Comment on “Absence of Spin Liquid in Nonfrustrated Correlated Systems”

Abstract

In a recent Letter, Hassan and Sénéchal [1] discussed the existence of a spin-liquid phase of the half-filled Hubbard model on the honeycomb lattice. Using schemes, such as the variational cluster approximation (VCA) and the cluster dynamical mean field theory (CDMFT) in combination with exact diagonalization (ED), they argued that a single bath orbital per site of the six-atom unit cell is insufficient and leads to the erroneous conclusion that the system is gapped for all nonzero values of the onsite Coulomb interaction U . In contrast, we point out here that, in the case of the honeycomb lattice, six bath levels per six-site unit cell are perfectly adequate for the description of short-range correlations. Instead, we demonstrate that it is the violation of long-range translation symmetry inherent in CDMFT-like schemes which opens a gap at Dirac points. The gap found at small U therefore does not correspond to a Mott gap. As a result, present CDMFT schemes are not suitable for the identification of a spin-liquid phase on the honeycomb lattice.As shown in Ref.[2], the cluster self-energy obtained within ED CDMFT [3] using six bath levels is in nearly quantitative agreement with results derived within continuous-time quantum Monte Carlo (QMC) CDMFT [4]. As the bath in QMC is infinite, the selfenergy is not subject to finite-size effects. The reason for the good agreement is that, because of the semimetallic nature of the honeycomb lattice, the projection of the infinite-lattice bath Green's function onto a finitecluster Anderson Green's function is not plagued by the low-energy-low-temperature disparities that arise in the case of correlated metals and the square-lattice Hubbard model. Moreover, because of the hexagonal symmetry, this projection can be performed in the diagonal molecular-orbital basis with non-symmetrical density of states components [3]. The main issue in Ref.[1] concerning the symmetry of bath levels then does not arise and the two independent bath Green's function components are fitted accurately using a total of six parameters (see Fig. 24 of Ref.[2]), while in the site basis only one fit parameter is available.Within CDMFT, the self-energy at Dirac points K exhibits the low-energy behavior [3], where a is the initial slope of Im [Σ 11 (iω n ) − Σ 13 (iω n )] and b the low-energy limit of Re [Σ 12 (iω n ) − Σ 14 (iω n )]. Here, Σ ij is the self-energy in the site basis and ω n are Matsubara frequencies. The excitation gap at temperatures T → 0 is given by ∆ ≈ 2|b|/ √ 1 − a (see also Ref.[5]). Using the site notation: a 1 = (0, 0), a 2 = (0, 1), a 3 = ( √ 3/2, 3/2), a 4 = ( √ 3, 1), a 5 = ( √ 3, 0), and a 6 = ( √ 3/2, −1/2), the elements Σ 12 and Σ 14 within this cell are independent, so that ∆ = 0. However, site 1 is also connected to site 4 at (− √ 3/2, −1/2) in the neighboring cell, requiring Σ 12 = Σ 14 which is not obeyed in CDMFT. Therefore, this violation of translation symmetry is responsible for the insulating contribution ∼ 1/iω n to the self-energy at K. Clearly, this term is not ...

Phases of correlated spinless fermions on the honeycomb lattice

“…Our use of twisted boundary conditions reduces finitesize effects by enhancing the resolution in momentum space, but can induce small spurious gaps as a result of the breaking of translational symmetry. 47 Nevertheless, the results in Fig. 2(b) suggest the existence of a singleparticle gap for V 2 2.5t, in accordance with the phase diagram shown in Fig.…”

Correlated Dirac semimetal by periodized cluster dynamical mean-field theory

“…Later, this result was reproduced by Seki and Ohta 10 , who used variational cluster approximation 16,17 (VCA) with also ED as impurity solver. Since it is known that the Dirac semimetal is stable for weak interactions 18,19 , the validity of the application of CDMFT-like methods to the present model is questioned 11,[20][21][22] . In all works using quantum cluster methods mentioned above, the impurity clusters consist of six sites forming a ring with the same rotation symmetries as the honeycomb lattice.…”

“…In contrast, when they adopted two-or four-site impurity clusters with two bath sites per boundary impurity site, a first-order transition at a finite interaction from the semimetal to the antiferromagnetic insulator was obtained. However, this argument was refuted by Liebsch et al 20,21 , who showed that CTQMC CDMFT, which uses a continuum of bath sites, yields the same results as the ED CDMFT at finite but low temperatures for the honeycomb lattice. Moreover, with the detailed analysis 20,21 on the ED CDMFT calculations, they attributed the absence of the semimetallic phase to the translation symmetry breaking in the CDMFT.…”

Correlated Dirac semimetal by periodized cluster dynamical mean-field theory

“…Later, this result was reproduced by Seki and Ohta 10 , who used variational cluster approximation 16,17 (VCA) with also ED as impurity solver. Since it is known that the Dirac semimetal is stable for weak interactions 18,19 , the validity of the application of CDMFT-like methods to the present model is questioned 11,[20][21][22] .…”

“…In contrast, when they adopted two-or four-site impurity clusters with two bath sites per boundary impurity site, a first-order transition at a finite interaction from the semimetal to the antiferromagnetic insulator was obtained. However, this argument was refuted by Liebsch et al 20,21 , who showed that CTQMC CDMFT, which uses a continuum of bath sites, yields the same results as the ED CDMFT at finite but low temperatures for the honeycomb lattice. Moreover, with the detailed analysis 20,21 on the ED CDMFT calculations, they attributed the absence of the semimetallic phase to the translation symmetry breaking in the CDMFT.…”