Carriers such as electrons and holes inside the Brillouin zone of complex semiconducting materials can form bound states (excitons, biexcitons, etc.). For obtaining the corresponding eigenstates (e.g., through the Wannier or Bethe-Salpeter equation) and dynamics (e.g., cluster expansion) the number of involved electrons and holes as well as the accuracy is limited by the appearing high-dimensional tensors (i.e., wave functions or correlations). These tensors can be efficiently represented and manipulated via tensor network methods. We show how tensor networks formulated via classic logic gates can be used to treat electron-hole complexes inside the Brillouin zone. The method is illustrated for the exciton and biexciton states of a single-layer transition-metal dichalcogenide ${\mathrm{MoS}}_{2}$-like model system.
Carriers such as electrons and holes inside the Brillouin zone of complex semiconducting materials can form bound states (excitons, biexcitons, etc.). For obtaining the corresponding eigenstates (e.g., through the Wannier or Bethe-Salpeter equation) and dynamics (e.g., cluster expansion) the number of involved electrons and holes as well as the accuracy is limited by the appearing high-dimensional tensors (i.e., wave functions or correlations). These tensors can be efficiently represented and manipulated via tensor network methods. We show how tensor networks formulated via classic logic gates can be used to treat electron-hole complexes inside the Brillouin zone. The method is illustrated for the exciton and biexciton states of a single-layer transition-metal dichalcogenide MoS2-like model system.
Recently, in Kuhn and Richter [Phys. Rev. B 99, 241301(R) (2019)], tensor networks built on logical circuits were briefly introduced to retrieve exciton and biexciton states. Compared to a conventional approach the tensor network methods scales logarithmic instead of linear in the grid points of the Brioullin zone and linear instead of exponential in the number of electrons and holes. This enables calculations with higher precision on the full Brioullin zone than previously possible. In this paper extensive details for an efficient implementation and the corresponding mathematical background are presented. In particular, this includes applications and results for excitons, trions, and biexcitons (for monolayer MoS2 as an example), going beyond the initial brief introduction. Furthermore strategies for calculating selective excited bound states and tests of common approximations are discussed making use of the high-accuracy full Brioullin zone treatment of the tensor network method.
In semiconductor nanostructures, optical excitation typically creates bound electron-hole states, such as excitons, trions, and larger complexes. Their relative motion is described by the Wannier equation, which is valid only for spatially extended motion in the Coulomb-dominated, weak-confinement limit. Other small nanostructures, such as quantum dots, are in the confinement-dominated strong confinement regime, where the wavefunction factorizes into independent electron and hole parts. Nanoplatelets are in between the two regimes and require solving an unfactorized higher-dimensional Schrödinger equation, which is computationally expensive. This work demonstrates how tensor networks can partially overcome this problem, using CdSe nanoplatelets as an example. The method is also applicable to related two-dimensional systems. As a demonstration, we calculate the excitonic and trionic ground states, as well as several excited states, for nanoplatelets of varying sizes, including their energies and oscillator strengths. More importantly, overall strategies for using tensor networks in real space for systems under intermediate confinement have been developed.
Computing excitonic spectra in quasicrystal and super-moiré systems constitutes a formidable challenge due to the exceptional size of the excitonic Hilbert space. Here, we demonstrate a tensor-network method for the real-space Bethe-Salpeter Hamiltonian, allowing us to access the spectra of an excitonic 1018-dimensional Hamiltonian, and enabling the direct computation of bound-exciton spectral functions for systems exceeding one billion lattice sites, several orders of magnitude beyond the capabilities of conventional approaches. Our method combines a tensor-network encoding of the real-space Bethe-Salpeter Hamiltonian with a Chebyshev tensor network algorithm. This strategy bypasses explicit storage of the Hamiltonian while preserving full real-space resolution across widely different length scales. We demonstrate our methodology for one- and two-dimensional super-moiré systems, achieving the simultaneous resolution of atomistic and mesoscopic structures in the excitonic spectra in billion-size systems, showing exciton miniband formation and moiré-induced spatial confinement. Our results establish a real-space methodology enabling the simulation of excitonic physics in large-scale quasicrystal and super-moiré quantum matter.
