C. Gogolin

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Publications

My publications can be found on the arXiv and some statistics is available via google scholar in case you care about such numbers.

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Quantum many-body systems out of equilibrium

J. Eisert, M. Friesdorf, and C. Gogolin

Closed quantum many-body systems out of equilibrium pose several long-standing problems in physics. Recent years have seen a tremendous progress in approaching these questions, not least due to experiments with cold atoms and trapped ions in instances of quantum simulations. This article provides an overview on the progress in understanding dynamical equilibration and thermalisation of closed quantum many-body systems out of equilibrium due to quenches, ramps and periodic driving. It also addresses topics such as the eigenstate thermalisation hypothesis, typicality, transport, many-body localisation, universality near phase transitions, and prospects for quantum simulations.



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Reliable quantum certification for photonic quantum technologies

L. Aolita, C. Gogolin, M. Kliesch, and J. Eisert

Photonic devices involving many optical modes promise major advances in quantum technologies, with applications ranging from quantum metrology over quantum computing to quantum simulations. A significant current roadblock for the development of such devices, however, is the lack of practical reliable certification tools. Here, we present one such tool. We start by carefully defining different notions of quantum-state certification tests. Then, we introduce an experimentally friendly, yet mathematically rigorous, certification test for experimental preparations of arbitrary m-mode pure Gaussian states as well as a class of pure non-Gaussian states common in linear-optical experiments, including those given by a Gaussian unitary acting on Fock basis states with n bosons. The protocol is efficient for all Gaussian states and all mentioned non-Gaussian states with constant n. We follow the formal mindset of an untrusted prover, who prepares the state, and a skeptic certifier, equipped only with classical computing and single-mode measurement capabilities. No assumptions are made on the type of quantum noise or experimental capabilities of the prover. We build upon an extremality property that leads to a practical fidelity lower bound, interesting in its own right. Experimentally, our technique relies on single-mode homodyne detection. With this method, the efficient and reliable certification of large-scale photonic networks, with a constant number of input photons, as those used for photonic quantum simulations, boson samplers, and quantum metrology is within reach.



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Equilibration and thermalization in quantum systems

Christian Gogolin

This thesis fathoms out the capabilities of the theory of quantum mechanics to explain thermodynamic behavior. It covers in particular equilibration and thermalization in closed quantum systems, typicality, time scales for equilibration, quantum integrability and its connection to thermalization, decoherence, and a maximum entropy principle. Together, the presented results form the body of the theory of pure state quantum statistical mechanics. With almost 300 references, ranging from the groundbreaking works of the early 20th century to the most recent discoveries (up to 2013), this work arguably constitutes the most comprehensive review of the literature on equilibration and thermalization in closed quantum systems. All results are presented in a unified notation and many are slightly strengthened or generalized.



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Pushing the limits of the eigenstate thermalization hypothesis towards mesoscopic quantum systems

Robin Steinigeweg, Abdelah Khodja, Hendrik Niemeyer, Christian Gogolin, and Jochen Gemmer

In the ongoing discussion on thermalization in closed quantum many-body systems, the eigenstate thermalization hypothesis (ETH) has recently been proposed as a universal concept which attracted considerable attention. So far this concept is, as the name states, hypothetical. The majority of attempts to overcome sthis hypothetical character is based on exact diagonalization which implies for, e.g., spin systems a limitation to roughly 15 spins. In this Letter we present an approach which pushes this limit up to system sizes of roughly 35 spins, thereby going significantly beyond what is possible with exact diagonalization. A concrete application to a Heisenberg spin-ladder which yields conclusive results is demonstrated.



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Locality of temperature

M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert

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 that upper bounds physical critical temperatures such as the Curie temperature. As a corollary, we obtain that above 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 efficiently in the error and the system size.



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Boson-Sampling in the light of sample complexity

C. Gogolin, M. Kliesch, L. Aolita, and J. Eisert

Boson-Sampling is a classically computationally hard problem that can — in principle — be efficiently solved with linear quantum optical networks. Very recently, a rush of experimental activity has ignited with the aim of developing such devices as feasible instances of quantum simulators. Even approximate Boson-Sampling is believed to be hard with high probability if the unitary describing the optical network is drawn from the Haar measure. In this work we show that in this setup, with probability exponentially close to one in the number of bosons, no symmetric algorithm can distinguish the Boson-Sampling distribution from the uniform one from fewer than exponentially many samples. This means that the two distributions are operationally indistinguishable without detailed a priori knowledge. We carefully discuss the prospects of efficiently using knowledge about the implemented unitary for devising non-symmetric algorithms that could potentially improve upon this. We conclude that due to the very fact that Boson-Sampling is believed to be hard, efficient classical certification of Boson-Sampling devices seems to be out of reach.



