Michael Knap

Collective Quantum Dynamics

Technical University of Munich

TUM School of Natural Sciences and Institute for Advanced Study

James-Franck-Str. 1

85748 Garching


Research webpage


Research focus: ultracold quantum gases, strongly correlated quantum matter, quantum dynamics, many-body localization

Collective Quantum Dynamics

The research in our group focuses on condensed matter theory and clusters around a variety of questions on non-equilibrium dynamics in ultracold quantum gases, trapped ions, superconducting qubits, and correlated quantum materials. Interactions and correlations in such systems often manifest in striking and novel properties, which emerge from the collective behavior of the quantum particles. Our group develops both analytical and numerical techniques to elucidate the effects of strong interactions. An important factor of our research is also its immediate relevance for experiments, which leads to a close collaboration with various experimental groups.

Correlated quantum systems out of equilibrium

Recent conceptional and technical progress makes it possible to prepare and explore strongly-correlated non-equilibrium quantum states of matter. The tremendous level of control and favorable time scales achieved in experiments with synthetic quantum matter, such as ultracold atoms, trapped ions, or superconducting qubits renders these systems as ideal candidates to explore non-equilibrium quantum dynamics. Furthermore, very powerful experimental techniques have been developed to study dynamic processes in condensed matter systems as well. These techniques are based on pump-probe spectroscopy on time scales reaching down to sub-femtoseconds. Such technology therefore makes it possible to manipulate and control material properties.

We develop both analytical and numerical techniques to explore the far-from-equilibrium quantum dynamics of these systems. The techniques range from non-equilibrium field theories to exact numerical calculations based on matrix product states. We study fundamental questions including thermalization in closed quantum systems, emergent phenomena in periodically driven Floquet systems, dynamic phase transitions, intertwined order far-from-equilibrium, and the competition between coherence and dissipation.

Selected Publications

  • Floquet prethermalization and regimes of heating in a periodically driven, interacting quantum system. Simon A. Weidinger, Michael Knap, Sci. Rep. 7, 45382 (2017). DOI: 10.1038/srep45382
  • Scrambling and thermalization in a diffusive quantum many-body system. A. Bohrdt, C. B. Mendl, M. Endres, M. Knap. [arXiv:1612.02434]
  • Ultrafast many-body interferometry of impurities coupled to a Fermi sea. M. Cetina, M. Jag, R. S. Lous, I. Fritsche, J. T. M. Walraven, R. Grimm, J. Levinsen, M. M. Parish, R. Schmidt, M. Knap, E. Demler, Science 354, 96 (2016). DOI: 10.1126/science.aaf5134
  • Far-from-equilibrium field theory of many-body quantum spin systems: Prethermalization and relaxation of spin spiral states in three dimensions. Mehrtash Babadi, Eugene Demler, Michael Knap, Phys. Rev. X 5, 041005 (2015). DOI: https://doi.org/10.1103/PhysRevX.5.041005

Disordered many-body systems

Disorder has a drastic influence on transport properties. In the presence of a random potential a system of interacting electrons can become insulating; a phenomenon known as many-body localization. However, even beyond the vanishing transport such systems have very intriguing properties. For example, many-body localization describes an exotic phase of matter, which is robust to small changes in the microscopic Hamiltonian. Moreover, fundamental concepts of thermodynamics break down in the many-body localized phase. We study how these particular properties can be characterized by interferometric techniques, explore distinct experimental signatures of disordered systems, and analyze the transition from the localized to the delocalized phase.

Selected Publications

  • Periodically Driving a Many-Body Localized Quantum System. Pranjal Bordia, Henrik Lüschen, Ulrich Schneider, Michael Knap, Immanuel Bloch Nature Phys. AOP (2017). DOI: 10.1038/nphys4020
  • Noise-induced subdiffusion in strongly localized quantum systems. Sarang Gopalakrishnan, K. Ranjibul Islam, Michael Knap. [arXiv:1609.04818]
  • Anomalous diffusion and Griffiths effects near the many-body localization transition. Kartiek Agarwal, Sarang Gopalakrishnan, Michael Knap, Markus Mueller, Eugene Demler, Phys. Rev. Lett. 114, 160401 (2015). DOI: 10.1103/PhysRevLett.114.160401


