scholarly journals Generalized eigenproblem without fermion doubling for Dirac fermions on a lattice

2021 ◽  
Vol 11 (6) ◽  
Author(s):  
Michał Pacholski ◽  
Gal Lemut ◽  
J. Tworzydło ◽  
Carlo Beenakker
Keyword(s):  
Lattice Gauge Theory ◽  
Tight Binding ◽  
Staggered Grid ◽  
Dirac Fermions ◽  
Symmetry Classes ◽  
Lattice Gauge ◽  
Single Cone ◽  
Spectral Statistics ◽  
Particle Current ◽  
Dirac Hamiltonian

The spatial discretization of the single-cone Dirac Hamiltonian on the surface of a topological insulator or superconductor needs a special ``staggered’’ grid, to avoid the appearance of a spurious second cone in the Brillouin zone. We adapt the Stacey discretization from lattice gauge theory to produce a generalized eigenvalue problem, of the form \bm{\mathcal H}\bm{\psi}=\bm{E}\bm{\mathcal P}\bm{\psi}ℋ𝛙=𝐄𝒫𝛙, with Hermitian tight-binding operators \bm{\mathcal H}ℋ, \bm{\mathcal P}𝒫, a locally conserved particle current, and preserved chiral and symplectic symmetries. This permits the study of the spectral statistics of Dirac fermions in each of the four symmetry classes A, AII, AIII, and D.

scholarly journals Lattice gauge theory for the Haldane conjecture and central-branch Wilson fermion

2020 ◽  
Vol 2020 (3) ◽  
Author(s):  
Tatsuhiro Misumi ◽  
Yuya Tanizaki
Keyword(s):  
Gauge Theory ◽  
Lattice Gauge Theory ◽  
Fine Tuning ◽  
Lattice Rotation ◽  
Dirac Fermions ◽  
Wilson Fermion ◽  
Schwinger Model ◽  
Lattice Gauge ◽  
Sign Problem ◽  
The Continuum

Abstract We develop a $(1+1)$D lattice $U(1)$ gauge theory in order to define the two-flavor massless Schwinger model, and discuss its connection with the Haldane conjecture. We propose to use the central-branch Wilson fermion, which is defined by relating the mass, $m$, and the Wilson parameter, $r$, by $m+2r=0$. This setup gives two massless Dirac fermions in the continuum limit, and it turns out that no fine-tuning of $m$ is required because the extra $U(1)$ symmetry at the central branch, $U(1)_{\overline{V}}$, prohibits additive mass renormalization. Moreover, we show that the Dirac determinant is positive semi-definite and this formulation is free from the sign problem, so a Monte Carlo simulation of the path integral is possible. By identifying the symmetry at low energy, we show that this lattice model has a mixed ’t Hooft anomaly between $U(1)_{\overline{V}}$, lattice translation, and lattice rotation. We discuss its relation to the anomaly of half-integer anti-ferromagnetic spin chains, so our lattice gauge theory is suitable for numerical simulation of the Haldane conjecture. Furthermore, it gives a new and strict understanding on the parity-broken phase (Aoki phase) of the $2$D Wilson fermion.

scholarly journals Meson spectrum of Sp(4) lattice gauge theory with two fundamental Dirac fermions

2019 ◽  
Author(s):  
Jong-Wan Lee ◽  
Ed Bennett ◽  
Deog Ki Hong ◽  
C.-J. David Lin ◽  
Biagio Lucini ◽  
...  
Keyword(s):  
Gauge Theory ◽  
Lattice Gauge Theory ◽  
Dirac Fermions ◽  
Meson Spectrum ◽  
Lattice Gauge

scholarly journals Localization of Dirac Fermions in Finite-Temperature Gauge Theory

Universe ◽  
2021 ◽  
Vol 7 (6) ◽  
pp. 194
Author(s):  
Matteo Giordano ◽  
Tamás Kovács
Keyword(s):  
Gauge Theory ◽  
Finite Temperature ◽  
Lattice Gauge Theory ◽  
Dirac Fermions ◽  
Abelian Gauge ◽  
Localization Transition ◽  
Gauge Groups ◽  
Topological Excitations ◽  
Dirac Spectrum ◽  
Lattice Gauge

It is by now well established that Dirac fermions coupled to non-Abelian gauge theories can undergo an Anderson-type localization transition. This transition affects eigenmodes in the lowest part of the Dirac spectrum, the ones most relevant to the low-energy physics of these models. Here we review several aspects of this phenomenon, mostly using the tools of lattice gauge theory. In particular, we discuss how the transition is related to the finite-temperature transitions leading to the deconfinement of fermions, as well as to the restoration of chiral symmetry that is spontaneously broken at low temperature. Other topics we touch upon are the universality of the transition, and its connection to topological excitations (instantons) of the gauge field and the associated fermionic zero modes. While the main focus is on Quantum Chromodynamics, we also discuss how the localization transition appears in other related models with different fermionic contents (including the quenched approximation), gauge groups, and in different space-time dimensions. Finally, we offer some speculations about the physical relevance of the localization transition in these models.

Interpolation and extrapolation

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Public Transportation ◽  
Spline Interpolation ◽  
Lattice Gauge Theory ◽  
Human Vision ◽  
Physical Sciences ◽  
One Dimensional ◽  
Lattice Gauge ◽  
Multidimensional Problems ◽  
Data Points ◽  
Lagrange Interpolating Polynomial

This chapter deals with two related problems occurring frequently in the physical sciences: first, the problem of estimating the value of a function from a limited number of data points; and second, the problem of calculating its value from a series approximation. Numerical methods for interpolating and extrapolating data are presented. The famous Lagrange interpolating polynomial is introduced and applied to one-dimensional and multidimensional problems. Cubic spline interpolation is introduced and an implementation in terms of Eigen classes is given. Several techniques for improving the convergence of Taylor series are discussed, including Shank’s transformation, Richardson extrapolation, and the use of Padé approximants. Conversion between representations with the quotient-difference algorithm is discussed. The exercises explore public transportation, human vision, the wine market, and SU(2) lattice gauge theory, among other topics.

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