non-Abelian Lattice Gauge Theories Research Articles

Non-Abelian string breaking phenomena with matrix product states

Using matrix product states, we explore numerically the phenomenology of string breaking in a non-Abelian lattice gauge theory, namely 1+1 dimensional SU(2). The technique allows us to study the static potential between external heavy charges, as traditionally explored by Monte Carlo simulations, but also to simulate the real-time dynamics of both static and dynamical fermions, as the latter are fully included in the formalism. We propose a number of observables that are sensitive to the presence or breaking of the flux string, and use them to detect and characterize the phenomenon in each of these setups.

Entanglement entropy and nonabelian gauge symmetry

Entanglement entropy has proven to be an extremely useful concept in quantum field theory. Gauge theories are of particular interest, but for these systems the entanglement entropy is not clearly defined because the physical Hilbert space does not factor as a tensor product according to regions of space. Here we review a definition of entanglement entropy that applies to abelian and nonabelian lattice gauge theories. This entanglement entropy is obtained by embedding the physical Hilbert space into a product of Hilbert spaces associated to regions with boundary. The latter Hilbert spaces include degrees of freedom on the entangling surface that transform like surface charges under the gauge symmetry. These degrees of freedom are shown to contribute to the entanglement entropy, and the form of this contribution is determined by the gauge symmetry. We test our definition using the example of two-dimensional Yang–Mills theory, and find that it agrees with the thermal entropy in de Sitter space, and with the results of the Euclidean replica trick. We discuss the possible implications of this result for more complicated gauge theories, including quantum gravity.

Towards quantum simulating QCD

Quantum link models provide an alternative non-perturbative formulation of Abelian and non-Abelian lattice gauge theories. They are ideally suited for quantum simulation, for example, using ultracold atoms in an optical lattice. This holds the promise to address currently unsolvable problems, such as the real-time and high-density dynamics of strongly interacting matter, first in toy-model gauge theories, and ultimately in QCD.

Constrained dynamics via the Zeno effect in quantum simulation: implementing non-Abelian lattice gauge theories with cold atoms.

We show how engineered classical noise can be used to generate constrained Hamiltonian dynamics in atomic quantum simulators of many-body systems, taking advantage of the continuous Zeno effect. After discussing the general theoretical framework, we focus on applications in the context of lattice gauge theories, where imposing exotic, quasilocal constraints is usually challenging. We demonstrate the effectiveness of the scheme for both Abelian and non-Abelian gauge theories, and discuss how engineering dissipative constraints substitutes complicated, nonlocal interaction patterns by global coupling to laser fields.

Simulation of non-Abelian gauge theories with optical lattices

Many phenomena occurring in strongly correlated quantum systems still await conclusive explanations. The absence of isolated free quarks in nature is an example. It is attributed to quark confinement, whose origin is not yet understood. The phase diagram for nuclear matter at general temperatures and densities, studied in heavy-ion collisions, is not settled. Finally, we have no definitive theory of high-temperature superconductivity. Though we have theories that could underlie such physics, we lack the tools to determine the experimental consequences of these theories. Quantum simulators may provide such tools. Here we show how to engineer quantum simulators of non-Abelian lattice gauge theories. The systems we consider have several applications: they can be used to mimic quark confinement or to study dimer and valence-bond states (which may be relevant for high-temperature superconductors).

Schwinger-Dyson equations in large-Nquantum field theories and nonlinear random processes

We propose a stochastic method for solving Schwinger-Dyson equations in large-$N$ quantum field theories. Expectation values of single-trace operators are sampled by stationary probability distributions of the so-called nonlinear random processes. The set of all the histories of such processes corresponds to the set of all planar diagrams in the perturbative expansions of the expectation values of singlet operators. We illustrate the method on examples of the matrix-valued scalar field theory and the Weingarten model of random planar surfaces on the lattice. For theories with compact field variables, such as sigma models or non-Abelian lattice gauge theories, the method does not converge in the physically most interesting weak-coupling limit. In this case one can absorb the divergences into a self-consistent redefinition of expansion parameters. A stochastic solution of the self-consistency conditions can be implemented as a ``memory'' of the random process, so that some parameters of the process are estimated from its previous history. We illustrate this idea on the two-dimensional $O(N)$ sigma model. The extension to non-Abelian lattice gauge theories is discussed.

