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arXiv:1607.08631 · Journal of Statistical Mechanics Theory and Experiment

Entanglement entropy of local operators in quantum Lifshitz theory

Tianci Zhou

Published Sep 1, 2016• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/2016/09/093106

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•Aug 16, 2016•67 citations
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 the time evolution of the entanglement entropy after quantum quenches in Lifshitz free scalar theories, with the dynamical exponent z > 1, by using the correlator method. For quantum quenches we consider two types of time-dependent mass functions: end-critical-protocol (ECP) and cis-critical-protocol (CCP). In both cases, at early times the entanglement entropy is independent of the subsystem size. After a critical time ( t c ), the entanglement entropy starts depending on the subsystem size significantly. This critical time t c for z = 1 in the fast ECP and CCP has been explained well by the fast quasi-particle of the quasi-particle picture. However, we find that for z > 1 this explanation does not work and t c is delayed. We explain why t c is delayed for z > 1 based on the quasiparticle picture: in essence, it is due to the competition between the fast and slow quasiparticles. At late times, in the ECP, the entanglement entropy slowly increases while, in the CCP, it is oscillating with a well defined period by the final mass scale, independently of z . We give an interpretation of this phenomena by the correlator method. As z increases, the entanglement entropy increases, which can be understood by long-range interactions due to z .
In recent years, the study of entanglement properties of quantum field theories has led to deep insights in fields as diverse as quantum gravity and condensed matter physics. Originating as effective field theories for certain quantum dimer models, the Quantum Lifshitz Model (QLM) and its generalizations are bosonic quantum field theories with anisotropic scaling symmetry between space and time. Being closely related to conformal field theories, they provide a fruitful playground, where diverse entanglement calculations can be performed analytically. \n \nIn this thesis, we concentrate on two entanglement measures, the entanglement entropy and logarithmic negativity. Motivated to extract sub-leading universal behavior, we perform analytic calculations in two and higher even dimensions. In order to make the calculations tractable, we put the QLM on compact manifolds, such as spheres and tori, where the spectrum of a certain operator appearing in the ground state of the theory is explicitly known. Mostly by means of the replica method, we then find analytic expressions for the finite sub-leading terms of the entanglement entropy and logarithmic negativity of the ground state, as well as the entanglement entropy of the excited states of the QLM. In the case of the ground state entanglement entropy, we provide analytic expressions for the sub-leading terms as functions of the dimension and the dynamical critical exponent. For the excited states we provide analytic formulae of the sub-leading coefficients as functions of the excitation numbers.
In this paper we study the time evolution of (Rényi) entanglement entropies for locally excited states in four dimensional free massless fermionic field theory. Locally excited states are defined by being acted by various local operators on the ground state. Their excesses are defined by subtracting (Rényi) entanglement entropy for the ground state from those for locally excited states. They finally approach some constant if the subsystem is given by half of the total space. They have spin dependence. They can be interpreted in terms of quasi-particles.
Abstract We investigate the entanglement properties of an infinite class of excited states in the quantum Lifshitz model (QLM). The presence of a conformal quantum critical point in the QLM makes it unusually tractable for a model above one spatial dimension, enabling the ground state entanglement entropy for an arbitrary domain to be expressed in terms of geometrical and topological quantities. Here we extend this result to excited states and find that the entanglement can be naturally written in terms of quantities which we dub ‘entanglement propagator amplitudes’ (EPAs). EPAs are geometrical probabilities that we explicitly calculate and interpret. A comparison of lattice and continuum results demonstrates that EPAs are universal. This work shows that the QLM is an example of a 2 + 1d field theory where the universal behavior of excited-state entanglement may be computed analytically.
We discuss and compute entanglement entropy (EE) in (1+1)-dimensional free Lifshitz scalar field theories with arbitrary dynamical exponents. We consider both the subinterval and periodic sublattices in the discretized theory as subsystems. In both cases, we are able to analytically demonstrate that the EE grows linearly as a function of the dynamical exponent. Furthermore, for the subinterval case, we determine that as the dynamical exponent increases, there is a crossover from an area law to a volume law. Lastly, we deform Lifshitz field theories with certain relevant operators and show that the EE decreases from the ultraviolet to the infrared fixed point, giving evidence for a possible c c -theorem for deformed Lifshitz theories.
This is an expanded version of the short report arXiv:1401.0539 , where we studied the time evolution of (Renyi) entanglement entropies for the excited state defined by acting a given local operator on the ground state. In the present paper, we introduce (Renyi) entanglement entropies of given local operators which are defined by late time values of excesses of (Renyi) entanglement entropies. They measure the degrees of freedom of local operators and characterize them in conformal field theories from the viewpoint of quantum entanglement. We explain how to compute them in free massless scalar field theories and we also investigate their time evolution. Our results can be interpreted in terms of the relativistic propagation of entangled pairs. The main new results which we acquire in the present paper are as follows. Firstly, we provide an explanation which shows that (Renyi) entanglement entropies of a specific operator are given by (Renyi) entanglement entropies whose reduced density matrices are given by the binomial distribution. That operator is constructed of only the scalar field. Secondly, we found the sum rule which (Renyi) entanglement entropies of those local operators obey. Those local operators are located separately. Moreover we argue that (Renyi) entanglement entropies of specific operators in conformal field theories are given by (Renyi) entanglement entropies whose reduced density matrices are given by the binomial distribution. These specific operators are constructed of single-species operators. We also argue that general operators obey the sum rule which we mentioned above.