Realizing topologically ordered states on a quantum processor

K. J. Satzinger, Y. Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth, C. Gidney, I. Aleiner, F. Arute, K. Arya, J. Atalaya, R. Babbush, J. C. Bardin, R. Barends, J. Basso, A. Bengtsson, A. Bilmes, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. R. Derk, D. Eppens, C. Erickson, E. Farhi, L. Foaro, A. G. Fowler, B. Foxen, M. Giustina, A. Greene, J. A. Gross, M. P. Harrigan, S. D. Harrington, J. Hilton, S. Hong, T. Huang, W. J. Huggins, L. B. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, T. Khattar, S. Kim, P. V. Klimov, A.N. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev, A. Locharla, E. Lucero, O. Martin, J. R. McClean, M. McEwen, K. C. Miao, M. Mohseni, S. Montazeri, W. Mruczkiewicz, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, T. E. O'Brien, A. Opremcak, B. Pató, A. Petukhov, N. C. Rubin, D. Sank, V. Shvarts, D. Strain, M. Szalay, B. Villalonga, T. C. White, Z. Yao, P. Yeh, J. Yoo, A. Zalcman, H. Neven, S. Boixo, A. Megrant, Y. Chen, J. Kelly, V. Smelyanskiy, A. Kitaev, M. Knap, F. Pollmann, P. Roushan

Science 374, 1237-1241 (2021).

Show Abstract

The discovery of topological order has revised the understanding of quantum matter and provided the theoretical foundation for many quantum error-correcting codes. Realizing topologically ordered states has proven to be challenging in both condensed matter and synthetic quantum systems. We prepared the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measured a topological entanglement entropy near the expected value of -ln2 and simulated anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigated key aspects of the surface code, including logical state injection and the decay of the nonlocal order parameter. Our results demonstrate the potential for quantum processors to provide insights into topological quantum matter and quantum error correction.

Critically slow operator dynamics in constrained many-body systems

J. Feldmeier, M. Knap

Physical Review Letters 127, 235301 (2021).

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The far-from-equilibrium dynamics of generic interacting quantum systems is characterized by a handful of universal guiding principles, among them the ballistic spreading of initially local operators. Here, we show that in certain constrained many-body systems the structure of conservation laws can cause a drastic modification of this universal behavior. As an example, we study operator growth characterized by out-of-time-order correlations (OTOCs) in a dipole-conserving fracton chain. We identify a critical point with sub-ballistically moving OTOC front, that separates a ballistic from a dynamically frozen phase. This critical point is tied to an underlying localization transition and we use its associated scaling properties to derive an effective description of the moving operator front via a biased random walk with long waiting times. We support our arguments numerically using classically simulable automaton circuits.

DOI: 10.1103/PhysRevLett.127.235301

Dynamics of Negativity of a Wannier-Stark Many-Body Localized System Coupled to a Bath

E. Wybo, M. Knap, F. Pollmann

Physica Status Solidi B-Basic Solid State Physics 2100161 (2021).

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An interacting system subjected to a strong linear potential can host a many-body localized (MBL) phase when being slightly perturbed. This so-called Wannier-Stark or "tilted-field" MBL phase inherits many properties from the well-investigated disordered MBL phase, and provides an alternative route to experimentally engineer interacting localized systems without quenched disorder. Herein, the dynamics of entanglement in a Wannier-Stark MBL system coupled to a dephasing environment is investigated. As an accessible entanglement proxy, the third Renyi negativity R 3 is used, which reduces to the third Renyi entropy in case the system is isolated from the environment. This measure captures the characteristic logarithmic growth of interacting localized phases in the intermediate-time regime, where the effects of the coupling to the environment are not yet dominating the dynamics. Thus, it forms a tool to distinguish Wannier-Stark MBL from noninteracting Wannier-Stark localization up to intermediate time-scales, and to quantify quantum correlations in mixed-state dynamics.

DOI: 10.1002/pssb.202100161

Electrically tunable Feshbach resonances in twisted bilayer semiconductors

I. Schwart, Y. Shimazaki, C. Kuhlenkamp, K. Watanabe, T. Taniguchi, M. Kroner, A. Imamoglu

Science 374, 336-340 (2021).

Show Abstract

Moiré superlattices in transition metal dichalcogenide bilayers provide a platform for exploring strong correlations with optical spectroscopy. Despite the observation of rich Mott-Wigner physics stemming from an interplay between the periodic potential and Coulomb interactions, the absence of tunnel coupling–induced hybridization of electronic states has ensured a classical layer degree of freedom. We investigated a MoSe2 homobilayer structure where interlayer coherent tunneling allows for electric field–controlled manipulation and measurement of the ground-state hole-layer pseudospin. We observed an electrically tunable two-dimensional Feshbach resonance in exciton-hole scattering, which allowed us to control the strength of interactions between excitons and holes located in different layers. Our results may enable the realization of degenerate Bose-Fermi mixtures with tunable interactions.

DOI: 10.1126/science.abj3831

Analyzing non-equilibrium quantum states through snapshots with artificial neural networks

A. Bohrdt, S. Kim, A. Lukin, M. Rispoli, R. Schittko, M. Knap, M. Greiner, J. Léonard

Physical Review Letters 127, 150504 (2021).