Homotopy and duality in non-Abelian lattice gauge theory

We propose an approach of lattice gauge theory based on a homotopic interpretation of its degrees of freedom. The basic idea is to dress the plaquettes of the lattice to view them as elementary homotopies between nearby paths. Instead of using a unique G-valued field to discretize the connection 1-form, A, we use an Aut( G)-valued field U on the edges, which discretizes the 1-form ad A , and a G-valued field V on the plaquettes, which corresponds to the Faraday tensor, F. The 1-connection, U, and the 2-connection, V, are then supposed to have a 2-curvature which vanishes. This constraint determines V as a function of U up to a phase in Z( G), the center of G. The 3-curvature around a cube is then Abelian and is interpreted as the magnetic charge contained inside this cube. Promoting the plaquettes to elementary homotopies induces a chiral splitting of their usual Boltzmann weight, w=v v ̄ , defined with the Wilson action. We compute the Fourier transform, v ̂ , of this chiral Boltzmann weight on G= SU 3 and we obtain a finite sum of generalized hypergeometric functions. The dual model describes the dynamics of three spin fields: λ P∈ G ̂ and m P∈ Z(G)≃ Z 3 , on each oriented plaquette P, and ε ab∈ Out(G) ≃ Z 2 , on each oriented edge ( ab). Finally, we sketch a geometric interpretation of this spin system in a fibered category modeled on the category of representations of G.

The dual of non-Abelian Lattice Gauge Theory

Non-Abelian Lattice Gauge Theory in Euclidean space-time of dimension d ≥ 2 whose gauge group is any compact Lie group is related to a Spin Foam Model by an exact strong-weak duality transformation. The group degrees of freedom are integrated out and replaced by combinatorial expressions involving irreducible representations and intertwiners of the gauge group. This transformation is available for the partition function, for the expectation value of observables (spin networks), and for the correlator of centre monopoles which is a ratio of partition functions in the original model and an ordinary expectation value in the dual formulation.

The dual of pure non-Abelian lattice gauge theory as a spin foam model

We derive an exact duality transformation for pure non-Abelian gauge theory regularized on a lattice. The duality transformation can be applied to gauge theory with an arbitrary compact Lie group G as the gauge group and on Euclidean space-time lattices of dimension d >= 2. It maps the partition function as well as the expectation values of generalized non-Abelian Wilson loops (spin networks) to expressions involving only finite-dimensional unitary representations, intertwiners and characters of G. In particular, all group integrations are explicitly performed. The transformation maps the strong coupling regime of non-Abelian gauge theory to the weak coupling regime of the dual model. This dual model is a system in statistical mechanics whose configurations are spin foams on the lattice.

Duals for Non-Abelian Lattice Gauge Theories by Categorical Methods

We introduce duals for non-Abelian lattice gauge theories in dimension at least three by using a categorical approach to the notion of duality in lattice theories. We first discuss the general concepts for the case of a dual-triangular lattice (i.e., the dual lattice is triangular) and find that the commutative tetrahedron condition of category theory can directly be used to define a gauge-invariant action for the dual theory. We then consider the cubic lattice (where the dual is cubic again). The case of the gauge group SU(2) is discussed in detail. We will find that in this case gauge connections of the dual theory correspond to SU(2) spin networks, suggesting that the dual is a discrete version of a quantum field theory of quantum simplicial complexes (i.e. the dual theory lives already on a quantized level in its classical form). We conclude by showing that our notion of duality leads to a hierarchy of extended lattice gauge theories closely resembling the one of extended topological quantum field theories. The appearance of this hierarchy can be understood by the quantum von Neumann hierarchy introduced by one of the authors in previous work.

Gauge invariant study of the monopole condensation in non-Abelian lattice gauge theories

We investigate the Abelian monopole condensation in finite temperature SU(2) and SU(3) pure lattice gauge theories. To this end we introduce a gauge invariant disorder parameter built up in terms of the lattice Schr\"odinger functional. Our numerical results show that the disorder parameter is different from zero and Abelian monopole condense in the confined phase. On the other hand our numerical data suggest that the disorder parameter tends to zero, in the thermodynamic limit, when the gauge coupling constant approaches the critical deconfinement value. In the case of SU(3) we also compare the different kinds of Abelian monopoles which can be defined according to the choice of the Abelian subgroups.

On a Trialgebraic Deformation of the Manin Plane

Quantum groups (or more precisely, function algebras on quantum groups), i.e. bialgebras with certain additional properties, can be gained by deforming an appropriate function algebra on a group. In a similar way, we show that a polynomial-like algebra on the (function algebra of the) Manin plane leads to a so-called trialgebra (as suggested by Crane and Frenkel), i.e. an algebraic structure possessing a coproduct and two products in a compatible way. We show how to deform this trialgebra to a noncocommutative and totally noncommutative (i.e. in both products) one. Trialgebras are of interest for various reasons, e.g. the search for topological invariants for four manifolds or the duality operation for non-Abelian lattice gauge theories, recently suggested by the authors.

