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1. Quantum dynamics in Krylov space: Methods and applications

   https://doi.org/10.1016/j.physrep.2025.05.001  

   Nandy, Pratik; Matsoukas-Roubeas, Apollonas S; Martínez-Azcona, Pablo; Dymarsky, Anatoly; del_Campo, Adolfo (June 2025, Physics Reports)

   The dynamics of quantum systems unfolds within a subspace of the state space or operator space, known as the Krylov space. This review presents the use of Krylov subspace methods to provide an efficient description of quantum evolution and quantum chaos, with emphasis on nonequilibrium phenomena of many-body systems with a large Hilbert space. It provides a comprehensive update of recent developments, focused on the quantum evolution of operators in the Heisenberg picture as well as pure and mixed states. It further explores the notion of Krylov complexity and associated metrics as tools for quantifying operator growth, their bounds by generalized quantum speed limits, the universal operator growth hypothesis, and its relation to quantum chaos, scrambling, and generalized coherent states. A comparison of several generalizations of the Krylov construction for open quantum systems is presented. A closing discussion addresses the application of Krylov subspace methods in quantum field theory, holog- raphy, integrability, quantum control, and quantum computing, as well as current open problems. 

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2. Quantum Many-Body Scars: A Quasiparticle Perspective

   https://doi.org/10.1146/annurev-conmatphys-031620-101617  

   Chandran, Anushya; Iadecola, Thomas; Khemani, Vedika; Moessner, Roderich (March 2023, Annual Review of Condensed Matter Physics)

   Weakly interacting quasiparticles play a central role in the low-energy description of many phases of quantum matter. At higher energies, however, quasiparticles cease to be well defined in generic many-body systems owing to a proliferation of decay channels. In this review, we discuss the phenomenon of quantum many-body scars, which can give rise to certain species of stable quasiparticles throughout the energy spectrum. This goes along with a set of unusual nonequilibrium phenomena including many-body revivals and nonthermal stationary states. We provide a pedagogical exposition of this physics via a simple yet comprehensive example, that of a spin-1 XY model. We place our discussion in the broader context of symmetry-based constructions of many-body scar states, projector embeddings, and Hilbert space fragmentation. We conclude with a summary of experimental progress and theoretical puzzles. 

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3. Data-driven low-rank approximation for the electron-hole kernel and acceleration of time-dependent GW calculations

   https://doi.org/10.1038/s41524-025-01680-9  

   Hou, Bowen; Wu, Jinyuan; Chang_Lee, Victor; Guo, Jiaxuan; Liu, Luna Y; Qiu, Diana Y (December 2025, npj Computational Materials)

   Many-body interactions are essential for understanding non-linear optics and ultrafast spectroscopy of materials. Recent first principles approaches based on nonequilibrium Green’s function formalisms, such as the time-dependent adiabatic GW (TD-aGW) approach, can predict nonequilibrium dynamics of excited states including electron-hole interactions. However, the high-dimensionality of the electron-hole kernel poses significant computational challenges. Here, we develop a data-driven low-rank approximation for the electron-hole kernel, leveraging localized excitonic effects in the Hilbert space of crystalline systems to achieve significant data compression through singular value decomposition (SVD). We show that the subspace of non-zero singular values remains small even as the k-grid grows, ensuring computational tractability with extremely dense k-grids. This low-rank property enables at least 95% data compression and an order-of-magnitude speedup of TD-aGW calculations. Our approach avoids intensive training processes and eliminates time-accumulated errors, seen in previous approaches, providing a general framework for high-throughput, nonequilibrium simulation of light-driven dynamics in materials. 

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4. Screening the Coulomb interaction leads to a prethermal regime in two-dimensional bad conductors

   https://doi.org/10.1038/s41467-023-42778-2  

   Stanley, L. J.; Lin, Ping V.; Jaroszyński, J.; Popović, Dragana (November 2023, Nature Communications)

   Abstract The absence of thermalization in certain isolated many-body systems is of great fundamental interest. Many-body localization (MBL) is a widely studied mechanism for thermalization to fail in strongly disordered quantum systems, but it is still not understood precisely how the range of interactions affects the dynamical behavior and the existence of MBL, especially in dimensionsD > 1. By investigating nonequilibrium dynamics in strongly disorderedD = 2 electron systems with power-law interactions ∝ 1/r^αand poor coupling to a thermal bath, here we observe MBL-like, prethermal dynamics forα = 3. In contrast, forα = 1, the system thermalizes, although the dynamics is glassy. Our results provide important insights for theory, especially since we obtained them on systems that are much closer to the thermodynamic limit than synthetic quantum systems employed in previous studies of MBL. Thus, our work is a key step towards further studies of ergodicity breaking and quantum entanglement in real materials. 

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5. Dissipative Floquet Dynamics: from Steady State to Measurement Induced Criticality in Trapped-ion Chains

   https://doi.org/10.22331/q-2022-02-02-638  

   Sierant, Piotr; Chiriacò, Giuliano; Surace, Federica M.; Sharma, Shraddha; Turkeshi, Xhek; Dalmonte, Marcello; Fazio, Rosario; Pagano, Guido (February 2022, Quantum)

   Quantum systems evolving unitarily and subject to quantum measurements exhibit various types of non-equilibrium phase transitions, arising from the competition between unitary evolution and measurements. Dissipative phase transitions in steady states of time-independent Liouvillians and measurement induced phase transitions at the level of quantum trajectories are two primary examples of such transitions. Investigating a many-body spin system subject to periodic resetting measurements, we argue that many-body dissipative Floquet dynamics provides a natural framework to analyze both types of transitions. We show that a dissipative phase transition between a ferromagnetic ordered phase and a paramagnetic disordered phase emerges for long-range systems as a function of measurement probabilities. A measurement induced transition of the entanglement entropy between volume law scaling and sub-volume law scaling is also present, and is distinct from the ordering transition. The two phases correspond to an error-correcting and a quantum-Zeno regimes, respectively. The ferromagnetic phase is lost for short range interactions, while the volume law phase of the entanglement is enhanced. An analysis of multifractal properties of wave function in Hilbert space provides a common perspective on both types of transitions in the system. Our findings are immediately relevant to trapped ion experiments, for which we detail a blueprint proposal based on currently available platforms. 

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