The non-equilibrium quantum thermodynamics of many-body open systems is notoriously rich, especially in the strong-coupling regime where strong system-bath correlations develop and interactions produce non-perturbative effects. Such systems can be described by `quantum impurity models' where the system and bath are treated on an equal footing as a single composite. Paradigmatic examples are the spin-boson and Anderson impurity models, in which system degrees of freedom interact with either bosonic or fermionic gapless baths. Here we study the quantum work distribution function (WDF) of such systems following a quench. Capturing the full continuum of many-body excitations in the WDF requires a non-perturbative solution of the underlying non-equilibrium quantum impurity problem. For this purpose, we extend the time-dependent numerical renormalization group (TDNRG) approach to the calculation of the WDF, which applies directly in the thermodynamic limit, can be used for arbitrary quench amplitudes at zero or finite temperature, and provides exponentially fine low-energy resolution. Our numerically-exact solution reveals power-law threshold behavior due to the Anderson orthogonality catastrophe when the work approaches its minimum value, with universal scaling collapse of the distribution below an emergent low-energy scale induced by system-bath correlations.
The non-equilibrium quantum thermodynamics of many-body open systems is notoriously rich, especially in the strong-coupling regime where strong system-bath correlations develop and interactions produce non-perturbative effects. Such systems can be described by `quantum impurity models' where the system and bath are treated on an equal footing as a single composite. Paradigmatic examples are the spin-boson and Anderson impurity models, in which system degrees of freedom interact with either bosonic or fermionic gapless baths. Here we study the quantum work distribution function (WDF) of such systems following a quench. Capturing the full continuum of many-body excitations in the WDF requires a non-perturbative solution of the underlying non-equilibrium quantum impurity problem. For this purpose, we extend the time-dependent numerical renormalization group (TDNRG) approach to the calculation of the WDF, which applies directly in the thermodynamic limit, can be used for arbitrary quench amplitudes at zero or finite temperature, and provides exponentially fine low-energy resolution. Our numerically-exact solution reveals power-law threshold behavior due to the Anderson orthogonality catastrophe when the work approaches its minimum value, with universal scaling collapse of the distribution below an emergent low-energy scale induced by system-bath correlations.
Using the non-perturbative numerical renormalisation group (NRG) we analyse purely fermionic as well as more complex Bose-Fermi quantum impurity systems on their equilibrium properties and beyond: We present an extensive exploration of real-time dynamics for various quenches using the time-dependent NRG. For the purely fermionic system, we use the Single-impurity Anderson model (SIAM) in which the impurity is a localised orbital hybridising with a surrounding conduction band. The model enables charge scattering with a pseudo-gap scattering rate and features a quantum critical point separating the local moment (LM) phase with an unscreened spin moment from a symmetric strong coupling (SSC) phase with a fractionally screened spin moment. The real-time dynamics of the local double occupancy nicely thermalises for quenches regarding the SSC phase. Whereas for quenches within or into the LM phase the double occupancy systematically deviates from thermal equilibrium for long times. A comparison with results of a recently published time-dependent Gutwiller approach reveals clear discrepancies since this approach restricts the formation of the local moment only on the impurity site. We extend the system by an additional bosonic bath leading to the Bose-Fermi Anderson model (BFAM) and study the influence of the additional bosonic degrees of freedom on the real-time dynamics. In equilibrium the bosonic bath features an additional quantum critical point leading to a localised phase once the coupling to a sub-ohmic bosonic bath exceeds a critical value. In general the coupled bosonic bath leads to a renormalisation of the Coulomb interaction. We show that the dynamics is influenced by the specific choice of the bath exponents. For a F-type combination the non-equilibrium dynamics can be completely reproduced by an effective purely fermionic SIAM while for a B-type combination the nature of the bosonic bath and the coupling strength has a profound impact on the electron dynamics which can only be partially understood by an effective SIAM. Furthermore, we investigate the spin dynamics of an initially polarised impurity spin which is coupled to a fermionic and a bosonic bath leading to the Bose-Fermi Kondo model (BFKM). Through the coupling to the fermionic bath the polarisation vanishes with time. We show that in analogy to the spin-boson model (SBM) an initially coupling to the bosonic bath affects the real-time dynamics fundamentally leading to a remaining polarisation for long times.
Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.
Quantum impurities can host exotic many-body states that serve as sensitive probes of bath correlations. However, quantitative and non-perturbative methods for determining impurity thermodynamics in such settings remain scarce. Here, we introduce an impurity renormalization group approach that merges the tensor-network representation with the numerical renormalization group cutoff scheme. This method overcomes conventional limitations by treating bath correlations and impurity interactions on an equal footing. Applying our approach to the finite-temperature quantum critical regime of quantum spin systems, we uncover striking impurity-induced phenomena. In a coupled Heisenberg ladder, the impurity triggers a fractionalization of the local magnetic moment. Moreover, the derivative of the impurity susceptibility develops cusps that mark the crossover into the quantum critical regime. We also observe an exotic evolution of the spin correlation function driven by the interplay between bath correlations and the impurity. Our results demonstrate that this method can efficiently solve correlated systems with defects, opening new pathways to discovering novel impurity physics beyond those in non-interacting thermal baths.
