Renormalization group flow

In this appendix, we present details of the renormalization group (RG) analysis of the continuum field theories. Due to the lack of Lorentz and continuous spatial rotational symmetries in the low-energy models, the Fermi and bosonic velocities, as well as their anisotropies, will in general receive different loop corrections. In order to appropriately take this multiple dynamics [60, 61] into account, it is useful to employ a regularization in the frequency only, which preserves the property that the different momentum components can be rescaled independently. This allows us to keep the boson velocities \(c \equiv c_+ = c_-\) fixed, i.e., we measure the Fermi velocities in units of \(c=1\). Integrating over the “frequency shell” \(\Lambda/b \leq |\omega| \leq \Lambda\) with \(b>1\) and all momenta causes the velocities and couplings to flow at criticality \(r=0\) as

(1)\[\begin{split} \frac{\du v_\parallel}{\du \ln b} & = \frac12 (\eta_\phi - \eta_+ - 2 \eta_\psi) v_\parallel - F(v_\parallel,v_\perp) g^2, % -\frac{5g^2}{3} v_\parallel - \frac{2g^2}{15} (5 N_\sigma-7 v_\parallel) (v_\perp-1) % \frac{g^2}{15} \left[v_\parallel \left(14 v_\perp - 39 \right)-10 N_\sigma \left(v_\perp-1\right)\right] % \\ % \frac{\du v_\perp}{\du \ln b} & = \frac12 (\eta_\phi - \eta_- - 2 \eta_\psi) v_\perp + F(v_\perp,v_\parallel) g^2, % - \frac{g^2}{15 v_\parallel} (10 N_\sigma+7 v_\parallel) (v_\perp-1)-\frac{g^2}{3} % -\frac{g^2}{15 v_\parallel} \left[10 N_\sigma (v_\perp-1)+v_\parallel (7 v_\perp-2)\right] % \\ % \frac{\du g^2}{\du \ln b} & = \left(\epsilon - \frac{\eta_++\eta_-}{2} - 2 \eta_\psi\right) g^2 - 2 G(v_\parallel, v_\perp) g^4, % \\ % \frac{\du \lambda}{\du \ln b} & = \left(\epsilon - \frac{\eta_++\eta_-}{2} - \eta_\phi\right) \lambda - 18 \lambda^2 + \frac{N' g^4}{16 v_\parallel v_\perp},\end{split}\]

with the anomalous dimensions \(\eta_\psi = g^2 H(v_\parallel,v_\perp)\), \(\eta_\phi = N' g^2/(12v_\parallel v_\perp)\), and \(\eta_\pm = a_\pm N' g^2 v_\perp/(12v_\parallel)\), to the one-loop order. Here, the angular integrals are performed in \(d=2\), while the dimensions of the couplings are counted in general \(d\) [51, 117]. At the present order, the flows of the two models differ only in the definition of the coefficients \(a_\pm\), with \(a_+ = 0\), \(a_- = 2\) (\(a_+=a_-=1\)) in the \(C_{2v}\) (\(C_{4v}\)) model, and the number of spinor components \(N' = 4N_\sigma\) (\(N' = 8N_\sigma\)). Our regularization scheme allows the evaluation of the one-loop integrals in closed form, leading to the functions

