Localization: Emergent Platforms and Novel Trends

Much ink has been spilled on exactly how and when the (nearly) flat bands in twisted bilayer graphene are "topological," which has implications for the construction of Wannier orbitals and strong correlation effects. We point out that the chiral Bistritzer-MacDonald model is precisely equivalent the Dirac description of a class-CI topological superconductor (TSC) surface-state theory; the case with weakly broken chirality corresponds to the same surface subject to time-reversal symmetry breaking. Quenched disorder, which respects the chiral symmetry, leads to remarkable results for class-CI topological Dirac surface states, including a power-law vanishing density of states and critically Anderson delocalized low-energy wave functions with universal multifractal statistics. Recent results have demonstrated that a more radical kind of universal wave-function criticality can extend over a wide range of the energy spectrum for these dirty Dirac models (“spectrum-wide quantum criticality"). I will discuss our numerical results [1] which show that a quasiperiodic version of the chiral BM model reproduces the same phenomenology as dirty class CI TSC surface states, except along "critical filaments" that link magic angles of the moire potentials, where wave function fractality and the DOS are further enhanced. We demonstrate that these fractal states strongly boost superconductivity without any magic-angle fine-tuning. We expect our approach to be directly relevant to trilayer graphene. I will also discuss a related recent work wherein we identify a surprising limitation of the Dirac description of topological surface states [2]. We show that 3D TSCs generally lack spectral flow, and that surface states are vulnerable to a certain type of lattice perturbation ("UFO") that can (1) detach the surface states from the bulk and (2) Anderson localize them. [1] Zhang, Wilson, Foster, arXiv:2406.06676 [2] Altland, Brouwer, Dieplinger, Foster, Moreno-Gonzalez, Trifunovic Phys. Rev. X 14, 011057 (2024).

Bulk-Edge Correspondence of Non-Hermitian Topological Phases in Junction Systems : Towards Localization-Delocalization Transition in One Dimension In these days, the research on non-Hermitian systems has attracted great interest, since non-Hermitian systems can describe phenomena that emerged from open quantum systems and have rich topological properties. The point-gap topological phase in non-Hermitian systems (non-Hermitian topological phase, in short) gives rise to unique phenomena, such as skin effects, which cannot be observed in Hermitian systems. The bulk-edge correspondence (BEC) for the point-gap topological phase has been established and successfully explains the relation of spectrum and eigenstates for the system with periodic boundary conditions and open boundary conditions. The most typical example of a system possessing the point-gap topological phase is a one-dimensional system with non-reciprocal hopping, known as the Hatano-Nelson model [1]. This model was originally introduced as a system showing the localization-delocalization transition in the one-dimensional disordered system contradicting the scaling theory. In this talk, first, we introduce our recent work to extend the BEC of non-Hermtitian topological phases to junction systems[2]. We consider a one-dimensional ring geometry system with two sub-systems, where each sub-system has its own point-gap topological invariant so that we can regard this system as the junction system. We find that the eigenspectrum for this junction system appears on the complex-energy plane in the regions where the point-gap topological invariants for the subsystems are different, which corresponds to the BEC for junction systems. The corresponding eigenstates are exponentially localized at the interface, called non-Hermitian proximity effects. However, as an exception, we find that the eigenenergy for the junction systems also appears at common gapless points of the point gaps for the two systems, and the corresponding eigenstates are delocalized. Next, we explain our recent study to apply the above BEC to the system with sub-systems of more than two so that the system approaches the Hatano-Nelson model as increasing the number of subsystems. Then, we try to understand the localization-delocalization transition from the perspective of the bulk-edge correspondence. [1] N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996). [2] G. Hwang and H. Obuse, Phys. Rev. B 108, L121302 (2023).

I will talk about a novel phase and phase transition in three-dimensional (3D) disordered systems in the chiral classes (AIII and BDI) with and without 1D weak topological indices [1]. Our recent works [1,2] showed that the chiral symmetric systems with a nontrivial 1D weak topological index universally exhibit an emergent thermodynamic phase where wave functions are delocalized along one spatial direction but exponentially localized in the other spatial directions, which we call the quasilocalized phase. Our extensive numerical study in the 3D systems clarifies that the critical exponent of the Anderson transition between the metallic and quasilocalized phases, as well as that between the quasilocalized and localized phases, are different from that with no weak topological index, signaling the new universality classes induced by topology. The quasilocalized phase and concomitant topological Anderson transition manifest themselves in the anisotropic transport phenomena of disordered weak topological insulators and nodal-line semimetals, which exhibit the metallic behavior in one direction but the insulating behavior in the other directions. [1] Z. Y. Xiao, et. al. Phys. Rev. Lett. 131, 056301 (2023) [2] P. Zhao, et. al., arXiv:2402.02310

Breaking Lorentz reciprocity (or time-reversal symmetry) is hard for photons because they don’t carry charge and therefore don’t respond directly to an external magnetic field. Here, I’ll present our experimental results on how straining a photonic crystal with a Dirac point generates a strong artificial magnetic field; we directly observe photonic Landau levels as a result. I will then present a theoretical result showing how a non-reciprocal photonic Chern insulator can be used to make a wide-bandwidth slow-light waveguide that is resistant to backscattering.

