In this thesis we study problems relating the the structure and simulation of entangled many body quantum systems, their utility in adiabatic quantum computation, and the influence of the environment in thermalizing the system and degrading the usefulness of quantum dynamics in this setting. We then study a particular strongly coupled many body quantum system in order to better understand when quantum systems do not thermalize in this manner. In chapter 2 of this thesis we study the properties of quantum dynamics restricted to an efficiently representable sub-manifold of quantum states both the finite and infinite chain of spin- 1=2 subsystems. We investigate the trade-off between gains in efficiency due to this restriction against losses in fidelity. We find the integration to be very stable and shows significant gains in efficiency compared to the naively related matrix product states. However, much of this advantage is offset by a significant reduction in fidelity. We investigate the effect of explicit symmetry breaking in the ansatz and formulate the principles for determining when correlator product states may be a useful tool. We find that scaling with overlap/bond order may be more stable with correlator product states allowing a more efficient extraction of critical exponents and present an example in which the use of correlator product states is orders of magnitude quicker than matrix product states. In chapters 3, 4 and 5 we extend this picture to allow for the study of the dissipative and decohering dynamics of a quantum system interacting with a bath, and pay particular reference to its effect on adiabatic quantum computation. In chapter 3 we consider a system of mutually interacting superconducting flux qubits coupled to a thermal bath that generalises the dissipative model of Landau-Lifschitz-Gilbert to the case of anisotropic bath couplings. We show that the dissipation acts to bias the quantum trajectories towards a reduced phase space. We study the model in the context of the D-Wave computing device and recover dynamics closely related to several models proposed on phenomenological grounds. In chapter 4 we extend this analysis to study explicitly the influence of dissipative dynamics on the lifetime of entanglement. In chapter 5 we apply this understanding to develop a methodology for benchmarking the quantum correlations harnessed by an adiabatic computation and apply this process to the D-Wave Vesuvius machine. Further developing this interest in the effect of thermalisation of quantum dynamics in chapter 6 we consider systems which fail to thermalise even in the presence of strong coupling to their surroundings. This many body localised behaviour has been recently established to be a robust phase of matter in the presence of strong disorder in one dimension. Here we show the the low lying energy states of a many body system contain immobile excitations, this immobility results in an transition in the character of low lying eigenstates at arbitrarily weak disorder. This represents a novel appearance of localising behaviour in many body systems. Finally we consider possible avenues for future work stemming from this thesis.
In this thesis we study problems relating the the structure and simulation of entangled many body quantum systems, their utility in adiabatic quantum computation, and the influence of the environment in thermalizing the system and degrading the usefulness of quantum dynamics in this setting. We then study a particular strongly coupled many body quantum system in order to better understand when quantum systems do not thermalize in this manner. In chapter 2 of this thesis we study the properties of quantum dynamics restricted to an efficiently representable sub-manifold of quantum states both the finite and infinite chain of spin- 1=2 subsystems. We investigate the trade-off between gains in efficiency due to this restriction against losses in fidelity. We find the integration to be very stable and shows significant gains in efficiency compared to the naively related matrix product states. However, much of this advantage is offset by a significant reduction in fidelity. We investigate the effect of explicit symmetry breaking in the ansatz and formulate the principles for determining when correlator product states may be a useful tool. We find that scaling with overlap/bond order may be more stable with correlator product states allowing a more efficient extraction of critical exponents and present an example in which the use of correlator product states is orders of magnitude quicker than matrix product states. In chapters 3, 4 and 5 we extend this picture to allow for the study of the dissipative and decohering dynamics of a quantum system interacting with a bath, and pay particular reference to its effect on adiabatic quantum computation. In chapter 3 we consider a system of mutually interacting superconducting flux qubits coupled to a thermal bath that generalises the dissipative model of Landau-Lifschitz-Gilbert to the case of anisotropic bath couplings. We show that the dissipation acts to bias the quantum trajectories towards a reduced phase space. We study the model in the context of the D-Wave computing device and recover dynamics closely related to several models proposed on phenomenological grounds. In chapter 4 we extend this analysis to study explicitly the influence of dissipative dynamics on the lifetime of entanglement. In chapter 5 we apply this understanding to develop a methodology for benchmarking the quantum correlations harnessed by an adiabatic computation and apply this process to the D-Wave Vesuvius machine. Further developing this interest in the effect of thermalisation of quantum dynamics in chapter 6 we consider systems which fail to thermalise even in the presence of strong coupling to their surroundings. This many body localised behaviour has been recently established to be a robust phase of matter in the presence of strong disorder in one dimension. Here we show the the low lying energy states of a many body system contain immobile excitations, this immobility results in an transition in the character of low lying eigenstates at arbitrarily weak disorder. This represents a novel appearance of localising behaviour in many body systems. Finally we consider possible avenues for future work stemming from this thesis.
