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Quantum Parrondo Games in Low-Dimensional Hilbert Spaces

Andreas Mielke. Quantum Parrondo Games in Low-Dimensional Hilbert Spaces.arxiv.org: 2306.16845, (2023).

doi.org/10.48550/arXiv.2306.16845.

We consider quantum variants of Parrondo games on low-dimensional Hilbert spaces. The two games which form the Parrondo game are implemented as quantum walks on a small cycle of length M. The dimension of the Hilbert space is 2M. We investigate a random sequence of these two games which is realized by a quantum coin, so that the total Hilbert space dimension is 4M. We show that in the quantum Parrondo game constructed in this way a systematic win or loss occurs in the long time limit. Due to entaglement and self-interference on the cycle, the game yields a rather complex structure for the win or loss depending on the parameters.

Multi player Parrondo games with rigid coupling

Sandro Breuer, Andreas Mielke. Multi player Parrondo games with rigid coupling.Physica A 622, 128890 (2023).

10.48550/arXiv.2303.01361.

In a Parrondo game, a single player combines two losing strategies to a winning strategy. In this paper we investigate the question what happens, if two or more players play Parrondo games in a coordinated way. We introduce a strong coupling between the players such that the gain or loss of all players in one round is the same. We investigate two possible realizations of such a coupling. For both we show that the coupling increases the gain per player. The dependency of the gain on the various parameters of the games is determined. The coupling can not only lead to a larger gain, but it can also dominate the driving mechanism of the uncoupled games. Which driving mechanism dominates, depends on the type of coupling. Both couplings are set side by side and the main similarities and differences are emphasised.

Localised pair formation in bosonic flat-band Hubbard models

Jacob Fronk, Andreas Mielke. Localised pair formation in bosonic flat-band Hubbard models.Journal of Statistical Physics 185:14 , (2021).

Flat-band systems are ideal model systems to study strong correlations. In a large class of one or two dimensional bosonic systems with a lowest flat-band it has been shown that at a critical density the ground states are Wigner crystals. Under very special conditions it has been shown that pair formation occurs if one adds another particle to the system. The present paper extends this result to a much larger class of lattices and to a much broader region in the parameter space. Further, a lower bound for the energy gap between these pair states and the rest of the spectrum is established. The pair states are dominated by a subspace spanned by states containing a compactly localised pair. Overall, this strongly suggests localised pair formation in the ground states of the broad class of flat-band systems and rigorously proves it for some of the graphs in it, including the inhomogeneous chequerboard chain as well as two novel examples of regular two dimensional graphs. Physically, this means that the Wigner crystal remains intact if one adds a particle to it.

Flat-band ferromagnetism in twisted bilayer graphene

Rebecca Pons, Andreas Mielke, Tobias Stauber. Flat-band ferromagnetism in twisted bilayer graphene.Phys. Rev. B 102, 235101 (2020).

We discuss twisted bilayer graphene (TBG) based on a theorem of flat band ferromagnetism put forward by Mielke and Tasaki. According to this theorem, ferromagnetism occurs if the single particle density matrix of the flat band states is irreducible and we argue that this result can be applied to the quasi-flat bands of TBG that emerge around the charge-neutrality point for twist angles around the magic angle $\theta\sim1.05^\circ$. We show that the density matrix is irreducible in this case, thus predicting a ferromagnetic ground state for neutral TBG ($n=0$). We then show that the theorem can also be applied only to the flat conduction or valence bands, if the substrate induces a single-particle gap at charge neutrality. Also in this case, the corresponding density matrix turns out to be irreducible, leading to ferromagnetism at half filling ($n=\pm2$).

Pair formation of hard core bosons in flat band systems

Andreas Mielke. Pair formation of hard core bosons in flat band systems.J Stat Phys. 171, 679-695 (2018).

Hard core bosons in a large class of one or two dimensional flat band systems have an upper critical density, below which the ground states can be described completely. At the critical density, the ground states are Wigner crystals. If one adds a particle to the system at the critical density, the ground state and the low lying multi particle states of the system can be described as a Wigner crystal with an additional pair of particles. The energy band for the pair is separated from the rest of the multi-particle spectrum. The proofs use a Gerschgorin type of argument for block diagonally dominant matrices. In certain one-dimensional or tree-like structures one can show that the pair is localised, for example in the chequerboard chain. For this one-dimensional system with periodic boundary condition the energy band for the pair is flat, the pair is localised.

