| Summary: | Abstract. Many-body localization is a phase of matter, which can occur in systems with strong disorder and interactions. One of the main characteristics of the many-body localized phase is that it cannot thermalize because localization slows or stops the propagation of information inside the system. Here, we study many-body localization in a one-dimensional transmon qubit array theoretically and numerically. Differing from frequently used approaches to many-body localization, we study the dynamical phase transition between a thermalized phase and the many-body localized phase by using a numerical method that is based on Fermi’s golden rule. This method makes it possible to distinguish the thermalized phase from the many-body localized phase and allows one to estimate how much disorder is required for the many-body localization phase transition. The distinction between the phases is made by a “soft” gap which appears at the zero-frequency and is known as a universal sign of localization. Unlike many other methods, which are used to recognize the many-body localized phase, the method in question can be easily applied to experiments. In the system that we have chosen the transmon qubits are capacitively coupled to each other, and the system is driven by a harmonic external magnetic flux that induces transitions between the energy eigenstates of the system.
We calculated the numerical results for a system with eight transmon qubits, and to make calculations simpler, we assumed that the temperature of the system is infinite. The size of the system is limited by the computational expensiveness of our method. Somewhat surprisingly, the most laborious part of the calculations proved to be the Fermi’s golden rule, which is used to calculate the transition rate spectrum.
The results show that the method in question can distinguish the many-body localized phase if the strength of disorder in the system is large, but near the many-body localized phase transition, it is challenging to observe the difference between the thermalized phase and the many-body localized phase. We calculated the transition rate spectra using different values of on-site interaction strength. The results show us that the disorder strength needed for the phase transition is smaller if the on-site interaction strength is weak or very strong. Lastly, we demonstrated the many-body localization phase transition by calculating the energy eigenstates as a function of disorder strength and observing how strongly the eigenenergies repel each other. This method proved to be relatively imprecise to specify the critical disorder strength needed for the many-body localized phase. However, one can clearly notice that the repulsion between energy levels grows weaker as the disorder strength grows stronger.
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