Precursors of the insulating state in the square lattice Hubbard model
Erik G. C. P. van Loon, Hartmut Hafermann, Mikhail Katsnelson
PPrecursors of the insulating state in the square lattice Hubbard model
Erik G. C. P. van Loon, Hartmut Hafermann, and Mikhail I. Katsnelson Radboud University, Institute for Molecules and Materials, NL-6525 AJ Nijmegen, The Netherlands Mathematical and Algorithmic Sciences Lab, Paris Research Center,Huawei Technologies France SASU, 92100 Boulogne Billancourt, France
We study the two-dimensional square lattice Hubbard model for small to moderate interactionstrengths 1 ≤ U/t ≤ U/t = 1.
PACS numbers: 71.30.+h71.10.-w,71.10.Fd,
I. INTRODUCTION
The Hubbard model captures numerous phenomenaof strongly correlated electron physics, in particular theMott metal-insulator transition . Despite various efforts,the nature of the change from metal to insulator in thetwo-dimensional (2D) Hubbard model at half-filling andintermediate to small interaction remains elusive.At large interaction strength U , spin and charge de-grees of freedom are decoupled and a gap opens in thedensity of states already at finite temperature. TheHubbard model approximately maps to a Heisenbergmodel in this regime with an effective exchange coupling J = − t /U , where t is the nearest-neighbor hopping.At zero temperature the local moments order due to thesuperexchange mechanism.At small interaction, the nature of the phase connect-ing the weakly correlated Fermi liquid at high temper-ature with the antiferromagnet (AF) at zero tempera-ture is much less clear. At U = 0, the Fermi surfaceis perfectly nested with antiferromagnetic nesting vec-tor q = ( π, π ). In one possible scenario, for small fi-nite U a spin-density wave instability develops at T = 0and the MIT is a consequence of the backfolding of theBrillouin-zone due to the magnetic ordering. This iscalled the Slater transition . Kyung et al. have arguedfor this scenario based on the two-particle self-consistentapproach (TPSC).In an alternative scenario propagated by Anderson ,the Hubbard model exhibits strong-coupling behavior forboth strong and weak coupling, so that a Mott gap ispresent at any finite value of U as in 1D . As the tem-perature is lowered, local moments develop first becauseof the MIT and finally order at T = 0. In this sce-nario, AF is a consequence of the MIT, contrary to theSlater scenario. Moukouri et al. have argued in favorof this scenario based on the double occupancy and den-sity of states in large cluster DCA calculations. Sch¨aferet al. identified a U c ( T ) based on a downturn in theelectronic self-energy at the lowest Matsubara frequen- cies within the dynamical vertex approximation (DΓA)and Quantum Monte Carlo (QMC). This downturn hasbeen interpreted as a destruction of the Fermi surfacedue to scattering of the electrons at Slater paramagnons,fluctuations with a very large correlation length whichcan extend over thousands of sites. The downturn inself-energy has been found to correlate with a decreasein potential energy. This is consistent with the cellularDMFT results of Fratino et al. , which show a decreasein potential energy when the finite cluster undergoes atransition to the antiferromagnetically ordered state.In this work, we contribute to the current physicalpicture by studying the small interaction, low temper-ature region using the ladder dual fermion approxima-tion (LDFA) . The LDFA, as well as DΓA belongto a class of methods known as diagrammatic extensionsof dynamical mean-field theory . Contrary to clustermethods, they include correlations over length scales cov-ering hundreds of lattice sites. While the self-energy isapproximate at any scale, good agreement of the LDFAwith benchmarks has been found over a wide parameterrange . Despite similarities in the LDFA and DΓA, theydiffer in how they treat the long-range fluctuations whichare essential to respect the Mermin-Wagner theorem .Because the physical question discussed here is closelyrelated to the presence of these fluctuations, we do notnecessarily expect the same results in these methods.In ladder DΓA the Mermin-Wagner theorem is a con-sequence of the so-called Moriya- λ correction . Its pur-pose is to ensure the correct leading asymptotic behaviorof the self-energy and can be understood as a mass termin the two-particle propagator (susceptibility) which cutsoff the divergence that leads to spurious long-range or-der. In the LDFA, the diagrammatic corrections donot alter the leading term in the asymptotic behavior .Here the long-range antiferromagnetic fluctuations areincluded through a self-consistent renormalization pro-cedure which ensures the expected exponential scaling ofthe susceptibility at low temperature .In the following section II we discuss the model andsome additional properties of the LDFA. Our numer- a r X i v : . [ c ond - m a t . s t r- e l ] D ec ical results and their implications for the small inter-action and low temperature phase diagram of the two-dimensional Hubbard model are discussed in Sec. III. Weconclude in Sec. IV. II. MODEL AND METHOD
