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"Lattice Quantum Field Theory"
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Modern perspectives in lattice QCD : quantum field theory and high performance computing
\"The book is based on the lectures delivered at the XCIII Session of the Ecole de Physique des Houches, held in August, 2009. The aim of the event was to familiarize the new generation of PhD students and postdoctoral fellows with the principles and methods of modern lattice field theory, which aims to resolve fundamental, non-perturbative questions about QCD without uncontrolled approximations. The emphasis of the book is on the theoretical developments that have shaped the field in the last two decades and that have turned lattice gauge theory into a robust approach to the determination of low energy hadronic quantities and of fundamental parameters of the Standard Model. By way of introduction, the lectures begin by covering lattice theory basics, lattice renormalization and improvement, and the many faces of chirality. A later course introduces QCD at finite temperature and density. A broad view of lattice computation from the basics to recent developments was offered in a corresponding course. Extrapolations to physical quark masses and a framework for the parameterization of the low-energy physics by means of effective coupling constants is covered in a lecture on chiral perturbation theory. Heavy-quark effective theories, an essential tool for performing the relevant lattice calculations, is covered from its basics to recent advances. A number of shorter courses round out the book and broaden its purview. These included recent applications to the nucleon--nucleon interation and a course on physics beyond the Standard Model\"--Provided by publisher.
Book
Fermion masses through four-fermion condensates
A
bstract
Fermion masses can be generated through four-fermion condensates when symmetries prevent fermion bilinear condensates from forming. This less explored mechanism of fermion mass generation is responsible for making four reduced staggered lattice fermions massive at strong couplings in a lattice model with a local four-fermion coupling. The model has a massless fermion phase at weak couplings and a massive fermion phase at strong couplings. In particular there is no spontaneous symmetry breaking of any lattice symmetries in both these phases. Recently it was discovered that in three space-time dimensions there is a direct second order phase transition between the two phases. Here we study the same model in four space-time dimensions and find results consistent with the existence of a narrow intermediate phase with fermion bilinear condensates, that separates the two asymptotic phases by continuous phase transitions.
Journal Article
Circuit complexity in quantum field theory
A
bstract
Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.
Journal Article
On the order of the QCD chiral phase transition for different numbers of quark flavours
A
bstract
The nature of the QCD chiral phase transition in the limit of vanishing quark masses has remained elusive for a long time, since it cannot be simulated directly on the lattice and is strongly cutoff-dependent. We report on a comprehensive ongoing study using unimproved staggered fermions with
N
f
∈ [2, 8] mass-degenerate flavours on
N
τ
∈ {4
,
6
,
8} lattices, in which we locate the chiral critical surface separating regions with first-order transitions from crossover regions in the bare parameter space of the lattice theory. Employing the fact that it terminates in a tricritical line, this surface can be extrapolated to the chiral limit using tricritical scaling with known exponents. Knowing the order of the transitions in the lattice parameter space, conclusions for approaching the continuum chiral limit in the proper order can be drawn. While a narrow first-order region cannot be ruled out, we find initial evidence consistent with a second-order chiral transition in all massless theories with
N
f
≤ 6, and possibly up to the onset of the conformal window at 9 ≲
N
f
∗
≲ 12. A reanalysis of already published
O
(
a
)-improved
N
f
= 3 Wilson data on
N
τ
∈ [4
,
12] is also consistent with tricritical scaling, and the associated change from first to second-order on the way to the continuum chiral limit. We discuss a modified Columbia plot and a phase diagram for many-flavour QCD that reflect these possible features.
Journal Article
Walking, weak first-order transitions, and complex CFTs
A
bstract
We discuss walking behavior in gauge theories and weak first-order phase transitions in statistical physics. Despite appearing in very different systems (QCD below the conformal window, the Potts model, deconfined criticality) these two phenomena both imply approximate scale invariance in a range of energies and have the same RG interpretation: a flow passing between pairs of fixed point at complex coupling. We discuss what distinguishes a real theory from a complex theory and call these fixed points complex CFTs. By using conformal perturbation theory we show how observables of the walking theory are computable by perturbing the complex CFTs. This paper discusses the general mechanism while a companion paper [
1
] will treat a specific and computable example: the two-dimensional
Q
-state Potts model with
Q
> 4. Concerning walking in 4d gauge theories, we also comment on the (un)likelihood of the light pseudo-dilaton, and on non-minimal scenarios of the conformal window termination.