Hard-core bosons in flat band systems above the critical density

Moritz Drescher, Andreas Mielke. Hard-core bosons in flat band systems above the critical density.Eur. Phys. J. B 90, 217 (2017).

10.1140/epjb/e2017-80218-1.

We investigate the behaviour of hard-core bosons in one- and two-dimensional flat band systems, the chequerboard and the kagom\'e lattice and one-dimensional analogues thereof. The one dimensional systems have an exact local reflection symmetry which allows for exact results. We show that above the critical density an additional particle forms a pair with one of the other bosons and that the pair is localised. In the two-dimensional systems exact results are not available but variational results indicate a similar physical behaviour.

Interacting bosons in two-dimensional flat band systems

Petra Pudleiner, Andreas Mielke. Interacting bosons in two-dimensional flat band systems.Eur. Phys. J. B 88, 207 (2015).

10.1140/epjb/e2015-60371-3.

The Hubbard model of bosons on two dimensional lattices with a lowest flat band is discussed. In these systems there is a critical density, where the ground state is known exactly and can be represented as a charge density wave. Above this critical filling, depending on the lattice structure and the interaction strength, the additional particles are either delocalised and condensate in the ground state, or they form pairs. Pairs occur at strong interactions, e.g., on the chequerboard lattice. The general mechanism behind this phenomenon is discussed.

The Hubbard Model and its Properties

Andreas Mielke. The Hubbard Model and its Properties.Modeling and Simulation 5, 1-26 (2015).

Accurate effective Hamiltonians via unitary flow in Floquet space

Albert Verdeny, Andreas Mielke, Florian Mintert. Accurate effective Hamiltonians via unitary flow in Floquet space.Phys. Rev. Lett. 111, 175301 (2013).

We present a systematic construction of effective Hamiltonians of periodically driven quantum systems. Due to an equivalence between the time dependence of a Hamiltonian and an interaction in its Floquet operator, flow equations, that permit to decouple interacting quantum systems, allow us to identify time-independent Hamiltonians for driven systems. With this approach, we explain the experimentally observed deviation of expected suppression of tunneling in ultra-cold atoms.

Bose-Hubbard model on two-dimensional line graphs

Johannes Motruk, Andreas Mielke. Bose-Hubbard model on two-dimensional line graphs.J. Phys. A: Math. Gen 45, 225206 (2012).

We construct a basis for the many-particle ground states of the positive hopping Bose-Hubbard model on line graphs of finite 2-connected planar bipartite graphs at sufficiently low filling factors. The particles in these states are localized on non-intersecting vertex-disjoint cycles of the line graph which correspond to non-intersecting edge-disjoint cycles of the original graph. The construction works up to a critical filling factor at which the cycles are close-packed.

Properties of Hubbard models with degenerate localised single particle eigenstates

Andreas Mielke. Properties of Hubbard models with degenerate localised single particle eigenstates.Eur. Phys. J. B 85, (2012).

We consider the repulsive Hubbard model on a class of lattices or graphs for which there is a large degeneracy of the single particle ground states and where the projector onto the space of single particle ground states is highly reducible. This means that one can find a basis in the space of the single particle ground states such that the support of each single particle ground state belongs to some small cluster and these clusters do not overlap. We show how such lattices can be constructed in arbitrary dimensions. We construct all multi-particle ground states of these models for electron numbers not larger than the number of localised single particle eigenstates. We derive some of the ground state properties, esp. the residual entropy, i.e. the finite entropy density at zero temperature.

Naturwissenschaft und Innovation - Zehn Thesen zur Wissen(schaft)sgesellschaft

Norbert Arnold, Peter Bursch, Wolfgang Frosch, Hans-Joachim Fuchs, Martin Gröger, Klaus Hantelmann, Andreas Mielke, Klemens Störtkuhl, Rupert Wagner. Naturwissenschaft und Innovation - Zehn Thesen zur Wissen(schaft)sgesellschaft.http://www.kas.de/wf/de/33.32622/, (2012).

