Emergent strongly coupled ultraviolet fixed point in four dimensions with 8 Kähler-Dirac fermions

I use a gauge action that includes fundamental and adjoint plaquette terms with couplings $\beta\equiv\beta_{F}$ and $\beta_{A}$, related by $\beta_{A}/\beta_{F}=-0.25$ [33]. The gauge links in the staggered fermion operator are nHYP-smeared [44, 45] with smearing parameters $\alpha=(0.5,0.5,0.4)$. This lattice action has been used in multiple 8-, 4+8, and 12-flavor works [33, 17, 20, 46, 47, 48, 49], including the large-scale spectrum studies of the LSD collaboration [50, 51].

In the simulations I set the fermion mass to $am_{f}=0$ and use symmetric $L^{4}$ volumes with $L/a=8$, 10, 12, 16, and 20. The fermion boundary condition is antiperiodic in all four directions to control the fermion zero modes. The gauge fields obey periodic boundary conditions. Simulations with $am_{f}=0$ are well behaved in conformal systems or in the small volume deconfined regime of QCD-like systems. The S4 phase is confining but chirally symmetric, and the pions are not particularly light [33]. While numerical simulations are increasingly difficult at strong gauge coupling, the Hybrid Monte Carlo (HMC) updates are well controlled even in the chiral limit of the S4 phase.

For each staggered fermion field, I include $N_{PV}$ PV bosons. The PV bosons have the same action as the fermions but with bosonic statistics [37]. They are degenerate in mass and I choose their lattice mass $am_{PV}$ to be much larger than any IR scale of the system. When integrated out, the PV fields generate an effective gauge action. This is not an ultra-local action, but still local, with localization governed by the mass of the PV “pions”. With heavy PV mass the bound state meson masses are approximately $2am_{PV}$, i.e. the range of the effective gauge action is a few lattice spacings if $2am_{PV}\gtrsim 1.0$. The leading term of the induced gauge action is a plaquette built from smeared links with coefficient

where $N_{s}=2$ is the number of staggered fermion fields. Since $\beta^{(p)}_{\text{ind}}$ is negative, the bare gauge coupling $\beta$ has to increase to keep the effective gauge coupling $\beta_{\text{eff}}=\beta+\beta_{\text{ind}}$ and the lattice spacing constant. In general, more PV bosons and/or smaller $am_{PV}$ values generate a larger induced gauge action. The constant physics regime is pushed to weaker bare gauge coupling, which reduces the ultraviolet fluctuations as evidenced by larger plaquette expectation values. The effect of the PV bosons was investigated in detail in Ref. [37] for the 12-flavor SU(3) system. I want to emphasize that the effect of the PV bosons in the simulations is the generation of a local effective gauge action. The corresponding system has the same infrared properties at the Gaussian, as well as other possible fixed points as the original action, as long as $am_{PV}\gg 1/L$ and $am_{PV}\gg am_{f}$.

The nonperturbative UVFP in the 8-flavor SU(3) gauge system, if it exists, has at least two relevant parameters, the fermion mass and the new operator emerging at the UVFP. The scaling forms I consider below assumes the simplest scenario, i.e. that there are exactly two relevant parameters. If the fermion mass is set to zero, the RG flow is restricted to the critical surface of the massless theory. In that case, the UVFP has only one relevant operator, and finite size scaling methods can be used to study its properties. This is the scenario depicted in the sketches of Fig. 1.

Finite size scaling studies usually consider susceptibilities or bound state masses obtained from 2-point functions. Here I use the gradient flow coupling $g^{2}_{GF}$. The gradient flow (GF) [52, 53]. is a smoothing transformation that has been used extensively in lattice studies GF is not an RG transformation as it lacks the essential coarse-graining step. However, coarse-graining can be implemented at the level of expectation values. If one relates the lattice GF time $t/a^{2}$ to the RG scale change as $b\propto\sqrt{t/a^{2}}$, at large $t/a^{2}\gg 1$ the GF can be interpreted as a well-defined real-space RG transformation [54]. Observables at finite flow time correspond to RG flowed observables at energy scale $\mu\propto 1/\sqrt{8t}$, and exhibit hyperscaling in the vicinity of an RG fixed point.