The Hubbard model on the 2D square lattice is de-scribed by the Hamiltonian H = − t (cid:88) (cid:104) ij (cid:105) c † jσ c iσ + U (cid:88) i n i ↑ n i ↓ , (1)where t is the nearest-neighbor hopping and our unit ofenergy. We are interested in small to intermediate cou-pling U up to half of the bandwidth given by W = 8 t .The LDFA is a particular diagram resummationscheme in the dual fermion (DF) approach. In DF thelattice model is replaced by a lattice of quantum impu-rity models which interact via auxiliary, so-called dualfermions. The strong local DMFT-like correlations aretreated at the level of the impurity model, while non-localcorrelations are included diagrammatically. For a spe-cific choice of self-consistency condition on the impurities,DMFT corresponds to non-interacting dual fermions .The latter couple to the physical fermions of the sameflavor locally. Diagrams in terms of dual fermions cantherefore be constructed based on physical considera-tions. In particular, we expect long-range particle-holefluctuations to be dominant. The corresponding LDFAself-energy has the form (cid:101) Σ k ν = − T N (cid:88) k q (cid:88) r A r F νν (cid:48) ωr (cid:101) G k (cid:48) ν (cid:48) (cid:101) G k (cid:48) + q ,ν (cid:48) + ω (cid:101) G k + q ,ν + ω × [ (cid:101) F νν (cid:48) ω lad ,r, q − F νν (cid:48) ωr ] , (2)where the ladder diagrams are generated by the Bethe-Salpeter equation (BSE) for the dual vertex (cid:101) F νν (cid:48) ω lad ,r, q ofthe lattice in the particle-hole channel, (cid:101) F νν (cid:48) ω lad ,r, q = F νν (cid:48) ωr − TN (cid:88) k F νν (cid:48)(cid:48) ωr (cid:101) G k (cid:48)(cid:48) ν (cid:48)(cid:48) (cid:101) G k (cid:48)(cid:48) + q ,ν (cid:48)(cid:48) + ω (cid:101) F ν (cid:48)(cid:48) ν (cid:48) ω lad ,r, q . (3)Here we have introduced sums over four-momenta k =( k , ν ), q = ( q , ω ) and the spin and charge channels r = sp , ch. ν and ω denote the discrete fermionic andbosonic Matsubara frequencies, respectively. We furtherhave A ch = 1, A sp = 3, where the latter accounts for thedegeneracy of the spin bosonic excitations. T denotestemperature and N is the total number of lattice sites.The second term in angular brackets avoids over-countingof the second-order diagram. F νν (cid:48) ωr is the exact local re-ducible vertex of the impurity model. If we approximatethe lattice vertex by the local one, (cid:101) F νν (cid:48) ω lad ,r, q ≈ F νν (cid:48) ωr , weobtain the second-order approximation DF (2) . Note that the LDFA includes diagrams from both the horizon-tal and vertical particle-hole channels (see for examplethe discussion in Ref. 21). In the spin channel, these dia-grams generate the collective paramagnon excitations .Remarkably, the LDFA reproduces non-mean-field criti-cal exponents . We refer the reader to Refs. 18 and20 for further details on the method.Below the DMFT N´eel temperature T DMFTN we haveto include the long-range fluctuations which destroy theAF order of the underlying mean-field. To this end, thedual Green’s functions ˜ G k ν in above equations are self-consistently renormalized: The self-energy ˜Σ k ν is calcu-lated starting from an initial guess for the dual Green’sfunction (typically the bare dual Green’s function). Anew dual Green’s function is obtained via Dyson’s equa-tion, which in turn is inserted into (3) and (2) to calcu-late a new self-energy (the impurity vertex is fixed). Thisprocess is repeated until self-consistency.Since below T DMFTN the BSE (3) initially diverges, wecut off the AF fluctuations in the initial iterations of thisinner self-consistency by restricting the eigenvalues of theBSE to values strictly smaller than 1. Once the iterationsconverge the cutoff is removed. When all BSE eigenval-ues are smaller than unity in the final iteration the solu-tion is well-defined and independent of the cutoff . Thescheme is not guaranteed to converge and the number ofiterations may diverge, which ultimately limits the acces-sible temperature range. The impurity model hybridiza-tion function is adjusted in an outer self-consistency loopbased on the condition that the lowest-order dual dia-gram vanishes . At the values of U we are interested in,its effect is merely a small enhancement of the imaginarypart of the impurity self-energy.The calculations are carried out on lattices of finitesize subject to periodic boundary conditions imposedby the discrete Fourier transform. Once the calcula-tion is converged, we compute the physical self-energyΣ k ν and momentum-resolved susceptibility , χ ( ω, q ) = (cid:104) ˆ S z ˆ S z (cid:105) ω, q with S z = (ˆ n ↑ − ˆ n ↓ ) / III. RESULTSA. Energetics
