UV fixed-point structure of the three-dimensional Thirring model

We investigate the UV fixed-point structure of the three-dimensional Thirring model by means of the functional renormalization group. We classify all possible 4-fermi interactions compatible with the present chiral and discrete symmetries and calculate the purely fermionic renormalization group flow using a full basis of fermionic four-point functions in the pointlike limit. Our results show that the UV complete (asymptotically safe) Thirring model lies in a two-dimensional coupling plane which reduces to the conventional Thirring coupling only in the strict large-$N_{\mathrm{f}}$ limit. In addition to the Thirring universality class, which is characterized by one relevant direction (also at finite $N_{\mathrm{f}}$), two further interacting fixed points occur which may define new universality classes of second-order phase transitions also involving parity-broken phases. The $N_{\mathrm{f}}$ dependence of the Thirring fixed point sheds further light on the existence of an $N_{\mathrm{f}}$-controlled quantum phase transition above which chiral symmetry remains unbroken for arbitrary large coupling, even though a definite answer will require a direct computation of competing orders.

Figure 1

Positions of RG fixed points and (ir-)relevant directions for flavor numbers $N_{\mathrm{f}}=1$, 2, 4, 10, 100. Arrows denote the RG flow toward the IR. In agreement with Eq. (35), one relevant direction of every interacting fixed point points toward the Gaussian fixed point $\mathcal{O}$.

Figure 2

Critical exponents $\Theta$ as a function of flavor number $N_{\mathrm{f}}$ for the interacting fixed points $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$. The value of the corresponding other critical exponent is $\Theta =1$ for all $N_{\mathrm{f}}$. Note that the scale of the vertical axis changes at $\Theta =-1$.

Figure 3

Classification of Thirring-like 4-fermi theories determined by the fixed-point positions and the corresponding RG trajectories for $N_{\mathrm{f}}=1$ (arrows denote the flow towards the IR). Red (solid) lines depict separatrices that interpolate between fixed points and separate different regions. The vertical axis $\tilde{g}=0$ corresponds to models with a pure Thirring coupling. The dashed line ($g=\tilde{g}/2$) marks a generalized NJL model, see text.

Figure 4

Fixed-point positions and separatrices in the $(g,\tilde{g})$ coupling plane as a function of the flavor number for $1\le N_{\mathrm{f}}\le 7$. The horizontal slice at $N_{\mathrm{f}}=1$ is equivalent to Fig. 3. For larger $N_{\mathrm{f}}$, the IR attractive line $\mathcal{O}\mathcal{C}$ approaches the Thirring axis $\tilde{g}=0$, where the vector channel is expected to dominate toward the infrared, inhibiting chiral symmetry breaking.

Figure 5

Comparison of nonuniversal critical Thirring couplings from different methods. Solid lines display the critical couplings from this work taken as the $g$ coordinate of the intersection point of the separatrix $\mathcal{B}C$ with the Thirring axis $\tilde{g}=0$ for the sharp cutoff (upper solid/brown line) or the linear regulator (lower solid/red line). Dotted lines: Monte Carlo results (magenta with circles: 30, purple with squares: 31). Dashed lines: DSE approaches (from right to left: green/right [27], cyan/middle [28], and blue/left [29]).