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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 $L/xi simeq 6$.
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The 2d O(3) model is widely used as a toy model for ferromagnetism and for Quantum Chromodynamics. With the latter it shares --- among other basic aspects --- the property that the continuum functional integral splits into topological sectors. Topology can also be defined in its lattice regularised version, but semi-classical arguments suggest that the topological susceptibility $chi_{rm t}$ does not scale towards a finite continuum limit. Previous numerical studies confirmed that the quantity $chi_{rm t}, xi^{2}$ diverges at large correlation length $xi$. Here we investigate the question whether or not this divergence persists when the configurations are smoothened by the Gradient Flow (GF). The GF destroys part of the topological windings; on fine lattices this strongly reduces $chi_{rm t}$. However, even when the flow time is so long that the GF impact range --- or smoothing radius --- attains $xi/2$, we do still not observe evidence of continuum scaling.
We study temperature dependence of the topological susceptibility with the $N_{f}=2+1$ flavors Wilson fermion. We have two major interests in this paper. One is a comparison of gluonic and fermionic definitions of the topological susceptibility. Two definitions are related by the chiral Ward-Takahashi identity but their coincidence is highly non-trivial for the Wilson fermion. By applying the gradient flow both for the gauge and quark fields we find a good agreement of these two measurements. The other is a verification of a prediction of the dilute instanton gas approximation at low temperature region $T_{pc}< T<1.5T_{pc}$, for which we confirm the prediction that the topological susceptibility decays with power $chi_{t}propto(T/T_{pc})^{-8}$ for three flavors QCD.
We compute the topological charge and its susceptibility in finite temperature (2+1)-flavor QCD on the lattice applying a gradient flow method. With the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson quarks, we perform simulations on a fine lattice with~$asimeq0.07,mathrm{fm}$ at a heavy $u$, $d$ quark mass with $m_pi/m_rhosimeq0.63$ but approximately physical $s$ quark mass with $m_{eta_{ss}}/m_phisimeq0.74$. In a temperature range from~$Tsimeq174,mathrm{MeV}$ ($N_t=16$) to $697,mathrm{MeV}$ ($N_t=4$), we study two topics on the topological susceptibility. One is a comparison of gluonic and fermionic definitions of the topological susceptibility. Because the two definitions are related by chiral Ward-Takahashi identities, their equivalence is not trivial for lattice quarks which violate the chiral symmetry explicitly at finite lattice spacings. The gradient flow method enables us to compute them without being bothered by the chiral violation. We find a good agreement between the two definitions with Wilson quarks. The other is a comparison with a prediction of the dilute instanton gas approximation, which is relevant in a study of axions as a candidate of the dark matter in the evolution of the Universe. We find that the topological susceptibility shows a decrease in $T$ which is consistent with the predicted $chi_mathrm{t}(T) propto (T/T_{rm pc})^{-8}$ for three-flavor QCD even at low temperature $T_{rm pc} < Tle1.5 T_{rm pc}$.
In lattice QCD with Wilson-type quarks, the chiral symmetry is explicitly broken by the Wilson term on finite lattices. Though the symmetry is guaranteed to recover in the continuum limit, a series of non-trivial procedures are required to recover the correct renormalized theory in the continuum limit. Recently, a new use of the gradient flow technique was proposed, in which correctly renormalized quantities are evaluated in the vanishing flow-time limit. This enables us to directly study the chiral condensate and its susceptibility with Wilson-type quarks. Extending our previous study of the chiral condensate and its disconnected susceptibility in (2+1)-flavor QCD at a heavy $u$, $d$ quark mass ($m_{pi}/m_{rho}simeq0.63$) and approximately physical $s$ quark mass, we compute the connected contributions to the chiral susceptibility in the temperature range of 178--348 MeV on a fine lattice with $asimeq0.07$ fm.
We compare lattice QCD determinations of topological susceptibility using a gluonic definition from the gradient flow and a fermionic definition from the spectral projector method. We use ensembles with dynamical light, strange and charm flavors of maximally twisted mass fermions. For both definitions of the susceptibility we employ ensembles at three values of the lattice spacing and several quark masses at each spacing. The data are fitted to chiral perturbation theory predictions with a discretization term to determine the continuum chiral condensate in the massless limit and estimate the overall discretization errors. We find that both approaches lead to compatible results in the continuum limit, but the gluonic ones are much more affected by cut-off effects. This finally yields a much smaller total error in the spectral projector results. We show that there exists, in principle, a value of the spectral cutoff which would completely eliminate discretization effects in the topological susceptibility.
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