Anderson Localization and Interactions

I will report on our work on the MBL-ETH transition. On the one hand, there is numerics that clearly supports the idea that finite-size thermal Girffiths regions can destabilize the MBL phase. On the other hand, building on this premisse, one can present a clean theory of the MBL transition, with quantitative prediction.

Using the time-dependent variational principle, we study the disordered XXZ Heisenberg chain for chains up to a length of 100 sites, around the regime where a transition from an ergodic to a many-body localized phase is expected. We find that finite-size effects lead to an underestimation of transport and the associated power law describing subdiffusion, suggesting that previous studies for smaller systems have underestimated the extent of the ergodic regime. Elmer V. H. Doggen, Konstantin S. Tikhonov, Dmitry G. Polyakov, Alexander D. Mirlin, Igor V. Gornyi

I will report on recent results about dynamical properties of many-body localization transition in both ergodic and localized phase. We investigate the charge relaxation in quantum-wires of spin-less disordered fermions (t−V-model). Our observable is the time-dependent density propagator at different disorder strengths and at finite interaction. In the ergodic phase, we observe slow charge relaxation, however with strong indication that it could only be transient. In the localized phase, we observe even slower, possibly logarithmic, decay of charge, which suggests that the many-body localized phase is of different kind compare to conventional Anderson localized phase.

Classical many body interacting systems are typically chaotic and their microcanonical dynamics ensures that time averages and phase space averages are identical. In proximity to an integrable limit the properties of the network of nonintegrable action space perturbations help decide whether i) ergodization time scales stay on the order of the Lyapunov times, or whether ii) the system fragments into regular and chaotic parts and enters a dynamical glass (DG) phase at a finite distance to the integrable limit. This DG phase is induced by coherent localized excitations - generalized discrete breathers - with anomalously large lifetimes. We use a set of observables to quantify the properties of the DG phase. A sectioning of a typical trajectory by equilibrium Poincare manifolds defines events, whose statistics signals the onset of the DG phase. Distributions of finite time averages allow to extract time scales of ergodization, and their dependence on control parameters as compared to the Lyapunov times. I will show results for a classical version of Josephson junction chains, first studied more than 30 years ago, and recently readdressed due to the prediction of an MBL phase in the full quantum version of the system.

Topological materials are characterized by a special band structure a with characteristic Dirac band dispersion and therefore offer a lot of new and interesting properties, especially with respect to future device applications. In this presentation, the material system of mercury-telluride (HgTe) is introduced as a proto-type for the investigation of transport properties in two-dimensional (2D) as well as in three-dimensional (3D) topological systems. The realization of a 2D topological insulator (TI) structure in HgTe quantum wells has already demonstrated the potential for spin injection and detection purposes in spintronic applications without any magnetic materials or applied magnetic fields. However, transport experiments on 3D TI surface states are rare, even though numerous TI materials have been identified and fabricated worldwide. Here, the magneto-transport characterization data of strained HgTe bulk layers are presented which show characteristic Dirac-like quantum Hall-effect sequences. The quantum Hall features originate from those two 2D surfaces states which are oriented perpendicular to the applied magnetic field in case of tensil strained HgTe. Using top and back-gated sample structures it is possible to distinguish their specific contributions. For compressively strained HgTe so-called Weyl nodes occure and give rise for a chiral anomaly in transport.

The realization of a quantum Spin Hall insulator (QSHI) in InAs/GaSb and InAs/GaInSb materials consisting of coupled electron and hole double quantum wells has offered an unique system for experimental studies of transport properties of a 1-D helical liquid, in which the Luttinger parameters may be tuned by several knobs, such as well width, gates, and lattice strain. We first report on the observation of a helical Luttinger liquid in the edge of an InAs/GaSb quantum spin Hall insulator, which shows characteristic suppression of conductance at low temperature and low bias voltage. Moreover, the conductance shows power-law behavior as a function of temperature and bias voltage. The results underscore the strong electron-electron interaction effect in transport of InAs/GaSb edge states. Because of the fact that the Fermi velocity of the edge modes is controlled by gates, the Luttinger parameter can be fine-tuned. We then report the findings in strained-layer InAs/GaInSbs, in which the bulk gaps are enhanced up to five folds as compared to the binary InAs/GaSb QSHI. Remarkably, with consequently increasing edge Fermi velocity, the edge conductance at zero and applied magnetic fields manifests weakly interacting helical edge states. We will focus on systematic data measured in InAs/Ga0.68In0.32Sb quantum wells, where the temperature and bias voltage dependence of the helical edge conductance for devices of various sizes will be presented. Moreover, we found that the magnetoresistance of the helical edge states is related to the edge interaction effect and the disorder strength. References: 1.Observation of a Helical Luttinger Liquid in InAs/GaSb Quantum Spin Hall Edges, Tingxin Li, et al, Phys. Rev. Lett. 115, 136804 (2015) 2.Tuning Edge States in Strained-Layer InAs/GaInSb Quantum Spin Hall Insulators, Lingjie Du et al, Phys. Rev. Lett. 119, 056803 (2017) 3.Low-temperature conductivity of weakly interacting quantum spin Hall edges in strained-layer InAs/GaInSb, Tingxin Li et al, Phys. Rev. B 96, 241406(R) (2017)

