Momentum-resolved spectral functions of super-moiré systems using tensor networks

TL;DR

The paper tackles the computational barrier to momentum-resolved spectral functions in giant, non-periodic super-moiré systems. It introduces a tensor-network workflow that encodes tight-binding Hamiltonians as MPOs, applies a tensor-network quantum Fourier transform to access momentum space, and handles interactions via a self-consistent mean-field loop. The method is demonstrated in 1D and 2D non-periodic settings, including inhomogeneous strain and a quasicrystalline modulation, with the ability to compute region-restricted spectra A_P(R,k,ω) to image local band structure and minigaps. This provides a scalable tool for modeling momentum-resolved observables in QTM-inspired experiments on twisted van der Waals heterostructures.

Abstract

Computing spectral functions in large, non-periodic super-moiré systems remains an open problem due to the exceptionally large system size that must be considered. Here, we establish a tensor network methodology that allows computing momentum-resolved spectral functions of non-interacting and interacting super-moiré systems at an atomistic level. Our methodology relies on encoding an exponentially large tight-binding problem as an auxiliary quantum many-body problem, solved with a many-body kernel polynomial tensor network algorithm combined with a quantum Fourier transform tensor network. We demonstrate the method for one and two-dimensional super-moiré systems, including super-moiré with non-uniform strain, interactions treated at the mean-field level, and quasicrystalline super-moiré patterns. Furthermore, we demonstrate that our methodology allows us to compute momentum-resolved spectral functions restricted to selected regions of a super-moiré, enabling direct imaging of position-dependent electronic structure and minigaps in super-moiré systems with non-uniform strain. Our results establish a powerful methodology to compute momentum-resolved spectral functions in exceptionally large super-moiré systems, providing a tool to directly model scanning twisting microscope tunneling experiments in twisted van der Waals heterostructures.

Figures (3)

• Figure 1: (a) Schematic of the mapping between a single particle problem with $N=2^L=8$ sites, and a many-body pseudo-spin chain of length $L=3$. While a sparse $2^L\times2^L$ matrix represents the Hamiltonian in the real-space representation, it is represented by an $L$-site MPO in the tensor-network representation. (b) Tensor network algorithm to compute the momentum-resolved spectral function. The purple MPSs at the top and bottom of this network represent momentum basis states $\ket{\mathbf{k}}$. The green MPOs represent the Quantum Fourier Transform $\hat{\mathcal{F}}$ and its inverse $\hat{\mathcal{F}}^{-1}$, which are acting on the operator $\delta (\omega-\hat{\mathcal{H}} )$. (c) The same tensor network as in (b), but augmented with projection MPOs $\hat{\mathcal{P}}_\mathbf{R}$ to resolve the spectral functions locally. (d) The tensor-network SCF loop. Here, $\mathcal{T_{\alpha\beta}}$ represents the non-interacting part of the tensorized Hamiltonian and $\chi_{\alpha\beta}$ represents the one-body operator in tensorized form that results from iteratively computing $\langle c_\alpha^\dagger c_\beta\rangle$ in the Hubbard term in \ref{['Eq: Hubbard term']}.
• Figure 2: (a) Local density of states $\rho(x_i,\omega)$ and (b) the total momentum-space spectral function $A(k,\omega)$. (c,d) projected spectral function $A_P(k,\omega)$ for a 1D chain featuring a hopping modulation with a linearly increasing frequency and amplitude. (e) The hopping modulation shows that the amplitude, the average, the atomic-scale and the moiré wavelengths increase. In (b) and (c), the splitting of the band due to the incommensurate modulation is not as obvious as in (d). The system size is $N=2^{24}$ and the projected regions have sizes $N_X=N/16$.
• Figure 3: (a) The hopping function in real space and a zoom into the central region, where the two moiré modulations at the different scales are clearly visible. (b) The total momentum-resolved spectral function $A(\mathbf{k},\omega)$ along the line $k_x=k_y$. (c) The LDOS $\rho(\mathbf{r}.\omega=0)$ and a zoom into the same region as (a), where the spatial pattern closely follows that of the hopping modulation. (d) The projected momentum-resolved spectral function $A_\text{P}(\mathbf{R}.\mathbf{k},\omega)$ in the two regions shown in (c), with insets showing pronounced differences between them. In particular, moiré minigaps are more clearly resolved in the top inset. The system size is $N=2^{24}$, with $N_x=N_y=2^{12}$, and the projected regions have linear sizes $N_{\mathbf{R}}\approx N_x/16=N_y/16$.