The thesis presents research related to the dynamics of quantum systems, both isolated and in the presence of interactions with their environment. Generally, I employ matrix product state (MPS) techniques to explore quantum dynamics in open and closed systems. In the first part I present a study of quantum chaos and how mapping to a MPS variational manifold allow the use of techniques developed in the study of classical many-body systems. Using code developed for this project the Lyapunov spectrum is extracted to give an alternative perspective on eigenstate thermalization, pre-thermalization and integrability. In the second part, I present a novel combination of MPS methods with a Langevin description of the open system. I use this to show how coupling to the environment restricts the growth of entanglement. The consequences of this are relevant for simulations of open quantum systems and their use in in quantum technologies. Finally I discuss applications of these ideas to quantum search. I consider adiabatic and quantum walk algorithms for optimal scaling quantum search algorithms, and hybridizations between the two. The robustness of the different underlying physical mechanisms is investigated in a simple infinite-temperature model, and in a low-temperature limit using the MPS Langevin equation.
This thesis treats the classical simulation of strongly-interacting many-body quantum-mechanical systems in more than one dimension using matrix product states and the more general tensor product states. Contrary to classical systems, quantum many-body systems possess an exponentially larger number of degrees of freedom, thereby significantly complicating their numerical treatment on a classical computer. For this thesis two different representations of quantum many-body states were employed. The first, the so-called matrix product states (MPS) form the basis for the extremely successful density matrix renormalization group (DMRG) algorithm. While originally conceived for one-dimensional systems, MPS are in principle capable of describing arbitrary quantum many-body states. Using concepts from quantum information theory it is possible to show that MPS provide a representation of one-dimensional quantum systems that scales polynomially in the number of particles, therefore allowing an efficient simulation of one-dimensional systems on a classical computer. One of the key results of this thesis is that MPS representations are indeed efficient enough to describe even large systems in two dimensions, thereby enabling the simulation of such systems using DMRG. As a demonstration of the power of the DMRG algorithm, it is applied to the Heisenberg antiferromagnet with spin S=1/2 on the kagome lattice. This model's ground state has long been under debate, with proposals ranging from static spin configurations to so-called quantum spin liquids, states where quantum fluctuations destroy conventional order and give rise to exotic quantum orders. Using a fully SU(2)-symmetric implementation allowed us to handle the exponential growth of entanglement and to perform a large-scale study of this system, finding the ground state for cylinders of up to 700 sites. Despite employing a one-dimensional algorithm for a two-dimensional system, we were able to compute the spin gap (i.e. the energy gap to the first spinful excitation) and study the ground state properties, such as the decay of correlation functions, the static spin structure factors, and the structure and distribution of the nearest-neighbor spin-spin correlations. Additionally, by applying a new tool from quantum information theory, the topological entanglement entropy, we could also with high confidence demonstrate the ground state of this model to be the elusive gapped Z2 quantum spin liquid with topological order. To complement this study, we also considered the extension of MPS to higher dimensions, known as tensor product states (TPS). We implemented an optimization algorithm exploiting symmetries for this class of states and applied it to the bilinear-biquadratic-bicubic Heisenberg model with spin S=3/2 on the z=3 Bethe lattice. By carefully analyzing the simulation data we were able to determine the presence of both conventional and symmetry-protected topological order in this model, thereby demonstrating the analytically predicted existence of the Haldane phase in higher dimensions within an extended region of the phase diagram. Key properties of this symmetry-protected topological order include a doubling of the levels in the entanglement spectrum and the presence of edge spins, both of which were confirmed in our simulations. This finding simultaneously validated the applicability of the novel TPS algorithms to the search for exotic order.