Periodically driven quantum systems, aka Floquet systems, have been recently identified as a prime platform for the study of topologically nontrivial quantum dynamics. Absent the notion of ground state(s), many-body localization (MBL) plays a crucial role in the definition of such dynamical phases. Strikingly, some of the discovered phases are intrinsically dynamical, in the sense that they do not admit static counterparts. In this talk, I will describe our results on the study of chiral phases in this context. These chiral phases are compatible with the notion of MBL---an impossibility in static systems. I will first discuss the analogy of the integer quantum Hall phases, where instead of charge or heat, quantum information is pumped along the edge in a unidirectional manner. Next, I will present an extension of this phase into one featuring intrinsic topological order, where the pumping of emergent Majorana fermions along the edge is accompanied by a necessary dynamical anyon transmutation in the bulk.

A fundamental assumption in statistical physics is that generic closed quantum many-body systems thermalise under their own dynamics. Recently, the emergence of many-body localised (MBL) systems has questioned this concept, challenging our understanding of the connection between statistical physics and quantum mechanics. In my talk, I will report on several recent experiments carried out in our group on the observation of Many-Body Localisation in different scenarios, ranging from 1D fermionic quantum gas mixtures in driven and undriven Aubry-André type disorder potentials and 2D systems of interacting bosons in 2D random potentials. It is shown that the memory of the system on its initial non-equilibrium state can serve as a useful indicator for a non-ergodic, MBL phase. Furthermore, I will present new results on the slow relaxation dynamics in the ergodic phase below the MBL transition and experiments that explore the resilience of a 2D MBL phase when coupled to a finite thermal bath. Our experiments represent a demonstration and in-depth characterisation of many-body localisation, often in regimes not accessible with state-of-the-art simulations on classical computers.

We introduce a simple, exactly solvable strong-randomness renormalization group (RG) model for the many-body localization (MBL) transition in one dimension. Our approach relies on a family of RG flows parametrized by the asymmetry between thermal and localized phases. We identify the physical MBL transition in the limit of maximal asymmetry, reflecting the instability of MBL against rare thermal inclusions. We find a critical point that is localized with power-law distributed thermal inclusions. The typical size of critical inclusions remains finite at the transition, while the average size is logarithmically diverging. We propose a two-parameter scaling theory for the many-body localization transition that falls into the Kosterlitz-Thouless universality class, with the MBL phase corresponding to a stable line of fixed points with multifractal behavior.

While generic interacting isolated quantum systems typically satisfy the eigenstate thermalization hypothesis (ETH) and thermalize, this is not the case for many-body localized systems. Using two examples, a static and a periodically driven one, I will show that there are corrections to ETH also in the thermal phase at weak disorder of disordered interacting spin chains, which are related to slow subdiffusive spin transport. They lead to a different scaling of matrix elements of the local $S_z$ operator with system size compared to the standard ETH and are accompanied by non-Gaussian distributions of these matrix elements. References: Anomalous thermalization and transport in disordered interacting Floquet systems Sthitadhi Roy, Yevgeny Bar Lev, and David J. Luitz Phys. Rev. B 98, 060201(R) Anomalous Thermalization in Ergodic Systems David J. Luitz and Yevgeny Bar Lev Phys. Rev. Lett. 117, 170404

