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arXiv:1608.06659 · Physical review. B./Physical review. B

Quenching a quantum critical state by the order parameter: Dynamical quantum phase transitions and quantum speed limits

Markus Heyl

Published Feb 9, 2017• 69 citations
arXivHTMLPDF

Abstract

The dynamics in the vicinity of quantum phase transitions entails a variety of important and universal phenomena, such as critical slowing down or the Kibble-Zurek mechanism, which are associated with the low-energy and long-time properties at criticality. Here, the author reveals that the opposite regime of transient nonequilibrium time evolution at criticality can similarly host universal temporal behavior. By strongly perturbing a quantum critical state through quenching the associated order parameter, the system experiences a dynamical quantum phase transition with physical quantities becoming nonanalytic at critical times. These dynamical quantum phase transitions are caused by superextensive energy fluctuations induced by the particular protocol, which turn out to have profound consequences for quantum speed limits and potential restricted thermalization despite the nonintegrability of the system.

DOI
10.1103/physrevb.95.060504

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The dynamics in the vicinity of quantum phase transitions entails a variety of important and universal phenomena, such as critical slowing down or the Kibble-Zurek mechanism, which are associated with the low-energy and long-time properties at criticality. Here, the author reveals that the opposite regime of transient nonequilibrium time evolution at criticality can similarly host universal temporal behavior. By strongly perturbing a quantum critical state through quenching the associated order parameter, the system experiences a dynamical quantum phase transition with physical quantities becoming nonanalytic at critical times. These dynamical quantum phase transitions are caused by superextensive energy fluctuations induced by the particular protocol, which turn out to have profound consequences for quantum speed limits and potential restricted thermalization despite the nonintegrability of the system.
Taking the quantum Kitaev chain as an example, we have studied the universal dynamical behaviors resulting from quantum criticality under the condition of environmental temperature quench. Our findings reveal that when the quantum parameter is at its critical value, both the excess excitation density at the end of linear quench and the free relaxation exhibit universal scaling behaviors. The scaling laws observed upon quenching to the zero-temperature quantum critical point and nonzero temperature points exhibit distinct scaling exponents, which are all intimately related to the dynamical critical exponents of the quantum phase transition. Additionally, for the case of linear quench to finite temperatures, we have also discovered an intrinsic universal dynamical behavior that is independent of quantum criticality. Our research offers profound insights into the relationship between quantum criticality and nonequilibrium dynamics from two perspectives: Kibble-Zurek-like scaling behavior and free relaxation dynamics. Notably, the Kibble-Zurek-like scaling behavior in this context differs from the standard Kibble-Zurek mechanism. These two aspects jointly open up a new avenue for us to understand quantum criticality through real-time dynamical behavior, even at finite temperatures.
Near a critical point, the equilibrium relaxation time of a system diverges and any change of control/thermodynamic parameters leads to nonequilibrium behavior. The Kibble-Zurek problem is to determine the dynamical evolution of the system parametrically close to its critical point when the change is parametrically slow. The nonequilibrium behavior in this limit is controlled entirely by the critical point and the details of the trajectory of the system in parameter space (the protocol) close to the critical point. Together, they define a universality class consisting of critical exponents, discussed in the seminal work by Kibble and Zurek, and scaling functions for physical quantities, which have not been discussed hitherto. In this article, we give an extended and pedagogical discussion of the universal content in the Kibble-Zurek problem. We formally define a scaling limit for physical quantities near classical and quantum transitions for different sets of protocols. We report computations of a few scaling functions in model Gaussian and large-NNN problems and prove their universality with respect to protocol choice. We also introduce a protocol in which the critical point is approached asymptotically at late times with the system marginally out of equilibrium, wherein logarithmic violations to scaling and anomalous dimensions occur even in the simple Gaussian problem.
