Critical behavior of the ($2+1$)-dimensional Thirring model

We investigate chiral symmetry breaking in the ($2+1$)-dimensional Thirring model as a function of the coupling as well as the Dirac flavor number $N_{\mathrm{f}}$ with the aid of the functional renormalization group. For small enough flavor number $N_{\mathrm{f}}<N_{\mathrm{f}}^{\mathrm{cr}}$, the model exhibits a chiral quantum phase transition for sufficiently large coupling. We compute the critical exponents of this second order transition as well as the fermionic and bosonic mass spectrum inside the broken phase within a next-to-leading order derivative expansion. We also determine the quantum critical behavior of the many-flavor transition which arises due to a competition between vector and chiral-scalar channels and which is of second order as well. Due to the problem of competing channels, our results rely crucially on the RG technique of dynamical bosonization. For the critical flavor number, we find $N_{\mathrm{f}}^{\mathrm{cr}}\simeq 5.1$ with an estimated systematic error of approximately one flavor.

Figure 1

Fermionic RG flow in the Gross-Neveu (GN)—Thirring (Th) coupling plane, compiled from Refs. 49, 50. The angle bisector in the third quadrant defines the theories with pure Nambu-Jona-Lasinio (NJL) interactions. Left panel: For $N_{\mathrm{f}}=1.75$ the Thirring fixed point $\mathcal{C}$ lies exactly on the NJL axis with ${\mathcal{L}}_{\mathrm{int}}\propto (S{)}^2$. Right panel: In the limit $N_{\mathrm{f}}\to \infty$, the Thirring fixed point $\mathcal{C}$ lies on the Thirring axis with ${\mathcal{L}}_{\mathrm{int}}\propto (V{)}^2$. Middle panel: For intermediate flavor number (here $N_{\mathrm{f}}\simeq 5$), we expect a competition between the scalar-type NJL channel $(S{)}^2$ and the vector-type Thirring channel $(V{)}^2$.

Figure 2

Log plot of the dynamically generated renormalized fermion mass ${\overline{m}}_{\mathrm{R},\mathrm{f}}^2$ in units of the renormalized mass ${\overline{m}}_{\mathrm{R},\rho }^2$ of a radial bosonic excitation on top of the chiral condensate. Our estimates for the critical flavor number, $N_{\mathrm{f}}^{\mathrm{cr}}\simeq 5.1$ for a linear regulator, is amended by an error bar (gray shaded area) obtained by varying the regularization scheme (e.g., $N_{\mathrm{f}}^{\mathrm{cr}}\simeq 5.8$ for the sharp cutoff).

Figure 3

Order parameter (black dots) versus $N_{\mathrm{f}}^{\mathrm{cr}}/N_{\mathrm{f}}-1$ in a double-log plot, showing very good compatibility with power-law scaling behavior $⟨\phi ⟩\propto (N_{\mathrm{f}}^{\mathrm{cr}}/N_{\mathrm{f}}-1{)}^b$. The slope of the regression line (gray) is $b\simeq 0.44$.

Figure 4

Left panel: Nonuniversal UV fixed-point values of the Thirring fixed point for the bosonic (black) and fermionic RG flow (gray, compiled from Ref. 49) for the linear regulator. In the bosonic language already for $N_{\mathrm{f}}\ge 4$ the fixed point is located in the pure fermion-vector sector, while in the fermionic language this is the case only for large $N_{\mathrm{f}}$. Fermionic and bosonic descriptions coincide only for $N_{\mathrm{f}}\to \infty$ but approach each other again for small $N_{\mathrm{f}}$. Right panel: Largest critical exponent ${\Theta }_1$ and subleading exponent ${\Theta }_2$ (inset).

Figure 5

Box diagrams contributing to the flow of $g_{\varphi }$ and $g_V$. Solid lines are fermions, dashed lines are scalar fields, and wiggly lines vector fields. Doubled inner lines denote full propagators $G_k=({\Gamma }_{k,0}^{(2)}+R_k{)}^{-1}$.

