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Entanglement Entropy of a One-Dimensional Scalar Field

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Entanglement Entropy of a One-Dimensional Scalar Field Master's thesis Entanglement entropy of a one-dimensional scalar field Author: Supervisor: K.R. de Ruiter BSc. Prof. dr. ir. H.T.C. Stoof Theoretical Physics Utrecht University June 2020 ii Abstract In this thesis, we study the entanglement entropy of a chain of coupled harmonic oscillators, which is used to model a one-dimensional bosonic scalar field. Initially, a model for a massless scalar field is discussed. When calculating the entanglement entropy from the reduced density matrix we run into a problem due to the presence of a zero mode, and we are only able to compute the entanglement entropy of one particular case. These difficulties are avoided when we consider a massive scalar field instead, which we accomplish by adding a mass term to the dispersion relation. We numerically study the entanglement entropy of the massive scalar field model as a function of the number of coordinates and the mass parameter and compare with analytical results. By fitting our numerical data we construct a single cross-over function, which describes the entanglement entropy as a function of both the number of coordinates and the mass parameter. The cross-over function agrees well with the numerical data, especially for small values of the mass parameter. Taking then the massless limit, the entanglement entropy of a massless scalar field can nevertheless be determined and reproduces the prediction of conformal field theory. Title page image taken from: https://www.symmetrymagazine.org/article/gravitys-waterfall iii Contents 1 Introduction 1 1.1 Motivation . .3 2 Physical background 7 2.1 Entropy . .7 2.2 Density matrix . .8 2.3 Entanglement entropy . .9 3 Massless scalar field 11 3.1 Three coupled harmonic oscillators . 11 3.1.1 Ground state wave function . 11 3.1.2 Density matrix of the ground state . 15 3.1.3 Reduced density matrix with two coordinates integrated out . 16 ′ 3.1.4 Entanglement entropy of reduced density matrix ρred x1; x1 .... 17 3.1.5 Reduced density matrix with one coordinate integrated out . 18 ( ′ ) ′ 3.1.6 Entanglement entropy of reduced density matrix ρred q1; q1; q2; q2 . 21 3.2 Chain of N harmonic oscillators . 22 ( ) 3.2.1 Ground state wave function . 22 3.2.2 Density matrix of the ground state . 23 3.2.3 Reduced density matrix . 23 3.2.4 Numerical results for the entanglement entropy . 28 4 Massive scalar field 31 4.1 Ground state wave function and density matrix . 31 4.2 Rewriting the reduced density matrix . 32 4.3 Numerical results for the entanglement entropy . 33 4.4 Dimensional analysis . 36 4.5 Determining a cross-over function . 38 4.5.1 First order fit . 41 4.5.2 Pad´efit . 41 4.6 Testing the cross-over function . 43 4.7 The massless limit . 44 5 Discussion and outlook 45 iv 5.1 Discussion . 45 5.2 Outlook . 47 A Proof of dispersion relation product 51 1 Chapter 1 Introduction In 1915, Einstein published his theory of general relativity in which he unified the theory of special relativity and gravity in terms of the curvature of spacetime. One year later, Schwarzschild found an exact solution to Einstein's field equations of general relativity, which contained a coordinate singularity. At that time, Schwarzschild did not understand the physical meaning of this singularity. Nowadays, we know that it describes a black hole. Black holes have an extremely high mass and density, resulting in a very strong curvature of the spacetime around them. The gravitational pull of a black hole is so strong that nothing can escape from it. Not even light particles, photons, can escape, which are the fastest moving particles according to Einstein's relativity theory. The strong gravitational field of a black hole marks a surface beyond which nothing can escape, known as the event horizon of the black hole. This horizon effectively seals off the interior of the black hole from the rest of the universe, making it impossible to observe. Hawking proved that the surface area of a black hole can never decrease as a function of time [1]. Bekenstein noted a strong analogy to the thermodynamical entropy, which also is a quantity that can never decrease in time, according to the second law of thermodynamics. In 1973, this inspired Bekenstein to write a paper in which he tried to unify thermodynam- ics with black hole dynamics [2]. Thermodynamical systems are typically characterised by a handful of macroscopic quantities, such as its temperature, pressure, and volume. The entropy of a system is a measure of the number of different possible internal states, called microstates, that correspond to the same macrostate. Analogously, there exists a theorem for black holes called the no-hair theorem, which states that any black hole that