Imaginary Field Research Articles

Information geometry and parameter sensitivity of non-Hermitian Hamiltonians

Information geometry is the application of differential geometry in statistics, where the Fisher-Rao metric serves as the Riemannian metric on the statistical manifold, providing an intrinsic property for parameter sensitivity. In this paper, we explore the application of information geometry in the realm of non-Hermitian quantum systems, focusing on the Fisher-Rao metric as a measure of parameter sensitivity. We approximate the Lindblad master equation for non-Hermitian Hamiltonians to analyze the temporal evolution of the quantum geometric metric. Utilizing the quantum spin Ising model with an imaginary magnetic field as an exemplar, we investigate the energy spectrum and geometric metric evolution within PT-symmetry Hamiltonians. We demonstrate that the detrimental effects of dissipation can be counteracted by introducing a control Hamiltonian, leading to improved accuracy in parameter estimation. Our work provides insights into the role of quantum control in mitigating dissipative impacts and enhancing the precision of quantum metrological tasks.

Study of the effects of external imaginary electric field and chiral chemical potential on quark matter

The behavior of quark matter with both external electric field and chiral chemical potential is theoretically and experimentally interesting to consider. In this paper, the case of simultaneous presence of imaginary electric field and chiral chemical potential is investigated using the lattice QCD approach with Nf=1+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_f=1+1$$\\end{document} dynamical staggered fermions. We find that overall both the imaginary electric field and the chiral chemical potential can exacerbate chiral symmetry breaking, which is consistent with theoretical predictions. However we also find a non-monotonic behavior of chiral condensation at specific electric field strengths and chiral chemical potentials. The behavior of Polyakov loop in the complex plane is not significantly affected by chiral chemical potential in the region of the parameters considered.

Maximizing the symmetry of Maxwell’s equations

Maxwell’s equations can be successfully extended to electromagnetic fields having three complex-valued components rather than their usual three real-valued components. Here the implications of interpreting the imaginary-valued components as extending into time rather than space are explored. The complex-valued Maxwell equations remain consistent with the original Maxwell equations and the experimental results that they predict. Further, the extended equations predict novel phenomena such as the existence of electromagnetic waves that propagate not only through regular space but also through a separate temporal space (time) that is implied by the three imaginary components of the fields. In a vacuum, part of these imaginary valued waves propagates through time at the same rate as an observer stationary in space. While the imaginary valued field components are not directly observable, analysis indicates that they should be indirectly detectable experimentally based on secondary effects that occur under special circumstances. Experimental investigation attempting to falsify or support the existence of complex valued electromagnetic fields extending into time is merited due to the substantial theoretical and practical implications involved.

Class numbers, Ono invariants and some interesting primes

Our aim is to find all the prime numbers p such that p+x2 has at most two different prime factors, for all the odd integers x such that x2≤p. We solve entirely the cases p≡1,3,5(mod8), using the knowledge of the quadratic imaginary number fields with class numbers 4, 1 and 2 respectively. The case p≡7(mod8) is not completely solved. Taking into account a result of Stéphane Louboutin, we prove that there is at most one value p≡7(mod8) besides our list. Assuming a Restricted Riemann Hypothesis, the list is complete. In the last section of the paper we give a short sketch for the general problem: find all odd integers n>1 such that n+x2 has at most two different prime factors, for all the odd integers x such that x2≤n.

Topological phase transition in fluctuating imaginary gauge fields

Topological phase transition in fluctuating imaginary gauge fields

On Some Multipliers Related to Discrete Fractional Integrals

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler’s identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ℓp→ℓq bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators.

Experimental Observation of the Yang-Lee Quantum Criticality in Open Quantum Systems.

The Yang-Lee edge singularity was originally studied from the standpoint of mathematical foundations of phase transitions. However, direct observation of anomalous scaling with the negative scaling dimension has remained elusive due to an imaginary magnetic field required for the nonunitary criticality. We experimentally implement an imaginary magnetic field with an open quantum system of heralded single photons, directly measure the partition function, and demonstrate the Yang-Lee edge singularity via the quantum-classical correspondence. We also demonstrate unconventional scaling laws for finite-temperature quantum dynamics.

