Localization: Emergent Platforms and Novel Trends

The interplay between electron-electron interactions and weak localization (or anti-localization) phenomena in two-dimensional systems can significantly enhance the superconducting transition temperature. We develop the theory of quantum fluctuations within such multifractally-enhanced superconducting states in thin films. Under conditions of weak disorder, we employ the Finkel’stein nonlinear sigma model to derive an effective action for the superconducting order parameter and the quasiclassical Green’s function, meticulously accounting for the influence of quantum fluctuations. This effective action, applicable for interactions of any strength, reveals the critical role of well-known collective modes in a dirty superconductor, and its saddle point analysis leads to modified Usadel and gap equations. We demonstrate an intimate relation between contributions from collective modes to the effective action for the order parameter and quasiclassical Green’s function in a superconducting phase, on the one hand, and the modified Usadel and self-consistent equations, on the other hand. Finally, we derive the expressions for both gapless and gapped modes in the dirty limit, revealing an intriguing logarithmic dispersion for the Schmidt-Higgs mode.

Abstract of the talk/poster: Many-body localization (MBL) hinders the thermalization of quantum many-body systems in strong disorder, resulting in numerous experimental and numerical studies. Although the status of MBL as a dynamical phase of matter is to be addressed, this does not diminish the importance of the regime of (nearly) arrested dynamics at strong disorder strengths. Typical studies on MBL consider the presence of on-site disorder; the phenomenological properties in the MBL regime can then be described within the framework of local integrals of motions (LIOMs) identified as dressed single-site operators. This talk/poster will focus on a bond-disordered Hamiltonian, a model relevant for experiments in Rydberg atom platforms, which shows nonparadigmatic features in the MBL regime that are not captured within a standard LIOM description. Instead, a simple renormalization group-based scheme will be used to elucidate the eigenstate properties and reveal appropriate probes for experiments. It will also illustrate how to extend this scheme to more generic Hamiltonians.

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)

Transport plays a key role in characterizing topological insulators and semimetals. Understanding the effect of disorder is crucial to assess the robustness of experimental signatures for topology. In this work, we find the absence of localization in nodal line semimetals for long-range scalar disorder and a large range of disorder strengths. Using a continuum transfer matrix approach, we find that the conductivity in the plane and out of the plane of the nodal line increases with system size and disorder strength. We substantiate these findings by a perturbative calculation and show that the conductivity increases with disorder strength using the Kubo formula in the self-consistent Born approximation. We also find that the system remains metallic for vector disorder and that vector disorder can drive a transition from an insulating to a metallic regime. Our results demonstrate the absence of localization in a bulk system.

Lattice gauge theories, the discretized cousins of gauge theories underpinning the Standard Model of particle physics, have been pivotal in our understanding of the universe. However, they are also a rich source of novel physics in their own right, especially when taken away from the continuum limit. Recent works have reported the possible existence of a many-body localized phase in the the $U(1)$ Schwinger model despite the absence of quenched disorder. We characterize this putative MBL phase through exact diagonalization methods and locate the ergodic-MBL transition point using various metrics of chaos such as the Spectral Form Factor, Eigenstate entanglement fluctuations and the system's response to local perturbations. We further report on the strong-coupling limit of the model, in which an unusual double-log growth in the configurational entropy emerges, and the Hilbert space fragments due to the emergence of a dynamical constraint on hopping. We argue that these features are connected and elucidate the mechanism that leads to a deviation from the predictions of random-matrix theory. Our results highlight the potential of lattice gauge theories to host MBL physics in the strong-coupling regimes, readily realizable in current-age quantum simulators.

The Rosenzweig-Porter model is a single-parameter random matrix ensemble that supports an ergodic, fractal, and localized phase. Introduced over sixty years ago, this model recently gained renewed interest as a toy model for the many-body localization transition. We construct a unitary (Floquet) equivalent of this model, for which we numerically study the long-range spectral statistics [1,2]. The construction is based on interpreting the Rosenzweig-Porter model as a Brownian quantum system [3]. Our main result is the observation that the transition between the ergodic and fractal phases can be probed through the spectral form factor. Complementing previous results on the level spacing distribution, this establishes that spectral statistics are sufficient to fully map out the phase diagram of the model. We quantitatively discuss the scaling of the Thouless time, and point out the possible universality of the spectral form factor at the transition between the fractal and the localized phases. [1] W. Buijsman and Y. Bar Lev, Circular Rosenzweig-Porter random matrix ensemble, SciPost Phys. 12, 082 (2022). [2] W. Buijsman, Long-range spectral statistics of the Rosenzweig-Porter model, Phys. Rev. B 109, 024205 (2024). [3] W. Buijsman, Efficient circular Dyson Brownian motion algorithm, arXiv: 2309.07457 (2023).

The investigation of the spectral and dynamical delocalization-localization (DL) transitions have revealed intriguing features in a wide expanse of non-Hermitian systems. The present study aims at exploring the spectral and the dynamical properties in a non-Hermitian quasiperiodic system with asymmetric hopping, in the presence of Rashba spin-orbit (RSO) interaction. In particular, in such systems, we have identified that the DL transition is associated with a concurrent change in the energy spectrum, where the eigenstates always break the time-reversal symmetry for all strengths of the quasiperiodic potential, contrary to the systems without RSO interaction. Furthermore, in this work, we have demonstrated that the open boundary energy spectrum in the prototypical 1D nonreciprocal lattice remains real up to a certain system size and forms complex spectral loops with an increase in the size of the lattice. We find that the skin effect remains unaltered irrespective of the nature of the spectrum. In addition, it is illustrated that the spin-flip term in the RSO interaction possesses a tendency to diminish the directionality of the skin effect. On scrutinizing the dynamical attributes in our non-Hermitian system, we unveil that in spite of the fact that the spectral DL transition accords with the dynamical phase transition, interestingly, the system acquires a nonzero transport behavior, and in fact comes across hyperdiffusive and negative diffusion dynamical regimes depending upon the strength of the RSO interaction, in the spectrally localized regime.

