Fundamental Aspects of Statistical Mechanics and
the Emergence of Thermodynamics in Non-Equilibrium Systems

Recent years have seen a great deal of effort to understand quantum thermalization: the question of whether closed quantum systems, evolving under unitary dynamics, reach a state of thermal equilibrium. In my talk, I will discuss how the presence of conservation laws affects the dynamics of thermalization. First, we investigate the dynamics of quantum entanglement in systems with conservation laws and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Renyi entropies. We argue that the latter generically grow sub-ballistically in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic non-integrable spin chain, where energy is the sole conserved quantity. Second, we introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators.

To characterize the generic behavior of open quantum systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size ℓ with up to n-body Lindblad operators, which are n local in the complexity-theory sense. Without locality (n=ℓ), the complex Liouvillian spectrum densely covers a “lemon”-shaped support, in agreement with recent findings [S. Denisov et al., Phys. Rev. Lett. 123, 140403 (2019)]. However, for local Liouvillians (n<ℓ), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to n-body decay modes. This implies a hierarchy of relaxation timescales of n-body observables, which we verify to be robust in the thermodynamic limit. Our findings for n locality generalize immediately to the case of spatial locality, introducing further splitting of timescales due to the additional structure.
Kevin Wang, Francesco Piazza, and David J. Luitz, Phys. Rev. Lett. 124, 100604 (2020)

In standard quantum mechanics, an adiabatic theorem follows from a spectral gap (or, alternatively, some smoothness assumptions). In many-body theory, it turns out that the adiabatic theorem does not automatically derive from a spectral gap assumption and I will describe how one can state and prove an appropriate adiabatic theorem. Furthermore, I ask whether one actually needs a global spectral gap or whether it can be replaced by a local one, in a well-defined sense. The answer turns out to be that a local gap is sufficient and I elaborate on this point, leading to a certain type of prethermalization result. This result is then related to some recent work and to the occurrence of fractional transport in Floquet systems. This is based on joint work with Sven Bachmann, Martin Fraas and Wen Wei Ho.

The active time-periodic driving of quantum systems has enjoyed great interest, since it opens the possibility to "Floquet engineer" novel interactions and phases in modern experiments. The theoretical technique will be illustrated using the well-known example of the 1D Hubbard model with modulated interactions at high frequencies. We then go beyond the high frequency expansion, where the numerical Floquet solution shows strong resonances, which can no longer be described by an effective model in the original Hilbert space. Therefore, such resonances pave the road for Floquet engineering of new internal states as well as tunable dissipation, which will be illustrated for selected examples.

The Floquet engineering, or controlling material properties and functionalities by time-periodic drives, is one of the forefronts of quantum physics of light-matter interaction, but often limited to ideal dissipationless systems. For the Floquet engineering extended to a broader class of materials, it is vital to understand the quantum states emerging in a balance of the periodic drive and energy dissipation. This talk consists of two topics: (1) a general description of nonequilibrium steady states (NESS) for high-frequency drives (Ref.[1]) and (2) time-crystalline NESS protected by "symmetry" (Ref.[2]). In Topic (1), we discuss the NESS in time-periodic Lindblad equation solved by the high-frequency expansion technique. In Topic (2), we introduce a new "symmetry" termed the Floquet dynamical symmetry and show it leads to time-crystalline states.
[1] T. N. Ikeda and M. Sato, Sci. Adv. 6, eabb4019 (2020). [2] K. Chinzei and T. N. Ikeda, Phys. Rev. Lett. 125, 060601 (2020)

While much work has provided strong evidence for the existence of a many-body localized phase in one-dimensional systems of fermions or spin-degree of freedoms, there are still open questions concerning bosonic systems. The larger local state space allows for multiple-site occupancies which may already favor localization. Then, bosonic systems with disorder exhibit a superfluid phase in the ground state in a larger parameter space than in clean systems, leading to the presence of an inverted mobility edge. Moreover, many experimental efforts studied bose gases in lattices rather than fermions.
In our work [1], we study the one-dimensional Bose-Hubbard model and put a particular focus on one-particle measures, that is, the one-particle density matrix (OPDM) and the full distributions of local densities computed in many-body eigenstates. The analysis of occupations provides information about the degree of Fock-space localization while the OPDM eigenstates are sensitive to real-space localization [2]. We introduce a new quantitative measure for Fock-space localization that exhibits a favorably small system-size dependence in the MBL phase, rendering it potentially useful for the analysis of future experimetns in extension of [3].
[1] M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020)
[2] S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, Phys. Rev. Lett. 115, 046603 (2015)
[3] See, e.g., Choi et al, Science 352, 1547 (2016), Lukin et al. Science 364, 256 (2019).

Generic eigenstates of non-integrable local Hamiltonians are expected to fulfill an entanglement area law, a necessary condition for their resembling (locally) thermal equilibrium. For arbitrary pure states, which can be written as linear combinations of these eigenstates, we may ask how much entanglement is needed to reduce the energy variance, and how small do we need this to be, in order to have local thermal properties. We have explored the relation between entanglement and energy variance in a pure state, as the system size increases, for local one dimensional Hamiltonians. Using a systematic construction for matrix product states, we have found that a polynomially increasing bond dimension is enough to construct states with energy variance that vanishes with the inverse of the logarithm of the system size. Our numerical results suggest that these states, which can be constructed efficiently, do converge to the thermal equilibrium in the thermodynamic limit, while the same is not true if the variance remains constant.

