Department of Physics • University of Cologne

Quantum Dynamics in many-body systems is nowadays well understood via the Eigenstate Thermalization Hypothesis (ETH). According to ETH, local observables in the energy eigenbasis are pseudorandom matrices, whose statistical properties are smooth thermodynamic functions. In this talk, I will show how the full version of ETH, which encompasses correlations among matrix elements, can be rationalized theoretically using the language of Free Probability. The latter is a generalization of probability theory to non-commuting variables, with applications in combinatorics and random matrix theory. The mathematical structure of Free Probability also allows us to provide a link between quantum dynamics and k-designs, an important concept with applications in quantum information theory.

Quantum cryptography offers a remarkable advantage compared to classical cryptography, because we can prove its security against all possible attacks an adversary can perform. Such proofs are always built on various assumptions, for example on the quantum states and measurements or on the isolation of the laboratories. The validity of the proof depends on the fulfilment of these assumptions, necessitating careful selection. Consequently, device-independent quantum key distribution (QKD) represents the ``holy grail’’ as it requires no assumptions about the quantum devices. Unfortunately, achieving practical implementation of these protocols poses substantial challenges. In this talk, I will delve into the ongoing quest to strike a balance between assumptions and practicality on QKD protocols. Throughout the talk, I will provide a comprehensive overview of QKD, ensuring that all the essential background information is covered.

A two-channel model with Hubbard-Stratonovich field is used to study a spin-balanced Fermi gas at finite temperature in the normal and superfluid phase. A Dyson-Schwinger approach is used to compute full Green's functions iteratively. Using a spectral representation, the Matsubara sums are computed and analytically continued to real frequencies. In this way, the full spectral functions are calculated directly. Finally, these results are used to study real-time and transport properties of ultracold atomic gases.

We explore the use of methods from statistical inference to study Bell nonlocality

Certain phase transitions in a graphene sheet are known to be described at criticality by an underlying continuum quantum field theory which is the Gross-Neveu model. While the Ising Gross-Neveu model has been widely studied for many years, perturbative and large N computations in the related chiral Heisenberg Gross-Neveu model have not been carried out to the same level of precision. In the talk we will review recent four loop renormalization of the Ising Gross-Neveu model before developing the large N expansion for the chiral Heisenberg Gross-Neveu case. In the latter case the first three terms of various critical exponents are determined including eta at O(1/N^3). To achieve this accuracy the large N conformal bootstrap method is used. Finally estimates of exponents are given for the values of N relevant to the graphene transitions and compared to Monte Carlo and functional renormalization group results as well as the epsilon expansion of four dimensional perturbation theory.

In der Veranstaltung wird das Lernmanagement-System Ilias, auf das jeder Dozent und Studierender der UzK Zugriff hat, vorgestellt. An Hand vonkonkreten Beispielen werden Einsatzmöglichkeiten und Funktionsumfang des Systems illustriert. Zur Demonstration der Handhabung von Ilias empfiehlt es sich, einen eigenen Laptop mitzubringen! Zielgruppe sind neben den Dozenten auch die Veranwortlichen für den Übungsbetrieb.

It is a well-known fact that the spreading of information in lattice quantum systems is not instantaneous, but it rather exists a maximum velocity dictated by the Lieb-Robinson bound. The existence of such a lightcone deeply affects equilibrium properties as well as out-of-equilibrium ones, which have been a subject of outstanding interest in the recent years. In this talk, a novel out-of-equilibrium protocol critically affected by the presence of a maximum velocity is proposed and discussed. Specifically, a one dimensional lattice model is considered, where a localized impurity is suddenly created and then dragged at a constant velocity. Focussing on a simple, but far from trivial, free model the response of the system at late times is analyzed, with emphasis on its transport properties. The finite maximum velocity is responsible for a rich phenomenology, for which exact results are provided. Taking into account the experience acquired so far, more general models are discussed and unpublished results presented, with exact predictions in completely generic (non integrable) one dimensional lattice systems. Reference paper: A. Bastianello, A. De Luca, Phys. Rev. Lett. 120, 060602 (2018).

Quantum Hall effect is one of the most interesting examples quantum many-particles systems. It occurs in certain two-dimensional electron systems at low temperatures and in high magnetic fields, which exhibit plateaux with the quantized values of the Hall conductance. The Fractional Quantum Hall effect (FQHE), when the Hall conductance takes on fractional values, is an example of the strongly-interacting quantum many-particles. The standard physics approach to the FQHE is to assign a certain many-body wave function to each plateaux. I will talk about a program as to how one can use a combination of probabilistic, asymptotic and geometric methods to learn more about the physics and mathematics of the FQHE wave functions, in particular, describe the electromagnetic and gravitational responses, asymptotics for a large number particles, novel quantized coefficients for the adiabatic transport.

In recent years, considerable experimental efforts have been devoted to dynamically generate complex states and monitor their evolution. Despite remarkable advances, the theoretical principles behind the non-equilibrium dynamics of strongly correlated quantum matter are still far from being fully understood. In particular, very few studies have sought to clarify the influence of environmental couplings on the propagation of correlations. We attempt here to fill this gap. Considering first an interaction quench in the Bose-Hubbard model under the effect of dephasing, we observe that dissipation effectively speeds up the propagation of single-particle correlations while reducing their coherence. In contrast, for two-point density correlations, the initial ballistic propagation regime gives way to diffusion at intermediate times. In a second time, considering the XXZ spin-1/2 model in contact with a similar environment, we find this system to display aging. A dynamical phenomenon characterized by a breakdown of time-translation invariance, a slow non-exponential relaxation of two-time correlations and the presence of dynamical scaling.

I will discuss several results (also in the mathematical sense) about the dynamics of quantum spin systems. The main theme will be 'localization' which is a term that has been much in vogue in the last years. Usually, it is understood as 'absence of dissipative behaviour'. I will explain what this means and how it comes about. The phenomenon that is probably to be expected to apply more broadly than 'localization' is 'sparsity of resonances', sometimes also called 'asymptotic localization' or 'quasi-localization'. It manifests itself in quantum and classical systems, where transport and thermalization take place only due to effects that are non-perturbative (smaller than any power) in the natural parameters of the system.

Recent developments in diverse areas - ranging from cold atomic gases over light driven semiconductors to microcavity arrays - move systems into the focus, which are located on the interface of quantum optics, many-body physics and statistical mechanics. These "driven open quantum systems" share in common that coherent and driven-dissipative dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in equilibrium condensed matter physics. A case in point are so-called exciton-polaritons in two spatial dimensions. We briefly explain the physical basis, their description, and how to detect the violation of thermal equilibrium conditions in the formalism. We then show that a paradigmatic hallmark of low-temperature equilibrium systems -- the presence of quasi-long range order, i.e. the algebraic decay of spatial correlation functions -- must be absent due to non-equilibrium conditions. This conclusion is drawn based on a connection to the problem of surface roughening.

We demonstrate that the exact nonequilibrium steady state of the one-dimensional Heisenberg XXZ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the nonequilibrium density matrix. For spin 1/2 the matrices satisfy the quantum algebra U_q[sl(2)]. For the isotropic Heisenberg chain, coupled at the ends to boundary reservoirs polarized in different directions with twist angle theta, we calculate the exact magnetization profiles and magnetization currents in the nonequilibrium steady state of a chain N sites. For large N the in-plane steady-state magnetization profiles are harmonic functions with a frequency proportional to the twist angle. In-plane steady-state magnetization currents are subdiffusive, while the transverse current saturates when the coupling strength is sufficiently large. For the anisotropic chain we find a current resonance at the specific values of the anisotropic interaction strength where the transverse current is independent of system size, even for non-integrable higher-spin chains.

In physics, there is the prevailing intuition that we are part of a unique external world, and that the goal of physics is to understand and describe this world. This assumption of the fundamentality of objective reality is often seen as a major prerequisite of any kind of scientific reasoning, delineating science from pseudoscience, and explaining why successful empirical science is possible in the first place. However, here I argue that we should consider relaxing this assumption in a specific way in some contexts. Namely, there is a collection of open questions in and around physics that can arguably be addressed in a substantially more consistent and rigorous way if we consider the possibility that the first-person perspective is ultimately more fundamental than our usual notion of external world. These are questions like: which probabilities should an observer assign to future experiences if she is told that she will be simulated on a computer? How should we think of cosmology’s Boltzmann brain problem or assign probabilities to properties of ‘possible worlds’? What can we learn from the fact that measurements in quantum theory seem to do more than just reveal preexisting properties? Why are there simple computable laws of physics in the first place? In the talk, I sketch a mathematically rigorous approach along these lines, suggesting a simple and unified framework (rooted in algorithmic information theory) to address questions like those above. It is not meant as a ‘theory of everything’ (in fact, it predicts its own limitations), but it shows how a notion of objective external world, looking very much like our own, can provably emerge from a starting point in which (only) the first-person perspective is primary. Based on arXiv:1712.01826 (full version) and arXiv:1712.01816 (short version).

