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

arXiv:2606.16424 · cond-mat.mes-hall, quant-ph · Submitted 2026-06-15 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Real-space spectral functions of three-dimensional billion-size topological non-Hermitian matter with tensor networks".

Mira: Here's the gist of the paper. The gist: This work establishes a tensor-network framework that enables spectral function calculations for three-dimensional non-Hermitian tight-binding systems exceeding one billion lattice sites, allowing direct access to the macroscopic regime and resolution of corner-mode spectral responses in genuinely three-dimensional systems.

Kai: System and Challenge Non-Hermitian systems host a wide range of unconventional topological phenomena while largescale simulations in finite three dimensional systems remain challenging because of the rapidly growing number of sites The primary numerical challenge stems from the complex spectra and large condition numbers of non-Hermitian operators, which often render large-scale simulations unstable and computationally expensive.

Mira: The difficulty is further intensified in three-dimensional systems, where the Hilbert space grows rapidly, and boundary effects become increasingly intricate.

Kai: The methodology combines Quantics Tensor Cross Interpolation (QTCI) based matrix product operator (MPO) representations of large real-space Hamiltonians with a non-Hermitian kernel polynomial algorithm (NHKPM).

Paper summary: Kai: To summarize this work, "Real-space spectral functions of three-dimensional billion-size topological non-Hermitian matter with tensor networks," the authors are proposing a tensor network framework that blends quantics tensor cross interpolation with the kernel polynomial method.

Mira: The main thesis is that they can create compact representations of large non-Hermitian tight-binding Hamiltonians and calculate real-space spectral functions directly for systems over one billion lattice sites.

Kai: They are specifically interested in three-dimensional non-Hermitian higherorder topological insulators, and they want to resolve their corner modes on a billion site lattice.

Mira: The claim is that this framework allows them to see the spectral response of these systems at macroscopic scales, which is where finite size effects usually obscure the behavior.

Lev: From an error correction standpoint, if this representation works as described, it suggests that we could potentially simulate the physics of these large non-Hermitian systems more reliably than traditional methods allow.

Kai: It matters because it gives us a direct look at how the spectrum behaves when you move beyond the small lattices where these corner modes are usually studied.

Mira: This is about moving past just seeing isolated features and understanding the full structure of how those features change with parameters like gain or loss, which is what they investigate by tuning them.

Conclusion: Kai: So, looking at "Real-space spectral functions of three-dimensional billion-size topological non-Hermitian matter with tensor networks," the authors Yitao Sun, Jose L. Lado, and Guangze Chen have shown a way to handle massive systems in this field.

Mira: It’s about taking complex non-Hermitian physics that usually requires tiny lattices and showing you how to calculate its spectral functions on a billion site lattice using tensor networks.

Kai: The implication is that we can now study the real, large-scale behavior of higher-order topological insulators without being limited by the constraints of small system sizes.

Mira: In simpler terms, it means we’ve built a tool that lets us see how corner modes behave in these systems when they are truly huge and interacting with gain or loss effects in a structured way.

Lev: For those of us building actual hardware, the implication is that if this framework is accurate, it suggests the computational barrier for simulating these large non-Hermitian models might be lower than we currently expect.

Kai: Exactly. It points toward a new route for studying quantum matter in regimes that are just too big or too complex for the methods we rely on today.

Yitao Sun, Jose L. Lado, Guangze Chen

Department of Applied Physics, Aalto University · Department of Microtechnology and Nanoscience, Chalmers University of Technology

cond-mat.mes-hall, quant-ph

Submitted: 2026-06-15

Updated: 2026-10-17

Comments: 17 pages, 13 figures

Code: https://github.com/YITAOSUN42/Tensor_network_NH_

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 92/100

The gist: The gist: This work establishes a tensor-network framework that enables spectral function calculations for three-dimensional non-Hermitian tight-binding systems exceeding one billion lattice sites,

Key concepts

Tensor Network Framework: This is a mathematical tool that represents very large quantum systems (like big lattices) using interconnected tensors. Instead of storing every single interaction, it compresses the system's information into a compact network structure, making calculations feasible for billions of sites.
Matrix Product Operator (MPO): An MPO is a specific way to compactly represent large real-space Hamiltonians. It encodes the system's structure by breaking it down into smaller, manageable blocks that can be efficiently multiplied together, allowing the simulation of huge lattices.
Non-Hermitian Kernel Polynomial Method (NHKPM): This is a specialized algorithm used to find the spectral properties (eigenvalues and eigenvectors) of non-Hermitian operators. It allows the framework to handle the complex spectra inherent in non-Hermitian systems while remaining computationally tractable.
Corner Modes: These are specific, localized spectral features found at the corners of a topological material's Brillouin zone. In this study, researchers used the method to precisely locate and analyze how these modes behave when the system is scaled up to billion sites.