Open quantum system offers promising applications in quantum information processing devices. By navigating the interplay between entangling unitary dynamics and disentangling measurements, we can induce novel entanglement phase transitions. In this talk, I will discuss our recent findings on unique critical properties associated with measurement-induced entanglement phase transitions in monitored quantum circuits, along with the potential for their realization in quantum devices. We predicted a new multi-fractal universality class in presence of U(1) symmetry, different from that without symmetry and percolation transition. I will also discuss possibility of observing this symmetric hybrid dynamics in a multi- qudit system in the presence of conserved quantities in an interesting experimental platform, namely superconducting circuit coupled to multi-mode cavities. Refs: 1. Charge and Entanglement Criticality in a U(1)-Symmetric Hybrid Circuit of Qubits by Ahana Chakraborty Kun Chen, Aidan Zabalo, Justin H Wilson, and JH Pixley; arXiv:2307.13038, 2. Realizing measurement-induced phase transitions in multimode circuit QED systems by Shivam Patel, Ahana Chakraborty, Jordan Huang, Thomas Dinapoli, J. H. Pixley and Srivatsan Chakram, to appear in arXiv.

The myriad manifestations of topological crystalline phases (TCP), from anomalous higher-order boundary states to obstructed atomic limits, are well understood. A comprehensive understanding of their stability to disorder however remains an open question. We classify such generalized phases (with order-two symmetries) for disorder that breaks the crystalline symmetry, while preserving it on average. We detail novel disordered phases; statistical higher-order phases whose hinge states are subject to backscattering but remain protected from Anderson localization at zero-energy and a novel superconducting phase which concurrently hosts the aforementioned critical zero-energy hinge state along with a stable chiral Majorana hinge mode. We also discover that obstructed atomic limits are stable to disorder if and only if they have a filling anomaly: An observation that results, in contrast to clean TCPs, in disordered TCPs satisfying a complete bulk-boundary correspondence.

Due to their amenability to analytical treatment, random-matrix ensembles serve as robust platforms for investigating exotic phenomena in systems that are computationally expensive. In this work, we investigate two random-matrix ensembles designed to capture an exotic multifractal behavior termed ``logarithmic multifractality", through both analytical techniques and extensive numerical simulations. In contrast to the conventional multifractality, the logarithmic multifractality is characterized by eigenstate moments which scale as algebraic laws in the logarithm of the system size--an exotic scaling behavior that was argued to emerge in infinite-dimensional Anderson transitions. Through perturbative approaches, we demonstrate the existence of logarithmic multifractality in our models. Additionally, we explore the manifestation of novel scaling behaviors in the time dynamics and the spatial correlation functions. Our models offer a valuable playground for the investigation of infinite-dimensional quantum disordered systems, and the universality of our findings allows for broad applicability to systems exhibiting strong finite-size effects and slow dynamics.

Motivated by many contemporary problems in condensed matter physics where matter particles experience random gauge fields, we investigate the physics of fermions on a square lattice with $\pi$-flux (that realizes Dirac fermions at low energies) subjected to flux disorder arising from a random $\mathbb{Z}_2$ gauge field. At half-filling where the system possesses BDI symmetry, we show that a critical phase is realized with the states at the chemical potential (zero energy) showing a multifractal character. The multifractal properties depend on the concentration $c$ of the $\pi$-flux defects and are characterized by the singularity spectrum, Lyapunov exponents, and transport properties. For any concentration of flux defects, we find that the multi-fractal spectrum shows termination, but {\em not freezing}. Away from half-filling (at finite energies), we show that the fermionic states are localized, for any value of $c$. We formulate a field theory that can describe this physics, and point to the intrinsically non-perturbative nature of the flux-perturbations. Using supersymmetry and CFT we have also shown that the correlation between the disorders can prevent the zero energy state from freezing. These results can throw light on a class of problems where fermions experience random gauge fields.

We investigate many-body localization (MBL) in Floquet random circuits and methods for their compression into shallow circuits. In particular, we address the question how the different characteristics of entanglement spreading in the localized and ergodic regimes affect the compressibility of the circuits. Besides serving as a possible probe for localization, compressed Floquet random circuits might open a practical route to observe dynamical signatures of localization in digital quantum simulations on near-term quantum processors.

Disordered quantum systems have become an important research topic in modern condensed matter physics ever since the discovery of Anderson localization. The investigation of many-body localization in quantum interacting systems has received much recent attention following the increase of computational power and improvement in numerical methods. We focus on a Heisenberg spin chain with SU(2) symmetry where the exchange couplings between neighboring spins are considered disordered. Both exact diagonalization and sparse matrix diagonalization methods are applied when calculating eigenvalues and eigenvectors of the Hamiltonian matrix. By understanding the structure of eigenvalues and eigenvectors in terms of spin symmetry, we investigate the consecutive gap ratio, participation ratio, and entanglement entropy as a function of disorder strengths. We average over many disorder realizations and compare the results for different disorder distributions. We find, for small system sizes, a clear distinction between the SU(2)-invariant random exchange model and the more often studied random field model. In particular, the regime of seemingly localized behavior is much less pronounced in the random exchange model than in the field model case.