Peter Schilke (Köln): Das unsichtbare Universum mit ALMA sichtbar gemacht Norbert Wex (Bonn): 50 Jahre Pulsare, 50 Jahre kosmische Uhren für Einstein Andreas Schulz (Köln, Bonn): Die Bewohnbarkeit unseres Planeten Alexander Küpper (Köln): Erfahrungen mit einer inklusiven astron. Lernumgebung Martin Quast (Bonn): Wo sind die Aliens? – Exoplaneten und ihre Bewohnbarkeit

If you are given a black box unitary U, can you build the inverse U^\dagger out of it? In this talk we are going to discuss this problem, and more generally which transformations on unitary channels can be done according to the rules of quantum mechanics

Quantum computation offers a novel way of processing information and promises solution of some classically intractable problems. However, if the component of the quantum computor has some errors, it does not output a correct computation outcome. Since the quantum computor is composed of several components, the unexpected correlation causes unexpected error that cannot be corrected by error correction. To resolve this problem, we need to verify the quantum computor. If a problem is in NP, we can verify the correctness of a solution, but a problem that we want to solve with a quantum computer is not necessarily in NP. We need to verify quantum computer. Usually a quantum computer is composed of a combination of so many quantum circuits. It is not easy to predict the outcome of the combination of so many quantum circuits. That is, since we do not know the computation outcome, we cannot verify the computation outcome by itself. To resolve this problem, we employ measurement-based quantum computation (MBQC). MBQC is composed of graph state and local measurements, which are known to us. In particular, the computation resorce is given as the quantum correlation of the graph state, which is in a known form. Hence, we can verify these components by using the method of statistical hypothesis testing. In this talk, we consider the following three settings. (1) We perfectly trust our local measurement. So, we need to verify only the graph state. (2) We trust local measurement, but it is noisy. The noise can be converted to noise of graph state. So, we need to verify only the noisy graph state. This protocol works with noisy graph state. (3) We do not trust measurement as well as graph state. However, it accept only the case when the measurement and the graph state are noiseless. Setting (3) has weakest assumption, but it works with ideal case. Setting (2) has stronger assumption, but it works with realistic case. Further, our verification scheme can be applied to the blind quantum computation. This talk is based on joint works with T. Morimae, K. Fujii, M. Hajdusek, Y. Takeuchi. The detail is available in arXiv:1701.05688, arXiv:1610.05216, arXiv:1603.02195, Phys. Rev. Lett. 115, 220502 (2015).

Abrikosov proposed in 1974 that a 3D electronic system with its fermi level at the point of quadratic band crossing, as in the (spin-orbit coupled) gray tin or mercury telluride, should represent the simplest non-fermi liquids, due to the common long-range Coulomb interaction between the particles. I will review this idea and discuss how a non-fermi liquid ground state may be unstable to ordering via the mechanism of "fixed-point collision". This mechanism, with its concomitant hierarchy of scales, may also be identified as the culprit behind the notorious dynamical chiral symmetry breaking in the three dimensional electrodynamics, known as the effective theory of several strongly interacting condensed matter systems and as a favorite toy model of particle theorists. Some further applications of the nematic order parameters, possibly relevant  to gray tin or iridates, in the current search for  higher-dimensional interacting field theories will also be mentioned. 

Low-rank tensor recovery is an interesting subject from both theoretical and application point of view. On one side, it is a natural extension of the sparse vector and low-rank matrix recovery problem. On the other side, estimating a low-rank tensor has applications in many different areas such as machine learning, video compression, and seismic data interpolation. In this talk, an iterative approach to low rank tensor recovery is introduced. This approach is a generalization of the iterative hard thresholding algorithm for sparse vector and low-rank matrix recovery to tensor scenario (tensor IHT or TIHT algorithm). Here, we consider the tensor train decomposition (or TT-decomposition), also called the matrix product states (or MPS). The a! nalysis of the algorithm is based on a version of the restricted isometry property (tensor RIP or TRIP). We show that subgaussian measurement ensembles satisfy TRIP with high probability under an almost optimal condition on the number of measurements. Additionally, we show that partial Fourier maps combined with random sign flips of the tensor entries satisfy TRIP with high probability. Under the assumption that the linear operator satisfies TRIP and under an additional assumption on the thresholding operator, we provide a linear convergence result for the TIHT algorithm. Similar results also hold for HOSVD (Tucker) and Hierarchical Tucker decomposition (HT-decomposition). Finally, we illustrate the performance of iterative hard thresholding algorithm for tensor recovery via numerical experiments where we consider recovery from Gaussian measurement ensembles and Fourier measurements for third order tensors.

Fully characterizing the steerability of a quantum state of a bipartite system has remained an open problem since the concept of steerability was defined. In this talk, we discuss a geometric approach to steerability and suggest a necessary and sufficient condition for a two-qubit state to be steerable with respect to projective measurements. To this end, we define the critical radius of local models and show that a state of two qubits is steerable with respect to projective measurements from Alice's side if and only if her critical radius of local models is less than 1. We then calculate the critical radius of local models for the so-called T-states by proving the optimality of a recently suggested local hidden state model.

Certain single-photon sources are capable of emitting strings of entangled photons. In particular, photonic cluster state sources could enable measurement-based quantum computing and are therefore under intense investigation. While experiments are progressing, the issue of benchmarking comes to the fore. The challenge is to quantify the long-range entanglement of the photonic state using only the few-photon correlation measurements that are feasible given limited emission/detection efficiencies. We present a lower bound on localizable entanglement that requires nothing but simple three-qubit correlation measurements. This method therefore enables direct demonstration of computationally powerful multi-photon entanglement with currently available experimental capabilities.

A recent framework for studying causal structures introduced the notion of a process: a resource shared between some parties that is compatible with standard quantum mechanics in each party’s laboratory, but that does not presuppose the existence of a global causal order between them. These processes can be used to perform several tasks that are impossible in standard quantum mechanics: they allow for the violation of causal inequalities, and provide an advantage for computational and communication complexity. Nonetheless, no physical process is known that allows for the violation of a causal inequality. There is therefore considerable interest in determining which processes are physical and which are just mathematical artefacts of the framework. Here we make the first step in this direction, by proposing a purification postulate: processes are physical only if they are purifiable. We derive necessary conditions for a process to be purifiable, and show that several known processes do not satisfy them.

This work is concerned with thermal quantum states of Hamiltonians on spin and fermionic lattice systems with short range interactions. We provide results leading to a local definition of temperature, thereby extending the notion of "intensivity of temperature'' to interacting quantum models. More precisely, we derive a perturbation formula for thermal states. The influence of the perturbation is exactly given in terms of a generalized covariance. For this covariance we prove exponential clustering of correlations above a universal critical temperature, which upper bounds physical critical temperatures such as the Curie temperature. As a corollary, we obtain that above a the critical temperature, thermal states are stable against distant Hamiltonian perturbations. Moreover, our results imply that above the critical temperature local expectation values can be approximated with a computational cost independent of the system size.

The non-equilibrium dynamics of an isolated quantum system after a sudden quench to a dynamical critical point is expected to be characterized by scaling and universal exponents due to the absence of time scales. We explore these features for a quench of the parameters of a Hamiltonian with O(N) symmetry, starting from the ground state of the disordered phase. In the limit of large N, the exponents and scaling forms of the relevant two-time correlation functions can be calculated exactly. For generic N, we use a perturbative renormalization-group approach to compute such exponent, to leading order in a dimensional expansion.

The decoupling technique has proven a useful tool in quantum information theory, with proofs of results as different as channel coding theorems and thermodynamical work extraction relying on it. I will show how decoupling is used in quantum source coding problems such as state merging and state redistribution and discuss the one shot case.

The Kronecker coefficients are among the most fascinating objects in classical representation theory. While many of their fundamental properties remain mysterious, the past decade has brought some significant advances, in part through new and surprising connections to quantum physics and computational complexity theory. In this talk, I will give an introduction to this subject and its motivations, and discuss some recent results. Specifically, I will explain that the problem of deciding positivity of Kronecker coefficients is NP-hard in general, while their asymptotic non-vanishing, as encoded by certain moment polytopes, can be decided in NP and coNP (arXiv:1410.8144, 1507.02955, 1511.03675). This is in marked contrast to celebrated results on the Littlewood-Richardson coefficients and the Horn inequalities, and it suggests that representation theory can be strictly easier in the asymptotic regime. Lastly, and somewhat counter-intuitively, our hardness result can be turned into an efficient algorithm for studying the failure of the saturation property for Kronecker coefficients, which is at present only poorly understood.

How single units interact in a complex network fundamentally underlies its collective dynamics. Yet, identifying the physical structure of interactions from recorded time series still poses a great challenge. Up-to-date methods either require (i) a detailed prior knowledge of the units' dynamical features, (ii) to externally drive the network or (iii) the network dynamics to be at stable states, such as fixed points or limit cycles. Here we develop a theory to reveal physical interactions of networks that relies on recorded time series only. By decomposing the dynamics of single units in terms of network interactions of different orders (pairs, triplets, quadruplets,...), we pose network reconstruction as an error minimization problem. We propose a greedy algorithm to solve such minimization problems. Our approach is principally model independent, ensuring its generality and applicability in different fields and making it particularly suitable when structural connections are desired, dynamical features are unknown and perturbing the network is infeasible. Thus, our approach may serve as a key stepping stone for the expanding field of model-independent network reconstruction.

The field of algebraic statistics is concerned with the interplay of techniques from commutative algebra, combinatorics, (real) algebraic geometry, and related fields with problems arising in data analysis. Since its inception in the late 1990s, it has brought forward both new results in statistics as well as challenging research directions in pure mathematics. We survey some of these developments.

The robust supply with electric power is fundamental for economy, industry and our daily life. In periods of high loads the breakdown of a single transmission line can cause a global cascade of failures implying large-scale outages with potentially catastrophic consequences. Such periods of high loads are expected to become more likely in the future, as electric power from renewable sources is often generated far away from the consumers. In this talk I will discuss the stability of power grids to transmission line failures from a network science perspective, addressing the following questions: Which links are indispensable for network operation? How does a failure spread, i.e. which parts of a network are affected by the initial failure? How can we efficiently detect those failures with few measurements?

