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arXiv:1811.11470 · Journal of High Energy Physics

Entanglement evolution in Lifshitz-type scalar theories

M. Reza Mohammadi Mozaffar, Ali Mollabashi

Published Jan 1, 2019• 47 citations
arXivHTMLPDF

Abstract

Abstract We study propagation of entanglement after a mass quench in free scalar Lifshitz theories. We show that entanglement entropy goes across three distinct growth regimes before relaxing to a generalized Gibbs ensemble, namely, initial rapid growth, main linear growth and tortoise saturation. We show that although a wide spectrum of quasi-particles are responsible for entanglement propagation, as long as the occupation number of the zero mode is not divergent, the linear main growth regime is dominated by the fastest quasi-particle propagating on the edges of a widen light-cone. We present strong evidences in support of effective causality and therefore define an effective notion of saturation time in these theories. The larger the dynamical exponent is, the shorter the linear main growth regime becomes. Due to a pile of tortoise modes which become dominant after saturation of fast modes, exact saturation time is postponed to infinity.

DOI
10.1007/jhep01(2019)137

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Abstract We study propagation of entanglement after a mass quench in free scalar Lifshitz theories. We show that entanglement entropy goes across three distinct growth regimes before relaxing to a generalized Gibbs ensemble, namely, initial rapid growth, main linear growth and tortoise saturation. We show that although a wide spectrum of quasi-particles are responsible for entanglement propagation, as long as the occupation number of the zero mode is not divergent, the linear main growth regime is dominated by the fastest quasi-particle propagating on the edges of a widen light-cone. We present strong evidences in support of effective causality and therefore define an effective notion of saturation time in these theories. The larger the dynamical exponent is, the shorter the linear main growth regime becomes. Due to a pile of tortoise modes which become dominant after saturation of fast modes, exact saturation time is postponed to infinity.
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Abstract We study the growth of entanglement entropy (EE) of local operator excitation in the quantum Lifshitz model with dynamic exponent z = 2. Specifically, we apply a local vertex operator to the groundstate at a distance l to the entanglement cut and calculate the EE as a function of time for the state’s subsequent time evolution. We find that the excess EE compared with the groundstate is a monotonically increasing function which is vanishingly small before the onset at t ∼ l 2 and eventually saturates at a constant value proportional to the scaling dimension of the vertex operator. The quasi-particle picture can interpret the final saturation as the exhaustion of the quasi-particle pairs, while the diffusive nature of the time scale t ∼ l 2 replaces the common causality constraint in CFT calculations. To further understand this property, we compute the excess EE of a small disk probe far from the excitation point and find chromatography patterns in EE generated by quasi-particles of different propagation speeds.
Abstract We study dynamics of quantum entanglement in smooth global quenches with a finite rate, by computing the time evolution of entanglement entropy in 1 + 1 dimensional free scalar theory with time-dependent masses which start from a nonzero value at early time and either crosses or approaches zero. The time-dependence is chosen so that the quantum dynamics is exactly solvable. If the quenches asymptotically approach a critical point at late time, the early-time and late-time entropies are proportional to the time and subsystem size respectively. Their proportionality coefficients are determined by scales: in a fast limit, an initial correlation length; in a slow limit, an effective scale defined when adiabaticity breaks down. If the quenches cross a critical point, the time evolution of entropy is characterized by the scales: the initial correlation length in the fast limit and the effective correlation length in the slow limit. The entropy oscillates, and the entanglement oscillation comes from a coherence between right-moving and left-moving waves if we measure the entropy after time characterized by the quench rate. The periodicity of the late-time oscillation is consistent with the periodicity of the oscillation of zero modes which are zero-momentum value of two point functions of a fundamental field and its conjugate momentum.
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Abstract We study Krylov complexity in Lifshitz-type Dirac field theories with a generic dynamical critical exponent z . By computing the Lanczos coefficients for massless and massive cases, we analyze the growth and saturation behavior of Krylov complexity in different regimes. We incorporate a hard UV cutoff and investigate the effects of lattice discretization, revealing fundamental differences between continuum and lattice models. In the presence of a UV cutoff, Krylov complexity exhibits an initial exponential growth followed by a linear regime, with saturation values dictated by the cutoff scale. For the lattice model, we find a fundamental departure from the continuum case: due to the finite Krylov basis, Krylov complexity saturates rather than growing indefinitely. Our findings suggest that Lifshitz scaling influences operator growth and information spreading in quantum systems. We further find that increasing the Lifshitz exponent z suppresses Krylov complexity, entropy, and Lanczos growth in both massless and massive cases, while enhancing K-variance. This trend reverses under a hard UV cutoff, where complexity and entropy increase with z . In lattice models, early-time complexity and bnb_nbn​ b n decay shift with z , echoing the continuum behavior of massive and massless regimes.
We study the time evolution of the entanglement entropy in the short- and long-range-coupled harmonic oscillators that have well-defined continuum limit field theories. We first introduce a method to calculate the entanglement evolution in generic coupled harmonic oscillators after quantum quench. Then we study the entanglement evolution after quantum quench in harmonic systems in which the couplings decay effectively as $1/{r}^{d+\ensuremath{\alpha}}$ with the distance rrr. After quenching the mass from a nonzero value to zero we calculate numerically the time evolution of von Neumann and R\'enyi entropies. We show that for $1<\ensuremath{\alpha}<2$ we have a linear growth of entanglement and then saturation independent of the initial state. For $0<\ensuremath{\alpha}<1$ depending on the initial state we can have logarithmic growth or just fluctuation of entanglement. We also calculate the mutual information dynamics of two separated individual harmonic oscillators. Our findings suggest that in our system there is no particular connection between having a linear growth of entanglement after quantum quench and having a maximum group velocity or generalized Lieb-Robinson bound.