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Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation. Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated. Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance. It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood. Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods. Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).
This book is an introduction to a comprehensive and unified dynamic transition theory for dissipative systems and to applications of the theory to a range of problems in the nonlinear sciences. The main objectives of this book are to introduce a general principle of dynamic transitions for dissipative systems, to establish a systematic dynamic transition theory, and to explore the physical implications of applications of the theory to a range of problems in the nonlinear sciences. The basic philosophy of the theory is to search for a complete set of transition states, and the general principle states that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic and random. The audience for this book includes advanced graduate students and researchers in mathematics and physics as well as in other related fields. This second edition introduces a unified theory for topological phase transitions, provides a first-principle approach to statistical and quantum physics, and offers a microscopic mechanism of quantum condensates (Bose-Einstein condensation, superfluidity, and superconductivity). Reviews of first edition: “The goals of this interesting book are to derive a general principle of dynamic transitions for dissipative systems and to establish a systematic dynamic transition theory for a wide range of problems in the nonlinear sciences. ... The intended audience for this book includes students and researchers working on nonlinear problems in physics, meteorology, oceanography, biology, chemistry, and the social sciences.” (Carlo Bianca, Mathematical Reviews, December, 2014) “This is a clearly written book on numerous types of phase transitions taken in a broad sense when a dynamical dissipative system transforms from one physical state into another. ... The book is a very useful literature not only for the professionals in the field of dynamic systems and phase transitions but also for graduate students due to its interdisciplinary coverage and state-of-the-art level.” (Vladimir Čadež, zbMATH, Vol. 1285, 2014)
Quantum systems with many degrees of freedom are inherently difficult to describe and simulate quantitatively. The space of possible states is, in general, exponentially large in the number of degrees of freedom such as the number of particles it contains. Standard digital high-performance computing is generally too weak to capture all the necessary details, such that alternative quantum simulation devices have been proposed as a solution. Artificial neural networks, with their high non-local connectivity between the neuron degrees of freedom, may soon gain importance in simulating static and dynamical behavior of quantum systems. Particularly promising candidates are neuromorphic realizations based on analog electronic circuits which are being developed to capture, e.g., the functioning of biologically relevant networks. In turn, such neuromorphic systems may be used to measure and control real quantum many-body systems online. This thesis lays an important foundation for the realization of quantum simulations by means of neuromorphic hardware, for using quantum physics as an input to classical neural nets and, in turn, for using network results to be fed back to quantum systems. The necessary foundations on both sides, quantum physics and artificial neural networks, are described, providing a valuable reference for researchers from these different communities who need to understand the foundations of both.
Quantum phase transitions (QPTs) offer wonderful examples of the radical macroscopic effects inherent in quantum physics: phase changes between different forms of matter driven by quantum rather than thermal fluctuations, typically at very low temperatures. QPTs provide new insight into outstanding problems such as high-temperature superconductivit
This book constitutes the thoroughly refereed post-proceedings of the 5th International Conference on Large-Scale Scientific Computations, LSSC 2005, held in Sozopol, Bulgaria in June 2005. The 75 revised full papers presented together with five invited papers were carefully reviewed and selected for inclusion in the book. The papers are organized in topical sections.
This thesis is concerned with the numerical study of one-dimensional (1D) spin-1/2 quantum magnets and related method development. Its focus is on the calculation of dynamical spin correlation functions both at zero and finite temperature. This is motivated by the accessibility of dynamical quantities in experiments such as inelastic neutron scattering (INS) and electron spin resonance (ESR). The numerical methods used in this thesis are based on extensions of the density-matrix renormalization group (DMRG) and are formulated in the framework of matrix product states (MPS). While zero-tempe...
This book discusses some scaling properties and characterizes two-phase transitions for chaotic dynamics in nonlinear systems described by mappings. The chaotic dynamics is determined by the unpredictability of the time evolution of two very close initial conditions in the phase space. It yields in an exponential divergence from each other as time passes. The chaotic diffusion is investigated, leading to a scaling invariance, a characteristic of a continuous phase transition. Two different types of transitions are considered in the book. One of them considers a transition from integrability to non-integrability observed in a two-dimensional, nonlinear, and area-preserving mapping, hence a conservative dynamics, in the variables action and angle. The other transition considers too the dynamics given by the use of nonlinear mappings and describes a suppression of the unlimited chaotic diffusion for a dissipative standard mapping and an equivalent transition in the suppression of Fermi acceleration in time-dependent billiards. This book allows the readers to understand some of the applicability of scaling theory to phase transitions and other critical dynamics commonly observed in nonlinear systems. That includes a transition from integrability to non-integrability and a transition from limited to unlimited diffusion, and that may also be applied to diffusion in energy, hence in Fermi acceleration. The latter is a hot topic investigated in billiard dynamics that led to many important publications in the last few years. It is a good reference book for senior- or graduate-level students or researchers in dynamical systems and control engineering, mathematics, physics, mechanical and electrical engineering.
This thesis is structured around three main chapters. In the first chapter I present new results which have been obtained for the out-of-equilibrium critical S\phi^4S-theory. Its dynamics are described by a Langevin equation driven by a colored noise. The temporal correlation of this noise features a power-law decrease which is governed by a certain exponent S\alphaS. It turns out that there exists a crossover S\alpha_cS which depends on the dimension SDS of the system and separates the S(\alpha, D)S-plane into a region where the color of the noise alters the critical behaviour and a region where the color is non relevant. I also discuss the scaling bahaviour of the non equilibrium correlation functions. In the second chapter I introduce a path integral formalism to describe non equilibrium quantum Brownian motion. I present the results which have been obtained during my PhD on the evolution of the non equilibrium correlation functions after a quantum quench. The third part of my thesis focuses on the impurity diffusion in one-dimensional quantum liquids which are commonly called Luttinger liquids. After an introductory part which covers the main issues related to such a system, I present a novel description of the impurity dynamics in the case where an external trapping potential is present. The non-homogeneous density profile of the Luttinger liquid then strongly influences on the impurity dynamics in a fascinating way. Such systems are currently being studied in cold atoms experiments.
Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic way, from the basic definitions to the important applications. This book is also useful to those who apply tensor networks in areas beyond physics, such as machine learning and the big-data analysis. Tensor network originates from the numerical renormalization group approach proposed by K. G. Wilson in 1975. Through a rapid development in the last two decades, tensor network has become a powerful numerical tool that can efficiently simulate a wide range of scientific problems, with particular success in quantum many-body physics. Varieties of tensor network algorithms have been proposed for different problems. However, the connections among different algorithms are not well discussed or reviewed. To fill this gap, this book explains the fundamental concepts and basic ideas that connect and/or unify different strategies of the tensor network contraction algorithms. In addition, some of the recent progresses in dealing with tensor decomposition techniques and quantum simulations are also represented in this book to help the readers to better understand tensor network. This open access book is intended for graduated students, but can also be used as a professional book for researchers in the related fields. To understand most of the contents in the book, only basic knowledge of quantum mechanics and linear algebra is required. In order to fully understand some advanced parts, the reader will need to be familiar with notion of condensed matter physics and quantum information, that however are not necessary to understand the main parts of the book. This book is a good source for non-specialists on quantum physics to understand tensor network algorithms and the related mathematics.