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Current quantum simulation experiments are starting to explore nonequilibrium many-body dynamics in previously inaccessible regimes in terms of system sizes and timescales. Therefore, the question emerges as to which observables are best suited to study the dynamics in such quantum many-body systems. Using machine learning techniques, we investigate the dynamics and, in particular, the thermalization behavior of an interacting quantum system that undergoes a nonequilibrium phase transition from an ergodic to a many-body localized phase. We employ supervised and unsupervised training methods to distinguish nonequilibrium from equilibrium data, using the network performance as a probe for the thermalization behavior of the system. We test our methods with experimental snapshots of ultracold atoms taken with a quantum gas microscope. Our results provide a path to analyze highly entangled large-scale quantum states for system sizes where numerical calculations of conventional observables become challenging.

DOI: 10.1103/PhysRevLett.127.150504

Correlator convolutional neural networks as an interpretable architecture for image-like quantum matter data

C. Miles, A. Bohrdt, R. Wu, C. Chiu, M. Xu, G. Ji, M. Greiner, K.Q. Weinberger, E. Demler, E.-A. Kim

Nature Communications 12, 3905 (2021).

Show Abstract

Image-like data from quantum systems promises to offer greater insight into the physics of correlated quantum matter. However, the traditional framework of condensed matter physics lacks principled approaches for analyzing such data. Machine learning models are a powerful theoretical tool for analyzing image-like data including many-body snapshots from quantum simulators. Recently, they have successfully distinguished between simulated snapshots that are indistinguishable from one and two point correlation functions. Thus far, the complexity of these models has inhibited new physical insights from such approaches. Here, we develop a set of nonlinearities for use in a neural network architecture that discovers features in the data which are directly interpretable in terms of physical observables. Applied to simulated snapshots produced by two candidate theories approximating the doped Fermi-Hubbard model, we uncover that the key distinguishing features are fourth-order spin-charge correlators. Our approach lends itself well to the construction of simple, versatile, end-to-end interpretable architectures, thus paving the way for new physical insights from machine learning studies of experimental and numerical data. Physical principles underlying machine learning analysis of quantum gas microscopy data are not well understood. Here the authors develop a neural network based approach to classify image data in terms of multi-site correlation functions and reveal the role of fourth-order correlations in the Fermi-Hubbard model.

DOI: 10.1038/s41467-021-23952-w

Optical Signatures of Periodic Charge Distribution in a Mott-like Correlated Insulator State

Y. Shimazaki, C. Kuhlenkamp, I. Schwartz, T. Smoleński, K. Watanabe, T. Taniguchi, M. Kroner, R. Schmidt, M. Knap, A. Imamoğlu

Physical Review X 11 (2), 21027 (2021).

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The elementary optical excitations in two-dimensional semiconductors hosting itinerant electrons are attractive and repulsive polarons—excitons that are dynamically screened by electrons. Exciton polarons have hitherto been studied in translationally invariant degenerate Fermi systems. Here, we show that periodic distribution of electrons breaks the excitonic translational invariance and leads to a direct optical signature in the exciton-polaron spectrum. Specifically, we demonstrate that new optical resonances appear due to spatially modulated interactions between excitons and electrons in an incompressible Mott-like correlated state. Our observations demonstrate that resonant optical spectroscopy provides an invaluable tool for studying strongly correlated states, such as Wigner crystals and density waves, where exciton-electron interactions are modified by the emergence of charge order.

DOI: 10.1103/PhysRevX.11.021027

Tunable Feshbach resonances and their spectral signatures in bilayer semiconductors

C. Kuhlenkamp, M. Knap, M. Wagner, R. Schmidt, A. Imamoglu

Show Abstract

Feshbach resonances are an invaluable tool in atomic physics, enabling precise control of interactions and the preparation of complex quantum phases of matter. Here, we theoretically analyze a solid-state analogue of a Feshbach resonance in two dimensional semiconductor heterostructures. In the presence of inter-layer electron tunneling, the scattering of excitons and electrons occupying different layers can be resonantly enhanced by tuning an applied electric field. The emergence of an inter-layer Feshbach molecule modifies the optical excitation spectrum, and can be understood in terms of Fermi polaron formation. We discuss potential implications for the realization of correlated Bose-Fermi mixtures in bilayer semiconductors.


Adiabatic formation of bound states in the one-dimensional Bose gas

R. Koch, A. Bastianello, J.S. Caux

Physical Review B 103, 165121 (2021).

