Critical flavor number in the $2+1\mathrm{D}$ Thirring model

The Thirring model in $2+1$ spacetime dimensions, in which $N$ flavors of relativistic fermion interact via a contact interaction between conserved fermion currents, is studied using lattice field theory simulations employing domain wall fermions, which furnish the correct $\mathrm{U}(2N)$ global symmetry in the limit that the wall separation $L_s\to \infty$. Attention is focussed on the issue of spontaneous symmetry breakdown via a nonvanishing fermion bilinear condensate $⟨\overline{\psi }\psi ⟩\ne 0$. Results from quenched simulations are presented demonstrating that a nonzero condensate does indeed form over a range of couplings, provided simulation results are first extrapolated to the $L_s\to \infty$ limit. Next, results from simulations with $N=1$ using an RHMC algorithm demonstrate that U(2) symmetry is unbroken at weak coupling but plausibly broken at strong coupling. Correlators of mesons with spin zero are consistent with the Goldstone spectrum expected from $\mathrm{U}(2)\to \mathrm{U}(1)\otimes \mathrm{U}(1)$. We infer the existence of a symmetry-breaking phase transition at some finite coupling, and combine this with previous simulation results to deduce that the critical number of flavors for the existence of a quantum critical point in the Thirring model satisfies $0<N_c<2$, with strong evidence that in fact $N_c>1$.

Figure 1

Bilinear condensate $⟨\overline{\psi }\psi ⟩$ vs $g^{-2}$ for $N=1$, 2 on a $12^3\times L_s$ lattice with $m=0.01$.

Figure 2

Bose action density $\frac{N}{2g^2}⟨A_{\mu }^2⟩$ vs $g^{-2}$ for $N=1$, 2.

Figure 3

$⟨\overline{\psi }\psi ⟩$ vs $g^{-2}$ for $N=0$, 1, 2 on $L^3\times 16$ with $m=0.01$.

Figure 4

$⟨\overline{\psi }\psi ⟩$ vs $L_s$ for various $g^{-2}$ with $m=0.05$.

Figure 5

$⟨\overline{\psi }\psi ⟩$ vs $L_s$ for $m=0.01,\dots 0.05$ with $g^{-2}=0.4$, 0.8.

Figure 6

$⟨\overline{\psi }\psi {⟩}_{\infty }$ vs $m$ for $g^{-2}\in [0.2,1.0]$. For clarity some points with large error bars have been horizontally displaced.

Figure 7

$⟨\overline{\psi }\psi {⟩}_{L_s}$ vs $m$ for $g^{-2}=0.4$.

Figure 8

$⟨\overline{\psi }\psi ⟩$ vs $m$ for various $g^{-2}$,$L_s$ for the quenched surface model. The differing symbols for $g^{-2}=0.2$ denote $L_s=8,16,\dots 40$.

Figure 9

$⟨\overline{\psi }\psi ⟩$ vs $L_s$ for $g^{-2}=0.3$, 0.6. Open symbols correspond to $16^3$, and dashed lines fits to (21). The inset plots $⟨(\overline{\psi }\psi {)}^2⟩-⟨\overline{\psi }\psi {⟩}^2$ vs $L_s$ for various $g^{-2}$; the color code is that of Fig. 10.

Figure 10

Bilinear condensate $⟨\overline{\psi }\psi (m)⟩$ obtained for various $g^{-2}$ in the $L_s\to \infty$ limit. Open circles denote data from $16^3$. Open triangles denote data from the Thirring model with $N=2$.

Figure 11

The decay constant $\mathrm{\Delta }(m)$ obtained from fits to (21) for $g^{-2}=0.3$, 0.4.

Figure 12

The residual ${\delta }_h(L_s)$, defined in (14), for various $g^{-2}$ and $m=0.01$. For $g^{-2}=0.3$ results for all 5 mass values are shown, along with $16^3$, $m=0.01$ (open circles). For $g^{-2}=0.6$ results for $m=0.05$ are shown as open circles.

Figure 13

Time slice correlator for the Goldstone meson $C_G(\tau )$ on $12^3\times 40$ for various $g^{-2}$ with $m=0.01$ (filled symbols) and $m=0.05$ (open symbols).

Figure 14

Time slice correlator for the non-Goldstone meson $C_{NG}(\tau )$ on $12^3\times 40$ for various $g^{-2}$ with $m=0.01$ (filled symbols) and $m=0.05$ (open symbols).

Figure 15

The ratio $⟨\overline{\psi }\psi ⟩/m{\chi }_{\pi }$ vs $g^{-2}$ on $12^3\times 40$ for various $m$, together with corresponding data taken at $N=2$ on $12^3\times 16$ and $m=0.01$ [17].