The one-dimensional quantum Ising model is one of the most studied quantum systems. In a transeverse field, it possesses a quantum critical point. For non-critical transverse field it can be described by free massive Majorana fermions. With critical transverse field, in the presence of finite longitudinal field it can be described by the $E_8$ integrable model. I will present our studies on the time evolution of entanglement entropy after a global (mass) quench in both cases, comparing scaling field theory results to exact and numerical spin chain calculations extrapolated to the scaling limit. I will discuss the linear in time growth of entanglement and its suppression, and long living entanglement oscillations.

The talk is based on the recent works:
O. A. Castro-Alvaredo, M. Lencsés, I. M. Szécsényi, J. Viti: Entanglement Dynamics after a Quench in Ising Field Theory: A Branch Point Twist Field Approach, JHEP 1912 (2019) 079, JHEP 2019 (2020) 079, arXiv:1907.11735
O. A. Castro-Alvaredo, M. Lencsés, I. M. Szécsényi, J. Viti: Entanglement Oscillations near a Quantum Critical Point, arXiv:2001.10007

We perform tunneling measurements to study the low-energy states of quantum dots formed in semiconductor nanowires and coupled strongly to a superconductor. Although non-topological in nature, these states, commonly known as Andreev bound states (ABSs) or Yu-Shiba-Rusinov states, hold relevance in the context of topological superconductivity. Indeed, ABSs are expected to evolve towards Majorana modes across a topological phase transition when in the long wire limit. In addition, understanding the ABS spectra of hybrid nanowire devices is important for the interpretation of experiments directed at the observation of Majorana modes. In this work, we perform a detailed study of the sub-gap states of a tunable spin-½ quantum dot. We first exploit the ability to control the coupling between the dot and the superconductor to explore the phase diagram of the possible ground states of the system: a spin singlet or a magnetic doublet. By applying external magnetic fields, we study the spin texture of the Andreev states and demonstrate zero-bias crossings resulting from parity-changing phase transitions.  Finally, we evaluate the impact of mesoscopic tunnel probes in the detection of ABSs. We show that the non-trivial density of states of such probes yields numerous replicas of the sub-gap states, thereby leading to crowded Andreev spectra.

Quantum information stored in local operators spreads over other degrees of freedom of the system during time evolution, known as scrambling. This process is conveniently characterized by the out-of-time-order commutators (OTOC), whose time dependence reveals salient aspects of the system's dynamics. Here we study the spatially local spin correlation function i.e., the expectation value of spin commutator and the corresponding OTOC of Dirac--Weyl systems in 1, 2 and 3 spatial dimensions. The OTOC can be written as the square of the expectation value of the commutator and the variance of the commutator. The problem features only two energy scales, the chemical potential, and the high energy cutoff, therefore the time evolution is separated into three different regions. The spin correlation function grows linearly with time initially and decays in a power-law fashion for intermediate and late times. The OTOC reveals a universal t^2 initial growth from both the commutator and the variance. Its intermediate and late time power-law decays are identical and originate from the variance of the commutator. These results indicate that Dirac--Weyl systems are slow information scramblers and are essential when additional channels for scrambling, i.e. interaction or disorder are analyzed.

Reference: https://arxiv.org/abs/1909.09376

Az előadáson a matematika sokszínűségéből, változatos alkalmazási lehetőségeiből szeretnénk egy kis ízelítőt nyújtani. Lesz szó egyebek között valószínűségről, algoritmusokról, hibamentes üzenetküldésről, a szavak geometriájáról, a matematikai és a hétköznapi gondolkodás kapcsolatáról, régi és újabb számokról, játékos és komoly kérdésekről is.

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