Entanglement and separability in continuum Rokhsar-Kivelson states

Abstract

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.

Universal finite-size amplitude and anomalous entangment entropy of $z=2$ quantum Lifshitz criticalities in topological chains

“…They emerge in condensed matter models, e.g., topological chains 23 and Rydberg atoms 24 . Unlike z = 1 CFTs, quantum statistics are important for LCs 25,26 , Interactions are essential to LCs. At a system with a finite-size L, the fundamental scale for kinetic energy is K ∼ t/L z .…”

Entanglement of skeletal regions

“…In d = 2, the celebrated Fradkin-Moore formula [37] gives the bipartite (Rényi) EE for the Lifshitz groundstate, S = − log(Z A Z B /Z), up to nonuniversal terms, where Z X is the lower-dimensional CFT 2 partition function on X with Dirichlet boundary conditions, and Z is that on the entire system A ∪ B. In flat space, this formula generalizes to all dimensions and positive even z (see [36,38]), and can be straightforwardly applied to skeletal regions.…”