Probing the conformal Calabrese-Cardy scaling with cold atoms

Abstract

We demonstrate that current experiments using cold bosonic atoms trapped in one-dimensional optical lattices and designed to measure the second-order Rényi entanglement entropy S2, can be used to verify detailed predictions of conformal field theory (CFT) and estimate the central charge c. We discuss the adiabatic preparation of the ground state at half-filling where we expect a CFT with c = 1. This can be accomplished with a very small hopping parameter J, in contrast to existing studies with density one where a much larger J is needed. We provide two complementary methods to estimate and subtract the classical entropy generated by the experimental preparation and imaging processes. We compare numerical calculations for the classical O(2) model with a chemical potential on a 1+1 dimensional lattice, and the quantum Bose-Hubbard Hamiltonian implemented in the experiments. S2 is very similar for the two models and follows closely the Calabrese-Cardy scaling, (c/8) ln(Ns), for Ns sites with open boundary conditions, provided that the large subleading corrections are taken into account.PACS numbers: 05.10. Cc, 11.15.Ha, 11.25.Hf, 37.10.Jk, 67.85.Hj, 75.10.Hk The concept of universality provides a unified approach to the critical behavior of lattice models studied in condensed matter, lattice gauge theory (LGT) and experimentally accessible systems of cold atoms trapped in optical lattices. Conformal field theory (CFT) [1, 2] offers many interesting examples of universal behavior that can be observed for lattice models in two [3][4][5], three [6], and four [7,8] dimensions. Practical simulations for these models unavoidably involve a finite volume that breaks explicitly the conformal invariance. However, this symmetry breaking follows definite patterns dictated by the restoration of the symmetry at infinite volume and allows us to identify the universality class. In view of the rich collection of interesting CFTs, it would be highly desirable to study their universality classes using quantum simulations. In order to start this ambitious program, one needs a simple concrete example to demonstrate the feasibility of the idea.In this Letter, we propose to use the setup of ongoing cold atom experiments to quantum simulate the O(2) model with a chemical potential and check the predictions of CFT for the growth of the entanglement entropy with the size of the system corresponding to the universality class of the superfluid (SF) phase. The O(2) model is an extension of the Ising model where the spin is allowed to move on a circle, making an angle θ with respect to a direction of reference. This model can be used to describe easy plane ferromagnetism and the compactness of θ leads to topological configurations called vortices. Their unbinding provides a prime example of a Berezinski-Kosterlitz-Thouless transition [9,10] in a way that has also been advocated to apply for gauge theories near the boundary of the conformal window [11]. When space and Euclidean time are treated isotropically, this model has important common ...

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