Shailesh Chandrasekharan

Meron-Cluster Solution of Fermion Sign Problems

We present a general strategy to solve the notorious fermion sign problem using cluster algorithms. The method applies to various systems in the Hubbard model family as well as to relativistic fermions. Here it is illustrated for nonrelativistic lattice fermions. A configuration of fermion world lines is decomposed into clusters that contribute independently to the fermion permutation sign. A cluster whose flip changes the sign is referred to as a meron. Configurations containing meron clusters contribute 0 to the path integral, while all other configurations contribute 1 . The cluster representation describes the partition function as a gas of clusters in the zero-meron sector. {copyright} {ital 1999} {ital The American Physical Society}

Quantum link models: A discrete approach to gauge theories☆

We construct lattice gauge theories in which the elements of the link matrices are represented by non-commuting operators acting in a Hilbert space. These quantum link models are related to ordinary lattice gauge theories in the same way as quantum spin models are related to ordinary classical spin systems. Here U(1) and SU (2) quantum link models are constructed explicitly. As Hamiltonian theories quantum link models are non-relativistic gauge theories with potential applications in condensed matter physics. When formulated with a fifth Euclidean dimension, universality arguments suggest that dimensional reduction to four dimensions occurs. Hence, quantum link models are also reformulations of ordinary quantum field theories and are applicable to particle physics, for example to QCD. The configuration space of quantum link models is discrete and hence their numerical treatment should be simpler than that of ordinary lattice gauge theories with a continuous configuration space.

QCD as a quantum link model

QCD is constructed as a lattice gauge theory in which the elements of the link matrices are represented by non-commuting operators acting in a Hilbert space. The resulting quantum link model for QCD is formulated with a fifth Euclidean dimension, whose extent resembles the inverse gauge coupling of the resulting four-dimensional theory after dimensional reduction. The inclusion of quarks is natural in Shamir{close_quote}s variant of Kaplan{close_quote}s fermion method, which does not require fine-tuning to approach the chiral limit. A rishon representation in terms of fermionic constituents of the gluons is derived and the quantum link Hamiltonian for QCD with a U(N) gauge symmetry is expressed in terms of glueball, meson and constituent quark operators. The new formulation of QCD is promising both from an analytic and from a computational point of view. {copyright} {ital 1999} {ital The American Physical Society}

QCD at fixed topology

Abstract Since present Monte Carlo algorithms for lattice QCD may become trapped in a fixed topological charge sector, it is important to understand the effect of calculating at fixed topology. In this Letter, we show that although the restriction to a fixed topological sector becomes irrelevant in the infinite volume limit, it gives rise to characteristic finite-size effects due to contributions from all θ -vacua. We calculate these effects and show how to extract physical results from numerical data obtained at fixed topology.

Lattice QCD with Ginsparg-Wilson fermions

Lattice QCD using fermions whose Dirac operator obeys the Ginsparg-Wilson relation is perhaps the best known formulation of QCD with a finite cutoff. It reproduces all the low energy QCD phenomenology associated with chiral symmetry at finite lattice spacings. In particular it explains the origin of massless pions due to spontaneous chiral symmetry breaking and leads to new ways to approach the U(1) problem on the lattice. Here we show these results in the path integral formulation and derive for the first time in lattice QCD a known formal continuum relation between the chiral condensate and the topological susceptibility. This relation leads to predictions for the critical behavior of the topological susceptibility near the phase transition and can now be checked in Monte Carlo simulations even at finite lattice spacings. {copyright} {ital 1999} {ital The American Physical Society}

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 μ 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 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 μ-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–antiquark, and antiquark–antiquark potentials which show screening as expected for a system with non-zero baryon density.

Progress on perfect lattice actions for QCD

Abstract We describe a number of aspects in our attempt to construct an approximately perfect lattice action for QCD. Free quarks are made optimally local on the whole renormalized trajectory and their couplings are then truncated by imposing 3-periodicity. The spectra of these short ranged fermions are excellent approximations to continuum spectra. The same is true for free gluons. We evaluate the corresponding perfect quark-gluon vertex function, identifying in particular the “perfect clover term”. First simulations for heavy quarks show that the mass is strongly renormalized, but again the renormalized theory agrees very well with continuum physics. Furthermore we describe the multigrid formulation for the non-perturbative perfect action and we present the concept of an exactly (quantum) perfect topological charge on the lattice.

A new computational approach to lattice quantum field theories

Developments in algorithms over the past decade suggest that there is a new computational approach to a class of quantum field theories. This approach is b ased on rewriting the partition function in a representation similar to the world-line repr esentation and hence we shall call it the “WL-approach”. This approach is likely to be more powerful than the conventional approach in some regions of parameter space, especially in the presence of chemical potentials or massless fermions. While world-line representations are natural in t he Hamiltonian formulation, they can also be constructed directly in Euclidean space. We first des cribe the approach and its advantages by considering the classical XY model in the presence of a chemical potential. We then argue that, CP N−1 models, models of pions on the lattice and the lattice massless Thirring model, can all be formulated and solved using the WL-approach. In particular, we discover that the WL-approach to the Thirring model leads to a novel determinantal Monte-Carlo algorithm which we call the “dynamical-bag” algorithm. Finally, we argue that a simple extension of the WL-approach to gauge theories leads to a world-sheet, “WS-approach”, in Abelian Lattice Gauge theory.

An Introduction to chiral symmetry on the lattice

Abstract The SU(Nf)L⊗SU(Nf)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 Luscher’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.

Chiral limit of strongly coupled lattice gauge theories

Abstract We construct a new and efficient cluster algorithm for updating strongly coupled U ( N ) lattice gauge theories with staggered fermions in the chiral limit. The algorithm uses the constrained monomer–dimer representation of the theory and should also be of interest to researchers working on other models with similar constraints. Using the new algorithm we address questions related to the chiral limit of strongly coupled U ( N ) gauge theories beyond the mean field approximation. We show that the infinite volume chiral condensate is non-zero in three and four dimensions. However, on a square lattice of size L we find ∑ x 〈 ψ ψ(x) ψ ψ(0)〉∼L 2−η for large L where η =0.420(3)/ N +0.078(4)/ N 2 . These results differ from an earlier conclusion obtained using a different algorithm. Here we argue that the earlier calculations were misleading due to uncontrolled autocorrelation times encountered by the previous algorithm.