Abelian nature of cooled SU(2) lattice configurations

We introduce a gauge-invariant measure of the local ``Abelianicity'' of any given lattice configuration in non-Abelian lattice gauge theory; it is essentially a comparison of the magnitude of field strength commutators to the magnitude of the field strength itself. This measure, in conjunction with the cooling technique, is used to probe the SU(2) lattice vacuum for a possible large-scale Abelian background, underlying the local short-range field fluctuations. We do, in fact, find a substantial rise in the Abelianicity over ten cooling steps or so, after which the Abelianicity tends to drop again.

Duals of nonabelian gauge theories in D dimensions

The dual of an arbitrary D-dimensional nonabelian lattice gauge theory, obtained after character expansion and integration over the gauge group, is shown to be a local lattice theory in the eigenspace of the Casimir operators. For D ≤ 4 we also provide the explicit form of the action as a product of character expansion coefficients and Racah coefficients. The representation can be used to facilitate strong coupling expansions. Furthermore, the possibility of simulations, at weak coupling, in the dual representation, is also discussed.

Multigrid Monte Carlo algorithms for SU(2) lattice gauge theory: Two versus four dimensions.

We study a multigrid method for nonabelian lattice gauge theory, the time slice blocking, in two and four dimensions. For SU(2) gauge fields in two dimensions, critical slowing down is almost completely eliminated by this method. This result is in accordance with theoretical arguments based on the analysis of the scale dependence of acceptance rates for nonlocal Metropolis updates. The generalization of the time slice blocking to SU(2) in four dimensions is investigated analytically and by numerical simulations. Compared to two dimensions, the local disorder in the four dimensional gauge field leads to kinematical problems.

Analytical study of finite-temperature non-Abelian lattice gauge theory.

An effective action for spacelike links is given in this paper. Using this action, we can obtain the effective spin model by the variational cumulant expansion method. The study of the critical coupling for the SU(2) case shows that quite accurate results can be obtained. We expect that this approach may open an avenue for the investigation of the deconfining phase transition for large N τ lattices

Theoretical analysis of acceptance rates in multigrid Monte Carlo

We analyze the kinematics of multigrid Monte Carlo algorithms by investigating acceptance rates for nonclonal Metropolis updates. With the help of a simple criterion we can decide whether or not a multigrid algorithms will have a chance to overcome critical slowing down for a given model. Our method is introduced in the context of spin models. A multigrid Monte Carlo procedure for nonabelian lattice gauge theory is described, and its kinematics is analyzed in detail.

Gauge invariant extremization

Recently, Duncan and Mawhinney introduced a method to find saddle points of the action in simulations of non-abelian lattice gauge theory. The idea, called ‘extremization’, is to minimize ∝( δS/ δA μ ) 2 instead of the action S itself as in conventional ‘cooling’. The method was implemented in an explicitly gauge variant way, however, and gauge dependence showed up in the results. Here we present a gauge invariant formulation of extremization on the lattice, applicable to any gauge group and any lattice action. The procedure is worked out in detail for U(1) and SU(N) lattice gauge theory with the plaquette action.

Gauge invariant extremization on the lattice

Recently, a method was proposed and tested to find saddle points of the action in simulations of non-abelian lattice gauge theory. The idea, called “extremization”, is to minimize ∫( δS δ Aμ ) 2 . The method was implemented in an explicitly gauge variant way, however, and gauge dependence showed up in the results. Here we show how extremization can be formulated in a way that preserves gauge invariance on the lattice. The method applies to any gauge group and any lattice action. The procedure is worked out in detail for the standard plaquette action with gauge groups U(1) and SU( N).

Kinematics of multigrid Monte Carlo

We study the kinematics of multigrid Monte Carlo algorithms by means of acceptance rates for nonlocal Metropolis update proposals. An approximation formula for acceptance rates is derived. We present a comparison of different coarse-to-fine interpolation schemes in free field theory, where the formula is exact. The predictions of the approximation formula for several interacting models are well confirmed by Monte Carlo simulations. The following rule is found: For a critical model with fundamental Hamiltonian H(phi), absence of critical slowing down can only be expected if the expansion of <H(phi + psi)> in terms of the shift psi contains no relevant (mass) term. We also introduce a multigrid update procedure for nonabelian lattice gauge theory and study the acceptance rates for gauge group SU(2) in four dimensions.