Aufstrebende Länder - wie etwa China - entwickeln sich zunehmend zu erfolgreichen 'Innovatoren'. Deutschland muss sich deshalb im globalen Wettbewerb neu verorten. Dies geht nur mit einer effektiven Innovationspolitik. Die Naturwissenschaften spielen im Innovationsprozess eine zentrale Role und müssen daher gestärkt werden. Das vorliegende Papier benennt die kritischen Punkte und schlägt Lösungsansätze vor. Bildung und die Förderung des wissenschaftlichen Nachwuchses stehen dabei im Mittelpunkt.

Perspektivwechel. Performancebetrachtungen im SAP Betrieb

Andreas Mielke. Perspektivwechel. Performancebetrachtungen im SAP Betrieb.s@pport 6, 24-25 (2011).

Metriken für Eigenentwicklungen in SAP ERP Systemen

Andreas Mielke. Metriken für Eigenentwicklungen in SAP ERP Systemen.Publication in Conference Proceedings: Informatik 2011, LNI Band 192, 143 (2011) , 1-15 (2011).

Für den Betrieb von großen SAP ERP-Systemen spielen Eigenentwicklungen eine wichtige Rolle. Sie tragen zwischen 15% und 35% der kompletten Betriebskosten der SAP-Systeme. In Upgrade-Projekten entfallen zwischen 60% und 80% der Kosten auf das Testen von Eigenentwicklungen. Von daher spielen Mengengerüste für Eigenentwicklungen eine wichtige Rolle in der Beurteilung von SAP-Kosten. Entsprechende Metriken werden hier dargestellt, basierend auf Auswertungen von Nutzungsdaten aus SAP-Systemen von jeweils mindestens einem Jahr. Wir liefern Metriken zur Nutzung von Eigenentwicklungen, zu den Kosten von Eigenentwicklungen und zu ihrer Performance.

Elements for Response Time Statistics in ERP Transaction Systems

Andreas Mielke. Elements for Response Time Statistics in ERP Transaction Systems.Performance Evaluation 63, 635-653 (2006).

The aim of this work is to provide some insight into the response-time statistics of enterprise resource planning systems. We propose a simple mean-field model for the response-time distribution in such systems. This model yields a log-normal distribution of response-times. We present data from performance measurements to support the result. The data show that the response-time distribution of a given transaction in a given system is generically a log-normal distribution or, in some situations, a sum of two or more log-normal distributions. Deviations of the log-normal form can often be traced back to performance problems in the system. Consequences for the interpretation of response-time data and for service level agreements are discussed.

Flow equations for Hamiltonians: Contrasting different approaches by using a numerically solvable model

Tobias Stauber, Andreas Mielke. Flow equations for Hamiltonians: Contrasting different approaches by using a numerically solvable model.J. Phys. A 36, 2707-2736 (2003).

To contrast different generators for flow equations for Hamiltonians and to discuss the dependence of physical quantities on unitarily equivalent, but effectively different initial Hamiltonians, a numerically solvable model is considered which is structurally similar to impurity models. By this we discuss the question of optimization for the first time. A general truncation scheme is established that produces good results for the Hamiltonian flow as well as for the operator flow. Nevertheless, it is also pointed out that a systematic and feasible scheme for the operator flow on the operator level is missing. For this, an explicit analysis of the operator flow is given for the first time. We observe that truncation of the series of the observable flow after the linear or bilinear terms does not yield satisfactory results for the entire parameter regime as - especially close to resonances - even high orders of the exact series expansion carry considerable weight.

Equilibrium Correlation Functions of the Spin-Boson Model with Sub-Ohmic Bath

Tobias Stauber, Andreas Mielke. Equilibrium Correlation Functions of the Spin-Boson Model with Sub-Ohmic Bath.Phys. Lett. A 64, 275-280 (2002).

The spin-boson model is studied by means of flow equations for Hamiltonians. Our truncation scheme includes all coupling terms which are linear in the bosonic operators. Starting with the canonical generator eta_c=[H_0,H] with H_0 resembling the non-interacting bosonic bath, the flow equations exhibit a universal attractor for the Hamiltonian flow. This allows to calculate equilibrium correlation functions for super-Ohmic, Ohmic and sub-Ohmic baths within a uniform framework including finite bias. Results for sub-Ohmic baths might be relevant for the assessment of dissipation due to 1/f-related noise, recently found in solid-state qubits.

