Demos.ExcitonPhonon.Main

WavePacket / WaveTrain demo examples: Exciton-Phonon-Coupling

Here we study chains of exciton-bearing sites (2-level systems) with or without periodic boundary conditions, as a model for excitonic energy transport in one dimension, including coupling to (acoustic) phonons. For not too large number of particles in the chain, we strive to compare the results from the conventional numerical schemes implemented in WavePacket versus the tensor train based techniques implemented in WaveTrain.

For the case of homogeneous excitonic systems with local (on-site) excitation energies α and nearest neighbor (NN) coupling energies β, the time-independent Schrödinger equation (TISE) can be solved analytically, in close analogy to Hückel theory. Within the Fock space of singly excited states, the eigenenergies are given by

E_k=α+2βcos(ka)

where a is the lattice constant. For cyclic (periodic) systems, the (dimensionless) wavenumbers k_ja=2πj/N are restricted to discrete values −N/2+1 ≤ j ≤ N/2 (for N even) or −(N-1)/2 ≤ j ≤ (N-1)/2 (for N odd). For linear (non-periodic) systems, one has to choose k_ja = πj/(N+1) with 1≤j≤N instead.

Here we will show a few typical results for our standard parameter set with α=0.1 and β=−0.01, for details see here. Hence, for the purely excitonic system, the lower band edge is found at α−2|β|=0.08. In the absence of exciton-phonon coupling (EPC), one simply has to add the vibrational (phonon) zero point energy, here for ν=0.001 and ω=√2ν. In the following two sections, however, we will see that the EPC leads to slightly lower energies, i.e. the coupling of the two subsystems leads to a stabilization, here for the value of the EPC constant σ=2E−4.

Trimer

For the case of a homogeneous cyclic trimer (N=3) and for 16 discretization points for each of the vibrational coordinates, the set of three coupled Schrödinger equations can still be solved by direct diagonalization within half a minute. Both with WavePacket and WaveTrain, we obtain a lower band edge energy of 0.081726 for the couped system. Comparing with an excitonic energy of 0.08 and a vibrational ground state energy of 0.0025, see here, we obtain a stabilization energy of 7.74E−4 for states at the lower band edge.

WavePacket (Matlab)	WaveTrain (Python)
Animated wavepacket	Animated graphics
Input data file	Input script
Logfile output	Output file

Hexamer

For the case of a homogeneous cyclic hexamer (N=6) and for 8 discretization points for each of the vibrational coordinates, the set of six coupled Schrödinger equations cannot be solved by direct diagonalization any more. Using WavePacket, an imaginary time propagation gives an energy of 0.081750 after 2500 units of (imaginary) time. Using WaveTrain, the tensor-train based ALS (alternating linear scheme) solvers yield an energy of 0.081838, again for 8 discretization points for each vibrational degree of freedom. Comparing with an excitonic energy of 0.08 and a vibrational ground state energy of 0.005032, see here, we obtain a stabilization energy of 3.282E−3 or 3.194E−3 for states at the lower band edge, about four times stronger than for the trimer (see above).

The bar chart nicely shows the effect of mutual self-trapping of excitons and phonons: The exciton is centered around site 0 (green bars), with the sites to the right (indices 1,2) being diplaced to the left and vice versa (orange bars). This effect of self-focussing is a consequence of the positive sign of the EPC constant σ, as predicted in the framework of Davdov's soliton theory.

WavePacket (Matlab)	WaveTrain (Python)
Animated wavepacket
Input data file	Input script
Logfile output	Output file

Discussion

While the conventional, grid-based numerical schemes implemented in WavePacket were successful in finding the vibrational ground state for N=6 for the examples given here, it should be mentioned, however, that on a standard PC one cannot go beyond N=7 due to the exponential growth of the computational effort.

This is in marked contrast to the tensor-train based techniques implemented in WaveTrain which can handle chains comprising tens of sites (e.g. N=40, N=60) within CPU times not exceeding a few days. By employing efficient low-rank tensor-train decompositions, the required storage is found to increase only linearly with the chain length, and the CPU time is found to increase only slightly faster than linearly, thus mitigating the curse of dimensionality. For more details, see our publication on coupled excitons and phonons where also the effect of different values of the EPC constant σ is investigated.

Finally, the minor discrepancy of the hexamer results is likely to be solved by increasing the number of discretization points for each vibrational degree of freedom. However, this makes the computations much slower which is why we don't want to pursue that issue here any further. Note that there is also a conceptual difference between our EPC calculations using WavePacket versus WaveTrain: Within the former approach, we restrict ourselves to the manifold of N states with one exciton only. (Including also the ground state would result in imaginary time propagations to converge towards zero-exciton states which is formally correct but which is not what we want here). In contrast, the latter approach includes all excitonic states, i.e., from zero to N excitons. However, for the examples preented above, no admixtures of states outside the one-exciton manifold were found.