This thesis is focused on many-body localization (MBL) and the development of algorithms using the tensor networks representation for many-body localized systems. Chapter 1 is a detailed discussion on the MBL phenomenon. Chapter 2 gives an introduction to the numerical techniques that are frequently used in studying many-body quantum systems. Chapter 3 studies the shift-and-invert matrix product state method for obtaining highly excited states of MBL systems. Chapter 4 studies an efficient full diagonalization method using Wegner unitary gates. Chapter 5 studies the distribution of the entanglement entropy in the MBL-ergodic transition region. Chapter 6 studies the the influence of pseudo-spin SU (2) symmetry on the MBL phase. Chapter 7 is a side project that studies the influence of the duality-twisted boundary conditions on the critical twist defect chains. The MBL-ergodic transition is a dynamical phase transition which happens at finite energy density for a disordered and isolated many-body interacting system. It is a natural extension of Anderson localization where the localized phase survives the inter-particle interaction. In Chapter 1, we describe its relation to quantum thermalization, typical lattice models that harbor a MBL phase, MBL system’s phenomenological description, the characteristics of the MBL phase, experimental studies of MBL phenomenon in 1D and higher dimensions, the latest developments and the open questions on the nature of the MBL-ergodic transition. In Chapter 2, we provide a short but self-contained discussion of basic numerical methods that are frequently used in studying MBL physics. We mainly focus on the exact diagonalization methods and the matrix product state (MPS) representation, and explain their advantages and disadvantages. In Chapter 3, because obtaining the interior eigenstates of MBL Hamiltonians is a key step to studying the MBL phases and the transitions, we develop algorithms that can capture individual excited states to high fidelity. Using the fact that eigenstates of many-body localized systems have area-law entanglement and can therefore be efficiently represented as a matrix product state (MPS), we designed two algorithms that can generate excited states in the MPS representation, and use them to test the basic properties of MBL in the regime of large one-dimensional random field Heisenberg chains that were previously inaccessible due to the limitations of exact diagonalization (ED). In Chapter 4, we investigate an efficient way to diagonlize a fully many-body localized (FMBL) Hamiltonian. Using the Wegner flow technique for subsystems, we constructively build unitary gates that can greedily transform a local part of the FMBL Hamiltonian with the aim of lowering its average energy variance. Applying the Wegner unitary successively can make the system “flow” towards the diagonalized form. We find that the performance of this method is mainly controlled by the length of each individual Wegner unitary gate, and gets better with increasing length of the gate. We compare the performance of this constructive method with previous methods based on quasi-Newtonian optimization methods, and find that they have similar performance, while our constructive method is far more efficient. We also comment the possibility of constructing a tensor network state that resembles a multi-scale entanglement renormalization ansatz (MERA) using this constructive method. In Chapter 5, using strong subadditivity (SSA) theorem, we develop two order parameters – cut-averaged entanglement entropy (CAEE) and its slope (SCAEE), and show that they can be used to directly identify the volume or area law scaling in single eigenstates. We study the distribution of the SCAEE over disorder realizations. This distribution appears Gaussian at weak disorder, while on the MBL side, the distribution of the SCAEE is peaked at zero slope and has an exponential tail. In the critical regime we find that the distribution of the SCAEE is bimodal both over multiple disorder realizations as well as for single disorder realizations. The variance of the SCAEE distribution in the transition region seems to grow with system size when considered over disorder realizations. Our system sizes are too small to pin down its maximal value, but they are consistent with (among other possibilities) the maximal variance possible which would lead to half the states having zero SCAEE and half having maximal SCAEE. This scenario would lead to an entanglement entropy at the transition which scales as a volume law with half its thermal value. In Chapter 6, we consider the strongly disordered one-dimensional Hubbard model with spin disorder, which, as we argue, cannot be categorized as a conventional MBL phase. Though the spin rotation SU (2) symmetry is broken by the spin disorder, this model still preserves the pseudo-spin SU (2) symmetry under periodic boundary conditions. On the theoretical side, using the pseudo-spin algebra we show that a significant number of the excited states at any disorder strength have logarithmic correction to their von Neumann entanglement entropy, which violates the area-law entanglement for a typical MBL