Noise induced transport of two coupled particles

Stefan Klumpp, Andreas Mielke, Christian Wald. Noise induced transport of two coupled particles.Phys. Rev. E 63, 031914, 1-10 (2001).

We study the motion of two harmonically coupled particles in a sawtooth potential. The particles are subject to temporally correlated multiplicative noise. The stationary current is calculated in an expansion about the limit of rigid coupling. For two coupled particles a driving mechanism occurs which is different from the one occurring in the case of a single particle. In particular this mechanism does not need diffusion. Depending on the equilibrium distance of the particles, a current reversal occurs. Possible relevance as a model for motor proteins is discussed.

Effective rate equations for the over-damped motion in fluctuating potentials

Andreas Mielke. Effective rate equations for the over-damped motion in fluctuating potentials.Phys. Rev. E 64, 021106, 1-12 (2001).

We discuss physical and mathematical aspects of the over-damped motion of a Brownian particle in fluctuating potentials. It is shown that such a system can be described quantitatively by fluctuating rates if the potential fluctuations are slow compared to relaxation within the minima of the potential, and if the position of the minima does not fluctuate. Effective rates can be calculated; they describe the long-time dynamics of the system. Furthermore, we show the existence of a stationary solution of the Fokker-Planck equation that describes the motion within the fluctuating potential under some general conditions. We also show that a stationary solution of the rate equations with fluctuating rates exists.

Diagonalization of Dissipative Quantum Systems I: Exact solution of the Spin-Boson Model with an Ohmic bath at $\alpha =1/2$

Andreas Mielke. Diagonalization of Dissipative Quantum Systems I: Exact solution of the Spin-Boson Model with an Ohmic bath at $\alpha =1/2$.preprint , (2000).

This paper shows how flow equations can be used to diagonalize dissipative quantum systems. Applying a continuous unitary transformation to the spin-boson model, one obtains exact flow equations for the Hamiltonian and for an observable. They are solved exactly for the case of an Ohmic bath with a coupling $\alpha =1/2$. Using the explicite expression for the transformed observable one obtains dynamical correlation functions. This yields some new insight to the exactly solvable case $\alpha =1/2$. The main motivation of this work is to demonstrate, how the method of flow equations can be used to treat dissipative quantum systems in a new way. The approach can be used to construct controllable approximation schemes for other environments.

Noise induced stability in fluctuating, bistable potentials

Andreas Mielke. Noise induced stability in fluctuating, bistable potentials.Phys. Rev. Lett. 84, 818-821 (2000).

The over-damped motion of a Brownian particle in an asymmetric, bistable, fluctuating potential shows noise induced stability: For intermediate fluctuation rates the mean occupancy of minima with an energy above the absolute minimum is enhanced. The model works as a detector for potential fluctuations being not too fast and not too slow. This effect occurs due to the different time scales in the problem. We present a detailed analysis of this effect using the exact solution of the Fokker-Planck equation for a simple model. Further we show that for not too fast fluctuations the system can be well described by effective rate equations. The results of the rate equations agree quantitatively with the exact results.

Flow equations for the Hénon-Heiles Hamiltonian

Daniel Cremers, Andreas Mielke. Flow equations for the Hénon-Heiles Hamiltonian.Physica D 126, 123-135 (1999).

The Hénon–Heiles Hamiltonian was introduced in 1964 [M. Hénon, C. Heiles: Astron. J. 69, 73 (1964)] as a mathematical model to describe the chaotic motion of stars in a galaxy. By canonically transforming the classical Hamiltonian to a Birkhoff-Gustavson normalform Delos and Swimm obtained a discrete quantum mechanical energy spectrum. The aim of the present work is to first quantize the classical Hamiltonian and to then diagonalize it using different variants of flow equations, a method of continuous unitary transformations introduced by Wegner in 1994 [Ann. Physik (Leipzig) 3, 77 (1994)]. The results of the diagonalization via flow equations are comparable to those obtained by the classical transformation. In the case of commensurate frequencies the transformation turns out to be less lengthy. In addition, the dynamics of the quantum mechanical system are analyzed on the basis of the transformed observables.

Stability of ferromagnetism in Hubbard models with degenerate single-particle ground states

Andreas Mielke. Stability of ferromagnetism in Hubbard models with degenerate single-particle ground states.J. Phys. A, Math. Gen. 32, 8411-8418 (1999).

