Near-zero Modes Research Articles

Constraints on the Dirac spectrum from chiral symmetry restoration

I derive constraints on the Dirac spectrum in the chirally symmetric phase of a gauge theory with two massless fermion flavors. Using only general properties of correlation functions of scalar and pseudoscalar bilinears, I prove that in the chiral limit of vanishing fermion mass m the corresponding susceptibilities and all their derivatives with respect to m2 must be finite. I then use the resulting spectral constraints to show that effective breaking of the anomalous U(1)A symmetry is allowed in the SU(2)A symmetric phase in the chiral limit, and leads to distinctive spectral features: (i) the spectral density must develop a singular O(m4)/λ peak as m→0, (ii) the two-point eigenvalue correlator of near-zero modes must be singular, and (iii) near-zero modes cannot be localized. Moreover, in the symmetric phase the topological charge distribution must be indistinguishable from that of an ideal gas of instantons and anti-instantons of vanishing density, to leading order in m. Published by the American Physical Society 2024

Dirac spectral density in Nf=2+1 QCD at T=230 MeV

We compute the renormalized Dirac spectral density in Nf=2+1 QCD at physical quark masses, temperature T=230 MeV, and system size Ls=3.4 fm. To that end, we perform a pointwise continuum limit of the staggered density in lattice QCD with staggered quarks. We find, for the first time, that a clear infrared structure (IR peak) emerges in the density of the Dirac operator describing dynamical quarks. We also provide numerical evidence that a component of this peak, which becomes dominant in the thermodynamic limit, is due to a nontrivial accumulation of near-zero modes. Features of this structure are consistent with those previously attributed to the recently proposed IR phase of thermal QCD. Our results (i) provide the only complete first-principles evidence that these IR features exist and are physical, (ii) improve the upper bound for IR phase transition temperature TIR so that the new window is 200<TIR<230 MeV, and (iii) are consistent with the nonrestoration of anomalous UA(1) symmetry (chiral limit) below T=230 MeV. Published by the American Physical Society 2024

Formulating strain-based quadrilateral membrane finite elements with drilling rotations

Membrane finite elements with drilling degrees of freedom have sparked interest in many research works since they can be conveniently combined with plates to form shell elements. This study presents two non-conforming strain-based four-node quadrilateral membrane elements, SBQ13 and SBQ13E, for static analysis. SBQ13 partially satisfies the equilibrium equations, while SBQ13E completely fulfils the force balance equations. Both elements carry drilling rotations at each node. One difficulty when formulating quadrilateral elements is the singularity in the transformation matrix, which is addressed in this study by utilising the properties of singular matrices. Both quadrilateral elements were tested in several benchmark problems. It has been found that both elements passed the higher-order patch test but failed the constant stress patch test. Nevertheless, SBQ13 produced accurate responses in most numerical tests, but SBQ13E unreasonably overestimated the solution. Solving the eigenvalue problem revealed that the SBQ13E element has a near-zero energy deformation mode, which might explain the anomaly. Although fulfilling equilibrium does not always enhance solution accuracy, it is essential to overcome volumetric locking. Apart from the newly developed elements, this paper presents several new ideas that may apply to strain-based element formulations.

Optimizing Dynamic Mode Decomposition for Video Denoising via Plug-and-Play Alternating Direction Method of Multipliers

Dynamic mode decomposition (DMD) is a powerful tool for separating the background and foreground in videos. This algorithm decomposes a video into dynamic modes, called DMD modes, to facilitate the extraction of the near-zero mode, which represents the stationary background. Simultaneously, it captures the evolving motion in the remaining modes, which correspond to the moving foreground components. However, when applied to noisy video, this separation leads to degradation of the background and foreground components, primarily due to the noise-induced degradation of the DMD mode. This paper introduces a novel noise removal method for the DMD mode in noisy videos. Specifically, we formulate a minimization problem that reduces the noise in the DMD mode and the reconstructed video. The proposed problem is solved using an algorithm based on the plug-and-play alternating direction method of multipliers (PnP-ADMM). We applied the proposed method to several video datasets with different levels of artificially added Gaussian noise in the experiment. Our method consistently yielded superior results in quantitative evaluations using peak-signal-to-noise ratio (PSNR) and structural similarity (SSIM) compared to naive noise removal methods. In addition, qualitative comparisons confirmed that our method can restore higher-quality videos than the naive methods.

