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Master of Science Thesis

Thermalisation of inelastic dark matter in the Sun with a light mediator

Simon Israelsson

Particle and Astroparticle Physics, Department of Physics,
School of Engineering Sciences,
KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2018

A

Typeset in LT X
E

  • Examensarbetesuppsats for avlaggande av Masterexamen i Teknisk fysik, med in-
  • ¨

riktning mot Teoretisk fysik.
¨
Master’s thesis for a Master’s degree in Engineering Physics in the subject area of Theoretical physics.

TRITA-SCI-GRU 2018:308

  • c
  • ꢀ Simon Israelsson, August 2018

Printed in Sweden by Universitetsservice US AB

Abstract

Particle dark matter is a popular solution to the missing mass problem present in the Universe. If dark matter interacts with ordinary matter, even very weakly, it might be the case that it is captured and accumulated in the Sun, where it may then annihilate into particles that we can observe here on Earth. The interaction between dark matter and standard model particles may be mediated by a light dark sector particle. This would introduce an extra recoil energy suppression into the scattering cross section for collision events, which is of the form needed to possibly also alleviate some of the observed small scale structure issues of collisionless cold dark matter. In this work we perform numerical simulations of the capture and subsequent scattering of inelastic dark matter in the Sun, in the presence of a light mediator particle. We find that the presence of the mediator results in a narrower capture region than expected without it and that it mainly affects the scattering rate in the phase space region where the highest scattering rates are found. However, it is not seen to cause any noticeable difference to the radial distribution of dark matter in the Sun. No steady state is reached and the captured dark matter does not reach thermal equilibrium in the Sun.

Key words: dark matter, solar capture, inelastic dark matter, light mediator.

iii

Sammanfattning

  • Morka materiapartiklar ar en popular losning till avsaknaden av materia i uni-
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  • ¨
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versum. Om mork materia interagerar med vanlig materia, aven valdigt svagt,

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s˚a skulle det kunna vara fallet att de f˚angas in och ansamlas i solen dar de se-
¨dan forintas och resulterar i partiklar vi kan observera har p˚a jorden. Interaktio-

  • ¨
  • ¨

nen mellan mork materia och vanlig materia skulle kunna formedlas av en latt

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kraftbarare i den morka sektorn. Detta skulle introducera ett rekylenergi-beroende

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i spridningstvarsnittet for kollisioner av samma form som behovs for att potentiellt

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avhjalpa ett antal observerade problem som i dagslaget finns for kollisionslos kall

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mork materia. I detta arbete har vi utfort numeriska simuleringar av inf˚angningen

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och den efterfoljande spridningen av mork materia inuti solen d˚a en latt kraftbarare

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ar narvarande. Vi kommer fram till att den latta kraftbararen resulterar i ett mindre

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  • inf˚angningsomr˚ade an vad man vantar sig utan dess existens, samt att den p˚averkar
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  • spridningen som mest i omr˚aden av fasrummet dar den totala spridningen ar som
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  • storst. Daremot s˚a verkar den inte leda till n˚agon urskiljbar skillnad for den radi-
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  • ella distributionen av mork materia i solen. Inget stationart beteende n˚as och den
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  • inf˚angade morka materian uppn˚ar inte termisk jamvikt i solen.
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  • Nyckelord: mork materia, solinf˚angning, inelastisk mork materia, latt kraftbarare.
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iv

Preface

Acknowledgements

I would like to begin by thanking my supervisor Tommy Ohlsson for enabling me to work on this project, and Mattias Blennow for making me aware that the project was available. I would also like to thank Stefan Clementz for allowing me to pick up where he left off, for allowing me to inherit his simulation code, and for all the help and guidance he has provided throughout the project. Further I would like to thank my other office mates, Marcus and Anton, who together with Stefan provided an amusing working environment filled with interesting discussions. For their insights provided during the regularly held journal clubs and other discussions, I would also like to thank Florian and Sofiane, as well as Sandhya and Sushant.
Lastly I would like to thank my family and friends who have supported me throughout the semester and the entirety of my time at KTH.

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv

  • Preface
  • v

  • Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  • v

  • Contents
  • vii

1

2

  • Introduction
  • 1

  • 1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  • 2

  • Background
  • 3

3344566677
2.1 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Observational evidence . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Rotation curves . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Weak lensing observations . . . . . . . . . . . . . . . . . . . 2.2.3 The cosmic microwave background . . . . . . . . . . . . . .
2.3 Alternatives to dark matter . . . . . . . . . . . . . . . . . . . . . . 2.4 Structure issues of dark matter . . . . . . . . . . . . . . . . . . . .
2.4.1 Missing satellites . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Cusp versus core . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Too big to fail . . . . . . . . . . . . . . . . . . . . . . . . .

