Non-hermitian Bose-Hubbard-like Quantum Models Demonstrate Efficient Matrix Continued Fraction Forms for Green’s Functions
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Quantum systems that do not follow traditional rules of symmetry, known as non-Hermitian systems, present a fascinating challenge to physicists, and Miloslav Znojil from The Czech Academy of Sciences investigates a particularly tractable example of these systems. He focuses on a non-Hermitian model inspired by the well-known Bose-Hubbard model, a cornerstone of condensed matter physics, and demonstrates a novel way to understand its behaviour. This approach allows scientists to determine key properties of the system, specifically its singular values, through a process resembling the standard Schrödinger equation, offering a user-friendly method for analysis. The research further delivers compact mathematical formulas for calculating crucial quantities, paving the way for efficient numerical simulations and a deeper understanding of these complex quantum systems. A family of multibosonic complex-symmetric Hamiltonians, exhibiting both real and complex energy spectra, receives detailed investigation. This work treats resonances as eigenstates of an effective quantum Hamiltonian, acknowledging the challenges in determining their complex energy eigenvalues. Consequently, researchers focused on evaluating auxiliary real quantities called singular values as a simplified approach. Several representations of Green’s functions, expressed through potentially matrix continued fractions, prove to be an efficient method for this task. The investigation centers on Bose-Hubbard-like Hamiltonians, representing a standard, many-body quantum system. Singular Values from Continued Fraction Forms This research details a novel approach to solving the complex eigenvalue problems that arise in non-Hermitian quantum mechanics, particularly within the context of Bose-Hubbard models. Researchers demonstrated a method for defining these models in a way that simplifies the calculation of their singular values using a Schrödinger-like equation. This advancement allows for the efficient determination of key properties of the system, paving the way for more accurate simulations of complex quantum phenomena.
The team focused on tridiagonal matrix Hamiltonians, a class of models particularly suited to numerical analysis. They successfully expressed the analytic Green’s functions of these Hamiltonians using compact and numerically efficient matrix continued fraction forms. This mathematical technique provides a powerful tool for calculating energy spectra and understanding the behavior of quantum particles within the model. In the simplest case, a single-particle model yields an elementary doublet of energies, E± = ± √(1 − γ²), where γ represents a key parameter governing the system’s properties. Further investigation revealed that the Hamiltonian ceases to be diagonalizable at specific parameter values, known as exceptional points, where the spectrum transitions from real to complex. At a small system size, researchers proved this behavior explicitly and extended the proof to a slightly larger system, demonstrating the existence of exceptional points and the resulting changes in the system’s physical interpretation. For a larger model, the team identified exceptional points of high order and successfully constructed a model exhibiting the same degree of degeneracy even after weakening the model’s symmetries. To facilitate these calculations, the researchers developed a factorization approach for the tridiagonal eigenvalue problem, allowing them to express the Hamiltonian in terms of bidiagonal matrices and a diagonal factor. This approach enables the efficient calculation of Green’s functions using non-linear continued fraction recurrences, providing a convergent resummation of perturbation series and offering a powerful tool for analyzing complex quantum systems. The method’s effectiveness stems from the ability to express analytic Green’s functions in terms of continued fractions, a technique that simplifies calculations and improves numerical stability. Hermitized Green’s Functions for Non-Hermitian Systems This work presents a new approach to understanding non-Hermitian quantum systems, specifically focusing on large tridiagonal Hamiltonians. Researchers have demonstrated that a particular subclass of these Hamiltonians, structured to resemble the realistic Bose-Hubbard model, can be treated in a user-friendly manner. This involves defining the system’s singular values through a “Hermitized” Schrödinger-like equation, effectively bridging the gap between non-Hermitian and more conventional quantum descriptions.
The team successfully developed compact and numerically efficient matrix continued fraction forms to represent the associated “Hermitized” Green’s functions, providing a powerful tool for calculating key properties of these complex quantum systems and potentially simplifying analysis in areas like resonance dynamics. While the study acknowledges the limitations inherent in focusing on a specific class of non-Hermitian Hamiltonians, the results establish a promising framework for exploring a wider range of non-Hermitian quantum models. 👉 More information 🗞 Non-Hermitian Bose-Hubbard-like quantum models 🧠 ArXiv: https://arxiv.org/abs/2512.07250 Tags: