PD Dr. Andreas Alvermann
Institute of Physics
Phone +49 3834 420-4763
Fax +49 3834 420-4701
Publications (in scientific journals)
- B. Höckendorf, A. Alvermann, H. Fehske, Controlling the direction of topological transport in a non-Hermitian time-reversal symmetric Floquet ladder, APL Photonics 6, 010801 (2021) [ DOI ] [ arXiv ]
Abstract: We propose a one-dimensional Floquet ladder that possesses two distinct topological transport channels with opposite directionality. The transport channels occur due to a Z(2) non-Hermitian Floquet topological phase that is protected by time-reversal symmetry. The signatures of this phase are two pairs of Kramers degenerate Floquet quasienergy bands that are separated by an imaginary gap. We discuss how the Floquet ladder can be implemented in a photonic waveguide lattice and show that the direction of transport in the resulting waveguide structure can be externally controlled by focusing two light beams into adjacent waveguides. The relative phase between the two light beams selects which of the two transport channels is predominantly populated, while the angles of incidence of the two light beams determine which of the transport channels is suppressed by non-Hermitian losses. We identify the optimal lattice parameters for the external control of transport and demonstrate the robustness of this mechanism against disorder.
- A. Fritzsche, B. Höckendorf, A. Alvermann, H. Fehske, Real and imaginary edge states in stacked Floquet honeycomb lattices, Eur. Phys. J. B 93, 151 (2020) [ DOI ] [ arXiv ]
Abstract: We present a non-Hermitian Floquet model with topological edge states in real and imaginary band gaps. The model utilizes two stacked honeycomb lattices which can be related via four different types of non-Hermitian time-reversal symmetry. Implementing the correct time-reversal symmetry provides us with either two counterpropagating edge states in a real gap, or a single edge state in an imaginary gap. The counterpropagating edge states allow for either helical or chiral transport along the lattice perimeter. In stark contrast, we find that the edge state in the imaginary gap does not propagate. Instead, it remains spatially localized while its amplitude continuously increases. Our model is well-suited for realizing these edge states in photonic waveguide lattices.
- B. Höckendorf, A. Alvermann, H. Fehske, Cutting off the non-Hermitian boundary from an anomalous Floquet topological insulator, Europhys. Lett. 131, 30007 (2020) [ DOI ] [ arXiv ]
Abstract: In two-dimensional anomalous Floquet insulators, non-Hermitian boundary state engineering can be used to completely separate chiral boundary states from bulk bands in the quasienergy spectrum. The topological properties of such spectrally separated boundary states are no longer restricted by the strict bulk-boundary correspondence of Hermitian systems. We show that this additional topological freedom enables one to faithfully transfer the topological properties of a boundary attached to a Floquet insulator to a non-Hermitian Floquet chain obtained by physically cutting off the boundary from the bulk. We implement this scenario for a simple model of an anomalous Floquet insulator with Hermitian and non-Hermitian boundaries, and discuss the relevance of our construction for the experimental realization of non-Hermitian topological phases that connect dimensions one and two.
- B. Höckendorf, A. Alvermann, H. Fehske, Topological origin of quantized transport in non-Hermitian Floquet chains, Phys. Rev. Research 2, 023235 (2020) [ DOI ] [ arXiv ]
Abstract: We show that non-Hermiticity enables topological phases with unidirectional transport in one-dimensional Floquet chains. The topological signatures of these phases are noncontractible loops in the spectrum of the Floquet propagator that are separated by an imaginary gap. Such loops occur exclusively in non-Hermitian Floquet systems. We define the corresponding topological invariant as the winding number of the Floquet propagator relative to the imaginary gap. To relate topology to transport, we introduce the concept of regularized dynamics of non-Hermitian chains. We establish that, under the conditions of regularized dynamics, transport is quantized insofar as the charge transferred over one period equals the topological winding number. We illustrate these theoretical findings with the example of a Floquet chain that features a topological phase transition and acts as a charge pump in the nontrivial topological phase. We finally discuss whether these findings justify the notion that non-Hermitian Floquet chains support topological transport.
- L. J. Maczewsky, B. Höckendorf, M. Kremer, T. Biesenthal, M. Heinrich, A. Alvermann, H. Fehske, A. Szameit, Fermionic time-reversal symmetry in a photonic topological insulator, Nat. Mater. (2020) [ DOI ] [ arXiv ]
Abstract: Much of the recent attention directed towards topological insulators is motivated by their hallmark feature of protected chiral edge states. In electronic (or fermionic) topological insulators, these states originate from time-reversal symmetry and allow carriers with opposite spin-polarization to propagate in opposite directions at the edge of an insulating bulk. By contrast, photonic (or bosonic) systems are generally assumed to be precluded from supporting edge states that are intrinsically protected by time-reversal symmetry. Here, we experimentally demonstrate counter-propagating chiral states at the edge of a time-reversal-symmetric photonic waveguide structure. The pivotal step in our approach is the design of a Floquet driving protocol that incorporates effective fermionic time-reversal symmetry, enabling the realization of the photonic version of an electronic topological insulator. Our findings allow for fermionic properties to be harnessed in bosonic systems, thereby offering alternative opportunities for photonics as well as acoustics, mechanical waves and cold atoms.