We formulate a quantum version of the hard-disk problem on lattices, which exhibits a natural realization in systems of Rydberg atoms. We find that quantum hard disks exihibit unique dynamical quantum features. In 1D, the crystal melting process displays ballistic behavior as opposed to classical sub-diffusion. For 2D, crystal structures remain intact against most defects, whereas classically they are washed out completely. We link this peculiar quantum behavior to quantum many-body scars. Our study highlights the potential of constrained 2D quantum matter to display unique dynamical behaviors.

We study the evolution of the ground state entanglement entropy with disorder using numerical DMRG (Density-Matrix Renormalization Group) for two models with topological ground states - (a) The fractional quantum Hall state at filling $\nu$ = 1/3 ([1], [2]), and (b) The spin-1 chain in the Haldane phase[3]. We find that the entanglement entropy can be used to determine the transition to a disordered, non-topological state, and estimate the correlation length exponent at the transition. In the former case (a), we find this method uses much less computational resources than former studies using energy gaps, and provides more quantitative results. For the latter model (b), our results for the correlation length are in good agreement with estimates based on the strong disorder renormalization group (SDRG) method. [1] Zhao Liu and R. N. Bhatt, Phys. Rev. Lett. 117, 206801 (2016). [2] Zhao Liu and R. N. Bhatt, Phys. Rev. B 96, 115111 (2017). [3] Prashant Kumar and R. N. Bhatt, Phys. Rev. B 108, L241113 (2023).

Localization, or its absence, is a fundamental distinguishing feature of any many-body system, affecting the ways that mass, charge, energy, and momentum move and respond to inhomogeneity. Decades of work in both classical and quantum matter have classified circumstances in which transport is ballistic, diffusive, localized, or something stranger. A current frontier in this venerable field is the study of transport in driven matter, in which some control parameter is made explicitly time-dependent. Subjecting a quantum system to a time-dependent Hamiltonian can generate a rich array of localizing and delocalizing dynamics. I will discuss results from a sequence of recent cold-atom experiments on kicked and driven quantum matter, highlighting data on anomalous transport, the interplay between dynamical and disorder-induced localization, and quantum simulation of integer quantum Hall matter illuminated by light of arbitrary polarization.

Understanding how a system’s behavior extrapolates beyond the dimensionality of our 3D world is a fundamental question throughout physics – spanning from the pursuit of unification theories to the exploration of exotic materials and into the realms of critical phenomena. In statistical physics, the strength of the fluctuations is very sensitive to the dimensionality, which in turn plays a key role in the existence and nature of phase transitions. Most often, low-dimensional systems often exhibit suppression of phase transitions, while high-dimensional systems tend to display simpler, mean-field-like behavior. In a few outstanding cases, among which the celebrated Anderson localization-delocalization transition in disordered media, criticality has been predicted to remain highly non-trivial even in dimensions larger than three – posing great theoretical challenges to the existing frameworks. In this work, using a periodically-driven ultracold atomic gas to engineer both disorder and synthetic dimensions, we experimentally observe and investigate a phase transition between (dynamically) localized and delocalized phases. Our results clearly exhibit three fundamental characteristics that define the specific 4D nature of the observed phase transition: 1) the relevant observables perfectly follow the d=4 critical scale invariance, 2) the measured critical exponents are in excellent agreement with numerical predictions of the 4D Anderson transition and, 3) they show good agreement with Wegner’s relation in 4D. These results open a new paradigm for experimentally exploring complex critical phenomena, and physical theories in general, in higher dimensions.

The structure of the phase diagram of localization in the energy-disorder plane, for random tight-binding Hamiltonians exhibits several well known properties: below dimension $d=2$, all eigenfunctions are localized for non-vanishing disorder. Above $d=2$, a delocalized phase appears separated from the localized phase by a transition line called the “mobility edge,” predicted by the so-called self-consistent theory of localization in the case of uniform disorder. We will show that, with a closer examination, a more complicated structure emerges in which several types of mechanisms interplay to give rise to this diagram and its localization/delocalization transition. In particular, we will present the latest advances obtained using the localization landscape theory.

Previous theoretical research demonstrated that longitudinal electromagnetic fields impede Anderson localization of light in three-dimensional (3D) random ensembles of resonant point-like scatterers [1,2]. At the same time, experimental efforts to observe Anderson localization of light in 3D were unsuccessful for various reasons (see Ref. 3 for a summary). Our recent numerical simulations made possible by a new, highly efficient combination of software and hardware, demonstrate that Anderson localization of light is impossible in large random ensembles of overlapping dielectric spheres, suggesting a plausible explanation for the failure of previous experiments [4]. Motivated by the work on the detrimental role of longitudinal electromagnetic fields [1,2], we propose to look for Anderson localization of light in porous conducting (i.e., metallic) structures where longitudinal fields are suppressed. Our numerical results demonstrate that indeed, transmission of light through such structures exhibits signatures expected for Anderson localization: non-exponential decay of the time-dependent transmission, arrested expansion of the diffusive halo, enhanced fluctuations in the spectrum of light, etc. [4]. We suggest that future experimental and theoretical research on Anderson localization of light in 3D should focus on metallic structures with random, percolating pores. 1. S.E. Skipetrov & I.M. Sokolov, Absence of Anderson localization of light in a random ensemble of point scatterers, Phys. Rev. Lett. 112, 023905 (2014) 2. B.A. van Tiggelen & S.E. Skipetrov, Longitudinal modes in diffusion and localization of light, Phys. Rev. B 103, 174204 (2021) 3. S.E. Skipetrov & J.H. Page, Red light for Anderson localization, New J. Phys. 18, 021001 (2016) 4. A. Yamilov, S.E. Skipetrov, T.W. Hughes, M. Minkov, Z. Yu, & H. Cao, Anderson localization of electromagnetic waves in three dimensions, Nature Physics 19, 1308 (2023)

