# Internal Workshops

### Friday 29 May 2020, 2pm

Kiryl Piasotski: Rational quantization of the boundary charge and universal properties of its fluctuations: a low-energy description

In my talk, I'm going to discuss the universal properties of a rather intriguing physical quantity, the charge hosted by a boundary of a generic one-dimensional insulator with a single orbital per site and nearest-neighbor hopping. First I'm going to show that in the vicinity of an arbitrary energetic band gap, all 1D insulators of the above-mentioned type admit for a universal description in terms of a one-dimensional Dirac-like Hamiltonian with a composite gap mechanism and, as we are going to see, one may relate the effective gap parameters of the low energy Hamiltonian to the microscopic ones in a deterministic fashion by means of Brillouin-Wigner perturbation theory. With the help of the low-energy model, one may then elegantly analyze the properties of the boundary charge, in particular, I will demonstrate that by appropriately tuning the model parameters one can achieve arbitrary rational charges (electron charge) times p/q with p and q being natural number-valued. As we are also going to see, the previous studies of fractional charges hosted by the interfaces generated by the coupling of fermions to the bose fields with broken discrete symmetry (solitons), may be easily incorporated into our picture and are nothing but the sum of the boundary charges right and left to the interface. Further, we are going to analyze the effects of strong short-range Coulomb interactions on the boundary and interface charge quantization by means of bosonization and perturbative renormalization group analysis. Finally, I'm going to discuss our recent developments related to the sharpness of the boundary charge as a quantum observable. As I'm going to show, to the leading order in 1/(the scale on which the charge probe smoothly looses) the fluctuations of the boundary charge show a critical 1/(energy gap) behaviour signifying the topological quantum phase transition at a gap-closing point. In the presence of the electron-electron scattering, we are going to see that one obtains a critical power-law 1/(gap to the power x), where x is an interaction strength dependent constant.

--------------------------------------------------------------------------------------------------------------------

### Friday 15 May 2020, 2pm

Jonas Hauck.: Real-space Functional Renormalization Group - Application to Penrose Crystals

Quasi crystals are systems which have a long range order but no translational symmetry. These two properties make those systems on the one hand difficult to investigate numerically, as most of the standard methods rely on the translational invariance of the systems to get an efficient treatment, but on the other hand very interesting, as there exist states between Anderson localization and free states as well as fractional ordered states. One of the most well known examples is the five-fold rotationally symmetric Penrose lattice. We here expand the real-space FRG, recently developed for 1D Systems, to 2D by linking it to the Truncated Unity FRG approach. This new method is then applied to three different variations of the Penrose Model. The method we employ is capable of incorporating competing order instabilities on an equal footage and therefore examining the competition between Charge-Density wave and Spin-Density wave phases upon applying long range interactions. The analysis suggest that those phases can coexist in spatially separated parts of the lattice.

--------------------------------------------------------------------------------------------------------------------

### Thursday 09 April, 2020, 2pm

Lukas Weber: Stochastic Series Expansion Quantum Monte Carlo

It is important to know how to flip your spins in a way that obeys detailed balance to efficiently simulate quantum magnets. A powerful, unbiased approach is the so-called Stochastic Series Expansion which we will discuss along with some more general aspects of Quantum Monte Carlo such as the infamous “sign problem”.

--------------------------------------------------------------------------------------------------------------------

### Friday 14 February 2020, 2pm

Lennart Klebl: Ordering in Twisted Graphene Bilayers

Twisted bilayer graphene has become a very vivid field of research both theoretically and experimentally since superconductivity was found in an experiment in 2018. The band structure is highly controllable by experimental parameters which in theory makes this material an ideal testbed for the study of electronic correlations. In certain 'magic' twist angles, the bands closest to charge neutrality are very flat and should thus enhance interaction effects. The flat bands lead to several distinct magnetic ordering tendencies found in both an RPA study and FRG. The nature of superconductivity remains an open question we might be able to address in future works.