- B. Höckendorf, A. Alvermann, H. Fehske, Non-Hermitian Boundary State Engineering in Anomalous Floquet Topological Insulators, Phys. Rev. Lett. 123, 190403 (2019) [ DOI ] [ arXiv ]
Abstract: In Hermitian topological systems, the bulk-boundary correspondence strictly constrains boundary transport to values determined by the topological properties of the bulk. We demonstrate that this constraint can be lifted in non-Hermitian Floquet insulators. Provided that the insulator supports an anomalous topological phase, non-Hermiticity allows us to modify the boundary states independently of the bulk, without sacrificing their topological nature. We explore the ensuing possibilities for a Floquet topological insulator with non-Hermitian time-reversal symmetry, where the helical transport via counterpropagating boundary states can be tailored in ways that overcome the constraints imposed by Hermiticity. Non-Hermitian boundary state engineering specifically enables the enhancement of boundary transport relative to bulk motion, helical transport with a preferred direction, and chiral transport in the same direction on opposite boundaries. We explain the experimental relevance of our findings for the example of photonic waveguide lattices.
- A. Alvermann, A. Basermann, H. Bungartz, C. Carbogno, D. Ernst, H. Fehske, Y. Futamura, M. Galgon, G. Hager, S. Huber, T. Huckle, A. Ida, A. Imakura, M. Kawai, S. Köcher, M. Kreutzer, P. Kus, B. Lang, H. Lederer, V. Manin, A. Marek, K. Nakajima, L. Nemec, K. Reuter, M. Rippl, M. Röhrig-Zöllner, T. Sakurai, M. Scheffler, C. Scheurer, F. Shahzad, D. Simoes Brambila, J. Thies, G. Wellein, Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects, Japan Journal of Industrial and Applied Mathematics 36, 699 (2019) [ DOI ] [ arXiv ]
Abstract: We first briefly report on the status and recent achievements of the ELPA-AEO (Eigen value Solvers for Petaflop Applications---Algorithmic Extensions and Optimizations) and ESSEX II (Equipping Sparse Solvers for Exascale) projects. In both collaboratory efforts, scientists from the application areas, mathematicians, and computer scientists work together to develop and make available efficient highly parallel methods for the solution of eigenvalue problems. Then we focus on a topic addressed in both projects, the use of mixed precision computations to enhance efficiency. We give a more detailed description of our approaches for benefiting from either lower or higher precision in three selected contexts and of the results thus obtained.
- B. Höckendorf, A. Alvermann, H. Fehske, Universal driving protocol for symmetry-protected Floquet topological phases, Phys. Rev. B 99, 245102 (2019) [ DOI ] [ arXiv ]
Abstract: We propose a universal driving protocol for the realization of symmetry-protected topological phases in 2+1-dimensional Floquet systems. Our proposal is based on the theoretical analysis of the possible symmetries of a square lattice model with pairwise nearest-neighbor coupling terms. Among the eight possible symmetry operators we identify the two relevant choices for topological phases with either time-reversal, chiral, or particle-hole symmetry. From the corresponding symmetry conditions on the protocol parameters, we obtain the universal driving protocol where each of the symmetries can be realized or broken individually. We provide specific parameter values for the different cases, and demonstrate the existence of symmetry-protected copropagating and counterpropagating topological boundary states. The driving protocol especially allows us to switch between bosonic and fermionic time-reversal symmetry, and thus between a trivial and nontrivial symmetry-protected topological phase, through continuous variation of a parameter.
- A. Alvermann, H. Fehske, Exciton mass and exciton spectrum in the cuprous oxide, J. Phys. B 51, 044001 (2018) [ DOI ] [ arXiv ]
Abstract: Excitons with a radius of a few lattice constants can be affected by strong central-cell corrections, leading to significant deviations of the optical spectrum from the hydrogen-like Rydberg series, and also to an enhancement of the exciton mass. We present an approach to this situation based on a lattice model that incorporates the effects of a non-parabolic band structure, short distance corrections to the Coulomb interaction between electrons and holes, spin-orbit and exchange coupling. The lattice model allows for observation of the crossover from large radius Wannier to small radius Frenkel excitons without invoking a continuum approximation. We apply the lattice model approach especially to the yellow exciton series in the cuprous oxide, for which the optical spectrum and exciton mass enhancement are obtained through adaptation of only a few model parameters to material-specific values. Our results predict a strongly anisotropic ortho-exciton mass.
- B. Höckendorf, A. Alvermann, H. Fehske, Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry, Phys. Rev. B 97, 045140 (2018) [ DOI ] [ arXiv ]
Abstract: We introduce $\mathbb Z_2$-valued bulk invariants for symmetry-protected topological phases in $2+1$ dimensional driven quantum systems. These invariants adapt the $W_3$-invariant, expressed as a sum over degeneracy points of the propagator, to the respective symmetry class of the Floquet-Bloch Hamiltonian. The bulk-boundary correspondence that holds for each invariant relates a non-zero value of the bulk invariant to the existence of symmetry-protected topological boundary states. To demonstrate this correspondence we apply our invariants to a chiral Harper, time-reversal Kane-Mele, and particle-hole symmetric graphene model with periodic driving, where they successfully predict the appearance of boundary states that exist despite the trivial topological character of the Floquet bands. Especially for particle-hole symmetry, combination of the $W_3$ and the $\mathbb Z_2$-invariants allows us to distinguish between weak and strong topological phases.
- B. Höckendorf, A. Alvermann, H. Fehske, Efficient computation of the $W_3$ topological invariant and application to Floquet-Bloch systems, J. Phys. A 50, 295301 (2017) [ DOI ] [ arXiv ]
Abstract: We introduce an efficient algorithm for the computation of the $W_3$ invariant of general unitary maps, which converges rapidly even on coarse discretization grids. The algorithm does not require extensive manipulation of the unitary maps, identification of the precise positions of degeneracy points, or fixing the gauge of eigenvectors. After construction of the general algorithm, we explain its application to the 2 + 1 dimensional maps that arise in the Floquet-Bloch theory of periodically driven two-dimensional quantum systems. We demonstrate this application by computing the $W_3$ invariant for an irradiated graphene model with a continuously modulated Hamilton operator, where it predicts the number of anomalous edge states in each gap.