A quasiperiodic system is an intermediate state between periodic and disordered systems with a unique delocalization-localization transition driven by the quasiperiodic potential (QP). One of the intriguing questions is whether the universality class of the Anderson transition (AT) driven by QP is similar to that of the AT driven by the random potential in the same symmetry class. Here, we study the critical behavior of the ATs driven by QP in the three-dimensional (3D) Anderson model, the Peierls phase model, and the Ando model, which belong to the Wigner-Dyson symmetry classes. The localization length and two-terminal conductance have been calculated using the transfer-matrix method, and we argue that their error estimations in statistics suffer from the correlation of QP. With the correlation under control, the critical exponents ν of the ATs driven by QP are estimated by the finite-size scaling analysis of conductance, and they are consistent with the exponents ν of the ATs driven by the random potential. We also find that a convolutional neural network trained by the localized/delocalized wave functions in a disordered system predicts the localization/delocalization of the wave functions in quasiperiodic systems. Our numerical results strongly support the idea that the universality classes of the ATs driven by QP and the random potential are similar in the 3D Wigner-Dyson symmetry classes. We will also comment on the hyperuniformity in low-dimensional systems. [1] X. Luo and T. Ohtsuki, Phys. Rev. B 106, 104205 (2022). [2] S. Sakai, R. Arita and T. Ohtsuki, Phys. Rev. B 105, 054202 (2022), Phys. Rev. Res. 4, 033241 (2022).

We explore the concept of generalized multifractality (GMF), which characterizes eigenstate fluctuations and correlations in disordered systems. Utilizing the non-linear sigma model formalism, we construct pure-scaling composite operators and their positive eigenfunction observable counter part, enabling the efficient numerical determination of scaling exponents. These exponents determine the renormalization group (ir)relevance of interactions. Further, recent findings (PRL 131, 266401) show that conformal invariance leads to the generalized parabolicity of the GMF spectrum (i.e., proportionality to eigenvalues of the quadratic Casimir operator). On the one hand, we demonstrate that the fixed-point theory of the spin quantum Hall (SQH) transition violates conformal invariance, using a mapping to classical percolation for a specific set of GMF observables, and show analytically and numerically that generalized parabolicity does not hold. On the other hand, we examine disordered 3D topological superconductors, whose surface states in the bulk gap are described by Wess-Zumino models and exhibit full conformal invariance. These results indicate that the presence of full conformal invariance in Anderson critical systems depends on the (topological) universality class of the transition.

We identify the key features of Kardar-Parisi-Zhang (KPZ) universality class in the fluctuations of the wave density logarithm in a two-dimensional Anderson localized wave packet. In our numerical analysis, the fluctuations are found to exhibit an algebraic scaling with distance characterized by an exponent of 1/3, and a Tracy-Widom probability distribution of the fluctuations. Additionally, within a directed polymer picture of KPZ physics, we identify the dominant contribution of a directed path to the wave packet density and find that its transverse fluctuations are characterized by a roughness exponent 2/3. Leveraging on this connection with KPZ physics, we verify that an Anderson localized wave packet in two dimensions exhibits a stretched exponential correction to its well-known exponential localization. Ref: Phys. Rev. Lett. 132, 046301 (2024)

In this talk, I will present our recent findings on the Anderson transition in effectively infinite-dimensional systems, revealing an exotic critical behavior termed ”logarithmic multifractality” [1,2]. Unlike the conventional multifractal properties observed at finite-dimensional Anderson transitions, logarithmic multifractality features eigenstate statistics, spatial correlations, and wave packet dynamics that exhibit scaling laws algebraic in the logarithm of system size or time. I will compare this critical behavior with that we found in small-world networks of infinite effective dimensionality [3,4,5,6]. These insights offer a new framework for understanding the strong finite-size effects and slow dynamics in systems undergoing such Anderson transitions, analogous to the many-body localization transition. [1] Weitao Chen, Olivier Giraud, Jiangbin Gong, Gabriel Lemarié, arXiv:2312.17481, to be published in Phys. Rev. Res. (2024). [2] Weitao Chen, Olivier Giraud, Jiangbin Gong, Gabriel Lemarié, arXiv:2405.10975, to be published in Phys. Rev. B (2024). [3] I. García-Mata, O. Giraud, B. Georgeot, J. Martin, R. Dubertrand, and G. Lemarié, Phys. Rev. Lett. 118, 166801 (2017). [4] I. García-Mata, J. Martin, R. Dubertrand, O. Giraud, B. Georgeot, and G. Lemarié, Phys. Rev. Research 2, 012020(R) (2020). [5] I. García-Mata, J. Martin, O. Giraud, B. Georgeot, R. Dubertrand, and G. Lemarié, Phys. Rev. B 106, 214202 (2022). [6] Weitao Chen et al., in preparation (2024).

