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arXiv:1108.3568 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Momentum-space entanglement and renormalization in quantum field theory

Vijay Balasubramanian, Michael B. McDermott, Mark Van Raamsdonk B

Published Aug 6, 2012• 172 citations
arXivHTMLPDF

Abstract

The degrees of freedom of any interacting quantum field theory are entangled in momentum space. Thus, in the vacuum state, the infrared degrees of freedom are described by a density matrix with an entanglement entropy. We derive a relation between this density matrix and the Wilsonian effective action obtained by integrating out degrees of freedom with spatial momentum above some scale. We argue that the entanglement entropy of and mutual information between subsets of field theoretic degrees of freedom at different momentum scales are natural observables in quantum field theory and demonstrate how to compute these in perturbation theory. The results may be understood heuristically based on the scale dependence of the coupling strength and number of degrees of freedom. We measure the rate at which entanglement between degrees of freedom declines as their scales separate and suggest that this decay is related to the property of decoupling in quantum field theory.

DOI
10.1103/physrevd.86.045014

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The degrees of freedom of any interacting quantum field theory are entangled in momentum space. Thus, in the vacuum state, the infrared degrees of freedom are described by a density matrix with an entanglement entropy. We derive a relation between this density matrix and the Wilsonian effective action obtained by integrating out degrees of freedom with spatial momentum above some scale. We argue that the entanglement entropy of and mutual information between subsets of field theoretic degrees of freedom at different momentum scales are natural observables in quantum field theory and demonstrate how to compute these in perturbation theory. The results may be understood heuristically based on the scale dependence of the coupling strength and number of degrees of freedom. We measure the rate at which entanglement between degrees of freedom declines as their scales separate and suggest that this decay is related to the property of decoupling in quantum 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
The entanglement between momentum modes of a quantum field theory at different scales is not as well studied as its counterpart in real space, despite the natural connection with the Wilsonian idea of integrating out the high-momentum degrees of freedom. Here, we push such a connection further by developing a novel method to calculate the R\'enyi and entanglement entropies between slow and fast modes, which is based on the Wilsonian effective action at a given scale. This procedure is applied to the perturbative regime of some scalar theories, comparing the lowest-order results with those from the literature and interpreting them in terms of Feynman diagrams. This method is easily generalized to higher-order or nonperturbative calculations. It has the advantage of avoiding matrix diagonalizations of other techniques.
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.
Abstract Quantum field theory defined on a noncommutative space is a useful toy model of quantum gravity and is known to have several intriguing properties, such as nonlocality and UV/IR mixing. They suggest novel types of correlation among the degrees of freedom of different energy scales. In this paper, we investigate such correlations by the use of entanglement entropy in the momentum space. We explicitly evaluate the entanglement entropy of scalar field theory on a fuzzy sphere and find that it exhibits different behaviors from that on the usual continuous sphere. We argue that these differences would originate in different characteristics; non-planar contributions and matrix regularizations. It is also found that the mutual information between the low and the high momentum modes shows different scaling behaviors when the effect of a cutoff becomes important.
In the first part of this thesis we explore the entanglement structure of relativistic field theories in momentum space. We discuss a Wilsonian path integral formulation and a perturbative approach. Using perturbation theory we obtain results for specific quantum field theories. These are understood through scaling and decoupling properties of field theories. Convergence of the perturbation theory taking loop diagrams into account is also discussed. We then discuss the entanglement structure in systems where Lorentz invariance is broken by a Fermi surface. The Fermi surface helps the convergence of perturbation theory and entanglement of modes near the Fermi surface is shown to be amplified, even in the presence of a large momentum cutoff. In the second part of this thesis we explore the connection between entanglement and gravity in the context of the AdS/CFT correspondence. We show that there are certain thermodynamic-like relations common to all conformal field theories, which when mapped via the AdS/CFT correspondence to the bulk are tantamount to Einstein's equations, to lowest order in the metric.
Integrating out high energy degrees of freedom to yield a low energy effective field theory leads to a loss of information with a concomitant increase in entropy. We obtain the effective field theory of a light scalar field interacting with heavy fields after tracing out the heavy degrees of freedom from the time evolved density matrix. The initial density matrix describes the light field in its ground state and the heavy fields in equilibrium at a common temperature TTT. For T=0T=0T=0, we obtain the reduced density matrix in a perturbative expansion; it reveals an emergent mixed state as a consequence of the entanglement between light and heavy fields. We obtain the effective action that determines the time evolution of the reduced density matrix for the light field in a nonperturbative Dyson resummation of one-loop correlations of the heavy fields. The Von-Neumann entanglement entropy associated with the reduced density matrix is obtained for the nonresonant and resonant cases in the asymptotic long time limit. In the nonresonant case the reduced density matrix displays an incipient thermalization albeit with a wave-vector, time and coupling dependent effective temperature as a consequence of memory of initial conditions. The entanglement entropy is time independent and is the thermal entropy for this effective, nonequilibrium temperature. In the resonant case the light field fully thermalizes with the heavy fields, the reduced density matrix loses memory of the initial conditions and the entanglement entropy becomes the thermal entropy of the light field. We discuss the relation between the entanglement entropy ultraviolet divergences and renormalization.