Topological Dimensions from Disorder and Quantum Mechanics?

We have recently shown that the critical Anderson electron in D=3 dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension dIR≈8/3. Here, we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probabilities, such that the points comprising a region are of similar relevance, and calculate the IR scaling dimension d of each. This allows us to infer the probability density p(d) for dimension d to be accessed by the electron. We find that p(d) has a strong peak at d very close to two. In fact, our data suggest that p(d) is non-zero on the interval [dmin,dmax]≈[4/3,8/3] and may develop a discrete part (δ-function) at d=2 in the infinite-volume limit. The latter invokes the possibility that a combination of quantum mechanics and pure disorder can lead to the emergence of integer (topological) dimensions. Although dIR is based on effective counting, of which p(d) has no a priori knowledge, dIR≥dmax is an exact feature of the ensuing formalism. A possible connection of our results to the recent findings of dIR≈2 in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.

Plane vortex pairs, colorful structures and low-lying Dirac modes

In this paper, we investigate the influence of topological charges on nonstable zero and near-zero modes of the single uni-color plane vortex pairs. We combine the uni-color plane vortices with the spherical vortex and also construct the plane vortex pairs with two colorful vortices. The stability of zero and near-zero modes is analyzed where various boundary conditions for the fermions are checked. In addition, plane vortex pairs may show the role of the topological charges for changing the chirality of fermions. The results clearly indicate characteristic properties for spontaneous chiral symmetry breaking.

Localisation of Dirac modes in gauge theories and Goldstone’s theorem at finite temperature

I discuss the possible effects of a finite density of localised near-zero Dirac modes in the chiral limit of gauge theories with Nf degenerate fermions. I focus in particular on the fate of the massless quasi-particle excitations predicted by the finite-temperature version of Goldstone’s theorem, for which I provide an alternative and generalised proof based on a Euclidean SU(Nf )A Ward-Takahashi identity. I show that localised near-zero modes can lead to a divergent pseudoscalar-pseudoscalar correlator that modifies this identity in the chiral limit. As a consequence, massless quasi-particle excitations can disappear from the spectrum of the theory in spite of a non-zero chiral condensate. Three different scenarios are possible, depending on the detailed behaviour in the chiral limit of the ratio of the mobility edge and the fermion mass, which I prove to be a renormalisation-group invariant quantity.

Fluxoid-induced pairing suppression and near-zero modes in quantum dots coupled to full-shell nanowires

We analyze the subgap excitations and phase diagram of a quantum dot (QD) coupled to a semiconducting nanowire fully wrapped by a superconducting (S) shell. We take into account how a Little-Parks (LP) pairing fluxoid (a winding in the S phase around the shell) influences the proximity effect on the dot. We find that under axially symmetric QD-S coupling, shell fluxoids cause the induced pairing to vanish, producing instead a level renormalization that pushes subgap levels closer to zero energy and flattens fermionic parity crossings as the coupling strength increases. This fluxoid-induced stabilization mechanism has analoges in symmetric S-QD-S Josephson junctions at phase $\pi$, and can naturally lead to patterns of near-zero modes weakly dispersing with parameters in all but the zero-th lobe of the LP spectrum.

Localised Dirac eigenmodes, chiral symmetry breaking, and Goldstone’s theorem at finite temperature

I show that a finite density of near-zero localised Dirac modes in the chirally broken phase of a gauge theory can lead to the disappearance of the massless excitations predicted by the Goldstone theorem at finite temperature.

