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.

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 continuum gauge theories such as quantum elec- trodynamics, have become an important platform for the exploration of non-equilibrium many-body phenomena beyond their original scope in the Standard Model of particle physics. Recent works have reported the possible existence of many-body localization (MBL) in the U(1) Schwinger model in the absence of quenched disorder. Using degenerate perturbation theory and numerical simulations based on exact diagonalization and matrix product states, we provide a detailed characterization of thermalization breakdown in the U (1) Schwinger model including its spectral properties, the struc- ture of eigenstates, and out-of-equilibrium quench dynamics. We scrutinize the strong-coupling limit of the model, in which an intriguing, double-logarithmic-in-time, growth of entanglement was previously reported from the initial vacuum state [Brenes et al., Phys. Rev. Lett. 120, 030601 (2018)]. We identify the origin of this anomalously slow growth of entanglement due to an approxi- mate Hilbert space fragmentation and the emergence of a dynamical constraint on particle hopping, which gives rise to sharp jumps in the entanglement entropy dynamics within individual background charge sectors. Based on the statistics of jump times, we argue that the entanglement growth, av- eraged over charge sectors, is more naturally explained as either single-logarithmic or a weak power law in time. Our results thus suggest a single dynamical regime for all sufficiently large values of the coupling in the U (1) Schwinger model, whose properties are consistent with conventional MBL within the numerically accessible system sizes.

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]. 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, Phys. Rev. Research 6, 023264 (2024).

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.

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.

(arXiv:2408.15072) Elizaveta Safonova, Mikhail Feigelman, Vladimir Kravtsov

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 present an analytical calculation of the local density of state correlation function $ \beta(\omega) $ in the Lévy-Rosenzweig-Porter random matrix ensemble at energy scales larger than the level spacing but smaller than the bandwidth. The only relevant energy scale in this limit is the typical width of the level $\Gamma_0$. We show that $\beta(\omega \ll \Gamma_0) \sim const$ whereas $\beta(\omega \gg \Gamma_0) \sim \omega^{-\mu}$ where $\mu$ is an index characterizing the distribution of the matrix elements. We also provide an expression for the average return probability at long times: $R(t\gg\Gamma_0^{-1}) = \exp\{-(\Gamma_0 t)^{\mu/2}\}$. Numerical results based on the pool method and exact diagonalization are also provided and are in agreement with the analytical theory.

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.

Chiral symmetric systems in two dimensions lack conventional topological characteristics but exhibit weak topological phases induced by SSH-like staggering bonds. We explore the impact of this topology on the Anderson localization transition in the Chiral class. Topology induces anisotropy in the transport properties of these systems. To illustrate this, we examine a bond-dimer model on a two dimensional square lattice that features a weak-topological insulator phase, a trivial insulating phase and a metallic phase, creating a diverse phase diagram that effectively demonstrates the interplay between topology and the Anderson transition. We discuss the universality of this localization transition within the Chiral class. The work is built from our recent work [1]: [1] N. P. Nayak, S. Sarkar, K. Damle, and S. Bera, Band-center metal-insulator transition in bond-disordered graphene, Phys. Rev. B 109, 035109 (2024).

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.

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.

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.

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.

We consider a free-fermion chain undergoing dephasing, described by two different random-measurement protocols (unravelings): a quantum-state-diffusion and a quantum-jump one. Both protocols keep the state in a Slater-determinant form, allowing to address quite large system sizes. We find a bifurcation transition in the distribution of the measurement operators along the quantum trajectories, where it changes from unimodal to bimodal. The value of the measurement strength where such transition occurs is similar for the two unravelings, but the distributions and the transition have different properties reflecting the symmetries of the two measurement protocols. We also consider the scaling with the system size of the inverse participation ratio of the Slater-determinant components and find a power-law scaling that marks a multifractal behaviour, in both unravelings and for any nonvanishing measurement strength. We consider also the extension of our results to the non-integrable case.

For a disordered Su-Schriefer-Heeger model there is a phase transition between topological and trivial phase. At this point, it is known that the model delocalizes. This is thoroughly analyzed by studying a hyperbolic critical energy which leads to a Dyson peak, a characteristic novel feature of the localization length (inverse Lyapunov exponent) and wave packet spreading (absence of localization).

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.

Based upon arxiv:2405.05958, arxiv:2408.00743 and arxiv:2408.02016

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.

We propose a two-band tight-binding model of topological nodal-line Weyl semimetal on the cubic lattice, which belongs to the symmetry class D of the Altland-Zirnbauer classification [1]. By studying disorder-driven quantum phase transitions of the model, we observed signatures of a quasilocalized phase and topological anisotropic Anderson transition, which had been universally observed in the chiral classes with weak topological index [2]. Thereby the metal phase and the Anderson localized phase are intervened by the quasilocalized phase that shows metallic behavior in the topological direction while insulating behavior in the other directions [2]. We investigate the possibility of the two-step phase transition and associated criticalities in this class-D model compared to our previous results of another model [3], and discuss its relation to the topological index and the underlying mechanism in contrast with the chiral classes. [1] A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997). [2] Z. Xiao, K. Kawabata, X. Luo, T. Ohtsuki, and R. Shindou, Phys. Rev. Lett. 131, 056301 (2023). [3] T. Wang, T. Ohtsuki, and R. Shindou, Phys. Rev. B 104, 014206 (2021).

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).

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.