  • 3
  • General theory
  • 9

  • 9
  • 3.1 Inelastic dark matter . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Orbital mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Differential scattering cross section . . . . . . . . . . . . . . . . . . 3.5 Decay of the χ2 state . . . . . . . . . . . . . . . . . . . . . . . . . .
10 12 15 18

vii viii

Contents

  • 4
  • Dark matter in the Sun
  • 21

4.1 Solar capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Scattering in the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dark matter distribution in the Sun . . . . . . . . . . . . . . . . .
22 24 24

  • 5
  • Numerical simulations and results
  • 27

27 30 34 36
5.1 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Solar capture rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Solar scattering rate . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Time evolved distributions . . . . . . . . . . . . . . . . . . . . . . .

  • 6
  • Summary and conclusions
  • 45

  • 47
  • Bibliography

Chapter 1

Introduction

The first half of the 20th century resulted in some of the most successful descriptions of our universe as of yet in physics. The development of quantum mechanics, and later quantum field theory, provided excellent descriptions of physics on small scales explaining phenomena involving elementary particles, atoms, and molecules. Meanwhile, Einstein’s theory of general relativity (GR) allowed for us to make accurate predictions concerning gravitational effects. It was through gravitational interactions, albeit still working using Newtonian gravity, that scientists started noticing discrepancies in the behaviour of astronomical objects [1]. The conclusion was reached that additional matter had to be unaccounted for in order to describe the observed phenomena. The nature of this matter, other than the fact that it was not visible, was, and to a large extent still is, unknown. The name dark matter (DM), or “dunkle Materie” in German, was given to this missing matter [2]. Discrepancies between theory and observations were not limited to the astrophysical sector. Within the particle physics community clear signs of physics beyond the standard model (SM) had also appeared. For example neutrinos, extremely light and very weakly interacting particles, are predicted by the SM to be massless [3]. This has since been proven to be incorrect as the phenomenon of neutrino oscillations, which was experimentally verified in the late 90s [4] and later confirmed in the early 21st century [5], requires there to be a mass difference between the three neutrino mass eigenstates [6]. Astrophysics and particle physics now often go hand in hand attempting to resolve both issues in one go. Particle physicists attempt to extend the SM and resolve its issues while also incorporating candidates for DM and the dark sector in general, and the astrophysicists can deduce, as well as measure, astrophysical observables that constrain the particle physics models. Attempts to further probe this new dark sector have yielded no positive results and the nature of DM, beyond its gravitational interactions, is still unknown. Indirect searches for DM aim to explore this new sector by observing particle and radiation by-products due to DM interactions. A target for these types of observations is the Sun. Massive bodies, such as the Sun, that generate significant gravitational potentials may

1
2

Chapter 1. Introduction

capture DM particles on the occasion that they lose sufficient energy in a scattering event while passing through the body [7]. A class of DM known as inelastic DM was originally introduced to alleviate tension between the DAMA experiment [8] and the CDMS experiment [9] as they had obtained disagreeing results. Inelastic DM consists of at least two different DM states that are separated by a very small mass difference that may have significant effects on the scattering kinematics of the DM particles [10]. This thesis is concerned with the thermalisation of inelastic DM in the Sun, where the heavier state is unstable and subsequently decays. A light mediator particle is also included to mediate the scattering process between DM and target particles. For the case where both states are stable and the scattering events are point like it has previously been shown [11] that neither a steady state or equilibrium is reached in general.

1.1 Outline

This thesis is organized as follows: In chapter 2 we cover some general background material concerning DM. In chapter 3 we review the basics of the DM framework known as inelastic DM and we cover most of the necessary theory for this thesis. Moving on to chapter 4 we concern ourselves with the connection between DM and the Sun. Finally, we present the results of this project in chapter 5 and summarise and conclude the thesis in chapter 6.