A Hubbard model with a $ N_{d} $-fold degenerate single-particle ground state has ferromagnetic ground states if the number of electrons is less or equal to $ N_{d} $. It is shown rigorously that the local stability of ferromagnetism in such a model implies global stability: The model has only ferromagnetic ground states, if there are no single spin-flip ground states. If the number of electrons is equal to $ N_{d} $, it is well known that the ferromagnetic ground state is unique if and only if the single-particle density matrix is irreducible. We present a simplified proof for this result.

Ferromagnetism in single band Hubbard models with a partially flat band

Andreas Mielke. Ferromagnetism in single band Hubbard models with a partially flat band.Phys. Rev. Lett. 82, 4312-4315 (1999).

A Hubbard model with a single, partially flat band has ferromagnetic ground states. It is shown that local stability of ferromagnetism implies its global stability in such a model: The model has only ferromagnetic ground states if there are no single spin-flip ground states. Since a single-band Hubbard model away from half filling describes a metal, this result may open a route to metallic ferromagnetism in single band Hubbard models.

Brownian motion in fluctuating periodic potentials

Enrique Abad, Andreas Mielke. Brownian motion in fluctuating periodic potentials.Ann. Physik (Leipzig) 7, 9-23 (1998).

This work deals with the overdamped motion of a particle in a fluctuating one-dimensional periodic potential. If the potential has no inversion symmetry and its fluctuations are asymmetric and correlated in time, a net flow can be generated at finite temperatures. We present results for the stationary current for the case of a piecewise linear potential, especially for potentials being close to the case with inversion symmetry. The aim is to study the stationary current as a function of the potential. Depending on the form of the potential, the current changes sign once or even twice as a function of the correlation time of the potential fluctuations. To explain these current reversals, several mechanisms are proposed. Finally, we discuss to what extent the model is useful to understand the motion of biomolecular motors.

Diagonalization of system plus environment Hamiltonians

Stefan Kehrein, Andreas Mielke. Diagonalization of system plus environment Hamiltonians.J. Stat. Phys. 90, 889-898 (1998).

A new approach to dissipative quantum systems modelled by a system plus environment Hamiltonian is presented. Using a continuous sequence of infinitesimal unitary transformations the small quantum system is decoupled from its thermodynamically large environment. Dissipation enters through the observation that system observables generically 'decay' completely into a different structure when the Hamiltonian is transformed into diagonal form. The method is particularly suited for studying low–temperature properties. This is demonstrated explicitly for the super-Ohmic spin-boson model.

Noise induced transport at zero temperature

Heiner Kohler, Andreas Mielke. Noise induced transport at zero temperature.J. Phys. A.: Math. Gen. 31, 1929-1941 (1998).

We consider a particle in the over-damped regime at zero temperature under the influence of a sawtooth potential and of a noisy force, which is correlated in time. A current occurs, even if the mean of the noisy force vanishes. We calculate the stationary probability distribution and the stationary current. We discuss, how these items depend on the characteristic parameters of the underlying stochastic process. A formal expansion of the current around the white-noise limit not always gives the correct asymptotic behaviour. We improve the expansion for some simple but representative cases.

Flow equations for band-matrices

Andreas Mielke. Flow equations for band-matrices.Euro. Phys. Jour. B 5, 605-611 (1998).

Continuous unitary transformations can be used to diagonalize or approximately diagonalize a given Hamiltonian. In the last four years, this method has been applied to a variety of models of condensed matter physics and field theory. With a new generator for the continuous unitary transformation proposed in this paper one can avoid some of the problems of former applications. General properties of the new generator are derived. It turns out that the new generator is especially useful for Hamiltonians with a banded structure. Two examples, the Lipkin model, and the spin-boson model are discussed in detail.

Low temperature equilibrium correlation function in dissipative quantum systems

Stefan Kehrein, Andreas Mielke. Low temperature equilibrium correlation function in dissipative quantum systems.Ann. Physik (Leipzig) 6, 90-135 (1997).

We introduce a new theoretical approach to dissipative quantum systems. By means of a continuous sequence of infinitesimal unitary transformations, we decouple the small quantum system that one is interested in from its thermodynamically large environment. This yields a trivial final transformed Hamiltonian. Dissipation enters through the observation that generically observables 'decay' completely under these unitary transformations, i.e. are completely transformed into other terms. As a nontrivial example the spin-boson model is discussed in some detail. For the super-Ohmic bath we obtain a very satisfactory description of short, intermediate and long time scales at small temperatures. This can be tested from the generalized Shiba-relation that is fulfilled within numerical errors.