Resources and the problem of their efficient use form an essential part of engineering and sometimes even of everyday life. I will introduce a mathematical toolbox for dealing with problems of this form, and sketch some applications to information theory and graph theory. Given enough time and interest, I can explain the difficulties that one faces when trying to extend this framework to approximate resource conversions, as they are commonly considered in information theory.

We give restrictions on the locality-preserving unitary automorphisms U, which are protected gates, for topologically ordered systems. For arbitrary anyon models, we show that such unitaries only generate a finite group, and hence do not provide universality. For abelian anyon models, we find that the logical action of U is contained in a proper subgroup of the generalized Clifford group. In the case D(Z2), which describes Kitaev's toric code, this represents a tightening of statement previously obtained within the stabilizer framework (PRL 110:170503). For non-abelian models, we find that such automorphisms are very limited: for example, there is no non-trivial gate for Fibonacci anyons. For Ising anyons, protected gates are elements of the Pauli group. These results are derived by relating such automorphisms to symmetries of the underlying anyon model: protected gates realize automorphisms of the Verlinde algebra. We additionally use the compatibility with basis changes to characterize the logical action.

We investigate particular cosmological models, based either on tachyon fields or on perfect fluids, for which soft future singularities arise in a natural way. Our main result is the description of a smooth crossing of the soft singularity in models with an anti-Chaplygin gas or with a particular tachyon field in the presence of dust. Such a crossing is made possible by certain transformations of matter properties. Some of these cosmological evolutions involving tachyons are compatible with SNIa data. We compute numerically their dynamics involving a first soft singularity crossing, a turning point and a second soft singulatity crossing during recollapse, ending in a Big Crunch singularity.

In den letzten 30 Jahren kam es zu einer besonders aktiven Wechselwirkung zwischen der Mathematik und der theoretischen Physik. Dank dieser Interaktion sind neue mathematische Gebiete entstanden, die ihrerseits neue Methoden für verschiedene Bereiche der theoretischen Physik liefern. Die Theorie der Yang-Baxter-Gleichung ist ein Beispiel von solchen Gebieten. Lösungen dieser Gleichung besitzen eine reiche und tiefe mathematische Struktur und haben zahlreiche Anwendungen in der Physik. Im Vortrag wird ein Überblick über klassische Ergebnisse und neue Entwicklungen in der Theorie der Yang-Baxter-Gleichung gegeben.

Present knowledge about physical properties of an inflationary stage in the early Universe (curvature, its rate of change, inflaton mass, etc.) which follows from the most recent observational data including the Planck 2014 ones is reviewed and possibilities to find out anything new about it are discussed. In particular, features in the CMB temperature anisotropy power spectrum in the multipole range l=20-40 are of interest in this respect and may point to some new physics during inflation.

Based on string theory, gauge/gravity duality maps strongly coupled quantum field theories to weakly coupled gravity theories. It thus provides a new approach for studying physical phenomena in strongly coupled systems. We describe the main features of this approach and discuss selected examples. These include new results for relativistic hydrodynamics and for anomalous transport, with possible implications for Dirac matter and for Weyl semimetals. As a further example, we present a new Kondo model within gauge/gravity duality, which amounts to a Luttinger liquid coupled to a magnetic impurity.

In this presentation, we will discuss the notion of time and unitarity in the vicinity of a bounce in quantum cosmology, that is, a turning point for the scale factor. This starts from the Vilenkin approach to the interpretation of the solutions of the Wheeler-DeWitt equation. In this approach, unitarity is defined through the conserved current and is by nature an approximate concept. In minisuperspace it amounts to using the scale factor as a time variable. A unitary evolution is recovered when the latter becomes semi-classical enough. Unfortunately, WKB methods drastically fail near a turning point and the scale factor cannot play the role of time in scenarios with a bounce or a recollapsing phase for the universe. In this work, we extend the results of Vilenkin Massar and Parentani to momentum representation. For this, we investigate the dynamics of matter transitions when using its conjugate momentum as a time. In a first part, we describe the precise conditions so as to recover unitarity, and hence, a consistent notion of probability. In a second part, we discuss a concrete example in the vicinity of a bounce.

Our understanding of the structure of real genotype fitness landscapes at different scales is still incomplete and requires new methods and models. We present a model of long-term protein evolution that, once fitted to data from prokaryotic protein alignments, shed light on the large-scale epistatic structure of the protein universe. We also propose a natural measure of epistasis inspired by this model, namely the correlation of fitness differences, and show how it can be used both to measure local and global epistasis in experimental landscapes and to discriminate between theoretical models. Finally we discuss how chains, i.e. constrained paths in the landscape, are related to the epistasis and the structure of the interactions.

DNA of the human genome is two meters long and is folded inside a cell nucleus, which is just 5 microns in diameter. Revealing genome organization in 3D and understanding physical principles that govern its folding constitute the Genome Folding Problem. Recently developed Chromosome Conformation Capture technique, Hi-C, provides comprehensive information about spatial genome organization. Our analysis of these data and polymer simulations suggest that the genome is folded into a non-equilibrium polymer state, called the fractal globule. Yeast genome, on the contrary, resembles an equilibrium polymer brush. I will discuss intriguing properties of these polymer states and other major features of the human and yeast genome organization.

It is a challenge to explain biological phenomena that appear to arise from the complex interaction of many genes in response to natural selection acting through the mechanisms of mutation and recombination. The proportions of a large population that have various genotypes form a probability measure on the space of possible genotypes and, under suitable assumptions, it is reasonable to model this probability measure as evolving in time according to a relativelysimple, continuous-time, measure-valued dynamical system. I will explore the structure of this dynamical system and its equilibria with a view to shedding light on evolutionary explanations of the similar patterns of senescence and mortality seen in many multi-cellular organisms. In particular, I will point out how the non-linear nature of the dynamical system causes problems with a popular explanation of the Gompertz-Makeham mortality curve and requires a reformulation of Haldane's Principle, one of the cornerstones of the theory of mutation accumulation. This is joint work with David Steinsaltz (Oxford) and Kenneth W. Wachter (Berkeley).

Isolation of a single sheet of graphene indicated that strictly two-dimensional (2D) materials can exist at finite temperatures. Indeed, inorganic 2D systems such as hexagonal BN sheets, transition metal dichalcogenides (TMD) with a common structural formula MeX2, where Me stands for transition metals (Mo, W, Ti, etc.), X for chalcogens (S, Se, Te), and SiO2 layers were later manufactured by various methods. All these materials have defects, which naturally affect their properties. Moreover, defects can deliberately be introduced by ion and electron irradiation to tailor the properties of the system [1]. In my talk, I will summarize our knowledge about defects [2] in graphene and other 2D materials. I will also touch upon defect production in 2D systems under impacts of energetic ions [3] and electrons [4,7] and present theoretical data obtained in collaboration with several experimental groups [3,6]. I will also dwell upon defect-mediated engineering of the electronic structure of 2D materials such as BN [5] or TMDs [7]. Besides, I will discuss mixed TMDs, such as MoS2x Se2(1−x), which can be referred to as 2D random alloys [8]. Our simulations indicate that 2D mixed ternary MoS2/MoSe2/MoTe2 compounds are thermodynamically stable at room temperature, so that such materials can be manufactured by CVD or exfoliation techniques that the direct gap in these material can continuously be tuned. I will further present advanced first-principles calculations of the electronic structure of TMDs [9]. 1. A.V. Krasheninnikov and F. Banhart, Nature Materials, 6 (2007) 723. 2. F. Banhart, J. Kotakoski and A. V. Krasheninnikov, ACS Nano, 5 (2011) 26. 3. M. Kalbac, et al., Advanced Materials 25 (2013) 1004. 4. J.C. Meyer, et al., Phys. Rev. Lett. 108 (2012) 196102. 5. N. Berseneva, A. V. Krasheninnikov, and R.M. Nieminen, Phys. Rev. Lett. 107 (2011) 035501. 6. R. R. Nair, et al., Nature Physics 8 (2012) 199. 7. H.-P. Komsa, et al., Phys. Rev. Lett. 109 (2012) 035503. 8. H.-P. Komsa and A. V. Krasheninnikov, J. Phys. Chem. Lett. 3 (2012) 3652. 9. H.-P. Komsa and A. V. Krasheninnikov, Phys. Rev. B (Rapid Comm.) 86 (2012) 241201.

Opening Colloquium of the key profile area "Quantum Matter and Materials". After the talk there will be a reception in the foyer. Abstract: This talk will review recent experiments by several groups, including ours, aimed at identifying signatures of Majorana fermions in condensed matter systems. To date, the experiments are very suggestive, but uncertainties remain. What can future experiments do to make the case air-tight? As time allows, I will also discuss how such states might be valuable to topological quantum information processing.

This talk will review recent experiments by several groups, including ours, aimed at identifying signatures of Majorana fermions in condensed matter systems. To date, the experiments are very suggestive, but uncertainties remain. What can future experiments do to make the case air-tight? As time allows, I will also discuss how such states might be valuable to topological quantum information processing.