Due to the presence of phonons, many body localization (MBL) does not occur in disordered solids, even if disorder is strong. Local conservation laws characterizing an underlying MBL phase decay due to the coupling to phonons. I will show that this decay can be compensated when the system is driven out of equilibrium. The resulting steady state shows strong variations of the local temperature, which provide characteristic fingerprints of an underlying MBL phase. I will consider a one-dimensional disordered spin-chain which is weakly coupled to a phonon bath and weakly irradiated by white light. The irradiation has weak effects in the ergodic phase. However, if the system is in the MBL phase irradiation induces strong temperature variations of order one despite the coupling to phonons. Temperature variations can be used similar to an order parameter to detect MBL phases, the phase transition, an MBL correlation length and even the critical exponents. Z. Lenarčič, E. Altman, A. Rosch, arXiv:1806.04772 (2018)

We study spatially and frequency resolved correlations of wavefunctions (WF) and energy level statistics in Anderson model on the Random Regular Graph (RRG) at criticality and in the delocalized phase. We find a very good agreement between analytical results and numerical approaches, including exact diagonalization and numerical solution of the self-consistency equation for the probability distribution of the local Green functions. We observe correlation length directly in the spatial decay of WF correlations and in the spectral compressibility. Finally, we connect our results to properties of the Anderson model defined on finite dimensional (d) lattices at d >> 1, stressing interesting peculiarities of this limit.

Localization is a key mechanism for the emergence of non-ergodicity in disordered quantum systems. In this talk, I will present some numerical results which suggest that quantum localization shares a close analogy with the physics of spin glasses, another paradigm of non-ergodic behavior in classical disordered systems. I will focus on the quantum transport through a disordered two-dimensional sample in the localized regime and show that it verifies several important glassy properties: pinning, avalanches and chaos. This problem is addressed by following the well-known analogy between quantum localization and the physics of directed polymers, one of the simplest model of statistical physics in which disorder plays a non-trivial role, as in spin glasses. I will first recall how to observe the dominant paths taken by the transport and demonstrate that these paths verify pinning and avalanches. Then, I will characterize the spin glass chaos property: Two infinitesimally perturbed replicas of the same sample have their correlation which vanishes at the thermodynamic limit following a characteristic single parameter scaling law.

By using the single-particle Anderson model on the Bethe lattice as a toy model for the many-body quantum dynamics in the Fock space, we discuss the unusual slow dynamics that emerges in the bad metal delocalized phase preceding the many-body localization transition. By mapping the propagation of the imaginary part of the Green's functions onto the partition function of directed polymers in random media, we relate the unusual power-law behavior of the dynamical correlation functions to the freezing-glass transition of the directed polymers on few specific disorder-dependent delocalizing paths. We also discuss the relation of these phenomena with the non-ergodic features of the spectral statistics and highlight the difference between loop-less Cayley trees and random-regular graphs.

Periodically driven, or Floquet, disordered quantum systems have generated many unexpected discoveries of late, such as the anomalous Floquet Anderson insulator and the discrete time crystal. Here, we report the emergence of an entire band of multifractal wavefunctions in a periodically driven chain of non-interacting particles subject to spatially quasiperiodic disorder. Remarkably, this multifractality is robust in that it does not require any fine-tuning of the model parameters, which sets it apart from the known multifractality of critical wavefunctions. The multifractality arises as the periodic drive hybridises the localised and delocalised sectors of the undriven spectrum. We account for this phenomenon in a simple random matrix based theory. Finally, we discuss dynamical signatures of the multifractal states, which should betray their presence in cold atom experiments. Such a simple yet robust realisation of multifractality could advance this so far elusive phenomenon towards applications, such as the proposed disorder-induced enhancement of a superfluid transition. Work with Sthitadhi Roy, Ivan M. Khaymovich, Arnab Das.

I will discuss results relating to electrons in two spatial dimensions (2D), subject to the effects of quenched disorder (impurities) and quantum interference [Anderson (de)localization]. In both cases, the key physics is tied to classical geometric critical phenomena in 2D. I will first present numerical evidence that most surface states of 3D topological superconductors exhibit universal critical statistics in the presence of nonmagnetic disorder [1]. This is highly unusual, since critical (multifractal) statistics typically arise only by fine-tuning to a mobility edge. For a particular class (CI), we show that (almost) every state within the surface energy band shows universal statistics consistent with the plateau transition of the spin quantum Hall effect (class C, "equivalent" to critical percolation in 2D). Thus critical percolation can be robustly realized at the surface of a topological superconductor without fine tuning. Our results may facilitate new approaches to the logarithmic conformal field theories believed to describe plateau transitions, using perturbed WZNW models. Second, I will discuss a "hole" in the standard theory of the ergodic phase in 2D. I will demonstrate that the self-dephasing of weak localization for an isolated many-fermion system with short-ranged interactions is described by a strongly coupled auxiliary quantum field theory. The problem can be cast as a type of self-interacting walk. I will present analytical results of a controlled expansion which suggest that dephasing can fail at sufficiently low temperatures, and speculate on the possible connection to many-body localization. [2,3] [1] S. A. A. Ghorashi, Y. Liao, and MSF, arXiv:1711.03972 [2] Y. Liao and MSF, arXiv:1710.05037 [3] Y. Liao, A. Levchenko, and MSF, Ann. Phys. 386 (2017) 97-157

