By formulating the renormalization group as a quantum channel acting on density matrices in Quantum Field Theories (QFTs), we show that ground-state expectation values of observables supported on slow momentum modes can be approximated by their averages on the fixed-point theories to which the QFTs flow. This is done by studying the fidelity between ground states of different QFTs and arriving at certain hyperscaling relations satisfied at criticality. Our results allow for a clear identification of cases in which one can replace a QFT by its scale-invariant limit in the calculation of expectation values, opening the way for a range of applications, including the improvement of numerical and analytical methods used to tackle the costly computer simulation of critical models.
By formulating the renormalization group as a quantum channel acting on density matrices in Quantum Field Theories (QFTs), we show that ground-state expectation values of observables supported on slow momentum modes can be approximated by their averages on the fixed-point theories to which the QFTs flow. This is done by studying the fidelity between ground states of different QFTs and arriving at certain hyperscaling relations satisfied at criticality. Our results allow for a clear identification of cases in which one can replace a QFT by its scale-invariant limit in the calculation of expectation values, opening the way for a range of applications, including the improvement of numerical and analytical methods used to tackle the costly computer simulation of critical models.
We show that the Wilsonian formulation of the renormalization group (RG) defines a quantum channel acting on the momentum-space density matrices of a quantum field theory. This information theoretical property of the RG allows us to derive a remarkable consequence for the vacuum of theories at a fixed point: they have no entanglement between momentum scales. Our result can be understood as deriving from the scale symmetry of such theories and leads to constraints on the form of the ground state and on expectation values of momentum-space operators.
In this Thesis, the problem of a quantum quench in quantum field theories is analyzed. This involves studying the real time evolution of observables in a theory that undergoes a change in one of its couplings. These quenches are then characterized by two parameters: deltalambda, the magnitude of the quench and most importantly, deltat, the quench duration. In contrast to previous studies of abrupt quenches in the condensed matter theory community, we will be interested in smooth quenches with a finite deltat.\nMotivated by existing results in holographic theories, we studied the problem of a fast smooth quench in free field theories by quenching the mass of a free scalar and a free Dirac fermion. For specific mass profiles, exact analytic answers were found. We found that expectation values for the quenched operators obey universal scaling properties. We provide both numerical and analytic evidence for this scaling to hold in free field theories and argue that the same scaling properties should hold independently of the theory, the coupling or the quench profile. In fact, we show that the renormalized expectation value of an operator ${\\cal{O}}_\\Delta$ of dimension Delta should scale as $\\vev{{\\cal{O}}_\\Delta}_{ren} \\sim \\delta \\lambda/(\\delta t)^{2\\Delta -d}$, when the quench rate is fast compared to the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions.\nThis result suggests that, for Delta>d/2, expectation values will diverge in the limit of deltatto0, which seem to contradict previous studies of abrupt quenches. In this Thesis, we carefully analzye the relation between the two approaches and establish restrictions to the kind of objects that can be studied using the abrupt approximation.\nApart from the fast smooth quench, another regime of interest is the slow quench through a critical point. In free field theories we found that adiabatic behaviour breaks down when the system is close enough to the critical point and renormalized expectation values scale different, following expectations from the Kibble-Zurek argument. Given any finite fixed time, we are able to show how expectation values scale at any value of the quench duration deltat.
We analyze the effect of decoherence, modeled by local quantum channels, on quantum critical states and we find universal properties of the resulting mixed state's entanglement, both between system and environment and within the system. Renyi entropies exhibit volume law scaling with a subleading constant governed by a "g function" in conformal field theory, allowing us to define a notion of renormalization group (RG) flow (or "phase transitions") between quantum channels. We also find that the entropy of a subsystem in the decohered state has a subleading logarithmic scaling with subsystem size, and we relate it to correlation functions of boundary condition changing operators in the conformal field theory. Finally, we find that the subsystem entanglement negativity, a measure of quantum correlations within mixed states, can exhibit log scaling or area law based on the RG flow. When the channel corresponds to a marginal perturbation, the coefficient of the log scaling can change continuously with decoherence strength. We illustrate all these possibilities for the critical ground state of the transverse-field Ising model, in which we identify four RG fixed points of dephasing channels and verify the RG flow numerically. Our results are relevant to quantum critical states realized on noisy quantum simulators, in which our predicted entanglement scaling can be probed via shadow tomography methods.
