Topological Lattice Actions

Abstract

We consider lattice field theories with topological actions, which are invariant against small deformations of the fields. Some of these actions have infinite barriers separating different topological sectors. Topological actions do not have the correct classical continuum limit and they cannot be treated using perturbation theory, but they still yield the correct quantum continuum limit. To show this, we present analytic studies of the 1-d O(2) and O(3) model, as well as Monte Carlo simulations of the 2-d O(3) model using topological lattice actions. Some topological actions obey and others violate a lattice Schwarz inequality between the action and the topological charge Q. Irrespective of this, in the 2-d O(3) model the topological susceptibility χ_t = ł< Q^2 >/V is logarithmically divergent in the continuum limit. Still, at non-zero distance the correlator of the topological charge density has a finite continuum limit which is consistent with analytic predictions. Our study shows explicitly that some classically important features of an action are irrelevant for reaching the correct quantum continuum limit.

38 pages, 12 figures

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Cited by in corpus (27)

• Topological susceptibility and the sampling of field space in $N_f=2$ lattice QCD simulations
• Non-trivial θ-Vacuum Effects in the 2-d O(3) Model
• An ideal toy model for confining, walking and conformal gauge theories: the O(3) sigma model with theta-term
• Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology
• Lattice determinations of the strong coupling
• Critical behaviour of the O(3) nonlinear sigma model with topological term at theta=pi from numerical simulations
• Measuring the Topological Susceptibility in a Fixed Sector
• Interpreting Numerical Measurements in Fixed Topological Sectors
• Quasi-Monte Carlo methods for lattice systems: a first look
• Is $N=2$ Large?
• Berezinskii-Kosterlitz-Thouless Transition with a Constraint Lattice Action
• Lattice determination of the topological susceptibility slope ${\chi }^{\prime }$ of $2d{\mathrm{CP}}^{N-1}$ models at large $N$
• Topological Susceptibility from Slabs
• Asymptotic low-temperature behavior of two-dimensional RP${}^{N-1}$ models
• Topological Lattice Actions for the 2d XY Model
• On the efficient numerical solution of lattice systems with low-order couplings
• $\theta$-dependence in the small-$N$ limit of $2d$ $C{P}^{N-1}$ models
• Drastic Reduction of Cutoff Effects in 2-d Lattice O(N) Models
• Topological Susceptibility of the 2d O(3) Model under Gradient Flow
• An Improved Single-Plaquette Gauge Action
• Non-abelian lattice gauge theory with a topological action
• Topological Susceptibility under Gradient Flow
• Mean distribution approach to spin and gauge theories
• Topology in the 2d Heisenberg Model under Gradient Flow
• Homotopy data as part of the lattice field: A first study
• Lattice meets lattice: Application of lattice cubature to models in lattice gauge theory
• Semi-vortices and cluster-vorticity: new concepts in the Berezinskii-Kosterlitz-Thouless phase transition