### Friday 07 February 2020, 2pm

Jannis Ehrlich: Catch the error

Most of the time in theoretical physics you develop your code. While writing code, however, there will occur errors which may give rise to unphysical behavior. To avoid this, your code has to be thoroughly tested if the results are correct. Even more important, the results should not change when the code is refactored to increase speed or new

features are implemented. As errors occur rather on the low level of simple routines, it is useful to test them separately, without comparing files manually. You can delegate this work to testing frameworks - which can be run by a simple command and tell you about errors.

In my talk I will present some ideas about tests, test- or behavioral driven development, and show the usage of a testing framework based on Catch2 (C++). If you want to find a testing framework for your favored programming language you may have a look at the corresponding Wikipedia page https://en.wikipedia.org/wiki/List_of_unit_testing_frameworks

### Friday 24 January 2020, 3pm

Jacob Beyer: Using the ubiquitous and very useful version control system git

Would you like to talk about Git, our lord and saviour?

Yes? Well you are in luck. This talk will exclusively focus on version control using git. Some basic instructions and commands will be presented in the start, but some - in my experience - useful tips and tricks will form the core of the talk.

### Wednesday 03 July 2019, 1pm

Lukas Weber: Nonordinary Edge Criticality of 2D Quantum Critical Magnets

Konstantin Nestmann: Time-local generator as a fixed point of the memory kernel

I will give a talk about the general relation between the generator of time-local master equations, that are often used in quantum information, and the memory kernel that features in time-nonlocal master equations (Nakajima-Zwanzig). While it is possible to obtain very good approximate evolutions using memory-kernel formalism, there are unique insights about the evolution that can only be obtained from the more cumbersome time-local generator. I will show how to non-perturbatively obtain a "best possible" Lindblad approximation by "sampling" the memory kernel over a few finite-frequency contributions, which is perhaps surprising since Markovian dynamics is often associated with "coarse graining" and retaining "low frequency" components.

Jan Diekmann: Leading logarithmic approximation to the X-ray-edge model from a single-loop functional renormalization group approach

Numerous models for low-dimensional correlated systems (e.g. X-ray-edge model, Kondo model, one-dimensional metals) have logarithmic divergencies in different diagrammatic channels of the four-point function. Then the sum of all parquet diagrams with bare lines contains all leading logarithmic divergencies, cf. Ref. [1]. The functional renormalization group (FRG) in a single-loop truncation does not reproduce all parts of all parquet diagrams; recently, it was shown how a multi-loop extension of the FRG can achieve this [2]. However, in this talk we present a (purely Fermionic) single-loop FRG in zero-temperature formalism that reproduces identically the leading logarithmic result of Ref. [1] for the four-point function of the X-ray-edge model. Our FRG approximation accounts for the leading contribution of every parquet diagram and is closely related to the parquet approach of Ref. [1]. Indeed, certain technical steps in Ref. [1] can be understood as introducing an RG cut-off and solving the very single-loop FRG flow equation. Since the zero-temperature formalism is not widely used today, we also discuss how our single-loop FRG can be setup in Matsubara formalism to tackle the leading logarithms. We expect that our findings can be transferred to other models with logarithmic divergencies.

[1] B. Roulet, J. Gavoret, P. Nozières, Phys. Rev. 178, 1072 (1969)

[2] F. B. Kugler, J. von Delft, Phys. Rev. Lett. 120, 057403 (2018)

### Wednesday 12 June 2019, 1pm

Yen-Ting Lin: Functional renormalization group study on the interacting Su-Schrieffer-Heeger model

Feng Xiong: Electronic correlation effects on Weyl semimetals

We introduce the basic point about Weyl semimetals(Ws) without interaction. eg: Topological charge of a Weyl node and Fermi arc. Among Ws, there's one kind-Time reversal symmetry is broken due to the spin orbital coupling in magnetic materials like pyrochlore irritates, which also assumed to be strongly correlated for 5d electrons. Here, we're interested in Type I and II Ws, which have topological inequivalent Fermi surface. We've calculated its magnetic response using RPA method. In the near further, we'll use 1PI fRG to calculate its leading instabilities on a no bias way. We want to see whether superconductivity can exist in Ws and how the phase diagram is like possibly with hole doped or electron doped Weyl semimetals.

