Real-space spectral functions of three-dimensional billion-size topological non-Hermitian matter with tensor networks

Abstract: Non-Hermitian systems host a wide range of unconventional topological phenomena while large-scale simulations in finite three dimensional systems remain challenging because of the rapidly growing number of sites. In particular, higher-order topological corner modes are often studied only in small lattices, where strong finite-size effects can mask their intrinsic behavior. Here, we develop a tensor-network framework that combines quantics tensor cross interpolation with the kernel polynomial method, enabling compact representations of large non-Hermitian tight-binding Hamiltonians and direct calculations of real-space spectral functions for systems exceeding one billion lattice sites. Using this approach, we investigate three-dimensional non-Hermitian higher-order topological insulators with with structured real-space geometries. The unprecedented system size enables direct access to the macroscopic regime and allows corner-mode spectral responses to be resolved in genuinely three-dimensional systems. By tuning the loss strength, we identify distinct in-gap corner modes across weak- and strong-loss regimes. Our results establish tensor-network algorithms as a powerful strategy to perform real-space spectral calculations in exceptionally large non-Hermitian systems.

• The paper introduces an efficient tensor-network algorithm that combines QTCI with an NHKPM to compute real-space spectral functions in billion-site non-Hermitian systems.
• It uses MPO representations and Chebyshev polynomial expansions to accurately extract local density of states and resolve in-gap corner-mode splitting in 3D HOTIs.
• The work demonstrates scalable, finite-size independent techniques with broad implications for studying topological phenomena in open quantum systems.

Real-Space Spectral Functions in 3D Billion-Size Topological Non-Hermitian Systems via Tensor Networks

Introduction

The study addresses computational characterization of spectral functions in three-dimensional non-Hermitian quantum systems at the macroscopic scale. Non-Hermitian lattices, essential for open quantum models with gain/loss or dissipative dynamics, exhibit phenomena distinct from their Hermitian counterparts, including atypical boundary states, exceptional points, and the non-Hermitian skin effect (NHSE). However, direct numerical access to their real-space spectral features in large-scale geometries is sharply constrained by explosive Hilbert space growth and numerical instability inherent to non-Hermiticity.

The presented work introduces a tensor-network-based algorithm that synergizes Quantics Tensor Cross Interpolation (QTCI) with a non-Hermitian kernel polynomial method (NHKPM), enabling the computation of real-space spectral functions for tight-binding non-Hermitian Hamiltonians exceeding $10^9$ lattice sites. As a demonstration, the framework is applied to three-dimensional higher-order topological insulators (HOTIs) with tunable non-Hermitian loss, resolving in-gap corner-mode responses inaccessible by conventional approaches.

Tensor-Network Formalism and Non-Hermitian KPM

A key technical development is the tensorization of non-Hermitian tight-binding Hamiltonians using efficient QTCI-based matrix product operator (MPO) representations. Lattice site indices are mapped to binary pseudo-spin variables, organizing the Hilbert space as that of $L$ effective two-level systems for system size $N = 2^L$. This enables representation of the Hamiltonian and arbitrary real-space operators with controlled MPO bond dimension, leveraging the inherent data sparsity and local connectivity of tight-binding models. Both on-site complex potentials and hopping matrices are encoded with compact MPOs, and QTCI is used to optimize over spatially structured inhomogeneities.

To compute local or global spectral functions, the method employs an NHKPM for non-Hermitian operators. The central object is the real-space spectral function

$$f(\omega) = \langle \psi_L | \delta^2(\omega - H) | \psi_R \rangle,$$

which is recast using a Hermitized auxiliary system $\tilde H(\omega)$ to circumvent the complex eigenspectrum's sensitivity and to allow for stable Chebyshev polynomial recursion. The delta function is expanded in this space, and spectral weight extraction reduces to evaluating expectation values with respect to MPOs that are amenable to contraction with matrix product state (MPS) vectors. This procedure leverages scalable tensor contractions and avoids explicit diagonalization, which is infeasible at billion-site scales.

Benchmarking and Application to 3D Non-Hermitian HOTIs

Benchmarking against exact diagonalization for modest system sizes ($16^3$), the approach demonstrates numerical fidelity in resolving spectral features and associated local density of states (LDOS) patterns.

For the principal application, the method is deployed on a 3D HOTI lattice model with $1024^3$ ($\sim 10^9$) sites possessing a non-Hermitian loss pattern designed to spectrally segregate corner states. The Hermitian part implements a BBH-type model—realizing a second-order topology—while the anti-Hermitian term creates spatially patterned loss with tunable strength $\lambda$. In the weak-loss regime, the corner DOS reveals a single well-separated in-gap mode, with spatial LDOS localization manifesting tightly at the geometrical corner. As loss strength increases, the corner spectral sector splits into multiple distinct in-gap modes along the complex-energy axis, and their hierarchy and spatial LDOS manifestation coincide with finite-size exact results. The calculations robustly extract these features even in the presence of a massively gapped bulk continuum.

A notable outcome is the ability to resolve the fine structure of higher-order corner-mode manifolds, including their energy separation and spatial patterning, with high resolution in complex-energy space and spatial locality.

Theoretical and Practical Implications

The algorithmic architecture generalizes directly to arbitrary non-Hermitian lattice geometries, including quasi-periodic, moiré, or systems with spatially correlated disorder, contingent only upon an MPO compression of the Hamiltonian. The bottleneck is set by the bond dimension required for the MPO and the required Chebyshev expansion order for spectral resolution, both of which are tunable. This approach, therefore, evades the exponential cost of direct diagonalization and sparse-matrix methods for large open quantum systems.

From the standpoint of non-Hermitian topological matter, the ability to resolve real-space spectral properties in truly macroscopic 3D geometries addresses questions of finite-size scaling, boundary mode hybridization, and the fate of higher-order topology in the presence of gain/loss, without artifacts of small system size. The method enables studies of NHSE, topological phase transitions, and corner mode evolution in complex lattice geometries with physical relevance to large-scale photonic and phononic platforms where non-Hermitian topology has been pursued.

The presented claims are strongly supported by numerics: accurate extraction of spectral weights and LDOS for $\sim 1$ billion sites, the resolution of corner-sector splitting under strong loss, and spatial mapping of localized states in 3D. This considerably surpasses previous tensor network treatments, which have been limited either to Hermitian settings or sub-macroscopic scale non-Hermitian models.

Future Directions

Prospective research may extend the formalism to interacting or driven-dissipative non-Hermitian systems, disordered topological phases, and momentum-resolved spectroscopies via generalized MPO contractions. The approach is also directly transferable to quantum simulation and quantum chemistry settings, leveraging rapidly advancing tensor network toolkits. Finally, given the intrinsic scalability, one can anticipate systematic finite-size scaling studies, disorder averaging, and exploration of critical phenomena near non-Hermitian phase transitions in real-space lattices.

Conclusion

A tensor network framework based on QTCI-compressed MPO Hamiltonians and non-Hermitian KPM has been formulated for the computation of real-space spectral functions in three-dimensional non-Hermitian systems at the billion-site scale (2606.16424). This enables investigation of finite-size-independent topological behavior, such as corner modes in non-Hermitian HOTIs, providing access to parameter regimes that far exceed the capabilities of traditional numerical techniques. The broader utility encompasses a wide class of open quantum lattice platforms, marking the framework as a centerpiece for non-Hermitian quantum matter studies at macroscopic scales.