The representation of quantum **many**-**body** states as tensor networks is connected to White’s density-matrix renormalization group [ 8 ], and in the case of one dimensional spin lattices is known as matrix product states (MPS) [ 9 ]. Among **many** useful properties of tensor networks, one which makes them well suited to the description of states with symmetries, is the ability to encode the symmetry on the level of a single tensor (or a few) describing the state. In the case of global symmetries, both for MPS and for certain classes of PEPS in 2D (Projected Entangled Pair States — the generalization of MPS to higher dimensional lattices), the relation between the symmetry of the state and the properties of the tensor is well understood [ 10 ]. Tensor networks studies of lattice gauge theories have so far included numerical works (e.g., mass spectra, thermal states, real time dynamics and string breaking, phase diagrams etc. for the Schwinger model and others) [ 11 – 30 ], furthermore, several **theoretical** formulations of classes of gauge invariant tensor network states have been proposed [ 31 – 35 ]. In all of the latter the construction method follows the ones common to conventional gauge theory formulations: symmetric tensors are used to describe the matter degree of freedom, and later on a gauge field degree of freedom is added, or, alternatively - a pure gauge field theory is considered. While the usefulness of tensor networks in lattice gauge theories has certainly been demonstrated by the above mentioned works, so far there were few attempts (e.g. [ 13 ]) to generally classify tensor network states with local symmetry.

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Strongly correlated systems can be defined as **many**-**body** systems whose properties can not be mapped to non-interacting quasi particles [ 33 ]. In the early 1960s, the main motivation to study strong correlations in condensed matter **physics** came from experiments on transition metal oxides and the Mott metal-insulator transition [ 34 ]. The discovery of heavy fermion compounds [ 35 , 36 ] and high-temperature cuprate superconductors [ 3 ] has revived the interest in the 1980s, which continues up to the present days. These materials show unusual electronic and magnetic properties, which are often of technological interest, but their **theoretical** under- standing presents a considerable challenge already for decades. The **theoretical** progress in the field of strongly correlated systems has been slowed by the difficulties of applying approx- imations. Due to the non-perturbative nature of the problem, it is often hard to say whether a **theoretical** prediction is a real feature of the Hamiltonian studied or an artifact of the ap- plied approximations. Direct numerical studies are slowed by the exponential growth of the computations with system size [ 37 ], while Quantum Monte Carlo calculations suffer from the sign-problem at low temperatures [ 38 ].

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As a first **many**-**body** system, quantum dots placed inside a cavity are considered with focus on **many**-**body** effects. In analogy to the standard optical laser [80, 81], the concept of coherent amplification by stimulated emission is adapted to sound waves. The idea is to create coherent vibrations, which could lead to new types of non-demolishing measurement devices [82]. Quantized vibrations of the lattice ions in a solid form the quasi-particle called phonon. The generation of coherent phonon statistics is furthermore interesting for fundamental **physics** itself. The idea of the so called phonon laser has led to a variety of experimental and **theoretical** proposals to generate coherent phonons in e.g. trapped ions [83, 84], compound microcavities [85], NV-centers [86], electromagnetic resonators [87] and semiconductor devices [88, 89, 90, 91, 92, 93]. In order to achieve lasing, the necessary ingredients are the active medium, an external pump mechanism, inversion and a cavity to confine a single mode. Thus, the design of the phonon or acoustic cavities [94, 95, 96, 97, 98, 99] forms the basis for phonon lasing. Via a combination of different lattice constants of the surrounding solid, a superlattice allows to confine a single phonon mode inside the cavity [94, 95, 96]. Quality factors up to Q = 10 5 have been achieved in the past [97, 98, 99]. The investigated model setup is based on a single quantum dot as an active medium [100] embedded within an acoustic cavity. Via external coherent optical excitation of the quantum dot [101], the induced Raman process [91, 92] leads to a coherent phonon population inside the acoustic cavity. The focus within this chapter lies on a generalization of the single-emitter quantum dot to a **many**-emitter system [102, 103]. The quantum dots are assumed to be identical and not coupled directly with each other, but via the cavity phonon field. For optical cavities this is known as the Tavis-Cummings model [104, 105]. Due to the **many**-emitter setup, collective effects are present. One example is superradiance, discovered by Dicke [106, 107, 108, 109]. Similar collective phenomena have been found recently for phonons as well [110, 111]. These collective processes due to a **many**-**body** setup, combined with the generation of coherent phonons by stimulated emission will be the focus of this chapter.