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We consider the one-dimensional interacting Bose gas in the presence of time-dependent and spatially inhomogeneous contact interactions. Within its attractive phase, the gas allows for bound states of an arbitrary number of particles, which are eventually populated if the system is dynamically driven from the repulsive to the attractive regime. Building on the framework of generalized hydrodynamics, we analytically determine the formation of bound states in the limit of adiabatic changes in the interactions. Our results are valid for arbitrary initial thermal states and, more generally, generalized Gibbs ensembles.

DOI: 10.1103/PhysRevB.103.165121

Emergent fracton dynamics in a nonplanar dimer model

J. Feldmeier, F. Pollmann, M. Knap

Physical Review B 103 (9), 94303 (2021).

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We study the late time relaxation dynamics of a pure U(1) lattice gauge theory in the form of a dimer model on a bilayer geometry. To this end, we first develop a proper notion of hydrodynamic transport in such a system by constructing a global conservation law that can be attributed to the presence of topological solitons. The correlation functions of local objects charged under this conservation law can then be used to study the universal properties of the dynamics at late times, applicable to both quantum and classical systems. Performing the time evolution via classically simulable automata circuits unveils a rich phenomenology of the system's nonequilibrium properties: For a large class of relevant initial states, local charges are effectively restricted to move along one-dimensional “tubes” within the quasi-two-dimensional system, displaying fracton-like mobility constraints. The timescale on which these tubes are stable diverges with increasing systems size, yielding a novel mechanism for nonergodic behavior in the thermodynamic limit. We further explore the role of geometry by studying the system in a quasi-one-dimensional limit, where the Hilbert space is strongly fragmented due to the emergence of an extensive number of conserved quantities. This provides an instance of a recently introduced concept of “statistically localized integrals of motion,” whose universal anomalous hydrodynamics we determine by a mapping to a problem of classical tracer diffusion. We conclude by discussing how our approach might generalize to study transport in other lattice gauge theories.

DOI: 10.1103/PhysRevB.103.094303

Anomalous Diffusion in Dipole- and Higher-Moment-Conserving Systems

J. Feldmeier, P. Sala, G. De Tomasi, F. Pollmann, M. Knap

Physical Review Letters 125 (24), 245303 (2020).

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The presence of global conserved quantities in interacting systems generically leads to diffusive transport at late times. Here, we show that systems conserving the dipole moment of an associated global charge, or even higher-moment generalizations thereof, escape this scenario, displaying subdiffusive decay instead. Modeling the time evolution as cellular automata for specific cases of dipole- and quadrupole conservation, we numerically find distinct anomalous exponents of the late time relaxation. We explain these findings by analytically constructing a general hydrodynamic model that results in a series of exponents depending on the number of conserved moments, yielding an accurate description of the scaling form of charge correlation functions. We analyze the spatial profile of the correlations and discuss potential experimentally relevant signatures of higher-moment conservation.

DOI: 10.1103/PhysRevLett.125.245303

Probing eigenstate thermalization in quantum simulators via fluctuation-dissipation relations

A.Schuckert, M.Knap

Physical Review Research 2 (4), 43315 (2020).

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The eigenstate thermalization hypothesis (ETH) offers a universal mechanism for the approach to equilibrium of closed quantum many-body systems. So far, however, experimental studies have focused on the relaxation dynamics of observables as described by the diagonal part of ETH, whose verification requires substantial numerical input. This leaves many of the general assumptions of ETH untested. Here, we propose a theory-independent route to probe the full ETH in quantum simulators by observing the emergence of fluctuation-dissipation relations, which directly probe the off-diagonal part of ETH. We discuss and propose protocols to independently measure fluctuations and dissipations as well as higher order time-ordered correlation functions. We first show how the emergence of fluctuation-dissipation relations from a nonequilibrium initial state can be observed for the two-dimensional (2D) Bose-Hubbard model in superconducting qubits or quantum gas microscopes. Then we focus on the long-range transverse field Ising model (LTFI), which can be realized with trapped ions. The LTFI exhibits rich thermalization phenomena: For strong transverse fields, we observe prethermalization to an effective magnetization-conserving Hamiltonian in the fluctuation-dissipation relations. For weak transverse fields, confined excitations lead to nonthermal features, resulting in a violation of the fluctuation-dissipation relations up to long times. Moreover, in an integrable region of the LTFI, thermalization to a generalized Gibbs ensemble occurs and the fluctuation-dissipation relations enable an experimental diagonalization of the Hamiltonian. Our work presents a theory-independent way to characterize thermalization in quantum simulators and paves the way to quantum simulate condensed matter pump-probe experiments.

DOI: 10.1103/PhysRevResearch.2.043315

Dynamical formation of a magnetic polaron in a two-dimensional quantum antiferromagnet

A. Bohrdt, F. Grusdt, M. Knap

New Journal of Physics 22 (12), 123023 (2020).