Networks of interacting units dominate our everyday lives, from the dynamics of neural circuits that control our behaviors to the power grids that provide us with electric energy. Over the last 15 years, researchers furthered our understanding of complex networks by analyzing their connectivity structure and collective dynamics, mostly using perspectives from either graph theory, statistical physics or nonlinear dynamics. Recent progress shows that features from the statistical physics and the nonlinear dynamics of networks are often intimately related; I discuss two main examples: Many different neural circuits exhibit spatio-temporal activity patterns, but the dynamic origin of such patterns remains unknown. I link the nonlinear deterministic dynamics of such systems to a statistical description. The results indicate that non-additive coupling, recently found in neuro-physiological experiments, enable the generation of spike patterns in circuits consistent with anatomy. In a second example, I identify a type of percolation transition that was unknown so far. Analyzing the microscopic dynamics of link additions reveals that the transition is continuous in the thermodynamics limit but discontinuous features prevail even for macroscopic finite systems. Finally, I briefly mention a new concept of universal computation via switching in complex networks of states and the intriguing self-organized dynamics of power grids.

Population genetics provides a theoretical frame work to interpret genetic diversity and make inferences about the evolutionary process. Traditional population genetics models assume that adaptation, i.e. change for the better, is rare. However, adaptive evolution is pervasive in many organisms and other models are needed to study their evolution. I will show that many models of asexual adaptive evolution give rise to a universal genealogical process known as Bolthausen-Sznitman coalescent. The Bolthausen-Sznitman coalescent, first described in spin-glass physics, makes concrete predictions for patterns of genetic diversity that can be compared against data and used for inference. I will close by discussing how these results for asexual evolution carry over to sexual populations.

I will give an overview of recent work on understanding learning processes on networks, specifically inference of a function: imagine trying to predict biochemical reactivity of a protein based on noisy measurements for "similar" molecules, where similarity is represented by links on a network whose nodes are the various proteins. In a Bayesian framework and using flexible priors (Gaussian processes) on the function to be learned, I will show that the cavity or replica methods can be used to predict the learning curve (mean squared error versus number of data points), exactly so in the limit of large random networks with some imposed degree distribution. I will then trace an arc from this research to my work on dynamics of protein interaction networks, glasses and dynamical phase transitions, and soft matter physics.

Describing a given system as a network means reducing it to set of discrete nodes connected by links. This reduction often leads to a considerable simplification, but preserves the complex pattern of interactions among the system's constituents. Network models thus lend themselves naturally to the analysis of complex systems in which the interactions between constituents give rise to emergent properties and phenomena. Here, the hope is that a properly constructed network model can simplify the system such that these properties and phenomena can be understood analytically. In this talk I  illustrate this approach by several short examples from my work and a study on the stability of ecological food webs that I present in greater detail. I conclude with a forward looking discussion arguing that recent successes follow a certain pattern that could lead to a conceptual unification of complex systems research.

Electronic correlations have long been known to fundamentally affect a material's property. I will discuss situations, where they can induce topologically nontrivial electronic states. Interaction between itinerant carriers and localized spins on frustrated lattices can stabilize phases that are in many respects similar to a Landau level, as a non-coplanar spin background gives the electrons a complex phase analogous to that induced by a magnetic field. If the bands are nearly flat, longer-range Coulomb repulsion can then induce states that are like lattice-analogs of fractional Quantum-Hall states, but do not require an external magnetic field. I will also discuss Mott insulators with spin-orbit coupling, where excitations then have combined spin and orbital character. We have investigated effective classical spin model for spin-orbit coupled Mott insulators. The ground state has turned out to support a vortex lattice of topological defects, Z2 vortices, whose density is determined by the strength of the spin-orbit coupling.

The spreading of discrete entities occurs in many complex systems, and controls for instance the speed of chemical reactions, species invasions, epidemic outbreaks or biological adaptation. Unless population are well-mixed (like bacteria in a shaken test tube), the spreading dynamics not only depends on reaction rates but also on the dispersal behavior of the species. Spreading at a constant speed is generally predicted when dispersal is sufficiently short-ranged. I present a recent exactly solvable model for this wave-like spread and the associated genealogical trees, which are generated by a  coalescent process with frequent multiple mergers. Spreading in the presence of rare long-range dispersal (e.g. triggered by aviation) has been unresolved so far: While it is clear that even rare long-range jumps can lead to a drastic speedup, it has been difficult to quantify the ensuing stochastic growth process. I present a simple self-consistent argument supported by simulations that accurately predicts the asymptotic spreading dynamics for fat-tailed jump distributions.  I discuss our results in the light of population genetics, and show how they can be used to predict evolutionary change in simple microbial experiments. 

I will describe my work characterizing quantum entanglement in systems with a Fermi surface. This class includes everything from Fermi liquids to exotic spin liquids in frustrated magnets and perhaps even holographic systems. I review my original scaling argument and then describe in detail a number of new precise results on entanglement in Fermi liquids. I will discuss recent quantum Monte Carlo calculations of Renyi entropies and will argue that we now have a rather complete agreement between theory and numerics for Fermi liquid entanglement. I will also exhibit universal crossover functions between thermal and entanglement entropy and a class of solvable interacting models where I have proved the universality of the Widom formula for Fermi surface entanglement. I will conclude by discussing some of the lessons learned and by giving a glimpse of the wider world of many-body entanglement.

In ultracold atomic gases, the areas of quantum optics and many-body quantum physics increasingly merge together, giving rise to non-equilibrium scenarios where coherent and dissipative dynamics appear on an equal footing. Controlled dissipation allows for counter-intuitive effects such as dissipatively induced long-range phase coherence. We discuss several aspects of new many-body physics in such systems: This includes phases of lattice bosons with competing coherent and tailored dissipative dynamics, and a purely dissipative pairing mechanism for fermions, operative in the absence of attractive conservative forces. We show how this mechanism can be used for cooling into topologically ordered states and point out a possible physical realization. We analyze the nature of topological order and Majorana edge states in such non- equilibrium systems, highlighting properties without immediate counterpart in equilibrium. Finally, we report on first results on non-equilibrium criticality in driven open quantum systems subject to particle loss and pumping. We identify an additional, independent critical exponent witnessing the driven nature of these systems.

The discovery of superconductivity in Fe-based superconductors in 2008 with Tc reaching almost 60K is among the most significant breakthroughs in contemporary condensed matter physics. The experimental data on superconductivity show rich behavior, with potentially different symmetry of a superconducting state for different compositions of the same material. Furthermore, the phase diagram reveal a complex interplay between magnetism and superconductivity typical for other exotic superconductors such as cuprates or heavy fermions. In my talk I will present an introduction to various aspects of the theory of iron-based superconductors. In particular, I will discuss general aspects of the Cooper-pairing in Fe-based superconductors. I will argue that the physics is universal and that all pairing states obtained so far can be understood within the same universal pairing scenario. Then I will review the magnetic ordering in the iron-based parent compounds. I will first analyze the selection of the experimentally observed magnetic order out of the degenerate manifold which occurs due to frustration associated with multiple nesting wave vectors. In addition, in analogy to the J1-J2 model of the localized spins, I will discuss the possibility of an Ising nematic order to form above the actual Neel temperature and its coupling to orbital and lattice degrees of freedom.

Atomic gases cooled to Nanokelvin temperatures are a new exciting tool to study a broad range of quantum phenomena. In particular, an outstanding and rapid control over the fundamental parameters, such as interaction strength, spin composition, and dimensionality allows to realize and observe many different situations far from equilibrium. Long-standing questions such as the thermalization dynamics can be investigated. Theoretical studies of this question have a long history, in particular, in the classical Fermi-Pasta-Ulam-Tsingou problem, a coupled chain of harmonic oscillators under the influence of further higher order couplings. Even though a lot of progress has been made, up to day, a complete understanding is missing. In the first part of my talk, I will report on recent progress on the investigation of relaxation dynamics in isolated quantum systems in connection with experimental studies. In the second part of my talk, I will address the question of the influence of a coupling to an environment on the system dynamics. We find that the environment can cause complex relaxation dynamics often described by spatially dependent classical diffusion equations.

Cells contain billions of molecular machines that drive processes such as directed transport and processing of the genetic information. The talk will address theoretical approaches to the mechanisms of such machines and to their cellular economy. I will discuss how the stochastic dynamics of several molecular motors can be coordinated by mechanical forces through a tug-of-war mechanism and how a global picture of the machines that process the genetic information (RNA polymerases and ribosomes) reveals the different economic principles that underlie their use in a bacterial cell.

We present quantum Monte Carlo simulations on a sign-problem free Bose-Hubbard model on the kagome lattice. This model supports a quantum Z2 spin liquid phase with fractional excitations and topological order, which can be characterized definitively through calculation of the topological entanglement entropy. I will outline how the entanglement entropy can be measured in general using a direct implementation of the ``replica trick'' which allows for the study of entanglement scaling in a variety of other models amenable to study by QMC. Finally, I will examine the kagome model's superfluid/spin-liquid transition, which is an example of an exotic deconfined quantum critical point called XY*, mediated by the fractional charges.

Although human musical performances represent one of the most valuable achievements of mankind, the best musicians perform imperfectly. Musical rhythms are not entirely accurate and thus inevitably deviate from the ideal beat pattern. Nevertheless, computer generated perfect beat patterns are frequently devalued by listeners due to a perceived lack of human touch. Professional audio editing software therefore offers a humanizing feature which artificially generates rhythmic fluctuations. However, the built-in humanizing units are essentially random number generators producing only simple uncorrelated fluctuations. The nature of temporal fluctuations in complex human musical rhythms has never been scrutinized as yet. Here, I will show that long-range correlations are an inevitable natural companion of both simple and complex human rhythmic performances. Moreover, it will be demonstrated that listeners strongly prefer long-range correlated fluctuations in musical rhythms. Thus, the favorable fluctuation type for humanizing interbeat intervals coincides with the one generically inherent in human musical performances.