phase. A common feature of this group of excited states is that they usually have far more double occupancies than single occupancies. On the numerical side, we studied the time evolution after quantum quench from two extreme cases of product states – all single occupancies at quarter filling, and all double occupancies at half filling. We find in the former case entanglement entropy behaves as in a typical MBL system, while the latter is clearly delocalized, which suggests the existence of incomplete set of local integrals of motion. With the above evidence, it is convincing that this model provides a playground for studying a non-ergodic, non-MBL phase, and its relation to continuous SU (2) symmetry. Chapter 7 deviates from the common scene of MBL physics. In this chapter, we consider a series of one-dimensional critical twist defect models with various boundary conditions. These twist defects are embedded in the background of a generalization of Kitaev Z 2 toric code model, and can also be understood as a discrete Z k gauge theory in its deconfined phase. We analytically construct the duality transformation of these critical models, and use numerical methods to extract information on the central charges, sound velocities, and conformal dimensions. With the numerical data, we show the relation between the CFT contents of the critical chains with periodic boundary conditions and the critical chains with duality-twisted boundary conditions.
Experimental advances have made it possible to realize and control quantum many-body systems, allowing the experimental study of non-equilibrium phenomena. Due to the Hilbert space growing exponentially with the number of particles, advanced numerical techniques are usually required to understand many-body effects at the theoretical level. In this thesis, we address the question of how to efficiently simulate the non-equilibrium dynamics of many-body systems using matrix product states (MPS). In particular, we consider quantum systems in interaction with macroscopic environments, as they appear in Anderson impurity problems, quantum thermodynamics, or open quantum systems. We develop a low-entanglement representation of the environment with only short-range interactions, perfectly suitable for simulations with MPS. We further show that an interleaved ordering of tight-binding chains can significantly reduce the creation of entanglement, as opposed to an intuitive implementation of the Hamiltonian's geometry. Our approach allows long-time simulations and an analysis of the environmental dynamics, in which, after a sudden quench, we find clear signatures of many-body effects. We employ our techniques to compute spectral functions of impurity models, with possible applications to dynamical mean-field theory impurity solvers, and to calculate dissipation in the non-equilibrium Anderson model with explicit time-dependence, as relevant for ongoing experiments with oscillating tip atomic force microscopes.
From the motion of hurricanes to the cooling of a cup of tea, dissipative dynamics are ubiquitous in nature. Unfortunately, when studying lattice-based strongly correlated quantum systems, a common focus of experiments and highly relevant for material science, the curse of dimensionality prevents us from directly solving the exact equations describing the dissipative motion. We instead often use approximations and numerical analysis, as few analytical techniques currently exist which can provide useful insights. This thesis studies the constraints that non-abelian algebraic structures place on the dynamics of such non-equilibrium many-body quantum systems. In doing so, we develop novel algebraic methods that provide previously absent analytic insight into a range of problems. We apply these new techniques to study dynamical phases of quantum matter that are of both theoretical and experimental interest. We study how simple algebraic structures imply persistent non-stationarity in dissipative many-body quantum systems. These results enable us to develop a theory of quantum synchronisation based on symmetries and to explore a novel class of time crystals where order is induced by noise. These dynamical phases of matter have been gaining recent interest for their insights into thermalisation and their potential application in quantum technologies. We therefore use our results to explore experimentally realisable models. We also use the general principles of prethermalisation to propose a new theory to potentially explain recent experiments studying light-induced superconductivity in terms of eta-pairing and approximate symmetries. We discuss the merits and weaknesses of our proposal in relation to the experimental evidence. We also indicate techniques that, within our theory, stabilise superconducting states by harnessing the quantum Zeno effect For certain systems, known as integrable, there are enough symmetries to allow for exact solutions using the Bethe ansatz. We demonstrate that Bethe ansatz techniques can be extended and applied to non-equilibrium problems. As an example, we use this method to analytically study the dynamical phase transitions of an XXZ spin chain with boundary loss.