In this talk, I will describe a new mechanism for many-body localization, making use of ideas drawn from the field of fractons, based on recent work in collaboration with Shriya Pai and Rahul Nandkishore. Specifically, I will present results on the spreading of initially local operators under random unitary evolution in spin chains subject to fracton conservation laws, such as conservation of dipole moment. We find that fractons remain permanently localized at their initial positions, providing a crisp example of a non-ergodic dynamical phase of random unitary evolution. These results can be interpreted as a consequence of the properties of low-dimensional random walks. This mechanism for localization remains robust in one and two dimensions, but breaks down in three-dimensional fracton systems. We argue that these results extend to Floquet and Hamiltonian time evolution, even in the absence of disorder, thereby providing a mechanism for many-body localization in a translationally invariant system.

In a strongly disordered system local degrees of freedom can be quenched by disorder. It was discovered recently that this can lead to the prominent phenomenon of many-body localization (MBL) when an interacting system lacks ergodic dynamics even in the thermodynamic limit. Existence of MBL phase in certain 1D lattice models with short-range interactions was proven theoretically and went through extensive numerical testing in a number of recent works. While quenching of local degrees of freedom is easy to achieve at strong disorder it does not always lead to MBL. In certain cases a disordered system may look completely frozen at short scales but evade MBL (for arbitrarily strong disorder) in thermodynamic limit due to resonant interactions between distant degrees of freedom. We discuss this delocalization scenario and the consequences thereof for transport and thermalization on the two examples of (a) systems with power-law interactions and (b) $SU(2)$-symmetric Heisenberg spin chain.

Linear arrays of trapped and laser cooled atomic ions (171Yb+) are a versatile platform for studying strongly correlated many-body quantum systems. These systems are well suited to investigate out-of-equilibrium phenomena, ranging from time-crystalline phases to dynamical phase transitions. In particular I will show how time crystals can be realized in a Floquet setting, where the spin system exhibits persistent time-correlations either under Many-body-localized dynamics [1] or in a prethermal regime [2]. I will also present our recent observation of a new type of out-of-equilibrium dynamical phase transition in a spin system with long range power law interactions [3] and over 50 spins [4]. I will also show our efforts towards scaling up the trapped-ion quantum simulator [5] using a cryopumped vacuum chamber where we can trap more than 100 ions for hours. We anticipate that this reliable production and lifetime enhancement of large linear ion chains will enable quantum simulation of spin models that are intractable with classical computer modelling. References [1] J. Zhang et al., Nature 543, 217 (2017) [2] P.Hess et al., in preparation (2017) [3] Jurcevic et al., PRL 119, 080501 (2017) [4] J.Zhang, G.Pagano et al., Nature 551, 601 (2017) [5] G. Pagano et al., arXiv:1802.03118

We chart out the ground state phase diagram and also demonstrate the presence of a many-body localized (MBL) phase in an experimentally realizable one-dimensional (1D) dipole boson model in the presence of an Aubry-Andre (AA) potential whose strength λ_0 can be tuned to precipitate an ergodic-MBL transition. We discuss the signature of such a transition in the quantum dynamics of the model by computing its response subsequent to a sudden quench of λ_0. We show that both the MBL and the ergodic phases can be clearly distinguished using such a response and provide an estimate for minimal time up to which experiments need to track the response of the system to confirm the onset of the MBL phase. We suggest concrete experiments which can test our theory.