In the field of ergodicity-breaking phases, it has been recognized that quantum avalanches can destabilize many-body localization at a wide range of disorder strengths. This has in particular been demonstrated by the numerical study of a toy model, sometimes simply called the “avalanche model” or the “quantum sun model”, which consists of an ergodic seed coupled to a perfectly localized material. In this talk, we connect this toy model to a well-studied model in random matrix theory, the ultrametric ensemble. We conjecture that the models share the following features. (1) The location of the critical point may be predicted sharply by analytics. (2) On the localized site, both models exhibit Fock space localization. (3) There is a manifold of critical points. On the critical manifold, the eigenvectors exhibit nontrivial multifractal behavior that can be tuned by moving on the manifold. (4) The spectral statistics at criticality is intermediate between Poisson statistics and random matrix statistics, also tunable on the critical manifold. We confirm numerically these properties. J Šuntajs, L Vidmar, Physical Review Letters 129 (5), 060602 (2022) J Šuntajs, M Hopjan, W De Roeck, L Vidmar, Physical Review Research 6 (2), 023030 (2024)

We study a partially disordered one-dimensional system with interacting particles. Concretely, we impose a disorder potential to only every other site, followed by a clean site. Our numerical analysis of eigenstate properties is based on the entanglement entropy and density distributions. Most importantly, at large disorder, there exist eigenstates with large entanglement entropies and significant correlations between the clean sites. These states have volume-law scaling, embedded into a sea of area-law states, reminiscent of inverted quantum-scar states. These eigenstate features leave fingerprints in the nonequilibrium dynamics even in the large-disorder regime, with a strong initial-state dependence. We demonstrate that certain types of initial charge-density-wave states decay significantly, while others preserve their initial inhomogeneity, the latter being the typical behavior for many-body localized systems. This initial-condition dependent dynamics may give extra control over the delocalization dynamics at large disorder strength and should be experimentally feasible with ultracold atoms in optical lattices.

Geometrically frustrated (GF) magnets are the main class of materials in which quantum spin liquids (QSLs) are sought. Quenched disorder, ubiquitous in all realistic materials, can lead to spin freezing and prevent the formation of the widely sought QSL states. As disorder can be introduced in a controlled way, it can also be used as a powerful tool for investigating the properties of GF materials. I will present a microscopic theory of spin-glass freezing and magnetic susceptibility in GF magnets with quenched disorder. I will demonstrate that the properties of such materials are dominated by two distinct types of excitations, qualitatively similar to ring-exchange and spin-flip processes. I will show that the former are rather sensitive to quenched disorder and, for realistic amounts of disorder, lead to spin-glass-freezing transitions at temperatures of the order of a small “hidden energy scale” determined by the properties of the clean GF medium. Disorder also has a profound effect on the magnetic susceptibility of GF systems. Non-magnetic atoms (vacancies) replacing magnetic ones are known to act as effective spins, “quasispins”. I will describe the values of quasispins in GF materials, the interactions between them and their interplay with the spin freezing.

Superconducting qubits provide a promising platform for simulating intriguing phenomena in quantum many-body systems. The simulation efficiency depends on various intertwining factors including the underlying circuit architecture, the number of highly coherent qubits that can be precisely controlled and measured, and the upper-level simulation protocol. In this talk, I will introduce the multiqubit superconducting devices that are being developed at Zhejiang University. These devices are fabricated using the flip-chip recipe, with frequency-tunable transmon qubits and couplers located on the sapphire substrate and the control/readout wirings on the silicon substrate. The coupler is of transmon type, which is used to adjust the effective coupling strength between two neighboring qubits. For our devices, the median lifetime of the frequency-tunable qubits on a single chip is above 100 microseconds; the median single- and two-qubit gate fidelities can be above 99.96% and 99.5%, respectively. The decent device performance enables us to execute complex digital quantum circuits to simulate quantum many-body dynamics and other intriguing phenomena. In particular, we implement a quantum simulation protocol to realize a distinct type of non-equilibrium state of matter, a Floquet symmetry-protected topological phase, which breaks the time translational symmetry only at the boundaries and has trivial dynamics in the bulk.

I will discuss the challenges in providing evidence for MBL in the thermodynamic limit both for random and quasiperiodic on-site disorder. Other promising models such as Quantum Sun and Rybgerd atoms randomly placed will also be discussed.

