arXiv subjects

Chia-Yi Ju

Publications and source records attributed to Chia-Yi Ju.

14 recordsLinked to original sources

arXiv2026

Evolution Generators for Complex Parameters

Theoretical studies on how quantum systems are affected by external factors are often formulated through parameter changes in the system's Hamiltonian. Beyond Berry connections and phases, which focus on specific Hamiltonian eigenstates, recent studies inspired by non-Hermitian quantum formalisms provide a framework for obtaining general state evolution generators for real-valued parameters. This study extends the applicability of the evolution generator formalism to complex-valued parameters via Wirtinger derivatives. By treating a complex parameter and its conjugate as independent variables, the evolution equations are derived for both quantum states and the metric of the Hilbert space bundle. The analysis demonstrates that while state evolution with respect to a complex parameter is naturally governed by its corresponding evolution generator, metric evolution requires a coupled contribution from both the generator and its conjugate counterpart to preserve state normalization. Explicit examples are worked out to illustrate the implementation and physical consistency of the formalism.

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arXiv2026

Non-Hermitian Rayleigh-Schrödinger-like Perturbation Theory at Exceptional Point

We develop a Rayleigh--Schrödinger-like perturbation theory for non-Hermitian quantum systems at an exceptional point of order $N$. Working in the Jordan basis of the unperturbed Hamiltonian and employing a Puiseux expansion of the perturbed eigenvalues and eigenstates, we derive explicit recursion relations for the expansion coefficients. The corrections to the unperturbed eigenvalue in the Puiseux expansion govern the splitting near the exceptional point; the first two are obtained iteratively in two equivalent forms. One is given in terms of the perturbation Hamiltonian in the Jordan basis, and the other in terms of the generator that drives the eigenvalue evolution with respect to the perturbation. The latter constitutes the exceptional-point counterpart of a geometric perturbation method recently developed for the non-exceptional-point regime. Both representations are verified explicitly for the $N = 2$ and $N = 3$ cases.

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arXiv2026

Fidelity and quantum geometry approach to Dirac exceptional points in diamond nitrogen-vacancy centers

Dirac exceptional points (EPs) represent a novel class of non-Hermitian singularities that, unlike conventional EPs, reside entirely within the parity-time unbroken phase and exhibit linear energy dispersion. Here, we theoretically investigate the quantum geometry of Dirac EPs realized in nitrogen-vacancy centers in diamond, utilizing fidelity susceptibility as a probe. We demonstrate that despite the absence of a symmetry-breaking phase transition, the Dirac EP induces a pronounced geometric singularity, confirming the validity of the fidelity in characterizing non-Hermitian EPs. Specifically, the real part of the fidelity susceptibility diverges to negative infinity, which serves as a signature of non-Hermitian criticality. Crucially, however, we reveal that this divergence exhibits a distinct anisotropy, diverging along the non-reciprocal coupling direction while remaining finite along the detuning axis. Furthermore, we establish that this anisotropy, characterized by at least one exact dark direction coexisting with divergent directions, is a generic consequence of the Dirac EP structure whenever the parameter derivatives collectively span the off-diagonal operator space at the Dirac EP. This behavior stands in stark contrast to the omnidirectional divergence observed in conventional EPs. Our findings provide a comprehensive picture of the fidelity probe near the Dirac EP, highlighting the critical role of parameter directionality in exploiting Dirac EPs for quantum control and sensing applications.

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arXiv2025

Quantum State Evolution and Berry Potentials at Exceptional Points and Quantum Phase Transitions

The behavior of quantum states at exceptional points and at critical points associated with quantum phase transitions is intriguing yet puzzling. In this study, we present an alternative method for obtaining the Berry potentials using the evolution generator along the parameter induced dimension and demonstrate that they are singular at these critical points. Although these singularities may appear to indicate a breakdown in quantum state evolution, we show that the information carried by quantum states evolving across these critical points is not destroyed. Specifically, when the evolution generator of the full Hilbert space bundle is taken into account, the quantum states remain insensitive to the critical points. In physical terms, it's similar to the classical image of an object smoothly passing through a black hole's event horizon. Further similarities between exceptional points and quantum phase transitions are explored in this work.