The possibility to trap individual atoms in optical tweezers and manipulate their internal electronic state (including strongly interacting Rydberg states) provides a novel, powerful tool to study interacting quantum many-body systems in a coherent, controllable AMO setting. Int this talk I will focus on different theoretical aspects of such systems and discuss the equilibrium and out-of equilibrium properties of some of the models that can be realized experimetally. This includes the zero temperature phase diagram, where the so-called Rydberg blockade mechanism leads to different spatially ordered phases, the possibility to explore the universality classes of the phase boundaries via the Kibble-Zurek mechanism, as well as the observation of state-dependent thermalization dynamics, coined ‘quantum many-body scars’. Finally I will present possibilities to use this system of reconfigurable trapped atoms to realize an architecture for adiabatic quantum computing.

The thermodynamic description of many-particle systems rests on the assumption of ergodicity -- the ability of a system to explore all allowed configurations in the phase space. Recent studies on many-body localization have revealed the existence of systems that strongly violate ergodicity in the presence of quenched disorder. Here, we demonstrate that ergodicity can be weakly broken by a different mechanism, arising from the presence of special eigenstates in the many-body spectrum that are reminiscent of quantum scars in chaotic non-interacting systems. In the single-particle case, quantum scars correspond to wavefunctions that concentrate in the vicinity of unstable periodic classical trajectories. We show that many-body scars appear in the Fibonacci chain, a model with a constrained local Hilbert space that has recently been experimentally realized in a Rydberg-atom quantum simulator. The quantum scarred eigenstates are embedded throughout the otherwise thermalizing many-body spectrum but lead to direct experimental signatures, as we show for periodic recurrences that reproduce those observed in the experiment. Our results suggest that scarred many-body bands give rise to a new universality class of quantum dynamics, opening up opportunities for the creation of novel states with long-lived coherence in systems that are now experimentally realizable.

Thin-film superconductors exhibit a rich set of phenomena, much of it is not well understood. I will describe recent experimental results which indicate that these systems are extremely sensitive to external perturbations, which tend to drive them out of thermodynamic equilibrium.

I will overview recent advances in the fermionic RG approach (based on the sigma-model formalism) to the superconductor-insulator transition in homogeneously disordered films. Predictions of this theory have a number of important similarities with what is observed experimentally: (i) possibility of a direct transition between the superconducting and insulating phases with critical resistance of the order of quantum resistance; (ii) a non-monotonous dependence of the resistivity on temperature T and magnetic field; in particular, a giant non-monotonous magnetoresistance low T; (iii) relation between the temperature of the maximum in the T dependence of resistivity and the true (BKT) transition temperature; (iv) a strong depletion of the average DOS in the metallic state (i.e., above the critical temperature) caused by the interplay of superconducting fluctuations and Coulomb repulsion, as well as a pseudogap in the insulating phase; (v) strong mesoscopic fluctuations of the local density of states.

The superconductor-insulator transition (SIT) is a prototype of a quantum phase transition which is very versatile experimentally: varying a non-thermal tuning parameter such as disorder, thickness, composition, magnetic field or gate-voltage causes the system to switch from a superconductor to an insulator at zero temperature. Unlike their classic counterparts, quantum phase transitions are governed by quantum fluctuations rather than thermal fluctuations. The direct experimental study of such fluctuations close to the SIT is rather challenging. So far research has mainly concentrated on dc resistivity based measurements such as transport and magnetoresistance and on global and local tunneling spectroscopy. These provide only limited information on the critical behavior through the transition. In my talk I will describe thermal (specific heat and Nernst effect) experiments designed to measure direct signatures of quantum fluctuations and critical behavior close to the SIT. I will discuss the significance of the results and their contribution to understanding the electronic processes in the vicinity of the quantum phase transition.

Since the seminal work of Fisher [1], crossing points of the magnetoresistance $R(T,B)$ at constant temperature $T$ constitute a hallmark of a magnetic field-induced superconductor-insulator quantum phase transition (SIT), provided that $R(T,B)$ displays universal scaling behavior. We show that three such crossing points can be observed in the same TiN thin film that are associated with distinct plateaus in the resistance $R(T,B)$ vs. $T$ at certain magnetic fields $B_\mathrm{c}$. For each crossing point, we trace the behavior of $B_\mathrm{c}$ and the corresponding scaling parameters while increasing the level of disorder down to very small values of $B_\mathrm{c,x}$. Only one of the crossing points displays values of the critical parameters that are independent of the level of disorder, indicating a certain universality class. The other two crossing points can be traced back to an accidental compensation of competing contribution to $R(T)$ that are not connected to a phase transition. Our results show that an independent control parameter is needed, besides the magnetic field, in order to establish candidates for a quantum critical behavior in strongly disordered superconductors.\\[2mm] [1] M. P. A. Fisher, Phys. Rev. Lett {\bf 65}, 923 (1990).