Typical models for many-body localization (MBL) can be represented as tight-binding models over the many-body Hilbert space (Fock space), where sites correspond to non-interacting basis states, and possible hoppings depend on interaction. In this representation, the disordered model is fully specified by the joint distributions and correlations for on-site energies as well as hopping matrix elements. In this talk, I will present our conclusions regarding the role of these correlations in the scaling of the MBL critical disorder $W_c(n)$. For this purpose, we study five different disordered spin models which share the same distributions of diagonal (energy) and off-diagonal (hopping) Fock-space matrix elements but differ by their Fock-space correlations. These include a quantum-dot (QD) and one-dimensional (1D) MBL models, their modifications (uQD and u1D models) with removed correlations of off-diagonal matrix elements, as well a quantum random energy model (QREM) with no correlations at all. QD (resp. uQD) and 1D (resp. u1D) models differ essentially by the structure of their many-body energy correlations, which reflect the real-space spatial structure in the latter case. On the other hand, the absence of hopping correlations in the uQD and u1D models makes them ``maximally chaotic''. In full consistency with our predictions from analytical arguments, our numerical results from exact diagonalization indicate $W_c \sim n$ for the QD model, $W_c(n) \sim \text{const}$ for the 1D model, $W_c \sim n \ln n$ for the uQD and u1D models without off-diagonal correlations, and $W_c \sim n^{1/2} \ln n$ for QREM, where $n$ is the system size. A cross-comparison of these results for all five models demonstrate that the scaling of $W_c(n)$ for MBL transitions is governed by a combined effect of Fock-space correlations of diagonal and off-diagonal matrix elements. Finally, we extend our investigation to dynamical aspects of the MBL transition. By numerically computing the time-evolution of the imbalance for all models at various disorder, we extract their relaxation rates in the vicinity of the MBL transition, and observe the critical slowing down of the dynamics with increasing disorder strength. Comparing the scaling of these relaxation rates with increasing system size for all models reveals the intricate relationship between Fock-space correlations and the breaking down of ergodicity in MBL systems. \textbf{References:}\\ [1] T. Scoquart, I. Gornyi and A. Mirlin, Role of Fock-space correlations in many-body localization, arXiv:2402.10123, \textit{accepted for publication in Phys. Rev. B}, (2024)\\ [2] T. Scoquart, I. Gornyi and A. Mirlin, \textit{unpublished}, (2024)

A central theoretical issue at the core of the current research on many-body localization (MBL) consists in characterizing the statistics of rare long-range resonances in many-body eigenstates. This is of paramount importance to understand: (i) the critical properties of the MBL transition and the mechanism for its destabilization through quantum avalanches; (ii) the unusual transport and anomalously slow out-of-equilibrium relaxation when the transition is approached from the metallic side. In order to study and characterize such long-range rare resonances, we develop a large-deviations approach based on an analogy with the physics of directed polymers in random media, and in particular with their freezing glass transition on infinite-dimensional graphs. The basic idea is to enlarge the parameter space by adding an auxiliary parameter (which plays the role of the inverse temperature in the directed polymer formulation) which allows us to fine-tune the effect of anomalously large outliers in the far-tails of the probability distributions of the transmission amplitudes between far-away many-body configurations in the Hilbert space. We apply this approach to the study of a class of disordered quantum spin chains in a transverse field. This analysis shows the existence of a broad disorder range in which rare, long-distance resonances, that in finite-size systems may only form for a few specific realizations of the disorder and a few specific choice of the random initial state, are expected to destabilize the MBL phase in the thermodynamic limit, while the genuine MBL transition is shifted to much larger values of the disorder than originally thought.

Many-body localization impedes the spread of information encoded in initial conditions, blocking (or at least radically slowing) thermalization of an isolated quantum system. We examine the potential to tailor the growth of entanglement in the Fermi Hubbard model by tuning disorder in both the charge and spin degrees of freedom. We begin by expressing the Hamiltonian in terms of a set of optimally localized conserved quantities, and examine in detail the growth of entanglement entropy and its connection with the coupling between these local integrals of motion. We demonstrate how the strength of the disorder in charge and in spin controls the time scales seen in entanglement growth. We also show a shift in behaviour between the weakly and strongly interacting limit in which local integrals of motion lose their close association with Anderson localized single-particle states.

We will discuss the effect of strong sample-to-sample fluctuations in a generic one-dimensional disordered interacting chain at relatively weak disorder potential. It leads to creep dynamics in several observables, which implies a lack of a generic time scale. We propose entanglement as a measure of time to reparametrize the ensemble average dynamics. This approach evidences a remarkably uniform parametrization of the dynamical many-body evolution within the otherwise highly heterogeneous ergodic regime, independent of the strength of the disorder.

In superconducting thin films, the superconductor-insulator transition (SIT) is a paradigmatic example of a quantum phase transition: With increasing disorder the critical temperature of the superconductor is suppressed towards zero until an insulating ground state that is expected at a critical level of disorder with normal state resistance $R_N ≃ h/4e^2$. Notably, in many materials the mechanism of the SIT is not entirely clear, with competing explanations based on suppression of the order parameter modulus or proliferation of phase fluctuations. Measuring both superfluid stiffness and dc transport properties, we investigate ultra-thin NbN films and trace the evolution of the superfluid transition as a function of disorder close to the SIT. We observe a sharp Berezinskii-Kosterlitz-Thouless transition that persists up to $R_N \simeq h/e^2$. All characteristic energies approach zero near the localization threshold.