(2)\[\begin{align} F(v_1, v_2) & = \frac{1}{\pi} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{v_1 q_1^2 \du q_1 \du q_2}{\left(1+q_1^2+q_2^2\right)^2 \left(1+v_1^2 q_1^2+v_2^2 q_2^2\right)} % \nonumber \\ & = \frac{v_1 \left[v_1 \left(v_2^2-1\right) \sqrt{\frac{1-v_1^2}{v_2^2-1}}+(v_1+v_2) \sin ^{-1}\left({v_1}{\sqrt{\frac{v_2^2-1}{v_2^2-v_1^2}}}\right)-(v_1+v_2) \csc ^{-1}\left(\sqrt{\frac{v_2^2-v_1^2}{v_2^2-1}}\right)\right]}{\left(v_1^2-1\right) \left(v_2^2-1\right) (v_1+v_2) \sqrt{\frac{1-v_1^2}{v_2^2-1}}}\,, \\ % % G(v_1, v_2) & = \frac{1}{2\pi} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{1-v_1^2 q_1^2+v_2^2 q_2^2}{\left(1+q_1^2+q_2^2\right) \left(1+v_1^2 q_1^2+v_2^2 q_2^2\right)^2}\du q_1 \du q_2 % \nonumber \\ % & = \frac{\left(v_2^2-1\right) (v_1 v_2-1) \sqrt{\frac{1-v_1^2}{v_2^2-1}}+\left(v_1^2+v_2^2-2\right) \csc ^{-1}\left(\frac{1}{v_1}\sqrt{\frac{v_2^2-v_1^2}{v_2^2-1}}\right)-\left(v_1^2+v_2^2-2\right) \csc ^{-1}\left(\sqrt{\frac{v_2^2-v_1^2}{v_2^2-1}}\right)}{2 \left(v_1^2-1\right) \left(v_2^2-1\right)^2 \sqrt{\frac{1-v_1^2}{v_2^2-1}}} % \nonumber \\ & \quad % +\frac{v_1 \left[\left(v_2^2-1\right) \sqrt{\frac{1-v_1^2}{v_2^2-1}}-v_1 (v_1+v_2) \sin ^{-1}\left({\sqrt{\frac{v_2^2-1}{v_2^2-v_1^2}}}\right)+v_1 (v_1+v_2) \csc ^{-1}\left(\frac{1}{v_1}\sqrt{\frac{v_2^2-v_1^2}{v_2^2-1}}\right)\right]}{2 \left(v_1^2-1\right) \left(v_2^2-1\right) (v_1+v_2) \sqrt{\frac{1-v_1^2}{v_2^2-1}}} % \nonumber \\ & \quad % +\frac{v_2^4 (v_1+v_2) \sin ^{-1}\left({\sqrt{\frac{v_2^2-1}{v_2^2-v_1^2}}}\right)}{2 v_2^2 \left(v_2^2-1\right)^2 (v_1+v_2) \sqrt{\frac{1-v_1^2}{v_2^2-1}}} % % \nonumber \\ & \quad % +\frac{\left(v_1^3+v_1^2 v_2+v_1 v_2^2+v_2^3\right) \sin ^{-1}\left({v_1}{\sqrt{\frac{v_2^2-1}{v_2^2-v_1^2}}}\right)- 2 \left(v_1^2-1\right) v_2^3 \sqrt{\frac{1-v_1^2}{v_2^2-1}}}{4 \left(v_1^2-1\right) v_2^2 \left(v_2^2-1\right) (v_1+v_2) \sqrt{\frac{1-v_1^2}{v_2^2-1}}} % \nonumber \\ & \quad % - \frac{\left(v_2^2+1\right) \left[v_1^3 \left(2 v_2^2-1\right)+v_1^2 v_2 \left(2 v_2^2-1\right)-v_1 v_2^2-v_2^3\right] \csc ^{-1}\left(\frac{1}{v_1}\sqrt{\frac{v_2^2-v_1^2}{v_2^2-1}}\right)}{4 \left(v_1^2-1\right) v_2^2 \left(v_2^2-1\right)^2 (v_1+v_2) \sqrt{\frac{1-v_1^2}{v_2^2-1}}}\,, \nonumber \\ % % H(v_1, v_2) & = \frac{1}{\pi} \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{\du q_1 \du q_2}{\left(1+q_1^2+q_2^2\right)^2 \left(1+v_1^2 q_1^2+v_2^2 q_2^2\right)} % \nonumber \\ & = \frac{\left(v_1^2+v_2^2-2 v_1^2 v_2^2\right) \left[\csc ^{-1}\left(\sqrt{\frac{v_2^2-v_1^2}{v_2^2-1}}\right) - \csc ^{-1}\left(\frac{1}{v_1}\sqrt{\frac{v_2^2-v_1^2}{v_2^2-1}}\right)\right]}{\left(v_1^2-1\right) \left(v_2^2-1\right)^2 \sqrt{\frac{1-v_1^2}{v_2^2-1}}} % % \nonumber \\ & \quad % -\frac{(v_1 v_2-1) \sqrt{\frac{1-v_1^2}{v_2^2-1}}}{\left(v_1^2-1\right) \left(v_2^2-1\right) \sqrt{\frac{1-v_1^2}{v_2^2-1}}}\,. \end{align}\]

The above one-loop flow equations admit a nontrivial fixed point that is characterized by anisotropic Fermi velocities \(v_\parallel^* = 0\) and \(v_\perp^* = 1/\sqrt{a_-}>0\), and vanishing \(g^2_*\) and \(\lambda^*\), but finite ratio \((g^2/v_\parallel)_* = 12\sqrt{a_-}\epsilon/N' + \mathcal O(\epsilon^2)\). Perturbations of the couplings \(g^2\) and \(\lambda\) and the Fermi velocity \(v_\perp\) away from this fixed point turn out to be irrelevant; however, the flow of \(v_\parallel\) near the fixed point is

