A note on the solutions of the Ginsparg-Wilson relation
The role of R in the solutions of the Ginsparg-Wilson relation is discussed.
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arXiv subjects
Ting-Wai Chiu
Publications and source records attributed to Ting-Wai Chiu.
Chiral Anomaly and Index Theorem on a finite lattice
The condition for a lattice Dirac operator D to reproduce correct chiral anomaly at each site of a finite lattice for smooth background gauge fields is that D possesses exact zero modes satisfying the Atiyah-Singer index theorem. This is also the necessary condition for D to have correct fermion determinant (ratio) which plays the important role of incorporating dynamical fermions in the functional integral.
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A construction of chiral fermion action
According to the necessary requirements for a chirally symmetric Dirac operator, we present a systematic construction of such operators. We formulate a criterion for the hermitian operator which enters the construction such that the doubled modes are decoupled even at finite lattice spacing.
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Topological Phases in Neuberger-Dirac operator
The response of the Neuberger-Dirac fermion operator $D=\Id + V$ in the topologically nontrivial background gauge field depends on the negative mass parameter $m_0$ in the Wilson-Dirac fermion operator $D_w$ which enters $D$ through the unitary operator $V = D_w (D_w^{\dagger} D_w)^{-1/2}$. We classify the topological phases of $D$ by comparing its index to the topological charge of the smooth background gauge field. An exact discrete symmetry in the topological phase diagram is proved for any gauge configurations. A formula for the index of D in each topological phase is derived by obtaining the total chiral charge of the zero modes in the exact solution of the free fermion propagator.
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Fermion determinant and chiral anomaly on a finite lattice
The fermion determinant and the chiral anomaly of lattice Dirac operator D on a finite lattice are investigated. The condition for D to reproduce correct chiral anomaly at each site of a finite lattice for smooth background gauge fields is that D possesses exact zero modes satisfying the Atiyah-Singer index theorem. This is also the necessary condition for D to have correct fermion determinant (ratio) which plays the important role of incorporating dynamical fermions in the functional integral. We outline a scheme for dynamical fermion simulation of lattice QCD with topologically proper D.
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GW fermion propagators and chiral condensate
By exploiting the chiral symmetry, we derive analytic formulas for the massless and the massive Ginsparg-Wilson fermion propagators. Using these formulas, we derive an expression for the chiral condensate which is the order parameter for spontaneous chiral symmetry breaking in QCD. These formulas provide the proper way to compute the fermion propagators and may save a significant amount of computing time especially for large lattices in four dimensions.
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Perturbation calculation of the axial anomaly of Ginsparg-Wilson fermion
We evaluate the axial anomaly for the general Ginsparg-Wilson fermion operator $D = D_c (\Id + R D_c)^{-1}$ with $R = r \Id$. For any chirally symmetric $D_c$ which in the free fermion limit, is free of species doubling and behaves like $i γ_μp_μ$ as $p \to 0$, the axial anomaly $\tr[γ_5 (R D) (x,x)]$ for U(1) lattice gauge theory with single fermion flavor is equal to $e^2/(32 π^2) ε_{μνλσ} F_{μν}(x) F_{λσ}(x+\hatμ+\hatν)$ plus terms which are higher orders and/or non-perturbative contribtuons. The $F \tilde{F}$ term is r-invariant and has the correct continuum limit.
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The axial anomaly of Ginsparg-Wilson fermion
The axial anomaly of Ginsparg-Wilson fermion operator $D$ is discussed in general for the operator $R$ which enters the chiral symmetry breaking part in the Ginsparg-Wilson relation. The axial anomaly and the index of $D$ as well as the exact realization of the Atiyah-Singer index theorem on the lattice are determined solely by the topological characteristics of the chirally symmetric operator $D_c$ in the chiral limit $ R \to 0 $.
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Solutions of the Ginsparg-Wilson Relation
We analyze general solutions of the Ginsparg-Wilson relation for lattice Dirac operators and formulate a necessary condition for such operators to have non-zero index in the topologically nontrivial background gauge fields.
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Exactly massless fermions on the lattice
The salient features of the Ginsparg-Wilson fermion in topologically nontrivial background gauge fields are outlined. The R-invariance of the zero modes, the indices and the index theorem on a finite lattice are illustrated. The role of R in converting the nonlocal D_c into a sequence of highly local D is demonstrated.
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Chiral structure of the solutions of the Ginsparg-Wilson relation
We analyse the structure of solutions of the Ginsparg-Wilson relation for lattice Dirac operator in topologically trivial gauge sector. We show that the properties of such solutions relating to the perturbative stability of the pole of the fermion propagator as well as to the structure of the Yukawa models based on these solutions are solely determined by the non-local chirally invariant part of these Dirac operators. Depending on the structure of this part, the pole in the fermion propagator may or may not be stable under radiative corrections. We illustrate this by explicit examples.
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Topological Charge and The Spectrum of Exactly Massless Fermions on the Lattice
The square root of the positive definite hermitian operator $D_w^{\dagger} D_w$ in Neuberger's proposal of exactly massless quarks on the lattice is implemented by the recursion formula $Y_{k+1} = {1/2} (Y_k + D_w^{\dagger} D_w Y_k^{-1})$ with $Y_0 = \Id$, where $Y_k^2$ converges to $D_w^{\dagger} D_w$ quadratically. The spectrum of the lattice Dirac operator for single massless fermion in two dimensional background U(1) gauge fields is investigated. For smooth background gauge fields with non-zero topological charge, the exact zero modes with definite chirality are reproduced to a very high precision on a finite lattice and the Index Theorem is satisfied exactly. The fermionic determinants are also computed and they are in good agreement with the continuum exact solution.
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Dynamical Fermion on Random-Block Lattice
Massless fermion field interacting with abelian dynamical gauge field on 2-dimensional random-block lattices are investigated using Hybrid Monte Carlo simulations. Preliminary results of the Wilson loop and the chiral correlation function are in agreement with the continuum Schwinger model.
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