Recent experimental studies of strongly disordered Indium Oxide films revealed an unusual first-order quantum phase transition between superconducting and insulating state (SIT), with the jump between nonzero and zero values of superfluid stiffness at the transition. This finding is in sharp contradiction with a ”scaling scenario” discussed usually in relation to SIT. In the present paper we propose a simple theory of this first-order transition. It is based upon idea of competition between two intrinsically different ground states that can be formed by initially localized (due to strong disorder) electron pairs: superconducting state and Coulomb glass insulator. These two ground states are characterized by two crucially different order parameters, thus it is natural to expect a discontinuous transition between them at T = 0. The transition happens when magnitudes of superconducting gap ∆ and Coulomb gap EC are comparable. We also extend our analysis to low nonzero temperatures and provide a mean-field ”phase diagram” in the plane (T /∆, EC /∆).

I will report the observation of two-threshold voltage-current characteristics accompanied by a peak of broadband noise in the insulating state in two-dimensional electron systems in silicon metal-oxide-semiconductor field-effect transistors and ultrahigh mobility SiGe/Si/SiGe heterostructures. The two-threshold V − I characteristics are strikingly similar to those known for the collective depinning of the vortex lattice in type-II superconductors, with voltage and current axes interchanged. The observed results can be described by a phenomenological theory of the collective depinning of elastic structures, which naturally generates a peak of a broadband current noise between the dynamic and static thresholds and changes to sliding of the crystal over a pinning barrier above the static threshold. This gives compelling evidence for the electron solid formation in these structures and shows the generality of the effect for different classes of strongly correlated two-dimensional electron systems.

Free fermions provide a setting where the effect of continual monitoring on a many-body system, and the competition between wavefunction collapse and entangling dynamics, can be understood in detail. I will discuss how effective replica field theories can be used to characterise distinct phases (strong versus weak monitoring) and to compute entanglement entropies. I will comment on the analogies and differences with putative field theories for monitored interacting systems.

We develop a theory of measurement-induced phase transitions (MIPT) for $d$-dimensional lattice free fermions subject to random projective measurements of local site occupation numbers. Our analytical approach is based on the Keldysh path-integral formalism and replica trick. In the limit of rare measurements, $\gamma \ll J$ (where $\gamma$ is measurement rate per site and $J$ is hopping constant), we derive a non-linear sigma model (NLSM) as an effective field theory of the problem. Its replica-symmetric sector is a $U(2)/U(1) \times U(1)$ NLSM describing diffusive behavior of average density fluctuations. The replica-asymmetric sector, which describes propagation of quantum information in a system, is a $(d+1)$-dimensional isotropic NLSM defined on $SU(R)$ manifold with the replica limit $R \to 1$, establishing close relation between MIPT and Anderson transitions. On the Gaussian level, valid in the limit $\gamma / J \to 0$, this model predicts "critical" (i.e. logarithmic enhancement of area law) behavior for the entanglement entropy. However, one-loop renormalization group analysis shows [1] that for $d=1$, the logarithmic growth saturates at a finite value $\sim (J / \gamma)^2$ even for rare measurements, implying existence of a single area-law phase. The crossover between logarithmic growth and saturation, however, happens at an exponentially large scale, $\ln l_{\text{corr}}\sim J/\gamma$, thus making it easy to confuse with a transition in a finite-size system. Furthermore, utilizing $\varepsilon$-expansion, we demonstrate [2] that the "critical" phase is stabilized for $d > 1$ with a transition to the area-law phase at a finite value of $\gamma / J$. The analytical calculations are supported by and are in excellent agreement with the extensive numerical simulations [1,2] for $d=1,2$. For $d=2$ we determine numerically the position of the transition and estimate the value of correlation length critical exponent $\nu$. [1] I.P., P. Pöpperl, I. Gornyi, A.D. Mirlin, Phys. Rev. X 13, 041046 (2023) [2] I.P., I. Gornyi, A.D. Mirlin, Phys. Rev. Lett. 132, 110403 (2024)

We will discuss the universal nature of measurement induced phase transitions (MIPTs) in random quantum circuits when the measurement profile follows a static profile or a dynamic but spatially uniform pattern. First, the measurement induced transition is shown to be unstable to static but spatially random perturbations and the transition flows off to an infinite randomness fixed point. Second, the nature of several distinct quasiperiodic profiles will be studied and non-Pisot structures will be used to tune between irrelevant and relevant perturbations at the MIPT. In the latter case, the transition flows to an infinite quasiperiodic fixed point where the entanglement scaling is dictated by the wandering exponent of the quasiperiodic profile. We then apply a space time rotation to this class of problems to discover a measurement induced transition with ultrafast dynamics characterized by a vanishing dynamical exponent. The nature of this unique transition will be analyzed with a focus towards implications for quantum information propagation.