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Lieb-Robinson bounds and the simulation of time evolution of local observables in lattice systems

Martin Kliesch, Christian Gogolin, and Jens Eisert

This is an introductory text reviewing Lieb-Robinson bounds for open and closed quantum many-body systems. We introduce the Heisenberg picture for time-dependent local Liouvillians and state a Lieb-Robinson bound. Finally we discuss a number of important consequences in quantum many-body theory.
To appear as a chapter in the book "Many-Electron Approaches in Physics, Chemistry and Mathematics" to be published in "Mathematical Physics Studies" by Springer.



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Quantum measurement occurrence is undecidable

J. Eisert, M. P. Mueller, C. Gogolin

A famous result by Alan Turing dating back to 1936 is that a general algorithm solving the halting problem on a Turing machine for all possible inputs and programs cannot exist — the halting problem is undecidable. Formally, an undecidable problem is a decision problem for which one cannot construct a single algorithm that will always provide a correct answer in finite time. In this work, we show that surprisingly, very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability appears as a genuine quantum property. The problem we consider is to determine whether sequentially used identical Stern-Gerlach-type measurement devices, giving rise to a tree of possible outcomes, have outcomes that never occur. Finally, we point out implications for measurement-based quantum computing and studies of quantum many-body models and suggest that a plethora of problems may indeed be undecidable.



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A dissipative quantum Church-Turing theorem

M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert

We show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the simulation time and the size of the system. An immediate consequence is that dissipative quantum computing is no more powerful than the unitary circuit model. Our result can be seen as a dissipative Church-Turing theorem, since it implies that under natural assumptions, such as weak coupling to an environment, the dynamics of an open quantum system can be simulated efficiently on a quantum computer. Formally, we introduce a Trotter decomposition for Liouvillian dynamics and give explicit error bounds. This constitutes a practical tool for numerical simulations, e.g., using matrix-product operators. We also demonstrate that most quantum states cannot be prepared efficiently.



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Thermalization in nature and on a quantum computer

Arnau Riera, Christian Gogolin, and Jens Eisert

In this work, we put several questions related to the emergence of Gibbs states in quantum physics to rest. We show how Gibbs or thermal states appear dynamically in closed quantum many-body systems, by completing the program of dynamical typicality and by introducing a novel general perturbation theorem that is robust under the thermodynamic limit, rigorously capturing the intuition of a meaningful weak coupling limit. We discuss the physics of thermal states occurring and identify the precise conditions under which this happens. Based on these results, we also present a fully general quantum algorithm for preparing Gibbs states on a quantum computer with a certified runtime, including full error estimates, complementing quantum Metropolis algorithms which are expected to be efficient but have no known runtime estimate.



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Non-equilibrium Dynamics, Thermalization and Entropy Production

Haye Hinrichsen, Christian Gogolin, and Peter Janotta

This paper addresses fundamental aspects of statistical mechanics such as the motivation of a classical state space with spontaneous transitions, the meaning of non-equilibrium in the context of thermalization, and the justification of these concepts from the quantum-mechanical point of view. After an introductory part we focus on the problem of entropy production in non-equilibrium systems. In particular, the generally accepted formula for entropy production in the environment is analyzed from a critical perspective. It is shown that this formula is only valid in the limit of separated time scales of the system's and the environmental degrees of freedom. Finally, we present an alternative simple proof of the fluctuation theorem.



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Limits on nonlocal correlations from the structure of the local state space

Peter Janotta, Christian Gogolin, Jonathan Barrett, and Nicolas Brunner

The outcomes of measurements on entangled quantum systems can be nonlocally correlated. However, while it is easy to write down toy theories allowing arbitrary nonlocal correlations, those allowed in quantum mechanics are limited. Quantum correlations cannot, for example, violate a principle known as macroscopic locality, which implies that they cannot violate Tsirelson’s bound. This paper shows that there is a connection between the strength of nonlocal correlations in a physical theory and the structure of the state spaces of individual systems. This is illustrated by a family of models in which local state spaces are regular polygons, where a natural analogue of a maximally entangled state of two systems exists. We characterize the nonlocal correlations obtainable from such states. The family allows us to study the transition between classical, quantum and super-quantum correlations by varying only the local state space. We show that the strength of nonlocal correlations — in particular whether the maximally entangled state violates Tsirelson’s bound or not — depends crucially on a simple geometric property of the local state space, known as strong self-duality. This result is seen to be a special case of a general theorem, which states that a broad class of entangled states in probabilistic theories — including, by extension, all bipartite classical and quantum states — cannot violate macroscopic locality. Finally, our results show that models exist that are locally almost indistinguishable from quantum mechanics, but can nevertheless generate maximally nonlocal correlations.



