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arXiv:1611.05016 · Physical Review A

Probing the conformal Calabrese-Cardy scaling with cold atoms

Judah Unmuth-Yockey, Jin Zhang, Philipp M. Preiss, Li‐Ping Yang, Shan-Wen Tsai, Yannick Meurice

Published Aug 1, 2017• 20 citations
arXivHTMLPDF

Abstract

We demonstrate that current experiments using cold bosonic atoms trapped in one-dimensional optical lattices and designed to measure the second-order R\'enyi entanglement entropy ${S}_{2}$ 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 and small hopping parameter $J/U$, where we expect a CFT with $c=1$. 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. ${S}_{2}$ is very similar for the two models and follows closely the Calabrese-Cardy scaling, $(c/8)ln({N}_{s})$, for ${N}_{s}$ sites with open boundary conditions, provided that the large subleading corrections are taken into account.

DOI
10.1103/physreva.96.023603

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We demonstrate that current experiments using cold bosonic atoms trapped in one-dimensional optical lattices and designed to measure the second-order R'enyi entanglement entropy S2{S}_{2}S2​ can be used to verify detailed predictions of conformal field theory (CFT) and estimate the central charge ccc. We discuss the adiabatic preparation of the ground state at half filling and small hopping parameter J/UJ/UJ/U, where we expect a CFT with c=1c=1c=1. 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+11+11+1)-dimensional lattice, and the quantum Bose-Hubbard Hamiltonian implemented in the experiments. S2{S}_{2}S2​ is very similar for the two models and follows closely the Calabrese-Cardy scaling, (c/8)ln(Ns)(c/8)ln({N}_{s})(c/8)ln(Ns​), for Ns{N}_{s}Ns​ sites with open boundary conditions, provided that the large subleading corrections are taken into account.
We calculate the von Neumann and R\'enyi bipartite entanglement entropy of the O(2)O(2)O(2) model with a chemical potential on a 1+11+11+1-dimensional Euclidean lattice with open and periodic boundary conditions. We show that the Calabrese-Cardy conformal field theory predictions for the leading logarithmic scaling of these entropies are consistent with a central charge c=1c=1c=1. This scaling survives the time-continuum limit and truncations of the microscopic degrees of freedom, modifications which allow us to connect the Lagrangian formulation to quantum Hamiltonians. At half-filling, the forms of the subleading corrections imposed by conformal field theory allow the determination of the central charge with an accuracy better than 2% for moderately sized lattices. We briefly discuss the possibility of estimating the central charge using quantum simulators.
We study the scaling of the Renyi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge c=1. We provide the analytic conformal field theory result for the second order Renyi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor network techniques that allowed to obtain the reduced density matrices of disjoint blocks of the spin-chain and to check the correctness of the predictions for Renyi and entanglement entropies from conformal field theory. In order to match these predictions, we have extrapolated the numerical results by properly taking into account the corrections induced by the finite length of the blocks to the leading scaling behavior.
We discuss the reformulation of the O(2) model with a chemical potential and the Abelian Higgs model on a 1+1 dimensional space-time lattice using the Tensor Renormalization Group (TRG) method. The TRG allows exact blocking and connects smoothly the classical Lagrangian approach to the quantum Hamiltonian approach. We calculate the entanglement entropy in the superfluid phase of the O(2) model and show that it approximately obeys the logarithmic Calabrese-Cardy scaling obtained from Conformal Field Theory (CFT). We calculate the Polyakov loop in the Abelian Higgs model and discuss the possibility of a deconfinement transition at finite volume. We propose Bose-Hubbard Hamiltonians implementable on optical lattices as quantum simulators for CFT models.
Abstract : In this project, we have developed a scheme for using cold atom systems on optical lattices as quantum simulators for lattice gauge theories. We designed cold atom experimental setups that could be used to learn about the phase diagrams of Lattice Gauge Theory models and real time evolution, which cannot be studied using conventional methods. We first constructed a sequence of theoretical steps connecting the classical O(2) model in 1+1 dimensions to a boson model that can be implemented on optical lattices, showing a proof-of-principle that quantum computing via optical lattices is possible for classical lattice models. We then extended this protocol to limits of the Abelian Higgs model, a model that exhibits important features of QCD such as confinement. Our results include construction of effective actions, testing with Monte Carlo simulations, tensor renormalization group technique development and application, mappings to quantum spin and boson Hamiltonians, and analysis of candidate systems for experimental realization. We verified that in the superfluid phase the central charge can be extracted from the entanglement entropy as predicted by conformal symmetry. We calculated and compared the Polyakov loop, an order parameter for confinement, in the two formulations, and propose that it can be a useful quantity to probe experimentally. The results from our project opens the door for exciting new research directions, such as quantum simulation of the Schwinger model and of non-Abelian models.
We study the time evolution of single interval Rényi and entanglement entropies following local quantum quenches in two dimensional conformal field theories at finite temperature for which the locally excited states have a finite temporal width ϵ. We show that, for local quenches produced by the action of a conformal primary field, the time dependence of Rényi and entanglement entropies at order ϵ2 is universal. It is determined by the expectation value of the stress tensor in the replica geometry and proportional to the conformal dimension of the primary field generating the local excitation. We also show that in CFTs with a gravity dual, the ϵ2 correction to the holographic entanglement entropy following a local quench precisely agrees with the CFT prediction. We then consider CFTs admitting a higher spin symmetry and turn on a higher spin chemical potential μ. We calculate the time dependence of the order ϵ2 correction to the entanglement entropy for small μ, and show that the contribution at order μ 2 is universal. We verify our arguments against exact results for minimal models and the free fermion theory.
Experimental realizations of topologically ordered states of matter, such as fractional quantum Hall states, with cold atoms are now within reach. In particular, optical lattices provide a promising platform for the realization and characterization of such states, where novel detection schemes enable an unprecedented microscopic understanding. Here we show that the central charge can be directly measured in current cold atom experiments using the number entropy as a proxy for the entanglement entropy. We perform density-matrix renormalization-group simulations of Hubbard-interacting bosons on coupled chains subject to a magnetic field with $\ensuremath{\alpha}=1/4$ flux quanta per plaquette. Tuning the interchain hopping, we find a transition from a trivial quasi-one-dimensional phase to the topologically ordered Laughlin state at magnetic filling factor $\ensuremath{\nu}=1/2$ for systems of three or more chains. We resolve the transition using the central charge, on-site correlations, momentum distributions, and the many-body Chern number. Additionally, we propose a scheme to experimentally estimate the central charge from Fock basis snapshots. The model studied here is experimentally realizable with existing cold atom techniques and the proposed observables pave the way for the detection and classification of a larger class of interacting topological states of matter.