Dirac Operator Research Articles

Monopoles in Dirac spin liquids and their symmetries from instanton calculus

The Dirac spin liquid (DSL) is a two-dimensional (2D) fractionalized Mott insulator featuring massless Dirac spinon excitations coupled to a compact U(1) gauge field, which allows for flux-tunneling instanton events described by magnetic monopoles in (2+1)D Euclidean spacetime. The state-operator correspondence of conformal field theory has been used recently to define associated monopole operators and determine their quantum numbers, which encode the microscopic symmetries of conventional ordered phases proximate to the DSL. In this work, we utilize semiclassical instanton methods not relying on conformal invariance to construct monopole operators directly in (2+1)D spacetime as instanton-induced ’t Hooft vertices, i.e., fermion-number-violating effective interactions originating from zero modes of the Euclidean Dirac operator in an instanton background. In the presence of a flavor-adjoint fermion mass, resummation of the instanton gas is shown to select the correct monopole to be proliferated, in accordance with predictions of the state-operator correspondence. We also show that our instanton-based approach is able to determine monopole quantum numbers on bipartite lattices.

Half inverse problem and interior inverse problem for the Dirac operators with discontinuity

Half inverse problem and interior inverse problem for the Dirac operators with discontinuity

Ordered Patterns of (3+1)-Dimensional Hadronic Gauged Solitons in the Low-Energy Limit of Quantum Chromodynamics at a Finite Baryon Density, Their Magnetic Fields and Novel BPS Bounds

In this paper, we will review two analytical approaches to the construction of non-homogeneous Baryonic condensates in the low-energy limit of QCD in (3+1) dimensions. In both cases, the minimal coupling with the Maxwell U(1) gauge field can be taken explicitly into account. The first approach (which is related to the generalization of the usual spherical hedgehog ansatz to situations without spherical symmetry at a finite Baryon density) allows for the construction of ordered arrays of Baryonic tubes and layers. When the minimal coupling of the Pions to the U(1) Maxwell gauge field is taken into account, one can show that the electromagnetic field generated by these inhomogeneous Baryonic condensates is of a force-free type (in which the electric and magnetic components have the same size). Thus, it is natural to wonder whether it is also possible to analytically describe magnetized hadronic condensates (namely, Hadronic distributions generating only a magnetic field). The idea of the second approach is to construct a novel BPS bound in the low-energy limit of QCD using the theory of the Hamilton–Jacobi equation. Such an approach allows us to derive a new topological bound which (unlike the usual one in the Skyrme model in terms of the Baryonic charge) can actually be saturated. The nicest example of this phenomenon is a BPS magnetized Baryonic layer. However, the topological charge appearing naturally in the BPS bound is a non-linear function of the Baryonic charge. Such an approach allows us to derive important physical quantities (which would be very difficult to compute with other methods), such as how much one should increase the magnetic flux in order to increase the Baryonic charge by one unit. The novel results of this work include an analysis of the extension of the Hamilton–Jacobi approach to the case in which Skyrme coupling is not negligible. We also discuss some relevant properties of the Dirac operator for quarks coupled to magnetized BPS layers.

Dirac operator associated to a quantum metric

Dirac operator associated to a quantum metric

∂¯‐problem for a second‐order elliptic system in Clifford analysis

In the framework of Clifford analysis, we study a second‐order elliptic (generally nonstrongly elliptic) system of partial differential equations of the form: , where stands for the Dirac operator with respect to a structural set . The solutions of this system are known as ‐inframonogenic functions. Our main purpose is to describe necessary and sufficient conditions for the solvability of a ‐problem associated with the sandwich operator .

Bohr-Sommerfeld Quantization Condition for Self-Adjoint Dirac Operators

Bohr-Sommerfeld Quantization Condition for Self-Adjoint Dirac Operators

Boundary Value Problems for the Perturbed Dirac Equation

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F˜λ, we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F˜λ.

Random Green’s Function Method for Large-Scale Electronic Structure Calculation

We report a linear-scaling random Green’s function (rGF) method for large-scale electronic structure calculation. In this method, the rGF is defined on a set of random states and is efficiently calculated by projecting onto Krylov subspace. With the rGF method, the Fermi–Dirac operator can be obtained directly, avoiding the polynomial expansion to Fermi–Dirac function. To demonstrate the applicability, we implement the rGF method with the density-functional tight-binding method. It is shown that the Krylov subspace can maintain at small size for materials with different gaps at zero temperature, including H2O and Si clusters. We find with a simple deflation technique that the rGF self-consistent calculation of H2O clusters at T = 0 K can reach an error of ∼ 1 meV per H2O molecule in total energy, compared to deterministic calculations. The rGF method provides an effective stochastic method for large-scale electronic structure simulation.

Fate of Chiral Symmetries in the Quark-Gluon Plasma from an Instanton-Based Random Matrix Model of QCD.

We propose a new way of understanding how chiral symmetry is realized in the high temperature phase of QCD. Based on the finding that a simple free instanton gas precisely describes the details of the lowest part of the spectrum of the lattice overlap Dirac operator, we propose an instanton-based random matrix model of QCD with dynamical quarks. Simulations of this model reveal that even for small quark mass the Dirac spectral density has a singularity at the origin, caused by a dilute gas of free instantons. Even though the interaction, mediated by light dynamical quarks, creates small instanton-anti-instanton molecules, those do not influence the singular part of the spectrum, and this singular part is shown to dominate Banks-Casher type sums in the chiral limit. By generalizing the Banks-Casher formula for the singular spectrum, we show that in the chiral limit the chiral condensate vanishes if there are at least two massless flavors. Our model also indicates a possible way of resolving a long-standing debate, as it suggests that for two massless quark flavors the U(1)_{A} symmetry is likely to remain broken up to arbitrarily high finite temperatures.