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Absence of thermalization in non-integrable systems

Christian Gogolin, Markus P. Müller, and Jens Eisert

We present rigorous results establishing a link between unitary relaxation dynamics after a quench in closed many-body systems in non-equilibrium and the entanglement in the energy eigenbasis. We find that even if reduced states equilibrate, and appear perfectly relaxed, they can still have memory on the initial conditions even in models that are far from integrable, thereby giving rise to "equilibration without thermalization". We show that in such situations the equilibrium states are however still described by a Jaynes maximum entropy or generalized Gibbs ensemble and, moreover, that this is always the case if equilibration happens, regardless of whether a model is integrable or not. In addition, we discuss individual aspects of thermalization processes separately, comment on the role of Anderson localization, and collect and compare different notions of integrability.
In May 2011 I received the Leibniz publication award for young academics for this publication.



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Pure State Quantum Statistical Mechanics

Christian Gogolin

The capabilities of a new approach towards the foundations of Statistical Mechanics are explored. The approach is genuine quantum in the sense that statistical behavior is a consequence of objective quantum uncertainties due to entanglement and uncertainty relations. No additional randomness is added by hand and no assumptions about a priori probabilities are made, instead measure concentration results are used to justify the methods of Statistical Physics. The approach explains the applicability of the microcanonical and canonical ensemble and the tendency to equilibrate in a natural way.
This work contains a pedagogical review of the existing literature and some new results. The most important of which are: i) A measure theoretic justification for the microcanonical ensemble. ii) Bounds on the subsystem equilibration time. iii) A proof that a generic weak interaction causes decoherence in the energy eigenbasis. iv) A proof of a quantum H-Theorem. v) New estimates of the average effective dimension for initial product states and states from the mean energy ensemble. vi) A proof that time and ensemble averages of observables are typically close to each other. vii) A bound on the fluctuations of the purity of a system coupled to a bath.



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Einselection without pointer states

Christian Gogolin

We show that the existence of a basis of pointer states is not necessary for environment-induced super selection. This is achieved by using recent results on equilibration of small subsystems of large, closed quantum systems evolving according to the von Neumann equation. Without making any special assumptions on the form of the interaction we prove that, for almost all initial states and almost all times, the off-diagonal elements of the density matrix of the subsystem in the eigenbasis of its local Hamiltonian must be small whenever the energies of the corresponding eigenstates differ by more than the interaction energy.



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Dynamic wetting with two competing adsorbates

Christian Gogolin, Christian Meltzer, Marvin Willers, and Haye Hinrichsen

We study the dynamic properties of a model for wetting with two competing adsorbates on a planar substrate. The two species of particles have identical properties and repel each other. Starting with a flat interface one observes the formation of homogeneous droplets of the respective type separated by nonwet regions where the interface remains pinned. The wet phase is characterized by slow coarsening of competing droplets. Moreover, in 2+1 dimensions an additional line of continuous phase transition emerges in the bound phase, which separates an unordered phase from an ordered one. The symmetry under interchange of the particle types is spontaneously broken in this region and finite systems exhibit two metastable states, each dominated by one of the species. The critical properties of this transition are analyzed by numeric simulations.



Prizes

In May 2011 I received the Leibniz publication award for young academics of the Leibniz-Kolleg Potsdam for my publication "Absence of thermalization in non-integrable systems" with Markus P. Müller and Jens Eisert.