Niclas Müller: Transport through the tip of a 1-D Floquet topological insulator via topological- and non-topological edge states

Floquet topological insulators are driven systems with a complex quasi-energy spectrum featuring many gaps due to anticrossings of the Floquet bands. These gaps can potentially host topological edge states (TES) which lie at the gap-center and are exponentially localized near the boundary of the system. We study a 1-D toy model of such a system and, in addition to the TES, find that it features a continuum of what we call non-topological edge states (NTES) in special regions of the phase space. These are still exponentially localized but are shifted in energy away from the gap-center and contain some bulk-state contribution, i.e. some finite weight in the bulk of the wire. We discuss how both TES and NTES contribute to electronic transport through the tip of such a system and discuss the origin of the NTES.

### Monday 22 January 2018, 10am

Michael Voigt (University of Postdam): Stochastic dynamics of mixing-induced patterns in reaction-advection-diffusion systems

I will introduce the theory describing how a convectively unstable active field in an open flow is transformed into absolutely unstable by local mixing. I review basic properties of the KPPF-equation and line out how noise can be included. By adapting the well-known Gillespie-Algorithm we create a stochastic simulation algorithm for discrete particles, which enables us to study the effect of intrinsic (demographic) noise on the stability of the population. I conclude by relating our results to the KPPF-equation with a cutoff.

### Wednesday 05 July 2017, 1pm

Iris Kleinjohann (TU Dortmund): Quantum mechanical simulation of a periodically pulsed semiconductor quantum dot

Andreas Fischer (TU Dortmund): Semiclassical simulation of two weakly coupled quantum dots

### Monday 08 May 2017, 1pm

Heike Eisenlohr (University of Göttingen): Reduced density-matrix functionals from Green's functions

Maximilian Buser (TU Berlin): A hierarchy of equations of motion approach to full counting statistics in the non-Markovian and strong-coupling regime

### Wednesday 18 January 2017, 2pm

Julian Mußhoff: Susceptibility calculations using a multi-orbital general CT- QMC solver for DMFT

Susceptibilities describe the response of a system to an external perturbation, and are therefore essential to compare theoretical calculations with experiments. We use a general continuous-time quantum Monte Carlo solver for dynamical mean-field theory to calculate generalized local susceptibilities. The method is applicable to strongly correlated materials with multi-orbital Hamiltonians. We calculate and store the susceptibilities in a compact form by using a Legendre polynomial representation. Furthermore we extend the local susceptibilities to lattice susceptibilities using the Bethe-Salpeter equation. In the talk we show magnetic susceptibility results for a representative system, VOMoO_{4}.

Andishe Khedri: The spinless Anderson-Holstein impurity model

We consider a spinless resonant level in a wide conduction band which is coupled to a phonon mode. This coupling induces an effective retarded and attractive electron-electron interaction which leads to a suppression of the tunneling rate from the local level to the conduction sea. Conventional perturbation theory in the coupling strength does neither capture the underlying polaron physics nor the power-law like renormalization as known from the purely fermionic interacting resonant level model (anti-adiabatic limit). We use the functional renormalization group to study the renormalization of the tunneling rate for arbitrary bare rate and phonon frequency in the limit of small to intermediate electron-phonon coupling.

### Wednesday 16 November 2016, 3pm

Qian Zhang: Modeling correlated systems: from atoms to materials

The study of strongly correlated materials requires careful treatment of electron-electron interactions. Our starting point is density functional calculations for atoms and ions to obtain realistic basis functions. For individual atoms and ions, we collect the Slater-Condon and spin-orbit parameters from the resulting self-consistent radial wave functions and potentials. We analyze the trends of the parameters systematically across the periodic table, which allows us to calculate atomic open-shell spectra in LS-, intermediate-, and jj-coupling schemes. Bringing localized atomic orbitals to lattices requires multi-center integral techniques. We will discuss orbital overlap, hopping elements, and long-range Coulomb interactions, which are the essential ingredients for lattice problems. A particularly important aspect on the lattice is the orthogonalization of localized orbitals. We will discuss this for a simple two-site model, where we show the importance of the deformation of localized orbitals when the lattice constant is changed.

