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arXiv:1709.00421 · Physical review. B./Physical review. B

Dynamical quantum phase transitions in systems with continuous symmetry breaking

Simon A. Weidinger, Markus Heyl, Alessandro Silva, Michael Knap

Published Oct 30, 2017• 61 citations
arXivHTMLPDF

Abstract

Interacting many-body systems that are driven far away from equilibrium can exhibit phase transitions between dynamically emerging quantum phases, which manifest as singularities in the Loschmidt echo. Whether and under which conditions such dynamical transitions occur in higher-dimensional systems with spontaneously broken continuous symmetries is largely elusive thus far. Here, we study the dynamics of the Loschmidt echo in the three-dimensional O($N$) model following a quantum quench from a symmetry-breaking initial state. The O($N$) model exhibits a dynamical transition in the asymptotic steady state, separating two phases with a finite and vanishing order parameter, that is associated with the broken symmetry. We analytically calculate the rate function of the Loschmidt echo and find that it exhibits periodic kink singularities when this dynamical steady-state transition is crossed. The singularities arise exactly at the zero crossings of the oscillating order parameter. As a consequence, the appearance of the kink singularities in the transient dynamics is directly linked to a dynamical transition in the order parameter. Furthermore, we argue, that our results for dynamical quantum phase transitions in the O($N$) model are general and apply to generic systems with continuous symmetry breaking.

DOI
10.1103/physrevb.96.134313

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Interacting many-body systems that are driven far away from equilibrium can exhibit phase transitions between dynamically emerging quantum phases, which manifest as singularities in the Loschmidt echo. Whether and under which conditions such dynamical transitions occur in higher-dimensional systems with spontaneously broken continuous symmetries is largely elusive thus far. Here, we study the dynamics of the Loschmidt echo in the three-dimensional O(NNN) model following a quantum quench from a symmetry-breaking initial state. The O(NNN) model exhibits a dynamical transition in the asymptotic steady state, separating two phases with a finite and vanishing order parameter, that is associated with the broken symmetry. We analytically calculate the rate function of the Loschmidt echo and find that it exhibits periodic kink singularities when this dynamical steady-state transition is crossed. The singularities arise exactly at the zero crossings of the oscillating order parameter. As a consequence, the appearance of the kink singularities in the transient dynamics is directly linked to a dynamical transition in the order parameter. Furthermore, we argue, that our results for dynamical quantum phase transitions in the O(NNN) model are general and apply to generic systems with continuous symmetry breaking.
We investigate dynamical quantum phase transitions (DQPTs) in the extended Agassi model following sudden quenches of same level pairing (ΣΣΣ), monopole-monopole (χχχ) and cross level pairing (ΛΛΛ) strengths. Using exact diagonalization, we characterize the non-equilibrium dynamics through the Loschmidt echo and intensive return-rate function. Quenches across equilibrium phase boundaries produce cusp-like structures in the return rate that shows non-analyticities with increasing system size. Equilibrium finite size to large scaling identifies the χχχ-driven transition as a second order phase transition similar to mean-field, Dicke, and Lipkin-Meshkov-Glick (LMG) models critical behaviour with the exponent of the order parameter β≃1/2β\simeq 1/2β≃1/2, while the ΣΣΣ- and ΛΛΛ-driven transitions are first order. We further characterize the associated dynamical singularities through a local scaling exponent ααα, with αχ≃2/3α_χ\simeq 2/3αχ​≃2/3, αΛ≃1/3α_Λ\simeq 1/3αΛ​≃1/3, and αΣ≃3/2α_Σ\simeq 3/2αΣ​≃3/2 of all three parameters. The distinct scaling behaviours across the parameter channels show that dynamical criticality is not determined solely by the equilibrium order of the transition, revealing multiple dynamical scaling regimes. These non-analyticities are associated with the Fisher zeros of the Loschmidt amplitude, which cross the real time axis at critical times and the time derivatives of the return-rate function to characterize the order of DQPTs within the extended Agassi model.
In this Letter it is shown that dynamical quantum phase transitions in Loschmidt echos control the nonequilibrium dynamics of the order parameter after particular quantum quenches in systems with broken-symmetry phases. A direct connection between Loschmidt echos and the order parameter dynamics is established which links nonequilibrium microscopic probabilities to the system's macroscopic dynamical properties. These concepts are illustrated numerically using exact diagonalization for quantum quenches in the XXZ chain with initial Néel states. An outlook is given on how to explore these predictions experimentally with ultracold gases in optical lattices.
Dynamical quantum phase transitions reveal singularities in quench dynamics, characterized by the emergence of Loschmidt echo zeros at critical times, which usually exist only in the thermodynamic limit but are absent in finite-size quantum systems. In this Letter, we propose a theoretical scheme to probe Loschmidt echo zeros in finite-size systems by applying a two-step quenching protocol, which offers an experimentally feasible approach to study Loschmidt echo zeros. Using the transverse Ising model as a test bed, we identify that the exact Loschmidt echo zeros can be always accessed by tuning the quench rate, when the quench is across the phase transition point. The associated rate function displays divergence at critical times, accompanying with the change of the dynamical topological order parameter. The critical times are influenced by the quench rate, system size, and momentum modes, embodying the interplay between finite-size effects and critical dynamics. Moreover, the generality of these observations is further confirmed in the XY and Haldane models.
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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.
Quantum states beyond thermodynamic equilibrium represent fascinating and cutting-edge research. However, the behavior of dynamical quantum phase transitions in complex open quantum systems remains poorly understood. Here, using state-of-the-art numerical approaches, we show that by quenching the qubits-oscillator coupling in a dissipative two-qubit Rabi model, the system undergoes dynamical quantum phase transitions. These transitions are characterized by kinks in the Loschmidt echo rate function at parameter values close to a thermodynamic quantum phase transition and are associated with distinct entanglement features. The two classes of critical phenomena depend on qubit interactions and entanglement, revealing different behaviors of the critical exponent of the first kink of the Loschmidt echo for interacting versus non-interacting qubits. This research enhances our understanding of non-equilibrium quantum systems and offers potential applications in quantum sensing and metrology, as it examines how dynamical transitions can enhance the sensitivity of the Loschmidt echo to the quench parameters. Dynamical quantum phase transitions can be observed when time is treated as a control parameter in non-equilibrium quantum systems. The authors show that quenching the qubits-oscillator coupling in a dissipative two-qubit system leads to different transitions depending on interactions and entanglement, with promising applications in quantum sensing and metrology.