Viscoacoustic VTI media reverse time migration method in the complex-domain based on the staining algorithm

Underground media exhibit characteristics of viscosity and anisotropy, with vertical transversely isotropic (VTI) media being a common example of anisotropic media. Consequently, it is crucial to address the viscoacoustic VTI wave equation during forward modeling and migration imaging processes to attain highly precise imaging outcomes. However, conventional subsalt imaging techniques yield low accuracy and suffer from a low signal-to-noise ratio. To overcome this challenge, this study proposes a staining algorithm-based viscoacoustic vertical transversely isotropic reverse time migration (VTI RTM) in the complex-domain method by deriving the complex-domain viscoacoustic VTI equation. This method requires both the traditional real velocity field and the imaginary velocity field as inputs. The imaginary velocity field contains stained geological bodies and significant areas without data. Seismic waves propagate normally in the real wavefield, while in the imaginary wavefield, reflections occur only when the wavefront encounters stained geological bodies. The proposed method is assessed through testing on both a horizontally layered model and a salt model. Through comparative analysis, it has been verified that the method fully incorporates reflected waves with distinct structural features into the imaginary wavefield. Numerical experiments demonstrate that this method can accurately image subsalt structures.

Experimental Investigation of Lee–Yang Criticality Using Non-Hermitian Quantum System

Lee–Yang theory clearly demonstrates where the phase transition of many-body systems occurs and the asymptotic behavior near the phase transition using the partition function under complex parameters. The complex parameters make the direct investigation of Lee–Yang theory in practical systems challenging. Here we construct a non-Hermitian quantum system that can correspond to the one-dimensional Ising model with imaginary parameters through the equality of partition functions. By adjusting the non-Hermitian parameter, we successfully obtain the partition function under different imaginary magnetic fields and observe the Lee–Yang zeros. We also observe the critical behavior of free energy in vicinity of Lee–Yang zero that is consistent with theoretical prediction. Our work provides a protocol to study Lee–Yang zeros of the one-dimensional Ising model using a single-qubit non-Hermitian system.

A criterion for nondensity of integral points

AbstractWe give a general criterion for Zariski degeneration of integral points in the complement of a divisor with components in a variety of dimension defined over or over a quadratic imaginary field. The key condition is that the intersection of the components of is not well approximated by rational points, and we discuss several cases where this assumption is satisfied. We also prove a greatest common divisor (GCD) bound for algebraic points in varieties, which can be of independent interest.

Estimates of Picard modular cusp forms

Abstract In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

Computing a group action from the class field theory of imaginary hyperelliptic function fields

We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over Fq acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We report on an explicit computation done with our proof-of-concept C++/NTL implementation; it took a fraction of a second on a standard computer. We prove that the problem of inverting the group action reduces to the problem of finding isogenies of fixed τ-degree between Drinfeld Fq[X]-modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski. We give asymptotic complexity bounds for all algorithms presented in this paper.

Two-dimensional lattice with an imaginary magnetic field

Two-dimensional lattice with an imaginary magnetic field

QCD phase diagram and equation of state in background electric fields

The phase diagram and the equation of state of quantum chromodynamics (QCD) is investigated in the presence of weak background electric fields by means of continuum extrapolated lattice simulations. The complex action problem at nonzero electric field is circumvented by a novel Taylor expansion, enabling the determination of the linear response of the thermal QCD medium to constant electric fields—in contrast to simulations at imaginary electric fields, which, as we demonstrate, involve an infrared singularity. Besides the electric susceptibility of QCD matter, we determine the dependence of the Polyakov loop on the field strength to leading order. Our results indicate a plasma-type behavior with a negative susceptibility at all temperatures, as well as an increase in the transition temperature as the electric field grows. Published by the American Physical Society 2024

The reciprocating and bipolar non-Hermitian skin effect engineered by spin–orbit coupling