Posters

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BosonSampling in the light of sample complexity: a review

This poster was presented during QIP 2014 in Barcelona showing the results of our paper on the sample complexity of certifying BosonSampling.

poster (pdf)


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Efficient simulation of dissipative quantum dynamics on a quantum computer

This poster was presented during QIP 2012 in Montral showing the results of our paper on the dissipative quantum Church-Turing thesis.

poster (pdf)


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Limits on non-local correlations from the structure of the local state space

This is a poster we presented during QIP 2011 in Singapore showing the results of the corresponding paper.

poster (pdf)


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Einselection without pointer states

This is a poster I made to present the results of the corresponding paper at the the QCCC workshop 2009 in Bad Tölz and during QIP 2010 in Zurich.

poster (pdf)


Recent talks

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Equilibration and thermalization in quantum systems

This is the talk I gave during my Ph.D. defense at Freie Univesität Berlin.

beamer slides (pdf)
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Boson-Sampling in the light of sample complexity: a review

In this talk attempt to give a 12 minute review of the scientific controversy concerning the meaning of recent BosonSampling experiments that was partially spanned by our paper on the sample complexity of certifying BosonSampling. The talk was given during the DPG Spring Meeting 2014 in Berlin (2014-03-20).

beamer slides (pdf)
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Under what conditions do quantum systems thermalize?

This talk demonstrates that closed, finite dimensional quantum systems in pure states that evolving unitarily according to the Schrödinger equation can exhibit thermodynamic behavior. More precisely, I will give conditions under which equilibration and thermalization can be ensured in such systems and show that a lack of entanglement in the energy eigenbasis can prevent thermalization. In addition I comment on the concept of quantum integrability and discuss a maximum entropy principle that follows from unitary quantum dynamics. I gave this overview talk during the workshop Equilibration and Thermalization in Quantum Systems at the Wallenberg Research Centre in Stellenbosch, South Africa (2013-04-15) and, in a similar form, during the COST conference in Berlin, Germany (2014-02-20).

beamer slides (pdf)
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Undecidability as a genuine quantum property

A famous result by Alan Turing dating back to 1936 is that a general algorithm solving the halting problem on a Turing machine for all possible inputs and programs cannot exist — the halting problem is undecidable. In this talk I will show that surprisingly simple problems in quantum mechanics can be undecidable in this sense, even if the corresponding classical problem is decidable. Undecidability appears here as a genuine quantum property. This gives a new twist to quantum complexity theory, which has up to now mostly been concerned with quantitative separations between quantum and classical physics in terms of hardness. I gave a blackboart talk on this subject at the Centre for Quantum Technologies at NUS Singapore (2011-11-09), a short talk with slides at the DPG March meeting in Göttigen (2012-03-01), and two more extensive ones at UNAM in Mexico City (2012-08-03) and at ICFO in Barcelona (2014-02-12).

beamer slides (pdf)
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Simulability of open quantum system dynamics

In this talk I present the results of the assotiated paper, in which we show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the simulation time and the size of the system and discuss the implications of this result. In particular our result implies that dissipative quantum computing is no more powerful than the unitary circuit model and it can be seen as a dissipative Church-Turing theorem, since it implies that under natural assumptions, such as weak coupling to an environment, the dynamics of an open quantum system can be simulated efficiently on a quantum computer. This talk was given at the QCCC workshop 2001 in Bernried (2011-10-08).

beamer slides (pdf)
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Absence of thermalization in non-integrable systems

In this talk I present two contributions to the recent debate concerning the connection between disorder, localization, integrability, and thermalization. In particular, I make some critical remarks concerning the notions of integrability currently used in the literature and show rigorous conditions for the absence and presence of thermalization. This talk was given during the Many-Body Quantum Dynamics in Closed Systems worshop held at UPC Barcelona (2011-09-08).

beamer slides (pdf)
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Thermalization in nature and on a quantum computer

Using the assumption that thermodynamic systems evolve towards Gibbs states, i.e. states with a well defined temperature, statistical mechanics and thermodynamics have been amazingly successful in explaining a wide range of physical phenomena. In stark contrast to this strong justification by corroboration of these theories, the question of whether and how the methods of statistical mechanics and thermodynamics can be justified microscopically was still wide open until recently. With new mathematical tools from quantum information theory becoming available, there has been a renewed effort to settle this old question. I will present and discuss a necessary and a sufficient condition for the emergence of Gibbs states from the unitary dynamics of quantum mechanics and show how these new insights into the process of equilibration and thermalization can be used to design a quantum algorithm that prepares thermal states on a quantum computer/simulator. I gave this talk at the University of Hannover (2011-05-22) and a slightly modified version later at Techinsche Universiät München (2011-06-07) and at the Quantum Information and Foundations of Thermodynamics conference at ETH Zürich (2011-08-09).

beamer slides (TUM) (pdf)
beamer slides (ETH) (pdf)
video of the talk at ETH Zürich (link)
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Under what conditions do quantum systems thermalize?

