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arXiv:1403.5238 · Physical Review A

Progress towards quantum simulating the classical O(2) model

Haiyuan Zou, Yuzhi Liu A, Chen-Yen Lai, Judah Unmuth-Yockey, Li-Ping Yang, Alexei Bazavov, Z. Y. Xie, Tao Xiang, Shailesh Chandrasekharan A, Shan-Wen Tsai, Yannick Meurice

Published Dec 1, 2014• 41 citations
arXivHTMLPDF

Abstract

We connect explicitly the classical $\text{O}(2)$ model in $1+1$ dimensions, a model sharing important features with $\text{U}(1)$ lattice gauge theory, to physical models potentially implementable on optical lattices and evolving at physical time. Using the tensor renormalization-group formulation, we take the time continuum limit and check that finite-dimensional projections used in recent proposals for quantum simulators provide controllable approximations of the original model. We propose two-species Bose-Hubbard models corresponding to these finite-dimensional projections at strong coupling and discuss their possible implementations on optical lattices using a $^{87}\mathrm{Rb}$ and $^{41}\mathrm{K}$ Bose-Bose mixture.

DOI
10.1103/physreva.90.063603

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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.
This thesis presents our recent studies on quantum simulation of the Abelian Higgs lattice gauge theory with quantum systems engineered with ultracold atoms in optical lattices and quantum equilibration and thermalization dynamics in the context of Joule expansion in cold atom systems.In the first part, we give proposals for the analog quantum simulation of lattice gauge theories with cold atoms in optical lattices with current experimental techniques, aiming at maximal simplicity on both the theoretical and experimental side. We search for the suitable platform for quantum simulation of the (1+1)(1+1)(1+1)-dimensional Abelian Higgs model, which is the scalar quantum electrodynamics replacing the fermionic fields by complex scalar fields. We use a discrete tensor reformulation to smoothly connect the space-time isotropic version used in most numerical lattice simulations to the continuous-time limit corresponding to the Hamiltonian formulation. We propose to use the Bose Hubbard model as a quantum simulator to probe the conformal Calabrese-Cardy scaling of its O(2) limit with a chemical potential. We also propose to use a physical multileg ladder of atoms trapped in optical lattices and interacting with Rydberg-dressed interactions to quantum simulate the model and measure the Polyakov loop. Numeric results from the Monte Carlo, the tensor renormalization group and the density matrix renormalization group techniques are cross-checked to support our proposals.In the second part of this thesis, we investigate the Joule expansion of nonintegrable quantum systems that contain bosons or spinless fermions in one-dimensional lattices. Initially a barrier confines the particles to be in half of the system in a thermal state described by the canonical ensemble and is removed at time t=0t = 0t=0. We study the properties of the time-evolved density matrix, the diagonal ensemble density matrix and the corresponding canonical ensemble density matrix with an effective temperature determined by the total energy conservation. We discuss the equilibration and thermalization for the R\\'enyi entropy of subsystems, few-body observables and reduced density matrices in subsystems. We finally discuss the difference between bosons and fermions in finite size scaling of the effective temperature. We propose the Joule expansion as a way to dynamically create negative temperature states for fermion systems with repulsive interactions.
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We discuss the U(1)-Higgs model in two dimensions in the strongly coupled regime. If we neglect the plaquette interactions, we generate an effective theory where link variables are integrated out, producing 4-field operators. Plaquette interactions can be restored order by order as in recent calculations with staggered fermions. In the case of a SU(2) gauge theory with fermions, this strong coupling expansion can be related to the strong coupling expansion of Fermi-Hubbard models possibly implementable on optical lattice. We would like to provide a similar construction relating the U(1)-Higgs model to some Bose-Hubbard model. As a first step in this direction, we discuss a recent proposal to implement the O(2) model on optical lattices using a 87Rb and 41K Bose-Bose mixture of cold atoms.
Towards 2+12+12+1D quantum electrodynamics on a cold-atom quantum simulator
Cold atoms have become a powerful platform for quantum-simulating lattice gauge theories in higher spatial dimensions. However, such realizations have been restricted to the lowest possible truncations of the gauge field, which limit the connections one can make to lattice quantum electrodynamics. Here, we propose a feasible cold-atom quantum simulator of a (2+1)(2+1)(2+1)-dimensional U(1)(1)(1) lattice gauge theory in a spin S=1S=1S=1 truncation, featuring dynamical matter and gauge fields. We derive a mapping of this theory onto a bosonic computational basis, stabilized by an emergent gauge-protection mechanism through quantum Zeno dynamics. The implementation is based on a single-species Bose--Hubbard model realized in a tilted optical superlattice. This approach requires only moderate experimental resources already available in current ultracold-atom platforms. Using infinite matrix product state simulations, we benchmark real-time dynamics under global quenches. The results demonstrate faithful evolution of the target gauge theory and robust preservation of the gauge constraints. Our work significantly advances the experimental prospects for simulating higher-dimensional lattice gauge theories using larger gauge-field truncations.