Dynamical quantum phase transitions from random matrix theory

David Pérez-García1, Leonardo Santilli2,3, and Miguel Tierz1

1Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
3Departamento de Matemática, Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal

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Abstract

We uncover a novel dynamical quantum phase transition, using random matrix theory and its associated notion of planar limit. We study it for the isotropic XY Heisenberg spin chain. For this, we probe its real-time dynamics through the Loschmidt echo. This leads to the study of a random matrix ensemble with a complex weight, whose analysis requires novel technical considerations, that we develop. We obtain three main results: 1) There is a third order phase transition at a rescaled critical time, that we determine. 2) The third order phase transition persists away from the thermodynamic limit. 3) For times below the critical value, the difference between the thermodynamic limit and a finite chain decreases exponentially with the system size. All these results depend in a rich manner on the parity of the number of flipped spins of the quantum state conforming the fidelity.

The great scientific achievements of recent years, such as the confirmation of the Higgs boson and gravitational waves, have been the result of experimental confirmation of theoretical predictions. The success of an experiment is more likely when the predicted numbers are more precise. Our work on quantum phase transitions aligns with this approach. We have discovered a quantum phase transition in a spin chain and have demonstrated its experimental accessibility. The technical novelty we introduce is the application of random matrix theory techniques to detect a new phase transition.

Currently, dynamical quantum phase transitions are attracting an enormous amount of effort from both the theoretical and experimental communities. These transitions cause certain measurable physical quantities in a spin chain to be discontinuous in time. We present a new example of a dynamical phase transition that exhibits several exotic features, distinguishing it from previously observed transitions. Our results are obtained from the Heisenberg XY model, a well-known and extensively studied spin chain. Two strengths of our study are its mathematical soundness and experimental verifiability. We develop tailor-made tools inspired by the discipline of random matrix theory and argue quantitatively that the transition should be detectable in a quantum device of modest size.

This work opens up two clear avenues: on the one hand, setting up an experiment to observe the dynamical phase transition, and on the other hand, extending our techniques to predict new dynamical phase transitions.

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[2] Gilles Parez, "Symmetry-resolved Rényi fidelities and quantum phase transitions", arXiv:2208.09457, (2022).

[3] Gilles Parez, "Symmetry-resolved Rényi fidelities and quantum phase transitions", Physical Review B 106 23, 235101 (2022).

[4] Ward L. Vleeshouwers and Vladimir Gritsev, "Unitary matrix integrals, symmetric polynomials, and long-range random walks", Journal of Physics A Mathematical General 56 18, 185002 (2023).

[5] Elliott Gesteau and Leonardo Santilli, "Explicit large $N$ von Neumann algebras from matrix models", arXiv:2402.10262, (2024).

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