The saddle points of a conventional Feynman path integral are not entangled, since they comprise a sequence of classical field configurations. We combine insights from field theory and tensor networks by constructing a Feynman path integral over a sequence of matrix product states. The paths that dominate this path integral include some degree of entanglement. This new feature allows several insights and applications: i. A Ginzburg-Landau description of deconfined phase transitions. ii. The emergence of new classical collective variables in states that are not adiabatically continuous with product states. iii. Features that are captured in product-state field theories by proliferation of instantons are encoded in perturbative fluctuations about entangled saddles. We develop a general formalism for such path integrals and a couple of simple examples to illustrate their utility.
The saddle points of a conventional Feynman path integral are not entangled, since they comprise a sequence of classical field configurations. We combine insights from field theory and tensor networks by constructing a Feynman path integral over a sequence of matrix product states. The paths that dominate this path integral include some degree of entanglement. This new feature allows several insights and applications: i. A Ginzburg-Landau description of deconfined phase transitions. ii. The emergence of new classical collective variables in states that are not adiabatically continuous with product states. iii. Features that are captured in product-state field theories by proliferation of instantons are encoded in perturbative fluctuations about entangled saddles. We develop a general formalism for such path integrals and a couple of simple examples to illustrate their utility.
One of the major ways in which quantum mechanics differs from classical mechanics is the existence of special quantum correlations - entanglement. Typical quantum states are highly entangled, making them complex and inefficient to represent. Physically interesting states are unusual, they are only weakly entangled. By restricting ourselves to weak entanglement, efficient representations of quantum states can be found. A tensor network is constructed by taking objects called tensors that encode spatially local information and gluing them together to create a large network that describes a complex quantum state. The manner in which the tensors are connected defines the entanglement structure of the quantum state. Tensors networks are therefore a natural framework for describing physical behaviour of complex quantum systems. In this thesis we utilize tensor networks to solve a number of interesting problems. Firstly, we study a Feynman path integral written over tensor network states. As a sum over classical trajectories, a Feynman path integral can struggle to capture entanglement. Combining the path integral with tensor networks overcomes this.We consider the effect of quadratic fluctuations on the tensor network path integral and calculate corrections to observables numerically and analytically. We also study the time evolution of complex quantum systems. By projecting quantum dynamics onto a classical phase space defined using tensor networks, we relate thermal behaviour of quantum systems to classical chaos. In doing so we demonstrate a relationship between entanglement growth and chaos. By studying the dynamics of coupled quantum chains we also gain insight into how quantum correlations spread over time. As noted, tensor networks are remarkably efficient. In the final section of this thesis we use tensor networks to create compressed machine learning algorithms. Their efficiency means that tensor networks can use 50 times fewer parameters with no significant decrease in performance.
We argue that the natural way to generalise a tensor network variational class to a continuous quantum system is to use the Feynman path integral to implement a continuous tensor contraction. This approach is illustrated for the case of a recently introduced class of quantum field states known as continuous matrix-product states (cMPS). As an example of the utility of the path-integral representation we argue that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. An argument that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity is also provided.
Tensor networks have historically proven to be of great utility in providing compressed representations of wave functions that can be used for the calculation of eigenstates. Recently, it has been shown that a variety of these networks can be leveraged to make real time non-equilibrium simulations of dynamics involving the Feynman-Vernon influence functional more efficient. In this work, a tensor network is developed for non-perturbatively calculating the equilibrium correlation function for open quantum systems using the path integral methodology. These correlation functions are of fundamental importance in calculations of rates of reactions, simulations of response functions and susceptibilities, spectra of systems, etc. The influence of the solvent on the quantum system is incorporated through an influence functional, whose unconventional structure motivates the design of a new optimal matrix product-like operator that can be applied to the so-called path amplitude matrix product state. This complex-time tensor network path integral approach provides an exceptionally efficient representation of the path integral, enabling simulations for larger systems strongly interacting with baths and at lower temperatures out to longer time. The derivation, design, and implementation of this method are discussed along with a wide range of illustrations ranging from rate theory and symmetrized spin correlation functions to simulation of response of the Fenna-Matthews-Olson complex to light.
A natural way to generalize tensor network variational classes to quantum field systems is via a continuous tensor contraction. This approach is first illustrated for the class of quantum field states known as continuous matrix-product states (cMPS). As a simple example of the path-integral representation we show that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. A completeness argument is also provided that shows that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity. Beyond this, we obtain a well-behaved field limit of projected entangled-pair states (PEPS) in two dimensions that provide an abstract class of quantum field states with natural symmetries. We demonstrate how symmetries of the physical field state are encoded within the dynamics of an auxiliary field system of one dimension less. In particular, the imposition of Euclidean symmetries on the physical system requires that the auxiliary system involved in the class' definition must be Lorentz-invariant. The physical field states automatically inherit entropy area laws from the PEPS class, and are fully described by the dissipative dynamics of a lower dimensional virtual field system. Our results lie at the intersection many-body physics, quantum field theory and quantum information theory, and facilitate future exchanges of ideas and insights between these disciplines.
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics, and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications. As well as the canonical examples of a quantum particle in a single and a double potential well, we discuss the generalization of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we will temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of functional field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various techniques and concepts of field theory, namely stationary phase analyses of functional integrals, the Euclidean formulation of field theory, instanton techniques, and the role of topological concepts in field theory, will be motivated and introduced in this chapter.
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics, and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications. As well as the canonical examples of a quantum particle in a single and a double potential well, we discuss the generalization of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various concepts of field theory, namely stationary phase analysis, the Euclidean formulation of field theory, instanton techniques, and the role of topology in field theory, are introduced in this chapter. The path integral: general formalism Broadly speaking, there are two approaches to the formulation of quantum mechanics: the “operator approach” based on the canonical quantization of physical observables and the associated operator algebra, and the Feynman path integral.