Three regimes of QCD

While the QCD Lagrangian as the whole is only chirally symmetric, its electric part has larger chiral-spin [Formula: see text] and [Formula: see text] symmetries. This allows separation of the electric and magnetic interactions in a given reference frame. Artificial truncation of the near-zero modes of the Dirac operator results in the emergence of the [Formula: see text] and [Formula: see text] symmetries in hadron spectrum. This implies that while the confining electric interaction is distributed among all modes of the Dirac operator, the magnetic interaction is located at least predominantly in the near-zero modes. Given this observation one could anticipate that above the pseudocritical temperature, where the near-zero modes of the Dirac operator are suppressed, QCD is [Formula: see text] and [Formula: see text] symmetric, which means absence of deconfinement in this regime. Solution of the [Formula: see text] QCD on the lattice with a chirally symmetric Dirac operator reveals that indeed in the interval [Formula: see text] QCD is approximately [Formula: see text] and [Formula: see text] symmetric which implies that degrees of freedom are chirally symmetric quarks bound by the chromoelectric field into color-singlet objects without the chromomagnetic effects. This regime is referred to as a Stringy Fluid. At larger temperatures this emergent symmetry smoothly disappears and QCD approaches the Quark–Gluon Plasma regime with quasifree quarks. The Hadron Gas, the Stringy Fluid and the Quark–Gluon Plasma differ by symmetries, degrees of freedom and properties.

Transitioning from equal-time to light-front quantization in ϕ24 theory

We use the interpolating coordinates studied by Hornbostel to investigate a transition from equal-time quantization to light-front quantization, in the context of two-dimensional $\phi^4$ theory. A consistent treatment is found to require careful consideration of vacuum bubbles, in a nonperturbative extension of the analysis by Collins. Numerical calculations of the spectrum at fixed box size are shown to yield results equivalent to those of equal-time quantization, except when the interpolating coordinates are pressed toward the light-front limit. In that regime, a fixed box size is inconsistent with an accurate representation of vacuum-bubble contributions and causes a spurious divergence in the spectrum. The light-front limit instead requires the continuum momentum-space limit of infinite box size. The calculation of the vacuum energy density is then shown to be independent of the interpolation parameter, which implies that the light-front limit yields the same spectrum as an equal-time calculation. This emphasizes the importance of zero modes and near-zero modes in a light-front analysis of any theory with nontrivial vacuum structure.

Plasmon coupling in topological insulator multilayers

Topological insulators (TIs) house two-dimensional Dirac plasmons on their surfaces. In TI thin films, plasmons on the top and bottom surfaces couple electrostatically, giving rise to an acoustic and an optical plasmon mode. By extension, a superlattice comprising TI layers and trivially insulating layers could house multiple complex plasmon modes which could be used to create a new type of Dirac metamaterial. In this paper, we synthesize TI superlattices, fabricate them into stripe arrays, characterize their optical and plasmonic properties, and model the samples using transfer matrices. We excite plasmon modes that couple to the phonons in the superlattice, resulting in hybrid plasmon-phonon polaritons. These modes are modeled using an analytical Fano resonance model and the extracted resonant positions are reproduced by transfer matrix modeling. We also excite an epsilon near-zero mode in the top dielectric material ${({\mathrm{Bi}}_{0.5}{\mathrm{In}}_{0.5})}_{2}{\mathrm{Se}}_{3}$. Understanding the behavior of the polariton modes in this complex system will lend insight into the many-body interaction in low-dimensional systems.

Magnetoplasmonic manipulation of nanoscale thermal radiation using twisted graphene gratings