Voltage measurements in four-terminal configurations are susceptible to quantum interference in electronic transport experiments. We study the voltage fluctuations in disordered graphene nanoribbons with zigzag and armchair edge terminations in a four-terminal configuration. We show that the average and standard deviation of the voltage oscillates with the separation of the attached voltage probes and depend on the coupling strength of the probes. The voltage fluctuations can be large enough for weakly coupled probes to observe negative voltages. The voltage fluctuations are described within a random matrix approach for weakly disorder at energies away from the Fermi energy. We show that near the Fermi energy, the voltage statistics of zigzag and armchair nanoribbons are different due to Anderson and anomalous localizations.

Despite considerable studies of non-Hermitian skin effect in many-body systems, the genuine characterization of skin mode has yet to be accomplished. Here we elucidate that the non-Hermitian skin effect exhibits multifractality in many-body Hilbert space. Since multifractality is absent in single-particle systems, our characterization reveals the unique feature of skin modes in many-body systems. Furthermore, we demonstrate that the many-body skin effect can coexist with spectral statistics of random matrices, in contrast to multifractality associated with the many-body localization, which necessitates the absence of ergodicity. We also show multifractality caused by the Liouvillian skin effect in Markovian open quantum systems. Our work establishes a defining characterization of the non-Hermitian skin effect and uncovers a fundamental relationship between multifractality and ergodicity in open quantum many-body systems.

In this work, we first argue why we expect the imbalance to become indistinguishable from the single-particle survival probability for typical Slater determinant. We then check the conjecture numerically for two paradigmatic models of localization: 3D Anderson model and 1D Aubry–André model. Additionally, by interpreting the imbalance as sum of auto-correlation functions, we generalize the conjecture to the correlation functions of sites of distance $d$ and the corresponding single-particle transition probabilities of the same distance. Finally, we also discuss equal-time connected correlation functions, which exhibit certain qualitative analogies to the single-particle survival and transition probabilities. Our work gives affirmative answer to the question whether it is possible to measure features of the single-particle survival and transition probabilities by observable dynamics in many-body states.

Kardar-Parisi-Zhang (KPZ) universality, traditionally associated with the growth of interfaces, manifests itself in many systems including quantum ones. In particular this physics can be related to quantum localization (Anderson localization) in two dimensions. In this study we use extended numerical simulations to show that fluctuations of localized eigenstates are well-described by KPZ characteristic exponents.

At zero temperature, symmetry-protected topological (SPT) order can encode quantum information in an edge strong zero mode, robust to perturbations respecting some symmetry. On the other hand, phenomena like many-body localisation (MBL) and quantum scarring can arrest the approach to thermal equilibrium, contrary to the ergodic dynamics expected of generic quantum systems. This raises the possibility that by combining SPT order with such ergodicity breaking phenomena, one might be able to construct a quantum memory that is robust at finite temperature. In this talk, I will focus on an SPT transition between two MBL phases. Through a renormalisation group approach, we identify many-body resonances in the basis of localised eigenstates, showing that these proliferate in the vicinity of the transition and cause delocalisation. Additionally, we characterise the SPT strong zero mode. This has important implications for the stability of MBL and transitions between MBL phases with different topological orders. I will also discuss how we may capture the dynamics of MBL systems using a similar approach.

The localisation landscape theory of M. Filoche and S. Mayboroda is a method to efficiently and intuitively understand the eigenstates in random potentials. To address the restrictions on the Hamiltonian in the original theory, several generalizations have been proposed, the L2 localisation landscape of L. Herviou and J. H. Bardarson being one of them. The L2 landscape can be applied to a wider range of systems, but the efficient computational method using the sparse matrix linear algebra, which is an advantage in the original theory, is not known. We propose a stochastic method to obtain the L2 landscape, in which the sparse matrix method is applicable. We also propose an energy filtering of the L2 landscape which can be used to focus on eigenstates with energies in any chosen range of the energy spectrum. We apply this method to Anderson’s model of localisation in one and two dimensions, and also to two-dimensional disordered system in a strong magnetic field that puts the system in the regime of the quantum Hall effect. M. Kakoi and K. Slevin (2023) “A stochastic method to compute the L2 localisation landscape” Journal of the Physical Society of Japan 92(5), 054707

Non-hermiticity provides novel aspects to the physics, both classical and quantum [1]. In particular, the interplay of non-Hermiticity and translational invariance leads to non-Hermitian topological phenomena. One of the central concepts of non-Hermitian topology is the non-Hermitian skin effect (NHSE), which is the exponential localization of the bulk wavefunctions to the boundary [2]. Exponentially localized wavefunctions represent a breakdown of conventional Bloch band theory and prevent accurate calculation of the spectrum and eigenstate under open boundary conditions (OBC). However, the asymptotic behavior of localization in the thermodynamic limit can be characterized in terms of complex-valued wavenumbers. This fact motivates a generalization of the conventional Bloch band theory and the first Brillouin zone using complex numbers. These generalizations are known as the non-Bloch band theory and the generalized Brillouin zone [3, 4]. Non-Bloch band theory defines the OBC band structure through the analytical continuation of the Bloch Hamiltonian, leading to a subvariety of the complicated Riemann surface. However, this complex Riemann surface makes the definition of non-Bloch band theory in higher dimensions both theoretically and numerically difficult. Recent work has partially overcome this problem using the algebraic geometric concepts of the Amobea and the Ronkin functions [5]. For Class-A systems with single bands in arbitrary dimensional systems, for reference energy, the corresponding Amoeba can determine whether the energy is the spectrum through the presence or absence of its “hole,” and the minimum point of the Ronkin function contains information about the localization length. However, the multiband nature resulting from the transpose-type Time-reversal symmetry embeds the holes in the Amoeba and shifts the point of the Ronkin function. We generalize this amoeba formulation to the class AII$^Dagger$ system; we show that the Monge-Amper\'e measurement of the Ronkin function, which is the generalization of the Hessian to functions with singularity [6], can decompose the Ronkin function into its bands and recover the above correspondence. [1] A. Yuto, G. Zongping, and U. Masahito, Advances in Physics 69, 249435 (2020) [2] S. Yao, F. Song, and Z. Wang, Pys. Rev. Lett. 121, 086803 (2018) [3] S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018) [4] K. Yokomizo and S. Murakami, Phys. Rev. Lett. 123, 066404 (2019) [5] H. Wang, F. Song, and Z. Wang, Phys. Rev. X 14, 021011 (2024) [6] J. Rauch, A. Taylor: Rocky Mountain J. Math. 7 (1977), 345 - 364.

