the multifractal scenario to quantify the temporal scaling character…
We generalize universal relations between the multifractal exponent α_0 for the scaling of the typical wave function magnitude at a (Anderson) localization-delocalization transition in two dimensions and the corresponding critical finite size scaling (FSS) amplitude Λ_c of the typical localization length in quasi-one-dimensional (Q1D) geometry: (i) When open boundary conditions are imposed in the transverse direction of Q1D samples (strip geometry), we show that the corresponding critical FSS amplitude Λ_c^o is universally related to the boundary multifractal exponent α_0^s for the typical wav…
We present a comprehensive study of the destruction of quantum multifractality in the presence of perturbations. We study diverse representative models displaying multifractality, including a pseudointegrable system, the Anderson model and a random matrix model. We apply several types of natural perturbations which can be relevant for experimental implementations. We construct an analytical theory for certain cases, and perform extensive large-scale numerical simulations in other cases. The data are analyzed through refined methods including double scaling analysis. Our results confirm the rec…
Describing the critical behavior of the Anderson transition in infinite dimension by random-matrix ensembles: logarithmic multifractality and critical localization
In many systems, the electronic energy spectrum is a continuous or singular continuous multifractal set with a distribution of scaling exponents. Here, we show that for a quasiperiodic potential, the multifractal energy spectrum can have a minimal dispersion of the scaling exponents. This is made by tuning the ratio between self-energies in a tigth-binding Hamiltonian defined in a quasiperiodic Fibonacci chain. The tuning is calculated numerically from a trace map, and coincides with the place where the scaling exponents of the map, obtained from cycles with period six and two, are equal. The …
Multifractal systems usually have singularity spectra defined on bounded sets of Hölder exponents. As a consequence, their associated multifractal scaling exponents are expected to depend linearly upon statistical moment orders at high enough orders -- a phenomenon referred to as the {\it{linearization effect}}. Motivated by general ideas taken from models of turbulent intermittency and focusing on the case of two-dimensional systems, we investigate the issue within the framework of Gaussian multiplicative chaos. As verified by means of Monte Carlo simulations, it turns out that the linearizat…
There is growing evidence, from experiments and numerical simulations, that a key feature of sufficiently disordered superconductors is the spatial inhomogeneity of the order parameter. However not much is known analytically about the details of its spatial distribution or the associated global critical temperature that signals the breaking of long-range order. Here we address this problem for disordered systems around an Anderson transition characterized by multifractal one-body eigenstates. In the limit of weak multifractality and for weakly coupled superconductors we compute the superconduc…
There has been a revival of interest in localization phenomena in quasiperiodic systems with a view to examining how they differ fundamentally from such phenomena in random systems. Mo- tivated by this, we study transport in the quasiperiodic, one-dimentional (1d) Aubry-Andre model and its generalizations to 2d and 3d. We study the conductance of open systems, connected to leads, as well as the Thouless conductance, which measures the response of a closed system to boundary perturbations. We find that these conductances show signatures of a metal-insulator transition from an insulator, with lo…
We present a nonperturbative calculation of all multifractal scaling exponents at strong disorder for critical wavefunctions of Dirac fermions interacting with a non-Abelian random vector potential in two dimensions. The results, valid for an arbitrary number of fermionic flavours, are obtained by deriving from Conformal Field Theory an effective Gaussian model for the wavefunction amplitudes and mapping to the thermodynamics of a single particle in a random potential. Our spectrum confirms that the wavefunctions remain delocalized in the presence of strong disorder.