We first discuss the energetics of the model. En-tropy disfavors antiferromagnetism, so the AF transitionis driven by the interplay of kinetic and potential energy.In the picture of nearly free electrons at small interaction,AF ordering reduces the double occupancy. Magnetic or-dering is hence potential energy driven. This is the Slaterregime. At strong interaction on the other hand, doubleoccupancy is largely suppressed. AF ordering promoteshopping processes and the (negative) kinetic energy islowered. Magnetic ordering is stabilized through the re-duction in kinetic energy. This is the Heisenberg regime.It is true even in the Mott insulator, where the exchange
T /t E / t E t o t / E pot E k i n FIG. 1. Energetics at
U/t = 4 obtained on a 64 ×
64 lattice.The kinetic energy is measured with respect to the kineticenergy at T = 0 and U = 0, E = − /π . Dashed linesand circles are the DMFT results, plus signs are LDFA. T /t E / t E tot / E pot E kin FIG. 2. Energetics at
U/t = 1 obtained on a 128 ×
128 lattice.The kinetic energy is measured with respect to the kineticenergy at T = 0 and U = 0, E = − /π . Dashed linesand circles are the DMFT results, plus signs are LDFA. coupling J = − t /U is mediated through virtual hop-ping processes.In DMFT it is possible to compute the energy differ-ence between the ordered and unordered states. In theSlater regime, the ordered state has a lower potentialenergy, but higher kinetic energy compared to the un-ordered state at T = 0 . This implies that indeed thepotential energy stabilizes the ordered state. At large U ,in the Heisenberg regime, the situation is opposite. Theseconclusions remain true when short-range non-local cor-relations come into play. Both 2 × and pla-quette DCA calculations in the Slater regime show thatthe potential energy is lowered compared to DMFT. Thisis expected because the DCA includes antiferromagneticcorrelations. At large U , on the other hand, the four-site plaquette has a higher potential energy. Fratino etal. further argue that the interaction scale at whichthe system switches from Slater- to Heisenberg behavioris given by the critical U of the underlying normal-stateMott transition of the plaquette. ω/t A ( ω ) DMFTDF (2)
LDFA
T/t = 0 . U/t = 4 n = ∞ n = 0 n = 10 n = 20 FIG. 3. Maximum entropy local density of states obtainedwithin DMFT and ladder DF. For a given n , the self-energyincludes ladder diagrams up to order n + 2 in the local vertex.The case n = 0 corresponds to the second-order approxima-tion DF (2) . DΓA includes nonlocal correlations diagrammaticallyand up to significantly larger length scales compared tothe cluster calculations. The correlations increase the ki-netic energy at small U , while at large U they decreaseit . The potential energy was found to be reduced com-pared to DMFT at all studied values of U , both in theSlater and in the Heisenberg regime. However, the ambi-guity of the potential energy in DMFT complicates theanalysis .In Figs. 1 and 2 we show the energetics extractedfrom LDFA calculations as a function of temperature at U/t = 4 and
U/t = 1, respectively. In both cases wefind that the potential energy is lowered in LDFA com-pared to DMFT (albeit only sightly at
U/t = 1) in ac-cordance with the cluster DMFT and DΓA results andas expected in the Slater regime. At
U/t = 4 the kineticenergy is also somewhat higher than in DMFT, while itseems slightly lower at
U/t = 1 and low temperature.Here relatively large uncertainties however prevent adefinite statement (note the scale compared to U/t = 4).The total energy of the system is lower when nonlocalcorrelations are included.
B. Density of States
The non-local fluctuations have a drastic effect on thelocal density of states (DOS) as seen in Fig. 3. While theDMFT solution exhibits a quasiparticle peak, the spec-tral weight is reduced in DF. The reduction is small insecond-order DF (labeled DF (2) ), but increases with or-der of the ladder diagrams. Remarkably, diagrams at allorders contribute to the pseudogap. The dual Green’sfunction decays rather rapidly in real space, with ex-ponential decay on a length scale ξ G ≈ .
