Lee-Yang theory and large deviation statistics of interacting many-body systems

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School of Science | Doctoral thesis (article-based) | Defence date: 2020-12-14
Degree programme
68 + app. 51
Aalto University publication series DOCTORAL DISSERTATIONS, 197/2020
The collective behavior of large numbers of interacting particles may give rise to a phase transition. A continuing challenge is to identify the underlying principles of this phenomenon emerging in many important systems in nature and characterize the critical behavior of interacting many-body systems. In this thesis, we present a theoretical and methodological framework for predicting the phase properties of a macroscopic system based on the behavior of just a few of its constituents. To this end, we devise a direct pathway from the detection of partition function zeros by measuring or simulating fluctuating observables in systems of finite size to the characterization of criticality and large deviation statistics in interacting many-body systems. Our approach combines ideas and concepts from the finite-size scaling analysis with the Lee-Yang formalism and theories of high cumulants and large deviations, and it can be applied in a wide range of critical systems from physics, chemistry, and biology, both in theory and experiment. The thesis consists of four publications. In publications I and II, we report a novel method that makes it possible to extract the partition function zeros in interacting many-body systems of finite size solely from the fluctuations of thermodynamic observables without any prior knowledge of the partition function. To illustrate the feasibility of our approach, we use the Fisher zeros and their relation to the energy fluctuations as a tool for probing criticality in Ising models in two and three dimensions. In particular, we suggest an alternative way of extracting the universal critical exponents from measured fluctuations in finite-size systems away from the phase transition. In publications III and IV, we develop a scaling analysis of the partition function zeros to investigate the criticality in higher dimensions where the hyperscaling breaks down. We also show that even if the system does not exhibit a sharp phase transition, the partition function zeros carry important information about the large-deviation statistics of the system and its symmetry properties. To this end, we determine the rare magnetization fluctuations from the asymptotic behavior of the Lee-Yang zeros, i.e., from the Yang-Lee edge singularities. This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.
Supervising professor
Flindt, Christian, Prof., Aalto University, Department of Applied Physics, Finland
Lee-Yang theory, Fisher zeros, high-order cumulants, fluctuations, large-deviation theory, critical phenomena, phase transitions, Yang-Lee edge singularity, rare event statistics, Ising model, zipper model
Other note
  • [Publication 1]: Aydin Deger, Kay Brandner and Christian Flindt. Lee-Yang Zeros and large-deviation statistics of a molecular zipper. Physical Review E, 97, 012115, Editors’ Suggestion, January 2018.
    Full text in Acris/Aaltodoc: http://urn.fi/URN:NBN:fi:aalto-201808014287
    DOI: 10.1103/PhysRevE.97.012115 View at publisher
  • [Publication 2]: Aydin Deger and Christian Flindt. Determination of Universal Critical Exponents Using Lee-Yang Theory. Physical Review Research, 1, 023004, September 2019.
    Full text in Acris/Aaltodoc: http://urn.fi/URN:NBN:fi:aalto-201911076090
    DOI: 10.1103/PhysRevResearch.1.023004 View at publisher
  • [Publication 3]: Aydin Deger and Christian Flindt. Lee-Yang theory of the Curie-Weiss model and its rare fluctuations. Physical Review Research, 2, 033009, July 2020.
    Full text in Acris/Aaltodoc: http://urn.fi/URN:NBN:fi:aalto-202008285230
    DOI: 10.1103/physrevresearch.2.033009 View at publisher
  • [Publication 4]: Aydin Deger, Fredrik Brange and Christian Flindt. Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model. Physical Review B, 102, 174418, Editors’ Suggestion, November 2020.
    DOI: 10.1103/PhysRevB.102.174418 View at publisher