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Tremendous recent progress in the quantum simulation of the Hubbard model paves the way to controllably study doped antiferromagnetic Mott insulators. Motivated by these experimental advancements, we numerically study the real-time dynamics of a single hole created in an antiferromagnet on a square lattice, as described by the t-J model. Initially, the hole spreads ballistically with a velocity proportional to the hopping matrix element. At intermediate to long times, the dimensionality as well as the spin background determine the hole dynamics. A hole created in the ground state of a two dimensional (2D) quantum antiferromagnet propagates again ballistically at long times but with a velocity proportional to the spin exchange coupling, showing the formation of a magnetic polaron. We provide an intuitive explanation of this dynamics in terms of a parton construction, which leads to a good quantitative agreement with the numerical tensor network state simulations. In the limit of infinite temperature and no spin exchange couplings, the dynamics can be approximated by a quantum random walk on a Bethe lattice with coordination number




Adding Ising interactions corresponds to an effective disordered potential, which can dramatically slow down the hole propagation, consistent with subdiffusive dynamics. The study of the hole dynamics paves the way for understanding the microscopic constituents of this strongly correlated quantum state.

DOI: 10.1088/1367-2630/abcfee

Local probes for charge-neutral edge states in two-dimensional quantum magnets

J. Feldmeier, W. Natori, M. Knap, J. Knolle

Physical Review B 102 (13), 134423 (2020).

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The bulk-boundary correspondence is a defining feature of topological states of matter. However, for quantum magnets in two dimensions such as spin liquids or topological magnon insulators, a direct observation of topological surface states has proven challenging because of the charge-neutral character of the excitations. Here we propose spin-polarized scanning tunneling microscopy as a spin-sensitive local probe to provide direct information about charge-neutral topological edge states. We show how their signatures, imprinted in the local structure factor, can be extracted by specifically employing the strengths of existing technologies. As our main example, we determine the dynamical spin correlations of the Kitaev honeycomb model with open boundaries. We show that by contrasting conductance measurements of bulk and edge locations, one can extract direct signatures of the existence of fractionalized excitations and nontrivial topology. The broad applicability of this approach is corroborated by a second example of a kagome topological magnon insulator.

DOI: 10.1103/PhysRevB.102.134423

Entanglement dynamics of a many-body localized system coupled to a bath

E. Wybo, M. Knap, F. Pollmann

Physical Review B 102 (6), 064303 (2020).

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The combination of strong disorder and interactions in closed quantum systems can lead to many-body localization (MBL). However, this quantum phase is not stable when the system is coupled to a thermal environment. We investigate how MBL is destroyed in systems that are weakly coupled to a dephasive Markovian environment by focusing on their entanglement dynamics. We numerically study the third Renyi negativity R-3, a recently proposed entanglement proxy based on the negativity that captures the unbounded logarithmic growth in the closed case and that can be computed efficiently with tensor networks. We also show that the decay of R-3 follows a stretched exponential law, similarly to the imbalance, with, however, a smaller stretching exponent.

DOI: 10.1103/PhysRevB.102.064304

Lattice modulation spectroscopy of one-dimensional quantum gases: Universal scaling of the absorbed energy

R. Citro, E. Demler, T. Giamarchi, M. Knap, E. Orignac

Physical Review Research 2 (3), 33187 (2020).

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Lattice modulation spectroscopy is a powerful tool for probing low-energy excitations of interacting many-body systems. By means of bosonization we analyze the absorbed power in a one-dimensional interacting quantum gas of bosons or fermions, subjected to a periodic drive of the optical lattice. For these Tomonaga-Luttinger liquids we find a universal ω3 scaling of the absorbed power, which at very low-frequency turns into an ω2 scaling when scattering processes at the boundary of the system are taken into account. We confirm this behavior numerically by simulations based on time-dependent matrix product states. Furthermore, in the presence of impurities, the theory predicts an ω2 bulk scaling. While typical response functions of Tomonaga-Luttinger liquids are characterized by exponents that depend on the interaction strength, modulation spectroscopy of cold atoms leads to a universal power-law exponent of the absorbed power. Our findings can be readily demonstrated in ultracold atoms in optical lattices with current experimental technology.

DOI: 10.1103/PhysRevResearch.2.033187

Parton theory of angle-resolved photoemission spectroscopy spectra in antiferromagnetic Mott insulators

A. Bohrdt, E. Demler, F. Pollmann, M. Knap, F. Grusdt

Physical Review B 102 (3), 035139 (2020).

