arXiv subjects

Katja Lindenberg

Publications and source records attributed to Katja Lindenberg.

100 records · Page 6Linked to original sources

arXiv1999

On the Generalized Kramers Problem with Oscillatory Memory Friction

The time-dependent transmission coefficient for the Kramers problem exhibits different behaviors in different parameter regimes. In the high friction regime it decays monotonically ("non-adiabatic"), and in the low friction regime it decays in an oscillatory fashion ("energy-diffusion-limited"). The generalized Kramers problem with an exponential memory friction exhibits an additional oscillatory behavior in the high friction regime ("caging"). In this paper we consider an oscillatory memory kernel, which can be associated with a model in which the reaction coordinate is linearly coupled to a nonreactive coordinate, which is in turn coupled to a heat bath. We recover the non-adiabatic and energy-diffusion-limited behaviors of the transmission coefficient in appropriate parameter regimes, and find that caging is not observed with an oscillatory memory kernel. Most interestingly, we identify a new regime in which the time-dependent transmission coefficient decays via a series of rather sharp steps followed by plateaus ("stair-like"). We explain this regime and its dependence on the various parameters of the system.

cond-mat.stat-mech↗

arXiv1998

The breakdown of adiabatic polaron theory in one, two, and three dimensions and the reformation of the large polaron concept

We prove variationally that at weak coupling in one, two, and three dimensions there exist correlated electron-phonon states below the approximate ground states characteristically found by adiabatic polaron theory. Besides differing non-trivially in quantitative aspects such as the value of the ground state energy, these improved ground states are found to differ significantly in qualitative aspects such as correlation structure and scaling behavior. These differences are sufficiently severe as to require a reevaluation of the physical meaning attached to such widely used terms as "large polaron" and "self-trapping transition".

cond-mat.str-el↗

arXiv1998

On the Generalized Kramers Problem with Exponential Memory Friction

The time-dependent transmission coefficient for the generalized Kramers problem with exponential memory friction has recently been calculated by Kohen and Tannor [D. Kohen and D. J. Tannor, J. Chem. Phys. Vol. 103, 6013 (1995)] using a procedure based on the method of reactive flux and the phase space distribution function. Their analysis is restricted to the high friction regime or diffusion-limited regime. We recently developed a complementary theory for the low-friction energy-diffusion-limited regime in the Markovian limit [Sancho et al., cond-mat/9806001, to appear in J. Chem. Phys.]. Here we generalize our method to the case of an exponential dissipative memory kernel. We test our results, as well as those of Kohen and Tannor, against numerical simulations.

cond-mat.stat-mech↗

arXiv1998

Polaron Effective Mass, Band Distortion, and Self-Trapping in the Holstein Molecular Crystal Model

We present polaron effective masses and selected polaron band structures of the Holstein molecular crystal model in 1-D as computed by the Global-Local variational method over a wide range of parameters. These results are augmented and supported by leading orders of both weak- and strong-coupling perturbation theory. The description of the polaron effective mass and polaron band distortion that emerges from this work is comprehensive, spanning weak, intermediate, and strong electron-phonon coupling, and non-adiabatic, weakly adiabatic, and strongly adiabatic regimes. Using the effective mass as the primary criterion, the self-trapping transition is precisely defined and located. Using related band-shape criteria at the Brillouin zone edge, the onset of band narrowing is also precisely defined and located. These two lines divide the polaron parameter space into three regimes of distinct polaron structure, essentially constituting a polaron phase diagram. Though the self-trapping transition is thusly shown to be a broad and smooth phenomenon at finite parameter values, consistency with notion of self-trapping as a critical phenomenon in the adiabatic limit is demonstrated. Generalizations to higher dimensions are considered, and resolutions of apparent conflicts with well-known expectations of adiabatic theory are suggested.

cond-mat.str-el↗

arXiv1998

Exact weak-coupling radius of the Holstein polaron in one, two, and three dimensions

We apply weak-coupling perturbation theory to the Holstein molecular crystal model in order to compute an electron-phonon correlation function characterizing the shape and size of the polaron lattice distortion in one, two, and three dimensions. This correlation function is computed exactly to leading order in the electron-phonon coupling constant, permitting a complete description of correlations in any dimension for both isotropic and arbitrarily anisotropic cases. Using this exact result, the width of the polaron is characterized along arbitrary directions. The width of the polaron thus determined disagrees in every dimension with some well-known characterizations of polarons, signalling in particular the breakdown of the adiabatic approximation and the characterizations of self-trapping associated with it.

