Abstract We investigate the entanglement entropy (EE) of circular entangling cuts in the 2 + 1-dimensional quantum Lifshitz model. The ground state in this model is a spatially conformal invariant state of the Rokhsar–Kivelson type, whose amplitude is the Gibbs weight of 2D Euclidean free boson. We show that the finite subleading corrections of EE to the area-law term, as well as the mutual information, are conformal invariants and calculate them for cylinder, disk-like and spherical manifolds with various spatial cuts. The subtlety due to the boson compactification in the replica trick is carefully addressed. We find that in the geometry of a punctured plane with many small holes, the constant piece of EE is proportional to the number of holes, indicating the ability of entanglement to detect topological information of the configuration. Finally, we compare the mutual information of two small distant disks with Cardy’s relativistic CFT scaling proposal. We find that in the quantum Lifshitz model, the mutual information also scales at long distance with a power determined by the lowest scaling dimension local operator in the theory.
Abstract We compute universal finite corrections to entanglement entropy for generalised quantum Lifshitz models in arbitrary odd spacetime dimensions. These are generalised free field theories with Lifshitz scaling symmetry, where the dynamical critical exponent z equals the number of spatial dimensions d, and which generalise the 2+1-dimensional quantum Lifshitz model to higher dimensions. We analyse two cases: one where the spatial manifold is a d-dimensional sphere and the entanglement entropy is evaluated for a hemisphere, and another where a d-dimensional flat torus is divided into two cylinders. In both examples the finite universal terms in the entanglement entropy are scale invariant and depend on the compactification radius of the scalar field.
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.
Abstract In this work we calculate the entanglement entropy of certain excited states of the quantum Lifshitz model (QLM). The QLM is a 2 + 1-dimensional bosonic quantum field theory with an anisotropic scaling symmetry between space and time that belongs to the universality class of the quantum dimer model and its generalizations. The states we consider are constructed by exciting the eigenmodes of the Laplace–Beltrami operator on the spatial manifold of the model. We perform a replica calculation and find that, whenever a simple assumption is satisfied, the bipartite entanglement entropy of any such excited state can be evaluated analytically. We show that the assumption is satisfied for all excited states on the rectangle and for almost all excited states on the sphere and provide explicit examples in both geometries. We find that the excited state entanglement entropy obeys an area law and is related to the entanglement entropy of the ground state by two universal constants. We observe a logarithmic dependence on the excitation number when all excitations are put onto the same eigenmode.
Abstract We study the von Neumann and Rényi entanglement entropy (EE) of the scale-invariant theories defined on the tori in 2 + 1 and 3 + 1 spacetime dimensions. We focus on the spatial bi-partitions of the torus into two cylinders, and allow for twisted boundary conditions along the non-contractible cycles. Various analytical and numerical results are obtained for the universal EE of the relativistic boson and Dirac fermion conformal field theories (CFTs), the fermionic quadratic band touching and the boson with z = 2 Lifshitz scaling. The shape dependence of the EE clearly distinguishes these theories, although intriguing similarities are found in certain limits. We also study the evolution of the EE when a mass is introduced to detune the system from its scale-invariant point, by employing a renormalized EE that goes beyond a naive subtraction of the area law. In certain cases we find the non-monotonic behavior of the torus EE under RG flow, which distinguishes it from the EE of a disk.
The entanglement entropy (EE) can measure the entanglement between a spatial subregion and its complement, which provides key information about quantum states. Here, rather than focusing on specific regions, we study how the entanglement entropy changes with small deformations of the entangling surface. This leads to the notion of entanglement susceptibilities. These relate the variation of the EE to the geometric variation of the subregion. We determine the form of the leading entanglement susceptibilities for a large class of scale invariant states, such as the ground states of conformal field theories, and systems with Lifshitz scaling, which includes fixed points governed by disorder. We then use the susceptibilities to derive the universal contributions that arise due to nonsmooth features in the entangling surface: corners in two dimensions, as well as cones and trihedral vertices in three dimensions. We finally discuss the generalization to R\'enyi entropies.
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.