8. Low-order diagrams hence mediate short-range correlations,
T /t χ s p i n ( ω = , q = ) QMC (Moreo)DMFTDF 8 × × T DMFT N FIG. 4. Uniform spin susceptibility as a function of tempera-ture compute with different methods, all at
U/t = 4. In QMCand LDFA, χ sp exhibits a maximum at the effective exchangeenergy scale. The QMC results are taken from Ref. 38 for an8 × while long-range correlations require high diagram or-ders. They nevertheless contribute to short-range corre-lations as well.The pseudogap develops in a regime where magneticfluctuations are very strong and is linked to a reduc-tion in potential energy, in accordance with DCA . Thetemperature here is slightly below the DMFT N´eel tem-perature. The AF correlation length is of the order of ξ AF ∼ . These results areconsistent with those of Ref. 15, where a gap fully opensat U/t = 4 and
T /t = 0 . U . On theother hand, the opening of the gap can be interpreted asa consequence of the scattering of quasiparticles off theAF fluctuations, which we include by construction in theLDFA. This interpretation and our numerical results areconsistent with the DΓA study of Ref. 16, which pointedout the importance of Slater paramagnons. The qualita-tive DOS predicted by TPSC at crossover temperatures is consistent with the LDFA results. C. Uniform susceptibility
Figure 4 shows the temperature dependence of the uni-form spin susceptibility χ sp at U/t = 4. We find a de-creasing susceptibility at high temperatures, even thoughCurie’s law χ ∝ /T only sets in at higher tempera-tures, T ≈ W = 8 t . While the results of different meth-ods agree at high temperature, they are qualitatively dif-ferent at low T . In DMFT, χ continues to increase up T /t χ s p i n ( ω = , q = ) DF 32 × × T DMFT N FIG. 5. Uniform spin susceptibility as a function of temper-ature, at
U/t = 1. to the point where the antiferromagnetic susceptibilitydiverges due to the second-order transition to the mean-field antiferromagnetic state at T DMFTN (not shown). InLDFA the susceptibility decreases at low temperatureand exhibits a maximum. AF correlations that buildup reduce the uniform susceptibility. This occurs at theenergy scale of the effective exchange interaction J be-tween neighboring sites. DMFT does not include suchnonlocal correlations, so that the maximum is absent.Note that the maximum occurs at a slightly lower tem-perature than T DMFTN . At T DMFTN the ladder diagramseries (3) diverges, which in turn causes large effects inthe self-consistent renormalization of the Green’s func-tions. The result are strong magnetic fluctuations whichdestroy the mean-field long-range order in the underly-ing DMFT solution. At the same time they lead to spincorrelations between sites.The LDFA susceptibility is in excellent agreement withlattice QMC results for the same lattice size. The com-parison with results for a larger lattice reveals that finitesize effects play a role. The magnitude of the susceptibil-ity is significantly reduced (by about 10%) in the largersystem. The position of the maximum however is notaffected by finite size effects, which is consistent with itsinterpretation as an effective exchange energy scale.The uniform spin susceptibility for U/t = 1 is shown inFig. 5. As for
U/t = 4, the nonlocal correlations reducethe susceptibility compared to DMFT, especially at lowertemperatures. In LDFA, the magnitude of the suscepti-bility is reduced on the larger lattice, even more stronglythan for
U/t = 4. However we do not find a maximum inthe susceptibility at the accessible temperatures down to
T /t = 1 / ≈ . U = 0, there isa Van Hove singularity in the electron density of statesexactly at the Fermi level and the uniform susceptibilityshows a logarithmic divergence as T →
0. Based on ourresults, it is not possible to distinguish between a maxi-mum at a finite but very small temperature, or at T = 0.The DMFT N´eel temperature is between T /t = 0 .
025 and
T /t = 0 . U/t = 4.In Fig. 6 we show results for several values of U . Thelocation of the maxima for U/t = 2 , . The location of the max-imum in the susceptibility decreases with decreasing U ,showing that the physics is clearly not Heisenberg-like.For the U values where we obtain a maximum, it is closeto the DMFT N´eel temperature. D. Finite-size effects
Figure 7 shows the leading eigenvalue of the BSE at theAF wave vector for
U/t = 1 and
U/t = 4 respectively,for different lattice sizes. At low temperature the antifer-romagnetic susceptibility is expected to exhibit an expo-nential scaling χ AF ∼ e ∆ /T . It follows that the lead-ing eigenvalue behaves as ( χ AF ) − ∼ − λ ∼ exp( − ∆ /T ).1 − λ remains non-zero for finite T as required by theMermin-Wagner theorem. We observe the scaling for U/t = 4 for the larger lattice, and not at all for
U/t = 1.Figure 8 shows the spin correlation function in realspace. At
U/t = 4 and
T /t = 0 . is short, ξ AF ≈
4. This is consistent with the factthat no finite-size effects are visible at this temperaturein Fig. 7. At
U/t = 1 and
T /t = 0 .