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The chapter is organized as follows. In Section 2.2 we analyze the low-energy 2D scattering of the polar molecules due to the dipole-dipole interaction. We obtain the scattering amplitude for all scattering channels with odd orbital angular momenta. The leading part of the amplitude comes from the so-called anomalous scattering, that is the scattering related to the interaction between particles at distances of the order of their de Broglie wavelength. This part of the amplitude corresponds to the first Born approximation and, due to the long-range 1/r 3 character of the dipole- dipole interaction, it is proportional to the relative momentum k of colliding particles for any orbital angular momentum l. We then take into account the second Born correction, which gives a contribution proportional to k 2 . For the p-wave scattering channel it is necessary to include the short-range contribution, which together with the second Born correction leads to the term behaving as k 2 ln k. In Section 2.3 , after reviewing the Landau Fermi liquid theory for 2D systems, we specify two- **body** (mean field) and **many**-**body** (beyond mean field) contributions to the ground state energy for 2D fermionic polar molecules in the weakly interacting regime. We then calculate the interaction function of quasiparticles on the Fermi surface and, following the idea of Abrikosov-Khalatnikov [ 67 ], obtain the compressibility, ground state energy, and effective mass of quasiparticles. In Section 2.4 we calculate the zero sound velocity and stress that the **many**-**body** contribution to the interaction function of quasiparticles is necessary for finding the undamped zero sound. In Section 2.5 , we conclude and emphasize that the 2D gas of fermionic polar molecules represents a novel Fermi liquid, which is promising for revealing **many**-**body** effects. Moreover, we show that with present facilities it is feasible to obtain this system in both collisionless and hydrodynamic regimes. We also summarize some recent development of this subject based on various **theoretical** tools, such as quantum Monte-Carlo, variational methods, etc.

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u, d or s quark content, but because of its “charming” quark it can only decay through the weak interaction, extending its lifetime substantially. Then, there is the last generation of bottom (b) and top (t) quarks, whereof the top quark is too heavy for bound states and directly decays after production. In Table 1.1 the quarks and their electric charges, flavor quantum numbers and masses are listed. Besides the electric charge the quarks hold one of three color charges (red, green, blue) whereby they interact with gluons. This color interaction is described by Quantum ChromoDynamics (QCD) which is the theory of the strong interaction, and builds with the electroweak interaction the standard model of particle **physics**. QCD is a non-abelian gauge theory with symmetry SU(3) where “3” stands for the number of color charges. The gluon in QCD is the equivalent of the photon in electrodynamics: the interaction mediator, but with the difference that the gluon itself carries color charge — unlike the photon that does not carry electric charge. The gluons interact with each other through the color charge already in first order of the coupling. An important feature of QCD is that the quarks are confined in hadrons since the running coupling of QCD becomes large for low energies or large distances. Only in the asymptotic limit of high energies the quarks become asymptotically free. This limits perturbative methods to high energies/temperatures, where the coupling becomes small. Thus, non-perturbative **theoretical** methods are needed to investigate QCD at lower energies or temperatures.

