# Out-of-time-order correlators in many-body quantum systems

**Advisor:** Pavel Stránský (IPNP MFF CUNI)

**Funding:** Fully funded

**Website:** http://www.pavelstransky.cz/

**Contact:** stransky@ipnp.troja.mff.cuni.cz

The Out-of-time-order correlator (OTOC) is a four-point correlation function of two suitably chosen probe operators, each taken at a different time in Heisenberg picture. This rather simple quantity has received a lot of attention in recent years, since it serves as a powerful indicator used in various areas of physics, such as (i) in the condensed-matter physics (scrambling of quantum information) [1], (ii) in the theory of quantum gravitation (holographic duality of black holes on AdS spaces and strongly coupled many-body systems) [2], (iii)in the quantum chaos theory due to its tight connection with classical Lyapunov exponent [3-4], and (iv) in the theory of quantum and thermal phase transitions [5-6].

In this project, you will perform a theoretical analysis of both short-time and long-time behaviour of the OTOC, especially in connection with quantum chaos, quantum phase transitions and excited-state quantum phase transitions. You will demonstrate your theoretical conclusions by a numerical study in simple many-body models (Lipkin model, Dicke model, Bose-Hubbard model etc.), which, despite their simplicity, exhibit many fundamental physical phenomena. These models are nowdays avaliable as the so called quantum simulators, so your conclusions can make a proposal for an experimental verification.

References:

[1] B. Swingle, Unscrambling the physics of out-of-time-order correlators, Nature Physics 14, 988 (2018).

[2] J. Maldacena, S.H. Shenker and D.Stanford, A bound on chaos, JHEP08, 106 (2016).

[3] K. Hashimoto, K. Murata, R. Yoshii, Out-of-time-order correlators in quantum mechanics, JHEP10, 138 (2017).

[4] J. Chávez-Carlos et al., Quantum and classical Lyapunov exponents in atom-field interaction systems, Phys. Rev. Lett. 122, 024101 (2019).

[5] R.J. Lewis-Swan, A. Safavi-Naini, J.J. Bollinger, A.M. Rey, Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model, Nature Communications 10, 1581 (2019).

[6] S. Pilatowsky-Cameo et al., Positive quantum Lyapunov exponents in experimental systems with a regular classical limit, Phys. Rev. E 101, 010202(R) (2020).