Critical flavor number in the three dimensional Thirring model

Abstract

We present results of a Monte Carlo simulation of the three dimensional Thirring model with the number of fermion flavors N f varied between 2 and 18. By identifying the lattice coupling at which the chiral condensate peaks, simulations are be performed at couplings g 2 (N f ) corresponding to the strong coupling limit of the continuum theory. The chiral symmetry restoring phase transition is studied as N f is increased, and the critical number of flavors estimated as N f c = 6.6(1). The critical exponents measured at the transition do not agree with self-consistent solutions of the Schwinger-Dyson equations; in particular there is no evidence for the transition being of infinite order. Implications for the critical flavor number in QED3 are briefly discussed.PACS numbers: PACS: 11.10. Kk, 11.30.Rd, 11.15.Ha The study of quantum field theories in which the ground state shows a sensitivity to the number of fermion flavors N f is intrinsically interesting. According to certain approximate solutions of Schwinger-Dyson equations (SDEs), in d = 3 spacetime dimensions both quantum electrodynamics (QED 3 ) and the Thirring model display this phenomenon. Both models have been proposed as effective theories describing different regions of the cuprate phase diagram, Thirring describing the superconducting phase, while QED 3 supposedly describes the nonsuperconducting "pseudogap" behaviour seen in the underdoped regime [1,2]. The Thirring model is a theory of fermions interacting via a current contact interaction:where ψ i ,ψ i are four-component spinors, m is a parityconserving bare mass, and the index i runs over N f distinct fermion flavors. In the chiral limit m → 0 the Lagrangian (1) shares the same global U(1) chiral symmetry ψ → e iαγ5 ψ,ψ →ψe iαγ5 as QED 3 . Since the coupling g 2 has mass dimension −1, naive power-counting suggests that the model is non-renormalisable. However [3,4,5], an expansion in powers of 1/N f , rather than g 2

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

“…By contrast, in the DSE studies [24,25] an essential scaling behavior of the Kosterlitz-Thouless type has been found, that is to say, a phase transition of infinite order. As argued in [31], the nature of the transition in these studies appears to depend on whether the strong-coupling limit is taken before or after N f N cr f . The scaling analysis on the lattice [31,32] is consistent with a power-law behavior corresponding to a secondorder phase transition which qualitatively confirms, but quantitatively deviates from the Kondo scenario [26].…”

“…By constructing an effective potential for the chiral order parameter, up to leading order of the 1/N f expansion N cr f = 2 has been found [26]. Extensive Monte Carlo simulations point to N cr f 6.6 [31]. The true value of N cr f is of decisive relevance for the applicability of such relativistic fermion effective theories for condensed matter systems, where the number of flavors is typically N f = 2.…”

“…Since the advent of graphene [1] there has been an enormous amount of renewed interest in (2+1)-dimensional relativistic fermion systems. Various variants of (2+1)d quantum electrodynamics [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the (2+1)d Thirring model [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] are actively discussed as effective descriptions of graphene's exceptional electronic properties [34][35][36][37][38][39][40][41]. Moreover, they are likewise intrinsically interesting: in these models the number of fermion flavors N f serves as a control parameter for a quantum phase transition at a critical value N cr f .…”

Renormalization group flow of quartic perturbations in graphene: Strong coupling and large-Nlimits

“…Of course, this is only an extrapolation of our leading-order result to finite N. Nevertheless, it is quite interesting that N crit is close to the real-world value of N = 4. (For comparison, for the Lorentz-invariant Thirring model the critical number of four-component Dirac fermions is quoted as 6.6(1) [10], to be compared with N crit /2 ≈ 2.) Due to the limitation of our calculations, we cannot determine whether the exact N crit is smaller or larger than 4.…”

Strong interaction effects at a Fermi surface in a model for voltage-biased bilayer graphene

“…2 would be needed to distinguish these two possibilities unambiguously. Another route would be to perform a "biased bilayer" study for a related 2 + 1d theory, the Thirring model [17], whose behavior as a function of N f and μ is qualitatively similar to the model here [18], but whose continuum action is manifestly covariant implying a t ≡ a s throughout.…”