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Register a new userEfficient computation of low-lying eigenmodes of non-Hermitian Wilson-Dirac type matrices
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Authors
H. Neff
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A polynomial transformation for non-Hermitian matrices is presented, which provides access to wedge-shaped spectral windows. For Wilson-Dirac type matrices this procedure not only allows the determination of the physically interesting low-lying eigenmodes but also provides a substantial acceleration of the eigenmode algorithm employed.
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In this paper, we present a method for the computation of the low-lying real eigenvalues of the Wilson-Dirac operator based on the Arnoldi algorithm. These eigenvalues contain information about several observables. We used them to calculate the sign of the fermion determinant in one-flavor QCD and the sign of the Pfaffian in N=1 super Yang-Mills theory. The method is based on polynomial transformations of the Wilson-Dirac operator, leading to considerable improvements of the computation of eigenvalues. We introduce an iterative procedure for the construction of the polynomials and demonstrate the improvement in the efficiency of the computation. In general, the method can be applied to operators with a symmetric and bounded eigenspectrum.
In situations where the low lying eigenmodes of the Dirac operator are suppressed one observed degeneracies of some meson masses. Based on these results a hidden symmetry was conjectured, which is not a symmetry of the Lagrangian but emerges in the quantization process. We show here how the difference between classes of meson propagators is governed by the low modes and shrinks when they disappear.
We study the isoscalar and isovector $J=0,1$ mesons with the overlap operator within two flavour lattice QCD. After subtraction of the lowest-lying Dirac eigenmodes from the valence quark propagator all disconnected contributions vanish and all possible point-to-point $J=0$ correlators become identical, signaling a simultaneous restoration of both $SU(2)_L times SU(2)_R$ and $U(1)_A$ symmetries. The ground states of the $pi,sigma,a_0,eta$ mesons do not survive this truncation. All possible $J=1$ states have a very clean exponential decay and become degenerate, demonstrating a $SU(4)$ symmetry of a dynamical QCD-like string.
We compute the low-lying spectrum of the staggered Dirac operator above and below the finite temperature phase transition in both quenched QCD and in dynamical four flavor QCD. In both cases we find, in the high temperature phase, a density with close to square root behavior, $rho(lambda) sim (lambda-lambda_0)^{1/2}$. In the quenched simulations we find, in addition, a volume independent tail of small eigenvalues extending down to zero. In the dynamical simulations we also find a tail, decreasing with decreasing mass, at the small end of the spectrum. However, the tail falls off quite quickly and does not seem to extend to zero at these couplings. We find that the distribution of the smallest Dirac operator eigenvalues provides an efficient observable for an accurate determination of the location of the chiral phase transition, as first suggested by Jackson and Verbaarschot.
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