The GF coupling was proposed in Ref. [39] for scale setting and in Ref. [40] to calculate the finite volume step scaling function in QCD-like or conformal system. It is defined as

where $E(t)$ is the energy density at bare gauge coupling $\beta$ and flow time $t$ on lattice volume $(L/a)^{4}$ ²²2If $\mathcal{N}=128\pi^{2}/(3(N_{c}^{2}-1))$, where $N_{c}$ refers to color, $g^{2}_{\text{GF}}$ matches the ${\overline{MS}}$ coupling at tree level perturbation theory [39]. In the non-perturbative phase this normalization is arbitrary, but I keep it for consistency.. In lattice calculations various local operators like the plaquette or the clover operator can be used to approximate $E$. A frequent choice to estimate $E$ is

where $\text{Re\,tr}\langle U_{\Box}\rangle$ and $\text{Re\,tr}\langle V_{\Box}(t)\rangle$ denote the expectation value of the plaquette before GF flow and at flow time $t$. The GF defines a non-linear RG transformation, therefore $\left\langle E(t)\right\rangle_{\beta,L/a}$ does not have an anomalous dimension, i.e. the $\eta$ exponent of the RG does not enter. Thus, the scaling dimension of the gradient flow coupling vanishes, its RG evolution depends only on the RG flow in the action parameter space.³³3The use of $t^{2}(3-\langle\text{Re\,tr}\langle V_{\Box}(t)\rangle)$ to approximate $g^{2}_{GF}$ is reminiscent of the operators used in Monte Carlo Renormalization Group (MCRG) studies [55, 56, 57, 58]. The main difference is that GF is continuous, so one is not restricted to scale change of 2. Also, the lattice volume does not decrease during GF, so larger flow time/ RG scale change is possible. . This property greatly simplifies the RG scaling relations, as only the scaling exponent related to the correlation length enters.

One can think of $g^{2}_{GF}$ as a scalar quantity that tracks the renormalized trajectory [59, 11]. It has several advantages, most important is that it is a simple expectation value that is easy to measure with high precision. Finite size scaling relations for $g^{2}_{GF}$ are derived in the usual way. At a second order phase transition one expects the correlation length to scale as

where $\beta_{\star}$ denotes the critical coupling. The combination

is a dimensionless scaling variable. Renormalization group scaling in finite volume therefore implies that $g^{2}_{\text{GF}}(\beta,L;c)$, $c=\sqrt{8t}/L$, depends on $(\beta/\beta_{\star}-1)\,L^{1/\nu}$, i.e. follows a unique but a priori unknown scaling function

First order phase transitions are expected to show similar scaling behavior but with discontinuity exponent $\nu=1/d=0.25$ in 4 dimensions.

If the RG $\beta$-function behaves as

in the vicinity of the critical point, the phase transition is continuous but “walking” or BKT-type. The correlation length scales as

and the corresponding scaling function depends on $L\,\text{exp}(-\zeta|\beta/\beta_{\star}-1|^{-\nu})$,

The 2-dimensional XY model has $\nu=1/2$, while the quadratic “walking” $\beta$ function in Eq. 8 corresponds to $\nu=1$. Both Eqs. 7 and 10 are leading order relations valid only in the vicinity of the critical point, $|\beta-\beta_{\star}|\ll 1$. One also has to assure that the flow does not drive the RG blocked correlation length below the lattice spacing, $\sqrt{t/a^{2}}\lesssim\xi$. At the same time $\sqrt{t/a^{2}}$ has to be large enough for the RG flow to reach the renormalized trajectory. This latter condition could limit the use of the smallest volumes, since $t=(cL)^{2}/8$ and $c\leq 0.5$ is commonly preferred⁴⁴4At larger $c$ values the GF “wraps around” the finite volume. That does not invalidate the use of $g^{2}_{GF}$, though the statistical errors increase with increasing $c$. A full investigation on the dependence on $c$, similar to Ref.[60], is a worthwhile project for the future. . The RT is approached according to the largest non-leading exponent, but corrections to scaling could be important. This, however, is beyond what my present data set can describe.

The scaling functions $f_{2nd}^{(c)}$ and $f_{BKT}^{(c)}$ depend on the parameter $c$, but should not depend on $\nu$, $\zeta$, and $\beta_{\star}$. In addition, different lattice actions should show scaling with the same $\nu$ exponent, though $\beta_{\star}$ and $\zeta$ are dependent on the action. Thus testing different $c$ values and different actions provides a consistency check and verifies the scaling form.

I investigated several different PV boson actions, two of them in detail. The first one contains 8 PV fields for each staggered flavor with mass $am_{PV}=0.75$ (8PV-m0.75 action), and the second one contains 4 PV fields per staggered flavor with mass $am_{PV}=0.5$ (4PV-m0.5 action). I have also looked at the action without PV bosons (0PV action). The 0PV action was studied previously in Ref. [17] at small but finite mass, and in the chiral limit but at weaker couplings in Ref. [20]. No previous work considered the 8-flavor systems at the phase transition directly in the chiral limit. In contrast, all simulations I present here are performed with $am_{f}=0$.