In this paper, magnetoplasmonic manipulation of near-field radiative heat transfer (NFRHT) is realized using two twisted graphene gratings. As a result of the quantum Hall regime of magneto-optical graphene and the grating effect, three types of graphene surface plasmon polaritons (SPPs) modes are observed in the system: near-zero modes, high-frequency hyperbolic modes, and elliptic modes. The elliptic SPPs modes, which are caused by the combined effect of magnetic field and grating, are observed in the graphene grating system for the first time. In addition, the near-zero modes can be greatly enhanced by the combined effect grating and magnetic field, rendering graphene devices promising for thermal communication at ultra-low frequency. In particular, the near-zero modes result in a unique enhancement region of heat transfer, no matter for any twisted angle between gratings. The combined effect of grating and magnetic field is investigated simultaneously. By changing the strength of magnetic field, the positions and intensities of the modes can be modulated, and hence the NFRHT can be tuned accordingly, no matter for parallel or twisted graphene gratings. The magnetic field endows the grating action (graphene filling factors and twisted angles) with a higher modulation ability to modulate the NFRHT compared with zero-field. Moreover, the modulation ability of twist can be tuned by the magnetic field at different twisted angles. In sum, the combined effect of magnetic field and grating provides a tunable way to realize the energy modulation or multi-frequency thermal communications related to graphene devices.

Asymmetric near-zero edge mode in topological photonic lattice without chiral or particle-hole symmetries

Generally speaking, a one-dimensional system requires chiral or particle-hole symmetry to be topologically nontrivial. In this Letter, we show that topological behavior can also be observed in a quasi-one-dimensional photonic system without the aforementioned symmetries. The results indicate that the quantized Zak phase is achieved in such a system even though the chiral and particle-hole symmetries are still absent. Further study shows that the system can support a topologically protected asymmetric near-zero mode on the right edge. Our work enriches the concepts of design of topological photonics and may have important applications in future quantum computations.

Zero-energy pinning of topologically trivial bound states in multiband semiconductor-superconductor nanowires

Recent tunneling conductance measurements on semiconductor-superconductor nanowires find zero-bias peaks to be ubiquitous across wide ranges of chemical potential and Zeeman energy. Motivated by this, we demonstrate that topologically-trivial Andreev abound states (ABSs) pinned near zero energy are produced rather generically in inhomogeneous systems with multi-band occupancy in the presence of inter-band coupling. We first investigate the inter-band coupling mechanism responsible for the pinning within a multi-band 1D toy model, then we confirm the findings using a 3D Schr\"{o}dinger-Poisson approach that incorporates the geometric and electrostatic details of the actual device. Our analysis shows that level-repulsion generated by inter-band coupling can lead to a rather spectacular pinning of the lowest-energy mode near zero energy in systems (or regions) characterized by very-short length scales ($\sim100~$nm).We show that level repulsion between the lowest energy levels can mimic the gap opening feature (simultaneous with the emergence of a near-zero energy mode) predicted to occur in Majorana hybrid systems. We also demonstrate that nearly-zero bias differential conductance features exhibiting particle-hole asymmetry are due to the presence of (topologically-trivial) ABSs pinned near zero-energy by level repulsion, not to Majorana zero modes, quasi-Majoranas, or any other low-energy mode that involves (partially) separated Majorana bound states. Our findings demonstrate the importance of understanding in detail multi-band physics and electrostatic effects in the context of the ongoing search for Majorana modes in semiconductor-superconductor heterostructures.

Chiralspin Symmetry and Its Implications for QCD

In a local gauge-invariant theory with massless Dirac fermions, a symmetry of the Lorentz-invariant fermion charge is larger than a symmetry of the Lagrangian as a whole. While the Dirac Lagrangian exhibits only a chiral symmetry, the fermion charge operator is invariant under a larger symmetry group, S U ( 2 N F ) , that includes chiral transformations as well as S U ( 2 ) C S chiralspin transformations that mix the right- and left-handed components of fermions. Consequently, a symmetry of the electric interaction, which is driven by the charge density, is larger than a symmetry of the magnetic interaction and of the kinetic term. This allows separating in some situations electric and magnetic contributions. In particular, in QCD, the chromo-magnetic interaction contributes only to the near-zero modes of the Dirac operator, while confining chromo-electric interaction contributes to all modes. At high temperatures, above the chiral restoration crossover, QCD exhibits approximate S U ( 2 ) C S and S U ( 2 N F ) symmetries that are incompatible with free deconfined quarks. Consequently, elementary objects in QCD in this regime are quarks with a definite chirality bound by the chromo-electric field, without the chromo-magnetic effects. In this regime, QCD can be described as a stringy fluid.

