Non-Abelian gauge potentials driven localization transition in quasiperiodic optical lattices

Derived from quantum waves immersed in an Abelian gauge potential, the quasiperiodic Aubry-André-Harper (AAH) model is a simple yet powerful Hamiltonian to study the Anderson localization of ultracold atoms. Here, we investigate the localization properties of ultracold atoms in quasiperiodic optical lattices subject to a non-Abelian gauge potential, which are depicted by non-Abelian AAH models. We identify that the non-Abelian AAH models can bear the self-duality. We analyze the localization of such non-Abelian self-dual optical lattices, revealing a rich phase diagram driven by the non-Abelian gauge potential involved: a transition from a pure delocalization phase, then to coexistence phases, and finally to a pure localization phase. This is in stark contrast to the Abelian counterpart that does not support the coexistence phases. Our results establish the connection between localization and gauge symmetry, and thus comprise a new insight on the fundamental aspects of localization in quasiperiodic systems, from the perspective of non-Abelian gauge potential.     

Vacuum mass spectra for SU(N) self-dual Chern-Simons-Higgs systems

We study the SU(N) self-dual Chem-Simons-Higgs systems with adjoint matter coupling, and show that the sixth-order self-dual potential has p(N) gauge inequivalent degenerate minima, where p(N) is the number of partitions of N. We compute the masses of the gauge and scalar excitations in these different vacua, revealing an intricate mass structure which reflects the self-dual nature of the model.     

Self-dual Vortices in Schrödinger-Chern-Simons Model

By making use of the U(1) gauge potential decomposition theory and the φ-mapping topological current theory, we investigate the Schrödinger-Chern-Simons model in the thin-film superconductor system and obtain an exact Bogomolny self-dual equation with a topological term. It is revealed that there exist self-dual vortices in the system. We study the inner topological structure of the self-dual vortices and show that their topological charges are topologically quantized and labeled by Hopf indices and Brouwer degrees. Furthermore, the vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the vector field φ.     

Dynamics of chiral (self-dual) p-forms

An action principle describing the dynamics of a p-form gauge field whose field strength is self-dual is given. The action is local, Lorentz invariant and also invariant under the standard gauge transformation of a p-form. The coupling to gravitation is described. The proposed action permits a consistent passage to quantum mechanics. The path integral is briefly discussed.     

Supersymmetric instantons and topological quantum field theory

We present an action which generates the supersymmetric self-dual equations corresponding to euclidean super Yang-Mills theory in four dimensions. By adding additional constraint fields with new local symmetries, the classical equations of this system are the usual super self-dual equations when a gauge is chosen for the constraint fields. This construction is a supersymmetric generalization of the Labastida-Pernici action which corresponds to a gauge unfixed version of Witten's topological quantum field theory. We discuss some topological prospects for this model, and the role of supersymmetric instantons in Donaldson theory.     

Vortices, tunneling, and deconfinement in bilayer quantum Hall excitonic superfluid

The physics of vortices, instantons, and deconfinement is studied for layered superfluids in connection to bilayer quantum Hall systems at filling fraction nu = 1. We develop an effective gauge theory taking into account both vortices and instantons induced by interlayer tunneling. The renormalization group flow of the gauge charge and the instanton fugacity shows that the coupling of the gauge field to vortex matter produces a continuous transition between the confining phase of free instantons and condensed vortices and a deconfined gapless superfluid where magnetic charges are bound into dipoles. The interlayer tunneling conductance and the layer-imbalance induced inhomogeneous exciton condensate are discussed in connection to experiments.     

ZN Gauge theories on a lattice and quantum memory

In the present paper we shall study (2+1)-dimensional Z_N gauge theories on a lattice. It is shown that the gauge theories have two phases, one is a Higgs phase and the other is a confinement phase. We investigate low-energy excitation modes in the Higgs phase and clarify relationship between the Z_N gauge theories and Kitaev’s model for quantum memory and quantum computations. Then we study effects of random gauge couplings (RGC) which are identified with noise and errors in quantum computations by Kitaev’s model. By using a duality transformation, it is shown that time-independent RGC give no significant effects on the phase structure and the stability of quantum memory and computations. Then by using the replica methods, we study Z_N gauge theories with time-dependent RGC and show that nontrivial phase transitions occur by the RGC.     

The model for self-dual chiral bosons as a Hodge theory

We consider (1+1) dimensional theory for a single self-dual chiral boson as a classical model for gauge theory. Using the Batalin–Fradkin–Vilkovisky (BFV) technique, the nilpotent BRST and anti-BRST symmetry transformations for this theory have been studied. In this model other forms of nilpotent symmetry transformations like co-BRST and anti-co-BRST, which leave the gauge-fixing part of the action invariant, are also explored. We show that the nilpotent charges for these symmetry transformations satisfy the algebra of the de Rham cohomological operators in differential geometry. The Hodge decomposition theorem on compact manifold is also studied in the context of conserved charges.     

