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International Symposium on Lattice Field Theory (LATTICE)

Finite-size scaling of Polyakov's loop in the 2D Abelian Higgs model

Judah Unmuth-Yockey, Jin Zhang, Johannes Zeiher, Alexei Bazavov, Shan-Wen Tsai, Yannick Meurice

Published May 29, 2019• 0 citations
Publisher

Abstract

Starting with the 2D Abelian Higgs model with the quartic self-coupling taken infinitely large we study the finite-size scaling of the Polyakov loop. We find an exponential decay for large temporal extents which is dictated by the energy gap between the ground states of a system with the Polyakov loop inserted, and one without. We study this system using the tensor renormalization group, and we take the continuous-time limit to obtain a quantum Hamiltonian where gauge invariance has been maintained exactly. Comparing with numerical results from the density matrix renormalization group we find universal features of the finite-size scaling of the energy gap survive this continuous-time limit. We propose an optical-lattice ladder to quantum simulate this model, and observe the universal features of the energy gap scaling.

DOI
10.22323/1.334.0228

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Starting with the 2D Abelian Higgs model with the quartic self-coupling taken infinitely large we study the finite-size scaling of the Polyakov loop. We find an exponential decay for large temporal extents which is dictated by the energy gap between the ground states of a system with the Polyakov loop inserted, and one without. We study this system using the tensor renormalization group, and we take the continuous-time limit to obtain a quantum Hamiltonian where gauge invariance has been maintained exactly. Comparing with numerical results from the density matrix renormalization group we find universal features of the finite-size scaling of the energy gap survive this continuous-time limit. We propose an optical-lattice ladder to quantum simulate this model, and observe the universal features of the energy gap scaling.
Lattice gauge theories are fundamental to our understanding of high-energy physics. Nevertheless, the search for suitable platforms for their quantum simulation has proven difficult. We show that the Abelian Higgs model in 1+1 dimensions is a prime candidate for an experimental quantum simulation of a lattice gauge theory. To this end, 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. The eigenstates of the Hamiltonian are neutral for periodic boundary conditions, but we probe the nonzero charge sectors by introducing either a Polyakov loop or an external electric field. In both cases we obtain universal functions relating the mass gap, the gauge coupling, and the spatial size, which are invariant under the deformation of the temporal lattice spacing. We 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 check the universal features. Our results provide a path to the analog quantum simulation of lattice gauge theories with atoms in optical lattices.
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.
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.
We study the SU(2)SU(2)SU(2) gauge-Higgs model in two Euclidean dimensions using the tensor renormalization group (TRG) approach. We derive a tensor formulation for this model in the unitary gauge and compare the expectation values of different observables between TRG and Monte Carlo simulations finding excellent agreement between the two methods. In practice we find the TRG method to be far superior to Monte Carlo simulation for calculations of the Polyakov loop correlation function which is used to extract the static quark potential.
We propose a method to construct universal order parameters for quantum phase transitions in many-body lattice systems. The method exploits the H-orthogonality of a few near-degenerate lowest states of the Hamiltonian describing a given finite-size system, which makes it possible to perform finite-size scaling and take full advantage of currently available numerical algorithms. An explicit connection is established between the fidelity per site between two H-orthogonal states and the energy gap between the ground state and low-lying excited states in the finite-size system. The physical information encoded in this gap arising from finite-size fluctuations clarifies the origin of the universal order parameter. We demonstrate the procedure for the one-dimensional quantum formulation of the q-state Potts model, for q = 2, 3, 4 and 5, as prototypical examples, using finite-size data obtained from the density matrix renormalization group algorithm.
We present a gauge-invariant effective action for the Abelian-Higgs model (scalar electrodynamics) with a chemical potential $\ensuremath{\mu}$ on a (1+11+11+1)-dimensional lattice. This formulation provides an expansion in the hopping parameter $\ensuremath{\kappa}$ which we test with Monte Carlo simulations for a broad range of the inverse gauge coupling ${\ensuremath{\beta}}_{pl}=1/{g}^{2}$ and small values of the scalar self-coupling $\ensuremath{\lambda}$. In the opposite limit of infinitely large $\ensuremath{\lambda}$, the partition function can be written as a traced product of local tensors which allows us to write exact blocking formulas. Gauss's law is automatically satisfied and the introduction of $\ensuremath{\mu}$ has consequences only if we have an external electric field, g2=0{g}^{2}=0g2=0 or an explicit gauge symmetry breaking. The time-continuum limit of the blocked transfer matrix can be obtained numerically and, for g2=0{g}^{2}=0g2=0 and a spin-1 truncation, the small volume energy spectrum is identical to the low energy spectrum of a two-species Bose-Hubbard model in the limit of large on-site repulsion. We extend this procedure for finite ${\ensuremath{\beta}}_{pl}$ and derive a spin-1 approximation of the Hamiltonian. It involves new terms corresponding to transitions among the two species in the Bose-Hubbard model. We propose an optical lattice implementation involving a ladder structure.