An Introduction to Chiral Symmetry on the Lattice

The $SU(N_f)_L \otimes SU(N_f)_R$ chiral symmetry of QCD is of central importance for the nonperturbative low-energy dynamics of light quarks and gluons. Lattice field theory provides a theoretical framework in which these dynamics can be studied from first principles. The implementation of chiral symmetry on the lattice is a nontrivial issue. In particular, local lattice fermion actions with the chiral symmetry of the continuum theory suffer from the fermion doubling problem. The Ginsparg-Wilson relation implies L\"uscher's lattice variant of chiral symmetry which agrees with the usual one in the continuum limit. Local lattice fermion actions that obey the Ginsparg-Wilson relation have an exact chiral symmetry, the correct axial anomaly, they obey a lattice version of the Atiyah-Singer index theorem, and still they do not suffer from the notorious doubling problem. The Ginsparg-Wilson relation is satisfied exactly by Neuberger's overlap fermions which are a limit of Kaplan's domain wall fermions, as well as by Hasenfratz and Niedermayer's classically perfect lattice fermion actions. When chiral symmetry is nonlinearly realized in effective field theories on the lattice, the doubling problem again does not arise. This review provides an introduction to chiral symmetry on the lattice with an emphasis on the basic theoretical framework.Comment: (41 pages, to be published in Prog. Part. Nucl. Phys. Vol. 53, issue 1 (2004)

Meron-Cluster Simulation of a Chiral Phase Transition with Staggered Fermions

We examine a (3+1)-dimensional model of staggered lattice fermions with a four-fermion interaction and Z(2) chiral symmetry using the Hamiltonian formulation. This model cannot be simulated with standard fermion algorithms because those suffer from a very severe sign problem. We use a new fermion simulation technique - the meron-cluster algorithm - which solves the sign problem and leads to high-precision numerical data. We investigate the finite temperature chiral phase transition and verify that it is in the universality class of the 3-d Ising model using finite-size scaling.Comment: 21 pages, 6 figure

Green's Functions from Quantum Cluster Algorithms

We show that cluster algorithms for quantum models have a meaning independent of the basis chosen to construct them. Using this idea, we propose a new method for measuring with little effort a whole class of Green's functions, once a cluster algorithm for the partition function has been constructed. To explain the idea, we consider the quantum XY model and compute its two point Green's function in various ways, showing that all of them are equivalent. We also provide numerical evidence confirming the analytic arguments. Similar techniques are applicable to other models. In particular, in the recently constructed quantum link models, the new technique allows us to construct improved estimators for Wilson loops and may lead to a very precise determination of the glueball spectrum.Comment: 15 pages, LaTeX, with four figures. Added preprint numbe

From Spin Ladders to the 2-d O(3) Model at Non-Zero Density

The numerical simulation of various field theories at non-zero chemical potential suffers from severe complex action problems. In particular, QCD at non-zero quark density can presently not be simulated for that reason. A similar complex action problem arises in the 2-d O(3) model -- a toy model for QCD. Here we construct the 2-d O(3) model at non-zero density via dimensional reduction of an antiferromagnetic quantum spin ladder in a magnetic field. The complex action problem of the 2-d O(3) model manifests itself as a sign problem of the ladder system. This sign problem is solved completely with a meron-cluster algorithm.Comment: Based on a talk by U.-J. Wiese, 6 pages, 12 figures, to be published in computer physics communication

Rotor Spectra, Berry Phases, and Monopole Fields: from Antiferromagnets to QCD

The order parameter of a finite system with a spontaneously broken continuous global symmetry acts as a quantum mechanical rotor. Both antiferromagnets with a spontaneously broken $SU(2)_s$ spin symmetry and massless QCD with a broken $SU(2)_L \times SU(2)_R$ chiral symmetry have rotor spectra when considered in a finite volume. When an electron or hole is doped into an antiferromagnet or when a nucleon is propagating through the QCD vacuum, a Berry phase arises from a monopole field and the angular momentum of the rotor is quantized in half-integer units.Comment: 4 page

Solution of the Complex Action Problem in the Potts Model for Dense QCD

Monte Carlo simulations of lattice QCD at non-zero baryon chemical potential $\mu$ suffer from the notorious complex action problem. We consider QCD with static quarks coupled to a large chemical potential. This leaves us with an SU(3) Yang-Mills theory with a complex action containing the Polyakov loop. Close to the deconfinement phase transition the qualitative features of this theory, in particular its Z(3) symmetry properties, are captured by the 3-d 3-state Potts model. We solve the complex action problem in the Potts model by using a cluster algorithm. The improved estimator for the $\mu$-dependent part of the Boltzmann factor is real and positive and is used for importance sampling. We localize the critical endpoint of the first order deconfinement phase transition line and find consistency with universal 3-d Ising behavior. We also calculate the static quark-quark, quark-anti-quark, and anti-quark-anti-quark potentials which show screening as expected for a system with non-zero baryon density.Comment: 28 pages, 7 figure

Meron-Cluster Approach to Systems of Strongly Correlated Electrons

Numerical simulations of strongly correlated electron systems suffer from the notorious fermion sign problem which has prevented progress in understanding if systems like the Hubbard model display high-temperature superconductivity. Here we show how the fermion sign problem can be solved completely with meron-cluster methods in a large class of models of strongly correlated electron systems, some of which are in the extended Hubbard model family and show s-wave superconductivity. In these models we also find that on-site repulsion can even coexist with a weak chemical potential without introducing sign problems. We argue that since these models can be simulated efficiently using cluster algorithms they are ideal for studying many of the interesting phenomena in strongly correlated electron systems.Comment: 36 Pages, 13 figures, plain Late