Comment on "Absence of spin liquid in non-frustrated correlated systems"
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Comment on “Absence of spin liquid in non-frustrated correlated systems”
Ansgar Liebsch
Peter Gr¨unberg Institut, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany
In a recent Letter, Hassan and S´en´echal [1] discussedthe existence of a spin-liquid phase of the half-filled Hub-bard model on the honeycomb lattice. Using schemes,such as the variational cluster approximation (VCA) andthe cluster dynamical mean field theory (CDMFT) incombination with exact diagonalization (ED), they ar-gued that a single bath orbital per site of the six-atomunit cell is insufficient and leads to the erroneous conclu-sion that the system is gapped for all nonzero values ofthe onsite Coulomb interaction U . In contrast, we pointout here that, in the case of the honeycomb lattice, sixbath levels per six-site unit cell are perfectly adequate forthe description of short-range correlations. Instead, wedemonstrate that it is the violation of long-range trans-lation symmetry inherent in CDMFT-like schemes whichopens a gap at Dirac points. The gap found at small U therefore does not correspond to a Mott gap. As aresult, present CDMFT schemes are not suitable for theidentification of a spin-liquid phase on the honeycomblattice.As shown in Ref. [2], the cluster self-energy ob-tained within ED CDMFT [3] using six bath levels isin nearly quantitative agreement with results derivedwithin continuous-time quantum Monte Carlo (QMC)CDMFT [4]. As the bath in QMC is infinite, the self-energy is not subject to finite-size effects. The reasonfor the good agreement is that, because of the semi-metallic nature of the honeycomb lattice, the projectionof the infinite-lattice bath Green’s function onto a finite-cluster Anderson Green’s function is not plagued by thelow-energy-low-temperature disparities that arise in thecase of correlated metals and the square-lattice Hub-bard model. Moreover, because of the hexagonal sym-metry, this projection can be performed in the diagonalmolecular-orbital basis with non-symmetrical density ofstates components [3]. The main issue in Ref. [1] con-cerning the symmetry of bath levels then does not ariseand the two independent bath Green’s function compo-nents are fitted accurately using a total of six parameters(see Fig. 24 of Ref. [2]), while in the site basis only onefit parameter is available.Within CDMFT, the self-energy at Dirac points K exhibits the low-energy behavior [3]: Σ( K, iω n ) ≈ aiω n + b / [ iω n (1 − a )], where a is the initial slope ofIm [Σ ( iω n ) − Σ ( iω n )] and b the low-energy limit ofRe [Σ ( iω n ) − Σ ( iω n )]. Here, Σ ij is the self-energyin the site basis and ω n are Matsubara frequencies.The excitation gap at temperatures T → ≈ | b | / √ − a (see also Ref. [5]). Using the sitenotation: a = (0 , a = (0 , a = ( √ / , / a = ( √ , a = ( √ , a = ( √ / , − / and Σ within this cell are indepen- dent, so that ∆ = 0. However, site 1 is also connected tosite 4 at ( −√ / , − /
2) in the neighboring cell, requiringΣ = Σ which is not obeyed in CDMFT. Therefore,this violation of translation symmetry is responsible forthe insulating contribution ∼ /iω n to the self-energy at K . Clearly, this term is not caused by the finite size northe symmetry properties of the bath employed in ED. Infact, in view of the good agreement with ED, the self-energy in QMC CDMFT [4] exhibits the same insulatingcontribution, so that the density of states should alsoreveal a gap at small U and low T .To restore translation symmetry, we have performedED calculations within the dynamical cluster approxi-mation (DCA). This approach ensures Σ = Σ , givingsemi-metallic rather than insulating behavior at small U [6]. The condition Σ = Σ , however, cannot be gen-erally correct for correlations within the unit cell. Thecritical Coulomb interaction U c ≈ U ≈ . . .
5, ED and QMCCDMFT [3,4] yield excitation gaps which agree remark-ably well with the charge gap derived in large-scale QMCcalculations [7]. Evidently, this gap corresponds to aMott gap induced by short-range correlations and is onlyweakly affected by the lack of long-range translation sym-metry. On the other hand, the key question as to how thisgap closes, i.e., how the semi-metallic phase with weaklydistorted Dirac cones is recovered at small U , cannot bestudied adequately within CDMFT. The gap obtainedat small U is an artifact caused by the violation of long-range lattice symmetry and does not represent a trueMott gap. As a result, the identification of a spin-liquidphase on the honeycomb lattice is presently not feasiblewithin CDMFT-like methods.In conclusion, the origin of the narrow excitation gap atsmall U found in ED CDMFT for the honeycomb latticeis not the finite bath as stated in Ref. [1] but the lackof translation symmetry. This problem is also shared byQMC CDMFT and can be avoided, for instance, withinED/QMC DCA.[1] S.R. Hassan and D. S´en´echal, Phys. Rev. Lett. ,096402 (2013).[2] A. Liebsch and H. Ishida, J. Phys. Condensed Matter , 053201 (2012).[3] A. Liebsch, Phys. Rev. B , 035113 (2011).[4] W. Wu, Y.H. Chen, H.Sh. Tao, N.H. Tong, and W.M.Liu, Phys. Rev. B , 245102 (2010).[5] K. Seki and Y. Ohta, arXiv:1209.2101.[6] A. Liebsch, to be published.[7] Z.Y. Meng, T.C. Lang, S. Wessel, F.F. Assaad, andA. Muramatsu, Nature464