Chapter 2

Background

2.1 Dark matter

Even though the evidence for the existence of DM, covered briefly below, is rather convincing, the true nature of what exactly it is remains unknown. The list of candidates that have been proposed is very long and include, amongst many other, SM neutrinos, particles from supersymmetry, and axions. Popular candidates have historically been weakly interacting massive particles, usually referred to as WIMPs. These DM candidates would be produced in the hot early Universe and “freeze out” such that their abundance today is rather predictable. It turns out that if one assumes DM to have a weak scale interaction cross section as well as just above weak scale mass, the predicted abundance match observed values rather nicely, something that have been dubbed the “WIMP miracle” [12, 13]. Originally WIMPs were those candidates that interacted with the SM through the weak interaction, but the term is now used in a wider sense. Based on cosmological simulations of structure formation in the Universe, DM, whatever it is, is “cold”, i.e., it was non-relativistic after “freeze out”. This in contrast to “hot” DM which would be relativistic instead [1]. Regardless of where the search end up, determining the details and specifics of DM is one of the remaining key discoveries to better understanding the Universe around us.

2.2 Observational evidence

Quite some time has passed since scientist first noticed that something was off with the matter content of the Universe. During the late 19th and early 20th century, galaxy clusters were observed and their masses were estimated [1]. This could be done in several ways. One could simply try to estimate the number of galaxies present in a cluster and multiply it by some estimated galaxy mass, or one could try to apply the virial theorem to the clusters, relating the velocity dispersion of the

3
4

Chapter 2. Background

galaxies with the total mass of the cluster. This was famously done by Fritz Zwicky who compared these methods and found that the expected velocity dispersion was an order of magnitude smaller than what was observed. He concluded that an unseen additional mass had to be missing somewhere [2].

2.2.1 Rotation curves

Galactic rotation curves, i.e., plots of a galaxy’s circular velocity as a function of radial distance from the galactic centre, provided some of the first convincing evidence for DM [1, 13, 14]. Following classical Newtonian gravity we have the following relation for a spherically symmetric mass distribution

r

GM(r)

vc(r) =

,

(2.1)

r

where vc is the circular velocity of the galaxy, G the gravitational constant, and M(r) the enclosed mass within a radius r. If we were to move far out enough that we include all the visible stars and gas in the galaxy we would expect M(r) → MG, with MG being the mass of our galaxy. If we continue to move outwards past this point, we see from the expression above that we expect the velocity to fall off as ∼ r−1/2 . What was actually seen in observations was that the circular velocity seemed to flatten out at large radii, tending to a constant value [15, 16]. From this one can imagine that the mass has to continue to increase beyond what we can actually see. This observation led people to consider the possibility that there could be a halo of DM that extends far beyond the radius of the visible matter. By considering contributions from the added halo, as well as from the observed matter, one can obtain quite nice fits to observational data [17]. It also seems possible to match both a universal DM density profile [18] as well as a universal velocity profile [19] to observed data, speaking in favour of the DM hypothesis.

2.2.2 Weak lensing observations

Further evidence can be gathered with the help of galactic clusters from a phenomenon known as gravitational lensing. Gravity does not only affect massive objects, it also changes the trajectory of light as it passes by, or through, volumes with significant gravitational potentials. This can give rise to distortions in images of bright objects, or even multiple phantoms of the same object, should there be a region with strong gravitational fields between the observer and the observed object [13, 14]. This effect on light can be accurately calculated using GR and under certain circumstances, that are very commonly satisfied in astrophysical scenarios, the general shape of the gravitational potential can be recreated from the distortions found in images of the observed objects [20]. These types of measurements are commonly referred to as weak lensing experiments or weak lensing observations.
One of the stronger arguments in favour of DM over versions of modified gravity is based in part on weak lensing observations. Observations of 1E0657-558, more

2.2. Observational evidence

5commonly referred to as the Bullet Cluster, provide an excellent opportunity to observe a system where a large part of the baryonic content is expected to be separated from the accompanying DM [21]. The system consist of two galaxy clusters that have collided. Galaxies pass right through each other, behaving essentially as collisionless particles, while the intra-galactic plasma is slowed down. It is known that for clusters in general the stellar content make up roughly 1-2 % of the mass and the plasma make up roughly 5-15 % [22, 23]. The plasma can be observed using X-ray astronomy and, if no DM is present, one would expect the gravitational potential to overlap significantly with the areas of high plasma abundance. Using weak lensing, the gravitational potential was reconstructed and it was found that the two peaks in the potential were significantly spatially separate from the plasma [21]. This speaks heavily in favour of DM that would, due to its very weak interactions, also pass unhindered through the collision, tracing the trajectory of the galaxies.

