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arXiv:1404.2079 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Zero modes and divergence of entanglement entropy

Krishnanand Mallayya, Rakesh Tibrewala, S. Shankaranarayanan, Τ. Padmanabhan

Published Aug 21, 2014• 18 citations
arXivHTMLPDF

Abstract

We investigate the cause of the divergence of the entanglement entropy for the free scalar fields in $(1+1)$ and $(D+1)$ dimensional space-times. In a canonically equivalent set of variables, we show explicitly that the divergence in the entanglement entropy of the continuum field in $(1+1)$ dimensions is due to the accumulation of a large number of near-zero frequency modes as opposed to the commonly held view of divergence having UV origin. The feature revealing the divergence in zero modes is related to the observation that the entropy is invariant under a hidden scaling transformation even when the Hamiltonian is not. We discuss the role of dispersion relations and the dimensionality of the space-time on the behavior of entanglement entropy.

DOI
10.1103/physrevd.90.044058

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We investigate the cause of the divergence of the entanglement entropy for the free scalar fields in (1+1)(1+1)(1+1) and (D+1)(D+1)(D+1) dimensional space-times. In a canonically equivalent set of variables, we show explicitly that the divergence in the entanglement entropy of the continuum field in (1+1)(1+1)(1+1) dimensions is due to the accumulation of a large number of near-zero frequency modes as opposed to the commonly held view of divergence having UV origin. The feature revealing the divergence in zero modes is related to the observation that the entropy is invariant under a hidden scaling transformation even when the Hamiltonian is not. We discuss the role of dispersion relations and the dimensionality of the space-time on the behavior of entanglement entropy.
Ultraviolet divergences are widely discussed in studies of entanglement entropy. Also present, but much less understood, are infrared divergences due to zero modes in the field theory. In this note, we discuss the importance of carefully handling zero modes in entanglement entropy. We give an explicit example for a chain of harmonic oscillators in 1D, where a mass regulator is necessary to avoid an infrared divergence due to a zero mode. We also comment on a surprising contribution of the zero mode to the UV-scaling of the entanglement entropy.
We study the momentum space entanglement between different energy modes of interacting scalar fields propagating in general (D+1D+1D+1)-dimensional flat space-time. As opposed to some of the recent works [V. Balasubramanian et al., Phys. Rev. D 86, 045014 (2012)], we use a Lorentz invariant normalized ground state to obtain the momentum space entanglement entropy. We show that the Lorenz invariant definition removes the spurious power-law behavior obtained in the earlier works. More specifically, we show that the cubic interacting scalar field in (1+11+11+1) dimensions leads to logarithmic divergence of the entanglement entropy and is consistent with the results from real space entanglement calculations. We study the effects of the introduction of the Lorentz violating higher derivative terms in the presence of a nonlinear self-interacting scalar field potential and show that the divergence structure of the entanglement entropy is improved in the presence of spatial higher derivative terms.
To this day, von Neumann definition of entropy remains the most popular measure of quantum entanglement. Much of the literature on entanglement entropy, particularly in the context of field theory, has focused on isolating the UV divergences. Zero-mode divergences of the entanglement entropy are less studied in this context, and apart from being easier to isolate, they offer an interesting insight into the physics of the system. To gain a better understanding of the system in this limit, we develop the free particle approximation of harmonic oscillator, with which we investigate the properties of entropy divergence in continuous bi-partite quantum systems such as the coupled harmonic oscillators and the hydrogen atom. We also show zero-mode divergence of the entropy of environment-induced entanglement in a tri-partite oscillator system. We discuss the implications of our result for field theory and IR structure of gravity.
We examine the correlations between divergences in ground state entanglement entropy and emergent zero-modes of the underlying Hamiltonian in the context of one-dimensional Bosonic and Fermionic chains. Starting with a pair of coupled Bosonic degrees of freedom, we show that zero modes are necessary, but not sufficient for entanglement entropy divergences. We then list sufficient conditions that identify divergences. Next, we extend our analysis to Bosonic chains, where we demonstrate that zero modes of the entanglement Hamiltonian provide a signature for divergences independent of the entanglement Hamiltonian. We then generalize our results to one-dimensional Fermionic lattices for a chain of staggered Fermions which is a discretized version of the Dirac field. We find that the methods detailed for Bosonic chains have Fermionic analogs and follow this up with a numerical study of the entanglement in the Fermionic chain. Finally, we discuss our results in light of the factorization algebra theorem.
To this day, von Neumann definition of entropy remains the most popular measure of quantum entanglement. Much of the literature on entanglement entropy, particularly in the context of field theory, has focused on isolating the UV divergences. IR divergences of the entanglement entropy are less studied in this context, and apart from being easier to isolate, they offer an interesting insight into the physics of the system, such as the role of zero modes in these divergences. To gain a better understanding of the system in this limit, we develop the free particle approximation of Harmonic oscillator, with which we investigate the properties of entropy divergence in continuous bi-partite quantum systems such as the coupled Harmonic oscillators and the Hydrogen atom. We also show IR divergence of the entropy of environment-induced entanglement in a tri-partite oscillator system. We discuss the implications of our result for field theory and IR structure of gravity.
Zero modes, understood here as degrees of freedom with vanishing confining frequency, play a central role in the nonequilibrium dynamics of bosonic systems. In Gaussian models, however, they lead to an unbounded, logarithmic growth of entanglement entropy. We show that this divergence is not an intrinsic property of zero modes themselves, but arises specifically for non-compact zero modes. Their non-compact configuration space allows unbounded spreading in position space, while their continuous spectra enable indefinite dephasing in momentum space. By contrast, compact zero modes in compact bosonic systems behave fundamentally differently: Spreading and dephasing are eventually halted, so that compactness caps the entanglement entropy at a finite value, making its dynamical role most transparent in the presence of a zero mode. We demonstrate this mechanism in a minimal setting by comparing two coupled harmonic oscillators with two coupled quantum rotors. We then show that the same physics persists in many-body systems by contrasting an N-site compact rotor chain with the non-compact harmonic chain. Finally, we relate these insights to ultra-cold-atom realizations of compact quantum field theories. In particular, we clarify when a compact free-boson (Tomonaga-Luttinger liquid) description is required and when the commonly used non-compact massless Klein-Gordon model breaks down. Even when the initial state is accurately captured by a non-compact Gaussian description, compactness ultimately governs the late-time quench dynamics, curbing entanglement growth rather than allowing a dynamical divergence.