Twisted Dirac Operators and General Kastler–Kalau–Walze Type Theorems for Manifolds with Boundary

In this paper, we establish some general Kastler–Kalau–Walze type theorems on any dimensional manifolds with boundary for twisted Dirac operators.

Low-mode deflation for twisted-mass and RHMC reweighting in lattice QCD

We propose improved estimators to compute the reweighting factors which are needed for lattice QCD calculations that rely on twisted-mass reweighting for the light quark contribution and the Rational Hybrid Monte Carlo (RHMC) algorithm for non-degenerate quark masses. This is the case for a number of modern large-scale simulations based on O(a) improved Wilson fermions. We find a significant reduction of uncertainties for the reweighting factors at similar computational cost compared to the conventional estimation. This leads to a significant increase in precision for phenomenologically relevant observables with high correlation to the low eigenmodes of the Wilson-Dirac operator in the presence of exceptionally small eigenvalues. Supplementary details regarding the spectral gap of the light quark Dirac operator on the 2+1 flavor large-volume ensembles explored in this study can be found in an accompanying appendix.

Boundary Value Problems for Dirac Operators on Graphs

We carry the index theory for manifolds with boundary of Bär and Ballmann over to first order differential operators on metric graphs. This approach results in a short proof for the index of such operators. Then the self-adjoint extensions and the spectrum of the Dirac operator on the complex line bundle are studied. We also introduce two types of boundary conditions for the Dirac operator, whose spectrum encodes information of the underlying topology of the graph.

Spectrum and q-index of the super q-deformed Dirac operator on the superquantum fuzzy two-sphere Sqμ(2|2)

In this paper, we have computed the spectrum and the [Formula: see text]-index of the super [Formula: see text]-deformed Ginsparg–Wilson Dirac operator in the different cases (fuzzy, non-fuzzy, gauged, and non-gauged) on the superquantum fuzzy two-sphere [Formula: see text]. We also presented the appropriate spin structure that this operator acts on the superquantum (Dirac) spinor bundle. Finally, it was shown that in the non-quantum limit, when [Formula: see text] tends to 1, in each case, there is a correct commutative limit for the spectrum and the [Formula: see text]-index.

On the reconstruction of an integro-differential Dirac operator with parameter-dependent nonlocal integral boundary conditions from the nodal data

On the reconstruction of an integro-differential Dirac operator with parameter-dependent nonlocal integral boundary conditions from the nodal data

On the completeness of root function system of the Dirac operator with two‐point boundary conditions

AbstractThe paper is concerned with the completeness property of root functions of the Dirac operator with summable complex‐valued potential and nonregular boundary conditions. We also obtain an explicit form for the fundamental solution system of the considered operator.

Direct and inverse resonance problems for the massless Dirac operator on the half line

For the direct problem, we give the asymptotic distribution of the resonances in the complex plane. For the inverse problem, we prove several uniqueness theorems of recovering the potential on the whole interval from partial resonances and the information of the potential on a subinterval.

Iterated generalized dirac operators of mixed sides

Iterated generalized dirac operators of mixed sides

Inverse problems for Dirac operators with constant delay less than half of the interval

In this work, we consider Dirac-type operators with a constant delay of less than half of the interval and not less than two-fifths of the interval. For our considered Dirac-type operators, two inverse spectral problems are studied. Specifically, reconstruction of two complex L2-potentials is studied from complete spectra of two boundary value problems with one common boundary condition y1(0) = 0 or y2(0) = 0. We give answers to the full range of questions usually raised in the inverse spectral theory. That is, we give uniqueness, necessary and sufficient conditions of the solvability, reconstruction algorithm and uniform stability for our considered inverse problems.

One-loop effective action up to dimension eight: Integrating out heavy fermion(s)

We present the universal one-loop effective action up to dimension eight after integrating out heavy fermion(s) using the Heat-Kernel method. We have discussed how the Dirac operator being a weak elliptic operator, the fermionic operator still can be written in the form of a strong elliptic one such that the Heat-Kernel coefficients can be used to compute the fermionic effective action. This action captures the footprint of both the CP conserving as well as violating UV interactions. As it does not rely on the specific forms of either UV or low energy theories, can be applicable for a very generic action. Our result encapsulates the effects of heavy fermion loop through Yukawa couplings only.

The Hodge-Dirac operator and Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary

Dabrowski et al. [Spectral metric and Einstein functionals for Hodge–Dirac operator, preprint (2023), arXiv:2307.14877] gave spectral Einstein bilinear functionals of differential forms for the Hodge–Dirac operator [Formula: see text] on an oriented even-dimensional Riemannian manifold. In this paper, we generalize the results of Dabrowski et al. to the cases of 4-dimensional oriented Riemannian manifolds with boundary. Furthermore, we give the proof of Dabrowski–Sitarz–Zalecki-type theorems associated with the Hodge–Dirac operator for manifolds with boundary.