Krzysztof Sacha - Time Crystals

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What are time crystals? I have been asked this question many times. Even if I did my best to explain the phenomena related to the time crystal research, there were further questions. Where can we read about them? Original scientific publications and even review articles are often written for specialists, is there a book on that subject? It was the main motivation why I decided to write this monograph.

The field of time crystals, despite being in its infancy, is developing quite rapidly and it is impossible to describe all interesting results related to the time crystal area. If I tried to do so, I would never finish writing this book. I have made a selection and the selection is obviously subjective. I concentrate on quantum aspects of time crystals and try to describe how the research initiated by the seminal articles by Alfred Shapere and Frank Wilczek is developing and in which directions.

Basic knowledge of the reader on quantum mechanics is assumed. If I feel, subjectively, that certain formalism which I have to use is more advanced or a considered problem needs an introduction, background information is provided. I am aware that this monograph is far from being perfect and try to excuse myself by saying that the primary purpose of the book is to support the field in its infancy and to introduce researchers, especially young scientists, to the new ideas: the ideas where the time dimension is as good to host crystalline structures as the space dimensions do.

I am deeply grateful to: Bryan Dalton, Alexandre Dauphin, Dominique Delande, Krzysztof Giergiel, Weronika Golletz, Peter Hannaford, Arkadiusz Kosior, Arkadiusz Kuro, Maciej Lewenstein, Pawe Matus, Bruno Mera, Marcin Mierzejewski, Florian Mintert, Artur Miroszewski, Luis Morales-Molina, Rick Mukherjee, Yasser Omar, Frederic Sauvage, Andrzej Syrwid, Jia Wang, and Jakub Zakrzewski for the collaboration on the time crystal research. It was my great pleasure.

I would also like to thank Jakub Zakrzewski for coming to my office in 2012 with the seminal papers on time crystals by Alfred Shapere and Frank Wilczek and asking for my opinion. It was the key moment when I got attracted to the time crystal problems which I did not have any idea about.

I am extremely grateful to Peter Hannaford for very close and intensive collaboration. His enthusiasm for time crystals and many difficult questions encouraged me to put a lot of effort into the research and allowed me to understand what I was actually doing.

A new research area needs a lot of support; otherwise, it will die out. I thank very much Maciej Lewenstein for his openness to novel ideas and his support of the time crystal field.

I am grateful to Arkadiusz Kosior for the scientific cooperation and for his invaluable help in organizing the workshop on time crystals which brought to Krakw enthusiasts of time crystallization in 2019.

I thank all my students and young collaborators, especially Krzysztof Giergiel without whom the progress of our research on time crystals, if any, would be much slower.

I acknowledge the support of the National Science Centre, Poland via Project No. 2018/31/B/ST2/00349.