Anexo 1. Resumen de eventos regulatorios claves en el Ecuador

We only started to understand how symmetry-protected edge modes influence the non-equilibrium time evolution of a FF system. In PBC, it is still unclear whether or not the relation between TPT and DPT holds in general, e.g. in systems which do not belong to the BDI symmetry class.

A more general approach than running simulations would require an analytical understanding of what is happening at the edges. A promising idea is to use the formalism of Fredholm determinants which is widely used in Bethe ansatz calculation and random matrix theory [27; 28].

Numerically, it might be interesting to calculate the EE in other systems than the SSH chain, which is only possible using the ED algorithm, since no analytic solutions are known for these systems in OBC. In the SSH chain, it should also be possible to derive an explicit expression for the EE, which would allow a more detailed understanding of the time scales of the oscillations. Further, one could search for an equivalent to the overlap with the initial state in systems without PHS, or to understand en effective model analytically. Our ED algorithm uses only matrix multiplication on GPU so far. It would also be interesting to put more parts of the algorithm on coprocessors like graphics processing units (GPU) or Xeon Phi accelerators (XPA). These devices have a different memory mapping than usual processors and achieve their perfor- mance by combining many cores at lower clock frequencies instead of few cores at very high frequencies. Therefore, they are an ideal environment for of parallel algorithms. Many standard matrix operations such as multiplication or diagonalization are known to be highly parallelizable.

Most current GPU implementations only support single precision calculations, some also work with double precision. If this is sufficient, we can in principle implement most of our ED algorithm on a GPU. Sometimes, our algorithm needs higher precision, as pointed out in section III.B.1. Double-double and quad-double libraries are not available for GPU yet, but in principle they compile on XPA, which, unlike GPU, support the x86 command set nearly completely. Being able to perform simulations completely on GPU or XPA would cut the needed calculation time drastically and allow us to run simulations in a much larger range of models and parameters.

Acknowledgements

I would like to acknowledge my advisers, Dr. Jesko Sirker and Dr. Sebastian Eggert for their support throughout the thesis project. Further, I thank the department of physics and astronomy for the invitation to University of Manitoba, the hospitality, and for financial support. My stay in Winnipeg was partly financed PROMOS programme of the German academic exchange service (Deutscher Akkademischer Austauschdienst, DAAD).

The thesis project was also supported by NVidia Corp. by donating a Tesla K40 GPU which was used for some calculations, and by Compute Canada and WestGrid, and the Regionales Hochschulrechenzentrum Kaiserslauern (RHRK) by providing high-performance computing (HPC) resources. I specially acknowledge the support of Dr. Grigory Shamov, who provided support setting up the GPU and installing all drivers correctly, as well as compiling various libraries for the use on GREX, the local HPC cluster at University of Manitoba. The same goes for the Guillimin Support team at McGill university in Montréal, QC. This thesis is partly based on previous works by two former group members, Dr. Nicholas Sedlmayr (now Michigan State University, US-MI) and Dr. Moitri Maiti (now Joint Institute for Nuclear Research, Dubna, RU). The development of the multi-precision exact diagonalization algorithm was supported by Craig McRae.

Further, I owe thanks to my colleagues Max Kiefer, Andrew Urichuk, Louis Chen, Amin Naseri, Laura Mihalceanu, Ellen Wiedemann, and Dr. Maxime Lanoy for many helpful discussions, and for their support during typesetting and proofreading this thesis. I also would like to thank Dr. Elroy Friesen and University of Manitoba Singers for keeping me in touch with the non-physics world throughout the year. Finally, I also would like to thank my family for supporting me during my programme and during the thesis project.

Hiermit erkläre ich, C. Philipp Jaeger, dass ich die Diplomarbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe. Wörtlich oder sinngemäß übernommenes Gedankengut wurde als Entlehnung kenntlich gemacht. Die Arbeit wurde keinem Prüfungsamt vorgelegt und bisher nicht veröffentlicht.

Correlation functions

Correlation functions (CF) are a very basic and widely used concept in physics. In many-particle physics, they are particularly interesting, because they contain all information about how particles interact. In this appendix, we will briefly introduce this concept. We also discuss the Wick theorem, which provides an easy way to calculate higher-order correlations, and how to use CF to obtain local quantities such as, for example, the susceptibility. Later on, we show some simulations of CF in the Su-Schrieffer-Heeger (SSH) model.

A.1

Green’s functions

In Quantum field theory (QFT), a common way of treating CF is the notion of Green’s functions (GF). In general, a GF G is the inverse of a differential operator L,

Ly = f (A.1.1) with parameters y. The fundamental solution of this equation is given by a right-inverse operator G, which fulfills LG = 1, and therefore

y = Gf. (A.1.2) Then,

Ly = L(Gf ) = (LG)f = f (A.1.3) To an arbitrary inhomogeneity f , the particular solution is then given by the convolution

yP= G ∗ f ≡ ˆ

The function G(x) is called a Green’s function. These functions are often used in QFT to calculate expectation values like for example CF.

Definition A.1 _{(Correlation function)}

Let ˆφ(~xm) be a scalar field depending on the parameters ~xm. Then, the correlation function is D ˆ_{φ(~x1)... ˆφ(~xn)}E

= hvac| N ˆφ(~x1)... ˆφ(~xn) |vaci , (A.1.5) where N is the normal-ordering operator.

For example, for fermionic operators cj, c†j on lattice sites j at T = 0, the two-point correlation

hcic†ji = δij (A.1.6)

is only non-zero if i and j are identical.

At zero temperature, (two-point) GF are related to two-point CF by

G(x − x′_{) = −ih ˆφ(x) ˆφ}†_(x′_{)i. (A.1.7)} For a detailed discussion we refer to ref. [74].