In this work, we study the non-equilibrium dynamics of low-dimensional strongly correlated quantum systems, which in our case are generated by sudden and time-dependent quantum quenches. Such non-equilibrium dynamics are a particularly difficult problem, because the time evolution of the initial many-body quantum state, which is governed by the post-quench Hamiltonian, is highly non-trivial. On top of that, applying an operator to this many-body quantum state in order to calculate its expectation values, typically creates an intricate set of excitations. Throughout the thesis, we furthermore contrast the results that we have obtained using exact numerical methods for two microscopic models, namely the Heisenberg and Fermi-Hubbard chains, with results derived from the Luttinger liquid theory, which, in equilibrium, describes the universal behavior of the two models.In the second part of the thesis, we reduce the dimensionality even further and employ complementary numerical and analytical tools to examine the unusual non-equilibrium behavior of a ring-shaped quantum impurity hosting interacting spin-less fermions.
Theoretical studies of electron and energy transport in nanostructures usually involve nonequilibrium quantum impurity models, which feature a locally interacting quantum system (impurity) coupled to noninteracting bosonic and/or electronic baths each at its own equilibrium. In these models, the interplay of strong correlation, nonequilibrium effects and dissipation leads to rich and complex phenomena. As a nonequilibrium strongly correlated many-body problem, these models, except at some rare cases, have no exact analytical solution, and hence their treatment heavily relies upon numerical techniques. The goal of this thesis is to develop numerically accurate and relatively low-cost techniques for treatment of strongly correlated open nonequilibirum quantum systems. Such impurity solvers are in high demand both as standalone methods for simulation of response in nanoscale junctions and as part of divide-and-conquer schemes such as dynamical mean field theory (DMFT).For nonequilibirum open quantum systems, two characteristic energy scales governing the physics are intra-system interactions and strength of the system-baths couplings. Thus, systems strongly coupled to their baths with weak intra-system interactions or weakly coupled systems with strong local interactions can be treated within developed perturbation theory approaches. My research focuses on development of nonequilibirum dual techniques, which are capable to accurately treat situations where perturbative techniques are inapplicable due to absence of a small parameter in the system and which reduce to standard perturbative expansions in the limits of weak coupling or weak intra-system interaction. \tDual techniques rely on ability to solve exactly a simplified reference system. The latter includes exactly all intra-system interactions and accounts approximately for system-bath couplings. Dual techniques (described below) allow to account for deviation from the correct hybridization function in a organized and controlled manner of superperturbative expansion. Here, I use Markovian Lindbald quantum master equation (QME) as a solver for the reference system. In the thesis I discuss possibility of mapping between true non-Markov dynamics and simpler Markov Lindblad QME (mapping between physical and reference systems), numerical methods for the mapping and for solution of the resulting Lindblad QME, and application of the mapping in novel dual formulations: auxiliary-QME dual fermion and dual boson approaches. \tFirst I analytically prove that the continuous fermionic baths that the impurity is coupled to can be mapped onto a set of dissipative auxiliary modes under Lindblad QME. Based on this mapping, complicated non-Markovian dynamics of the open fermionic system can be treated by a much simpler Markovian Lindblad QME. However, nice mapping generally requires many auxiliary modes, and numerically only a small number of modes can be exactly diagonalized due to exponential increase of the Hilbert space. To perturbatively treat the difference between the true bath and the small-size auxiliary one, we introduce dual-fermion method. It can directly target the steady state and avoid long-time propagation. Besides current-voltage relation, it can also provide spectral functions, indicating itself as a potential impurity solver for DMFT. Furthermore, we generalize dual-fermion method to consider additionally the effects of bosonic environment; hence it is called dual-boson method. It can be used for studying electron-phonon interaction and light-matter interaction in nanojunctions. By comparing with numerically exact results we argue that both methods are very accurate and relatively cheap. \tFinally, matrix product state representation of the Lindblad QME enables us to treat a large number of auxiliary modes, which provide an exponentially improved mapping. Adding counting field to the auxiliary QME, we can compute the generating function of time-dependent full-counting statistics of electron transport, which yields not only electronic current but also higher-order cumulants (e.g., zero-frequency noise).
Quantum impurity models with strong electron correlations, such as the paradigmatic Anderson Impurity Model (AIM), are central to our understanding of a range of physical phenomena including local moment formation, Coulomb blockade and Kondo screening. They describe magnetic atoms and molecules on surfaces, quantum dot circuits, and correlated materials through dynamical mean field theory. The physics of such systems in strongly non-equilibrium conditions is particularly complex and challenging to capture, whereas Gaussian models of free fermions can be easily solved. Here we show that the time-evolving dynamics of the AIM after a quench can be described by a completely non-interacting version of the model, at the expense of coupling to additional static auxiliary degrees of freedom. Starting from the full solution of the quenched AIM using ED and DMRG, we study the properties of this mapping using numerical optimization, and uncover intriguing structure in the auxiliary system. The method allows us to understand interacting non-equilibrium dynamics through the simpler lens of an effective non-interacting system of larger dimension.