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.
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.
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
Renormalization group (RG) is a core concept from statistical mechanics to quantum field theory, yet its schemes differ widely across literature. We formulate Wilsonian RG for generic, including finite-dimensional, quantum systems as a quantum channel combining coarse graining with isometric rescaling, whose Kraus representation yields pure conditional trajectories for pure inputs and recovers mixed-state flows upon averaging. We then develop an algebraic extension without assuming tensor-product factorization: the retained low-energy algebra is constructed from the one-particle spectrum, and coarse graining is a conditional expectation onto it. Identifying the channel's Cesàro mean with a resource-destroying map onto fixed states establishes a resource theory of RG and yields a relative-entropy monotone without variational optimization. For thermal states in the zero-temperature limit, the resulting Bogoliubov--Kubo--Mori metric per unit spacetime volume is proportional to the Zamolodchikov metric, directly relating the resource density to c−cIR near a two-dimensional IR fixed point. We also introduce the relative entropy of coarse graining to measure discarded UV information, prove its monotonicity along successive coarse grainings, and evaluate it holographically in global AdS3.
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.
Abstract We study the information content of the reduced density matrix of a region in quantum field theory that cannot be recovered from its subregion density matrices. We reconstruct the density matrix from its subregions using two approaches: scaling maps and recovery maps. The vacuum of a scale-invariant field theory is the fixed point of both transformations. We define the entanglement of scaling and the entanglement of recovery as measures of entanglement that are intrinsic to the continuum limit. Both measures increase monotonically under the renormalization group flow. This provides a unifying information-theoretic structure underlying the different approaches to the renormalization monotones in various dimensions. Our analysis applies to non-relativistic quantum field theories as well the relativistic ones, however, in relativistic case, the entanglement of scaling can diverge.
Abstract Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.
Critical transition points between symmetry-broken phases are usually characterized as fixed points in the renormalization group (RG) theory. We show that, following the standard Wilsonian procedure that iteratively traces out the large-momentum modes, this well-known fact can easily break down in non-Hermitian systems under open boundary conditions. Based on non-Hermitian Su-Schrieffer-Heeger-type models, we propose a RG scheme based on real-space decimation. We provide concrete examples and an analytic proof to show that the real-space scheme always respects the critical points under RG transformations. We investigate the correlation length of order parameters in the vicinity of critical points. Its divergence and scaling behavior remain intact under the real-space decimation, indicating that our method can be successfully applied to study the critical phenomena in non-Hermitian systems. These results suggest that the real-space RG method will find versatile applications in interacting non-Hermitian quantum systems.