- D. Pagel, A. Alvermann, H. Fehske, Dynamic Stark effect, light emission, and entanglement generation in a laser-driven quantum optical system, Phys. Rev. A 95, 013825 (2017) [ DOI ] [ arXiv ]
Abstract: We calculate the emission spectra, the Glauber $g_2$ function, and the entanglement of formation for two-level emitters coupled to a single cavity mode and subject to an external laser excitation. To evaluate these quantities we couple the system to environmental degrees of freedom, which leads to dissipative dynamics. Because of the periodic time dependence of the system Hamiltonian, the coefficients of the Markovian master equation are constant only if Floquet states are used as the computational basis. Studying the emission spectra, we show that the dynamic Stark effect first appears in second order of the laser intensity. For the Glauber function, we find clearly distinguished parameter regimes of super- and sub-Poissonian light emission and explain the additional features appearing for finite laser intensity in terms of the quasienergy spectrum of the driven emitter-cavity system. Finally, we analyze the temperature and emitter-cavity-coupling regimes where entanglement among the emitters is generated and show that the laser excitation leads to a decrease of entanglement.
- C. Wurl, A. Alvermann, H. Fehske, Symmetry-breaking oscillations in membrane optomechanics, Phys. Rev. A 94, 063860 (2016) [ DOI ] [ arXiv ]
Abstract: We study the classical dynamics of a membrane inside a cavity in the situation where this optomechanical system possesses a reflection symmetry. Symmetry breaking occurs through supercritical and subcritical pitchfork bifurcations of the static fixed-point solutions. Both bifurcations can be observed through variation of the laser-cavity detuning, which gives rise to a boomerang-like fixed-point pattern with hysteresis. The symmetry-breaking fixed points evolve into self-sustained oscillations when the laser intensity is increased. In addition to the analysis of the accompanying Hopf bifurcations we describe these oscillations at finite amplitudes with an ansatz that fully accounts for the frequency shift relative to the natural membrane frequency. We complete our study by following the route to chaos for the membrane dynamics.
- A. Pieper, M. Kreutzer, A. Alvermann, M. Galgon, H. Fehske, G. Hager, B. Lang, G. Wellein, High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations, J. of Comp. Phys. 325, 226 (2016) [ DOI ] [ arXiv ]
Abstract: We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps towards a parallel high-performance implementation. Because Chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers. To demonstrate the potential of Chebyshev filter diagonalization for large-scale problems of this kind we include as an example the computation of the $10^2$ innermost eigenpairs of a topological insulator matrix with dimension $10^9$ derived from quantum physics applications.
- C. Schulz, A. Alvermann, L. Bakemeier, H. Fehske, Optomechanical multistability in the quantum regime, Europhys. Lett. 113, 64002 (2016) [ DOI ] [ arXiv ]
Abstract: Classical optomechanical systems feature self-sustained oscillations, where multiple periodic orbits at different amplitudes coexist. We study how this multistability is realized in the quantum regime, where new dynamical patterns appear because quantum trajectories can move between different classical orbits. We explain the resulting quantum dynamics from the phase space point of view, and provide a quantitative description in terms of autocorrelation functions. In this way we can identify clear dynamical signatures of the crossover from classical to quantum mechanics in experimentally accessible quantities. Finally, we discuss a possible interpretation of our results in the sense that quantum mechanics protects optomechanical systems against the chaotic dynamics realized in the classical limit.
- L. Bakemeier, A. Alvermann, H. Fehske, Variational treatment of entanglement in the Dicke model, Physica Scripta T165, 014001 (2015) [ DOI ]
Abstract: We introduce a variational ansatz for the Dicke model that extends mean-field theory through the inclusion of spin-oscillator correlations. The correlated variational state is obtained from the mean-field product state via a unitary transformation. The ansatz becomes correct in the limit of large oscillator frequency and in the limit of a large spin, for which it captures the leading quantum corrections to the classical limit exactly including the spin-oscillator entanglement entropy. We explain the origin of the unitary transformation before we show that the ansatz improves substantially upon mean-field theory, giving near exact results for the ground state energy and very good results for other observables. We then discuss why the ansatz still encounters problems in the transition regime at moderate spin lengths, where it fails to capture the precursors of the superradiant quantum phase transition faithfully. This observation illustrates the principal limits of semi-classical formulations, even after they are extended with correlations and entanglement.
- D. Pagel, A. Alvermann, H. Fehske, Nonclassical light from few emitters in a cavity, Phys. Rev. A 91, 043814 (2015) [ DOI ] [ arXiv ]
Abstract: We study the characteristics of the light generated by a few emitters in a cavity at strong light-matter coupling. By means of the Glauber $g_2$ function we can identify clearly distinguished parameter regimes with super-Poissonian and sub-Poissonian photon statistics. We establish a relation between the emission characteristics for one and multiple emitters and explain its origin in terms of the photon-dressed emitter states. Cooperative effects lead to the generation of nonclassical light already at reduced light-matter coupling if the number of emitters is increased. Our results are obtained with a full input-output formalism and master equation valid also at strong light-matter coupling. We compare the behavior obtained with and without counterrotating light-matter-interaction terms in the Hamiltonian and find that the generation of nonclassical light is robust against such modifications. Finally, we contrast our findings with the predictions of the quantum optical master equation and find that it fails entirely at predicting regimes with different photon statistics.