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Angle-resolved photoemission spectroscopy (ARPES) has revealed peculiar properties of mobile dopants in correlated antiferromagnets (AFMs). But, describing them theoretically, even in simplified toy models, remains a challenge. Here, we study ARPES spectra of a single mobile hole in the t-J model. Recent progress in the microscopic description of mobile dopants allows us to use a geometric decoupling of spin and charge fluctuations at strong couplings, from which we conjecture a one-to-one relation of the one-dopant spectral function and the spectrum of a constituting spinon in the undoped parent AFM. We thoroughly test this hypothesis for a single hole doped into a two-dimensional Heisenberg AFM by comparing our semianalytical predictions to previous quantum Monte Carlo results and our large-scale time-dependent matrix product state calculations of the spectral function. Our conclusion is supported by a microscopic trial wave function describing spinon-chargon bound states, which captures the momentum and t/J dependence of the quasiparticle residue. From our conjecture we speculate that ARPES measurements in the pseudogap phase of cuprates may directly reveal the Dirac-fermion nature of the constituting spinons. Specifically, we demonstrate that our trial wave function provides a microscopic explanation for the sudden drop of spectral weight around the nodal point associated with the formation of Fermi arcs, assuming that additional frustration suppresses long-range AFM ordering. We benchmark our results by studying the crossover from two to one dimension, where spinons and chargons are confined and deconfined, respectively.

DOI: 10.1103/PhysRevB.102.035139

Theory of quantum work in metallic grains

I. Lovas, A. Grabarits, M. Kormos, G. Zaránd

Physical Review Research 2, 023224 (2020).

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We generalize Anderson's orthogonality determinant formula to describe the statistics of work performed on generic disordered, noninteracting fermionic nanograins during quantum quenches. The energy absorbed increases linearly with time, while its variance exhibits a superdiffusive behavior due to Pauli's exclusion principle. The probability of adiabatic evolution decays as a stretched exponential. In slowly driven systems, work statistics exhibit universal features and can be understood in terms of fermion diffusion in energy space, generated by Landau-Zener transitions. This diffusion is very well captured by a Markovian symmetrical exclusion process, with the diffusion constant identified as the energy absorption rate. The energy absorption rate shows an anomalous frequency dependence at small energies, reflecting the symmetry class of the underlying Hamiltonian. Our predictions can be experimentally verified by calorimetric measurements performed on nanoscale circuits.

DOI: 10.1103/PhysRevResearch.2.023224

Statistical localization: From strong fragmentation to strong edge modes

T. Rakovszky, P. Sala, R. Verresen, M. Knap, F. Pollmann

Physical Review B 101 (12), 125126 (2020).

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Certain disorder-free Hamiltonians can be nonergodic due to a strong fragmentation of the Hilbert space into disconnected sectors. Here, we characterize such systems by introducing the notion of "statistically localized integrals of motion" (SLIOM), whose eigenvalues label the connected components of the Hilbert space. SLIOMs are not spatially localized in the operator sense, but appear localized to subextensive regions when their expectation value is taken in typical states with a finite density of particles. We illustrate this general concept on several Hamiltonians, both with and without dipole conservation. Furthermore, we demonstrate that there exist perturbations which destroy these integrals of motion in the bulk of the system while keeping them on the boundary. This results in statistically localized strong zero modes, leading to infinitely long-lived edge magnetizations along with a thermalizing bulk, constituting the first example of such strong edge modes in a nonintegrable model. We also show that in a particular example, these edge modes lead to the appearance of topological string order in a certain subset of highly excited eigenstates. Some of our suggested models can be realized in Rydberg quantum simulators.

DOI: 10.1103/PhysRevB.101.125126

Periodically Driven Sachdev-Ye-Kitaev Models

C. Kuhlenkamp, M.Knap

Physical Review Letters 124 (10), 106401 (2020).

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Periodically driven quantum matter can realize exotic dynamical phases. In order to understand how ubiquitous and robust these phases are, it is pertinent to investigate the heating dynamics of generic interacting quantum systems. Here we study the thermalization in a periodically driven generalized Sachdev-Ye-Kitaev (SYK) model, which realizes a crossover from a heavy Fermi liquid (FL) to a non-Fermi liquid (NFL) at a tunable energy scale. Developing an exact field theoretic approach, we determine two distinct regimes in the heating dynamics. While the NFL heats exponentially and thermalizes rapidly, we report that the presence of quasiparticles in the heavy FL obstructs heating and thermalization over comparatively long timescales. Prethermal high-frequency dynamics and possible experimental realizations of nonequilibrium SYK physics are discussed as well.

DOI: 10.1103/PhysRevLett.124.106401

Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltonians

P. Sala, T. Rakovszky, R. Verresen, M. Knap, F. Pollmann

Physical Review X 10 (1), 011047 (2020).