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dardized concepts of statistical **physics** are applied to a wide range of physi- cal problems, but often consider a coupling to an idealized bath [248]. There has been much research, experimental as well as **theoretical**, about how con- cepts of thermodynamics can also be applied to closed quantum systems. For systems in thermal equilibrium, a small number of intensive parameters (typically including temperature and chemical potential) are sufficient to de- scribe the whole system accurately. In particular the precise initial state is unimportant and different initial states can lead to the same thermal state. It is an interesting question how these concepts can be applied to closed quan- tum systems as the information in a system is conserved under unitary time evolution. Since the unitary Hamiltonian can always be inverted in theory, the exact initial state can be recovered. However in experiments this informa- tion is not easily accessible anymore with local operators because it is rather hidden [39]. The basic notion of quantum thermalization arises already for a system which has no other conserved extensive quantity than energy, fol- lowing the ideas of [5, 39]. Here we consider a small subsystem A within the total closed system S (See Figure 6.1). We now consider initial states ρ ( t = 0 ) which thermalize to the temperature T and the expectation value of the total energy is given by h H iT . The density matrix for the system at thermal equi- librium with this temperature is given by ρ (eq) ( T ) . A closed quantum system thermalizes if locally, in the small subsystem A, and in the long time limit t ∞ the reduced density matrix ρ A ( t ∞) , calculated by tracing out the rest of the system S \ A, is the same as for the thermal equilibrium. Thus the follow- ing two equivalent equations need to be fulfilled for any choice of a small subsystem A:

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Over the last decade, **many** proof of principle experiments have been demonstrated on various platforms 6 . Here in this work, a particular Hamiltonian - namely the transverse field Ising Hamil- tionian with tunable spin-spin interaction range - is implemented on up to 15 ions. Moreover, the experiment presented in chapter 5 goes beyond proof of principle, as for the first time it was possible to observe how entanglement spreads out in a **many**-**body** interacting system after quenching [ 26 ]. The dependence of this spreading on the spin-spin interaction range was investigated and compared to so-called Lieb-Robinson bounds [ 27 ], i.e. upper bounds on the speed of correlation propagation in interacting systems. In addition, a spectroscopic technique is demonstrated for probing the low- lying energy states of the Ising Hamiltonian by creating superpositions of its eigenstates [ 28 ]. This allows the dispersion relation of the given Hamiltonian to be deduced, revealing information about the dynamical behaviour of the system under consideration. In fact, this spectroscopic method is generic and can be applied to other Hamiltonians that show certain symmetries in their eigenstates. In the quest to develop a quantum computer, not only experimental progress has been made, but also **theoretical** developments have led to new insights into quantum information theory over the past two decades. A prominent example is the realization that quantum information can be processed in ways other than in the standard model, or gate model, of quantum computation [ 12 ]. In 2001 Raussendorf and Briegel formulated their idea of a measurement-based quantum

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The exact solution of quantum mechanical **many**-**body** problems is only possible for few particles. There- fore, numerical methods were developed in the fields of quantum **physics** and quantum chemistry for larger particle numbers. Configuration Interaction (CI) methods or the No-Core Shell Model (NCSM) allow ab initio calculations for light and intermediate-mass nuclei, without resorting to phenomenology. An extension of the NCSM is the Importance-Truncated No-Core Shell Model, which uses an a priori selection of the most important basis states. The importance truncation was first developed and applied in quantum chemistry in the 1970s and latter successfully applied to models of light and intermediate- mass nuclei. Other numerical methods for calculations for ultra-cold fermionic **many**-**body** systems are the Fixed-Node Diffusion Monte Carlo method (FN-DMC) and the stochastic variational approach with Correlated Gaussian basis functions (CG). There are also such method as the Coupled-Cluster method, Green’s Function Monte Carlo (GFMC) method, et cetera, used for calculation of **many**-**body** systems.