The interplay of charge ordering with superconducting correlations in underdoped cuprates at high magnetic fields ($H$) is an open question, and even the value of the upper critical field has been the subject of a long-term debate. We combined three complementary transport techniques on underdoped La-214 cuprates with a ``striped'' charge order and a low $H=0$ transition temperature $T_{c}^{0}$, which opens a much larger energy scale window to explore the vortex phases compared to previous studies. We determine the $T-H$ phase diagram, directly detect the upper critical field, and probe deep into the normal state. Our results demonstrate a key role of disorder in the behavior of vortex matter as $T\rightarrow 0$. Furthermore, the agreement between our transport results in $H$ and spectroscopic data in $H=0$ provides a unified perspective on the superconducting phase transition in underdoped cuprates. Most strikingly, however, within the vortex glass region we discover the emergence of an unanticipated, insulatinglike state, but with strong superconducting correlations. This novel state is consistent with the ordering of localized Cooper pairs within the quasi-1D stripes, but other possible scenarios will be also discussed. The presence of stripes, however, plays a crucial role in the freezing of Cooper pairs in this novel state. This work was supported by NSF Grants No. DMR-1307075 and No. DMR-1707785, and the National High Magnetic Field Laboratory through the NSF Cooperative Agreement No. DMR-1157490 and the State of Florida.

We investigate the amplitude (Higgs) mode of a diluted quantum rotor model in two dimensions close to the superfluid-Mott glass quantum phase transition. After mapping the Hamiltonian onto a classical (2+1)-dimensional XY model, the scalar susceptibility is calculated in imaginary time by means of large-scale Monte Carlo simulations. Analytic continuation of the imaginary time data is performed via maximum entropy methods and yields the real-frequency spectral function. The spectral peak associated with the Higgs mode is identified and its fate upon approaching the disordered quantum phase transition is determined. This work is supported in part by the National Science Foundation under grant no. DMR-1506152.

In this talk I present the results for the quantum corrections to the conductivity of the two-dimensional disordered interacting electron system in the diffusive regime due to inelastic scattering off rare magnetic impurities. Contrary to recent studies I focus on inelastic corrections (alike Altshuler-Aronov type) rather than modification of the dephasing rate. In the case of very different g-factors for electrons and magnetic impurities such corrections appear within the Born approximation for the inelastic scattering off magnetic impurities. In the case of two dimensions, we found quantum corrections which are logarithmic in temperature. Our results demonstrate that the low temperature transport in interacting disordered electron systems with rare magnetic impurities is more interesting than it was commonly believed on the basis of treatment of magnetic impurity spins as classical ones.

The Sachdev-Ye-Kitaev (SYK) model is a system of N>>1 randomly interacting Majorana fermions. It stands in the tradition of the random k-body interaction models pioneered in nuclear physics and later applied in condensed matter contexts. In this talk we will discuss how the SYK model can be looked at from three interrelated perspectives: a) a systems showing many body chaos and random matrix correlations, b) a paradigm of strongly correlated (Majorana) quantum matter, and c) the holographic shadow of a two-dimensional AdS2 gravitational bulk. We will review the appearance of familiar signatures of quantum chaos which might lead to the expectation that the essentials of the system are well described by a RMT ansatz. This, however, is not so. The SYK model supports an approximately realized infinite dimensional symmetry group which at low excitation energies manifests itself in pronounced Goldstone mode fluctuations. While the many body collective fluctuations described by these modes are chaotic they are not described by RMT. Nonetheless they are amenable to analytic description, and they lend themselves to a holographic interpretation. The complementarity of the two approaches to the system is not well understood at present and a subject of ongoing research.

