Response to Comment on “The role of electron-electron interactions in two-dimensional Dirac fermions”

Abstract

Hesselmann et al. question one of our conclusions: the suppression of Fermi velocity at the Gross-Neveu critical point for the specific case of vanishing long-range interactions and at zero energy. The possibility they raise could occur in any finite-size extrapolation of numerical data. Although we cannot definitively rule out this possibility, we provide mathematical bounds on its likelihood.

Robust orbital diamagnetism in correlated Dirac fermions

“…This can be compared with the previous results for long-range Coulomb interacting Dirac fermions where the magnetic catalysis is dominant over the Fermi velocity enhancement at zero temperature and consequently the diamagnetism is suppressed [15]. Suppression of diamagnetism is expected to occur also in on-site interacting Hubbard models where the Fermi velocities are decreased by the interactions [51][52][53][54][55]. In figure 3, as the magnetic field becomes stronger, |M(V = 0.50t)| becomes even larger than |M(V = 0)| within the present model calculation.…”

“…Especially near the QCP, V V c , the CDW order parameter behaves as M CDW ∼ l −β/ν B ∼ B β/2ν with β 0.54, ν 0.80 corresponding to the N = 4 chiral Ising universality class. Note that, if present, the long-range part of the Coulomb interaction would be less important around the QCP and would not affect the criticality at zero magnetic field [42,47,[51][52][53].…”

“…The critical behaviors around a quantum critical point (QCP) of the semimetal-insulator phase transition have been well established mainly in absence of a magnetic field [35][36][37][38][39][40][41][42][43][44][45][46][47][48], and quantum criticality of magnetic catalysis can also be understood in a similar manner [49]. The scaling analysis shows that the Fermi velocity v F remains regular around a QCP [50], but numerical calculations demonstrate that v F decreases to some extent in interacting Dirac fermions which exhibit antiferromagnetic quantum phase transitions [51][52][53][54] and furthermore the reduced v F was observed in the molecular compound α-(BEDT-TTF) 2 I 3 [55]. Therefore, it is natural to expect suppression of diamagnetism in these systems similarly to the long-range Coulomb interacting graphene [15].…”

Robust orbital diamagnetism of correlated Dirac fermions in chiral Ising universality class

“…Based on this result, we suppose that the increase of v F tends to enhance M but at the same time the magnetic catalysis generating fermion mass does to suppress it, and as a result of this cancellation, M remains almost unchanged for small V . Note that a similar effect is expected in a real Dirac material, where a long-range part of the Coulomb interaction could suppress the diamagnetism via the magnetic catalysis at zero temperature with a partial cancellation by the enhanced Fermi velocity 12 , although it is expected to be less important when the short-range interaction is strong enough 38,43,[54][55][56] . As the magnetic field becomes stronger, |M (V = 0.5t)| becomes even larger than |M (V = 0)| within the present model calculation.…”

“…54, ν 0.80 corresponding to the N = 4 chiral Ising universality class. Note that, if present, the long-range part of the Coulomb interaction is less important around the QCP and does not affect the criticality at zero magnetic field 38,43,[54][55][56] . Because the system is gapped at B = 0 for any V > 0 due to the magnetic catalysis, the iDMRG numerical calculations with finite bond dimensions χ are stable and extrapolation χ → ∞ works well.…”