### Wednesday 29 June 2016, 2pm

Niklas Dittmann: A Superconducting Single-Electron Turnstile for Quantized Spin Currents -- Emitting quasiparticles one by one --

In recent years, single-charge turnstiles based on superconductor/normal-metal (S/N) nanostructures have been successfully implemented [1,2]. These devices allow the manipulation of single electrons at high frequencies. In contrast, the implementation of single-spin sources in solid-state devices is only weakly explored [3]. In this seminar, I will present a recent work [4] where we propose a clocked single-spin source for spintronic applications. The device is based on a superconducting island covered by a ferromagnetic-insulator layer through which it is coupled to superconducting contacts. The ferromagnetic insulator polarizes the island and provides spin-selective tunnel barriers. The working principle of the proposed single-spin source combines in a novel way aspects of single-particle transport in charge turnstiles with special spin properties of the superconductor/ferromagnetic-insulator combination. We show that a bias and a time-periodic gate voltage results in the clocked transport of single quasiparticles with defined spin through the nanostructure. Realistic material combinations and experimental requirements allow for a clocked spin current in the MHz regime.

[1] Pekola et al, Review of Modern Physics 85 (2013)

[2] Pekola et al, Nature Physics 4 (2008)

[3] Costache & Valenzuela, Science 330 (2010)

[4] Dittmann et al, arXiv:1602:09068

### Wednesday 16 December 2015, 3pm

Viktor Reimer: How to guarantee complete positivity in approximations to open-system dynamics: Kraus theorem on the Keldysh contour

A fundamental problem in the theory of open quantum systems is how to guarantee that the time-evolution within a nontrivial approximation scheme results in a valid physical mixed state. Simple approximations may result in completely positive evolution, e.g., through Lindblad equations, but neglect microscopic processes that are relevant beyond weak coupling. On the other hand, advanced approximations such as higher-order real-time perturbation theory or renormalization group flows are not known to strictly guarantee positivity. Spurred by a key insight of van Wonderen and Suttorp [EPL 102, 60001 (2013)], we present a framework based on the Kraus representation of the exact reduced system dynamics, in which positivity-, hermicity- and trace-preserving approximations can be systematically formulated. We relate the Kraus operators to the standard real-time Keldysh diagrammatic expansion and identify precisely which partial resummations of diagrams strictly enforce complete positivity of the density operator. This diagrammatic formulation of Kraus theorem essentially relies only on the initial factorization of the system-bath state and noninteracting, grandcanonical reservoirs. It thus applies also to time-dependent systems with both bosonic and fermionic excitations as well as arbitrary multilinear system-bath couplings.

### Wednesday 28 October 2015, 2pm

Mohsin Iqbal: Semionic resonating valence bond states

Johannes Mitscherling: Non-linear conductance in mesoscopic weakly disordered wires - Interaction and magnetic field asymmetry

### Wednesday 24 June 2015, 2pm

Jonas Becker: Finite Temperature Spin Dynamics of the Haldane Chain from Monte Carlo

Khaldoon Ghanem: Analytic Continuation and Stochastic Sampling Approach - Efficient Sampling, Functional Reformulation and Parameter Selection

### Wednesday 29 April 2015, 2pm

Jan Rentrop: First applications of two-particle irreducible functional renormalization group

Functional renormalization group (RG) is an established method for the investigation of non-relativistic correlated quantum many-body problems in low dimensions. Commonly, the functional RG flow equations are derived in generating functional formalism and formulated for one-particle irreducible (1PI) vertex functions. However, generating functionals have long been written in a 2PI form as well. Based on these, proposals for 2PI functional RG schemes have been made. These mainly differ from one another with respect to the introduction of the flow parameter. It is remarkable that hardly any calculations for particular systems employing these proposed schemes have been published.