The Sachdev-Ye-Kitaev model is a model of N fermions with all-to-all two-body random interactions. We show analytically and numerically that a generalized Sachdev-Ye-Kitaev model with an additional one-body infinite-range random interaction, which is a relevant perturbation in the infrared, is still quantum chaotic, and retains most of its holographic features, for a fixed value of the perturbation and sufficiently high temperature. However a chaotic-integrable transition, characterized by the vanishing of the Lyapunov exponent and spectral correlations given by Poisson statistics, occurs at a temperature that depends on the strength of the perturbation. [1] Moreover, we study the level statistics of the model with interactions of finite range. Tuning the range of the one-body term, while keeping the two-body interaction sufficiently long-ranged, does not alter substantially the spectral correlations, which are still given by the random matrix prediction typical of a quantum chaotic system. However a transition to an insulating state, characterized by Poisson statistics, is observed by reducing the range of the two-body interaction. Close to the many-body metal-insulator transition, we show that spectral correlations share features previously found in systems at the Anderson transition and in the proximity of the many-body localization transition. Our results suggest the potential relevance of SYK models in the context of chaotic-integrable transition and many-body localization. [2] [1] Antonio M. García-García, Bruno Loureiro, Aurelio Romero-Bermúdez, and Masaki Tezuka, "Chaotic-integrable transition in the Sachdev-Ye-Kitaev model", Phys. Rev. Lett. in press. (arXiv:1709.07189) [2] Antonio M. García-García and Masaki Tezuka, "Many-Body Localization in a finite-range Sachdev-Ye-Kitaev model", arXiv:1801.03204.

The three-dimensional Anderson model is a well-studied model of disordered electron systems that shows the delocalization--localization transition. As in the studies on two- and three-dimensional (2D, 3D) quantum phase transitions [J. Phys. Soc. Jpn. {¥bf 85}, 123706 (2016), {¥bf 86}, 044708 (2017)], we used an image recognition algorithm based on a multilayered convolutional neural network. In contrast to previous studies in which 2D image recognition was used, we applied 3D image recognition to analyze entire 3D wave functions. We show that a full phase diagram of the disorder-energy plane is obtained once the 3D convolutional neural network has been trained at the band center. We further demonstrate that the full phase diagram for 3D quantum bond and site percolations can be drawn by training the 3D Anderson model at the band center. References: T. Mano and T. Ohtsuki: Journal of the Physical Society of Japan 86, 113704 (2017).

Machine learning has been successfully applied to identify phases and phase transitions in condensed matter systems. However, quantitative characterization of the critical fluctuations near phase transitions is lacking. In this study we extract the critical behavior of a quantum Hall plateau transition with a convolutional neural network. We introduce a finite-size scaling approach and show that the localization length critical exponent learned by the neural network is consistent with the value obtained by conventional approaches. We illustrate the physics behind the approach by a cross-examination of the inverse participation ratios.

To date the most precise estimations of the critical exponent for the Anderson transition have been made using the transfer matrix method introduced into the field of Anderson localization by Pichard and Sarma [1], and MacKinnon and Kramer [2]. This method involves the simulation of extremely long quasi one-dimensional systems. The method is inherently sequential and is not well suited to modern massively parallel supercomputers. The obvious alternative is to simulate a large ensemble of hypercubic systems and average. While this permits taking full advantage of both OpenMP and MPI on massively parallel supercomputers, a straight forward implementation results in data that does not scale. We show that this problem can be avoided by generating random sets of orthogonal starting vectors with an appropriate stationary probability distribution. We have applied this method to the Anderson transition in the three-dimensional orthogonal universality class and been able to increase the largest L × L cross section simulated from L=24 in [3] to L=64 here. This permits an estimation of the critical exponent with improved precision and without the necessity of introducing an irrelevant scaling variable. In addition, this approach is better suited to simulations with correlated random potentials such as is needed in quantum Hall or cold atom systems. 1.Pichard, J.L. and G. Sarma, Finite size scaling approach to Anderson localisation. Journal of Physics C: Solid State Physics, 1981. 14(6): p. L127. 2.MacKinnon, A. and B. Kramer, One-parameter scaling of localization length and conductance in disordered systems. Physical Review Letters, 1981. 47(21): p. 1546-1549. 3.Slevin, K. and T. Ohtsuki, Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class. New Journal of Physics, 2014. 16(1): p. 015012.

It is well known that quantum particles when moving in a random potential may undergo Anderson localization transition. It has only recently become appreciated that in high enough dimensions a different transition may happen. It manifests itself in the singular disorder averaged density of states as the disorder strength is tuned to the value corresponding to this transition point, and in multifractal wave functions. A particle obeying Schrödinger equation has to be above four dimensions for this to happen, but for Dirac or Weyl particle this happens already in three dimensional space corresponding to the experimentally studied Weyl materials. For certain disordered ion chains this can happen already in one dimensional space. Some evidence in the literature points to this criticality being a sharp crossover instead of a genuine phase transition, but this issue has not yet been unambiguously resolved. I will discuss the theory behind these phenomena and existing open questions.