In Refs. [1,2] we have shown how a combination of modern linear-scaling DFT, together with a subsequent use of large, effective tight-binding Hamiltonians, allows to compute multifractal wave functions yielding the critical properties of the Anderson metal-insulator transition (MIT) in doped semiconductors. This combination allowed us to construct large and atomistically realistic samples of sulfur-doped silicon (Si:S). The critical properties of such systems and the existence of the MIT are well known, but experimentally determined values of the critical exponent $ν$ close to the transition h…
We analyze the critical behavior of the dephasing rate induced by short-range electron-electron interaction near an Anderson transition of metal-insulator or quantum Hall type. The corresponding exponent characterizes the scaling of the transition width with temperature. Assuming no spin degeneracy, the critical behavior can be studied by performing the scaling analysis in the vicinity of the non-interacting fixed point, since the latter is stable with respect to the interaction. We combine an analytical treatment (that includes the identification of operators responsible for dephasing in the …
The statistical properties of wave functions at the critical point of the spin quantum Hall transition are studied. The main emphasis is put onto determination of the spectrum of multifractal exponents $Δ_q$ governing the scaling of moments $<|ψ|^{2q}>\sim L^{-qd-Δ_q}$ with the system size $L$ and the spatial decay of wave function correlations. Two- and three-point correlation functions are calculated analytically by means of mapping onto the classical percolation, yielding the values $Δ_2=-1/4$ and $Δ_3=-3/4$. The multifractality spectrum obtained from numerical simulations is given with a g…
We propose to use the method introduced by Volchkov et al., based on state dependent disordered ultracold bosons, to address the critical state at the mobility edge of the Anderson localization transition, and to observe its intriguing multifractal structure. An optimally designed external radio frequency pulse can be applied to generate transitions to eigenstates in a narrow energy window close to the mobility edge, where critical scaling and multifractality emerge. Two-photon laser scanning microscopy will be used to address individual localized states even close to the transition. The proje…
We present an ultra-high-precision numerical study of the spectrum of multifractal exponents $Δ_q$ characterizing anomalous scaling of wave function moments $<|ψ|^{2q}>$ at the quantum Hall transition. The result reads $Δ_q = 2q(1-q)[b_0 + b_1(q-1/2)^2 + ...]$, with $b_0 = 0.1291\pm 0.0002$ and $b_1 = 0.0029\pm 0.0003$. The central finding is that the spectrum is not exactly parabolic, $b_1\ne 0$. This rules out a class of theories of Wess-Zumino-Witten type proposed recently as possible conformal field theories of the quantum Hall critical point.
We study the interaction driven localization transition, which a recent experiment in Ga_{1-x}Mn_xAs As has shown to come along with multifractal behavior of the local density of states (LDoS) and the intriguing persistence of critical correlations close to the Fermi level. We show that the bulk of these phenomena can be understood within a Hartree-Fock treatment of disordered, Coulomb-interacting spinless fermions. A scaling analysis of the LDoS correlation demonstrates multifractality with correlation dimension d_2=1.57, which is significantly larger than at a non-interacting Anderson transi…
The spin quantum Hall (or class C) transition represents one of the few localization-delocalization transitions for which some of the critical exponents are known exactly. Not known, however, is the multifractal spectrum, $τ_q$, which describes the system-size scaling of inverse participation ratios $P_q$, i.e., the $q$-moments of critical wavefunction amplitudes. We here report simulations based on the class C Chalker-Coddington network and demonstrate that $τ_q$ is (essentially) a quartic polynomial in $q$. Analytical results fix all prefactors except the quartic curvature that we obtain as …
We study a simple model of conducting polymers in which a single electron propagates through a randomly tangled chain. The model has the geometry of a small-world network, with a small density $p$ of crossings of the chain acting as shortcuts for the electron. We use numerical diagonalisation and simple analytical arguments to discuss the density of states, inverse participation ratios and wave functions. We suggest that there is a critical point at $p=0$ and demonstrate finite-size scaling of the energy and wave functions at the lower band edge. The wave functions are multifractal. The critic…
A Weyl semimetal is a three dimensional topological gapless phase. In the presence of strong enough disorder it undergoes a quantum transition towards a diffusive metal phase whose universality class depends on the range of disorder correlations. Similar to other quantum transitions driven by disorder, the critical wave functions at the semimetal-diffusive metal transition exhibit multifractality. Using renormalization group methods we study the corresponding multifractal spectrum as a function of the range of disorder correlations for generic disorder including random scalar and vector potent…
Mesoscopic Fluctuations and Multifractality at and across Measurement-Induced Phase Transition
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院量子材料与技术研究所）；Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院凝聚态理论研究所）
We study generalized multifractality characterizing fluctuations and correlations of eigenstates in disordered systems of symmetry classes AII, D, and DIII. Both metallic phases and Andersonlocalization transitions are considered. By using the non-linear sigma-model approach, we construct pure-scaling eigenfunction observables. The construction is verified by numerical simulations of appropriate microscopic models, which also yield numerical values of the corresponding exponents. In the metallic phases, the numerically obtained exponents satisfy Weyl symmetry relations as well as generalized p…
The statistics of critical wave functions at the Anderson transition in three and four dimensions are studied numerically. The distribution of the inverse participation ratios (IPR) $P_q$ is shown to acquire a scale-invariant form in the limit of large system size. Multifractality spectra governing the scaling of the ensemble-averaged IPRs are determined. Conjectures concerning the IPR statistics and the multifractality at the Anderson transition in a high spatial dimensionality are formulated.