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arXiv2025

Evidence for Exceptional Points as Topological Defects

Studies have shown that quantum states reside in a Hilbert space bundle. When a quantum system depends on continuous external parameters, these parameters define additional dimensions in the base space of the bundle. While much of the existing literature focuses on eigenstate subbundles, where geometric properties like Berry curvature arise, this work considers the entire Hilbert space bundle. Although the Hilbert space bundle has been found to be locally flat, suggesting that the system's geometry may appear trivial, we revisit this assumption. Specifically, we examine how an arbitrary quantum state evolves when transported along closed parameter loops, a phenomenon characterized by holonomy. Our results demonstrate that nontrivial holonomy can emerge in the presence of exceptional points. Consequently, the topology of the full Hilbert space bundle is nontrivial, with exceptional points acting as topological defects.

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arXiv2025

Heisenberg and Heisenberg-Like Representations via Hilbert Space Bundle Geometry in the Non-Hermitian Regime

The equivalence between the Schrödinger and Heisenberg representations is a cornerstone of quantum mechanics. However, this relationship remains unclear in the non-Hermitian regime, particularly when the Hamiltonian is time-dependent. In this study, we address this gap by establishing the connection between the two representations, incorporating the metric of the Hilbert space bundle. We not only demonstrate the consistency between the Schrödinger and Heisenberg representations but also present a Heisenberg-like representation grounded in the generalized vielbein formalism, which provides a clear and intuitive geometric interpretation. Unlike the standard Heisenberg representation, where the metric of the Hilbert space is encoded solely in the dual states, the Heisenberg-like representation distributes the metric information between both the states and the dual states. Despite this distinction, it retains the same Heisenberg equation of motion for operators. Within this formalism, the Hamiltonian is replaced by a Hermitian counterpart, while the "non-Hermiticity" is transferred to the operators. Moreover, this approach extends to regimes with a dynamical metric (beyond the pseudo-Hermitian framework) and to systems governed by time-dependent Hamiltonians.

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arXiv2024

Non-Hermitian Generalization of Rayleigh-Schrödinger Perturbation Theory

While perturbation theories constitute a significant foundation of modern quantum system analysis, extending them from the Hermitian to the non-Hermitian regime remains a non-trivial task. In this work, we generalize the Rayleigh-Schrödinger perturbation theory to the non-Hermitian regime by employing a geometric formalism. This framework allows us to compute perturbative corrections to eigenstates and eigenvalues of Hamiltonians iteratively to any order. Furthermore, we observe that the recursion equation for the eigenstates resembles the form of the Girard-Newton formulas, which helps us uncover the general solution to the recursion equation. Moreover, we demonstrate that the perturbation method proposed in this paper reduces to the standard Rayleigh-Schrödinger perturbation theory in the Hermitian regime.

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arXiv2024

Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics

Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.

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arXiv2023

Visually quantifying single-qubit quantum memory

To store quantum information, quantum memory plays a central intermediate ingredient in a network. The minimal criterion for a reliable quantum memory is the maintenance of the entangled state, which can be described by the non-entanglement-breaking (non-EB) channel. In this work, we show that all single-qubit quantum memory can be quantified without trusting input state generation. In other words, we provide a semi-device-independent approach to quantify all single-qubit quantum memory. More specifically, we apply the concept of the two-qubit quantum steering ellipsoids to a single-qubit quantum channel and define the channel ellipsoids. An ellipsoid can be constructed by visualizing finite output states within the Bloch sphere. Since the Choi-Jamiołkowski state of a channel can all be reconstructed from geometric data of the channel ellipsoid, a reliable quantum memory can be detected. Finally, we visually quantify the single-qubit quantum memory by observing the volume of the channel ellipsoid.

