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Register a new userDislocations under gradient flow and their effect on the renormalized coupling
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Non-zero topological charge is prohibited in the chiral limit of gauge-fermion systems because any instanton would create a zero mode of the Dirac operator. On the lattice, however, the geometric $Q_text{geom}=langle F{tilde F}rangle /32pi^2$ definition of the topological charge does not necessarily vanish even when the gauge fields are smoothed for example with gradient flow. Small vacuum fluctuations (dislocations) not seen by the fermions may be promoted to instanton-like objects by the gradient flow. We demonstrate that these artifacts of the flow cause the gradient flow renormalized gauge coupling to increase and run faster. In step-scaling studies such artifacts contribute a term which increases with volume. The usual $a/Lto 0$ continuum limit extrapolations can hence lead to incorrect results. In this paper we investigate these topological lattice artifacts in the SU(3) 10-flavor system with domain wall fermions and the 8-flavor system with staggered fermions. Both systems exhibit nonzero topological charge at the strong coupling, especially when using Symanzik gradient flow. We demonstrate how this artifact impacts the determination of the renormalized gauge coupling and the step-scaling $beta$ function.
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We study the impact of the Gradient Flow on the topology in various models of lattice field theory. The topological susceptibility $chi_{rm t}$ is measured directly, and by the slab method, which is based on the topological content of sub-volumes (slabs) and estimates $chi_{rm t}$ even when the system remains trapped in a fixed topological sector. The results obtained by both methods are essentially consistent, but the impact of the Gradient Flow on the characteristic quantity of the slab method seems to be different in 2-flavour QCD and in the 2d O(3) model. In the latter model, we further address the question whether or not the Gradient Flow leads to a finite continuum limit of the topological susceptibility (rescaled by the correlation length squared, $xi^{2}$). This ongoing study is based on direct measurements of $chi_{rm t}$ in $L times L$ lattices, at $L/xi simeq 6$.
To obtain the precise values of the bulk quantities and transport coefficients in quark-gluon-plasma phase, we propose that a direct calculation of the renormalized energy-momentum tensor (EMT) on the lattice using the gradient flow. From one-point function of EMT, authors in Ref.[1] obtained the interaction measure and thermal entropy. The results are consistent with the one obtained by the integral method. Based on the success, we try to measure the two-point function of EMT, which is related to the transport coefficients. Advantages of our method are (1) a clear signal because of the smearing effects of the gradient flow and (2) no need to calculate the wave function renormalization of EMT. In addition, we give a short remark on a comparison of the numerical cost between the positive- and adjoint-flow methods for fermions, needed to obtain the EMT in the (2+1) flavor QCD.
The energy density and the pressure of SU(3) gauge theory at finite temperature are studied by direct lattice measurements of the renormalized energy-momentum tensor obtained by the gradient flow. Numerical analyses are carried out with $beta=6.287$--$7.500$ corresponding to the lattice spacing $a= 0.013$--$0.061,mathrm{fm}$. The spatial (temporal) sizes are chosen to be $N_s= 64$, $96$, $128$ ($N_{tau}=12$, $16$, $20$, $22$, $24$) with the aspect ratio, $5.33 le N_s/N_{tau} le 8$. Double extrapolation, $arightarrow 0$ (the continuum limit) followed by $trightarrow 0$ (the zero flow-time limit), is taken using the numerical data. Above the critical temperature, the thermodynamic quantities are obtained with a few percent precision including statistical and systematic errors. The results are in good agreement with previous high-precision data obtained by using the integral method.
The 2d Heisenberg model --- or 2d O(3) model --- is popular in condensed matter physics, and in particle physics as a toy model for QCD. Along with other analogies, it shares with 4d Yang-Mills theories, and with QCD, the property that the configurations are divided in topological sectors. In the lattice regularisation the topological charge $Q$ can still be defined such that $Q in mathbb{Z}$. It has generally been observed, however, that the topological susceptibility $chi_{rm t} = langle Q^2 rangle / V$ does not scale properly in the continuum limit, i.e. that the quantity $chi_{rm t} xi^2$ diverges for $xi to infty$ (where $xi$ is the correlation length in lattice units). Here we address the question whether or not this divergence persists after the application of the Gradient Flow.
We report on a sub-percent scale determination using the omega baryon mass and gradient-flow methods. The calculations are performed on 22 ensembles of $N_f=2+1+1$ highly improved, rooted staggered sea-quark configurations generated by the MILC and CalLat Collaborations. The valence quark action used is Mobius Domain-Wall fermions solved on these configurations after a gradient-flow smearing is applied with a flowtime of $t_{rm gf}=1$ in lattice units. The ensembles span four lattice spacings in the range $0.06 lesssim a lesssim 0.15$ fm, six pion masses in the range $130 lesssim m_pi lesssim 400$ MeV and multiple lattice volumes. On each ensemble, the gradient-flow scales $t_0/a^2$ and $w_0/a$ and the omega baryon mass $a m_Omega$ are computed. The dimensionless product of these quantities is then extrapolated to the continuum and infinite volume limits and interpolated to the physical light, strange and charm quark mass point in the isospin limit, resulting in the determination of $sqrt{t_0}=0.1422(14)$ fm and $w_0 = 0.1709(11)$ fm with all sources of statistical and systematic uncertainty accounted for. The dominant uncertainty in this result is the stochastic uncertainty, providing a clear path for a few-per-mille uncertainty, as recently obtained by the Budapest-Marseille-Wuppertal Collaboration.
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