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Haldane conjectured in 1983 that integer-spin Heisenberg antiferromagnet chains dif- fer qualitatively from the half-integer case [ 104 , 105 ]. He suggested that integer-spin chains should have a finite excitation gap with an exponential decay of the ground state correlation function while the half-integer case should be gapless with power-law decay of the ground state spin correlations. This came as a surprise to **many** people working in the field because his conjecture marked a huge deviation from what Bethe had found in 1931 [ 120 ]. The Bethe ansatz solution had predicted gapless excitations for the spin-1/2 chain and it was thought that other spin chains would have similar behavior. This issue remained in large for a while because of the non-rigorous nature of Haldane’s argument involving topological terms in quantum-field theories. The numerical and experimental evidence of the existence of a finite gap for the case of s = 1 spin chain finally came [ 121 , 122 , 123 , 124 , 125 , 126 , 127 , 128 ] and further devel- opments made the situation more clear. For example, Lieb, Schultz and Mattis had already provided the rigorous proof of zero-gap in 1961 for spin s = 1/2 chain [ 129 ]. This proof was later extended for any arbitrary half-integer spin s but it was shown that it failed for integer spin s chain [ 130 ]. The Lieb-Schultz-Mattis theorem states that any half-integer spin chain with essentially any local Hamiltonian respecting the translational and rotational symmetry either has a zero gap or has degenerate ground states corresponding to spontaneously broken parity.

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The high-resolution technique can also be used in reverse to control individual atoms and to modify Hamiltonian parameters on the level of individual lattice sites. In particular, we experimentally demonstrated that the spin of individual atoms in a Mott insulator can be addressed [27]. We subsequently used this technique to study a single, mobile spin-impurity in a strongly interacting one-dimensional system [28]. One of the most advantageous features of high-resolution fluorescence detec- tion is its ability to resolve single particles in each experimental run. This allows for the direct measurement of correlations between fluctuations in different parts of the system, in addition to the average on-site parity. Notably, this is not limited only to correlations between two lattice sites. Rather, the detection of high-order correlation functions between an arbitrary set of sites became feasible. Such non-local correla- tion functions play an important role for the characterization of quantum phases in low-dimensional **many**-**body** systems. We used these possibilities to detect two-site and non-local correlations across the superfluid-Mott-insulator transition [29], which is the focus of Part II of this thesis.

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The last two decades have seen exceptional progress in the ability to engineer, manip- ulate and probe complex quantum systems. The concepts of quantum computation, quantum simulation or precision measurement beyond the classical limit have been validated in the laboratory and quantum sensing and metrological devices have been developed for speciﬁc applications [117, 118, 119, 120, 121, 122]. Another important challenge to meet in order to fully exploit the potential of complex quantum systems is to design more robust and eﬃcient experimental protocols. Most of the protocols developed so far in research laboratories rely on analytic or simple empirical solu- tions. In the paradigmatic example of a superﬂuid-to-Mott-insulator transition in a lattice, adiabatic manipulations are generally applied. Although maintaining adia- baticity is impossible in the thermodynamic limit, almost adiabatic transformations can become feasible for ﬁnite size systems. However, they are – by deﬁnition – slow compared to the typical timescales of the system. Thus, they are highly sensitive to decoherence and experimental imperfections. Speeding up the transformation can lead to a signiﬁcant gain in that regard. In another common case, the driving of a transition between two energy levels of a system, an adiabatic solution does not necessarily exist. The transition can, under certain constraints, be driven by a Rabi pulse at the frequency of the level splitting. However, in the presence of other ac- cessible levels or loss processes, this option has a strongly limited eﬃciency. It would therefore be desirable to have at our disposal a method to design fast and arbitrary complex manipulations. In addition, to be experimentally sound, such a method would have to be robust with respect to system perturbations. This challenge can be met by means of optimal control theory, that is, the automated search of an op- timal control ﬁeld to steer the system towards the desired goal [123, 124]. Quantum Optimal Control (QOC) has been applied very successfully in the case of (eﬀective) few-**body** quantum systems: it has been shown that QOC can steer the dynamics in the minimum allowed time and that the optimal protocols are robust with respect to noise [124]. In particular, it has been experimentally demonstrated, for quantum dynamics taking place in an eﬀective two-level system, that QOC allows to saturate the Quantum Speed Limit (QSL) – the minimal time necessary to transform one state into another for a given energy of the driving [125, 126, 127, 128, 129, 130]. However, only recently QOC has been extended to embrace **many**-**body** quantum dynamics in non-integrable quantum systems [96, 131, 132, 133, 134].