Calculating superconducting transition temperatures in a renormalized Hamiltonian framework

Andreas Mielke. Calculating superconducting transition temperatures in a renormalized Hamiltonian framework.Europhys. Lett. 40, 195-200 (1997).

It is shown that one can obtain quantitatively accurate values for the superconducting critical temperature within a Hamiltonian framework. This is possible if one uses a renormalized Hamiltonian that contains an attractive electron–electron interaction and renormalized single particle energies. It can be obtained by similarity renormalization or using flow equations for Hamiltonians. We calculate the critical temperature as a function of the coupling using the standard BCS-theory. For small coupling we rederive the McMillan formula for $T_c$. We compare our results with Eliashberg theory and with experimental data from various materials. The theoretical results agree with the experimental data within 10%. Renormalization theory of Hamiltonians provides a promising way to investigate electron–phonon interactions in strongly correlated systems.

Similarity renormalization of the electron–phonon coupling

Andreas Mielke. Similarity renormalization of the electron–phonon coupling.Ann. Physik (Leipzig) 6, 215-233 (1997).

We study the problem of the phonon-induced electron-electron interaction in a solid. Starting with a Hamiltonian that contains an electron-phonon interaction, we perform a similarity renormalization transformation to calculate an effective Hamiltonian. Using this transformation singularities due to degeneracies are avoided explicitely. The effective interactions are calculated to second order in the electron-phonon coupling. It is shown that the effective interaction between two electrons forming a Cooper pair is attractive in the whole parameter space. The final result is compared with effective interactions obtained using other approaches.

On the spin-boson model with a sub-Ohmic bath

Stefan Kehrein, Andreas Mielke. On the spin-boson model with a sub-Ohmic bath.Phys. Lett. A 219, 313-318 (1996).

We study the spin-boson model with a sub-Ohmic bath using infinitesimal unitary transformations. Contrary to some results reported in the literature we find a zero temperature transition from an untrapped state for small coupling to a trapped state for strong coupling. We obtain an explicit expression for the renormalized level spacing as a function of the bare parameters of the system. Furthermore we show that typical dynamical equilibrium correlation functions exhibit an algebaric decay at zero temperature.

Theory of the Anderson impurity model: The Schrieffer-Wolff transformation re-examined

Stefan Kehrein, Andreas Mielke. Theory of the Anderson impurity model: The Schrieffer-Wolff transformation re-examined.Ann. Physics (NY) 252, 1-32 (1996).

We apply the method of infinitesimal unitary transformations recently introduced by Wegner to the Anderson single impurity model. It is demonstrated that this method provides a good approximation scheme for all values of the on-site interaction $U$, it becomes exact for $U=0$. We are able to treat an arbitrary density of states, the only restriction being that the hybridization should not be the largest parameter in the system. Our approach constitutes a consistent framework to derive various results usually obtained by either perturbative renormalization in an expansion in the hybridization Anderson's 'poor man's' scaling approach or the Schrieffer-Wolff unitary transformation. In contrast to the Schrieffer-Wolff result we find the correct high-energy cutoff and avoid singularities in the induced couplings. An important characteristic of our method as compared to the 'poor man's' scaling approach is that we continuously decouple modes from the impurity that have a large energy difference from the impurity orbital energies. In the usual scaling approach this criterion is provided by the energy difference from the Fermi surface.

Flow equations for the spin-boson problem

Stefan Kehrein, Andreas Mielke, Peter Neu. Flow equations for the spin-boson problem.Z. Phys. B 99, 269-280 (1996).

Using continuous unitary transformations recently introduced by Wegner we obtain flow equations for the parameters of the spin-boson Hamiltonian. Interactions not contained in the original Hamiltonian are generated by this unitary transformation. Within an approximation that neglects additional interactions quadratic in the bath operators, we can close the flow equations. Applying this formalism to the case of Ohmic dissipation at zero temperature, we calculate the renormalized tunneling frequency. We find a transition from an untrapped to a trapped state at the critical coupling constant $\alpha= 1$. We also obtain the static susceptibility via the equilibrium spin correlation function. Our results are both consistent with results known from the Kondo problem and those obtained from mode coupling theories. Using this formalism at finite temperature, we find a transition from coherent to incoherent tunneling at $T_2\approx T_1$, where $T_1$ is the corssover temperature of the dynamics from underdamped to overdamped motion known from the NIBA.