The standard picture of the Anderson localization in a 3d single-particle system with short-range hoppings [1,2] is represented by the phase transition between ergodic and localized phases at a certain critical disorder strength with a sharp mobility edge separating ergodic and localized states at the intermediate disorder. Exactly at the Anderson localization transition (ALT) point non-ergodic (multifractal) extended states were proved to appear. This picture was generalized (see, e.g., a review [2]) to low-dimensional systems with long-range hoppings by applying the well-known criterion of the delocalization at the point of the breakdown of the locator expansion [3, 4]. The states with critical statistics emerge typically at ALT in various disordered systems [2]. The examples of the robust multifractality away from phase transitions has been found only recently. Here I first present such examples. In low-dimensional systems we have found a class of random matrix models with extremely long-range hoppings, possessing a whole phase of multifractal extended states, squeezed between Anderson-localized and ergodic phases [5]. Another example demonstrates a separation of phases with an entire multifractal subband emerging in a single-particle system with quasiperiodic "disorder" and short-range hoppings under a generic periodic drive [5]. In both cases multifractal states are robust to perturbations and are found at a whole range of parameters. Recently, the standard picture of ALT [3, 4] has been also argued from another perspective. There have been reported [7-10] several counterintuitive examples of single-particle systems with long-range hoppings where almost all states are (at least power-law) localized even in a nominally ergodic regime, where the standard locator expansion breaks down [3, 4]. Some of these "new" models demonstrate critical behavior [7, 8] for disorder strengths below the ones at the ALT [3, 4]. In the other ones [9, 10] wave-function spatial decay rates obey a "mysterious" duality [10] mapping different powers of power-law bending [4]. What is the striking difference between the standard long-range models [3-5] and the new ones [7-10]? The systems [7-10] belong to a new universality class where correlations of hoppings play a significant role in the localization properties. In the second part of the talk I address this intriguing question. I present a general approach applicable to all such models and uncover the role of correlations and the origin of the duality [10]. Based on an example of a system with translationary invariant (random) hoppings, I demonstrate a simple though powerful method of recovering the standard criterion [3, 4] and giving a clue to the calculation of wave-function spatial profile. In the end of my talk I show how uncorrelated random hoppings [3-5] added to the correlated ones [7-10] can recover the standard picture of ALT. [1] P. W. Anderson Phys. Rev. 109, 1492 (1958). [2] F. Evers and A. D. Mirlin Rev. Mod. Phys. 80, 1355 (2008). [3] L. S. Levitov, Europhy. Lett. 9, 83 (1989); Phys. Rev. Lett. 64, 547 (1990). [4] A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, Phys. Rev. E 54, 3221 (1996). [5] V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, M. Amini, New J. Phys. 17, 122002 (2015). [6] S. Roy, I. M. Khaymovich, A. Das, R. Moessner, SciPost Phys. 4, 025 (2018). [7] H. K. Owusu, K. Wagh, and E. A. Yuzbashyan, J. Phys. A: Math. Theor. 42, 035206 (2009). [8] A. Ossipov, J. Phys. A 46, 105001 (2013). [9] G. L. Celardo, R. Kaiser, and F. Borgonovi, Phys. Rev. B 94, 144206 (2016). [10] X. Deng, V. E. Kravtsov, G. V. Shlyapnikov and L. Santos, Phys. Rev. Lett. 120, 110602 (2018).

I will discuss transitions between distinct phases of one-dimensional periodically driven (Floquet) systems. I will argue on general grounds [1] that these are controlled by infinite-randomness fixed points of a strong-disorder renormalization group procedure [2] suitably adapted to treat Floquet systems. Working in the fermionic representation of the prototypical Floquet Ising chain, I will leverage infinite randomness physics to provide a simple description of Floquet (multi)criticality in terms of a new type of domain wall associated with time-translational symmetry-breaking and the formation of `Floquet time crystals’. Refs: [1] W. Berdanier, M. Kolodrubetz, SP, R. Vasseur, arXiv:1803.00019, [2] W. Berdanier, M. Kolodrubetz, SP, R. Vasseur, arXiv:1807.09767.

Multifractality