Journal Article
On lattice models of gapped phases with fusion category symmetries
A
bstract
We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction is based on two-dimensional state sum TQFT whose input datum is an
H
-simple left
H
-comodule algebra, where
H
is a finite dimensional semisimple Hopf algebra. We show that the actions of fusion category symmetries
C
on the boundary conditions of these state sum TQFTs are represented by module categories over
C
. This agrees with the classification of gapped phases with symmetry
C
. We also find that the commuting projector Hamiltonians for these state sum TQFTs have fusion category symmetries at the level of the lattice models and hence provide lattice realizations of gapped phases with fusion category symmetries. As an application, we discuss the edge modes of SPT phases based on these commuting projector Hamiltonians. Finally, we mention that we can extend the construction of topological field theories to the case of anomalous fusion category symmetries by replacing a semisimple Hopf algebra with a semisimple pseudo-unitary connected weak Hopf algebra.
Journal Article
The glueball spectrum of SU(3) gauge theory in 3 + 1 dimensions
A
bstract
We calculate the low-lying glueball spectrum of the SU(3) lattice gauge theory in 3 + 1 dimensions for the range
β ≤
6
.
50 using the standard plaquette action. We do so for states in all the representations
R
of the cubic rotation group, and for both values of parity
P
and charge conjugation
C
. We extrapolate these results to the continuum limit of the theory using the confining string tension
σ
as our energy scale. We also present our results in units of the
r
0 scale and, from that, in terms of physical ‘GeV’ units. For a number of these states we are able to identify their continuum spins
J
with very little ambiguity. We also calculate the topological charge
Q
of the lattice gauge fields so as to show that we have sufficient ergodicity throughout our range of
β
, and we calculate the multiplicative renormalisation of
Q
as a function of
β
. We also obtain the continuum limit of the SU(3) topological susceptibility.
Journal Article
Circuit complexity in interacting QFTs and RG flows
A
bstract
We consider circuit complexity in certain interacting scalar quantum field theories, mainly focusing on the
ϕ
4
theory. We work out the circuit complexity for evolving from a nearly Gaussian unentangled reference state to the entangled ground state of the theory. Our approach uses Nielsen’s geometric method, which translates into working out the geodesic equation arising from a certain cost functional. We present a general method, making use of integral transforms, to do the required lattice sums analytically and give explicit expressions for the
d
= 2
,
3 cases. Our method enables a study of circuit complexity in the epsilon expansion for the Wilson-Fisher fixed point. We find that with increasing dimensionality the circuit depth increases in the presence of the
ϕ
4
interaction eventually causing the perturbative calculation to breakdown. We discuss how circuit complexity relates with the renormalization group.
Journal Article
Circuit complexity for coherent states
A
bstract
We examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen’s geometric approach as in [
1
]. The complexity of the coherent states have the same UV divergences as the vacuum state complexity and so we consider the finite increase of the complexity of these states over the vacuum state. One observation is that generally, the optimal circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen’s approach with those found using the Fubini-Study method of [
2
]. For general coherent states, we find that the complexities, as well as the optimal circuits, derived from these two approaches, are different.
Journal Article
Parton distribution functions from Ioffe time pseudo-distributions
A
bstract
In this paper, we present a detailed study of the unpolarized nucleon parton distribution function (PDF) employing the approach of parton pseudo-distribution func- tions. We perform a systematic analysis using three lattice ensembles at two volumes, with lattice spacings
a
= 0.127 fm and
a
= 0.094 fm, for a pion mass of roughly 400 MeV. With two lattice spacings and two volumes, both continuum limit and infinite volume ex- trapolation systematic errors of the PDF are considered. In addition to the
x
dependence of the PDF, we compute their first two moments and compare them with the pertinent phenomenological determinations.
Journal Article