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arXiv:2005.06481 · Physical review. B./Physical review. B

Unconventional critical exponents at dynamical quantum phase transitions in a random Ising chain

Daniele Trapin, Jad C. Halimeh, Markus Heyl

Published Sep 27, 2021• 24 citations
arXivHTMLPDF

Abstract

Dynamical quantum phase transitions (DQPTs) feature singular temporal behavior in transient quantum states during nonequilibrium real-time evolution. In this work we show that DQPTs in random Ising chains exhibit critical behavior with nontrivial exponents that are not integer valued and not of mean-field type. By means of an exact renormalization group transformation we estimate the exponents with high accuracy eliminating largely any finite-size effects. We further discuss how the considered dynamical phenomena can be made accessible in current Rydberg atom platforms. In this context we explore signatures of the DQPTs in the statistics of spin configuration measurements available in such architectures. Specifically, we study the statistics of clusters of consecutively aligned spins and observe a marked influence of the DQPT on the corresponding distribution.

DOI
10.1103/physrevb.104.115159

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Dynamical quantum phase transitions (DQPTs) feature singular temporal behavior in transient quantum states during nonequilibrium real-time evolution. In this work we show that DQPTs in random Ising chains exhibit critical behavior with nontrivial exponents that are not integer valued and not of mean-field type. By means of an exact renormalization group transformation we estimate the exponents with high accuracy eliminating largely any finite-size effects. We further discuss how the considered dynamical phenomena can be made accessible in current Rydberg atom platforms. In this context we explore signatures of the DQPTs in the statistics of spin configuration measurements available in such architectures. Specifically, we study the statistics of clusters of consecutively aligned spins and observe a marked influence of the DQPT on the corresponding distribution.
Dynamical quantum phase transitions (DQPTs) at critical times appear as nonanalyticities during nonequilibrium quantum real-time evolution. Although there is evidence for a close relationship between DQPTs and equilibrium phase transitions, a major challenge is still to connect to fundamental concepts such as scaling and universality. In this work, renormalization group transformations in complex parameter space are formulated for quantum quenches in Ising models showing that the DQPTs are critical points associated with unstable fixed points of equilibrium Ising models. Therefore, these DQPTs obey scaling and universality. On the basis of numerical simulations, signatures of these DQPTs in the dynamical buildup of spin correlations are found with an associated power-law scaling determined solely by the fixed point's universality class. An outlook is given on how to explore this dynamical scaling experimentally in systems of trapped ions.
We investigate the nonequilibrium quench dynamics of the one-dimensional transverse-field Ising model in both integrable and nonintegrable regimes. In particular, we report on a novel type of dynamical quantum phase transition (DQPT) that is characterized by a singular multipartite entanglement signature occurring at critical times in the post-quench dynamics. We show that this behavior is fundamentally distinct from previously studied DQPTs characterized by a nonanalytic rate function. We quantify the multipartite entanglement of the state by the quantum Fisher information and demonstrate that the DQPT belongs to a different universality class than the ground-state phase transition. Furthermore, we perform a spectral analysis of the DQPT and demonstrate that it is a genuine nonequilibrium transition arising from the constructive interference of excited states of the system during the many-body dynamics. Finally, we discuss potential experimental realizations in Rydberg platforms as well as applications in the context of quantum metrology.
Using a renormalization group approach, we solve the time evolution of random Ising spin chains with generic interactions starting from initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system is tuned through a dynamical transition, similar to the ground-state critical point, at which the local spin correlations establish true long-range temporal order. In the state with a dominant transverse field, a spin that starts in an up state loses its orientation with time, while in the ``ordered'' state it never does. As in ground-state quantum phase transitions, the dynamical transition has unique signatures in the entanglement properties of the system. When the system is initialized in a product state, the entanglement entropy grows as log(t)\mathrm{log}(t)log(t) in the two ``phases,'' while at the critical point it grows as ${\mathrm{log}}^{\ensuremath{\alpha}}(t)$, with $\ensuremath{\alpha}$ a universal number. This universal entanglement growth requires generic (``integrability breaking'') interactions to be added to the pure transverse field Ising model.
Dynamical quantum phase transitions (DQPTs) occur at times when a quantum state exhibits a nonanalytic change in its return probability. This can be viewed as the probability of collapsing the evolved state to the initial state by quantum measurement. However, the initial wave function usually has exponentially small amplitude in the late time evolved state. Here we perform statistical characterization for all the possible post-measurement states distributed according to the Born's rule, by sampling a one-dimensional quantum Ising chain after a quantum quench dynamics. The statistical ensemble can also be viewed as a mixed state when the time evolved state is subjected to maximally dephasing noise in a certain basis. We map the distribution to a statistical model and characterize its effective "energy" spectrum, and introduce the average dynamical free energy, establishing a framework for the statistical DQPTs. We show the recovering of DQPT under high-moment average and a delocalized level distribution following critical times. Through analytic continuation into the complex time plane, we demonstrate the vanishing of Yang-Lee-Fisher zeros and the emergent level crossing near critical times. Finally, we propose a measurement-based quantum computation protocol to simulate the unitary evolution via single-qubit measurements on a two-dimensional cluster state. Our results provide a way for experimentally investigating statistical DQPTs in quantum devices, shedding light on the structured circuit sampling with insights from DQPT and generalizing the understanding of mixed state due to decoherence beyond equilibrium.
We describe a new universality class of dynamical quantum phase transitions of the Loschmidt amplitude of quantum spin chains off equilibrium criticality. We demonstrate that in many cases it is possible to change the conventional linear singularity of the Loschmidt rate function into a smooth peak by tuning one parameter of the quench protocol. Exactly at the point when this change-over occurs, the singularity of the Loschmidt rate function persists, with a critical exponent equal to 12\frac{1}{2}21​. The nonequilibrium renormalization group fixed point controlling this universality class is described. An asymptotically exact renormalization group recursion relation is derived around this fixed point to obtain the critical exponent.
Abstract Dynamical phase transition in quantum many body systems is usually studied by taking it in the ground state and then quenching a parameter to a new value. We investigate here the dynamics when one performs the time evolution of a generic state and observe that the rate function related to the Loschmidt echo shows non-analytic behavior of two types, one related, and the other unrelated, to the appearance of a quantum phase transition. Specifically, we consider a quantum Ising chain in an initial configuration which is a generic superposition of the eigenstates, and follow its dynamics under the transverse Ising Hamiltonian with a constant field. Depending on the the configuration of the initial state, some singularities appear in the rate function which do not correspond to the equilibrium phase transition of the Hamiltonian. However another class of singularity is found having a connection with the quantum critical point. Some features of the singularities of the rate function have been derived analytically and it is observed that the occupancy of quasiparticle eigenstates plays the key role in triggering the non-analyticities.