Ultra-cold Rydberg atoms offer a remarkable platform for realizing a non-linear optical medium. This platform holds significance both at the applied level, as it reintroduces the concept of nonlinear optical computation, and at a more fundamental level, where the strong photon interactions enable the exploration of exotic quantum many-body states of light, such as photon liquids or crystals. The initial step towards constructing an effective theoretical framework involves developing a convenient and consistent approach for describing existing experiments. Recent experimental advancements in measuring three-particle states within such a medium [1] not only raise pertinent questions for theoretical elucidation but also serve as a pivotal starting point for further inquiry in this domain. We have successfully constructed a simplified framework that facilitates a visually comprehensible theoretical exposition of the topological and localization effects delineated in recent experimental works[1]. [1] Drori, L., et al., Science, vol. 381, no. 6654, 2023.

A band with a nonzero Chern number cannot be fully localized by weak disorder. There must remain at least one extended state, which ``carries the Chern number.'' Here we show that a trivial band can behave in a similar way. Instead of fully localizing, arbitrarily weak disorder leads to the emergence of two sets of extended states, positioned at two different energy intervals, which carry opposite Chern numbers. Thus, a single trivial band can show the same behavior as two separate Chern bands. We show that this property is predicted by a topological invariant called a ``localizer index.'' Even though the band as a whole is trivial as far as the Chern number is concerned, the localizer index allows access to a topological fine structure. This index changes as a function of energy within the bandwidth of the trivial band, causing nontrivial extended states to appear as soon as disorder is introduced. Our work points to a previously overlooked manifestation of topology, which impacts the response of systems to impurities beyond the information included in conventional topological invariants.

We study FCS in open quantum systems using tensor network studies. The distinct profile of local quantities in non-equilbrium steady state in integrable & non-integrable systems are well studied. Recent investigations in quantum computing and many-body physics have investigated the emergence of quantum ensembles beyond the first moment of observables. We study the impact of integrability/ quasi-integrability on behaviour of higher moments of conserved quantities in systems under boundary driving.

The topological characterization of matter is among the new paradigms of modern physics. Indeed, nontrivial topological phases promise the existence of localized or propagating interface waves that are immune to fluctuations and defects. As such, these systems are perfect candidates for applications in advanced devices like robust sensors, information processors, and waveguides. In the linearized limit of disordered topological systems, the short-time dynamics of bulk-localized wave packets have been demonstrated to accurately act as a topological indicator. Here, we explore the long-time dynamics of these localized excitations in the presence of disorder and nonlinearity. In the linearized limit, we show that initial wave packets experience Anderson localization in both trivial and nontrivial topological phases, and delocalization at a topological transition. Such features are lost in the presence of nonlinearity due to nonlinear mode-mode interactions.

Quasicrystals are solids that break translational symmetry but instead exhibit a scale invariance. They retain some long-range order, characterized by the presence of Bragg peaks in their diffraction pattern. Thus, the electronic states of quasicrystals are neither fully delocalized like crytalline Bloch states, nor localized as in disordered solids. Falling in between those two limits, they are said to be critical. The quantum metric is sensitive to the localization of electronic states. Although it trivially vanishes in atomic insulators and diverges in normal metals, it provides an insightful information for any system that falls in between those two limits. In this work, we use the quantum metric to characterize the critical states of a Fibonacci chain, a 1D quasicrystal. We show that the electronic states are localized on groups of sites sharing similar local environments but well separated in space, hence enhancing the quantum metric. Studying different crystalline approximants for the Fibonacci chain, we also establish that the quantum metric strongly depends on the filling fraction of the electronic states but that it always remain greater than a lower bond depending on the gap label. This property is reminiscent of Chern insulators, where the quantum metric is greater than the Berry curvature. In quasicrystal, the filling fraction can thus be used as a tuning parameter for the quantum metric, enabling to study the emergence of correlated states with increasing quantum metric.

Generically isolated quantum many-body systems reach a thermal equilibrium state upon unitary time evolution, which is explained by the Eigenstate thermalization hypothesis. But, when disorder is added to these systems, the dynamics becomes extremely slow. These systems are believed to evade thermalization even after very long time evolution. Our work sheds light on the slow dynamics of these systems from a very different perspective, namely the internal clock perspective. Considering the entanglement entropy as an internal clock, we get an idea about the fate of these disordered systems (simulation done for an XXZ chain) which can not be predicted from the real time simulation. We extend this idea in a disordered floquet model where we study the relaxation dynamics of local (inverse-) temperature. The broad distribution of relaxation time of the local (inverse-) temperature even in the ergodic regime, suggests us a striking similarity of this system with classical glasses which also show an inhomogeneous relaxation dynamics. However, a unified perspective emerges when considering the system's diagonal entropy as an internal clock, revealing an underlying homogeneity in the temperature dynamics for a broad range of disorder strengths.

