Fixed points of scalar field theories with quartic interactions in
$d=4-\varepsilon$ dimensions are considered in full generality. For such
theories it is known that there exists a scalar function $A$ of the couplings
through which the leading-order beta-function can be expressed as a gradient.
It is here proved that the fixed-point value of $A$ is bounded from below by a
simple expression linear in the dimension of the vector order parameter, $N$.
Saturation of the bound requires a marginal deformation, and is shown to arise
when fixed points with the same global symmetry coincide in coupling space.
Several general results about scalar CFTs are discussed, and a review of known
fixed points is given.
We study spin systems which exhibit symmetries that act on a fractal subset
of sites, with fractal structures generated by linear cellular automata. In
addition to the trivial symmetric paramagnet and spontaneously symmetry broken
phases, we construct additional fractal symmetry protected topological (FSPT)
phases via a decorated defect approach. Such phases have edges along which
fractal symmetries are realized projectively, leading to a symmetry protected
degeneracy along the edge. Isolated excitations above the ground state are
symmetry protected fractons, which cannot be moved without breaking the
symmetry. In 3D, our construction leads additionally to FSPT phases protected
by higher form fractal symmetries and fracton topologically ordered phases
enriched by the additional fractal symmetries.
In these lecture notes we give a technical overview of tangent-space methods
for matrix product states in the thermodynamic limit. We introduce the manifold
of uniform matrix product states, show how to compute different types of
observables, and discuss the concept of a tangent space. We explain how to
variationally optimize ground-state approximations, implement real-time
evolution and describe elementary excitations for a given model Hamiltonian.
Also, we explain how matrix product states approximate fixed points of
one-dimensional transfer matrices. We show how all these methods can be
translated to the language of continuous matrix product states for
one-dimensional field theories. We conclude with some extensions of the
tangent-space formalism and with an outlook to new applications.
We discuss generalized global symmetries and their breaking. We extend
Goldstone's theorem to higher form symmetries by showing that a perimeter law
for an extended $p$-dimensional defect operator charged under a continuous
$p$-form generalized global symmetry necessarily results in a gapless mode in
the spectrum. We also show that a $p$-form symmetry in a conformal theory in
$2(p+1)$ dimensions has a free realization. In four dimensions this means any
1-form symmetry in a $CFT_4$ can be realized by free Maxwell electrodynamics,
i.e. the current can be photonized. The photonized theory has infinitely many
conserved 0-form charges that are constructed by integrating the symmetry
currents against suitable 1-forms. We study these charges by developing a
twistor-based formalism that is a 4d analogue of the usual holomorphic complex
analysis familiar in $CFT_2$. The charges are shown to obey an algebra with
central extension, which is an analogue of the 2d Abelian Kac-Moody algebra for
higher form symmetries.
Based on a generalized free energy we derive exact thermodynamic Bethe ansatz
formulas for the expectation value of the spin current, the spin
current-charge, charge-charge correlators, and consequently the Drude weight.
These formulas agree with recent conjectures within the generalized
hydrodynamics formalism. They follow, however, directly from a proper treatment
of the operator expression of the spin current. The result for the Drude weight
is identical to the one obtained 20 years ago based on the Kohn formula and
TBA. We numerically evaluate the Drude weight for anisotropies
$\Delta=\cos(\gamma)$ with $\gamma = n\pi/m$, $n\leq m$ integer and coprime. We
prove, furthermore, that the high-temperature asymptotics for general
$\gamma=\pi n/m$---obtained by analysis of the quantum transfer matrix
eigenvalues---agrees with the bound which has been obtained by the construction
of quasi-local charges.
We study quantum transport after an inhomogeneous quantum quench in a free
fermion lattice system in the presence of a localised defect. Using a new
rigorous analytical approach for the calculation of large time and distance
asymptotics of physical observables, we derive the exact profiles of particle
density and current. Our analysis shows that the predictions of a semiclassical
approach that has been extensively applied in similar problems match exactly
with the correct asymptotics, except for possible finite distance corrections
close to the defect. We generalise our formulas to an arbitrary non-interacting
particle-conserving defect, expressing them in terms of its scattering
A natural question about Quantum Field Theory is whether there is a
deformation to a trivial gapped phase. If the underlying theory has an anomaly,
then symmetric deformations can never lead to a trivial phase. We discuss such
discrete anomalies in Abelian Higgs models in 1+1 and 2+1 dimensions. We
emphasize the role of charge conjugation symmetry in these anomalies; for
example, we obtain nontrivial constraints on the degrees of freedom that live
on a domain wall in the VBS phase of the Abelian Higgs model in 2+1 dimensions.
In addition, as a byproduct of our analysis, we show that in 1+1 dimensions the
Abelian Higgs model is dual to the Ising model. We also study variations of the
Abelian Higgs model in 1+1 and 2+1 dimensions where there is no dynamical
particle of unit charge. These models have a center symmetry and additional
discrete anomalies. In the absence of a dynamical unit charge particle, the
Ising transition in the 1+1 dimensional Abelian Higgs model is removed. These
models without a unit charge particle exhibit a remarkably persistent order: we
prove that the system cannot be disordered by either quantum or thermal
fluctuations. Equivalently, when these theories are studied on a circle, no
matter how small or large the circle is, the ground state is non-trivial.
We construct gauge theories with a vector-multiplet and hypermultiplets of
$(1,0)$ supersymmetry on the six-sphere. The gauge coupling on the sphere
depends on the polar angle. This has a natural explanation in terms of the
tensor branch of $(1,0)$ theories on the six-sphere. For the vector-multiplet
we give an off-shell formulation for all supersymmetries. For hypermultiplets
we give an off-shell formulation for one supersymmetry. We show that the path
integral for the vector-multiplet localizes to solutions of the
Hermitian-Yang-Mills equation, which is a generalization of the (anti-)self
duality condition to higher dimensions. For the hypermultiplet, the path
integral localizes to configurations where the field strengths of two complex
scalars are related by an almost complex structure.
The space of n-point correlation functions, for all possible time-orderings
of operators, can be computed by a non-trivial path integral contour, which
depends on how many time-ordering violations are present in the correlator.
These contours, which have come to be known as timefolds, or out-of-time-order
(OTO) contours, are a natural generalization of the Schwinger-Keldysh contour
(which computes singly out-of-time-ordered correlation functions). We provide a
detailed discussion of such higher OTO functional integrals, explaining their
general structure, and the myriad ways in which a particular correlation
function may be encoded in such contours. Our discussion may be seen as a
natural generalization of the Schwinger-Keldysh formalism to higher OTO
correlation functions. We provide explicit illustration for low point
correlators (n=2,3,4) to exemplify the general statements.
We compute correlation functions of three-dimensional Landau-gauge Yang-Mills
theory with the Functional Renormalisation Group. Starting from the classical
action as only input, we calculate the non-perturbative ghost and gluon
propagators as well as the momentum-dependent ghost-gluon, three-gluon, and
four-gluon vertices in a comprehensive truncation scheme. Compared to the
physical case of four spacetime dimensions, we need more sophisticated
truncations due to significant contributions from non-classical tensor
structures. In particular, we apply a special technique to compute the tadpole
diagrams of the propagator equations, which captures also all perturbative
two-loop effects, and compare our correlators with lattice and Dyson-Schwinger