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The intrinsic complexity of quantum **many**-**body** systems – namely the exponential scaling of the number of variables not only for mixed, but also for pure states where full knowledge about the systems’ quantum state is available – is a feature which clearly distinguishes it from classical **physics**. This curse of dimensionality prohibits the desire to simulate the behavior of quantum **many**-**body** systems and the quest for exact solutions of these systems. The mathematical charac- terization of finitely correlated states [43] and the invention of the density matrix renormalization group (DMRG) methods [93, 139] have greatly influenced the development of analytical tech- niques and numerical algorithms approximating properties of quantum **many**-**body** systems in one-dimension [93, 95, 114]. These methods have in common that they truncate the full Hilbert space to reduce the number of parameters and hence to approximate the states within these mod- els. This allows not only to determine properties of low lying eigenfunctions of Hamiltonians with a finite interaction range but also the characterization of thermal mixtures at finite tempera- tures [25,93,104,107,131,133,134,142]. These states are well approximated by states which do not occupy the full Hilbert space but rather are restricted to a small subset characterized by the families of matrix product states and operators [25, 31, 95, 104, 114, 131, 142]. This restriction is motivated by applying results established in the field of quantum information theory where a wide range of analytical techniques have been exploited to justify the approximation of phys- ically relevant states by matrix product structures [5, 15, 41, 57–60, 72, 115, 130]. While these approximations are a powerful tool to describe and analyze one-dimensional systems on lattices such as distinguishable ions in an ion trap, we need to rely on different strategies to reduce the exponential scaling of the Hilbert space when it comes to higher-dimensional systems such as atomic clouds. For the latter system, we will exploit the indistinguishability of the constituents to restrict our description to the totally symmetric subspace of the full Hilbert space. This, again, reduces the number of parameters in the state representation from an exponential to an algebraic scaling. In this chapter we discuss the intrinsic complexity of quantum **many**-**body** systems to- gether with the notation used throughout this thesis. We review and motivate the families of matrix product states and operators and outline properties that help to simplify computations at later stages of this thesis. In the last section of this chapter, we discuss the restriction of quan- tum states describing permutation invariant systems to the totally symmetric subspace. This convenient representation arises naturally from the indistinguishability of the subsystems and is

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Another interesting observation is the fact that the tensor correlator of unrestricted range which is optimal in the two-**body** system actually gives less binding for the heavier nuclei than the correlator γ that is restricted in range. In the two-**body** system we only consider a L = 0 trial state but in the two-**body** densities of the heavier nuclei also higher relative angular momenta appear. The additional angular momentum dependence which comes with the tensor correlations gives then, increasing with the correlation range and decreasing with the radius, more repulsion which overcompensates the increased binding in the L=0 channels. This effect cannot be observed with the short ranged tensor correlator as the probability density at short distances is already strongly suppressed by the centrifugal barrier.

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In this thesis we have studied three different examples of unconventional order in quantum systems. We have first studied dissipative solid-state graphene driven by circularly polar- ized light. In contrast to the high-frequency non-dissipative limit we have found that the Hall conductivity has opposite sign for low-frequency driving and its magnitude depends on the interplay of dissipation and driving. Nevertheless a major contribution to the Hall conductivity is obtained by weighting the Berry curvature of Floquet bands with their respective occupations. Hence our **theoretical** formalism represents a new approach to the application of periodic driving to solids and presents an interpretation of the experiments presented in Ref. [86]. We have considered the limit of increasing the driving frequency by orders of magnitude while keeping the gap at the Dirac point fixed and find that the resulting steady state is a high-temperature state with almost equal occupation in the lower and upper Floquet band. Only for low dissipation or short times the contribution of the Dirac point recovers the result expected in the high-frequency limit.