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)

We explore the critical properties of a topological transition in a two-dimensional, amorphous lattice with randomly distributed points. The model intrinsically breaks the time-reversal symmetry without an external magnetic field, akin to a Chern insulator. Here, the topological transition is induced by varying the density of lattice points or adjusting the mass parameter. Using the two-terminal conductance and multifractality of the wave function, we found that the topological transition belongs to the same universality class as the integer quantum Hall transition. Regardless of the approach to the critical point across the phase boundary, the localization length exponent remains within $\nu \approx 2.55-2.61$. The irrelevant scaling exponent for both the observables is $y\approx 0.3(1)$, comparable to the values obtained using transfer matrix analysis in the Chalker-Coddigton network. Additionally, the investigation of the entire distribution function of the inverse participation ratio at the critical point shows possible deviations from the parabolic multifractal spectrum at the anomalous quantum Hall transition.

We investigate a model of Dirac fermions with Haldane type mass impurities which open a global topological gap even in the dilute limit. Surprisingly, we find that the chirality of this mass term, i.e., the sign of the Chern number, can be reversed by tuning the magnitude of the single-impurity scattering. Consequently, the disorder induces a phase disconnected from the clean topological phase, i.e., a genuine topological Anderson insulator. In seeming contradiction to the expectation that mass disorder is an irrelevant perturbation to the clean integer quantum Hall transition, the tri-critical point separating these two Chern insulating phases and a thermal metal phase is located at zero impurity density and connected to the appearance of a zero energy bound state in the continuum corresponding to a divergent Haldane mass impurity. Our conclusions based on the T-matrix expansion are substantiated by large scale Chebyshev-Polynomial-Green-Function numerics. We discuss possible experimental platforms.

Many-body localization (MBL) provides a generic mechanism to avoid thermalization in strongly disordered quantum systems. However, the existence of an MBL phase in the thermodynamic limit remains unclear. In this work, we study the properties of the MBL transition in the Kicked Ising model (KIM) with a quasiperiodic potential. We observe significantly fewer finite-size effects and strong evidence of localization, with a transition much sharper than in previous models. Additionally, the derivative of typical quantities, such as the average gap ratio or the entanglement entropy, across the transition decays exponentially with system size, indicating non-analytic behavior at finite length scales. While this behavior is not present in the KIM with random disorder, we observe it in the Heisenberg chain with a quasiperiodic potential, suggesting that the differences between the two disorder choices must also be shared by Floquet systems.

We study the superconducting instability of a two-dimensional disordered Fermi liquid weakly coupled to the soft fluctuations associated with proximity to an Ising-ferromagnetic quantum critical point. We derive interaction-induced corrections to the Usadel equation governing the superconducting gap function, and show that diffusion and localization effects drastically modify the interplay between fermionic incoherence and strong pairing interactions. In particular, we obtain the phase diagram, and demonstrate that: (i) there is an intermediate range of disorder strength where superconductivity is enhanced, eventually followed by a tendency towards the superconductor-insulator transition at stronger disorder; and (ii) diffusive particle-particle modes (so-called “Cooperons”) acquire anomalous dynamical scaling z=4, indicating strong non-Fermi liquid behavior.

The many-body localised phase of quantum systems is an unusual dynamical phase wherein the system fails to thermalise and yet, entanglement grows unboundedly albeit very slowly in time. We present a microscopic theory of this ultraslow growth of entanglement in terms of dynamical eigenstate correlations of strongly disordered, interacting quantum systems in the many-body lo-calised regime. These correlations involve sets of four or more eigenstates and hence, go beyond correlations involving pairs of eigenstates which are usually studied in the context of eigenstate thermalisation or lack thereof. We consider the minimal case, namely the second Renyi entropy of entanglement, wherein the correlations involve quartets of four eigenstates. We identify that the dynamics of the entanglement entropy is dominated by the spectral correlations within certain special quartets of eigenstates. We uncover the spatial structure of these special quartets and the ensuing statistics of the spectral correlations amongst the eigenstates therein, which reveals a hierarchy of timescales or equivalently, energyscales. We show that the hierarchy of these timescales along with their non-trivial distributions conspire to produce the logarithmic in time growth of entanglement, characteristic of the many-body localised regime. This microscopic theory therefore provides a much richer perspective on entanglement growth in strongly disordered systems compared to the commonly employed phenomenological approach based on the ℓ-bit picture.

Driven quantum many body systems constitute the building block in numerous experiments. Due to this, these systems have naturally attracted huge interests from the theoretical community. In this work, we provide analytical expressions for conductance at finite temperatures and for occupation at both zero and finite temperatures as extensions to the Greenwood-Kubo formula used for the noninteracting systems. This is possible due to the introduction of the concept of "embedding" to mimic semi-infinite baths, which is frequently used in systems with ultrafast dynamics. Beyond the analytical expressions, we also provide a general prescription for "embedding" depending on the system of interest to study its out of equilibrium properties. Finally, we apply this technique to study the effect of mean field corrections on the localization properties of the 1D Anderson chain at zero temperature. Both occupation and conductance clearly show that even at the mean field level, interaction brings in delocalization in the systems.

With random bonds of both signs, the classical two-dimensional Ising model undergoes a ferromagnet-to-paramagnet transition at low temperatures which is still little understood. The transition is governed by a zero-temperature fixed point separating ferromagnetic and spin-glass ground states. We use efficient evaluation of ground states of large systems to numerically study the critical behavior. To more fully characterize the transition we utilize free-fermionic representations of the renormalization group flows in the vicinity of the critical point, and point out connections with quantum infinite-randomness physics.

The fractional quantum Hall (FQH) effect gives rise to abundant topological phases, presenting an ultimate platform for studying the transport of edge states. Generic FQH edge contains multiple edge modes, commonly including the counter-propagating ones. A question of the influence of Anderson localization on transport through such edges arises. Recent experimental advances in engineering novel devices with interfaces of different FQH states enable transport measurements of FQH edges and edge junctions also featuring counter-propagating modes. These developments provide an additional strong motivation for the theoretical study of the effects of localization on generic edge states. We develop a general framework for analyzing transport in various regimes that also naturally includes localization. Using a reduced field theory of the edge after localization, we derive a general formula for the conductance. We apply this framework to analyze various experimentally relevant geometries of FQH edges and edge junctions.