- L. Bakemeier, A. Alvermann, H. Fehske, Route to Chaos in Optomechanics, Phys. Rev. Lett. 114, 013601 (2015) [ DOI ] [ arXiv ]
Abstract: We establish the emergence of chaotic motion in optomechanical systems. Chaos appears at negative detuning for experimentally accessible values of the pump power and other system parameters. We describe the sequence of period-doubling bifurcations that leads to chaos and state the experimentally observable signatures in the optical spectrum. In addition to the semiclassical dynamics, we analyze the possibility of chaotic motion in the quantum regime. We find that quantum mechanics protects the optomechanical system against irregular dynamics, such that simple periodic orbits reappear and replace the classically chaotic motion. In this way observation of the dynamical signatures makes it possible to pin down the crossover from quantum to classical mechanics.
- M. Röhrig-Zöllner, J. Thies, M. Kreutzer, A. Alvermann, A. Pieper, A. Basermann, G. Hager, G. Wellein, H. Fehske, Increasing the performance of the Jacobi-Davidson method by blocking, SIAM J. Sci. Comput. 37, (2015) [ DOI ]
Abstract: Block variants of the Jacobi-Davidson method for computing a few eigenpairs of a large sparse matrix are known to improve the robustness of the standard algorithm when it comes to computing multiple or clustered eigenvalues. In practice, however, they are typically avoided because the total number of matrix-vector operations increases. In this paper we present the implementation of a block Jacobi-Davidson solver. By detailed performance engineering and numerical experiments we demonstrate that the increase in operations is typically more than compensated by performance gains through better cache usage on modern CPUs, resulting in a method that is both more efficient and robust than its single vector counterpart. The steps to be taken to achieve a block speedup involve both kernel optimizations for sparse matrix and block vector operations, and algorithmic choices to allow using blocked operations in most parts of the computation. We discuss the aspect of avoiding synchronization in the algorithm and show by numerical experiments with our hybrid parallel implementation that a significant speedup through blocking can be achieved for a variety of matrices on up to 5120 CPU cores as long as at least about 20 eigenpairs are sought.
- L. Bakemeier, A. Alvermann, H. Fehske, Dynamics of the Dicke model close to the classical limit, Phys. Rev. A 88, 043835 (2013) [ DOI ] [ arXiv ]
Abstract: We study the dynamical properties of the Dicke model for increasing spin length, as the system approaches the limit of a classical spin. First, we describe the emergence of collective excitations above the ground state that converge to the coupled spin-oscillator oscillations found in the classical limit. The corresponding Green functions reveal quantum dynamical signatures close to the superradiant quantum phase transition. Second, we identify signatures of classical quasiperiodic orbits in the quantum time evolution using numerical time propagation of the wave function. The resulting phase-space plots are compared to the classical trajectories. We complete our study with the analysis of individual eigenstates close to the quasiperiodic orbits.
- D. Pagel, P. Nalbach, A. Alvermann, H. Fehske, M. Thorwart, Nonequilibrium quantum fluctuation relations for harmonic systems in nonthermal environments, New J. Phys. 15, 105008 (2013) [ DOI ] [ arXiv ]
Abstract: We formulate exact generalized nonequilibrium fluctuation relations for the quantum mechanical harmonic oscillator coupled to multiple harmonic baths. Each of the different baths is prepared in its own individual (in general nonthermal) state. Starting from the exact solution for the oscillator dynamics we study fluctuations of the oscillator position as well as of the energy current through the oscillator under general nonequilibrium conditions. In particular, we formulate a fluctuation-dissipation relation for the oscillator position autocorrelation function that generalizes the standard result for the case of a single bath at thermal equilibrium. Moreover, we show that the generating function for the position operator fulfils a generalized Gallavotti-Cohen-like relation. For the energy transfer through the oscillator, we determine the average energy current together with the current fluctuations. Finally, we discuss the generalization of the cumulant generating function for the energy transfer to nonthermal bath preparations.
- D. Pagel, A. Alvermann, H. Fehske, Equilibration and thermalization of the dissipative quantum harmonic oscillator in a nonthermal environment, Phys. Rev. E 87, 012127 (2013) [ DOI ] [ arXiv ]
Abstract: We study the dissipative quantum harmonic oscillator with general nonthermal preparations of the harmonic oscillator bath. The focus is on equilibration of the oscillator in the long-time limit and the additional requirements for thermalization. Our study is based on the exact solution of the microscopic model obtained by means of operator equations of motion, which provides us with the time evolution of the central oscillator density matrix in terms of the propagating function. We find a hierarchy of conditions for thermalization, together with the relation of the asymptotic temperature to the energy distribution in the initial bath state. We discuss the presence and absence of equilibration for the example of an inhomogeneous chain of harmonic oscillators, and we illustrate the general findings about thermalization for the nonthermal environment that results from a quench.
- A. Alvermann, H. Fehske, P. B. Littlewood, Numerical time propagation of quantum systems in radiation fields, New J. Phys. 14, 105008 (2012) [ DOI ] [ arXiv ]
Abstract: Atoms, molecules or excitonic quasiparticles, for which excitations are induced by external radiation fields and energy is dissipated through radiative decay, are examples of driven open quantum systems. We explain the use of commutator-free exponential time propagators for the numerical solution of the associated Schrodinger or master equations with a time-dependent Hamilton operator. These time propagators are based on the Magnus series but avoid the computation of commutators, which makes them suitable for the efficient propagation of systems with a large number of degrees of freedom. We present an optimized fourth-order propagator and demonstrate its efficiency in comparison to the direct Runge-Kutta computation. As an illustrative example we consider the parametrically driven dissipative Dicke model, for which we calculate the periodic steady state and the optical emission spectrum.