Measurement-induced entanglement transitions occur when entanglement production through unitary evolution is opposed by the disruptive effects of measurements. This leads to a transition from an ergodic volume law phase, where entanglement scales with the system size, to a localised area law phase, where entanglement scales as the boundary. The universality class of this novel transition depends crucially on the specific details of the unitary dynamics, for example, monitored free fermions are in a different class from hybrid random circuit models. Specifically, some free fermion systems are endowed with a conformal critical phase that exhibits entanglement logarithmic in the system size. In this work, we focus on integrable models, which despite being interacting share similarities with free models - such as an extensive number of integrals of motion. We employ the matrix product state approach to determine entanglement entropy, two-body correlation functions, and tripartite mutual information, and find strong signatures of three distinct phases - the volume law, the critical phase, and the area law phase. We show how the existence of the critical phase is linked to the particulars of the monitored integrable model and its non-Hermitian equivalent. Moreover, we provide an intuitive explanation for the emergence of the critical phase in these models. Our research pushes our understanding of how integrability impacts the nature of entanglement transitions in monitored systems, and more broadly, the thermalisation of open quantum systems.

We study the statistical properties of a single two-level system (qubit) subject to repetitive ancilla-based measurements. This setup is a fundamental minimal model for exploring the intricate interplay between the unitary dynamics of the system and the nonunitary stochasticity introduced by quantum measurements, which is central to the phenomenon of measurement-induced phase transitions. We demonstrate that this “toy model” harbors remarkably rich dynamics, manifesting in the distribution function of the qubit's quantum states in the long-time limit. We uncover a compelling analogy with the phenomenon of Anderson localization, albeit governed by distinct underlying mechanisms. Specifically, the state distribution function of the monitored qubit, parameterized by a single angle on the Bloch sphere, exhibits diverse types of behavior familiar from the theory of Anderson transitions, spanning from complete localization to almost uniform delocalization, with fractality occurring between the two limits. By combining analytical solutions for various special cases with numerical approaches, we achieve a comprehensive understanding of the structure delineating the “phase diagram” of the model. We categorize and quantify the emergent regimes and identify two distinct phases of the monitored qubit: ergodic and nonergodic. Furthermore, we identify a genuinely localized phase within the nonergodic phase, where the state distribution functions consist of delta peaks, as opposed to the delocalized phase characterized by extended distributions. Identification of these phases and demonstration of transitions between them in a monitored qubit are our main findings [1]. [1] P. Pöpperl, I.V. Gornyi, D.B. Saakian, and O.M. Yevtushenko, Phys. Rev. Research 6, 013313 (2024)

I will discuss correlations between eigenstates of the time-evolution operator that are associated with features of the dynamics at long times and large distances in spatially extended chaotic many-body quantum systems. These correlations are the counterpart for many-body systems of ones that are familiar in single-particle models of diffusive conductors, for example via the diffusion propagator. Based on joint work with Dominik Hahn and David Luitz, and with Takato Yoshimura and Sam Garratt: arXiv:2312.14234 arXiv:2309.12982

The evolution of complex random quantum systems is governed by the gradual buildup of entanglement and entropy. In this talk, we discuss real-time field theory describing how during this process, weakly entangled (‘single particle’) initial states turn into a fully entangled ergodic many-body state. To introduce the approach in the simplest nontrivial setting, we consider a tensor product of just two coupled random matrices, and compare to exact diagonalization. On this basis, we will discuss the dynamics of thermalization and entanglement spreading in spatially extended random matrix arrays.

I will discuss spectral structure and relaxation rates in ensembles of quantum master equations in which the Hamiltonian and dissipative processes are described by random matrices, banded random matrices, or local many-body Hamiltonians. I will discuss the spectral phase transitions that occur in each of these cases, as well as some unusual phenomena that arise in the weak-noise limit.

The nonequilibrium dynamics of disordered many-body quantum systems after a quantum quench unveils important insights about the competition between interactions and disorder, yielding, in particular, an interesting perspective toward the understanding of many-body localization. Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on their dynamical properties have hardly been investigated. Here, we present recent results on the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power $\alpha$[1]. Using a dynamical version of the strong disorder renormalization group, which we had been previously extended to long range coupled spin chains[2,3] we find for $\alpha > \alpha_c$ that the entanglement entropy grows logarithmically with time and becomes smaller with larger $\alpha$ as $S(t) = S_p \ln(t)/(2\alpha),$ where $S_p$ is a numerical factor which depends on the initial state. We present results of numerical exact diagonalization calculations for system sizes up to $N=16$ spins, in good agreement with the analytical results for sufficiently large $\alpha > \alpha_c ≈ 1.8$. For $\alpha < \alpha_c$ we find that the entanglement entropy grows as a power law with time, $S(t) ∼ t^{\gamma(\alpha)$ with $0 < \gamma(\alpha) < 1$ a decaying function of exponent $\alpha$. [1] Y. Mohdeb, J. Vahedi, S. Haas, R. N. Bhatt, S. Kettemann, Global Quench Dynamics and the Growth of Entanglement Entropy in Disordered Spin Chains with Tunable Range Interactions, Phys. Rev. B 108, L140203 (2023). [2] Y. Mohdeb, J. Vahedi, S. Kettemann, Excited-Eigenstate Entanglement Properties of XX Spin Chains with Random Long-Range Interactions, Phys. Rev. {\bf B 106}, 104201 (2022). [3] Y. Mohdeb, J. Vahedi, N. Moure, A. Roshani, H.Y. Lee, R. N. Bhatt, S. Haas, S. Kettemann, Entanglement Properties of Disordered Quantum Spin Chains with Long-Range Antiferromagnetic Interactions, Phys. Rev. {\bf B 102}, 214201 (2020).