We theoretically study a one-dimensional non-Hermitian Su–Schrieffer–Heeger model with an imaginary gauge field and spin–orbit coupling. We find that, under open boundary conditions, the dispersions possess the reciprocating real–complex–real transitions with increasing the strength of spin–orbit coupling. Correspondingly, the bulk energy eigenstates exhibit a reciprocating non-Hermitian skin effect. This mechanism can be characterized by the generalized Brillouin zone moduli, which approaches zero or infinity at the transition points. We further demonstrate the non-zero winding number inside the loops of generalized Brillouin zone when the strength of intra-cell spin–orbit coupling is larger than the inter-cell one, which results in the bipolar non-Hermitian skin effect. As the intra-cell and inter-cell spin–orbit coupling strength becomes comparable, the bipolar non-Hermitian skin effect degenerates to the conventional non-Hermitian skin effect. Our work may pave the way for the non-Hermitian optoelectronic devices utilizing the reciprocating non-Hermitian skin effect and the bipolar non-Hermitian skin effect by engineering the spin–orbit coupling.

Cotorsion of anti-cyclotomic Selmer groups on average

For an elliptic curve, we study how many Selmer groups are cotorsion over the anti-cyclotomic Z p \mathbb {Z}_p -extension as one varies the prime p p or the quadratic imaginary field in question.

Proposal of new crack-tip-opening-displacement as a mechanical driving force of elastic-plastic fracture mechanics

In this study, the current status of linear and nonlinear fracture mechanics was surveyed, and the conditions required for elastic–plastic fracture mechanics (EPFM) as the next step of fracture mechanics were listed. Then, the imaginary crack tip opening displacement (I-CTOD), expressed by the integral of the plastic strain in a two-dimensional plastic zone, was proposed as a driving force of EPFM. Elastic-plastic FEMs of cracked solids in various work-hardened materials were performed under both the plane stress and plane strain states to understand the singular field characteristics of elastic and plastic strains near a crack tip. The I-CTOD was found to represent the intensity of the plastic strain singular field, the elastic strain singular field, and the imaginary plastic deformation field near the crack tip in elastic–plastic solids under both the plane stress and plane strain states. This I-CTOD is promising as a mechanical driving force parameter of EPFM because the analyzable and visualizable characteristics of the behavior and its clear relationship to actual material behavior make it easy to develop prediction methods for the actual crack behavior.

Definability and decidability for rings of integers in totally imaginary fields

AbstractWe show that the ring of integers of is existentially definable in the ring of integers of , where denotes the field of all totally real numbers. This implies that the ring of integers of is undecidable and first‐order nondefinable in . More generally, when is a totally imaginary quadratic extension of a totally real field , we use the unit groups of orders to produce existentially definable totally real subsets . Under certain conditions on , including the so‐called ‐number of being the minimal value , we deduce the undecidability of . This extends previous work that proved an analogous result in the opposite case . In particular, unlike prior work, we do not require that contains only finitely many roots of unity.

Proposal for Observing Yang-Lee Criticality in Rydberg Atomic Arrays.

Yang-Lee edge singularities (YLES) are the edges of the partition function zeros of an interacting spin model in the space of complex control parameters. They play an important role in understanding non-Hermitian phase transitions in many-body physics, as well as characterizing the corresponding nonunitary criticality. Even though such partition function zeroes have been measured in dynamical experiments where time acts as the imaginary control field, experimentally demonstrating such YLES criticality with a physical imaginary field has remained elusive due to the difficulty of physically realizing non-Hermitian many-body models. We provide a protocol for observing the YLES by detecting kinked dynamical magnetization responses due to broken PT symmetry, thus enabling the physical probing of nonunitary phase transitions in nonequilibrium settings. In particular, scaling analyses based on our nonunitary time evolution circuit with matrix product states accurately recover the exponents uniquely associated with the corresponding nonunitary CFT. We provide an explicit proposal for observing YLES criticality in Floquet quenched Rydberg atomic arrays with laser-induced loss, which paves the way towards a universal platform for simulating non-Hermitian many-body dynamical phenomena.

Composition of Bhargava’s cubes over number fields

In this paper, the composition of Bhargava’s cubes is generalized to the ring of integers of a number field of narrow class number one, excluding the case of totally imaginary number fields. The exclusion of the latter case arises from the nonexistence of a bijection between (classes of) binary quadratic forms and an ideal class group. This problem, together with a related mistake in another paper of the author, is addressed in the appendix.