Approach to a rational rotation number in a piecewise isometric system

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we prove that in this region the area occupied by stable periodic orbits remains positive. The main device is the construction of an induced map on a domain with vanishing measure; this map is the product of two involutions, and each involution preserves all its atoms. Dynamically, the composition of these involutions represents linking together two sector maps; this dynamical system features an orderly array of stable periodic orbits having a smooth parameter dependence, plus irregular contributions which become negligible in the limit.Comment: LaTeX, 57 pages with 13 figure

Geometric representation of interval exchange maps over algebraic number fields

We consider the restriction of interval exchange transformations to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically, and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable interval exchange maps with zero and non-zero drift vector, and carry out some investigations of their properties. In particular, we look for evidence of the finite decomposition property: each lattice is the union of finitely many orbits.Comment: 34 pages, 8 postscript figure

On the trace identity in a model with broken symmetry

Considering the simple chiral fermion meson model when the chiral symmetry is explicitly broken, we show the validity of a trace identity -- to all orders of perturbation theory -- playing the role of a Callan-Symanzik equation and which allows us to identify directly the breaking of dilatations with the trace of the energy-momentum tensor. More precisely, by coupling the quantum field theory considered to a classical curved space background, represented by the non-propagating external vielbein field, we can express the conservation of the energy-momentum tensor through the Ward identity which characterizes the invariance of the theory under the diffeomorphisms. Our ``Callan-Symanzik equation'' then is the anomalous Ward identity for the trace of the energy-momentum tensor, the so-called ``trace identity''.Comment: 11 pages, Revtex file, final version to appear in Phys.Rev.

Exact solution (by algebraic methods) of the lattice Schwinger model in the strong-coupling regime

Using the monomer--dimer representation of the lattice Schwinger model, with $N_f =1$ Wilson fermions in the strong--coupling regime ($\beta=0$), we evaluate its partition function, $Z$, exactly on finite lattices. By studying the zeroes of $Z(k)$ in the complex plane $(Re(k),Im(k))$ for a large number of small lattices, we find the zeroes closest to the real axis for infinite stripes in temporal direction and spatial extent $S=2$ and 3. We find evidence for the existence of a critical value for the hopping parameter in the thermodynamic limit $S\rightarrow \infty$ on the real axis at about $k_c \simeq 0.39$. By looking at the behaviour of quantities, such as the chiral condensate, the chiral susceptibility and the third derivative of $Z$ with respect to $1/2k$, close to the critical point $k_c$, we find some indications for a continuous phase transition.Comment: 22 pages (6 figures

Ground State and Excitations of Spin Chain with Orbital Degeneracy

The one dimensional Heisenberg model in the presence of orbital degeneracy is studied at the SU(4) symmetric viewpoint by means of Bethe ansatz. Following Sutherland's previous work on an equivalent model, we discuss the ground state and the low-lying excitations more extensively in connection to the spin systems with orbital degeneracy. We show explicitly that the ground state is a SU(4) singlet. We study the degeneracies of the elementary excitations and the spectra of the generalized magnons consisting of these excitations. We also discuss the complex 2-strings in the context of the Bethe ansatz solutions.Comment: Revtex, 9 pages, 3 figures; typos correcte

Periodic ground state for the charged massive Schwinger model

It is shown that the charged massive Schwinger model supports a periodic vacuum structure for arbitrary charge density, similar to the common crystalline layout known in solid state physics. The dynamical origin of the inhomogeneity is identified in the framework of the bozonized model and in terms of the original fermionic variables.Comment: 19 pages, 10 figures, revised version, accepted in Phys. Rev.

Loop Variables for compact two-dimensional quantum electrodynamics

Variables parametrized by closed and open curves are defined to reformulate compact U(1) Quantum Electrodynamics in the circle with a massless fermion field. It is found that the gauge invariant nature of these variables accommodates into a regularization scheme for the Hamiltonian and current operators that is specially well suited for the study of the compact case. The zero mode energy spectrum, the value of the axial anomaly and the anomalous commutators this model presents are hence determined in a manifestly gauge invariant manner. Contrary to the non compact case, the zero mode spectrum is not equally spaced and consequently the theory does not lead to the spectrum of a free scalar boson. All the states are invariant under large gauge transformations. In particular, that is the case for the vacuum, and consequently the $\theta$-dependence does not appear.Comment: 24 pages, 1 figure, to be published in Phys. Rev.

The Strongly Coupled 't Hooft Model on the Lattice

We study the strong coupling limit of the one-flavor and two-flavor massless 't Hooft models, $large-{\cal N}_c$-color $QCD_2$, on a lattice. We use staggered fermions and the Hamiltonian approach to lattice gauge theories. We show that the one-flavor model is effectively described by the antiferromagnetic Ising model, whose ground state is the vacuum of the gauge model in the infinite coupling limit; expanding around this ground state we derive a strong coupling expansion and compute the lowest lying hadron masses as well as the chiral condensate of the gauge theory. Our lattice computation well reproduces the results of the continuum theory. Baryons are massless in the infinite coupling limit; they acquire a mass already at the second order in the strong coupling expansion in agreement with the Witten argument that baryons are the $QCD$ solitons. The spectrum and chiral condensate of the two-flavor model are effectively described in terms of observables of the quantum antiferromagnetic Heisenberg model. We explicitly write the lowest lying hadron masses and chiral condensate in terms of spin-spin correlators on the ground state of the spin model. We show that the planar limit (${\cal N}_c\longrightarrow \infty$) of the gauge model corresponds to the large spin limit ($S\longrightarrow \infty$) of the antiferromagnet and compute the hadron mass spectrum in this limit finding that, also in this model, the pattern of chiral symmetry breaking of the continuum theory is well reproduced on the lattice.Comment: LaTex, 25 pages, no figure

Chiral Symmetry Breaking on the Lattice: a Study of the Strongly Coupled Lattice Schwinger Model

We revisit the strong coupling limit of the Schwinger model on the lattice using staggered fermions and the hamiltonian approach to lattice gauge theories. Although staggered fermions have no continuous chiral symmetry, they posses a discrete axial invari ance which forbids fermion mass and which must be broken in order for the lattice Schwinger model to exhibit the features of the spectrum of the continuum theory. We show that this discrete symmetry is indeed broken spontaneously in the strong coupling li mit. Expanding around a gauge invariant ground state and carefully considering the normal ordering of the charge operator, we derive an improved strong coupling expansion and compute the masses of the low lying bosonic excitations as well as the chiral co ndensate of the model. We find very good agreement between our lattice calculations and known continuum values for these quantities already in the fourth order of strong coupling perturbation theory. We also find the exact ground state of the antiferromag netic Ising spin chain with long range Coulomb interaction, which determines the nature of the ground state in the strong coupling limit.Comment: 24 pages, Latex, no figure

Algebraic approach in unifying quantum integrable models

A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given $R$-matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known discrete and field models a new class of inhomogeneous and impurity models are obtained.Comment: Revtex, 6 pages, no figure, revised version to be published in Phys. Rev. Lett., 199