Recent experiments have realized a twisted-bilayer-like optical potential for ultracold atoms, which in contrast to solid-state setups may allow for an arbitrary ratio between the inter-and intralayer couplings. For commensurate moiré twistings, a large-enough interlayer coupling results in particle transport dominated by channel formation. For incommensurate twistings, the interlayer coupling acts as an effective disorder strength. Whereas for weak couplings the whole spectrum remains ergodic, at a critical value part of the eigenspectrum transitions into multifractal states. A similar transition may be observed as well as a function of an energy bias between the two layers. Our theoretical study reveals atoms in a twisted-bilayer system of square optical lattices as an interesting platform for the study of ergodicity breaking and multifractality.

In 2D superconducting thin films the unbinding of thermal vortex-antivortex pairs should lead to a finite resistance above the Berezinskii-Kosterlitz-Thouless (BKT) temperature [1]. However, experimental results often showed a broadened transition by correlated disorder in the samples. Very recently, this was avoided in homogeneously disordered 3nm NbN films grown by atomic layer deposition [2]. Finite size effects were expected to affect the transition for sizes smaller than $\Lambda_p = 2\lambda^2/d$, but surprisingly, the transition in our films agrees with theory down to a width of $10\mu m$ with $\Lambda_p$ ≪ 2mm. When further reducing the width to $w \leq 1\mu m$, we observe a finite resistance even for $T

We investigate the spectral statistics and ergodicity of quasimodes in random, non-Hermitian 3D ensembles of point scatterers. The so-called point-dipole model [1] has been successfully employed to investigate Anderson localization of light over the years and here we propose a further characterization that suggests the presence of multifractal wavefunctions not only at the critical point in the Anderson transition in 3D. Specifically, by investigating the mean level ratio, level spacing statistics in the complex plane, and multifractal properties of eigenstates of the Green's matrix, we provide evidence of the existence of a Non-Ergodic Extended (NEE) phase [2] in 3D disordered point-scatterer ensembles for both scalar and vectorial models, where the latter includes the polarization of light. We also investigate the possible connection of such a NEE phase to the onset of subdiffusive dynamics in recent light transport experiments in disordered media [3]. [1] F. A. Pinheiro, M. Rusek, A. Orlowski, and B. Van Tiggelen, “Probing Anderson localization of light via decay rate statistics”, Physical Review E 69, 026605 (2004). [2]P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov, “Correlation-induced localization”, Physical Review B 99, 104203 (2019). [3] L. A. Cobus, G. Maret, and A. Aubry, “Crossover from renormalized to conventional diffusion near the three-dimensional Anderson localization transition for light”, Physical Review B 106, 014208 (2022).

Uncorrelated disorder in generalized 3D Lieb models gives rise to the existence of bounded mobility edges, destroys the macroscopic degeneracy of the flat bands and breaks their compactly-localized states. We now introduce a mix of order and disorder such that this degeneracy remains and the compactly-localized states are preserved. We obtain the energy-disorder phase diagrams and identify mobility edges. Intriguingly, for large disorder the survival of the compactly-localized states induces the existence of delocalized eigenstates close to the original flat band energies – yielding seemingly divergent mobility edges. For small disorder, however, a change from extended to localized behavior can be found upon decreasing disorder — leading to an unconventional “inverse Anderson” behavior. We show that transfer matrix methods, computing the localization lengths, as well as sparse-matrix diagonalization, using spectral gap-ratio energy-level statistics, are in excellent quantitative agreement. The preservation of the compactly-localized states even in the presence of this disorder might be useful for envisaged storage applications.

In this talk, I will tell you about our recent work on the investigation of the quantum Hall effect in a single Landau level in the presence of a square superlattice of $\delta$-function potentials. The important length scales in this clean system are: the superlattice spacing and the magnetic length, the interplay of which leads to three interesting characteristic regimes. I will show that the intermediate regime, where the two length scales are comparable with each other, is really special since the continuous magnetic translation symmetry breaks down to discrete lattice symmetry unlike the other two regimes, where the same is hardly broken in the topological band despite the presence of the superlattice. I will further show that in presence of (weak) disorder, interestingly, one obtains a large fraction of extended states throughout the intermediate regime which maximizes at a specific point within the same regime. I will argue this superlattice induced percolation phenomenon requires both the breaking of the time reversal symmetry and the continuous magnetic translation symmetry. I will discuss its direct implication on the integer plateau transitions in both continuous quantum Hall systems and the lattice based anomalous quantum Hall effect. Also I will touch on what happens if one considers a single anyon, instead of an electron, in similar situations. Ref: arXiv:2403.17137 (2024)

Many-body chaos is an important characterization of dynamical systems. Recently, there has been a tremendous surge in the interest of studying chaos in many-body quantum or classical systems. In this talk, I shall discuss how chaos sets within the intermediate time in a symmetry-breaking phase of a classical interacting spin model. Starting with the recently developed formalism of the decorrelation function, we shall discuss how a localized quenched defect in the ordered phase can give rise to secondary lightcones. These are particularly interesting in the sense that in the long time limit, the scattering from the secondary lightcones gives rise to the onset of chaos and actual propagation of a primary chaotic front. We shall also discuss the dynamical effects on the defect and link the emergence of chaos with low-energy spin-wave excitations by solving an analytical mode-coupling equation.