02, the correlationlength is significantly longer, ξ AF ≈
17, but unlikely toexplain the absence of the scaling for the 128 ×
128 lattice.In addition to the length scale associated with two-particle fluctuations, we can introduce a scale related tosingle-particle properties obtained from the Green’s func-tion at τ = β/ ξ G . For U/t = 4 and
T /t = 0 .
2, we find ξ G ≈ . < ξ AF ≈
4, while for
U/t = 1 and
T /t = 0 . ξ G ≈
35, which is larger than ξ AF and thesmaller lattice and may explain why finite-size effects actin opposite directions for U = 4 and U = 1. Even though T /t χ s p i n ( ω = , q = ) U/t = 1
U/t = 4
U/t = 2
U/t = 1 . U/t = 3
FIG. 6. Uniform spin susceptibility as a function of temper-ature, for various values of U . Note that all simulations havebeen performed on a 32 ×
32 lattice, except for the
U/t = 4simulations which were performed on a 64 ×
64 lattice. Thevertical lines show the respective T DMFT N . /T − λ − λ × × × × U/t = 1
U/t = 4
FIG. 7. Leading LDFA antiferromagnetic eigenvalue as afunction of temperature. Note the different x-axes for
U/t = 1and
U/t = 4. The results for
U/t = 4 are reproduced fromRef. 25.
0 20 40 60 80 100 120 140 x | h S z S z i | x , ω = U/t = 1,
T /t = 0 . U/t = 4,
T /t = 0 . ξ ≈ ξ ≈ − − − − − FIG. 8. Spin correlation function in real space on a 256 ×
256 lattice along the line ( x, x isgiven in units of the lattice constant. The grey lines show anexponential fit with correlation length ξ . Finite-size becomeapparent at distance of roughly half the linear latte size. we cannot rule out that the exponential scaling of 1 − λ is obscured by finite size effects, we expect that it sets inat even lower, inaccessible temperatures.Figs. 10 and 11 show the local Green’s function forthe same U values. For U/t = 4, finite-size effects areclearly absent for large lattices. For
U/t = 1 small finite-size effects are visible even on the largest lattice. Notethat the finite-size effects are also visible in DMFT. Thisunderlines that they are related to the momentum dis-cretization. Fig. 9 illustrates this for the noninteractingDOS, which requires of the order of 64 ×
64 points at T = 0 .
02 to be accurately represented.
E. Self-energy
In Fig. 12, we plot the momentum-resolved differenceof the self-energy at the two lowest Matsubara frequen-cies at
U/t = 4 and
T /t = 0 .
2. The presence of positivevalues near the point k = ( π, indicates a downturnin the self-energy and signals a breakdown of Fermi liquidtheory. This feature is robust with respect to the latticesize and consistent with the presence of the pseudogap.At the studied temperature, the downturn only occursclose to k = ( π, T /t ≈ .
11 which is outside the accesible temperaturerange
T /t ≥ . U/t = 1and
T /t = 0 .
02. We do not observe a downturn, but val-ues close to zero (a flattening of the self-energy at low fre-quencies) in a very narrow strip along the Fermi surface(the diagonal from top left to bottom right). Comparedto Fig. 12, the tendency towards a downturn occurs ina much narrower part of the Brillouin Zone. This sug-gests that the phenomenon is related to AF fluctuationson very long length scales.Figure 14 shows that finite-size effects can change be-havior qualitatively. A downturn in the self-energy oc-curs on a 32 ×
32 lattice, while it is absent on the larger128 ×
128 lattice. If the scattering of electrons at mag-netic fluctuations induces the downturn, it is conceivablethat the downturn occurs because the antiferromagneticsusceptibility is overestimated, as visible in Figs. 5 and 7.The downturn, and ultimately an opening of the gap maystill appear at lower temperatures. However we do notobserve a downturn in the accessible temperature range.This excludes a crossover down to
T /t = 1 /
69, in contrastto the crossover temperature of
T /t ≈ /
38 obtained inDΓA . The discrepancy between both methods may liein the way they treat the magnetic fluctuations. E/t − I m G / π ×
32 64 × FIG. 9. Noninteracting density of states − π Im G ( E + iη ) at T /t = 0 .