The density matrix renormalization group (DMRG) is an extremely powerful technique for solving interacting many-body systems, but most applications of DMRG have been to lattice models on one-dimensional chains or ladder geometries. I will describe recent advances in DMRG that make it practical to explore richer classes of systems. In one dimension, we can now treat continuum models with long-range interactions with applications to electronic structure theory (especially density functional theory) and cold atom experiments. For two-dimensional systems, a new method for parallelizing DMRG means we can apply an order of magnitude more power to reach new system sizes. If time permits, I will also describe a promising extension of DMRG for finite temperature systems.

We study the non-equilibrium dynamics of a one-dimensional directional voter model with opinion exchange. The opinion of an individual at site i is denoted by σi which can take either +1 or -1. If σi =+1 and σi+1 =-1 (+- domain wall), the opinion of both individuals can be either ++ or -- with probability (1-α/2) [voter dynamics], but with probability α both individuals just exchange opinions without attaining a consensus [simple exclusion process]. Note that -+ domain wall does not induce any configurational change. When α < 1, the whole population eventually reaches a full consensus. The way how the consensus is reached is described by decaying behavior of the domain wall density ρ. For any α < 1, ρ is found to show a power-law decay behavior. Quite interestingly, however, the exponent varies continuously with α, from 0.5 (pure voter dynamics) to 0 (ASEP case). These results are obtained from both Monte-Carlo simulations and spectral analysis of the time evolution operator.

The Hubbard model is one of the most studied microscopic models in condensed matter physics. It describes on a very simple level the interplay between the kinetics and the Coulomb interaction of electrons in solid state systems. Generically, one expects at half filling a metallic phase for large kinetics while a Mott insulator is present for large interactions. In most cases the Mott phase shows additionally long-range spin order, since the Hubbard model can be mapped to a Heisenberg model in the strong-coupling limit. A direct transition between the metal and the long-range ordered Mott insulator was considered to be the standard case for a long time. But in recent years there are more and more evidences that especially on frustrated lattices there is the possibility of exotic and insulating intermediate phases without long range order. It is therefore an obviously relevant question what kind of effective low-energy theory describes such Mott phases and how to derive them. This is particular complicated when the spin liquid is located close to the metal-insulator transition as for the recently discovered spin liquid of the Hubbard model on the honeycomb lattice. In this talk we discuss these issues for the Mott phase of the Hubbard model on the triangular and on the honeycomb lattice.

Many physical phenomena such as quark confinement and the fractional quantum Hall effect arise at strong coupling, thus making a perturbative treatment impossible. Occasionally, a good description can nevertheless be found which involves effective degrees of freedom - different from the fundamental ones and adapted to the specific situation at hand. In such a situation one would speak about a duality. In 1+1 or 2 dimensions there is a wealth of dualities, some of which are extremely well understood. In higher dimensions, the best known duality is the celebrated AdS/CFT correspondence, a concrete realization of the holographic principle. The "conformal superspace sigma-models" featuring in the title are a special class of strongly coupled supersymmetric 2D quantum field theories which naturally arise in the context of the AdS/CFT correspondence (but also in the theory of disordered systems). As will be discussed, the underlying supergeometry allows for the derivation of exact partition functions but it also leads to a few unusual peculiarities. As an application, we conclude with arguments in favor of a supersymmetric duality which generalizes the well-known bosonization of a Luttinger liquid.

Some f-electron systems such as URu2Si2 and PrFe4P12 show ordered phases where the order parameter is difficult to identify. These systems are characterized by the non-Kramers configuration where the number of f -electrons per site is even. We have studied generalized Kondo lattice models for such non-Kramers systems, and have found a new kind of orders by combining the dynamical mean-field theory and the continuous-time quantum Monte Carlo calculation. In the case of two-channel Kondo lattice with two conduction electrons per site, for example, a strange order emerges where the one-body quantities such as magnetization and charge density remain the same as in the high- temperature phase. The (hidden) order parameter Ψ at each site is given by Ψ = ⟨S · (s1 − s2)⟩ where S is the localized pseudo spin and si (i = 1, 2) is the pseudo spin of conduction channel i at the same site. This order can be interpreted as itinerant higher multipoles of electrons, and also as analogue of the composite superconducting order with odd frequency pairing amplitude. We also discuss other exotic orders and implication of our results in actual f-electron systems.

We calculate the phase diagram of the Bose-Fermi Hubbard model on the 3d cubic lattice at fermionic half filling and bosonic unity filling by means of single-site dynamical mean-field theory (DMFT). For fast bosons, this is the Cooper problem in which the bosons can induce s-wave pairing, even between repulsively interacting fermions. Superfluid and supersolid phases are found, depending on the interspecies coupling strength. The modifications in the phase diagram compared to the purely fermionic case are well explained by renormalized hoppings and on-site interactions. Slow bosons favor fermionic charge density wave structures for attractive fermionic interactions.

Systems with long-range interactions have interaction potentials that decay slower than 1/r^d at large distances r in d dimensions. Common examples are self-gravitating systems, non-neutral plasmas, dipolar ferroelectrics and ferromagnets, etc. Long-range interacting systems are non-additive. This leads to unusual properties, both thermodynamic (e.g., negative microcanonical specific heat, ensemble inequivalence) and dynamic (e.g., slow relaxation, breaking of ergodicity). After a brief review, I will discuss a paradigmatic example, the Hamiltonian Mean-Field model, involving XY spins with mean-field interactions. For this model, relaxation of some initial states towards equilibrium proceeds through intermediate quasistationary states characterized by a slow variation of macroscopic observables over time. The life time of these states diverges with the system size. These states are observed in deterministic microcanonical evolution within a certain energy interval. Our recent study suggests that in presence of noise, quasistationary states occur only as a crossover phenomenon depending on the relative magnitudes of two timescales set by the system size and the level of noise in the evolution. Our proposed scaling form for the relaxation time to equilibrium is verified in simulations by a scheme of piecewise deterministic evolution for the model.

Beautiful spatial patterns can be formed in a cellular automaton known as Abelian Sandpile Model (ASM), by dropping large N number of particles at a single site and then redistributing them according to ASM toppling rules. The patterns are usually composed of large number of macroscopic regions with sharp boundaries, all of which grow at the same rate with N, keeping the overall shape unchanged. This proportionate growth makes the patterns important in the context of biology where such ability to adapt to size variations is common in growing animals. The patterns are also a good example of analytically tractable complex structures generated using simple local rules. I will present exact characterization of few asymptotic (N→∞) patterns on two dimensional lattices, which involve some intriguing connection to the mathematics of discrete analytic functions. I will also discuss robustness of the patterns against different types of noise.

One of the long term challenges in human health and disease is the control of pathogens, such as antibiotic-resistant forms of bacteria. In this talk, we will briefly describe two directions where materials science based approaches have been useful. 1) Bacterial biofilms are structured multi-cellular communities that are notoriously resistant to antibiotics. We translate bacteria movies into searchable databases of bacterial behavior and find an unexpected diversity in motility driven by Type IV pili across different bacterial species. The associated phenomena include ‘stick-slip’ motion analogous to earthquake dynamics, and self-organized distribution of resources in early biofilm development reminiscent of capitalist economies. 2) We examine the mechanism of a range of pore-forming polypeptides, including antimicrobial peptides, cell penetrating peptides, viral fusion peptides, and apoptosis proteins, and show how a combination of topology, coordination chemistry, and materials science can be used to approach a unified understanding.

Since the 90's random matrix theory (RMT) was successfully applied to quantum chromodynamics (QCD). In particular the low lying spectrum of the Dirac operator can be exactly mapped to chiral RMT. The same applies to lattice QCD. One particular modification of the Dirac operator on the lattice is the one introduced by Wilson. This modification explicitly breaks chiral symmetry even though the quark masses are set to zero. Recently a random matrix model was introduced which lies in the same universality class as the Wilson-Dirac operator. The aim is to investigate the lattice artifacts in data of lattice simulations with RMT as strongly simplified model. Quite recently we derived the joint probability density and, in particular, the level density of the so called Wilson random matrix ensemble. I will present these new results in my talk. Moreover I will show that the Wilson random matrix ensemble lies in a new class of ensembles related to a mixing of orthogonal and skew-orthogonal polynomials. Hence it is a new kind of interpolations between universality classes.

I discuss edge states in network models for Anderson localization. For the case of the spin quantum Hall effect, some transport properties maps to a classical localization problem formulated in terms of a loop model. Using conformal field theory and lattice algebras, critical exponents governing the decay of the conductance at higher plateaus transitions can be predicted, and are found to be in very good agreement with numerical simulations.

One of the defining properties of topological insulators and superconductors, which are topological phases with a full gap in the bulk, is the appearance of topologically protected zero-energy surface or edge states, some of which are of Majorana type. Boundary modes of topological nature can also occur in (semi-)metallic systems that exhibit topologically stable Fermi surfaces or in nodal superconductors with non-trivial topology. For instance, dispersionless boundary states exist on the (110) surface of a d_{x^2-y^2}-wave superconductor (e.g., in high-temperature superconductors), or at the zig-zag edge in graphene. In this talk I present a classification of both gapped and gapless topological phases in terms of symmetry and spatial dimension. As a consequence of the non-trivial bulk topology all of these systems exhibit topologically stable surface states, which are either dispersionless or show a linear dispersion as a function of edge or surface momentum. As a concrete example, superconductors without inversion symmetry are discussed in detail. Three different types of topologically protected surface states are identified in these systems, namely Kramers-degenerate Majorana modes, arc surface states, and surface flat bands, whose stability is guaranteed by the conservation of three different types of topological invariants. I examine the signatures of the Majorana modes, the arc surface states and the zero-energy surface flat bands in tunneling-conductance spectra and other experimental probes.