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The collective-coordinate representation is the keystone for the formulation of the boundary conditions in the **many**-**body** Hilbert space. It can be obtained from a mi- croscopic representation of the nuclear system, in which the nucleons are the fundamental components. The physical interpretation of the collective coordinate as the relative dis- tance between the fragments is, by construction, unique in the asymptotic region, but it can also be used for the interior region where the two clusters are close to each other and their components are indistinguishable. This representation allows the relative distance between the clusters to be defined in terms of valid observables for antisymmetrized sys- tems. Moreover, a wave function of the system in this representation can be defined, as well as its derivatives. The formalism is written in terms of operators and states defined in a model Hilbert space, which is spanned by the set of **many**-**body** states that build the reaction path.

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The most fundamental optical investigation scheme is the linear absorption spectrum, where the sample is excited with a very weak pulse. When weakly exciting the system, the shape of the spectra can already give important insights into the dispersion of the material or non-equilibrium **many**-**body** processes. **Many** **theoretical** and experimental studies have already been done to determine the influence of **many**- particle interactions on the ISB absorption [WWFK03, PDMK08, VVA + 09, GWS + 09, HAG + 11]. However, especially concerning the low-temperature regime there has been a deficit in the **theoretical** description, predicting line widths much smaller then experimentally observed. To gain a better under- standing of the ISB absorption line shape, this chapter will focus on C OULOMB interaction since the spectral width is basically determined by the fastest scattering process, which is the electron-electron (el- el) scattering [LN04]. In the first part of this chapter, the main el-el scattering contributions of the micro- scopic polarization are revisited. Also, the processes which are responsible for the deficit of the infinitely small line width are illuminated. Then, in the second part of this chapter, a modification of the theoreti- cal description is introduced via the C OULOMB -induced ground state correlations. Studies about the el- ph interaction in the linear regime can be found, for example, in [NAK99, WFL + 04, SRW + 05, BK06]. The sections are organized as follows: Starting from the free carrier dynamics in Sec. 5.1, the first and the second-order correlation contributions of the non-diagonal elements of the density operator will be derived in Sec. 5.2 and Sec. 5.3, where also the **theoretical** problem of the small absorption line width is specified. Beginning from Sec. 5.4 the electronic correlations in the ground state are investigated as a possible mechanism for an intrinsic line shape broadening in the ISB absorption. They are derived in Sec. 5.4.1 as a second-order correlation contribution of the microscopic density, caused by the C OULOMB interaction. The properties of the renormalized ground state distribution are examined in Sec. 5.4.2. Subsequently its influence on the absorption spectrum is investigated in Sec. 5.5, where the spectra including and neglecting ground state correlations are compared. It will be shown that correlations in the ground state yield an inherent line broadening of the absorption in the low temperature regime. The main results of this chapter have been published in [DWRK10].

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a) The vertical flows of information and energy supply the NES and AS. Otherwise the systems (being dissipative in nature) would disintegrate. As indicated above (section 2.2), the FES acts as intermediary in the transfer of information from the superior environment (demand for energy), and of energy from the inferior environment (supply of energy), although not at the same time. On the contrary, the effect follows the cause, i.e. the supply of energy follows the demand for it 4) . This takes place repeatedly, causing the system to oscillate around a centre level, thereby achieving a flow equilibrium as v. Bertalanffy expressed it in 1952 (L.v.BERTALANFFY, W.BEIER und R.LAUE 1952/77). The system regulates itself by means of feedback (Proc. p. 77 f.). An example of this in the field of **physics** is the flow of electric current between coil and capacitor, in ecology by the predator-prey relationship, and in economy by

