Periodic Reporting for period 1 - QuLeeYang (Lee-Yang theory of phase transitions in interacting quantum many-body systems)
Berichtszeitraum: 2020-04-01 bis 2022-03-31
Over the past few decades, the increasing ability to control and manipulate nanoscale systems has opened up a whole new world of opportunities to explore and exploit quantum physics. The overarching aim of this project was to investigate the connection between the phase-behavior of interacting many-body quantum systems, such as quantum computers, and their smaller individual constituents, such as single quantum bits (“qubits”). Besides fundamental aspects, the understanding of this connection is of great importance to predict – and design – the properties of future quantum devices that may be used to store, process, and transfer information in new ground-breaking ways.
To this end, the objective of the project was to develop a unified Lee-Yang theory to describe phase transitions in quantum many-body systems. The original Lee-Yang theory of classical equilibrium phase transitions connects the phase-behavior of large systems in the thermodynamic limit to the properties of small systems by considering the partition function zeros in the complex plane of an external control parameter. The crucial insight of Lee and Yang was that these complex zeros, with increasing system size, approach the real value of the control parameter for which a phase transition occurs in the thermodynamic limit. The main aim of the project was to develop a generalized theory for non-equilibrium quantum systems lacking a partition function.
During the five months that the project was ongoing, we found that generalized Lee-Yang zeros play an important role for predicting and describing both space-time phase transitions and dynamical phase transitions. By identifying appropriate generalized partition functions, many of the ideas and results of the Lee-Yang theory of classical equilibrium phase transitions can be extended to non-equilibrium phase transitions.
To this end, the objective of the project was to develop a unified Lee-Yang theory to describe phase transitions in quantum many-body systems. The original Lee-Yang theory of classical equilibrium phase transitions connects the phase-behavior of large systems in the thermodynamic limit to the properties of small systems by considering the partition function zeros in the complex plane of an external control parameter. The crucial insight of Lee and Yang was that these complex zeros, with increasing system size, approach the real value of the control parameter for which a phase transition occurs in the thermodynamic limit. The main aim of the project was to develop a generalized theory for non-equilibrium quantum systems lacking a partition function.
During the five months that the project was ongoing, we found that generalized Lee-Yang zeros play an important role for predicting and describing both space-time phase transitions and dynamical phase transitions. By identifying appropriate generalized partition functions, many of the ideas and results of the Lee-Yang theory of classical equilibrium phase transitions can be extended to non-equilibrium phase transitions.
The project started with an investigation of how Lee-Yang zeros can be used to describe and understand rare fluctuations in one of the most renowned models in statistical physics – the Ising model. We found that the zeros contain important information about the large deviation statistics of the magnetization, which describes the probability distribution of observing rare fluctuations in the thermodynamic limit.
We then investigated a type of non-equilibrium phase transitions called quantum trajectory phase transitions (or space-time phase transitions) in a micromaser, where excited atoms pump a microwave cavity. While the physical mechanisms behind these phase transitions are completely different from those in the Ising model, we found that the complex zeros of the moment-generating function describing the number of de-excited atoms play the analogue role of Lee-Yang zeros in equilibrium systems.
In addition to this, we also started to investigate how other kinds of phase transitions, such as dynamical phase transitions, can be understood in terms of generalized Lee-Yang zeros. These investigations will continue under a different grant arrangement (therefore the project was terminated after five months).
We then investigated a type of non-equilibrium phase transitions called quantum trajectory phase transitions (or space-time phase transitions) in a micromaser, where excited atoms pump a microwave cavity. While the physical mechanisms behind these phase transitions are completely different from those in the Ising model, we found that the complex zeros of the moment-generating function describing the number of de-excited atoms play the analogue role of Lee-Yang zeros in equilibrium systems.
In addition to this, we also started to investigate how other kinds of phase transitions, such as dynamical phase transitions, can be understood in terms of generalized Lee-Yang zeros. These investigations will continue under a different grant arrangement (therefore the project was terminated after five months).
The project has led to a deepened understanding of how generalized Lee-Yang zeros may be utilized to predict and describe phase transitions in quantum many-body systems.
For the Ising model, we have established a new connection between the large-deviation statistics of the magnetization and the properties of small system sizes. This connection is based on a relatively simple ansatz where the Lee-Yang zeros carry much of the information about rare fluctuations in the system.
Our findings for the space-time phase transitions in the micromaser indicate that the generalized Lee-Yang zeros of a moment-generating function may be used to predict when these phase transitions take place. Similar results have been found for predicting critical times of dynamical phase transitions.
Overall, the results constitute a promising start for continued investigations that will be carried out under a different grant arrangement, with the aim of developing a unified Lee-Yang theory for phase transitions in interacting quantum many-body systems.
For the Ising model, we have established a new connection between the large-deviation statistics of the magnetization and the properties of small system sizes. This connection is based on a relatively simple ansatz where the Lee-Yang zeros carry much of the information about rare fluctuations in the system.
Our findings for the space-time phase transitions in the micromaser indicate that the generalized Lee-Yang zeros of a moment-generating function may be used to predict when these phase transitions take place. Similar results have been found for predicting critical times of dynamical phase transitions.
Overall, the results constitute a promising start for continued investigations that will be carried out under a different grant arrangement, with the aim of developing a unified Lee-Yang theory for phase transitions in interacting quantum many-body systems.