- L. Bakemeier, A. Alvermann, H. Fehske, Quantum phase transition in the Dicke model with critical and noncritical entanglement, Phys. Rev. A 85, 043821 (2012) [ DOI ] [ arXiv ]
Abstract: We study the quantum phase transition of the Dicke model in the classical oscillator limit, where it occurs already for finite spin length. In contrast to the classical spin limit, for which spin-oscillator entanglement diverges at the transition, entanglement in the classical oscillator limit remains small. We derive the quantum phase transition with identical critical behavior in the two classical limits and explain the differences with respect to quantum fluctuations around the mean-field ground state through an effective model for the oscillator degrees of freedom. With numerical data for the full quantum model we study convergence to the classical limits. We contrast the classical oscillator limit with the dual limit of a high-frequency oscillator, where the spin degrees of freedom are described by the Lipkin-Meshkov-Glick model. An alternative limit can be defined for the Rabi case of spin length one-half, in which spin frequency renormalization replaces the quantum phase transition.
- A. Alvermann, L. Bakemeier, H. Fehske, Collapse-revival dynamics and atom-field entanglement in the nonresonant Dicke model, Phys. Rev. A 85, 043803 (2012) [ DOI ] [ arXiv ]
Abstract: We consider the dynamics of atomic and field coherent states in the nonresonant Dicke model. At weak coupling an initial product state evolves into a superposition of multiple field coherent states that are correlated with the atomic configuration. This process is accompanied by the buildup and decay of atom-field entanglement and leads to the periodic collapse and revival of Rabi oscillations. We provide a perturbative derivation of the underlying dynamical mechanism that complements the rotating wave approximation at resonance. The identification of two different time scales explains how the dynamical signatures depend on the sign of detuning between the atomic and field frequency and predicts the generation of either atomic or field cat states in the two opposite cases. We finally discuss the restrictions that the buildup of atom-field entanglement during the collapse of Rabi oscillations impose on the validity of semiclassical approximations that neglect entanglement.
- T. Koch, J. Loos, A. Alvermann, H. Fehske, Nonequilibrium transport through molecular junctions in the quantum regime, Phys. Rev. B 84, 125131 (2011) [ DOI ] [ arXiv ]
Abstract: We consider a quantum dot, affected by a local vibrational mode and contacted to macroscopic leads, in the nonequilibrium steady-state regime. We apply a variational Lang-Firsov transformation and solve the equations of motion of the Green functions in the Kadanoff-Baym formalism up to a second order in the interaction coefficients. The variational determination of the transformation parameter through minimization of the thermodynamic potential allows us to calculate the electron/polaron spectral function and conductance for adiabatic to antiadiabatic phonon frequencies and weak to strong electron-phonon couplings. We investigate the qualitative impact of the quasiparticle renormalization on the inelastic electron tunneling spectroscopy signatures and discuss the possibility of a polaron induced negative differential conductance. In the high-voltage regime, we find that the polaron level follows the lead chemical potential to enhance resonant transport.
- A. Alvermann, P. B. Littlewood, H. Fehske, Variational discrete variable representation for excitons on a lattice, Phys. Rev. B 84, 035126 (2011) [ DOI ] [ arXiv ]
Abstract: We construct numerical basis function sets on a lattice, whose spatial extension is scalable from single lattice sites to the continuum limit. They allow us to compute small and large bound states with comparable, moderate effort. Adopting concepts of discrete variable representations, a diagonal form of the potential term is achieved through a unitary transformation to Gaussian quadrature points. Thereby the computational effort in three dimensions scales as the fourth instead of the sixth power of the number of basis functions along each axis, such that it is reduced by two orders of magnitude in realistic examples. As an improvement over standard discrete variable representations, our construction preserves the variational principle. It allows for the calculation of binding energies, wave functions, and excitation spectra. We use this technique to study central-cell corrections for excitons beyond the continuum approximation. A discussion of the mass and spectrum of the yellow exciton series in the cuprous oxide, which does not follow the hydrogenic Rydberg series of Mott-Wannier excitons, is given on the basis of a simple lattice model.
- A. Alvermann, H. Fehske, High-order commutator-free exponential time-propagation of driven quantum systems, J. of Comp. Phys. 230, 5930 (2011) [ DOI ] [ arXiv ]
Abstract: We discuss the numerical solution of the Schrodinger equation with a time-dependent Hamilton operator using commutator-free time-propagators. These propagators are constructed as products of exponentials of simple weighted sums of the Hamilton operator. Owing to their exponential form they strictly preserve the unitarity of time-propagation. The absence of commutators or other computationally involved operations allows for straightforward implementation and application also to large scale and sparse matrix problems. We explain the derivation of commutator-free exponential time-propagators in the context of the Magnus expansion, and provide optimized propagators up to order eight. An extensive theoretical error analysis is presented together with practical efficiency tests for different problems. Issues of practical implementation, in particular the use of the Krylov technique for the calculation of exponentials, are discussed. We demonstrate for two advanced examples, the hydrogen atom in an electric field and pumped systems of multiple interacting two-level systems or spins that this approach enables fast and accurate computations.
- D. M. Edwards, S. Ejima, A. Alvermann, H. Fehske, A Green's function decoupling scheme for the Edwards fermion-boson model, J. Phys. Condens. Matter 22, 435601 (2010) [ DOI ] [ arXiv ]
Abstract: Holes in a Mott insulator are represented by spinless fermions in the fermion-boson model introduced by Edwards. Although the physically interesting regime is for low to moderate fermion density, the model has interesting properties over the whole density range. It has previously been studied at half-filling in the one-dimensional (1D) case by numerical methods, in particular using exact diagonalization and the density matrix renormalization group (DMRG). In the present study the one-particle Green's function is calculated analytically by means of a decoupling scheme for the equations of motion, valid for arbitrary density in 1D, 2D and 3D with fairly large boson energy and zero boson relaxation parameter. The Green's function is used to compute some ground state properties, and the one-fermion spectral function, for fermion densities n = 0.1, 0.5 and 0.9 in the 1D case. The results are generally in good agreement with numerical results obtained using the DMRG and dynamical DMRG, and new light is shed on the nature of the ground state at different fillings. The Green's function approximation is sufficiently successful in 1D to justify future application to the 2D and 3D cases.