We reveal a prethermal dynamical regime upon suddenly quenching to the vicinity of a quantum phase transition in the time evolution of one-dimensional spin chains. The prethermal regime is analytically found to be self-similar and its duration is governed by the ground-state energy gap. Based on analytical insights and numerical evidence, we show that this dynamical regime universally exists independently of the location of the probe site, the presence of weak interactions, or the initial state. The resulting prethermal dynamics leads to an out-of-equilibrium scaling function of the order parameter in the vicinity of the transition. Our theory suggests that sudden quench dynamics, besides probing quantum phase transitions, may give rise to a universal critical slowing down near the critical point.
We present a formulation for investigating quench dynamics across quantum phase transitions in the presence of decoherence. We formulate decoherent dynamics induced by continuous quantum non-demolition measurements of the instantaneous Hamiltonian. We generalize the well-studied universal Kibble-Zurek behavior for linear temporal drive across the critical point. We identify a strong decoherence regime wherein the decoherence time is shorter than the standard correlation time, which varies as the inverse gap above the groundstate. In this regime, we find that the freeze-out time tˉ∼τ2νz/(1+2νz)\bar{t}\simτ^{{2νz}/({1+2νz})}tˉ∼τ2νz/(1+2νz) for when the system falls out of equilibrium and the associated freeze-out length ξˉ∼τν/(1+2νz)\barξ\simτ^{ν/({1+2νz})}ξˉ​∼τν/(1+2νz) show power-law scaling with respect to the quench rate 1/τ1/τ1/τ, where the exponents depend on the correlation length exponent ννν and the dynamical exponent zzz associated with the transition. The universal exponents differ from those of standard Kibble-Zurek scaling. We explicitly demonstrate this scaling behavior in the instance of a topological transition in a Chern insulator system. We show that the freeze-out time scale can be probed from the relaxation of the Hall conductivity. Furthermore, on introducing disorder to break translational invariance, we demonstrate how quenching results in regions of imbalanced excitation density characterized by an emergent length scale which also shows universal scaling. We perform numerical simulations to confirm our analytical predictions and corroborate the scaling arguments that we postulate as universal to a host of systems.
The nonequilibrium dynamics of a system that is located in the vicinity of a quantum critical point is affected by the critical slowing down of order-parameter correlations with the potential for novel out-of-equilibrium universality. After a quantum quench, i.e., a sudden change of a parameter in the Hamiltonian, such a system is expected to almost instantly fall out of equilibrium and undergo aging dynamics, i.e., dynamics that depends on the time passed since the quench. Investigating the quantum dynamics of an NNN-component ${\ensuremath{\varphi}}^{4}$ model coupled to an external bath, we determine this universal aging and demonstrate that the system undergoes a coarsening, governed by a critical exponent that is unrelated to the equilibrium exponents of the system. We analyze this behavior in the large-NNN limit, which is complementary to our earlier renormalization-group analysis, allowing in particular the direct investigation of the order-parameter dynamics in the symmetry-broken phase and at the upper critical dimension. By connecting the long-time limit of fluctuations and response, we introduce a distribution function that shows that the system remains nonthermal and exhibits quantum coherence even on long time scales.
Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations. These fluctuations play a dominant role in the quantum critical region surrounding the transition point, where the dynamics are governed by the universal properties associated with the QPT. The resulting quantum criticality has been explored by probing linear response for systems near thermal equilibrium. While time dependent phenomena associated with classical phase transitions have been studied in various scientific fields, understanding critical real-time dynamics in isolated, non-equilibrium quantum systems is of fundamental importance both for exploring novel approaches to quantum information processing and realizing exotic new phases of matter. Here, we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations while crossing the QPT at variable speeds, we experimentally verify the quantum Kibble-Zurek mechanism (QKZM) for an Ising-type QPT, explore scaling universality, and observe corrections beyond simple QKZM predictions. This approach is subsequently used to investigate novel QPTs associated with chiral clock model providing new insights into exotic systems, and opening the door for precision studies of critical phenomena and applications to quantum optimization.