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We show that the combination of charge and dipole conservation-characteristic of fracton systems-leads to an extensive fragmentation of the Hilbert space, which, in turn, can lead to a breakdown of thermalization. As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total S-z) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature autocorrelation saturates to a finite value-showcasing nonthermal behavior. The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local S-z basis, arising from the interplay of dipole conservation and local interactions. Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: The system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states. More generally, for any finite range of interactions, the system still exhibits nonthermal eigenstates appearing throughout the entire spectrum. We compare our results to charge and dipole moment-conserving random unitary circuit models for which we reach identical conclusions.

DOI: 10.1103/PhysRevX.10.011047

Evaluation of time-dependent correlators after a local quench in iPEPS: hole motion in the t - J model

C. Hubig, A. Bohrdt, M. Knap, F. Grusdt, J.I. Cirac

Scipost Physics 8 (2), 021 (2020).

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Infinite projected entangled pair states (iPEPS) provide a convenient variational description of infinite, translationally-invariant two-dimensional quantum states. However, the simulation of local excitations is not directly possible due to the translationally-invariant ansatz. Furthermore, as iPEPS are either identical or orthogonal, expectation values between different states as required during the evaluation of non-equal-time correlators are ill-defined. Here, we show that by introducing auxiliary states on each site, it becomes possible to simulate both local excitations and evaluate non-equal-time correlators in an iPEPS setting under real-time evolution. We showcase the method by simulating the t - J model after a single hole has been placed in the half-filled antiferromagnetic background and evaluating both return probabilities and spin correlation functions, as accessible in quantum gas microscopes.

DOI: 10.21468/SciPostPhys.8.2.021

Nonlocal emergent hydrodynamics in a long-range quantum spin system

A. Schuckert, I. Lovas, M. Knap

Physical Review B 101 (2), 020416 (2020).

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Generic short-range interacting quantum systems with a conserved quantity exhibit universal diffusive transport at late times. We employ nonequilibrium quantum field theory and semiclassical phase-space simulations to show how this universality is replaced by a more general transport process in a long-range XY spin chain at infinite temperature with couplings decaying algebraically with distance as r(-alpha). While diffusion is recovered for alpha > 1.5, longer-ranged couplings with 0.5 < alpha <= 1.5 give rise to effective classical Levy flights, a random walk with step sizes drawn from a distribution with algebraic tails. We find that the space-time-dependent spin density profiles are self-similar, with scaling functions given by the stable symmetric distributions. As a consequence, for 0.5 < alpha <= 1.5, autocorrelations show hydrodynamic tails decaying in time as t(-1/(2 alpha-1)) and linear-response theory breaks down. Our findings can be readily verified with current trapped ion experiments.

DOI: 10.1103/PhysRevB.101.020416

Many-body chaos near a thermal phase transition

A. Schuckert, M. Knap.

SciPost Physics 7, 022 (2019).

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We study many-body chaos in a (2+1)D relativistic scalar field theory at high temperatures in the classical statistical approximation, which captures the quantum critical regime and the thermal phase transition from an ordered to a disordered phase. We evaluate out-of-time ordered correlation functions (OTOCs) and find that the associated Lyapunov exponent increases linearly with temperature in the quantum critical regime, and approaches the non-interacting limit algebraically in terms of a fluctuation parameter. OTOCs spread ballistically in all regimes, also at the thermal phase transition, where the butterfly velocity is maximal. Our work contributes to the understanding of the relation between quantum and classical many-body chaos and our method can be applied to other field theories dominated by classical modes at long wavelengths.

DOI: 10.21468/SciPostPhys.7.2.022

Emergent Glassy Dynamics in a Quantum Dimer Model

J. Feldmeier, F. Pollmann, and M. Knap.

Physical Review Letters 123, 040601 (2019).

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We consider the quench dynamics of a two-dimensional quantum dimer model and determine the role of its kinematic constraints. We interpret the nonequilibrium dynamics in terms of the underlying equilibrium phase transitions consisting of a Berezinskii-Kosterlitz-Thouless (BKT) transition between a columnar ordered valence bond solid (VBS) and a valence bond liquid (VBL), as well as a first-order transition between a staggered VBS and the VBL. We find that quenches from a columnar VBS are ergodic and both order parameters and spatial correlations quickly relax to their thermal equilibrium. By contrast, the staggered side of the first-order transition does not display thermalization on numerically accessible timescales. Based on the model’s kinematic constraints, we uncover a mechanism of relaxation that rests on emergent, highly detuned multidefect processes in a staggered background, which gives rise to slow, glassy dynamics at low temperatures even in the thermodynamic limit.

DOI: 10.1103/PhysRevLett.123.040601

String patterns in the doped Hubbard model

C. S. Chiu, G. Ji, A. Bohrdt, M. Xu, M. Knap, E. Demler, F. Grusdt, M. Greiner, D. Greif.

Science 365, 251-256 (2019).

