We consider the fermionic quantum criticality of anisotropic nodal point semimetals in $d=d_L+d_Q$ spatial dimensions that disperse linearly in $d_L$ dimensions, and quadratically in the remaining $d_Q$ dimensions. When subject to strong interactions, these systems are susceptible to semimetal-insulator transitions concurrent with spontaneous symmetry breaking. Such quantum critical points are described by effective field theories of anisotropic nodal fermions coupled to dynamical order parameter fields. We analyze the universal scaling in the physically relevant spatial dimensions, generalizing to a large number $N_f$ of fermion flavors for analytic control. Landau damping by gapless fermionic excitations gives rise to nonanalytic self-energy corrections to the bosonic order-parameter propagator that dominate the long-wavelength behavior. We show that perturbative momentum shell RG leads to nonuniversal, cutof-dependent results, as it does not correctly account for this nonanalytic structure. In turn, using a completely general soft cutoff formulation, we demonstrate that the correct IR scaling of the dressed bosonic propagator can be deduced by enforcing that results are independent of the cutoff scheme. Using the soft cutoff RG with the dressed dynamical RPA boson propagator, we compute the exact critical exponents for anisotropic semi-Dirac fermions $(d_L=1$, $d_Q=1)$ to leading order in $1/N_f$ and to all loop orders. Applying the same method to relativistic Dirac fermions, we reproduce the critical exponents obtained by other methods, such as conformal bootstrap. Unlike in the relativistic case, where the UV-IR connection is reestablished at the upper critical dimension, nonanalytic IR contributions persist near the upper critical line $2d_L+d_Q=4$ of anisotropic nodal fermions. We present $\epsilon$ expansions in both the number of linear and quadratic dimensions. The corrections to critical exponents are nonanalytic in $\epsilon$, with a functional form that depends on the starting point on the upper critical line.

中文翻译：

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各向异性节点半金属的费米离子临界距离上临界尺寸：1 Nf中精确到领先的指数

我们考虑了各向异性节点半金属的费米离子量子临界 $d=d_{\mathrm{大号}}+d_{问}$ 线性分散的空间尺寸 $d_{\mathrm{大号}}$ 尺寸，然后在剩余的面积上平方 $d_{问}$尺寸。当受到强烈的相互作用时，这些系统易受半金属-绝缘体转变的影响，并伴有自发的对称破坏。此类量子临界点由与动态阶数参数场耦合的各向异性节点费米子的有效场理论来描述。我们分析了与物理相关的空间维度中的通用缩放，并将其概括为大量${ñ}_F$用于分析控制的费米子味。无间隙铁电激发引起的Landau阻尼引起对玻色子阶跃参数传播器的非解析自能校正，该校正器主导了长波行为。我们表明，扰动动量壳RG导致非普遍的，依赖截止的结果，因为它不能正确地说明这种非解析结构。反过来，使用完全通用的软截断公式，我们证明可以通过强制结果独立于截断方案来推断穿戴式玻色子传播器的正确IR标度。使用软截止RG和经过修饰的动力学RPA玻色子传播子，我们计算各向异性半狄拉克费米子的精确临界指数$（d_{\mathrm{大号}}=\mathrm{1个}$， $d_{问}=\mathrm{1个}）$ 领先于 $\mathrm{1个}/{ñ}_F$以及所有循环订单。将相同的方法应用于相对论狄拉克费米子，我们重现了通过其他方法（例如共形引导程序）获得的临界指数。与相对论情况不同，UV-IR连接在上临界尺寸处重新建立，非分析型IR贡献在上临界线附近持续存在$2d_{\mathrm{大号}}+d_{问}=4$各向异性节点费米子。我们提出$\epsilon$线性和二次维数的扩展。关键指数的更正是非解析的$\epsilon$，其功能形式取决于上临界线的起点。