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arXiv2022

Flattening the Curve with Einstein's Quantum Elevator: Hermitization of Non-Hermitian Hamiltonians via a Generalized Vielbein Formalism

The formalism for non-Hermitian quantum systems sometimes blurs the underlying physics. We present a systematic study of the vielbein-like formalism which transforms the Hilbert space bundles of non-Hermitian systems into the conventional ones, rendering the induced Hamiltonian to be Hermitian. In other words, any non-Hermitian Hamiltonian can be "transformed" into a Hermitian one without altering the physics. Thus we show how to find a reference frame (corresponding to Einstein's quantum elevator) in which a non-Hermitian system, equipped with a non-trivial Hilbert space metric, reduces to a Hermitian system within the standard formalism of quantum mechanics.

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arXiv2020

Hunting for the non-Hermitian exceptional points with fidelity susceptibility

The fidelity susceptibility has been used to detect quantum phase transitions in the Hermitian quantum many-body systems over a decade, where the fidelity susceptibility density approaches $+\infty$ in the thermodynamic limits. Here the fidelity susceptibility $χ$ is generalized to non-Hermitian quantum systems by taking the geometric structure of the Hilbert space into consideration. Instead of solving the metric equation of motion from scratch, we chose a gauge where the fidelities are composed of biorthogonal eigenstates and can be worked out algebraically or numerically when not on the exceptional point (EP). Due to the properties of the Hilbert space geometry at EP, we found that EP can be found when $χ$ approaches $-\infty$. As examples, we investigate the simplest $\mathcal{PT}$ symmetric $2\times2$ Hamiltonian with a single tuning parameter and the non-Hermitian Su-Schriffer-Heeger model.

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arXiv2020

Simplifying 4d $\mathcal{N}=3$ Harmonic Superspace

We quantize super Yang-Mills action in $\mathcal{N}=3$ harmonic superspace using "Fermi-Feynman" gauge and also develop the background field formalism. This leads to simpler propagators and Feynman rules that are useful in performing explicit calculations. The superspace rules are used to show that divergences do not appear at 1-loop and beyond. We also compute a finite contribution to the effective action from a 4-point diagram at 1-loop, which matches the expected covariant result.

hep-th↗

arXiv2019

Non-Hermitian Hamiltonians and no-go theorems in quantum information

Recently, apparent nonphysical implications of non-Hermitian quantum mechanics (NHQM) have been discussed in the literature. In particular, the apparent violation of the no-signaling theorem, discrimination of nonorthogonal states, and the increase of quantum entanglement by local operations were reported, and therefore NHQM was not considered as a fundamental theory. Here we show that these and other no-go principles (including the no-cloning and no-deleting theorems) of conventional quantum mechanics still hold in finite-dimensional non-Hermitian quantum systems, including parity-time symmetric and pseudo-Hermitian cases, if its formalism is properly applied. We have developed a modified formulation of NHQM based on the geometry of Hilbert spaces which is consistent with the conventional quantum mechanics for Hermitian systems. Using this formulation the validity of these principles can be shown in a simple and uniform approach.

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arXiv2016

Gauging Unbroken Symmetries in F-theory

F-theory attempts to include all U-dualities manifestly. Unlike its T-dual manifest partner, which is based on string current algebra, F-theory is based on higher dimensional brane current algebra. Like the T-dual manifest theory, which has $O(D-1,1)^2$ unbroken symmetry, the F-theory vacuum also enjoys certain symmetries ("$H$"). One of its important and exotic properties is that worldvolume indices are also spacetime indices. This makes the global brane current algebra incompatible with $H$ symmetry currents. The solution is to introduce worldvolume covariant derivatives, which depend on the $H$ coordinates even in a "flat" background. We will also give as an explicit example the 5-brane case.

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