2.2.3 The cosmic microwave background

The cosmic microwave background (CMB) is the remnant radiation from a much younger universe. In the early Universe the temperature was high enough to separate electrons and baryons and together with the photons they formed what can best be described as a fluid. As the temperature decreased it was possible for electrons to bind together with protons to form neutral hydrogen, this period is referred to as recombination, and particle interactions with photons died off drastically. The photons left over from after recombination is what today is observed as the CMB [24, 25]. Today it is very well described by a black-body spectrum with a corresponding temperature of TCMB = 2.726 K [26]. It was initially seen as isotropic, i.e., equal in all directions, but it was soon discovered that it did contain some very slight anisotropies that revealed structure on smaller scales within it [27].
Today these anisotropies are the target of intense studies as they provide an excellent probe for cosmological parameters and thus help us test our theories about the Universe. The focus of these studies is the power spectrum of the CMB. The quantity Θ = ∆T/T is expanded in terms of spherical harmonics and one can analyse the different modes and the relation between them. Before recombination oscillations in the photon-baryon fluid are expected to appear as the photon pressure competes against gravity in areas of gravitational collapse. As recombination takes place and the photons no longer scatter, they are red-shifted while climbing out of the gravitational potential, altering their wavelengths, giving rise to some of the anisotropies we see in the spectrum today. These are called primary anisotropies. Secondary anisotropies arise due to gravitational effects from perturbations to the metric, the mathematical object describing the shape of space-time in general relativity, as well as rescattering, i.e., some photons scatter again against objects during the later stages of the evolution of the Universe [24, 25].
Working with the so called ΛCDM cosmological model, measurements from the
CMB lead us to believe that roughly 5 % of the energy density in the Universe
6

Chapter 2. Background

is baryonic matter, while the total matter density is roughly 31 % [28]. Thus DM should make up approximately 26 % of the energy density of the Universe, or roughly 80 % of all matter. The rest of the energy density is attributed to the so called cosmological constant, Λ, which is interpreted as a “vacuum energy”. This parameter is responsible for the accelerating expansion of the Universe [20].

2.3 Alternatives to dark matter

The different observed phenomena covered above, as well as numerical simulations of structure formation, paint a picture where DM is a very attractive description. It is however worth mentioning that some alternative descriptions have been suggested. One of the theories that appeared in contrast to DM is the theory of modified Newtonian dynamics (MOND) [29]. This theory was developed mainly to describe observations of galactic rotation curves without introducing any DM. It is based on the idea that Newton’s second law should be modified to take the form

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  • New Equilibrium Ensembles for Isolated Quantum Systems

    New Equilibrium Ensembles for Isolated Quantum Systems

    entropy Article New Equilibrium Ensembles for Isolated Quantum Systems Fabio Anza Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK; [email protected] Received: 16 June 2018; Accepted: 26 September 2018; Published: 29 September 2018 Abstract: The unitary dynamics of isolated quantum systems does not allow a pure state to thermalize. Because of that, if an isolated quantum system equilibrates, it will do so to the predictions of the so-called “diagonal ensemble” rDE. Building on the intuition provided by Jaynes’ maximum entropy principle, in this paper we present a novel technique to generate progressively better approximations to rDE. As an example, we write down a hierarchical set of ensembles which can be used to describe the equilibrium physics of small isolated quantum systems, going beyond the “thermal ansatz” of Gibbs ensembles. Keywords: thermalization; quantum thermodynamics; quantum information; entropy 1. Introduction The theory of Statistical mechanics is meant to address the equilibrium physics of macroscopic systems. Both at the classical and at the quantum level, the whole theory is based on the thermal equilibrium assumption. Such a physical condition is expressed, mathematically, by saying that the system is described by one of the so-called Gibbs ensembles [1,2]. The theory has enjoyed a marvelous success in explaining and predicting the phenomenology of large quantum systems and, nowadays, we use its results and tools well beyond the domain of physics. However, strictly speaking, this course of action is justified only in the thermodynamic limit. More realistically, the assumptions of Statistical Mechanics are believed to be justified on a scale of, say, an Avogadro number of particles N ∼ 1023.
  • Thermalisation in Few-Body Systems: Revealing Missing Charges with Quantum Fluctuation Relations

    Thermalisation in Few-Body Systems: Revealing Missing Charges with Quantum Fluctuation Relations