A note on ferromagnetism in the Hubbard model on the complete graph

Andreas Mielke, Hal Tasaki. A note on ferromagnetism in the Hubbard model on the complete graph.preprint cond-mat/9606115 , (1996).

Transport in a fluctuating potential

Andreas Mielke. Transport in a fluctuating potential.Ann. Physik (Leipzig) 4, 721-738 (1995).

We study the overdamped motion of a particle in a fluctuating one-dimensional periodic potential. The potential has no inversion symmetry, and the fluctuations are correlated in time. At finite temperatures, a stationary current is induced. The amplitude and the direction of the current depend on the details of the noise process that is responsible for the potential fluctuations. We discuss several limiting situations for a general case. Furthermore we calculate the current in the case of a piecewise linear potential for different noise processes and parameters. A detailed discussion of the results is given, including a discussion of the mechanism that is responsible for the current reversal. We compare the present results with results for transport in a ratchet-like potential due to a fluctuating force. We also discuss the biological relevance of the present models for molecular motors. We present a model for the motion of molecular motors that explains why similar molecular motors can move in different directions.

Noise induced transport

Andreas Mielke. Noise induced transport.Ann. Physik (Leipzig) 4, 476-500 (1995).

We study the overdamped motion of a particle in a one-dimensional periodic potential driven by a stochastic force. If the force is correlated in time (non-white), and if the potential has no inversion symmetry, a current is generated. In the case of a piecewise linaer potential we obtain a closed form for the current as a ratio of two determinants. This allows us to calculate the current as a function of the noise strength, the correlation time and the temperature of the system for several stochastic processes. We examin several limiting situations. Depending on the statistics of the noise process, the direction of the current may change. Two different mechanisms for this effect are discussed.

Flow equations for the Anderson Hamiltonian

Stefan Kehrein, Andreas Mielke. Flow equations for the Anderson Hamiltonian.J. Phys. A: Math. Gen. 27, 4259-4279, corrigendum 5705 (1994).

Using a continuous unitary transformation recently proposed by Wegner together with an approximation that neglects irrelevant contributions, we obtain flow equations for Hamiltonians. These flow equations finally yield a diagonal or almost diagonal Hamiltonian. As an example we investigate the Anderson Hamiltonian for dilute magnetic alloys. We study the different fixed points of the flow equations and the corresponding relevant, marginal or irrelevant contributions. Our results are comparable to results obtained with a numerical renormalization group method, but our approach is considerably simpler.

Ferromagnetism in the Hubbard model - Examples from Models with Degenerate Single-Electron Ground States

Andreas Mielke, Hal Tasaki. Ferromagnetism in the Hubbard model - Examples from Models with Degenerate Single-Electron Ground States.Commun. Math. Phys. 158, 341-371 (1993).

Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrödinger equation has $N_{\rm d}$-fold degenerate ground states. The degeneracy $N_{\rm d}$ is proportional to the total number of sites $\abs{\Lambda}$. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to $\rho_0=N_{\rm d}/(2\abs{\Lambda})$, and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.

Ferromagnetism in the Hubbard model and Hund's rule

Andreas Mielke. Ferromagnetism in the Hubbard model and Hund's rule.Phys. Lett. A 174, 443-448 (1993).

We investigate the Hubbard model with a $N_{\rm d}$-fold degenerate single particle ground state. If the number of electrons satisfies $N_{\rm e}<N_{\rm d}$, the model has ferromagnetic multiparticle ground states. We give a necessary and sufficient condition for the ground state to be unique $N_{\rm e}=N_{\rm d}$. It is ferromagnetic with spin $S=\frac12N_{\rm e}$. As a corollary, we obtain Hund's rule for the general Hubbard model with degenerate single particle eigenstates on translationally invariant lattices in the special case, where each of the degenerate single particle states if filled with one electron.

Exact results for the $U=\infty$ Hubbard model

Andreas Mielke. Exact results for the $U=\infty$ Hubbard model.J. Phys. A: Math. Gen. 25, 6507-6515 (1992).