02, with broadening η = πT , determined using threedifferent lattice sizes. The filled, green curve uses a 256 × N x < -1.2-1-0.8-0.6-0.4-0.2 0 0 0.5 1 1.5 2 2.5 3 ν/t G l o c DMFT 32 × × × × × × FIG. 10. Comparison of the local physics according to DMFTand LDFA, at
U/t = 1 and
T /t = 0 .
02 and three differentlattice sizes. There are significant finite-size effects both inDMFT and in LDFA separately, while both give similar re-sults for the same lattice size. -0.4-0.2 0 0 2 4 6 8 10 ν/t G l o c DMFT 8 × × × × × × FIG. 11. Comparison of the local Green’s function accordingto DMFT and LDFA, at
U/t = 4 and
T /t = 0 . Weak-coupling approximations can put these resultsinto perspective. The Hartree-Fock gap ∆ E =32 t exp( − π (cid:112) t/U ) is exponentially small in U , and pre-dicts a typical energy scale 0 . t at U/t = 1 due to thesignificant prefactor. The temperatures studied here areactually below this scale. Renormalization-group analy-sis , on the other hand, suggests a critical energy scale ofapproximately 0 . t for the formation of bound particle-hole pairs, which is exactly in the range studied here. IV. CONCLUSIONS
We have studied the half-filled 2D Hubbard model onthe square lattice in the interaction range 1 ≤ U/t ≤ and cellularDMFT results, non-local AF correlations reduce thepotential energy as expected in the Slater regime. StrongAF fluctuations develop in the vicinity of the DMFT N´eeltemperature. Scattering of electrons off these fluctua-tions leads to (presumably singlet-like) correlations and -0.25-0.2-0.15-0.1-0.05 0 0.05 π πk x k y FIG. 12. Difference in self-energy at the lowest two Matsub-ara frequencies, Im Σ ν − Im Σ ν , at U/t = 4 and
T /t = 0 . ×
256 lattice. Positive values indicate a downturn. Onlya quarter of the Brillouin Zone is shown. The black contoursindicate where the self-energy difference is zero. -0.005-0.0045-0.004-0.0035-0.003-0.0025 k x k y π π FIG. 13. Difference in self-energy at the lowest two Matsub-ara frequencies, Im Σ ν − Im Σ ν at U/t = 1,
T /t = 0 .
02, on a256 ×
256 lattice. Note the extremely sharp the feature at theFermi surface. However, it does not become positive, whichwould signal a downturn. to a pseudogap in the density of states. A concomitantdownturn is observed in the self-energy at
U/t = 4. Wefurther find a maximum in the uniform susceptibility ingood agreement to Monte Carlo results. It can be ex-plained through the buildup of AF correlations on an effective exchange energy scale J . This scale gets smallerwhen U decreases.We found that finite-size effects can alter the resultsqualitatively. While they play a minor role at U/t =4 due to relatively short correlations lengths, they leadto a spurious downturn in the self-energy at
U/t = 1.Nevertheless a tendency to a downturn in the self-energyis observed for
U/t = 1 in a very narrow region in theBrillouin zone along the Fermi surface, which suggeststhat a possible crossover would be associated with long-range AF fluctuations.We do not find a crossover to an insulator in the acces-sible temperature range for 1 ≤ U/t ≤
4. From our cal-culations we obtain an upper bound T downturn /t < . U/t = 1, which is sig-nificantly lower than the value reported in Ref. 16. Theorigin of this discrepancy between LDFA and DΓA mightlie in the way the methods treat the mean-field divergenceof the AF susceptibility.
ACKNOWLEDGMENTS
We thank Emanuel Gull, Georg Rohringer, ThomasSch¨afer and Alessandro Toschi for useful discussions.E.G.C.P. v. L. and M.I.K. acknowledge support fromERC Advanced Grant 338957 FEMTO/NANO. Our im-plementation is based on the ALPS framework and onthe impurity solver of Ref. 49. -0.03-0.02-0.01 0 0 0.5 1 1.5 2 2.5 ν/t I m Σ X: 32 × × × × FIG. 14. Finite-size effects in the self-energy, at
U/t = 1 and
T /t = 0 .
02. The self-energy is shown at the X= ( π,
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