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In this thesis, we also consider the thermalization and relaxation in closed quantum **many**- **body** systems. Experiments with ultracold atoms allow the observation of quantum dynamics over long time intervals and enable the study of strongly correlated states from a new per- spective [52, 53]. A simple initial state that is not an eigenstate of the system Hamiltonian can be prepared and the ensuing microscopic dynamics can then be investigated directly. This technique has been used in the study of non-equilibrium dynamics in Hubbard- and Heisenberg-type models [54–58]. The specific problem of the decay of N´eel order has so far been addressed in the non-interacting case in one dimension [59] and for a two-dimensional system [60]. Also, the decay of a spin spiral has been studied in one and two dimensions [61]. We study the real-time decay of the N´eel state in the one-dimensional Fermi-Hubbard model [62] (see also [63]). By that we extend previous studies [64] by the incorporation of charge dynamics additional to the spin dynamics present in the spin-1/2 XXZ-model. As a main result, we show that the relevant time scales for the relaxation of the double occupancy – a charge-related quantity – is set by the inverse of the hopping matrix element while for the staggered magnetization – a spin-related quantity – the dynamics is slower as the on-site repulsion U increases. This reflects the existence of two characteristic velocities in the low- energy, equilibrium **physics** of strongly interacting one-dimensional (1D) systems: spin and charge velocity related to spin-charge separation [65]. Also, we investigate if the steady-state value of the double occupancy is thermal in the framework of the eigenstate thermalization hypothesis (ETH) [66] for different choices of the on-site interaction U . The ETH gives an interpretation to the thermalization of a subsystem in closed quantum systems along with a few requirements on the Hamiltonian describing the system. One important requirement is ergodicity: the ability of the system to reach all states in the **many**-**body** Hilbert space of the system which have the same energy as the initial state in a finite time. We find that the steady-state value is close to the thermal one but different. This is not surprising since our system is integrable and thus not ergodic which is in line with Ref. [66]. We conclude from this that the double occupancy does not thermalize in our system.

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DOI: 10.1103/PhysRevX.5.011024 Subject Areas: Particles and Fields, Quantum **Physics**, Strongly Correlated Materials
The fascinating subject of gauge theories is omnipresent throughout **many**-**body** **physics**. The gauge principle, which states that the fundamental interactions of nature originate from gauging global symmetries of the free theory, is one of the cornerstones of the standard model, but quantized gauge fields also emerge as effective degrees of freedom in several models for strongly correlated condensed matter, alongside other effective interactions. Historically, the concept of gauging, i.e., transforming a global symmetry into a local symmetry, is based on a Lagrangian or Hamiltonian description of the system where operators of the original (matter) theory are transformed into gauged operators using the “minimal coupling rule,” which is not always unambiguous [1] . Indeed, while there is a unique way to ungauge a theory (by setting the gauge fields and gauge coupling constant equal to zero), the reverse process is not unique as new degrees of freedom are introduced and the Hilbert space is enlarged.

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This manuscript is separated into two parts. The first part serves as a more general introduction to the topic of nonequilibrium **physics**, related experiments and numerical methods. The second part deals with more specific problems. It is divided into three chapters, each studying a different quench setup. We predominantly study one- and two-dimensional spin models, perform local or global quenches and relate the time evolution of real-space observables to spectral properties of the respective systems and initial states. Key aspects of each project are summarized as follows: Firstly, we analytically estimate relaxation dynamics of magnetically-ordered initial states in terms of simple spectral properties, and compare to numerical results. In this way, we achieve an extensive description in one and two spatial dimensions. Relaxation dynamics is significantly altered by long- range interactions, spin anisotropy or thermal excitations. Secondly, we study the spreading of connected spin correlations after quenches to systems with a tunable integrability breaking. We analyze the correlation density matrix which enables us to determine the dominant spin correlation function. After a quench, correlations expand into the system by showing wave fronts which are measured and characterized with respect to the integrability breaking. Along these wave fronts, correlations decay faster and speed up as the nonintegrability of the post-quench Hamiltonian is increased. Thirdly, we describe modes of propagation observed after a local quench in a two-leg spin ladder, and analyze their behavior with respect to the coupling of the legs of the ladder. The modes of propagation are identified with the dynamics of various quasiparticles, which again are ascribed to different sub-Hilbert spaces of the system. Finally, we discuss a diffusive mode of propagation which has been described in various studies, but still is not very well-understood.

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