The study of the ground states of local Hamiltonians in quantum many-body and quantum field theoretic systems is a source of many research problems of great complexity. Almost in its entirety, this thesis deals with constructions related to the field of tensor networks, which are efficient parametrizations of mathematical objects from quantum many-body theory (such as quantum states and operators) that exploit our knowledge of their entanglement structures in order to obtain improvements in our computational abilities. In the past decade, the new research program of continuous tensor networks has arisen with the goal of reproducing, in the setting of quantum field theory, the success that tensor network techniques have enjoyed on the lattice. First, we present a number of results from research performed on non-interacting examples of the continuous Multiscale Entanglement Renormalization Ansatz, or cMERA, a continuous tensor network that provides a variational wavefunctional for a quantum field theoretic ground state. Initially, we study the correlations and entanglement structure of cMERA states, providing evidence that such states can be interpreted as UV regularized versions of the physical states they approximate. After that, we study what modifications of the formalism are necessary in order to consider systems with gauge symmetry, or in the presence of boundaries and defects. We obtain prescriptions for the treatment of these cases which, we expect, may also hold in the interacting setting. We then abandon briefly the area of continuous tensor networks to present a piece of research within a different though related research program: that of the study of manifestations of universal emergent behaviour in critical lattice systems. In particular we bring attention to the existence of an approximate representation of the Virasoro algebra supported on the eigenvectors of the reduced density matrices of critical lattice systems. We call this the entanglement algebra, and study its accuracy in the particular case of the Ising model. Lastly, we present a proposal for a continuous version of the Tensor Network Renormalization (TNR) algorithm, which we dub continuous TNR, or cTNR. cTNR operates on UV regularized classical partition functions or quantum Euclidean path integrals, and generates a real-space renormalization group flow via continuous coarse-graining and rescaling operations. We show that a UV regularized version of the free boson path integral can be made a fixed point of such a renormalization flow, in a way that allows for recovery of the conformal data associated to the free boson conformal field theory.
This thesis aims to establish the study of a somewhat novel form of entanglement in field-theoretical quantum systems: that between degrees of freedom in momentum space. Although scattered results can be found in the literature, with some even going back decades, this work, based on the papers published during the PhD, provides a systematic investigation of the matter and arrive at non-perturbative results, shedding light on unexplored structures present in Quantum Field Theories (QFTs) and their ground states. This line of research is motivated by the inherent connection between the notion of momentum modes of fields and the Wilsonian formulation of the Renormalization Group (RG), a cornerstone of the modern understanding of field theories and phases of matter which is naturally defined by the partition of the degrees of freedom into “fast' and “slow' modes, according to their momenta. Furthermore, by glancing at the achievements of techniques such as the Density Matrix Renormalization Group and Tensor Networks, which are based on ideas from renormalization in real space, there is reason to believe that similarly efficient methods may be derived by connecting the fundaments of the RG with quantum information in momentum-space. With this in mind, the thesis begins by introducing the main notions of information theory and renormalization used throughout, with the inclusion of a brief discussion on the use of von Neumann algebras in quantum physics, a formalism which is not directly applied here, but nevertheless provides a solid basis for many intuitions on the nature of entanglement. Afterwards, the idea of degrees of freedom in momentum-space is defined and its different subtleties clarified. A key insight, that any field theory possesses multiple definitions of partitioning its Hilbert space into momentum modes, based on a choice of a complete set of local and commuting observables or field variables in a path integral, is then obtained. Given this solid grasp on the fundaments of quantum information in momentum-space, two directions of further investigation are pursued. First, a path-integral method is developed for the calculation of the associated entanglement entropies in ground states and applied perturbatively to several models, which also allows for the quantitative confirmation of certain predictions. Secondly, the Wilsonian renormalization procedure is translated to this formalism and is shown to be equivalent to a so-called quantum channel which acts on momentum-space. While an interesting result on its own right, this also leads directly to a powerful constraint on the structure of ground states of renormalization group fixed points: they must be separable, and thus uncorrelated, when the degrees of freedom are partitioned based on their momentum scale. This opens the way for further analytical and numerical studies of these special QFTs beyond the scope of this thesis. Finally, the concept of fidelity of quantum states is invoked in order to better understand the information aspects of the renormalization group flow for theories in the critical manifold of a given fixed point. It is used to quantify the idea that such QFTs are indistinguishable from the fixed point itself when long-distance observables are measured and new “critical exponents' are introduced in this context, providing tight bounds on the differences of expectation values of operators in the critical and fixed-point theories. The results obtained in this work mark only the beginning of the in-depth research on different structures present in the momentum-space of Quantum Field Theories and may, hopefully, serve as a basis for further developments.:Contents Acknowledgements iii 1 Introduction 1 1.1 Notions of quantum information 5 1.2 Comments on von Neumann algebras 10 1.3 Basics of Wilsonian renormalization 14 2 Defining momentum-space entanglement 18 2.1 Momentum modes from the path integral 19 2.2 Non-uniqueness of partitions 24 3 The replica method for non-local actions 33 3.1 Construction of the replica trick 34 3.2 Entanglement between momentum scales in perturbation theory 38 3.2.1 Perturbation theory and the replica trick 39 3.2.2 Coupled Oscillators 40 3.2.3 Perturbative interacting QFT 42 3.2.4 Sine-Gordon and Thirring models 48 4 Wilsonian RG as a quantum channel 53 4.1 Quantum operations from the RG steps 54 4.2 Separability of ground states at fixed points 59 4.3 Further implications 63 4.3.1 The RG is a quantum dynamical semigroup 64 4.3.2 First-order phase transitions 67 5 Critical theories and operator estimations 74 5.1 Quantum state distinguishability 75 5.2 Fidelity in translation-invariant systems 77 5.3 Hyperscaling in the critical manifold 80 6 Conclusions and further directions 86 Bibliography 88
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.