Percolation is the best known example of geometrical phase transition. In two dimensions a plethora of exact results were derived at criticality (critical exponents, crossing probabilities) using Conformal Field Theory. In this talk I will discuss a possible field theoretical description for the scaling region of off-critical percolation, exploiting the $q\rightarrow 1$ limit of the integrable q-color Potts field theory. I will apply the formalism to the computation of universal amplitude ratios and off-critical crossing probabilities. All the predictions can be checked by numerical simulations.

Silicon-based complementary metal-oxide-semiconductor (CMOS) devices are the basic building blocks of integrated electronic logic and memory devices. Their continuous downscaling which, as described by Moore’s law, leads to a doubling of the number of transistors on a chip every two years, cannot go on indefinitely: Today’s smallest transistors function despite quantum mechanics; their metal-wire interconnects will also face scaling limits. New design paradigms that make use of quantum mechanics instead of trying to avoid its effects will be necessary. In this talk, I will present some current work on spin injection into SiGe-heterostructures, tunneling field effect transistors (FETs), and electrical detection of plasmons.

The prediction and discovery of topological insulators (TIs) generated a large interest in the physics community during the past years. This new state of mater is characterized by conducting Dirac type surface states while the bulk of the material remains insulating. These surface states have been observed so far in, e.g., Bi2Te3, Bi2Se3 and Sb2Te3. These materials do however exhibit large defect densities paired with low arrier mobilities. So far this prevented transport studies in the quantum Hall regime of 3D TIs. Recently, however, we succeeded in using strained bulk HgTe as 3D TI. This enabled us to measure the quantum Hall effect from the 3D TI surface state in transport experiments. The samples were fabricated by molecular beam epitaxy of HgTe layers on top of CdTe substrates. The strain induced by the lattice mismatch between HgTe and CdTe results in opening a gap in the HgTe band structure which makes the TI state observable.

The time evolution of a 1d quantum system after a quench is a challenging many-body problem which can be studied numerically using time-dependent density matrix renormalization group (DMRG) techniques. As shown by Lieb and Robinson, information spreads at a finite velocity resulting in an effective "light cone". In my talk I will present a very efficient DMRG algorithm based on this light cone structure. With the help of the DMRG data and analytical results for non-interacting cases I will discuss the relaxation and thermalization of doublon states in the fermionic Hubbard model.

The topological invariant of a topological insulator (or superconductor) is given by the number of symmetry-protected edge states present at the Fermi level. Despite this fact, established expressions for the topological invariant require knowledge of all states below the Fermi energy. Here, we propose a way to calculate the topological invariant employing solely its scattering matrix at the Fermi level without knowledge of the full spectrum. Since the approach based on scattering matrices requires much less information than the Hamiltonian-based approaches (surface versus bulk), it is numerically more efficient. In particular, is better-suited for studying disordered systems. Moreover, it directly connects the topological invariant to transport properties potentially providing a new way to probe topological phases.

Semiconductor spin qubits are promising candidates for quantum computation because of their slow decoherence and potential for scalability. All fundamental single qubit operations have been demonstrated for GaAs-based spin qubits, but they suffer from decoherence due to hyperfine coupling to nuclear spins. We have developed effective techniques to mitigate this decoherence channel for two-electron spin qubits in double quantum dots. By repeatedly inverting the qubit in a way that decouples its evolution from the fluctuations of the nuclear spin bath, its coherence time can be extended to more than 200 ms, two orders of magnitude longer than previously shown. Alternatively, operating the qubit as a feedback loop that controls the nuclear bath also enhances the coherence time and enables universal single qubit control with greatly improved gate fidelities. Having achieved good single qubit control, a fundamental milestones is to couple qubits in order to generate entanglement. We demonstrate a protocol to implement two-qubit gates using the Coulomb interaction between adjacent double dots while decoupling both qubits from slow electrical fluctuations. These results indicate that Bell states can be created with sufficient fidelity to demonstrate entanglement.

In this talk, two different strands of quantum field theory on non-commutative space will be presented. First, the scattering of the quantized Dirac field on Minkowski spacetime will be discussed, and it will be explained how this gives rise to observables of the quantized Dirac field on Moyal-deformed Minkowski spacetime. Furthermore, it will be argued that this construction is a model for obtaining a correspondence between more general Lorentzian non-commutative geometries in the sense of spectral geometry and observables of quantum field theories over such non-commutative Lorentzian geometries. In the second strand of the talk, the problem of Wick rotation, i.e. the relation between a quantum field theory on Euclidean space on one hand, and Minkowski spacetime on the other, will be generalized for the Moyal deformations of Euclidean space and Minkowski spacetime. It will be shown that there is a relation if the Moyal deformation leaves the time coordinate untouched. These latter results were obtained in joint work with H. Grosse, G. Lechner and T. Ludwig.

This talk will be on a topic related to mathematics and (particle) physics. Scattering amplitudes in particle physics are related to the probability with which a certain scattering process occurs. The scattering amplitudes are calculable in perturbation theory. Higher orders in the perturbative expansion are needed for precision predictions for the experiments at the LHC collider. Recent progress in the calculation of scattering amplitudes has shown that these scattering amplitudes have a much simpler structure than previously believed. This simplicity is directly related to mathematical structures hiding underneath. The complexity of scattering amplitudes increases with the number of external legs and with the number of internal loops. Simplicity with respect to the number of external legs is obtained by formulating the theory not in space-time, but in twistor space instead. Simplicity with respect to the number of internal loops is obtained by making use of the algebra of transcendental functions, like the algebra of multiple poly-logarithms. Here, particle physics touches the domain of the theory of motives.

The symmetry group on the Anti-de-Sitter space acts as a conformal group on its boundary. It is shown that quantum fields on Euclidean AdS – when restricted via suitable scaling – indeed fulfill conformal invariance on the boundary. Even in the case of non-interacting fields some new renormalization factors are required. In the case of interacting fields in AdS2 or fields with cut offs in arbitrary dimensions a new kind of somewhat intriguing infra-red problem arises. While the approach taken here does not reflect the richness of Maldacena’s AdS/CFT conjecture on supergravity and Yang-Mills theory, the AdS/CFT problem for Euclidean quantum fields allows the statement and proof of some rigorous results on the basis of what has been established in Constructive QFT.

The Batalin-Vilkovisky formalism for field theories with gauge symmetries is revisited. It is shown that the step from the classical theory to the quantum theory, which is usually discussed via the path integral, can be performed by the renormalized time ordering operator which is obtained by a generalization of Epstein-Glaser renormalization.

In 1986 Kardar, Parisi, and Zhang proposed a stochastic PDE for the motion of driven interfaces. Bertini and Giacomin (1997) explained how to approximate the solution to the 1D KPZ equation through the weakly asymmetric simple exclusion process. Based on work of Tracy and Widom on the PASEP, we report a formula for the one-point generating function of the KPZ equation in the case of sharp wedge initial data. The long time limit is given by the Tracy-Widom distribution from GUE random matrices. Of particular interest are the finite time corrections. This is joint work with Tomohiro Sasamoto.

We consider the quantum field theoretical description of quantum many-body systems. In fermionic gases at low densities, and in specific materials of solid-state physics, the Fermi surface becomes small or even degenerates to a point. A multiscale analysis can be used to determine the correlation functions by convergent expansions. In the talk I will discuss models for double-layer graphene and the dilute Fermi gas, and explain the main ideas in the analysis.

We consider certain functions of eigenvalues and eigenvectors (spectral statistics) of real symmetric and hermitian random matrices of large size. We first show that for these functions an analog of the Law of Large Numbers is valid as the size of matrices tends to infinity. We then discuss the scale and the form for limiting fluctuations laws of the statistics and show that the laws can be standard Gaussian (i.e., analogous to usual Central Limit Theorem for appropriately normalized sums of i.i.d. random variables) in non-standard asymptotic settings, certain non-Gaussian in seemingly standard asymptotic settings, and other non-Gaussian in non-standard asymptotic settings.

We explain the notation of a bicategory and its use in topological, classical and quantum field theory.

I will review some recent developments in string theory in a setting initiated by Klaus Pohlmeyer in the 1980s. Here, a Poisson algebra is assigned to surfaces of extremal area immersed in Minkowski space, e.g. to the world sheet of a string. Two deformations of this Poisson algebra (quantization schemes) have been proposed. One is based on the quantization of an auxiliary Lie algebra in terms of its universal enveloping algebra, the other is based on the deformation theory of quasi-Lie-bialgebras. Contrary to the ordinary setting of string theory, which is based on conformal field theory, these two quantization schemes do not require a critical dimension for consistency.

Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably, while their relevance has grown in practice as well. Here, the phenomenon of homometry shows various unexpected new facets, particularly so in the presence of disorder. After an introduction to the mathematical tools, we review pure point spectra, based on the Poisson summation formula for lattice Dirac combs, aiming at the diffraction formulas of perfect crystals and quasicrystals. We continue by considering classic deterministic examples with singular or absolutely continuous diffraction spectra, and we recall an isospectral family of structures with continuously varying entropy. We close with a summary of more recent results on the diffraction of dynamical systems of algebraic or stochastic origin.