Quantifying wave-function overlaps in inhomogeneous Majorana nanowires

A key property of Majorana zero modes is their protection against local perturbations. In the standard picture, this protection is guaranteed by a high degree of spatial nonlocality of the Majoranas, namely a suppressed wave-function overlap, in the topological phase. However, a careful characterisation of resilience to local noise goes beyond mere spatial separation, and must also take into account the projection of wave-function spin. By considering the susceptibility of a given zero mode to different local perturbations, we find the relevant forms of spin-resolved wave-function overlaps that measure its resilience. We quantify these overlaps and study their dependence with nanowire parameters in several classes of experimentally relevant configurations. These include nanowires with inhomogeneous depletion and induced pairing, barriers and quantum dots. Smooth inhomogeneities have been shown to produce near-zero modes, so-called pseudo-Majoranas, below the critical Zeeman field in the bulk. Surprisingly, their resilience is found to be comparable or better than that of topological Majoranas in realistic systems. We further study how accurately their overlaps can be estimated using a purely local measurement on one end of the nanowire, accessible through conventional transport experiments. In uniform nanowires this local estimator is remarkably accurate. In inhomogeneous cases it is less accurate but can still provide reasonable estimates for potential inhomogeneities of the order of the superconducting gap. We further analyse the zero mode wave-function structure, spin texture and spectral features associated with each type of inhomogeneity. All our results highlight the strong connection between internal wave-function degrees of freedom, nonlocality and protection in smoothly inhomogeneous nanowires.

Dual-wavelength fiber Fabry-Perot cavities with engineered birefringence.

We present a method to engineer the frequency splitting of polarization eigenmodes in fiber Fabry-Perot (FFP) cavities. Using specific patterns of multiple CO2 laser pulses, we machine paraboloidal micromirrors with controlled elliptical shape in a large range of radii of curvature. This method is versatile and can be used to produce cavities with maximized or near-zero polarization mode splitting. In addition, we realize dual-wavelength FFP cavities with finesse exceeding 40 000 at 780 nm and at 1559 nm in the telecom range. We provide direct evidence that the birefringent frequency splitting in FFP cavities is governed only by the geometrical shape of the mirrors, and that the astigmatism of the cavity modes needs to be taken into account for specific cavities.

Distribution law of the Dirac eigenmodes in QCD

The near-zero modes of the Dirac operator are connected to spontaneous breaking of chiral symmetry in QCD (SBCS) via the Banks–Casher relation. At the same time, the distribution of the near-zero modes is well described by the Random Matrix Theory (RMT) with the Gaussian Unitary Ensemble (GUE). Then, it has become a standard lore that a randomness, as observed through distributions of the near-zero modes of the Dirac operator, is a consequence of SBCS. The higher-lying modes of the Dirac operator are not affected by SBCS and are sensitive to confinement physics and related [Formula: see text] and [Formula: see text] symmetries. We study the distribution of the near-zero and higher-lying eigenmodes of the overlap Dirac operator within [Formula: see text] dynamical simulations. We find that both the distributions of the near-zero and higher-lying modes are perfectly described by GUE of RMT. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is linked to the confining chromo-electric field.

Distribution of the Dirac modes in QCD

It was established that distribution of the near-zero modes of the Dirac operator is consistent with the Chiral Random Matrix Theory (CRMT) and can be considered as a consequence of spontaneous breaking of chiral symmetry (SBCS) in QCD. The higherlying modes of the Dirac operator carry information about confinement physics and are not affected by SBCS. We study distributions of the near-zero and higher-lying modes of the overlap Dirac operator within NF = 2 dynamical simulations. We find that distributions of both near-zero and higher-lying modes are the same and follow the Gaussian Unitary Ensemble of Random Matrix Theory. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is related to some more general property of QCD in confinement regime.