Generalised actions for lattice gauge models

We consider simple modifications of the conventional Wilson action for lattice gauge theory. An SU(2) action is defined on “plaquettes” of 2×1 links. It is found to possess phase transitions in three- and four-dimensional realisations of the model. A similar model with gauge group Z(2) is also studied, and found to have two phases in three and four dimensions. We discuss the phase structure of Z(N) gauge models in four dimensions with several coupling constants and present phase diagrams for Z(4), Z(5) and Z(6).     

Interference in Quantum Field Theory: Detecting Ghosts with Phases

This study discusses the implications of the principle of locality for interference in quantum field theory. As an example, it considers the interaction of two charges via a mediating quantum field and the resulting interference pattern in the Lorenz gauge. Using the Heisenberg picture, it is proposed that detecting relative phases or entanglement between two charges in an interference experiment is equivalent to accessing empirically the gauge degrees of freedom associated with the so-called ghost (scalar) modes of the field in the Lorenz gauge. These results imply that ghost modes are measurable and hence physically relevant, contrary to what is usually thought. They also raise interesting questions about the relation between the principle of locality and the principle of gauge-invariance. This analysis also applies to linearized quantum gravity in the harmonic gauge, and hence has implications for the recently proposed entanglement-based witnesses of non-classicality in gravity.     

The Schwinger model: A view from the temporal gauge

We present a new operator solution of the Schwinger model, i.e., of massless quantum electrodynamics in 1 + 1 dimensions in the temporal gauge A⁰ = 0. This gauge is well-suited for the treatment of static external charges. The energy functional reflects the immediate onset of pair creation of massless fermions. We show that every point charge is screened completely by a Yukawa-like polarization charge cloud of radius $\text{π}\text{e}$, e the coupling constant.     

On the dual equivalence of the Born–Infeld–Chern–Simons model coupled to dynamical U(1) charged matter

We study the equivalence between a nonlinear self-dual model (NSD) with the Born–Infeld–Chern–Simons (BICS) theory using an iterative gauge embedding procedure that produces the duality mapping, including the case where the NSD model is minimally coupled to dynamical, U(1) charged fermionic matter. The duality mapping introduces a current–current interaction term while at the same time the minimal coupling of the original nonlinear self-dual model is replaced by a non-minimal magnetic like coupling in the BICS side.     

BRST Formalism in Self-Dual Chern-Simons Theory with Matter Fields

We apply BRST method to the self-dual Chern-Simons gauge theory with matter fields and the generators of symmetries of the system from an elegant Lie algebra structure under the operation of Poisson bracket. We discuss four different cases: abelian, nonabelian, relativistic, and nonrelativistic situations and extend the system to the whole phase space including ghost fields. In addition, we obtain the BRST charge of the field system and check its nilpotence of the BRST transformation which plays an important role such as in topological quantum field theory and string theory.     

Topological order and conformal quantum critical points

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary two-dimensional free boson, the two-dimensional Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously varying critical exponents separating phases with long-range order from a deconfined topologically ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang–Mills gauge theory with a Chern–Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.     

Born–Infeld–Chern–Simons theory: Hamiltonian embedding,duality and bosonization

In this paper we study in detail the equivalence of the recently introduced Born–Infeld self-dual model to the Abelian Born–Infeld–Chern–Simons model in 2+1 dimensions. We first apply the improved Batalin, Fradkin and Tyutin scheme, to embed the Born–Infeld self-dual model to a gauge system and show that the embedded model is equivalent to Abelian Born–Infeld–Chern–Simons theory. Next, using Buscher's duality procedure, we demonstrate this equivalence in a covariant Lagrangian formulation and also derive the mapping between the n-point correlators of the (dual) field strength in Born–Infeld–Chern–Simons theory and of basic field in Born–Infeld self-dual model. Using this equivalence, the bosonization of a massive Dirac theory with a non-polynomial Thirring type current–current coupling, to leading order in (inverse) fermion mass is also discussed. We also rederive it using a master Lagrangian. Finally, the operator equivalence between the fermionic current and (dual) field strength of Born–Infeld–Chern–Simons theory is deduced at the level of correlators and using this the current–current commutators are obtained.     

The phase structure of Einstein–Cartan theory

In the Einstein–Cartan theory of torsion-free gravity coupling to massless fermions, the four-fermion interaction is induced and its strength is a function of the gravitational and gauge couplings, as well as the Immirzi parameter. We study the dynamics of the four-fermion interaction to determine whether effective bilinear terms of massive fermion fields are generated. Calculating one-particle-irreducible two-point functions of fermion fields, we identify three different phases and two critical points for phase transitions characterized by the strength of four-fermion interaction: (1) chiral symmetric phase for massive fermions in strong coupling regime; (2) chiral symmetric broken phase for massive fermions in intermediate coupling regime; (3) chiral symmetric phase for massless fermions in weak coupling regime. We discuss the scaling-invariant region for an effective theory of massive fermions coupled to torsion-free gravity in the low-energy limit.     