Chirality, a geometrical concept in which the structure lacks both inversion and mirror symmetries, gives a new twist to modern condensed matter physics. Organic molecules and structures, which are the building blocks of all organisms, commonly exhibit a well-defined chirality. The chirality-induced spin selectivity (CISS), a spin filtering effect in chiral molecules, has been extensively studied over the past few decades since its discovery in the DNA double helix [B. Göhler et al., Science (2011)]. The striking feature of CISS is an achievement of a large spin polarization even at room temperature without breaking the time-reversal symmetry. Therefore, chirality is essential to emerge spin functionality in organic molecules which have longer spin relaxation lengths. The CISS effect has opened new possibilities for using them in spintronic applications and could provide a fundamental understanding on the role of electron spin in biology. Despite intensive efforts have been devoted to identifying the key mechanism of CISS, the underlying physics still remains a long-standing mystery. The temperature dependence of the conductivity often provides rich insights into the underlying mechanisms of these transport phenomena. Experiments have demonstrated a long-range charge migration along the DNA double helix, indicating that DNA is a candidate for a one-dimensional molecular wire. Because a DNA double helix is not a periodic system due to a random base-pair sequence, disorder effects essentially determine the electronic features of DNA. The investigation of the conductivity and its temperature dependence has revealed that electrons in DNA can be consistently described by the variable-range hopping (VRH) for the charge transport [P. Tran et al., PRL (2000), Z. G. Yu & X. Song, PRL (2001)]. In this VRH model, the electron transport is dominated by incoherent hopping between Anderson localized states with emission or absorption of a phonon that bridges the energy difference between them. In this talk, we apply VRH to the electronic spin transport along the DNA double helix. The CISS effect is commonly observed at room temperature; therefore, it is natural to expect that phonons play a crucial role. In DNA, phonons acquire chirality reflecting its chiral structure. Thus, it is essential to consider the coupling between chiral phonons and electron spins. This can be accomplished by the micropolar elasticity theory [J. Kishine et al., PRL (2020)], which captures the rotational nature of chiral phonons. The resultant temperature dependence of the spin polarization quantitatively explains observations in experiments and indicates the relevance of both disorder effects and chiral phonons to the spin transport along the DNA double helix, which in turn will provide clues to the origin of CISS.

We numerically investigate the dynamics of a many-body localized (MBL) disordered quantum spin chain subjected to noise at one edge. This noise initiates a thermalization front at infinite temperature, that propagates differently depending on the disorder amplitude. This effect is observed in both the propagation of information, measured through mutual information, and matter, observed through spin expectations. The logarithmic slowdown at high disorder indicates an impact on the propagation of the local integrals of motion in MBL, within the system sizes accessible through our numerical simulations.

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)

Many-body localization (MBL) provides a mechanism to avoid thermalization in interacting systems. It is well understood that the MBL phase can exist in closed one-dimensional systems subjected to random disorder, quasiperiodic modulations, or homogeneous electric fields. However, the fate of MBL in higher dimensions remains unclear. Although some experiments on randomly disordered two-dimensional (2D) systems observe a stable MBL phase on intermediate time scales, recent theoretical works show that the phenomenon cannot persist forever and in a thermodynamic limit due to the rare regions and the avalanche instability. On the other hand, quasiperiodic systems do not host rare regions, and the avalanche instability is avoided; yet, the existence of an MBL phase in these systems remains to date largely unexplored. In this talk, I will discuss the localization properties of the many-body 2D Aubry- André quasiperiodic model. By studying the out-of-equilibrium dynamics of the interacting model with a numerical method of time-dependent variational principle, I will show that quasiperiodic systems can host a stable localized phase on experimentally relevant timescales. The numerical calculations show strong evidence that this nonergodic phase can survive even in the thermodynamic limit, in contrast to random disorder. Furthermore, I will discuss how deterministic lines of weak potential, which appear in the 2D Aubry-André model, support some transport while keeping the localized parts of the system unchanged.

The Hermiticity of a Hamiltonian guarantees its real eigenvalues and orthogonal eigenstates. Last few years have witnessed a surge of interest in non-Hermitian physics of non-conservative systems which has led to novel and unprecedented phenomena and applications. One fascinating phenomenon that has no Hermitian counterparts is the non-Hermitian skin effect (NHSE) which describes the anomalous localization behaviour of eigenstates of a lattice with open boundaries. We investigate the interplay of NHSE with the localization of a spinless fermionic lattice in the presence of a quasiperiodic potential. We examine a set of eigenstate and dynamical properties to characterize the distinction between two localization behaviours. The MBL states residing in the low-energy spectrum remain robust to non-Hermiticity, whereas the suppression in entanglement entropy indicates a strong sensitivity for the extended states. Furthermore, the time dynamics of imbalance clearly separates the two localization behaviours of a non-Hermitian many-body system. Our work presents the experimentally accessible diagnostics to realize the intriguing phenomena of non-Hermitian systems in quantum many-body experiments.

Precisely computing highly excited many-body eigenstates using exact methods is a formidable challenge. Thus, accessing high-energy excited state physics via ground-state methods, for which efficient algorithms exist, is highly appealing. By using the ground state wavefunction of a given local Hamiltonian, one can construct a new Hamiltonian of the same local form through the eigenstate-to-Hamiltonian construction approach. The reliability of this method hinges on the vanishing of energy variance. We employ the stochastic series expansion quantum Monte Carlo method, along with the eigenstate-to-Hamiltonian construction, to develop a Heisenberg model with correlated disorder that exhibits many-body localization in its excited states. Similarly, we design a new spin Hamiltonian with spatially alternating exchange couplings that hosts quantum many-body scar states. Our approach offers a systematic way to create various families of designer Hamiltonians with non-ergodic excited state properties.