- A. Alvermann, H. Fehske, S. A. Trugman, Polarons and slow quantum phonons, Phys. Rev. B 81, 165113 (2010) [ DOI ] [ arXiv ]
Abstract: We describe the formation and properties of Holstein polarons in the entire parameter regime. Our presentation focuses on the polaron mass and radius, which we obtain with an improved numerical technique. It is based on the combination of variational exact diagonalization with an improved construction of phonon states, providing results even for the strong coupling adiabatic regime. In particular we can describe the formation of large and heavy adiabatic polarons. A comparison of the polaron mass for the one- and three-dimensional situation explains how the different properties in the static oscillator limit determine the behavior in the adiabatic regime. The transport properties of large and small polarons are characterized by the f-sum rule and the optical conductivity. Our calculations are approximation free and have negligible numerical error. This allows us to give a conclusive and impartial description of polaron formation. We finally discuss the implications of our results for situations beyond the Holstein model.
- A. Alvermann, H. Fehske, Non-equilibrium current and electron pumping in nanostructures, J. Phys.: Conf. Ser. 200, UNSP 012005 (2010) [ DOI ] [ arXiv ]
Abstract: We discuss a numerical method to study electron transport in mesoscopic devices out of equilibrium. The method is based on the solution of operator equations of motion, using efficient Chebyshev time propagation techniques. Its peculiar feature is the propagation of operators backwards in time. In this way the resource consumption scales linearly with the number of states used to represent the system. This allows us to calculate the current for non-interacting electrons in large one-, two- and three-dimensional lead-device configurations with time-dependent voltages or potentials. We discuss the technical aspects of the method and present results for an electron pump device and a disordered system, where we find transient behaviour that exists for a very long time and may be accessible to experiments.
- A. Alvermann, D. M. Edwards, H. Fehske, Analytical calculation of the Green's function and Drude weight for a correlated fermion-boson system, J. Phys.: Conf. Ser. 220, 012023 (2010) [ DOI ] [ arXiv ]
Abstract: In classical Drude theory the conductivity is determined by the mass of the propagating particles and the mean free path between two scattering events. For a quantum particle this simple picture of diffusive transport loses relevance if strong correlations dominate the particle motion. We study a situation where the propagation of a fermionic particle is possible only through creation and annihilation of local bosonic excitations. This correlated quantum transport process is outside the Drude picture, since one cannot distinguish between free propagation and intermittent scattering. The characterization of transport is possible using the Drude weight obtained from the f-sum rule, although its interpretation in terms of free mass and mean free path breaks down. For the situation studied we calculate the Green's function and Drude weight using a Green's functions expansion technique, and discuss their physical meaning.
- T. Koch, J. Loos, A. Alvermann, A. R. Bishop, H. Fehske, Transport through a vibrating quantum dot: Polaronic effects, J. Phys.: Conf. Ser. 220, 012014 (2010) [ DOI ] [ arXiv ]
Abstract: We present a Green's function based treatment of the effects of electron-phonon coupling on transport through a molecular quantum dot in the quantum limit. Thereby we combine an incomplete variational Lang-Firsov approach with a perturbative calculation of the electron-phonon self energy in the framework of generalised Matsubara Green functions and a Landauer-type transport description. Calculating the ground-state energy, the dot single-particle spectral function and the linear conductance at finite carrier density, we study the low-temperature transport properties of the vibrating quantum dot sandwiched between metallic leads in the whole electron-phonon coupling strength regime. We discuss corrections to the concept of an anti-adiabatic dot polaron and show how a deformable quantum dot can act as a molecular switch.
- J. Loos, T. Koch, A. Alvermann, A. R. Bishop, H. Fehske, Phonon affected transport through molecular quantum dots, J. Phys. Condens. Matter 21, 395601 (2009) [ DOI ] [ arXiv ]
Abstract: To describe the interaction of molecular vibrations with electrons at a quantum dot contacted to metallic leads, we extend an analytical approach that we previously developed for the many-polaron problem. Our scheme is based on an incomplete variational Lang-Firsov transformation, combined with a perturbative calculation of the electron-phonon self-energy in the framework of generalized Matsubara functions. This allows us to describe the system at weak-to-strong coupling and intermediate-to-large phonon frequencies. We present results for the quantum dot spectral function and for the kinetic coefficient that characterizes the electron transport through the dot. With these results we critically examine the strengths and limitations of our approach, and discuss the properties of the molecular quantum dot in the context of polaron physics. We place particular emphasis on the importance of corrections to the concept of an anti-adiabatic dot polaron suggested by the complete Lang-Firsov transformation.
- A. S. Mishchenko, N. Nagaosa, A. Alvermann, H. Fehske, G. De Filippis, V. Cataudella, O. P. Sushkov, Localization-delocalization transition of a polaron near an impurity, Phys. Rev. B 79, 180301 (2009) [ DOI ] [ arXiv ]
Abstract: We solve the problem of polaron localization on an attractive impurity by means of direct-space diagrammatic Monte Carlo implemented for the system in the thermodynamic limit. In particular we determine the ground-state phase diagram in dependence on the electron-phonon coupling and impurity potential strength for the whole phonon frequency range. Including the quantum phonon effects we find and characterize a phase where self-trapped polarons are not localized at shallow impurities, which is missing in the adiabatic approximation. We show that near the localization transition a region with a mixture of weak- and strong-coupling spectral responses is realized.