We investigate, both numerically and analytically, the time evolution of a particle in an initial plane-wave state as it is subject to elastic scattering in a two-dimensional disordered system with Rashba spin-orbit coupling (SOC). In the analytic calculation, we treat the SOC nonperturbatively and the disorder perturbatively using the diffuson and the cooperon. We calculate the time dependence of coherent backscattering as a function of the strength of the SOC. We identify weak and strong SOC regimes and give the relevant time and energy scales in each case. By studying the time dependence of the anisotropy of the disorder-averaged momentum distribution, we identify the spin-relaxation time. We find a crossover from D’yakonov-Perel’ spin relaxation for weak SOC to Elliot-Yafet-like behavior for strong SOC. Kakoi, M. and K. Slevin (2024). "Time evolution of coherent wave propagation and spin relaxation in spin-orbit-coupled systems." Physical Review A 109(3): 033303.

I will discuss a class of encoding-decoding random circuits subject to local coherent and incoherent errors [1]. I will analytically demonstrate the existence of a Hilbert space localization phase transition in the system when the error strength is varied. Similarities of the unraveled transition with ergodicity-breaking phenomena and measurement-induced phase transitions will be highlighted. The considered transition is accompanied by Rényi entropy transitions, by the onset of multifractal features in the system, and by singular changes in the non-stabilizerness of the system's state. This phenomenon can be interpreted as a transition between error-protecting and error-vulnerable phases occurring when the error strength is increased. The described results provide a perspective on storing and processing quantum information, while the employed framework enables analytic insights into dynamical critical phenomena in many-body systems. [1] X. Turkeshi, P. Sierant, Phys. Rev. Lett. 132, 140401 (2024)

Exploring and understanding topological phases in systems with strong distributed disorder requires developing fundamentally new approaches to replace traditional tools such as topological band theory. Here, we present a general real-space renormalization group (RG) approach for scattering models, which is capable of dealing with strong distributed disorder without relying on the renormalization of Hamiltonians or wave functions. Such scheme, based on a block-scattering transformation combined with a replica strategy, is applied for a comprehensive study of strongly disordered unitary scattering networks with localized bulk states, uncovering a connection between topological physics and critical behavior. Our RG scheme leads to topological flow diagrams that unveil how the microscopic competition between reflection and non-reciprocity leads to the large-scale emergence of macroscopic scattering attractors, corresponding to trivial and topological insulators. Our findings are confirmed by a scaling analysis of the localization length (LL) and critical exponents, and experimentally validated. The results not only shed light on the fundamental understanding of topological phase transitions and scaling properties in strongly disordered regimes, but also pave the way for practical applications in modern topological condensed-matter and photonics, where disorder may be seen as a useful design degree of freedom, and no longer as a hindrance.

While Anderson is a single-particle wave effect, guaranteeing a single excitation in the system can be experimentally challenging. We here discuss light localization in three-dimensional disordered atomic clouds, in presence of several photons. We show that the presence of these multiple excitations does not affect substantially the abnormal intensity fluctuations which characterize the Anderson localization transition, provided that the radiated light is frequency filtered. Due to their narrow linewidth, long-lived modes (also known as subradiant modes), and particularly the localized ones, are strongly saturated even for the weakest pump, leading to a large increase of the inelastic scattering and to reduced fluctuations in the total radiation. Yet the atomic coherences and the resulting elastic scattering remain a proper witness of the Anderson (single-photon) localization transition. Hence, frequency filtering allows one to investigate specifically the single-excitation sector, while the many-body effects will rather show up in the fluorescence spectrum. This is a first step toward the localization of multiple photons in cold atom clouds. [1] "Nonlinear effects in Anderson localization of light by two-level atoms", Noel Araujo Moreira, Robin Kaiser, and Romain Bachelard, Phys. Rev. A (L) 109, L031501 (2024) [2] "Subradiance with saturated atoms: population enhancement of the long-lived states", A. Cipris, N.A. Moreira, T.S. do Espirito Santo, P. Weiss, C.J. Villas-Boas, R. Kaiser, W. Guerin, R. Bachelard, Physical Review Letters 126 (10), 103604 (2021)

Experimentally, strong localization of electromagnetic waves in three dimensions has never been achieved despite extensive studies. We propose a workaround by returning to the study of 2D systems, where localization is easily observable, focusing on the role of spatial correlations. Moving away from the paradigm of uncorrelated disorder, we perform microwave transport experiments in different correlated disordered or aperiodic 2D arrays of cylinders with high dielectric permittivity. In practice, we investigate: disordered hyperuniform systems, hard-disks packings, or planar aperiodic Vogel spirals. In all cases, we measure long lived-modes and characterize them by using a well-established predictor of Anderson localization, namely the Thouless conductance. By increasing the height of the cavity, we gradually increase the number of transverse modes allowed to propagate in the cavity, and observe that only a certain class of modes found in Vogel spiral survive this 2D/quasi-2D transition, in contrast to what is observed in correlated disordered systems. This result reveals that correlations in the scattering media seem to have a great influence on localization properties, and therefore should be considered in the quest for electromagnetic localization in 3D.