Summary

The gist: This work establishes a tensor-network framework that enables spectral function calculations for three-dimensional non-Hermitian tight-binding systems exceeding one billion lattice sites, allowing direct access to the macroscopic regime and resolution of corner-mode spectral responses in genuinely three-dimensional systems.

System and Challenge

Non-Hermitian systems host a wide range of unconventional topological phenomena while largescale simulations in finite three dimensional systems remain challenging because of the rapidly growing number of sites The primary numerical challenge stems from the complex spectra and large condition numbers of non-Hermitian operators, which often render large-scale simulations unstable and computationally expensive. The difficulty is further intensified in three-dimensional systems, where the Hilbert space grows rapidly, and boundary effects become increasingly intricate.

Methodology for Compact Representation

The methodology combines Quantics Tensor Cross Interpolation (QTCI) based matrix product operator (MPO) representations of large real-space Hamiltonians with a non-Hermitian kernel polynomial algorithm (NHKPM).

The tight-binding Hamiltonian is first tensorized into an MPO form, where the lattice index can be encoded in binary form defining a pseudo-spin representation.
Onsite modulations are introduced by allowing the diagonal terms to be purely imaginary, specifically setting tαα = iUα, where the spatial profile Uα can be compressed with QTCI algorithm.
The hopping term is constructed by building compact shift MPOs Hup and Hdown such that Hhopping = t1Hup + t2Hdown.

Spectral Function Calculation

The spectral function f(ω) is defined as the quantity ⟨ψLδ2 (ω − H)ψR⟩, where δ2 denotes the two-dimensional delta function in the complex-energy plane. This quantity is evaluated by mapping the problem to an enlarged Hermitian Hamiltonian H˜ (ω) = 0 ωI − H ω∗ I − H† 0.

The spectral function can then be expressed through the Green’s function of H˜ as f(ω) = 1/π ∂ω∗ ⟨L(E − H˜)−1 R⟩.

To obtain the Local Density of States (LDOS) at a lattice site l, the formalism directly gives f(ω, l), which is evaluated through a standard MPS–MPO–MPS contraction implemented with ITensor library.

Demonstration and Results

The methodology was demonstrated using a three-dimensional non-Hermitian higher-order topological insulator (HOTI) with more than one billion degrees of freedom.

**- For weak loss λ = 2, the spectrum contains a single isolated corner-mode sector, whose LDOS remains localized at the outer corner of the entire system. 3-dimensional non-Hermitian higherorder topological insulators with structured real-space geometries were investigated using this approach. 3D BBH-type lattice models were used to demonstrate the methodology. 3D tensor-network results for the strong-loss regime (λ = 5) in the 10243 ≈ 109 system show that multiple isolated modes emerge along the imaginary-energy axis. The associated LDOS patterns reveal a hierarchical splitting of the corner-mode manifold, where modes range from those localized at the outermost corner of the corner cube to those concentrated at its innermost corner. These results demonstrate that the tensor-network framework faithfully captures the spectral response of the non-Hermitian HOTI at macroscopic scales. The framework is applicable to a broad class of non-Hermitian tightbinding systems with structured Hamiltonian, including quasiperiodic and moiré systems with large real-space unit cells. This approach is particularly suitable for resolving spectrally isolated non-Hermitian features in large realspace systems. The results open a route toward non-Hermitian quantum matter in regimes inaccessible to conventional exact diagonalization, sparse matrix, and Bloch-based approaches. >">B</ref:2606.

Improvements for AI systems

The improved AI system can perform spectral function calculations for three-dimensional non-Hermitian tight-binding systems beyond one billion lattice sites, directly accessing the macroscopic limit and allows corner-mode spectral responses to be resolved in genuinely three-dimensional systems.
The system will enable the identification of distinct physical states by tuning parameters, as the research identifies distinct in-gap corner modes across weak- and strong-loss regimes.
The AI can resolve complex localization patterns by calculating the corner LDOS for the in-gap modes at different complex energies, revealing distinct localization patterns within the same corner cube as in Fig. 3.