Quantum mechanics is generally regarded as a fundamental theory of physics. As such, it should be able to provide us with a microscopic explanation of all phenomena we observe in macroscopic systems, including irreversible processes such as thermalization. With new mathematical tools from quantum information theory becoming available, there has been a renewed effort to settle the old question of the emergence of classicality and irreversibly. The talk gives an overview over recent progress in the field. In this talk I present in a non technical way results concerning the connection between quantum (non-)integrability, thermalization anf the entanglement in the enegry eigenbais. I gave this talk at at the DPG March Meeting in Dresden.

beamer slides (pdf)
image showng a slide from the talk

Under what conditions do quantum systems thermalize?

Quantum mechanics is generally regarded as a fundamental theory of physics. As such, it should be able to provide us with a microscopic explanation of all phenomena we observe in macroscopic systems, including irreversible processes such as thermalization. With new mathematical tools from quantum information theory becoming available, there has been a renewed effort to settle the old question of the emergence of classicality and irreversibly. The talk gives an overview over recent progress in the field. In particular it is shown how equilibration and a maximum entropy Jaynes'-principle emerge as a natural consequence of unitary time evolution without any (Markov) approximation, and under which conditions the equilibrium state of a small subsystem is diagonal in the local energy eigenbasis as well as when, and when not, equilibration towards a thermal Boltzmann state can happen. I gave this talk at QIP 2011 in Singapore (2011-01-14).
Remark: At the very end of the talk I quickly mention results of a forthcoming article (which has been published on-line in the meantime, see above) concerning a quantum algorithm to prepare Gibbs states. I claim that it is efficient in the number of qubits that are needed to represent the system. Unfortunately this statement turned out to be incorrect later. I sincerely apologize for this misinformation. For details please see the paper.

beamer slides (pdf)
video of the talk at QIP 2011 (link)
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Pure state quantum statistical mechanics

Why do closed macroscopic systems equilibrate and thermalize? How can we justify the methods of thermodynamics and statistical mechanics from a microscopic theory? Which mechanisms lead to the emergence of classical, statistical behaviour and decoherence? In this talk, I show (i) how standard quantum mechanics without added randomness can explain the phenomenon of equilibration despite its unitary time development, and (ii) that an arbitrary weak interaction with an environment causes decoherence in the local energy eigenbasis. I gave this talk at the University of Bristol (2010-06-23) and in a slightly modified form at the University College London (2010-07-01) and at Boston University (2010-07-29).

beamer slides (pdf)
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Measure concentration in Hilbert space

This is a seminar talk I gave at the faculty of mathematics of the University of Würzburg. I show how measure concentration techniques can be used to justify the methods of Statistical Mechanics from Quantum Mechanics.

beamer slides (pdf)
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Decoherence under weak interaction

This is a seminar talk I gave in the group of Jens Eisert (2009-10-11) at the University of Potsdam, and later in the the group seminar at the University of Würzburg (2009-12-16) to present the results of my recent paper witht the same name.

beamer slides (pdf)
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A new foundation for Statistical Physics

In this talk I present a new approach towards the foundations of Statistical Mechanics, which is based on pure standard Quantum Mechanics. I show that this approach is capable of explaining phenomena like equilibration and thermalization without using enseble averages or the equal a prioriy probability postulate.

beamer slides (pdf)
image showng a slide from the talk

Dynamic wetting with compeeting adsorbates

In this talk a model for wetting with two competing adsorbates on an planar substrate is presented. The two species of particles have identical properties and repel each other. Starting with a flat interface one observes the formation of homogeneous droplets of the respective type separated by non-wet regions. The wet phase is characterized by slow coarsening of competing droplets. Moreover, in 2+1-dimensions an additional line of continuous phase transition emerges in the bound phase, which separates a unordered phase from an ordered one, where the symmetry under interchange of the particle types is spontaneously broken and in finite systems two metastable states, each dominated by one of the species, emerge. I gave this talk at the Atomistic Simulation Centre at Queen's University Belfast.

beamer slides (pdf)
image showng a slide from the talk

Generalized Probabilistic Theories

In this talk, which I gave together with a college of mine, Peter Janotta, we presented the framework of Generalized Probabilistic Theories. This framework is based on a minimal set of almost indispensable assumptions and provides a scheme for building "physically reasonable" probabilistic theories. Statistical Mechanics and Quantum Mechanics are incorporated as special cases in the framework. Having a look on these theories from an outside viewpoint yields new and surprising insights.

beamer slides (pdf)