    Clarendon Laboratory Thermalisation in few-body systems: Revealing missing charges with quantum fluctuation relations Jordi Mur-Petit J. Mur-Petit, A. Relaño, R. A. Molina & D. Jaksch, Nature Commun. 9, 2006 (2018) Dynamics? Achilles vs. tortoise Zeno (5th Cent. BCE) ibmathresources.com Dynamics vs. conservation laws [email protected] Dynamics? Achilles vs. tortoise Boris vs. Parliament UK (2019) HuffPostUK ibmathresources.com Dreamstime.com Dynamics vs. conservation laws [email protected] Quantum dynamics vs. Relaxation Single isolated spin: Bloch oscillations Many-body (closed): Single spin coupled to Eigenstate Thermalisation large bath: Dephasing Hypothesis (ETH) Trotzky et al., Nat. Phys. (2012) Dynamics vs. conservation laws [email protected] Quantum dynamics vs. Relaxation Single isolated spin: Bloch oscillations Many-body (closed): Single spin coupled to Eigenstate Thermalisation large bath: Dephasing Hypothesis (ETH) Trotzky et al., Nat. Phys. (2012) How does relaxation emerge as number of particles increases? Dynamics vs. conservation laws [email protected] Known exceptions to relaxation ⚫Decoherence free subspaces ⚫Reservoir engineering ⚫Integrable systems Dynamics vs. conservation laws [email protected] Integrable systems Strongly interacting bosons in 1D N~106 Energy & momentum conservation in each collision preclude relaxation Kinoshita et al., Nature (2006) Dynamics vs. conservation laws [email protected] Relaxation in an integrable system Strongly interacting bosons in 1D x 2 N~103 Hofferberth et al., Nature (2007); Gring et al., Science (2012); Langen et al., Science (2015) (Schmiedmayer lab) Dynamics vs. conservation laws [email protected] Relaxation in an integrable system Strongly interacting bosons in 1D x 2 N~103 (?) Hofferberth et al., Nature (2007); Gring et al., Science (2012); Langen et al., Science (2015) (Schmiedmayer lab) Dynamics vs.
  • Focus on Quantum Thermodynamics

    Focus on Quantum Thermodynamics

    Home Search Collections Journals About Contact us My IOPscience Focus on quantum thermodynamics This content has been downloaded from IOPscience. Please scroll down to see the full text. 2017 New J. Phys. 19 010201 (http://iopscience.iop.org/1367-2630/19/1/010201) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 95.181.183.47 This content was downloaded on 24/01/2017 at 11:20 Please note that terms and conditions apply. You may also be interested in: Quantum optomechanical piston engines powered by heat A Mari, A Farace and V Giovannetti Perspective on quantum thermodynamics James Millen and André Xuereb The role of quantum information in thermodynamics—a topical review John Goold, Marcus Huber, Arnau Riera et al. Coherence-assisted single-shot cooling by quantum absorption refrigerators Mark T Mitchison, Mischa P Woods, Javier Prior et al. Clock-driven quantum thermal engines Artur S L Malabarba, Anthony J Short and Philipp Kammerlander Out-of-equilibrium thermodynamics of quantum optomechanical systems M Brunelli, A Xuereb, A Ferraro et al. Work and entropy production in generalised Gibbs ensembles Martí Perarnau-Llobet, Arnau Riera, Rodrigo Gallego et al. Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems Christian Gogolin and Jens Eisert The smallest refrigerators can reach maximal efficiency Paul Skrzypczyk, Nicolas Brunner, Noah Linden et al. New J. Phys. 19 (2017) 010201 doi:10.1088/1367-2630/19/1/010201 EDITORIAL Focus on quantum thermodynamics OPEN ACCESS Janet Anders1 and Massimiliano Esposito2 PUBLISHED 6 January 2017 1 College of Engineering, Mathematics, and Physical Sciences, University of Exeter, Stocker Road, Exeter EX4 4QL, UK 2 Complex Systems and Statistical Mechanics, University of Luxembourg, L-1511 Luxembourg, Luxembourg Original content from this E-mail: [email protected] and [email protected] work may be used under the terms of the Creative Commons Attribution 3.0 licence.
  • Arxiv:1907.10061V2 [Hep-Th] 14 Aug 2019