- A. Alvermann, H. Fehske, Sparse Polynomial Space Approach to Dissipative Quantum Systems: Application to the Sub-Ohmic Spin-Boson Model, Phys. Rev. Lett. 102, 150601 (2009) [ DOI ] [ arXiv ]
Abstract: We propose a general numerical approach to open quantum systems with a coupling to bath degrees of freedom. The technique combines the methodology of polynomial expansions of spectral functions with the sparse grid concept from interpolation theory. Thereby we construct a Hilbert space of moderate dimension to represent the bath degrees of freedom, which allows us to perform highly accurate and efficient calculations of static, spectral, and dynamic quantities using standard exact diagonalization algorithms. The strength of the approach is demonstrated for the phase transition, critical behavior, and dissipative spin dynamics in the spin-boson model.
- A. Alvermann, H. Fehske, S. A. Trugman, Solution of the Holstein polaron anisotropy problem, Phys. Rev. B 78, 165106 (2008) [ DOI ] [ arXiv ]
Abstract: We study Holstein polarons in three-dimensional anisotropic materials. Using a variational exact diagonalization technique we provide highly accurate results for the polaron mass and polaron radius. With these data we discuss the differences between polaron formation in dimensions one and three and at small and large phonon frequencies. Varying the anisotropy we demonstrate how a polaron evolves from a one-dimensional to a three-dimensional quasiparticle. We thereby resolve the issue of polaron stability in quasi-one-dimensional substances and clarify to what extent such polarons can be described as one-dimensional objects. We finally show that even the local Holstein interaction leads to an enhancement of anisotropy in charge-carrier motion.
- G. Wellein, H. Fehske, A. Alvermann, D. M. Edwards, Correlation-induced metal insulator transition in a two-channel fermion-boson model, Phys. Rev. Lett. 101, 136402 (2008) [ DOI ] [ arXiv ]
Abstract: We investigate charge transport within some background medium by means of an effective lattice model with a novel form of fermion-boson coupling. The bosons describe fluctuations of a correlated background. By analyzing ground state and spectral properties of this transport model, we show how a metal-insulator quantum phase transition can occur for the half-filled band case. We discuss the evolution of a mass-asymmetric band structure in the insulating phase and establish connections to the Mott and Peierls transition scenarios.
- A. Alvermann, H. Fehske, Chebyshev approach to quantum systems coupled to a bath, Phys. Rev. B 77, 045125 (2008) [ DOI ] [ arXiv ]
Abstract: We propose an alternative concept for the dynamics of a quantum bath, the Chebyshev space, and a method based on this concept, the Chebyshev space method. The Chebyshev space is an abstract vector space that exactly represents the fermionic or bosonic bath degrees of freedom, without a discretization of the bath density of states. Relying on Chebyshev expansions, the Chebyshev space representation of a bath has very favorable properties with respect to extremely precise and efficient calculations of ground state properties, static and dynamical correlations, and time evolution for a great variety of quantum systems. The aim of the present work is to introduce the Chebyshev space in detail and to demonstrate the capabilities of the Chebyshev space method. Although the central idea is derived in full generality, the focus is on model systems coupled to fermionic baths. In particular, we address quantum impurity problems, such as an impurity in a host or a bosonic impurity with a static barrier, and the motion of a wave packet on a chain coupled to leads. For the bosonic impurity, the phase transition from a delocalized electron to a localized polaron in arbitrary dimension is detected. For the wave packet on a chain, we show how the Chebyshev space method implements different boundary conditions, including transparent boundary conditions replacing infinite leads. Furthermore, the self-consistent solution of the Holstein model in infinite dimension is calculated. With the examples, we demonstrate how highly accurate results for system energies, correlation and spectral functions, and time dependence of observables are obtained with modest computational effort.
- J. Loos, M. Hohenadler, A. Alvermann, H. Fehske, Optical conductivity of polaronic charge carriers, J. Phys. Condens. Matter 19, 236233 (2007) [ DOI ] [ arXiv ]
Abstract: The optical conductivity of charge carriers coupled to quantum phonons is studied in the framework of the one-dimensional spinless Holstein model. For one electron, variational diagonalization yields exact results in the thermodynamic limit, whereas at finite carrier density analytical approximations based on previous work on single-particle spectral functions are obtained. Particular emphasis is put on deviations from weak-coupling, small-polaron or one-electron theories occurring at intermediate coupling and/or finite carrier density. The analytical results are in surprisingly good agreement with exact data, and exhibit the characteristic polaronic excitations observed in experiments on manganites.
- A. Alvermann, D. M. Edwards, H. Fehske, Boson-controlled quantum transport, Phys. Rev. Lett. 98, 056602 (2007) [ DOI ] [ arXiv ]
Abstract: We study the interplay of collective dynamics and damping in the presence of correlations and bosonic fluctuations within the framework of a newly proposed model, which captures the principal transport mechanisms that apply to a variety of physical systems. We establish close connections to the transport of lattice and spin polarons, or the dynamics of a particle coupled to a bath. We analyze the model by exactly calculating the optical conductivity, Drude weight, spectral functions, ground state dispersion and particleboson correlation functions for a 1D infinite system.
- J. Loos, M. Hohenadler, A. Alvermann, H. Fehske, Phonon spectral function of the Holstein polaron, J. Phys. Condens. Matter 18, 7299 (2006) [ DOI ] [ arXiv ]
Abstract: The phonon spectral function of the one-dimensional Holsteinmodel is obtained within weak-coupling and strong-coupling approximations based on analytical self-energy calculations. The characteristic excitations found in the limit of small charge-carrier density are related to the known ( electronic) spectral properties of Holstein polarons such as the polaron band dispersion. Particular emphasis is laid on the different physics occurring in the adiabatic and antiadiabatic regimes, respectively. Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit. This method, similar to cluster perturbation theory, confirms the analytical findings, and also yields accurate results in the intermediate-coupling regime.
