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arXiv:2603.25439 · Physical review. B./Physical review. B

Tensor network methods for bound electron-hole complexes beyond strong and weak confinement in nanoplatelets

Bruno Hausmann, Marten Richter

cond-mat.mes-hallquant-phSubmitted Mar 26, 2026• Announced Mar 27, 2026• 0 citations
v1 · Mar 26, 2026v2 · Jul 14, 2026
arXivHTMLPDF

Abstract

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.

DOI
10.1103/mmwh-bjpn
Comments
12 pages, 8 figures, 8 tables

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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.
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{\mathrm{MoS}}_{2}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.
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 101810^{18}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.
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{\mathrm{MoS}}_{2}MoS2​-like model system.
Quantum dot structures confine electrons in a small region of space. Some properties of semiconductor quantum dots, such as the discrete energy levels and shell filling effects visible in addition spectra, have analogies to those of atoms and indeed dots are sometimes referred to as 'artificial atoms'. However, atoms and dots show some fundamental differences due to electron correlations. For real atoms, the kinetic energy of electrons dominates over their mutual Coulomb repulsion energy and for this reason the independent electron approximation works well. For quantum dots the confining potential may be shallower than that of real atoms leading to lower electron densities and a dominance of mutual Coulomb repulsion over kinetic energy. In this strongly correlated regime the independent electron picture leads to qualitatively incorrect results. This thesis concentrates on few-electron quantum dots in the strongly correlated regime both for quasi-one-dimensional and two-dimensional dots in a square confining potential. In this so-called 'Wigner' regime the ground-state electronic charge density is localised near positions of classical electrostatic minima and the interacting electronic spectrum consists of well separated spin multiplets. In the strongly correlated regime the structure of low-energy multiplets is explained by mapping onto lattice models with extended-Hubbard and Heisenberg effective Hamiltonians. The parameters for these effective models are calculated within a Hartree approximation and are shown to reproduce well the exact results obtained by numerical diagonalisation of the full interacting Hamiltonian. Comparison is made between square dots and quantum rings with full rotational symmetry. In the very low-density regime, direct diagonalisation becomes impractical due to excessive computer time for convergence. In this regime a numerical renormalisation group method is applied to one-dimensional dots, enabling effective spin-interactions to be pr
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.
Strongly correlated electrons have long been a central subject of condensed matter physics. In this thesis, we approach them through a real-space perspective: electrons defined in real space in two-dimensional crystals underpin the two themes developed here. First, the local, real-space description makes quantum states constructible. When a free-fermion ground state is spanned by exponentially localized Wannier functions, its tensor-network representation can be constructed directly. We present a stacked-tree algorithm that builds projected entangled pair states for such systems without variational optimization, and demonstrate it on obstructed atomic insulators in one and two dimensions. Second, the compact local orbitals serve as a route to strong cor-relation, because quenching the kinetic energy makes the interaction effectively strong. The kagome lattice reaches this limit by geometry. Destructive interference confines electrons to single hexagons, and the band spanned by these localized states is flat, so the interactions can no longer be neglected. In collaboration with scanning tunneling microscopy experiments on Fe-doped CoSn, we identify the inter-band nematic order parameter and the orbital-selective Mott state realized in its partially filled flat bands. We further map the competition among ferromagnetism, nematicity, and Fermi-surface-nesting-driven orders using self-consistent Hartree–Fock theory, and find nematicity to be the dominant phase across a wide range of fillings, consistent with the experimental observations.