Recently it was advocated by several groups that a variety of pertubations in condensed matter type systems may have "generic" effects on the dynamics of expectation values [1,2,3]. We investigate this approach numerically and to some extend analytically, scrutinizing various ways of modelling said generic effects.
[1] L. Dabelow, P. Reimann, Relaxation Theory for Perturbed Many-Body Quantum Systems versus Numerics and Experiment, PRL 124,120602
[2] J. Richter et al., Exponential damping induced by random and realistic perturbations, PRE 101, 062133
[3] L. Knipschild, J. Gemmer, Stability of quantum dynamics under constant Hamiltonian perturbations, PRE 98, 062103

We investigate decoherence of subsystems if the complete system is evolved according to the time-dependent Schrödinger equation. The focus of such research is on how to prevent decoherence or how to achieve long coherence times. One option is the use of clock transitions, i.e. transitions that are independent of an external magnetic field (at least to some order). In accordance with recent experiments [1], we show that such transitions decohere indeed much more slowly than other transitions [2]. In a second investigation we demonstrated that a combined treatment of a system of spins and a bath of phonons leads to entanglement phenomena such as avoided level crossings that could not be modelled by master equations [3].
[1] Y. Bae, K. Yang, P. Willke, T. Choi, A. J. Heinrich, and C. P. Lutz, Sci. Adv. 4, eaau4159 (2018);
[2] P. Vorndamme, J. Schnack, Phys. Rev. B 101, 075101 (2020); [3] K. Irländer, J. Schnack, Phys. Rev. B 102, 054407 (2020)

The phenomenon of many-body localization (MBL) usually is seen as a generalization of Anderson localization to interacting systems and thus is discussed to appear in disordered systems. There have been suggestions that MBL could also appear in clean interacting systems without disorder, if other mechanisms for localization are considered. One such possibility are flat band (FB) systems, in which localization appears due to the presence of dispersionless bands in the band structure. In this talk we will discuss a clean quasi 1D diamond ladder, which hosts a flat band. We present evidence that a flat band induced MBL appears by showing that the eigenstate thermalization hypothesis (ETH) is violated when FBs are present. The results are underpinned by the time evolution of observables that measure localization properties.

Random quantum circuits are known to exhibit a measurement-driven quantum phase transition from an ergodic thermal phase to a nonergodic localized phase. We show that this transition survives in an open quantum system setting with weak (generalized) measurements [1]. We obtain a consistent phase boundary in the space of the measurement strength and the measurement probability, clearly demonstrating a critical value of the measurement strength below which the system is always ergodic, irrespective of the measurement probability. While this entanglement transition is currently well established for stroboscopic (discrete) measurements in random quantum circuits, a crucial link to physical settings is its extension to continuous observations, where for an integrable model it has been shown [2] that the transition changes its nature and becomes immediate. We demonstrate that the entanglement transition at finite coupling persists if the continuously measured system is randomly nonintegrable, and show that it is smoothly connected to the transition in the stroboscopic models [3]. This provides a bridge between a wide range of experimental settings and the wealth of knowledge accumulated for the latter systems.
[1] M. Szyniszewski, A. Romito, and H. Schomerus, Phys. Rev. B 100(6), 064204 (2019). [2] X. Cao, A. Tilloy, and A. D. Luca, SciPost Phys. 7, 24 (2019). [3] M. Szyniszewski, A. Romito, and H. Schomerus, arXiv:2005.01863 (2020).

We study matrix elements of a local spin operator in the eigenbasis of different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with all standard indicators of the eigenstate thermalization hypothesis, the eigenvalue distribution still exhibits clear signatures of the original operator, implying correlations between matrix elements. Moreover, we demonstrate that at much smaller energy scales, the eigenvalue distribution approximately assumes the universal semicircle shape, indicating transition to the random-matrix behavior, and in particular that matrix elements become uncorrelated.
[1] J. Richter, A. Dymarsky, R. Steinigeweg, and J. Gemmer, arXiv:2007.15070

Recent progress in the realm of noisy, intermediate scale quantum (NISQ) devices represents an exciting opportunity for many-body physics, by introducing new laboratory platforms with unprecedented control and measurement capabilities. We explore the implications of NISQ platforms for many-body physics in a practical sense: we ask which physical phenomena, in the domain of quantum statistical mechanics, they may realize more readily than traditional experimental platforms. As a particularly well-suited target, we identify discrete time crystals (DTCs), novel non-equilibrium states of matter that break time translation symmetry. These can only be realized in the intrinsically out-of-equilibrium setting of periodically driven quantum systems stabilized by disorder induced many-body localization. We show that a new generation of quantum simulators can be programmed to realize the DTC phase and to experimentally detect its dynamical properties, a task requiring extensive capabilities for programmability, initialization and read-out.