    Arxiv:1907.10061V2 [Hep-Th] 14 Aug 2019

    Extended Eigenstate Thermalization and the role of FZZT branes in the Schwarzian theory Pranjal Nayaka, Julian Sonnerb & Manuel Vielmab aDepartment of Physics & Astronomy, University of Kentucky, 505 Rose St, Lexington, KY, USA bDepartment of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Gen`eve 4, Switzerland [email protected] b julian.sonner, manuel.vielma @unige.ch f g Abstract In this paper we provide a universal description of the behavior of the basic operators of the Schwarzian theory in pure states. When the pure states are energy eigenstates, expectation values of non-extensive operators are thermal. On the other hand, in coher- ent pure states, these same operators can exhibit ergodic or non-ergodic behavior, which is characterized by elliptic, parabolic or hyperbolic monodromy of an auxiliary equation; or equivalently, which coadjoint Virasoro orbit the state lies on. These results allow us to establish an extended version of the eigenstate thermalization hypothesis (ETH) in theories with a Schwarzian sector. We also elucidate the role of FZZT-type bound- ary conditions in the Schwarzian theory, shedding light on the physics of microstates associated with ZZ branes and FZZT branes in low dimensional holography. arXiv:1907.10061v2 [hep-th] 14 Aug 2019 1 Contents 1 Introduction2 1.1 Summary of results . .4 2 Pure states in Liouville and the Schwarzian8 2.1 Boundary Liouville theory and Schwarzian coherent states . .8 2.2 Descending to the Schwarzian . 12 2.3 Boundary conditions as States of the theory . 14 3 Semiclassical results 16 3.1 One-point function and effective temperature .
  • Equilibration, Thermalisation, and the Emergence of Statistical Mechanics in Closed Quantum Systems

    Equilibration, Thermalisation, and the Emergence of Statistical Mechanics in Closed Quantum Systems

    REVIEW ARTICLE Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems Christian Gogolin1,2,3 and Jens Eisert1 1Dahlem Center for Complex Quantum Systems, Freie Universitat Berlin, 14195 Berlin, Germany 2ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain 3Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany E-mail: [email protected] E-mail: [email protected] Abstract. We review selected advances in the theoretical understanding of complex quantum many-body systems with regard to emergent notions of quantum statistical mechanics. We cover topics such as equilibration and thermalisation in pure state statistical mechanics, the eigenstate thermalisation hypothesis, the equivalence of ensembles, non-equilibration dynamics following global and local quenches as well as ramps. We also address initial state independence, absence of thermalisation, and many-body localisation. We elucidate the role played by key concepts for these phenomena, such as Lieb-Robinson bounds, entanglement growth, typicality arguments, quantum maximum entropy principles and the generalised Gibbs ensembles, and quantum (non-)integrability. We put emphasis on rigorous approaches and present the most important results in a unified language. arXiv:1503.07538v5 [quant-ph] 13 Jun 2018 CONTENTS 2 Contents 1 Introduction 3 2 Preliminaries and notation 7 3 Equilibration 16 3.1 Notionsofequilibration. .. 17 3.2 Equilibrationonaverage . 18 3.3 Equilibration during intervals . .... 24 3.4 Othernotionsofequilibration . ... 25 3.5 Lieb-Robinsonbounds ............................ 26 3.6 Time scales for equilibration on average . ...... 28 3.7 Fidelitydecay................................. 33 4 Investigations of equilibration for specific models 33 4.1 Globalquenches................................ 34 4.2 Localandgeometricquenches .
  • 1 Sep 2015 Quantum Many-Body Systems out of Equilibrium

    1 Sep 2015 Quantum Many-Body Systems out of Equilibrium

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UPCommons. Portal del coneixement obert de la UPC Quantum many-body systems out of equilibrium J. Eisert, M. Friesdorf, and C. Gogolin Dahlem Center for Complex Quantum Systems, Freie Universitat¨ Berlin, 14195 Berlin, Germany Closed quantum many-body systems out of equilibrium pose several long-standing problems in physics. Recent years have seen a tremendous progress in approaching these questions, not least due to experiments with cold atoms and trapped ions in instances of quantum simulations. This article provides an overview on the progress in understanding dynamical equilibration and thermalisation of closed quantum many-body systems out of equilibrium due to quenches, ramps and periodic driving. It also addresses topics such as the eigenstate thermalisation hy- pothesis, typicality, transport, many-body localisation, universality near phase transitions, and prospects for quantum simulations. How do closed quantum many-body systems out of equi- such as the Ising model, as well as bosonic or fermionic lat- librium eventually equilibrate? How and in precisely what tice systems, prominently Fermi- and Bose-Hubbard models. way can such quantumsystems with many degrees of freedom In the latter case, the Hamiltonian takes the form thermalise? It was established early on how to capture equilib- † † rium states corresponding to some temperature in the frame- H = J (bj bk + bkbj )+ U nj (nj − 1), (2) work of quantum statistical mechanics, but it seems much less hXj,ki Xj clear how such states are eventually reached via the local dy- † namics following microscopic laws.