The main topic of this thesis is the study of many-body effects in strongly correlated one- or quasi one-dimensional condensed matter systems. These systems are characterized by strong quantum and thermal fluctuations, which make mean-field methods fail and demand for a fully numerical approach. Fortunately, a numerical method exist, which allows to treat unusually large one -dimensional system at very high precision. This method is the density-matrix renormalization group method (DMRG), introduced by Steve White in 1992. Originally limited to the study of static problems, time-dependent DMRG has been developed allowing one to investigate non-equilibrium phenomena in quantum mechanics. In this thesis I present the solution of three conceptionally different problems, which have been addressed using mostly the Krylov-subspace version of the time-dependent DMRG. My findings are directly relevant to recent experiments with ultracold atoms, also carried out at LMU in the group of Prof. Bloch. The first project aims the ultimate goal of atoms in optical lattices, namely, the possibility to act as a quantum simulator of more complicated condensed matter system. The underline idea is to simulate a magnetic model using ultracold bosonic atoms of two different hyperfine states in an optical superlattice. The system, which is captured by a two-species Bose-Hubbard model, realizes in a certain parameter range the physics of a spin-1/2 Heisenberg chain, where the spin exchange constant is given by second order processes. Tuning of the superlattice parameters allows one to controlling the effect of fast first order processes versus the slower second order ones. The analysis is motivated by recent experiments, %\cite{Folling2007,Trotzky2008} where coherent two-particle dynamics with ultracold bosonic atoms in isolated double wells were detected. My project investigates the coherent many-particle dynamics, which takes place after coupling the double well. I provide the theoretical background for the next step, the observation of coherent many-particle dynamics after coupling the double wells. The tunability between the Bose-Hubbard model and the Heisenberg model in this setup could be used to study experimentally the differences in equilibration processes for non-integrable and Bethe ansatz integrable models. It turns out that the relaxation in the Heisenberg model is connected to a phase averaging effect, which is in contrast to the typical scattering driven thermalization in nonintegrable models In the second project I study a many-body generalization of the original Landau-Zener formula. This formula gives the transition probability between the two states of a quantum mechanical two-level system, where the offset between the two levels is varying linearly in time. In a recent experiment this framework has been extended to a many-body system consisting of pairwise tunnel-coupled one-dimensional Bose liquids. It was found that the tunnel coupling between the tubes and the intertube interactions strongly modify the original Landau-Zener picture. After a introduction to the two-level and the three-level Landau-Zener problem I present my own results for the quantum dynamics of the microscopic model and the comparison to the experimental results. I have calculated both Landau-Zener sweeps as well as the time-evolution after sudden quenches of the energy offset. A major finding is that for sufficiently large initial density quenches can be efficiently used to create quasi-thermal states of arbitrary temperatures. The third project is more mathematical and connects the fields of quantum computation and of quantum information. Here, the main purpose is to analyse systematically the effects of decoherence on maximally entangled multi-partite states, which arise typically during quantum computation processes. The bigger the number of entangled qubits the more fragile is its entanglement under the influence decoherence. As starting point I consider first two entangled qubits, whereby one qubit interacts with an arbitrary environment. For this particular case I have derived a factorization law for the disentanglement. Next, I calculate the decrease of entanglement of two , three and four entangled qubits, general W- and general GHZ-state, coupled to a global spin-1/2 bath or several independent spin-1/2 baths , one for each qubit. Although there is no appropriate entanglement measure for three and more qubits, it turns out that this decrease is directly related to the increase of entanglement between the central system and the bath. This implies the formation of a much bigger multipartite entangled network. Thus, using the von Neumann entropy and the Wootters concurrence, I derive a simple upper bound for the bath-induced entanglement breaking power of the initially maximally entangled multi-partite states.