The author investigates the U= infinity Hubbard model on a large class of lattices which are line graphs. The most interesting lattices in this class are line graphs of regular bipartite lattices with Ns sites and coordination number k>or=4. The ground state energy and some ground states are given. If the number of electrons N satisfies Ns>or=N>or=2Ns/k-2, the ground state energy is -4 mod t mod (Ns-N). The ground states have no magnetic ordering, they are projections of the ground states at U=0 onto the subspace of states without doubly occupied sites.

Exact ground states for the Hubbard model on the kagomé lattice

Andreas Mielke. Exact ground states for the Hubbard model on the kagomé lattice.J. Phys. A: Math. Gen. 25, 4335-4345 (1992).

The author gives a complete and rigorous description of the ground states of the Hubbard model on the Kagome lattice for electron densities n>or=5/3 and U>0. If 11/6>n>or=5/3 the system shows a ferromagnetic behaviour at zero temperature. If n is above 11/6 the system is paramagnetic. The proof of these results uses some graph-theoretic methods. The results are applicable to all line graphs of planar lattices, of which the Kagome lattice is an example.

Ferromagnetism in the Hubbard model on line graphs and further considerations

Andreas Mielke. Ferromagnetism in the Hubbard model on line graphs and further considerations.J. Phys. A: Math. Gen. 24, 3311-3321 (1991).

Let L(G) be the line graph of a graph G=(V,E). The Hubbard model on L(G) has ferromagnetic ground states with a saturated spin if the interaction is repulsive (U>0) and if the number of electrons N satisfies N>or=M. M= mod E mod + mod V mod -1 if G is bipartite and M= mod E mod + mod V mod otherwise. The author shows that the ferromagnetic ground state is unique if N=M. Further he gives a sufficient condition for the existence of other ground states if N>M. The results are valid also for a multi-band Hubbard model on a bipartite graph. In the case of a periodic lattice, the results are related to the existence of a flat energy band.

Ferromagnetic ground states for the Hubbard model on line graphs

Andreas Mielke. Ferromagnetic ground states for the Hubbard model on line graphs.J. Phys. A: Math. Gen. 24, L73-L77 (1991).

The author discusses some of the properties of the Hubbard model on a line graph with n vertices. It is shown that the model has ferromagnetic ground states if the interaction is repulsive (U)0) and if the number of electrons N satisfies 2n>or=N>or=M. M is a natural number that depends on the line graph. For example, the Kagome lattice is a line graph, it has M=5n/3-1.

The One-Dimensional Hubbard Model for Large or Infinite $U$

Andreas Mielke. The One-Dimensional Hubbard Model for Large or Infinite $U$.J. Stat. Phys. 62, 509-528 (1991).

Disorder and the fractional quantum Hall effect: the reduction of the gap

Andreas Mielke. Disorder and the fractional quantum Hall effect: the reduction of the gap.J. Phys.: Condensed Matter 2, 9567-9576 (1990).

The author presents numerical as well as analytical results for the reduction of the energy gap in the fractional quantum Hall regime due to disorder. The disordered substrate is treated perturbatively. It consists of randomly distributed long-range scatterers. It turns out that the reduction of the energy gap is determined by the mean value of the fluctuations of the potential alone and that it is proportional to the fractional charge of the quasiparticles.

Disorder and the fractional quantum Hall effect: higher Landau levels

Andreas Mielke. Disorder and the fractional quantum Hall effect: higher Landau levels.Z. Phys. B - Condensed Matter 80, 57-62 (1990).

Disorder and the theory of the fractionally quantized Hall effect

Andreas Mielke. Disorder and the theory of the fractionally quantized Hall effect.Z. Phys. B - Condensed Matter 73, 191-199 (1988).

Two dimensional interacting electrons on a torus in a strong magnetic field: Symmetry properties and the effect of disorder

Andreas Mielke. Two dimensional interacting electrons on a torus in a strong magnetic field: Symmetry properties and the effect of disorder.Z. Phys. B - Condensed Matter 71, 179-185 (1988).

Scaling Behavior of One-Dimensional Weakly Disordered Models

Andreas Mielke, Franz Wegner. Scaling Behavior of One-Dimensional Weakly Disordered Models.Z. Phys. B - Condensed Matter 62, 1-8 (1985).

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