In summary, we investigated the role of Coulomb interactions in the nature of eigenfunction multifractality of an Anderson metal-insulator transition, based on the Hartree-Fock approximation and the Ewald summation technique. As a result, we showed that two types of mobility edges appear near the Fermi energy and at a high energy, respectively, where the low-energy mobility edge results from Coulomb interactions while the high-energy one is nothing but the mobility edge of the Anderson localization transition without electron correlations. Indeed, not only multifractal scaling exponents but al…
The delocalized non-ergodic phase existing in some random $N \times N$ matrix models is analyzed via the Wigner-Weisskopf approximation for the dynamics from an initial site $j_0$. The main output of this approach is the inverse $Γ_{j_0}(N)$ of the characteristic time to leave the state $j_0$ that provides some broadening $Γ_{j_0}(N) $ for the weights of the eigenvectors. In this framework, the localized phase corresponds to the region where the broadening $Γ_{j_0}(N) $ is smaller in scaling than the level spacing $Δ_{j_0}(N) \propto \frac{1}{N}$, while the delocalized non-ergodic phase corres…
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…
We study an asymptotic behavior of the return probability for the critical random matrix ensemble in the regime of strong multifractality. The return probability is expected to show critical scaling in the limit of large time or large system size. Using the supersymmetric virial expansion we confirm the scaling law and find analytical expressions for the fractal dimension of the wave functions $d_2$ and the dynamical scaling exponent $μ$. By comparing them we verify the validity of the Chalker's ansatz for dynamical scaling.
The dynamical scaling for statistics of critical multifractal eigenstates proposed by Chalker is analytically verified for the critical random matrix ensemble in the limit of strong multifractality controlled by the small parameter $b\ll 1$. The power law behavior of the quantum return probability $P_{N}(τ)$ as a function of the matrix size $N$ or time $τ$ is confirmed in the limits $τ/N\rightarrow\infty$ and $N/τ\rightarrow\infty$, respectively, and it is shown that the exponents characterizing these power laws are equal to each other up to the order $b^{2}$. The corresponding analytical expr…
We study the critical behavior near the integer quantum Hall plateau transition by focusing on the multifractal (MF) exponents $X_q$ describing the scaling of the disorder-average moments of the point contact conductance $T$ between two points of the sample, within the Chalker-Coddington network model. Past analytical work has related the exponents $X_q$ to the MF exponents $Δ_q$ of the local density of states (LDOS). To verify this relation, we numerically determine the exponents $X_q$ with high accuracy. We thereby provide, at the same time, independent numerical results for the MF exponents…
We investigate the wave-packet dynamics of the power-law bond disordered one-dimensional Anderson model with hopping amplitudes decreasing as $H_{nm}\propto |n-m|^{-α}$. We consider the critical case ($α=1$). Using an exact diagonalization scheme on finite chains, we compute the participation moments of all stationary energy eigenstates as well as the spreading of an initially localized wave-packet. The eigenstates multifractality is characterized by the set of fractal dimensions of the participation moments. The wave-packet shows a diffusive-like spread developing a power-law tail and achieve…
The multifractality of the critical eigenstate at the metal to insulator transition (MIT) in the three-dimensional Anderson model of localization is characterized by its associated singularity spectrum f(alpha). Recent works in 1D and 2D critical systems have suggested an exact symmetry relation in f(alpha). Here we show the validity of the symmetry at the Anderson MIT with high numerical accuracy and for very large system sizes. We discuss the necessary statistical analysis that supports this conclusion. We have obtained the f(alpha) from the box- and system-size scaling of the typical averag…
In this work we put forward an effective Gaussian free field description of critical wavefunctions at the transition between plateaus of the integer quantum Hall effect. To this end, we expound our earlier proposal that powers of critical wave intensities prepared via point contacts behave as pure scaling fields obeying an Abelian operator product expansion. Our arguments employ the framework of conformal field theory and, in particular, lead to a multifractality spectrum which is parabolic. We also derive a number of old and new identities that hold exactly at the lattice level and hinge on t…
We study effects of classical magnetic impurities on the Anderson metal-insulator transition numerically. We find that a small concentration of Heisenberg impurities enhances the critical disorder amplitude $W_{\rm c}$ with increasing exchange coupling strength $J$. The resulting scaling with $J$ is analyzed which supports an anomalous scaling prediction by Wegner due to the combined breaking of time-reversal and spin-rotational symmetry. Moreover, we find that the presence of magnetic impurities lowers the critical correlation length exponent $ν$ and enhances the multifractality parameter $α_…