is a solu- tion of the Einstein-Maxwell equations can be fully characterised by only three externally observable quantities: mass, charge, and angular momentum. This would imply, however, that information about the internal states of the black hole is lost. For example, it would be impossible to distinguish between a black hole that formed from collapsing matter or one that formed from collapsing antimatter, if their mass, charge, and angular momentum are equal. This means there is a lack of information about the internal configuration of black holes. Bekenstein noted the analogy to thermodynamical entropy and argued that 2 Chapter 1. Introduction there must be a black-hole entropy related to the lack of information about the interior of a black hole. With the theorem of the black-hole surface area of Hawking in mind, Bekenstein suggested that the black-hole entropy should be proportional to the surface area of the black hole. Previously, physicists had struggled to explain the apparent loss of information of an object with some entropy falling into a black hole. It seemed that the information of the infalling object was lost behind the event horizon, thus violating the second law of thermodynamics, which states that the entropy of an isolated system (the universe) can never decrease. The notion of black-hole entropy provided a solution to this problem, as the entropy of the infalling object is added to the entropy of the black hole. This led to the formulation of the generalised second law, which states that the entropy of the exterior of the black hole plus the entropy of the black hole itself never decreases [3]. One year later, in 1974, Stephen Hawking proved that black holes emit thermal radiation, and thus have a temperature [4]. This confirmed the earlier ideas of Bekenstein, and this quickly led to the famous Bekenstein-Hawking entropy of a black hole, c3k A S B : (1.1) BH 4Gh The fact that the entropy of a black hole indeed= ̵ scales with its area was a striking result because from thermodynamics entropy was known to be an extensive quantity, thus scaling with the volume of the system. This suggests that all the information of the interior of the black hole is somehow encoded on its surface, the event horizon. This insight is what inspired the formulation of the holographic principle, which states that a description of an n 1 -dimensional curved spacetime can be understood as an n-dimensional theory living on its boundary [5]. ( + ) Physicists have been trying to understand the area law behaviour of black-hole entropy ever since it was discovered. A promising candidate to explain the area law of black holes is entanglement entropy, which is a measure of the degree of entanglement in a system. To calculate the entanglement entropy of a system, one first needs to effectively remove a part of the system, which is done by taking the partial trace over a number of the degrees of freedom, thus essentially dividing the system into two subsystems. Tracing out coordinates results in a loss of information, and therefore the leftover state is a mixed state. The mixed state has some entropy associated with it, which is known as the entanglement entropy. To understand why entanglement entropy might help to explain the area law, let us consider an example of a black hole. In the case of a black hole formed by gravitational collapse, the degrees of freedom of the quantum field in the exterior and the interior of the black hole are entangled with each other [6]. Since an observer exterior to the black hole cannot access the interior, the interesting state for an external observer of a black hole is the state where the internal states are removed. Mathematically this can be done by tracing over the degrees of freedom of the interior of the black hole, resulting in a mixed state with some associated entropy. Entanglement entropy is defined as the entropy associated with a mixed state that describes a subsystem that is part of a larger system. Due to the analogy between the state of interest for an external observer of a black hole and the definition of entanglement entropy, it is thought that entanglement entropy might be a 1.1. Motivation 3 good candidate to try to explain the area law of black-hole entropy. In 1986, Bombelli et al. showed that the entanglement entropy of a system of coupled oscillators, which is used to model quantum fields, is indeed proportional to the area of the inaccessible traced out region [7]. Later, in 1993, Srednicki separately found similar results, showing that the entanglement entropy of a quantum field obeys an area law [8]. Bombelli and Srednicki argue that the entanglement of a massless free scalar field might be related to the entropy of a black hole and that entanglement entropy might, therefore, help us to better understand the area law of black-hole entropy.
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