We study different aspects of quantum entanglement and its measures, including entanglement entropy in the vacuum state of a certain Lifshitz free scalar theory. We present simple intuitive arguments based on “non-local” effects of this theory that the scaling of entanglement entropy depends on the dynamical exponent as a characteristic parameter of the theory. The scaling is such that in the massless theory for small entangling regions it leads to area law in the Lorentzian limit and volume law in the z → ∞ limit. We present strong numerical evidences in (1+1) and (2+1)-dimensions in support of this behavior. In (2 + 1)-dimensions we also study some shape dependent aspects of entanglement. We argue that in the massless limit corner contributions are no more additive for large enough dynamical exponent due to non-local effects of Lifshitz theories. We also comment on possible holographic duals of such theories based on the sign of tripartite information.
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.
Starting from the
(d+1)-dimensional Lifshitz critical boson with dynamical exponent
z=2, we propose two nontrivial deformations that preserve the Rokhsar-Kivelson structure where the groundstate is encoded in an underlying, local
d-dimensional QFT. Specializing to
d=1 spatial dimension, the first deformation maps the groundstate to the quantum harmonic oscillator, leading to a gap for the scalar. We study the resulting correlation functions, and find that Cluster Decomposition is restored. The special form of the groundstate allows to analytically compute the
c-function for the entanglement entropy along a renormalization group (RG) flow for the wavefunction, which is found to be strictly decreasing as in conformal field theories (CFTs). From the entropic
c-function, we obtain the corner term for the
z=2 Lifshitz critical boson in (2+1)D in the small angle limit. The second deformation is non-Gaussian and yields a groundstate described by
SL(2,R)-conformal quantum mechanics. This deformation preserves the conformal spatial symmetry of the groundstate, and constrains the form of the correlators and entanglement entropy. As a byproduct of our calculations, we obtain explicit results for the capacity of entanglement in Lifshitz theories and discuss their interpretation. We also prove the separability of the reduced density matrix of two disconnected subsystems for real-valued RK wavefunctions, implying the vanishing of logarithmic negativity. Finally, we comment on the relations to certain stoquastic quantum spin chains, the Motzkin and Fredkin chains.
The field space entanglement entropy of a quantum field theory is obtained by integrating out a subset of its fields. We study an interacting quantum field theory consisting of massless scalar fields on a closed compact manifold M. To this model we associate its Lifshitz dual model. The ground states of both models are invariant under constant shifts. We interpret this invariance as gauge symmetry and subject the models to proper gauge fixing. By applying the heat kernel regularization one can show that the field space entanglement entropies of the massless scalar field model and of its Lifshitz dual are agreeing.
We study the universal scaling behavior of the entanglement entropy of critical theories in 2 + 1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from (a) the quasi-1D conformal field theory, (b) the bosonic quantum Lifshitz model and (c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision.
We study the capacity of entanglement in certain integrable scale invariant theories that exhibit Lifshitz scaling symmetry with a generic dynamical exponent z at the critical point. This measure characterizes the width of the eigenvalue spectrum of the reduced density matrix and is a quantum informational counterpart of heat capacity. We explore various aspects of capacity of entanglement, such as the corresponding universal terms for the ground state, its z dependence, and its temporal evolution, after a global quantum quench in two dimensions. We carefully examine the existence of a convenient entropic c -function based on this quantity both in bosonic and fermionic theories. While in the relativistic case the corresponding c -function displays a monotonic behavior under the renormalization group flow, this is not the case for the nonrelativistic theories. We also investigate the dynamics of capacity of entanglement after a mass quench and show that it follows the quasiparticle interpretation for the spreading of entanglement. Finally, we discuss how these results are consistent with the behavior of other entanglement measures, including the entanglement entropy.