Figure 6

Left panel: Nonuniversal UV fixed-point values of dynamically bosonized RG flow for the linear regulator. The couplings have been multiplied with suitable $N_{\mathrm{f}}$ factors; for better visibility, the scalar mass has been multiplied with an additional factor 50. Right panel: Comparison of dynamically bosonized RG (black) with fermionic RG (gray). The improvement due to dynamical bosonization becomes obvious by comparing this plot with the left panel of Fig. 4.

Figure 7

Left panel: Correlation length exponent $\nu$ extracted from the linearized flow in the fixed-point regime with linear (solid) and sharp regulator (dashed). The difference (gray shaded band) serves as a rough error estimate. The critical exponent is to a large extent independent of the regulator, while the uncertainty increases for $N_{\mathrm{f}}\searrow 2$, as shown in the inset. For increasing $N_{\mathrm{f}}$ it approaches rapidly the fermionic result $1/{\Theta }_1=1$. Right panel: Corrections-to-scaling exponent $\omega$.

Figure 8

Anomalous dimension ${\eta }_{\varphi }$ at the fixed point, with linear regulator (solid) and sharp regulator (dashed).

Figure 9

RG evolution of renormalized masses in $\chi \mathrm{SB}$ regime for $k<k^{*}$ and $N_{\mathrm{f}}=2$. We stop the flow when the radial mass ${\overline{m}}_{\mathrm{R},\rho }^2$ approaches its maximum (vertical line).

Figure 10

Scalar mass ${\overline{m}}_{\mathrm{R},\tau }^2$ in units of the radial mass ${\overline{m}}_{\mathrm{R},\rho }^2$ on logarithmic scale (linear cutoff). Gray shaded area: Estimates for critical flavor number $N_{\mathrm{f}}^{\mathrm{cr}}\simeq 5.1$ (linear cutoff) and 5.8 (sharp cutoff).

Figure 11

RG evolution of dimensionless couplings for the “pure” Thirring model with $N_{\mathrm{f}}=2$ and initial couplings $m_{\varphi ,\Lambda }^2={10}^6$, ${\lambda }_{i,\Lambda }=0$, $h_{\varphi ,\Lambda }^2=0$, and two different $g_{V,\Lambda }=N_{\mathrm{f}}{h}_{V,\Lambda }^2/m_{V,\Lambda }^2$ just above (black) and below (gray) $g_{\mathrm{cr}}$. Near $\ln (k/\Lambda )\simeq -5$ the flow approaches the UV fixed-point regime. The black solid line corresponds to $m_{\varphi }^2$ above and $\kappa$ below the $\chi \mathrm{SB}$ scale $\ln (k^{*}/\Lambda )\simeq -15$, respectively.

Figure 12

Critical coupling $1/g_{\mathrm{cr}}$ for the pure Thirring model with initially decoupled scalar sector as a function of $N_{\mathrm{f}}$. For $N_{\mathrm{f}}\gtrsim 5.1$ (gray area) we find no signature of chiral symmetry breaking (dashed line).

Figure 13

Critical behavior of pure Thirring model with $N_{\mathrm{f}}=2$: order parameter $⟨\phi ⟩\propto |\delta g{|}^{\beta }$ (left panel) and inverse susceptibility ${\chi }^{-1}={\overline{m}}_{\rho }^2\propto |\delta g{|}^{\gamma }$ (right panel) in double-log plot, showing the expected power law. The slope of the regression line (gray dashed) is $\beta =2.769$ and $\gamma =1.553$, respectively. Left inset: For very large coupling on the pure Thirring axis the order parameter decreases again; see Sec. 10.

Figure 14

Critical exponents $\delta$ and $\nu$ for a theory defined at the fixed point $\mathcal{B}$ with more than one RG relevant direction; to be compared with Fig. 5 of Ref. 31.