(3)\[\begin{align} \left.\frac{\du v_\parallel}{\du \ln b}\right|_{g^2_*,v_\perp^*} & = \frac{\epsilon}{2} (1-a_+) v_\parallel - \frac{20\epsilon}{N'} v_\parallel^2 + \mathcal O(v_\parallel^3). \end{align}\]

Hence, in the \(C_{4v}\) model with \(a_+ = a_- = 1\), \(v_\parallel\) is marginally irrelevant, rendering the fixed point stable. The fixed point represents a quantum critical point with maximally anisotropic Fermi velocities \((v_\parallel^*, v_\perp^*) = (0,1)\) and boson anomalous dimensions, describing the temporal and spatial decays of the order-parameter correlations, as \(\eta_\phi = \epsilon\) and \(\eta_+ = \eta_- = \epsilon\), respectively. The fermion anomalous dimension becomes \(\eta_\psi = 0\). In the vicinity of this fixed point, the flow of \(v_\parallel\) can be integrated out analytically, reading

(4)\[\begin{align} v_\parallel(b) \simeq \frac{N'}{20\epsilon \ln b}, \end{align}\]

where we have assumed \(b \gg 1\) for simplicity. This demonstrates that the Fermi velocity flow in the vicinity of the \(C_{4v}\) fixed point is logarithmically slow, reflecting the fact that \(v_\parallel\) is marginally irrelevant at this fixed point. This indicates that exponentially large lattice sizes are needed to ultimate reach the fixed point. By contrast, in the \(C_{2v}\) model with \(a_+ = 0\) and \(a_- = 2\), \(v_\parallel\) is a relevant parameter near the maximal-anisotropy fixed point and flows to larger values. By numerically integrating out the flow, we find that the parameters \(v_\perp\), \(v_\parallel\), and \(g^2\) flow to a new nontrivial stable fixed point at which the boson anomalous dimensions satisfy a sum rule, \(\eta_++\eta_- + 2\eta_\phi = 2\epsilon\) with \(0 = \eta_+ < \eta_-, \eta_\phi < \epsilon\). The fixed point is located at \((v_\parallel^*, v_\perp^*) = (0.1611, 0.5942)\) and \((g^{2}_{*},\lambda^*) = (0.2123,0.1755)\epsilon + \mathcal O(\epsilon^2)\) for \(N' = 4N_\sigma = 4\). We find the corresponding anomalous dimensions as \((\eta_\phi,\eta_+,\eta_-,\eta_\psi) = (0.7391, 0, 0.5219, 0.1643)\epsilon + \mathcal O(\epsilon^2)\), reflecting again the fact that the character of the stable fixed point in the \(C_{2v}\) model is different from the one of the \(C_{4v}\) model. The different behaviors of the Fermi velocities in the two models is illustrated in Fig. 1, which shows the renormalization group flow in the \(v_\parallel\)-\(v_\perp\) plane. For visualization purposes, we have fixed the ratios \(g^2/(v_\perp v_\parallel)\) to their values at the respective stable fixed points in these plots. We have explicitly verified that \(g^2/(v_\perp v_\parallel)\) corresponds to an irrelevant parameter near these fixed points (marked as red dots in Fig. 1).

Fig. 1 Renormalization group flow in the \(v_\parallel\)-\(v_\perp\) plane for (a) the \(C_{2v}\) model and (b) the \(C_{4v}\) model. Arrows denote flow towards infrared. The fixed point at \((v_\parallel^*, v_\perp^*) = (0,1/\sqrt{a_-})\) and \((g^2/v_\parallel)_* = 12\sqrt{a_-}\epsilon/N'\) is unstable in the \(C_{2v}\) model [black dot in (a)], but stable in the \(C_{4v}\) model [red dot in (b)]. In the \(C_{2v}\) model, there is a nontrivial stable fixed point at \((v_\parallel^*, v_\perp^*) = (0.1611, 0.5942)\), with \(g^2_* = 0.2123\epsilon\) for \(N' = 4\) [red dot in (a)]. For visualization purposes, we have fixed the ratio \(g^2/(v_\perp v_\parallel)\) to its value at the respective stable fixed point (red dots) in these plots.

