Sommerfeld Theory Colloquium (ASC)

Quantum mechanical potentials with multiple classically degenerate minima lead to spectra that are determined by the pertaining tunneling amplitudes. For the strong interactions, these classical minima correspond to configurations of a given Chern-Simons number. The tunneling amplitudes are then given by instanton transitions, and the associated gauge invariant eigenstates are the theta-vacua. Under charge-parity (CP) reversal theta changes its sign, and so it is believed that CP-violating observables such as the electric dipole moment of the neutron or the decay of the eta-prime meson into two pions are proportional to theta. Here we argue that this is not the case. This conclusion is based on the assumption that the path integral is dominated by saddle points of finite action and fluctuations around these. In spacetimes of infinite volume, this leads to the requirement of vanishing physical fields at the boundaries. For the gauge fields, this implies topological quantization corresponding to homotopy classes for all integers. We consequently calculate quark correlations by first taking the spacetime volume to infinity and then summing over the sectors. This leads to an absence of CP violation in the quark correlations, in contrast to the conventional way of taking the limits the other way around. While there is an infinite number of homotopy classes in the strong interactions, there is only a finite number of classical vacua for quantum mechanical systems. For the latter the order of taking time to infinity and summing over the transitions is therefore immaterial.

The dynamics of the early universe and black holes are fundamental reflections of the interplay between general relativity and quantum fields. The essential physical processes occur in situations that are difficult to observe and impossible to experiment with: when gravitational interactions are strong, quantum effects are important, and theoretical predictions for these regimes are based on major extrapolations of laboratory-tested physics.

We will discuss the possibility to study these processes in experiments by employing analogue classical/quantum simulators. Their high degree of tunability, in terms of dynamics, effective geometry, and field theoretical description, allows one to emulate a wide range of elusive physical phenomena in a controlled laboratory setting. We will discuss recent developments in this area of research.

Topological phases were discovered in condensed matter physics and recently extended to classical physics such as topological mechanical metamaterials. Their study and realization in soft-matter and biological systems has only started to develop. In this talk we discuss how topological phases may determine the behavior of nonlinear dynamical systems that arise, for example, in population dynamics. We have shown that topological phases can be realized with the anti-symmetric Lotka-Volterra equation (ALVE). The ALVE is a paradigmatic model system in population dynamics and governs, for example, the evolutionary dynamics of zero-sum games, such as the rock-paper-scissors game [1], but also describes the condensation of non-interacting bosons in driven-dissipative set-ups [2]. We have shown that for the ALVE, defined on a one-dimensional chain of rock-paper-scissors cycles, robust polarization emerges at the chain’s edge [3]. The system undergoes a transition from left to right polarization as the control parameter passes through a critical value. At the critical point, solitary waves are observed. We found that the polarization states are topological phases and that this transition is indeed a topological phase transition. Remarkably, this phase transition falls into symmetry class D within the “ten-fold way” classification scheme of gapped free-fermion systems, which also applies, for example, to one-dimensional topological superconductors. Beyond the observation of topological phases in the ALVE, it might be possible to generalize the approach of our work to other dynamical systems in biological physics whose attractors are nonlinear oscillators or limit cycles.

[1] J. Knebel, T. Krüger, M. F. Weber, and E. Frey, Phys. Rev. Lett. 110, 168106 (2013).

[2] J. Knebel, M. F. Weber, T. Krüger, and E. Frey, Nature Communications 6, 6977 (2015).

[3] J. Knebel, P. M. Geiger, and E. Frey, Phys. Rev. Lett. (in press) [arXiv:2009.01780].

Particle physics is at the crossroads. The last particle firmly predicted by the Standard Model (SM) has been discovered. In recent years many of its interactions with other known particles have been experimentally studied, again confirming the SM's predictions in large detail. However, despite this impressive success, we are sure that the SM is incomplete. The Standard Model can not explain dark matter, neutrino masses, and matter-antimatter asymmetry of the Universe, and we do know what particles are responsible for these phenomena — their masses, interaction strength, life time etc. I will discuss how the synergy with Cosmology and Astrophysics can help to limit these uncertainties and discuss several concrete examples which can drastically change the situation in particle physics in the next 5-7 years.