- M. Hohenadler, G. Wellein, A. R. Bishop, A. Alvermann, H. Fehske, Spectral signatures of the Luttinger liquid to the charge-density-wave transition, Phys. Rev. B 73, 245120 (2006) [ DOI ] [ arXiv ]
Abstract: Electron- and phonon spectral functions of the one-dimensional, spinless-fermion Holstein model at half filling are calculated in the four distinct regimes of the phase diagram, corresponding to an attractive or repulsive Luttinger liquid at weak electron-phonon coupling, and a band- or polaronic insulator at strong coupling. The results obtained by means of kernel polynomial and systematic cluster approaches reveal substantially different physics in these regimes and further indicate that the size of the phonon frequency significantly affects the nature of the quantum Peierls phase transition, the latter being either of the soft-mode or central-peak type. The generic features observed are relevant to several classes of low-dimensional materials.
- M. Hohenadler, G. Wellein, A. Alvermann, H. Fehske, Many-polaron problem by cluster perturbation theory, Physica B 378-80, 64 (2006) [ DOI ] [ arXiv ]
Abstract: The carrier-density dependence of the photoemission spectrum of the Holstein many-polaron model is studied using cluster perturbation theory combined with an improved cluster diagonalization by Chebychev expansion.
- A. Alvermann, H. Fehske, Stochastic Green's function approach to disordered systems, J. Phys.: Conf. Ser. 35, 145 (2006) [ DOI ] [ arXiv ]
Abstract: Based on distributions of local Green's functions we present a stochastic approach to disordered systems. Specifically we address Anderson localisation and cluster effects in binary alloys. Taking Anderson localisation of Holstein polarons as an example we discuss how this stochastic approach can be used for the investigation of interacting disordered systems.
- A. Weiße, G. Wellein, A. Alvermann, H. Fehske, The kernel polynomial method, Rev. Mod. Phys. 78, 275 (2006) [ DOI ] [ arXiv ]
Abstract: Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed-matter physics. In this paper basic properties and recent developments of Chebyshev expansion based algorithms and the kernel polynomial method are reviewed. Characterized by a resource consumption that scales linearly with the problem dimension these methods enjoyed growing popularity over the last decade and found broad application not only in physics. Representative examples from the fields of disordered systems, strongly correlated electrons, electron-phonon interaction, and quantum spin systems are discussed in detail. In addition, an illustration on how the kernel polynomial method is successfully embedded into other numerical techniques, such as cluster perturbation theory or Monte Carlo simulation, is provided.
- A. Alvermann, H. Fehske, Local distribution approach to disordered binary alloys, Eur. Phys. J. B 48, 295 (2005) [ DOI ] [ arXiv ]
Abstract: We study the electronic structure of the binary alloy and (quantum) percolation model. Our study is based on a self-consistent scheme for the distribution of local Green functions. We obtain detailed results for the density of states, from which the phase diagram of the binary alloy model is constructed, and discuss the existence of a quantum percolation threshold.
- G. Schubert, G. Wellein, A. Weiße, A. Alvermann, H. Fehske, Optical absorption and activated transport in polaronic systems, Phys. Rev. B 72, 104304 (2005) [ DOI ] [ arXiv ]
Abstract: We present exact results for the optical response in the one-dimensional Holstein model. In particular, by means of a refined kernel polynomial method, we calculate the ac and dc electrical conductivities at finite temperatures for a wide parameter range of electron-phonon interaction. We analyze the deviations from the results of standard small polaron theory in the intermediate coupling regime and discuss nonadiabaticity effects in detail.
- A. Alvermann, G. Schubert, A. Weiße, F. X. Bronold, H. Fehske, Characterisation of Anderson localisation using distributions, Physica B 359, 789 (2005) [ DOI ] [ arXiv ]
Abstract: We examine the use of distributions in numerical treatments of Anderson localisation and supply evidence that treating exponential localisation on Bethe lattices recovers the overall picture known from a cubic lattice in 3d.
- F. X. Bronold, A. Alvermann, H. Fehske, Anderson localization in strongly coupled disordered electron-phonon systems, Philos. Mag. 84, 673 (2004) [ DOI ] [ arXiv ]
Abstract: Based on the statistical dynamic mean-field theory, we investigate, in a generic model for a strongly coupled disordered electron-phonon system, the competition between polaron formation and Anderson localization. The statistical dynamic mean-field approximation maps the lattice problem to an ensemble of self-consistently embedded impurity problems. It is a probabilistic approach, focusing on the distribution instead of the average values for observables of interest. We solve the self-consistent equations of the theory with a Monte Carlo sampling technique, representing distributions for random variables by random samples, and discuss various ways to determine mobility edges from the random sample for the local Green function. Specifically, we give, as a function of the `polaron parameters', such as adiabaticity and electron-phonon coupling constants, a detailed discussion of the localization properties of a single polaron, using a bare electron as a reference system.
- A. Alvermann, F. X. Bronold, H. Fehske, Electron transport in the Anderson model, Phys. Status Solidi C 1, 63 (2004) [ DOI ] [ arXiv ]
Abstract: Based on a selfconsistent theory of localization we study the electron transport properties of a disordered system in the framework of the Anderson model on a Bethe lattice. In the calculation of the dc conductivity we separately discuss the two contributions to the current-current correlation function dominating its behaviour for small and large disorder. The resulting conductivity abruptly vanishes at a critical disorder strength marking the localization transition.