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Journal ArticleDOI

Slow quenches in a quantum Ising chain: Dynamical phase transitions and topology

TL;DR: In this article, the authors studied the slow quenching dynamics of a one-dimensional transverse Ising chain with nearest neighbor ferromagentic interactions across the quantum critical point (QCP) and analyzed the Loschmidt overlap measured using the subsequent temporal evolution of the final wave function with the final time independent Hamiltonian.
Abstract: We study the slow quenching dynamics (characterized by an inverse rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$) of a one-dimensional transverse Ising chain with nearest neighbor ferromagentic interactions across the quantum critical point (QCP) and analyze the Loschmidt overlap measured using the subsequent temporal evolution of the final wave function (reached at the end of the quenching) with the final time-independent Hamiltonian. Studying the Fisher zeros of the corresponding generalized ``partition function,'' we probe nonanalyticities manifested in the rate function of the return probability known as dynamical phase transitions (DPTs). In contrast to the sudden quenching case, we show that DPTs survive in the subsequent temporal evolution following the quenching across two critical points of the model for a sufficiently slow rate; furthermore, an interesting ``lobe'' structure of Fisher zeros emerge. We have also made a connection to topological aspects studying the dynamical topological order parameter $[{\ensuremath{ u}}_{D}(t)]$ as a function of time $(t)$ measured from the instant when the quenching is complete. Remarkably, the time evolution of ${\ensuremath{ u}}_{D}(t)$ exhibits drastically different behavior following quenches across a single QCP and two QCPs. In the former case, ${\ensuremath{ u}}_{D}(t)$ increases stepwise by unity at every DPT (i.e., $\mathrm{\ensuremath{\Delta}}{\ensuremath{ u}}_{D}=1$). In the latter case, on the other hand, ${\ensuremath{ u}}_{D}(t)$ essentially oscillates between 0 and 1 (i.e., successive DPTs occur with $\mathrm{\ensuremath{\Delta}}{\ensuremath{ u}}_{D}=1$ and $\mathrm{\ensuremath{\Delta}}{\ensuremath{ u}}_{D}=\ensuremath{-}1$, respectively), except for instants where it shows a sudden jump by a factor of unity when two successive DPTs carry a topological charge of the same sign.
Citations
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Journal ArticleDOI
Markus Heyl1
TL;DR: The theory of dynamical quantum phase transitions as mentioned in this paper attempts to identify general principles by lifting the concept of phase transitions to coherent quantum real-time evolution, by defining phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times.
Abstract: Quantum theory provides an extensive framework for the description of the equilibrium properties of quantum matter Yet experiments in quantum simulators have now opened up a route towards generating quantum states beyond this equilibrium paradigm While these states promise to show properties not constrained by equilibrium principles such as the equal a priori probability of the microcanonical ensemble, identifying general properties of nonequilibrium quantum dynamics remains a major challenge especially in view of the lack of conventional concepts such as free energies The theory of dynamical quantum phase transitions attempts to identify such general principles by lifting the concept of phase transitions to coherent quantum real-time evolution This review provides a pedagogical introduction to this field Starting from the general setting of nonequilibrium dynamics in closed quantum many-body systems, we give the definition of dynamical quantum phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times We summarize the achieved theoretical advances as well as the first experimental observations, and furthermore provide an outlook onto major open questions as well as future directions of research

362 citations

Journal ArticleDOI
Markus Heyl1
TL;DR: This review provides a pedagogical introduction to the theory of dynamical quantum phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times.
Abstract: Quantum theory provides an extensive framework for the description of the equilibrium properties of quantum matter. Yet experiments in quantum simulators have now opened up a route towards the generation of quantum states beyond this equilibrium paradigm. While these states promise to show properties not constrained by equilibrium principles, such as the equal a priori probability of the microcanonical ensemble, identifying the general properties of nonequilibrium quantum dynamics remains a major challenge, especially in view of the lack of conventional concepts such as free energies. The theory of dynamical quantum phase transitions attempts to identify such general principles by lifting the concept of phase transitions to coherent quantum real-time evolution. This review provides a pedagogical introduction to this field. Starting from the general setting of nonequilibrium dynamics in closed quantum many-body systems, we give the definition of dynamical quantum phase transitions as phase transitions in time with physical quantities becoming nonanalytic at critical times. We summarize the achieved theoretical advances as well as the first experimental observations, and furthermore provide an outlook to major open questions as well as future directions of research.

301 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the nonunitary dynamics following quenches across exceptional points in a non-Hermitian lattice realizable by optical resonators, and provided a simple framework to study dynamical and topological quantum phase transitions in non-hermitian systems.
Abstract: In closed quantum systems, dynamical phase transitions are identified by the nonanalytic behavior of the return probability as a function of time. In this work, we study the nonunitary dynamics following quenches across exceptional points in a non-Hermitian lattice realizable by optical resonators. Dynamical quantum phase transitions with topological signatures are found when an isolated exceptional point is crossed during the quench. A winding number defined by a real, noncyclic geometric phase is introduced, whose value features quantized jumps at critical times of these phase transitions and remains constant elsewhere, playing the role of a topological order parameter. This work provides a simple framework to study dynamical and topological quantum phase transitions in non-Hermitian systems.

112 citations

Journal ArticleDOI
TL;DR: In this article, the authors considered two versions of the generalized Loschmidt overlap amplitude (GLOA) and showed that the GLOA constructed using the Uhlmann approach does not show any signature of DQPTs at any nonzero initial temperature.
Abstract: Preparing an integrable system in a mixed state described by a thermal density matrix, we subject it to a sudden quench and explore the subsequent unitary dynamics. To address the question of whether the nonanalyticities, namely, the dynamical quantum phase transitions (DQPTs), persist when the initial state is mixed, we consider two versions of the generalized Loschmidt overlap amplitude (GLOA). Our study shows that the GLOA constructed using the Uhlmann approach does not show any signature of DQPTs at any nonzero initial temperature. On the other hand, a GLOA defined in the interferometric phase approach through the purifications of the time-evolved density matrix, indeed shows that nonanalyiticies in the corresponding ``dynamical free-energy density'' persist, thereby establishing the existence of mixed state dynamical quantum phase transitions (MSDQPTs). Our work provides a framework that perfectly reproduces both the nonanalyticities and also the emergent topological structure in the pure state limit. These claims are corroborated by analyzing the nonequilibrium dynamics of a transverse Ising chain initially prepared in a thermal state and subjected to a sudden quench of the transverse field.

93 citations

Journal ArticleDOI
TL;DR: In this paper, a quantum simulation of the quench dynamics of a many-body system was performed using a single superconducting qubit, by varying the control parameter over the range of momenta as the qubit's spin-1/2 state evolves on the Bloch sphere.
Abstract: Quantum information helping condensed matter research: A dynamical quantum phase transition can occur during the time evolution of a suddenly quenched quantum system, in analogy to the nonanalyticity of the free-energy density at the critical temperature of a macroscopic system. The authors succeed in observing such a transition in a quantum simulation of the quench dynamics of a many-body system. This computational experiment shows that quantum phase transitions of many-body systems can be simulated successfully using a single superconducting qubit, by varying the control parameter over the range of momenta as the qubit's spin-1/2 state evolves on the Bloch sphere.

92 citations