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Understanding strongly correlated quantum many-body states is one of the most difficult challenges in modern physics. For example, there remain fundamental open questions on the phase diagram of the Hubbard model, which describes strongly correlated electrons in solids. In this work, we realize the Hubbard Hamiltonian and search for specific patterns within the individual images of many realizations of strongly correlated ultracold fermions in an optical lattice. Upon doping a cold-atom antiferromagnet, we find consistency with geometric strings, entities that may explain the relationship between hole motion and spin order, in both pattern-based and conventional observables. Our results demonstrate the potential for pattern recognition to provide key insights into cold-atom quantum many-body systems.

DOI: 10.1126/science.aav3587

Classifying snapshots of the doped Hubbard model with machine learning

A. Bohrdt, C. S. Chiu, G. Ji, M. Xu, D. Greif, M. Greiner, E. Demler, F. Grusdt, M. Knap.

Nature Physics 15, 921-924 (2019).

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Quantum gas microscopes for ultracold atoms can provide high-resolution real-space snapshots of complex many-body systems. We implement machine learning to analyse and classify such snapshots of ultracold atoms. Specifically, we compare the data from an experimental realization of the two-dimensional Fermi–Hubbard model to two theoretical approaches: a doped quantum spin liquid state of resonating valence bond type (1,2), and the geometric string theory (3,4), describing a state with hidden spin order. This technique considers all available information without a potential bias towards one particular theory by the choice of an observable and can therefore select the theory that is more predictive in general. Up to intermediate doping values, our algorithm tends to classify experimental snapshots as geometric-string-like, as compared to the doped spin liquid. Our results demonstrate the potential for machine learning in processing the wealth of data obtained through quantum gas microscopy for new physical insights.

DOI: 10.1038/s41567-019-0565-x

Site-selectively generated photon emitters in monolayer MoS2 via local helium ion irradiation

J. Klein, M. Lorke, M. Florian, F. Sigger, J. Wierzbowski, J. Cerne, K. Müller, T. Taniguchi, K. Watanabe, U. Wurstbauer, M. Kaniber, M. Knap, R. Schmidt, J. Finley, A. Holleitner.

Nature Communications 10, Article number: 2755 (2019).

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Quantum light sources in solid-state systems are of major interest as a basic ingredient for integrated quantum photonic technologies. The ability to tailor quantum emitters via site-selective defect engineering is essential for realizing scalable architectures. However, a major difficulty is that defects need to be controllably positioned within the material. Here, we overcome this challenge by controllably irradiating monolayer MoS2 using a sub-nm focused helium ion beam to deterministically create defects. Subsequent encapsulation of the ion exposed MoS2 flake with high-quality hBN reveals spectrally narrow emission lines that produce photons in the visible spectral range. Based on ab-initio calculations we interpret these emission lines as stemming from the recombination of highly localized electron–hole complexes at defect states generated by the local helium ion exposure. Our approach to deterministically write optically active defect states in a single transition metal dichalcogenide layer provides a platform for realizing exotic many-body systems, including coupled single-photon sources and interacting exciton lattices that may allow the exploration of Hubbard physics.

DOI: 0.1038/s41467-019-10632-z

Atomtronics with a spin: Statistics of spin transport and nonequilibrium orthogonality catastrophe in cold quantum gases

J.S. You, R. Schmidt, D.A. Ivanov, M. Knap, and E. Demler.

Physical Review B 99, 214505 (2019).

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We propose to investigate the full counting statistics of nonequilibrium spin transport with an ultracold atomic quantum gas. The setup makes use of the spin control available in atomic systems to generate spin transport induced by an impurity atom immersed in a spin-imbalanced two-component Fermi gas. In contrast to solid-state realizations, in ultracold atoms spin relaxation and the decoherence from external sources is largely suppressed. As a consequence, once the spin current is turned off by manipulating the internal spin degrees of freedom of the Fermi system, the nonequilibrium spin population remains constant. Thus one can directly count the number of spins in each reservoir to investigate the full counting statistics of spin flips, which is notoriously challenging in solid-state devices. Moreover, using Ramsey interferometry, the dynamical impurity response can be measured. Since the impurity interacts with a many-body environment that is out of equilibrium, our setup provides a way to realize the nonequilibrium orthogonality catastrophe. Here, even for spin reservoirs initially prepared in a zero-temperature state, the Ramsey response exhibits an exponential decay, which is in contrast to the conventional power-law decay of Anderson's orthogonality catastrophe. By mapping our system to a multistep Fermi sea, we are able to derive analytical expressions for the impurity response at late times. This allows us to reveal an intimate connection of the decay rate of the Ramsey contrast and the full counting statistics of spin flips.

DOI: 10.1103/PhysRevB.99.214505

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