I start by comparing the plaquette expectation value, the finite volume GF coupling, the S4 order parameter for the plaquette, and the topological susceptibility of the three actions. The top panel of Fig. 2 shows the plaquette as the function of the bare gauge coupling $\beta$ on volumes $L/a=8$, 10, 12, 16, and 20 for the 0PV action. The discontinuity at the phase transition is too small to be resolved on the plot. Nevertheless, I have observed tunneling and 2-state signals in several ensembles, consistent with a first order phase transition. The GF coupling $g^{2}_{\text{GF}}$ at $c\equiv\sqrt{8t}/L=0.45$ is shown on the second panel. It exhibits a rapid change around $\beta\approx 4.3-4.55$ on these volumes, consistent with a phase transition around $\beta_{1st}\approx 4.6$, the value predicted with small but finite fermion masses in Ref.[17]. The strong coupling phase exhibits spontaneous breaking of the staggered single-site shift symmetry (S4 phase). The S4 order parameter that measures the staggered nature of the plaquette (Eq. 3. in Ref. [33]) is shown on the third panel. It is fairly noisy but useful to verify that the new phase shows the S4 symmetry breaking pattern. The S4 phase is confining but chirally symmetric, and the existence of a local order parameter ensures that it is separated from both the conformal and the chirally broken, confining phases by a phase transition [33] The bottom panel of Fig. 2 shows the topological susceptibility $\chi_{Q}=\langle Q^{2}\rangle/L^{4}$. In the chiral $am_{f}=0$ limit of conformal and QCD-like systems one expects $\chi_{Q}=0$, because the fermion determinant has a zero mode on isolated instantons. Staggered fermion lattice artifacts lift $\chi_{Q}$ slightly at finite lattice spacing [61]. However, in the S4 phase, the topological charge is large and no longer suppressed. This is further evidence of the very different nature of this new phase.

The largest GF coupling in the weak coupling phase of the 0PV model is $g^{2}_{\text{GF}}\lesssim 26$ with $L=16$ and $c=0.45$. The plaquette is fairly small at the phase transition, $\text{Re\,tr}\langle U_{\Box}\rangle\approx 1.0$ (normalized to 3.0), suggesting large UV fluctuations. This system has another first order phase transition at an even stronger coupling that separates the S4 phase from a QCD-like chirally broken, confining regime [17]. I am not concerned with that transition in this work.

The panels of Figs. 3 and 4 show the same quantities as Fig. 2 but for the 8PV-m0.75 and 4PV-m0.5 actions and volumes $L\leq 24$ ⁵⁵5Simulations with the PV actions are computationally faster, similar to the observation presented in Table I of Ref. [37] for $N_{f}=12$, thus larger volumes are feasible.. The GF coupling now appears continuous. None of my numerical simulations showed 2-state signals or phase tunneling either. The S4 order parameter becomes non-zero at the same place where $g^{2}_{\text{GF}}$ shows a qualitative change of $\beta$ dependence, around $g^{2}_{\text{GF}}\gtrsim 35$, $c=0.45$. This is also where the S4 order parameter and topological susceptibility show a sudden increase. The plaquette is significantly larger at the phase transition, $\text{Re\,tr}\langle U_{\Box}\rangle\approx 1.75$ and $\text{Re\,tr}\langle U_{\Box}\rangle\approx 1.7$, respectively. The bare gauge coupling of the phase transition has also shifted to larger bare coupling, from $\beta_{\star}\approx 4.6$ of the 0PV action to $\beta_{\star}\approx 8.7$ and $\beta_{\star}\approx 8.1$. This is the consequence of the induced effective action.

The strongest GF coupling with the 0PV action is $g^{2}_{\text{GF}}\approx 26$ on the volumes considered, compared to $g^{2}_{\text{GF}}\approx 35$ with both PV actions. The 0PV action has large vacuum fluctuations. It appears that those trigger a first order phase transition before the GF coupling is strong enough to undergo the continuous transition seen with the 4PV-m0.5 and 8PV-m0.75 actions. This suggests that the first order phase transition could be only a lattice artifact. The picture is similar to the frequently observed first order bulk transition with Wilson fermions that are not considered physical.

If the phase transition is continuous, it should follow a scaling form like Eq. 7 or Eq. 10. A first order phase transition should follow the scaling of Eq. 7 with exponent $\nu=1/4$. The critical coupling $\beta_{\star}$, exponent $\nu$ and parameter $\zeta$ can be determined based on a curve collapse analysis of $g^{2}_{\text{GF}}(x;c)$ vs. $x=|\beta_{\star}-\beta|^{\nu}\,L$ or $x=L\text{exp}\left(-\zeta|\beta/\beta_{\star}-1|^{-\nu}\right)$, respectively. I perform this analysis similar the Nelder-Mead method as described in [62].

Specifically, I chose a reference volume $L_{0}$ and interpolate $g_{GF}^{2}(\beta,L_{0};c)$ versus $\beta$ with a (smoothed) cubic spline. In the analysis below I use $L_{0}=16$. I fit the parameters $\beta_{\star}$, $\nu$ (and $\zeta$) by minimizing the deviation of $g_{GF}^{2}(\beta,L;c)$ from this interpolating curve at $\beta_{0}$, where $\beta_{0}$ is the solution of the equation

if fitting Eq. 7 or

if fitting Eq. 10 The procedure can be iterated by refining the “ideal” interpolating curve using more volumes, once the fit parameters are known approximately.