We rigorously show that a local spin system giving rise to a slow Hamiltonian dynamics is stable against generic, even time-dependent, local perturbations. The sum of these perturbations can cover a significant amount of the system's size. The stability of the slow dynamics follows from proving that the Lieb-Robinson bound for the dynamics of the total Hamiltonian is the sum of two contributions: the Lieb-Robinson bound of the unperturbed dynamics and an additional term coming from the Lieb-Robinson bound of the perturbations with respect to the unperturbed Hamiltonian. Our results are particularly relevant in the context of the study of the stability of Many-Body-Localized systems, implying that if a so called ergodic region is present in the system, to spread across a certain distance it takes a time proportional to the exponential of such distance. The non-perturbative nature of our result allows us to develop a dual description of the dynamics of a system. As a consequence we are able to prove that the presence of a region of disorder in a ergodic system implies the slowing down of the dynamics in the vicinity of that region.

In Anderson localization, the mobility edge separates a phase where the wave function is localized in a disordered potential from another phase where it is delocalized. Usually, Anderson localization is understood as the result of destructive interferences of the wave function at long distance. Here, we show that, in the language of the localization landscape (LL) theory, this localization can be seen as a confinement in an effective potential deduced from the LL function. The mobility edge is interpreted as a percolation transition of the basins of the effective potential deduced from the LL. This transition arguably exists only in dimension . Surprisingly, the maps of the inverse participation ratio (IPR) in the energy-disorder diagram in dimension 2 and 3 still exhibit remarkably similar features, despite the absence of a genuine localization/delocalization transition in d=2. This suggests a possibly universal nature of the behavior of the IPR, and the existence of a quasi-transition in lower dimensions, opening the perspective to study critical exponents of the Anderson transition in large two-dimensional systems.

Transitions between distinct obstructed atomic insulators (OAIs) protected by crystalline symmetries, where electrons form molecular orbitals centering away from the atom positions, must go through an intermediate metallic phase. In this work, we find that the intermediate metals will become a scale invariant critical metal phase (CMP) under certain types of quenched disorder that respect the magnetic crystalline symmetries on average. We explicitly construct models respecting average C2zT, m, and C4zT and show their scale-invariance under chemical potential disorder by the finite-size scaling method. Conventional theories, such as weak anti-localization and topological phase transition, cannot explain the underlying mechanism. A quantitative mapping between lattice and network models shows that the CMP can be understood through a semi-classical percolation problem. Ultimately, we systematically classify all the OAI transitions protected by (magnetic) groups , and with and without spin orbit coupling, most of which can support CMP.

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. Using a tank circuit compatible with dc transport measurements, we investigate ultra-thin atomic layer deposited NbN films and trace the evolution of the superfluid stiffness as a function of disorder close to the SIT. We observe a sharp Berezinskii-Kosterlitz-Thouless transition in dc transport and in superfluid stiffness that persists even up to $R_N ≃ h/e^2$. In the vicinity of the SIT, phase fluctuations suppress the superfluid stiffness, consistent with a bosonic mechanism of SIT.

We propose a model of quantum walkers interacting through a Kondo-like interaction. Our final goal is to propose a model of quantum active \textit{matter} by introducing interactions between quantum active \textit{particles} [1]. In the present work, we first develop a description of quantum-walk dynamics in terms of scattering of a massless Dirac particle due to periodically located non-magnetic impurity potentials with the Hamiltonian of the form $H_{\mathrm{non-mag}}:=\epsilon p\sigma^z+m\delta(x)\sigma^y$, where $\epsilon$ and $m$ are positive parameters, while $\delta(x)$ is Dirac's delta function. We then replace one non-magnetic impurity potential with a magnetic impurity with the following Hamiltonian: $H_{\mathrm{mag}}:=\epsilon p\sigma^z\otimes s^0+H_{\mathrm{m}}\delta(x), H_{\mathrm{m}}:=J_x\sigma^x\otimes s^x+J_y\sigma^y\otimes s^y+J_z\sigma^z\otimes s^z$, where $J_x$, $J_y$ and $J_z$ are real parameters describing the coupling between the impurity and quantum walkers. The new degree of freedom, represented by a set of Pauli matrices $\{ s^x, s^y, s^z\}$, is a magnetic impurity localized at $x=0$, with $s^0$ being the identity operator in the same space. Each quantum walker is scattered by the spinful magnetic impurity, thus interacting with each other indirectly, analogous to the Kondo model. We present numerical results for two particles in the cases with and without the magnetic impurity and compare the cases of different interactions with the impurity spins. We have looked at dynamics of two quantum walkers with an impurity spin at the origin, focusing on differences between (a) a singlet state and (b) a direct product state as its initial state. We clearly observed a finite probability around the center in the case of the singlet initial state, whereas we did not in the other case of the direct-product initial state. This implies that in the second case, the walker at the origin is completely scattered away by the other incoming walker, while the walker at the origin in the first case partly screens the magnetic impurity, and hence the other walker cannot completely scatter it away. In the future, we aim to propose a multi-dimensional model using the results of Ref. [2] and introduce a quantum active matter in two and three dimensions. This work is under collaboration with N. Hatano (U. Tokyo), A. Nishino (Kanagawa U.), F. Nori (RIKEN and U. Michigan) and H. Obuse (Hokkaido U. and U. Tokyo). \\ \\ [1] M. Yamagishi, N. Hatano and H. Obuse, arXiv:2305.15319 (2023).\\ [2] M. Yamagishi, N. Hatano, K.-I. Imura and H. Obuse, Phys. Rev. A 107, 042206 (2023).

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)

Network models are a go-to tool for studying the localization transitions since the famous Chalker-Coddington model of the integer hall effect. To aid researchers in studying such systems using network models we have created an open source package which allows for flexibly creating different networks. The package supports both finite and infinite networks in multiple different dimensions and has methods for calculating energy spectra and scattering between leads.