To make further contact with the QMC data displayed in Fig. 2.14(e), we show in Fig. 2(a,b) the Fermi velocity ratio \(v_\perp/v_\parallel\) as function of RG scale \(1/b\) in the two models, assuming an isotropic ratio \(v_\perp/v_\parallel = 1\) at the ultraviolet scale \(b=1\), for different initial values of the interaction parameter \(g^2/(v_\parallel v_\perp)\). We emphasize that a sizable deviation between the two models is observable only at very low energies \(1/b \lesssim 0.01\), while the RG flows in the high-energy regime are very similar for the employed starting values. Identifying the RG energy scale \(1/b\) roughly with the inverse lattice size \(1/L\), this result explains why the lattice sizes available in our simulations are too small to detect a substantial difference in the finite-size scaling of \(v_\perp/v_\parallel\). This also implies that the estimates for the critical exponent obtained from the finite-size analysis of the QMC data describes only an intermediate regime, in which the RG flow is not yet fully integrated out. Let us illustrate this point further for the case of the \(C_{4v}\) model. In this case, we can define a scale-dependent effective correlation-length exponent by using the scaling relation

(5)\[ 1/\nu_\text{eff}(b) = 2 - \eta_\phi^\text{eff}(b),\]

where \(\eta_\phi^\text{eff}(b) = N' g^2(b) /[12 v_\perp(b) v_\perp(b)]\) is the effective boson anomalous dimension. This relation becomes exact in the vicinity of the \(C_{4v}\) fixed point, for which \(\lambda^\ast = 0\). The effective correlation-length exponent \(1/\nu_\text{eff}\) is plotted as function of the RG scale \(1/b\) in Fig. 2(c) for different values of the initial interaction parameter \(g^2/(v_\parallel v_\perp)\). We note that the approach to \(\nu_\text{eff} \to 1\) in the deep infrared is extremely slow, with sizable deviations from the fixed-point value at intermediate scales. Interestingly, while the behavior in the high-energy regime \(1/b \gtrsim 0.05\) is nonuniversal and strongly depends on the particular starting values of the RG flow, a quasiuniversal regime emerges at intermediate energy \(1/b \lesssim 0.05\), in which the exponents still drift, but have only a very weak dependence on the initial interaction parameters. This quasiuniversal behavior is a characteristic feature of systems with marginal or close-to-marginal operators [65, 66]. Here, it arises from the slow flow of the velocity anisotropy ratio \(v_\perp/v_\parallel\), which implies that the effective exponents will become functions of \(v_\perp/v_\parallel\) only, but not of the ultraviolet starting values of the interaction parameters. The quasiuniversality reflects the fact that there is only one slowly decaying perturbation to the fixed point (i.e., the leading irrelevant operator), whereas all other perturbations decay quickly, and hence have died out once \(1/b\lesssim 0.05\). Importantly, the largest lattice sizes available in the QMC simulations appear to be just large enough to approach the quasiuniversal regime, if we again identify \(1/b\) roughly with \(1/L\). Reassuringly, for \(L=20\), we therewith obtain the RG estimate \(1/\nu_\text{eff} \simeq 1.20 \dots 1.25\), which is in the same ballpark as the estimate from the finite-size scaling analysis of the QMC data discussed in the main text.

Fig. 2 (a,b) Ratio of Fermi velocities \(v_\perp/v_\parallel\) as function of RG scale \(1/b\) in the \(C_{2v}\) model (blue) and \(C_{4v}\) model (green) for different starting values of the interaction parameter \(g^2/(v_\parallel v_\perp)\) at the ultraviolet scale \(b=1\). Here, we have numerically integrated out the full RG flow in the \((v_\parallel, v_\perp, g^2)\) parameter space, assuming initial velocities

\(v_\parallel(b=1) = v_\perp(b=1) = 0.25\), and \(g^2/(v_\parallel v_\perp)(b=1)\) between 50% and 100% of the value at the respective stable fixed point. (a) Semilogarithmic plot, demonstrating the finite infrared anisotropy in the \(C_{2v}\) model and the logarithmic divergence in the \(C_{4v}\) model. (b) Same data as in (a), but using a linear plot, illustrating the similarity of the anisotropy flows in the two models on the high-energy scale, to be compared with the QMC data shown in Fig. 2.14. (c) Effective correlation-length exponent \(1/\nu_\text{eff}\) as function of RG scale \(1/b\) in the \(C_{4v}\) model in semilogarithmic plot, defined according to Eq. (5), illustrating the drifting of the exponents and the quasiuniversal behavior for \(1/b \lesssim 0.05\). We have used the same ultraviolet starting values as in (a,b).