cond-mat.str-el↗

arXiv1998

Solving the Holstein molecular crystal model

We present selected results for the Holstein molecular crystal model in one space dimension as determined by the Global-Local variational method, including complete polaron energy bands, ground state energies, and effective masses. We juxtapose our results with specific comparable results of numerous other methodologies of current interest, including quantum Monte Carlo, cluster diagonalization, dynamical mean field theory, density matrix renormalization group, semiclassical analysis, weak-coupling perturbation theory, and strong-coupling perturbation theory. This comparison confirms the Global-Local variational method as being highly accurate over a wide range of the polaron parameter space, from the non-adiabatic limit to the extremes of high adiabaticity, from weak coupling through intermediate coupling to strong coupling.

cond-mat↗

arXiv1998

Electronic Kinetic Energy in the Polaron Ground State and the Location of the Self-Trapping Transition

In this paper we discuss the electronic kinetic energy in the ground state of the Holstein Hamiltonian in one space dimension using the Global-Local variational method together with perturbation theory at weak and strong coupling. The electronic kinetic energy is detailed over a broad region of the polaron parameter space, and is analyzed to accurately determine the location of the self-trapping transition. The self-trapping line so determined separates the small polaron regime from the large polaron regime, constituting a polaron phase diagram.

cond-mat↗

arXiv1998

The Kramers Problem in the Energy-Diffusion Limited Regime

The Kramers problem in the energy-diffusion limited regime of very low friction is difficult to deal with analytically becasue of the repeated recrossings of the barrier that typically occur before an asymptotic rate constant is achieved. Thus, the transmission coefficient of particles over the potential barrier undergoes oscillatory behavior in time before settling into a steady state. Recently Kohen and Tannor (JCP Vol. 103, Pg. 6013, 1995) developed a method based on the phase space distribution function to calculate the transmission coefficient as a function of time in the high-friction regime. Here we formulate a parallel method for the low-friction regime. We find analytic results for the full time and temperature dependence of the rate coefficient in this regime. Our low-friction result at long times reproduces the equilibrium result of Kramers at very low friction and extends it to higher friction and lower temperatures below the turn-over region. Our results indicate that the single most important quantity in determining the entire time evolution of the transmission coefficient is the rate of energy loss of a particle that starts above the barrier. We test our resutls, as well as those of Kohen and Tannor for the Kramers problem, against detailed numerical simulations.

cond-mat.stat-mech↗

arXiv1997

Properties of resonant activation phenomena

The phenomenon of resonant activation of a Brownian particle over a fluctuating barrier is revisited. We discuss the important distinctions between barriers that can fluctuate among "up" and "down" configurations, and barriers that are always "up" but that can fluctuate among different heights. A resonance as a function of the barrier fluctuation rate is found in both cases, but the nature and physical description of these resonances is quite distinct. The nature of the resonances, the physical basis for the resonant behavior, and the importance of boundary conditions are discussed in some detail. We obtain analytic expressions for the escape time over the barrier that explicitly capture the minima as a function of the barrier fluctuation rate, and show that our analytic results are in excellent agreement with numerical results.

physics.chem-ph↗

arXiv1996

A Variational Approach to Nonlocal Exciton-Phonon Coupling

In this paper we apply variational energy band theory to a form of the Holstein Hamiltonian in which the influence of lattice vibrations (optical phonons) on both local site energies (local coupling) and transfers of electronic excitations between neighboring sites (nonlocal coupling) is taken into account. A flexible spanning set of orthonormal eigenfunctions of the joint exciton-phonon crystal momentum is used to arrive at a variational estimate (bound) of the ground state energy for every value of the joint crystal momentum, yielding a variational estimate of the lowest polaron energy band across the entire Brillouin zone, as well as the complete set of polaron Bloch functions associated with this band. The variation is implemented numerically, avoiding restrictive assumptions that have limited the scope of previous assaults on the same and similar problems. Polaron energy bands and the structure of the associated Bloch states are studied at general points in the three-dimensional parameter space of the model Hamiltonian (electronic tunneling, local coupling, nonlocal coupling), though our principal emphasis lay in under-studied area of nonlocal coupling and its interplay with electronic tunneling; a phase diagram summarizing the latter is presented. The common notion of a "self-trapping transition" is addressed and generalized.

cond-mat↗