An excited eigenvalue of an atom is believed to be unstable and turn into a resonance under a perturbation. In this talk the precise definition of such resonances is given and their existence and construction is outlined in case the atom is minimally coupled to the quantized radiation field. This model is infrared singular and notoriously difficult to treat. We review Sigal’s recent construction of resonances based on the “Feshbach map” and present a novel, alternative construction based on “Pizzo’s method”.

In this talk we will consider the density of states (DOS) of Wigner matrices on very small intervals. In such intervals, the fluctuations of the DOS are important and one cannot expect convergence to the semicircle law to hold in probability. Assuming the entries of the matrices to have a sufficiently regular law, we prove, nevertheless, that the average DOS still converges to the semicircle law, on arbitrarily small intervals.

Irreversible phenomena like diffusion and thermalization obviously occur in our world, which is described by deterministic and time-reversible equations (Newton’s equations of motion or the Schrödinger equation in quantum mechanics). However, up to today we seem to lack the tools to describe and derive these phenomena rigorously (apart from a few special models). In physics, one usually models them by stochastic evolution equations and the transition from deterministic equations to stochastic ones remains a leap of faith. Our work treats one of the simplest possible models: a quantum particle interacting with a gas of bosons. With the help of a time-dependent renormalization group analysis, we prove diffusion.

In a high dimensional genotype space, biological evolution in the limit of strong selection and rare mutations is studied. Adaptation processes can be treated as an uphill walk in fitness, which terminates at a fitness maximum. When fitness is assigned randomly, it is shown, that the mean walk length shows logarithmic behaviour in the number of initially available beneficial steps, with a prefactor, determined by the tail of the fitness distribution.

The inclusion of charging and spin-exchange interactions within the Universal Hamiltonian description of quantum dots is a highly non-trivial problem owing to the fact that it leads to a formulation of the problem in terms of a non-Abelian action. We present an exact analytical solution of the problem, in particular, in the vicinity of the Stoner instability point. We calculate exactly several physical observables, including the tunneling density of states (TDOS) and the spin susceptibility. We find that due to the presence of spin-exchange interaction, at the vicinity of the instability point the TDOS exhibits a non-monotonous behavior as function of the tunneling energy. This effect survives even at temperatures much higher than the exchange energy. Also, we find that in the same temperature range the single-particle level fluctuations enhances the spin-susceptibility in the vicinity of the Stoner instability. Experimentally, our results could be tested in nearly ferromagnetic materials. Some of the results are published in I.S. Burmistrov, Yu. Gefen, M.N. Kiselev, JETP Lett. 92, 766 (2010).

Spinless bosons in a one dimensional chain at integer filling can form two gapped phases, the Mott and Haldane insulators, which do not break any obvious symmetry. I will argue that the distinction between the two phases is protected by lattice inversion symmetry. Breaking this symmetry gaps out the transition and leaves the critical point as an isolated gapless point in the two-parameter space of the original tuning parameter and the symmetry breaking perturbation. I will show that encircling this point adiabatically entails pumping of exactly one boson across the Bose insulator. When multiple chains are coupled, the two insulating phases are no longer sharply distinct, but the pumping property survives and allows to associate a topological flux with gapless regions in the phase diagram. This leads to strict constraints on the topology of the phase diagram of systems of quasi-one dimensional interacting bosons. Finally I will use the pumping property to elucidate the topological invariant underlying the Haldane phase and to discuss possible extensions to interacting topological phases in higher dimensions.

"Raumschiff Enterprise" begründete ein TV-Universum, das mit Ablegern wie "Star Trek: The Next Generation" und "Deep Space Nine" bis heute weltweit Kultstatus genießt und seinen Fans als "Trekkies" einen Eintrag ins Oxford English Dictionary bescherte. Andreas Rauscher erörtert diese metaphorischen Weiten des Star-Trek-Universums aus medienwissenschaftlicher Sicht. Die Abenteuer im All spiegeln immer wieder Ereignisse aus unserer irdischen Realität. Standen die Anfänge noch im Zeichen der Kennedy-Ära und des Kalten Krieges, so sind die späteren Serien u.a. geprägt durch die Ideen der UNO und die Infragestellung von Mythologien und Utopien.

Wir sind das Zentrum des Universums – das glaubte man seit der Antike, bis ein bahnbrechender Denker damit aufräumte: Im 16. Jahrhundert erklärte Nicolaus Copernicus, dass sich die anderen Himmelskörper nicht um die Erde drehen. Vielmehr liege die Mitte des Universums “nahe der Sonne,” lautete der Gedanke, der einen Umbruch in der Geschichte des Weltbildes lostrat. Der Vortrag skizziert die Entwicklung der astronomischen Weltanschauung seit Copernicus und erläutert auch die Bedeutung von Galileo Galilei. Als weitere Phasen in der Entwicklung des heliozentrischen Weltbildes werden Johannes Keplers Gesetze der Planetenbewegung und Isaac Newtons Gravitationsgesetz einführend dargestellt. Schließlich verfolgt der Vortrag die Entwicklung weiter bis zur Gegenwart, in der wir längst erkannt haben, dass wir am Rand unserer Milchstraße leben – nur einer von unzähligen Galaxien.

We study the influence of the interaction range on the out-of-equilibrium behavior of both isolated and thermostated systems. In the case of short-range (extensive) systems, we investigate coarsening dynamics. There we find the existence of a short-time regime where the laws relating the rate of growth of domains' size with the interaction range are different in isolated and thermostated systems. In the case of long-range (non-extensive) systems, we focus on the "Quasi-stationary states" in which these systems are trapped before reaching the thermodynamic equilibrium. There, we demonstrate that one can in principle measure negative specific heat both in the case of isolated and thermostated systems. This is a surprising result since for classical systems at equilibrium negative specific heat can be found just in isolated systems, while it is generally recognized that a positive specific heat should always be measured in systems in contact with a thermal bath.

Etwa ab Mitte des 19. Jhs. führten verschiedene Entwicklungen innerhalb der Mathematik schließlich dazu, dass der klassische Raum der Euklidischen Geometrie - wie man ihn z.B. in Newtons absolutem Raum wieder findet - seine Monopolstellung verlor. In diesem Zusammenhang sind u.a. die Entstehung der verschiedenen nichteuklidischen Geometrien (hyperbolisch, sphärisch, elliptisch) und der Aufbruch in höhere Dimensionen (insbesondere natürlich in die vierte) zum nennen. Im Vortrag möchte ich hauptsächlich auf ein weniger bekanntes aber durchaus interessantes Thema eingehen, nämlich auf die sogenannten Clifford-Kleinschen Raumformen. Das sind Mannigfaltigkeiten, welche lokal wie ein klassisch-geometrischer Raum aussehen, die sich aber global von diesen unterscheiden. Da unsere Kentnis des Universums immer nur lokal sein kann, stellte sich somit - schon vor der Relativitätstheorie - die Frage, in welchem Raum wir denn leben.

We study the collective dynamics of the Skyrmion crystal (SkX) in thin films of ferromagnetic metals resulting from the nontrivial Skyrmion topology. We show that the current-driven motion of the crystal reduces the topological Hall effect and the Skyrmion trajectories bend away from the direction of the electric current (the Skyrmion Hall effect). We find a new dissipation mechanism in non-collinear spin textures that can lead to a much faster spin relaxation than Gilbert damping, calculate the dispersion of phonons in the SkX, and discuss effects of impurity pinning of Skyrmions. This work has been done in collaboration with Jiadong Zang, Maxim Mostovoy, and Jung Hoon Han.

In populations with a finite number of individuals, demographic fluctuations play an important role. The dynamics of such finite populations are analyzed in the framework of stochastic evolutionary game theory. In evolutionary biology, the probability that a mutant strategy spreads in the population and takes over is an important quantity that is well understood. Here, focus is on the mean exit time such a fixation event is expected to take (fixation time). Starting from a master equation for the distribution of fixation times of, e.g., a birth-death process, one derives and generally solves recursions for the moments. A concept that allows a deeper insight here is weak selection. This corresponds to a perturbation theory with neutral evolution as a reference, where all strategies perform equally well. Weak selection considers only small deviations from neutral evolution and is a highly relevant case in biology - most (point) mutations only provide a small change in the overall reproductive success. Various new aspects of the mean exit times in evolutionary games under weak selection can be addressed: How do the exit times depend on the parameters of the evolutionary game? How universal is the folklore of weak selection results when comparing the two most important classes of birth-death processes? How can the counter-intuitive effect be understood that a beneficial mutation can take over slower than a neutral mutation?

Evolutionary dynamics can proceed on two types of fitness landscapes, static or dynamic. Population movement on a static fitness landscape has been studied in great detail but usually the dimensions of the landscape are kept to a minimum. I address the time required for populations to move in a d dimensional fitness landscape from one fitness peak to another. If there are multiple paths via a fitness valley or a single ridge which is neutral then which is a faster route and when? As populations move over the fitness landscape they may change the landscape itself. To analyse such dynamic fitness landscapes I employ evolutionary game theory. Traditionally evolutionary game dynamics of two players with two strategies is studied. These games have been used to model many biologically relevant scenarios, from social dilemmas in mammals to microbial diversity. Some of these games take place between a number of individuals and not just two. We generalise the important results from two player, two strategy games to multiplayer games. Our results hold for any game and can easily be applied for specific cases, e.g. public goods games or multiplayer stag hunts.

