Lattice regularizations of $\theta$ vacua: Anomalies and qubit models

Abstract

Anomalies are a powerful way to gain insight into possible lattice regularizations of a quantum field theory. In this work, we argue that the continuum anomaly for a given symmetry can be matched by a manifestly-symmetric, local, lattice regularization in the same spacetime dimensionality only if (i) the symmetry action is offsite, or (ii) if the continuum anomaly is reproduced exactly on the lattice. We consider lattice regularizations of a class of prototype models of QCD: the (1+1)-dimensional asymptotically-free Grassmannian nonlinear sigma models (NLSMs) with a $\theta$ term. Using the Grassmannian NLSMs as a case study, we provide examples of lattice regularizations in which both possibilities are realized. For possibility (i), we argue that Grassmannian NLSMs can be obtained from $\mathrm{SU}(N)$ antiferromagnets with a well-defined continuum limit, reproducing both the infrared physics of $\theta$ vacua and the ultraviolet physics of asymptotic freedom. These results enable the application of new classical algorithms to lattice Monte Carlo studies of these quantum field theories, and provide a viable realization suited for their quantum simulation. On the other hand, we show that, perhaps surprisingly, the conventional lattice regularization of $\theta$ vacua due to Berg and Lüscher reproduces the anomaly exactly on the lattice, providing a realization of the second possibility.

10 pages + appendices

References in corpus (14)

• Atomic Quantum Simulation of U(N) and SU(N) Non-Abelian Lattice Gauge Theories
• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Néel and Spin-Peierls ground states of two-dimensional SU(N) quantum antiferromagnets
• Efficient Basis Formulation for (1+1)-Dimensional SU(2) Lattice Gauge Theory: Spectral calculations with matrix product states
• Vacuum structure of bifundamental gauge theories at finite topological angles
• Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles
• Global inconsistency, 't Hooft anomaly, and level crossing in quantum mechanics
• The spectrum of qubitized QCD: glueballs in a $S(1080)$ gauge theory
• From asymptotic freedom to $\theta$ vacua: Qubit embeddings of the O(3) nonlinear $\sigma$ model
• Phase structure of self-dual lattice gauge theories in 4d
• Universality of a truncated sigma-model
• Efficient Cluster Algorithm for CP(N-1) Models
• Space-time symmetric qubit regularization of the asymptotically free two-dimensional O(4) model
• Induced topological gravity and anomaly inflow from Kaehler-Dirac fermions in odd dimensions

Cited by in corpus (2)

• Towards a Quantum Simulation of Nonlinear Sigma Models with a Topological Term
• Transverse Field $\gamma$-Matrix Spin Chains