Understanding and optimizing the relaxation dynamics of many-body systems is essential both for foundational studies in quantum thermodynamics and for applications such as quantum simulation and quantum computing. Efficient preparation of thermal states of a many-body Hamiltonian is governed by the spectral properties of the associated Lindbladian, in particular its spectral gap, which determines the slowest relaxation rate. In this work, we develop a systematic framework for constructing Lindbladians that prepare thermal states. Our approach reveals a simple relation between the relaxation dynamics at finite and infinite temperatures. The framework is scalable to larger system sizes when implemented using tensor-network methods. We find that efficient thermalization requires that the relaxation dynamics respect the symmetries of the thermal state, which reduces the number of free parameters. By applying gradient-based optimization to the Lindbladians, we enhance the spectral gap and thereby boost thermalization. When applied to both classical and quantum spin models, our method demonstrates a substantial enhancement of the spectral gap. For larger system sizes, our approach provides a variational upper bound and enables a certified lower bound on the minimum relaxation rate.
Quantum matter emerges from the interplay of fluctuations, topology, and entanglement, which - in equilibrium - governs quantized transport, universal criticality, and topological classification. Non-Hermitian systems, widely explored in platforms ranging from electric circuits to photonics, are intrinsically out-of-equilibrium, and display fundamentally new phenomena, including complex spectra, spectral winding, exceptional topology, and non-unitary dynamics. A central challenge is understanding how the complex single-particle spectrum governs universal many-body behavior. We introduce a symmetry-protected dynamical topological index derived directly from the complex spectrum. Through the lens of algebraic topology, more specifically Morse theory, we identify critical points in the spectrum with topological defects, whose curvature and stability are protected under continuous deformations. This links spectral geometry to many-body observables, unifying non-Hermitian band topology, entanglement, and transport. We demonstrate that non-Hermitian quantum criticality in non-interacting systems is controlled by gain-and-loss-selected non-equilibrium steady states, which dynamically generate an emergent imaginary Fermi surface whose Fermi points host scale-invariant gapless modes with logarithmic entanglement scaling and algebraic correlations. Our work establishes a unified framework for non-Hermitian quantum matter, connecting spectral topology to Morse theory, revealing a topological foundation of non-equilibrium quantum criticality.
In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between phase-space structures, reflecting the distinction between chaotic, mixed or integrable classical dynamics, and the corresponding quantum properties. Well established techniques dealing with ergodic wave interference in the usual semiclassical limit
ℏ→0, where the classical limit is given by Hamiltonian mechanics of particles, constitute a now standard part of the toolkit of theoretical physics. During the last years, these ideas have been extended into the field theoretical domain of systems composed of
N indistinguishable particles, aka quantum fields, displaying a different type of semiclassical limit
ℏeff=1/N→0 and accounting for genuine many-body quantum interference. The foundational concept behind this idea of many-body interference, the many-body version of the van Vleck-Gutzwillers semiclassical propagator, is explained in detail. Based on this the corresponding semiclassical many-body theory is reviewed. It provides a unified framework for understanding a variety of quantum chaotic phenomena addressed, including random-matrix spectral correlations in many-body systems, the universal morphology of many-body eigenstates, interference effects kin to mesoscopic weak localization, and the key to the scrambling of many-body correlations characterized by out-of-time-order correlators.