In a recent breakthrough, a complete set of constraints on fermionic occupation numbers, extending Pauli’s original exclusion principle, has been found. We provide an introduction into this new research field. In particular, we show that those generalized Pauli constraints are approximately saturated in various few-fermion systems, i.e. the vector of occupation numbers lies close to the boundary of the allowed region. Striking implications of this quasipinning phenomenon are revealed and discussed: These are the concept of robust quantum information, a hierarchy of natural extensions of the Hartree-Fock ansatz and the existence of an exchange force emerging from the fermionic exchange symmetry. Finally, we explain how these novel research concepts offer a universal perspective on fermionic quantum systems.

Condensed matter is found in a variety of phases, the vast majority of which are characterized in terms of symmetry breaking. For example, magnets spontaneously break time-reversal and spin rotation symmetries. A notable exception was provided by the discovery of the quantum Hall effects which exhibit new kinds of topological orders not associated with any symmetry breaking. One of the characterizing features of topological order is the existence of excitations with

exotic properties. These so-called anyonic excitations might make topologically ordered systems ideal building blocks of fault-tolerant quantum computers. In this colloquium, I will start by giving a general introduction to the concept of topological order and then address some of the recent developments. In particular, I will introduce theoretical frameworks that allow us to classify topological phases and discuss dynamical signatures that are useful to experimentally detect them in experiments.

String theory seems to offer an enormous number of possibilities for low energy physics. The huge set of solutions is often known as the String Theory Landscape. In recent years, however, it has become clear that not all quantum field theories can be consistently coupled to gravity. Theories that cannot be ultraviolet completed in quantum gravity are said to be in the Swampland. In this talk, I'll discuss some conjectured properties of quantum gravity, evidences for them, and their applications to cosmology and particle physics.

The functional renormalization group (RG) is an ideal tool for dealing with the diversity of energy scales and competition of instabilities in interacting fermion systems. Starting point is an exact flow equation which yields the gradual evolution from a microscopic model action to the effective low-energy action as a function of a continuously decreasing energy scale. Expanding in powers of the fields yields an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes [1].

Applications reviewed in the colloqium include: (i) d-wave superconductivity and other instabilities in the two-dimensional Hubbard model, and (ii) transport through a barrier and resonant tunneling in a one-dimensional Luttinger liquid metal. Recently, the functional RG has been upgraded from a weak-coupling method to a computational tool for strongly interacting fermion systems [2,3].

[1] W. Metzner et al., Rev. Mod. Phys. 84, 299 (2012).

[2] C. Taranto et al., Phys. Rev. Lett. 112, 196402 (2014).

[3] D. Vilardi et al., Phys. Rev. B 99, 104501 (2019).

We shall review the attempts to extend the quantum

mechanical property of non-commutativity from phase space to

ordinary space. These attempts took a more precise form in the

case of gauge theories for which some concrete results have

been obtained. In flat space they amount to a reformulation of

the theory which looks interesting but they have not given so

far any novel physical results. However, the introduction of

gravity gives a richer structure and may offer some new

insights.

The construction of a quantum theory of gravity

remains an open problem despite decades of efforts.

In time, the very perspective on this problem evolved.

From quantising General Relativity, the goal is now

mostly understood to be unraveling a more

fundamental microstructure of spacetime, based on

non-geometric building blocks, and to show how

spacetime and matter emerge as effective, approximate

notions. Given some candidate building blocks, the task

becomes analogous to that of extracting the

macroscopic, collective behaviour of the atoms of a

condensed matter system, but even more challenging

since we cannot use the usual spacetime intuition and

no direct observational input is available to guide

theory construction.

Lacking a fundamental theory of quantum gravity,

existing cosmological models which have proven

extremely successful in accounting for the observed

features of the very early universe (via CMB data)

remain without a solid foundation, having to make a

number of assumptions about a physical regime (close

to the big bang), where the quantum nature of gravity

and spacetime is expected to be relevant. This is all the

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more unfortunate, since the very early universe is also

where any proposed quantum theory of gravity has the

highest chance of finding its observational test-bed.

The gap needs to be bridged.

In this talk I will first of all review the basic aspects of

the problem of quantum gravity, and of some current

approaches. I will then focus on one specific formalism

for quantum gravity, so-called group field theories

(strictly related to a number of other modern

approaches). I will introduce its main features, trying to

clarify the nature of the suggested building blocks of

spacetime and their mathematical description. Next, I

will outline a general strategy to extract an effective

cosmological dynamics from quantum gravity, within

this formalism. In this setting, the universe emerges as

a quantum condensate of the fundamental “atoms of

spacetime”, and cosmology is its corresponding

hydrodynamics. Finally, I will summarize the recent

results obtained along this research direction.