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arXiv:1810.03888 · Physical review. D/Physical review. D.

Divergence of entanglement entropy in quantum systems: Zero-modes

S. Mahesh Chandran, S. Shankaranarayanan

Published Feb 19, 2019• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.99.045010

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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.
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 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.
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.
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.
The thermodynamic entropy of an isolated system is given by its von Neumann entropy. Over the last few years, there has been an intense activity to understand the thermodynamic entropy from the principles of quantum mechanics. More specifically, is there a relation between the (von Neumann) entropy of entanglement between a system and some (separate) environment and the thermodynamic entropy? It is difficult to obtain the relation for many body systems, hence, most of the work in the literature has focused on small number systems. In this work, we consider black holes—which are simple yet macroscopic systems—and show that a direct connection could not be made between the entropy of entanglement and the Hawking temperature. In this work, within the adiabatic approximation, we explicitly show that the Hawking temperature is indeed given by the rate of change of the entropy of entanglement across a black hole’s horizon with regard to the system energy. This is yet other numerical evidence leading to understanding the key features of black-hole thermodynamics from the viewpoint of quantum information theory.
We study the quantum entanglement of systems of coupled harmonic oscillators on the basis of thermo-field dynamics (TFD).For coupled harmonic oscillators at equilibrium, the extended entanglement entropy is derived using the TFD method, and it is demonstrated to be controlled by temperature and coupling parameters.For non-equilibrium systems, in addition to temperature and coupling parameters, the time dependence of the extended entanglement entropy is calculated in accordance with the dissipative von Neumann equation, and the dissipative dynamics of the systems of coupled harmonic oscillators is discussed.Consequently, based on TFD, the physical parametrization of the entanglement entropies is confirmed in both the equilibrium and nonequilibrium cases of harmonic oscillator systems by means of the laws of thermodynamics.