We study low-dimensional quantum systems with analytical and computational methods. Firstly, the one-dimensional extended t-V model of fermions with interactions of a finite range is investigated. The model exhibits a phase transition between liquid and insulating regimes. We use various analytical approaches to generalise previous theoretical studies. We devise a strong coupling expansion to go beyond first-order perturbation theory. The method is insensitive to the presence or the lack of integrability of the system. We extract the ground state energy and critical parameters of the model near the Mott insulating commensurate density. We also study the possible charge-density-wave phases that exist when the model is at the critical density. Secondly, we investigate Mott-Wannier complexes of two (excitons), three (trions) and four (biexcitons) charge carriers in two-dimensional semiconductors. Our study also includes impurity-bound complexes. We provide a classification of trions and biexcitons in transition-metal dichalcogenides, which incorporates the difference of spin polarisation between molybdenum- and tungsten-based materials. Using the diffusion Monte Carlo method, which is statistically exact for these systems, we extract binding energies of the complexes for a complete set of parameters of the model. Our results are compared with theoretical and experimental work on transition-metal dichalcogenides. An agreement is found for excitonic and trionic results, but we also observe a large discrepancy in the theoretical biexcitonic binding energies as compared to the experimental values. Possible reasons for this are outlined. We also calculate contact pair densities, which in the future can be used in the determination of the contact interaction.
The first-principles quantum simulation of the full electronic Hamiltonian of a material model of realistic size appears daunting even in a future era of fully fault-tolerant quantum computers. In contrast, classical simulations of material excited states rely on heuristic quasiparticle frameworks, such as the GW approximation and Bethe-Salpeter equation, where only a few quasiparticles are explicitly considered. Here we give a detailed description of quasiparticle quantum simulation for multi-excitonic states within the BSE. We develop an explicit first-quantized, orbital-based block encoding of the multi-exciton BSE Hamiltonian for qubitization, and reduce its cost using integral factorization and crystal symmetries. For a materials model of size N, and a fixed number of quasiparticles, the resulting algorithms require O(N) Toffoli gates and O(logN) logical qubits, or, using space-time tradeoffs, O(N) Toffoli gates and O(N) logical qubits, achieving up to a sixth power reduction in Toffoli count, or up to exponential reduction in qubit count, versus standard full Hamiltonian simulation techniques that have been explored in materials simulation. Resource estimates for the classically challenging problem of triexcitonic eigenvalue estimation across a range of semiconductor models of a realistic physical size of O(103) atoms, show that our Toffoli-optimized implementation reduces the Toffoli count by 106−107 relative to full Hamiltonian simulation estimates, while a qubit-optimized implementation requires only a few hundred logical qubits. More broadly, these results show that classical heuristic frameworks will be essential in the design of future fault-tolerant quantum algorithms for materials.
The fermionic many-body problem in the strong correlation regime is notoriously difficult to tackle. In a previous work (Phys. Rev. B 101, 045109 (2020)), we have proposed to extend the single-reference coupled-cluster (SRCC) method to the strong correlation regime using low-rank tensor decompositions (LRTD) to express the cluster operator, without truncating it with respect to the number of excitations. For that purpose, we have proposed a new type of LRTD called ``superpositions of tree-tensor networks'' (STTN), which use the same set of building blocs to define all the tensors involved in the CC equations, and combine different ``channels'', i.e. different types of pairing among excited particles and holes, in the decomposition of a given tensor. Those two principles are aimed at globally minimizing the total number of free parameters required to accurately represent the ground state. In this work, we show that STTN can indeed be compact and accurate representations of strongly correlated ground states by using them to express the CC cluster operator amplitudes and wave function coefficients of exact grounds states of small two-dimensional Hubbard clusters, at half-filling, up to three particle-hole excitations. We show the compactness of STTN by using a number of free parameters smaller than the number of equations in the CCSD approximation, i.e. much smaller than the number of fitted tensor elements. We find that, for the systems considered, the STTN are more accurate as the size of the system increases and that combining different channels in the decompositions of the most strongly correlated tensors is crucial to obtain good accuracy.