In asexual populations, both clonal interference, competition for fixation among concurrent beneficial mutations, and genetic drift, waste of beneficial mutations purely by random events, limit the speed of adaptation. In sexual population, on the other hand, loss of beneficial mutations due to clonal interference can be alleviated by genetic recombination. By considering the simplest two-locus model with an infinite number of sites each, we study quantitatively how much the speed of adaptation can be enhanced by genetic recombination. We first consider the infinite population size limit which renders the population dynamics deterministic and tractable analytically to a certain extent. Having a hint from the deterministic dynamics, we investigate the finite population dynamics using numerical simulations.

Ultrathin superconducting wires (UTSCW) fabricated recently by the means of novel experimental techniques have been the subject of intense experimental and theoretical research. When the diameter is smaller than the coherence length of the bulk superconductor, these systems are effectively one-dimensional (1D), since the order parameter can only vary along the parallel direction. Interestingly, the theory predicts the vanishing of long-range order (LRO) due to strong quantum critical fluctuations, a fact that allows the use of UTSCW to address fundamental questions in Quantum Critical Phenomena. In this work, we focus on UTSCW under the effect of environment-induced dissipation. Presumably, this is a realistic situation in many experimental setups. Remarkably, we show that a weak coupling to a diffusive metallic film can reinforce superconductivity in the wire through the quench of order-parameter fl uctuations. We obtain the critical points and phases of the system at T=0, and in particular, we predict a quantum phase transition towards a superconducting phase with LRO as a function of the wire's superconducting stiffness and coupling parameter to the metal. We also discuss implications for experiments.

Detecting community structure from network topology is a well known problem with many possible applications. A large number of studies was conducted over the last decade, but a principal approach that would for instance output that a random graph does not have any community structure is still missing. Based on a random graph model for a community structure I will first show the existence of a phase transition between possible and impossible community inference. This phase transition is related to some known results from statistical physics of spin glasses, for optimal inference the partition function of a corresponding spin glass model needs to be computed. Then I will turn to real-world networks and inspired by the theoretical results I Introduce a new message passing algorithm which is able to learn parameters of the community structure (number of communities, ...), and to infer the most likely community assignment. As an application I will present some results on real-world net.

The newly discovered iron based superconductors with unusually high transition temperatures exhibit a rich phase diagram that includes structural, magnetic and superconducting transitions. As such, these materials are the latest playgrounds to study how in complex materials different phases compete, and how this unconventional setting eventually gives rise to superconductivity. The first part of the talk will be a review of this topic. From a theoretical point of view one of the goals at present is to identify the microscopic interactions that give rise to the rich phase diagram. This is the motivation to present in the second part of the talk a study of the effects of magneto-elastic couplings between the magnetic and the lattice degrees of freedom. We will argue that several unusual magneto-structural features of these materials can be understood from this point of view.

The effects of mutations in cancer can be analyzed with respect to the evolutionary conservation of the genomic target. I will briefly discuss, how to use Kimura's one locus model to infer fitness effects of mutations in a controlled manner and how to asses the significance of such scoring schemes when it comes to understand cancer evolution.

Helical magnets support topologically stable excitations characterized by a full twist of the magnetization on a length scale determined by spin-orbit coupling. Such magnetic whirls are known as Skyrmions, originally conceived by Tony Skyrme in a different context to model baryons. In the presence of a magnetic field, magnetic Skyrmions can even condense in the magnetic ground state forming a magnetic lattice similar as the vortex lattice in type II superconductors. Such a Skyrmion lattice has been recently identified by neutron scattering in MnSi. I give an overview of the helical magnet MnSi and report on recent spin-transfer torque experiments. The magnetization within the Skyrmion lattice phase can be manipulated upon applying an ultralow electric current, five orders of magnitudes smaller than those typically applied in experimental studies on current-driven magnetization dynamics on nanostructures. I explain the efficient spin-transfer torques arising from the coupling of spin currents to the magnetic whirls and discuss future perspectives.

In this talk we present a method to derive the primordial power spectrum of density fluctuations in the framework of quantum cosmology. For this purpose, we perform a Born-Oppenheimer approximation to the Wheeler-DeWitt equation for an inflationary universe with a scalar field. In this way we first recover the scale-invariant power spectrum that is found as an approximation in the simplest inflationary models. We then obtain quantum gravitational corrections to this spectrum and discuss whether they lead to measurable signatures in the CMB anisotropy spectrum.

One-dimensional systems allow for a detailed study of the competing eects of disor- der and interactions on the localization of a Fermi or Bose gas.[1] While non-interacting fermions are always localized in a random potential, a transition to a superconducting phase occurs for strong enough attractive interactions. Similarly, strong repulsive in- teractions will trigger the localization of a Bose gas.[1,2] During this talk, I will present the study of an interacting mixture of bosons and spinless fermions in a random po- tential. Super uid correlations are enhanced by interspecies interactions, while the disordered potential tries to pin each component of the gas. Using renormalization group methods, supplemented by a variational calculation, we have found several lo- calized or super uid phases, including a new insulating phase, similar to the Bose glass, in which both species are coupled and localized.[3] We also computed the dynamical structure factor for typical parameters of various ultra-cold atoms experiments. Each phase exhibits its own signature, that could be probed in Bragg scattering experiments. [1] T. Giamarchi and H. J. Schulz, Phys. Rev. B 37, 325 (1988) [2] M. P. A. Fisher et al., Phys. Rev. B 40, 546 (1989) [3] F. Crépin, G. Zaránd, P. Simon, Phys. Rev. Lett. 105, 115301 (2010)

F-theory can be thought of the correct way to describe Type IIB theory with 7-branes in generic situations where the string coupling cannot be assumed to be small. It geometrizes the variation of the string coupling and the backreaction of the 7-branes on the ambient space by introducing two additional auxiliary dimensions. Thus, it is formulated using twelve dimensions. For string phenomenology it is very interesting because one the one hand exceptional gauge groups are available, which are highly relevant in the context of unification and are heavily exploited in heterotic string theory, and on the other hand the particle physics degrees of freedom localize on submanifolds of the compactification, as is the case in Type II theories with intersecting D-branes. Given this localization, a considerable number of phenomenological questions can be answered in a so-called local model where large parts of the compactification space are decoupled and particle physics essentially takes place in eight space-time dimensions. I will briefly introduce the mathematical description of the theory and then focus on local model building with explicit examples in the framework of SU(5) GUTs.

Ongoing theoretical and experimental developments in the area of protein evolution emphasise the importance of functionally promiscuous, or multi-functional, ancestors and the possibility of proteins that can assume more than one stable structural state. A recently popularised model of protein evolution is the `Escape from an Adaptive Conflict' (EAC), a type of subfunctionalisation. This conflict of a single gene to adapt to more than one selection pressure can lead to a certain degree of multi-functionality prior to a gene duplication event. After duplication, the resulting paralogs are thought to specialise on different sub-functions of their ancestor. Here, using a simple biophysical protein model and taking structural stability as an indicator for biological activity, we demonstrate under which conditions adaptive conflicts on a dual fitness landscape can be resolved by multifunctionality, gene duplication and subfunctionalisation.

We investigate the properties of the Heisenberg S=1 chain with bilinear and biquadratic interactions in a magnetic field using the Density Matrix Renormalization Group, Bethe ansatz and field theoretical considerations. In a large region of the parameter space, we identify a magnetized ferroquadrupolar Luttinger liquid consisting of a quasi-condensate of bound magnon pairs. This liquid undergoes a continuous pair unbinding transition to a more conventional Luttinger liquid region obtained by polarizing the system above the Haldane gap region. This pair unbinding transition is shown to be in the Ising universality class on top of a Luttinger liquid, leading to an effective central charge 3/2. We also revisit the nature of the partially polarized Luttinger liquid around and above the Uimin-Lai-Sutherland point. Our results confirm that this is a two-component liquid and rule out the formation of a single-component vector chiral phase.

A two-day workshop bringing together theoretical and bioinformatic approaches to problems of viral evolution. Speakers include Thomas Lengauer, Alejandro Pironti, Bastian Beggel, Stephan Schiffels, Kasia Bozek, Glenn Lawyer, Michael Lässig, Alexander Thielen, Ville Mustonen, Sven-Eric Schelhorn and Jasmina Bogojeska

Heavy fermions exhibit strong correlations and variety of ordered states stemming from strong hybridization and Kondo screening. Nature of some of these ordered states like Hidden Order in URu2Si2 is not settled. Recent advances in local scanning spectroscopies shed a new light on these correlated states and reveal significant suprises in the electronics of Kondo lattices. I focus on the hybridization wave as the order parameter in URu2Si2 and possibly other materials with similar band structures. I will introduce a simple phenomenological model for the 'hidden order' transition in the heavy Fermion material URu2Si2. The 'hidden order' is identified as an incommensurate, momentum-carrying hybridization between the light hole band and the heavy electron band. The model is qualitatively consistent with numerous experimental results obtained from e.g. neutron scattering and scanning tunneling microscopy. I also will present predictions which are amenable to current experiments.

The vortex in any O(2) order parameter on graphene's honeycomb lattice is known to produce zero-energy states. I will discus how these rather special states in the middle of the spectrum induce local order in the core of the vortex. The local order parameters in general close the algebra Cl(3)x U(1), where Cl(3) is the three dimensional Clifford algebra, with the members of this algebra depending on the type of order supporting the vortex. The superconducting order will be discussed in some detail, and the curious duality between the topological insulation and the superconductivity in graphene will be pointed out.