In this talk, I will introduce 2PI formalism coming from a 1PI perspective and explain how a functional RG flow parameter can be introduced in this setting. Depending on the details of how the flow parameter is introduced and how the set of flow equations is truncated, a number of 2PI functional RG methods is obtained. These were applied to the toy model of the quantum anharmonic oscillator as well as to the single impurity Anderson model in equilibrium. I will present the results obtained for these two models, focussing on placing the 2PI schemes into a broader context (how they relate to other methods) and evaluating their performance in comparison to other schemes, in particular 1PI ones.

Julian Lichtenstein: Towards high-performance renormalization group calculations for interacting fermions

### Wednesday 10 December 2014, 2pm

Manuel Schmidt: Introduction on the history of edge magnetism in graphene

Cornelie Koop: Derivation of effective low-energy theories for edge magnetism in graphene

Michael Golor: Quantum Monte-Carlo meets edge magnetism

### Wednesday 29 October 2014, 3.15pm

Thilo Plücker: Open System Geometric Phases in Adiabatic Quantum Pumping

Adiabatic pumping of charge can be realized by the slow and cyclic variation of at least two parameters for a single level quantum dot coupled to metallic electron leads. Here, we are interested in charge-pumping by modulating in time, e.g., the bias voltage and the gate voltage applied to the quantum dot. In this particular case, it can be shown that it is the strong onsite-interaction of the electrons, which entirely generates the pumping current, coexisting with the non- linear transport current induced by the bias voltage [1, 2].

In this talk I will identify the geometric nature of this interaction-induced pumped charge by using an extension of Berry’s approach to geometric phases in open quantum systems[3]. I will relate this approach to previously used methods, namely to counting statistics for adiabatic pumping [4] and to a real-time diagrammatic approach [5]. This will show, that due to the dependence of the current on the environment the pumped charge is given by an open system geometric phase.

[1] F. Reckermann, J. Splettstoesser, and M. R. Wegewijs, “Interaction-induced adiabatic nonlinear transport,”

Phys. Rev. Lett., vol. 104, p. 226803, 2010.

[2] H. L. Calvo, L. Classen, J. Splettstoesser, and M. R. Wegewijs, “Interaction-induced charge and spin pumping through a quantum dot at finite bias,” Phys. Rev. B, vol. 86, p. 245308, 2012.

[3] M. S. Sarandy and D. A. Lidar, “Abelian and non-abelian geometric phases in adiabatic open quantum systems,”

Phys. Rev. A, vol. 73, p. 062101, 2006.

[4] R. Yoshii and H. Hayakawa, “arxiv:1312.3772,”

[5] R. B. Saptsov and M. R. Wegewijs, “Fermionic superoperators for zero-temperature nonlinear transport: Real-time perturbation theory and renor- malization group for anderson quantum dots,” Phys. Rev. B, vol. 86, p. 235432, 2012.

Katharina Eissing: Functional renormalization group in floquet space and its application to periodically driven quantum dots

The functional renormalization group (RG) was recently extended to study interacting, low-dimensional systems out of equilibrium. This includes correlated quantum dot setups with explicitly time-dependent Hamiltonians as e.g. realized in quantum quenches or in the presence of time-dependent bias voltages[1,2] . However, following this route periodic pumping processes, which are of particular interest in e.g. nanoelectronics and quantum information science, can only be described in an inefficient way. Taking advantage of the periodicity, we combine the Floquet theorem with the functional RG. It allows us to transform the double-time self-energy and Green functions in the Floquet basis[3] and the functional RG treatment resembles the stationary formalism. This makes it feasible to study transport in periodically driven systems.

In my talk, I will introduce the basic ideas of the functional RG, explain how we can set it up in the Floquet space and present first results on transport through a quantum dot described by the interacting resonant level model.

[1] Phys. Rev. B 85, 085113 (2012),

[2] Phys. Rev. B 85, 245101 (2012)

[3] J.Phys.: Condens. Matter 20 085224