Remarks on the Quantization of Self-Dual Fields

We investigate the Srivastava's and the Floreanini-Jackiw's formalisms of self-dual fields. By imposing an additional constraint on phase space we overcome the former's non-unitarity at the quantum level and simultaneously translate the former into the latter. Through introducing a parameter to extend the self-duality condition we show that both formalisms guarantee that it is only the self-dual field that exists in a consistent Hamiltonian quantum theory.     

Electric-magnetic duality in non-abelian gauge theories

The duality transformation of the vacuum expectation value of the operator which creates magnetic vortices (the 't Hooft loop operator in the Higgs phase), is performed in the radial gauge (x_uA_u^a(x) = 0). It is found that in the weak coupling region (small g) of a pure Yang-Mills theory the dual operator creates electric vortices whose strength is $\text{1}\text{g}$. The theory is self-dual in this region, and the effective coupling of the dual Lagrangian is $\text{1}\text{g}$. (It is self-dual also in the extreme strong coupling region.) Thus the above duality transformation reduces to electric-magnetic duality where the electric field in the 't Hooft loop operators transforms into a magnetic field in the dual operator. In a spontaneously broken gauge theory these results are valid only within the region where the vortices (or the monopoles) are concentrated, or in directions of the algebra space of unbroken symmetry, as self-duality holds only for this subset of fields. Noting that the 't Hooft loop operator project into the subspace of these field configurations we find that it is an electric-magnetic duality for the spontaneously broken theory as well. In the strong coupling region a strong coupling expansion in powers $\text{1}\text{g}$ is suggested.     

Non-perturbative aspects in quantum field theory: Proceedings of Les Houches Winter Advanced Study Institute,March 1978

Until recent years field theories were only studied from standard perturbation series. These are equivalent to small fluctuation expansions around the classical ground state configurations in which the fields are space-time independent. It has been gradually realized that this procedure may miss some crucial physical features of the theory. In particular the absence of free quarks cannot be explained in quantum chromodynamics from this perturbative viewpoint. One is led to deal with the non-linearities in an essential way. This issue aims at covering the recent advances in this direction typically since 1975 and the issue 23C of Physics Reports.One way of going beyond perturbation theory is to make use of non-perturbative classical solutions which are in general very difficult to obtain. In Yang-Mills theories remarkable advances have been made in the search for self-dual solutions. They are described in the lectures (I, II) by E. Corrigan and R. Stora et al. The quantum meaning of these solutions and the technical problems raised by quantizing fluctuations around them are discussed by J.L. Gervais (III) and D. Gross (IV). The most recent topics of this subject deal with imaginary time solutions, the so-called instantons, which describe quantum tunneling in the semi-classical approximation. Other field configurations such as merons, are also discussed in relation with quark confinement. An attempt to build a general picture of strong interactions on these grounds is displayed in (IV).Recent advances concerning the solitons which are real time, classical Minkowski solutions, are discussed by D. Olive (V). These solutions describe generalized Dirac magnetic monopoles in non-Abelian gauge theories. The resulting model contains confined magnetic charges and unconfined electric charges. If the magnetic and elastic characters can be interchanged by a dual transformation, as discussed by F. Englert et al. in (VI), this would provide a model for quark confinement. This duality transformation has been spelled out in (IX) by C. Itzykson who proves the existence of a confinement phase transition in a lattice model with Z₂ gauge symmetry.Another non-linear feature of Yang-Mills theory is that no single gauge condition can be implemented over large field fluctuations. This is discussed in (VII) by Sciuto and also in (III).Another mechanism of confinement is proposed by McCoy and Wu (VIII) in which the propagator has a cut instead of a pole, as it does happen in the two-dimensional Ising model.The lectures of J. Zinn-Justin and G. Parisi, (X) and (XI) deal with the asymptotic estimation of the large order behaviour of perturbation theory, which one can obtain by semi-classical methods. The use of these methods is either to provide a way of improving practical perturbative calculations, or to characterize possible ambiguities due to vacuum tunneling. Additional problems raised in renormalizable theories are discussed in (XI).Another non-perturbative approach is to examine the large N limit of an SU(N) gauge theory. At leading order, only planar Feynman diagrams contribute. They are discussed in (XII) by E. Brézin. There are very subtle questions about the corresponding exact summation in two dimensions which are studied by T.T. Wu in (XV).In two-dimensional field theories exact solutions for the S-matrix have been recently discovered. They are discussed by Karowski in (XIII). In connection with this problem, Lüscher (XIV) has shown the existence of non-linear conserved charges which ensure the absence of particle production. These models are particularly interesting since the mass spectrum cannot be obtained from coupling constant expansion.Finally several related topics in statistical physics are discussed. D. Nelson (XVI) reviews the beautiful properties of the two-dimensional X-Y model which is an ideal example of mechanisms invoked in hadronic field theories. A. Luther (XVII) discussed an attempt to extend to a higher number of dimensions the bosonization of fermion theories which is so powerful in two dimensions. G. Toulouse (XVIII) describes modern ideas in spin glass phase transitions, a possible testing ground for gauge field theories besides its obvious intrinsic interest.