Mott physics beyond Brinkman-Rice scenario Marcin M. Wysokin´ski1,2,∗ and Michele Fabrizio1,† 1International School for Advanced Studies (SISSA), via Bonomea 265, IT-34136, Trieste, Italy 2Marian Smoluchowski Institute of Physics, Jagiellonian University, ulica prof. S. L ojasiewicza 11, PL-30-348 Krak´ow, Poland (Dated: January 10, 2017) The main flaw of the well-known Brinkman–Rice description, obtained through the Gutzwiller approximation, of the paramagnetic Mott transition in the Hubbard model is in neglecting high- energy virtual processes that generate for instance theantiferromagnetic exchange J ∼t2/U. Here 7 we propose a way to capture those processes by combining the Brinkman–Rice approach with a 1 variational Schrieffer-Wolff transformation, and apply this method to study the single-band metal- 0 to-insulator transition in a Bethe lattice with infinite coordination number, where the Gutzwiller 2 approximation becomes exact. We indeed find for the Mott transition a description very close to n the real one provided by dynamical mean-field theory; an encouraging result in view of possible a applications to more involved models. J 7 A metal to insulator transitiondrivenby the electron- inclusion of the dynamical processes in a semi-analytic ] electronrepulsionwasenvisionedbyMottmorethanfifty manner. In order to achieve this goal, we construct a l e years ago [1]. Since then, the underlying physics of this method that combines the Gutzwiller’s variational ap- - phenomenon has been studied by large variety of quan- proachwithavariationalSchrieffer-Wolfftransformation. r t tum many-body tools in models for strongly correlated As acase study,we apply ourtechnique to the half-filled s . systems [2–5]. Hubbard model in the paramagnetic phase on the in- t a OneoftheearliestmicroscopicdescriptionsoftheMott finitely coordinatedBethe lattice. The energyfunctional m localisation is owned to Brinkman and Rice [2], and ob- to be minimised can be obtained fully analytically. Its - tained through the Gutzwiller approximation applied to minimisation leads to a significantly improved descrip- d thehalf-filledHubbardmodel. Intheirscenariothetran- tion of the Mott transition as compared to the standard n o sition to the insulating state occurs when the hopping is Brinkman–Rice scenario, and much closer to the exact c fully hamperedby repulsion,i.e. its expectationvalue in DMFT one [3]. The improvement is in particular high- [ the variational wavefunction strictly vanishes. This re- lighted in: (i) a sizeable lowering of the critical interac- 1 sult is elegant in many ways. It is fully analytical and tionstrengthfor atransition; (ii)a lowervalue ofthe in- v provides a very intuitive and physically transparent, al- sulatorenergythatincludesanon-zeroexpectationvalue 9 most classical, interpretation of the Mott phenomenon. ofthe hopping t2/U;and(iii)aproperbalanceofki- 1 ∼− Nonetheless, this description, frequently called netic and potential energies at the transition. 8 1 Brinkman–Rice transition, has a severe drawback: the The starting point of our analysis is the half-filled 0 expectation value of the hopping cannot be zero, and single-bandHubbardmodelontheinfinitelycoordinated . it is so in the Gutzwiller approximation only because Bethe lattice, 1 0 there is a complete, static and dynamic, locking of 7 charge degrees of freedom. In reality, dynamical charge H = t T + U n +n 1 2, (1) ij i↑ i↓ 1 fluctuations do play a role even deep in the Mott phase, −√z 2 − v: and in particular they mediate the antiferromagnetic Xhiji Xi (cid:0) (cid:1) i spin-exchange,as clear by the large U mapping onto the where z is the coordination number of the Bethe X Heisenberg model that can be formally derived through lattice. T→∞=T (c† c +c† c ) is the hermitian r ij ji ≡ σ iσ jσ jσ iσ a the Schrieffer-Wolff transformation [6]. hoppingoperatorbetweenneighbouringsitesiandj,and Since the result of Brinkman and Rice, a variety of niσ = c†iσciσ the locPal density of spin σ =↑,↓ electrons. quantum many-body tools have been constructed that We rewrite the interaction, last term in Eq. (1), as are generically able to sensibly capture those dynamical U processes [3, 5, 7, 8]. In particular, when applied to the Hint = (n 1)2 Pi(n), (2) Hubbard model, they provide satisfying descriptions of 2 − i n the Mott transition, though relying on heavy numerical XX computation. Nowadays, the scenario provided by dy- where P(n) is the projector at site i onto the subspace i namical mean-field theory (DMFT) [3], which becomes with n electrons. exact in infinite dimensions [9], has become an invalu- In order to construct a partial Schrieffer-Wolff trans- able benchmark to compare with. formation [6] that accounts for not complete projection In the present work we revisit the problem of the of double occupancies, we separately define components Brinkman–Rice transition, and complement it with the of the hopping operator, T projectedon the right or on ij 2 the left onto the configurations where both sites i and j ground state of is approximated by a variational H are singly occupied, Gutzwiller wave function ψ constructed from the un- G | i correlated Fermi sea ψ through 0 T˜ P(2)P (0)+P (0)P (2) T P(1)P (1) , | i ij i j i j ij i j ≡ T˜† (cid:16)P(1)P (1) T P (2)P ((cid:17)0)+(cid:16)P(0)P (2)(cid:17). (3) |ψGi≡ Pi |ψ0i. (9) ij ≡ i j ij i j i j Yi (cid:16) (cid:17) (cid:16) (cid:17) i is a linear operator that, in the presence of particle- P Theirsumisgatheredundertheformofthenewoperator hole symmetry, can be parametrised as T T˜ +T˜†, while the remaining part of the hopping opijer≡atoirjundiejr ij Tij Tij. We construct the partial =√2 sinθP (0)+cosθP(1)+sinθP(2) , (10) T ≡ − i i i i Schrieffer-Wolff transformation P (cid:16) (cid:17) ǫ where θ is a variational parameter bounded by (ǫ)=exp S , (4) U √z θ 0,π/4 , where θ = π/4 corresponds to the uncor- (cid:18) (cid:19) ∈{ } related (metallic) state, whereas θ = 0 projects out of through the anti-hermitian operator ψ all configurations with doubly occupied and empty 0 | i sites. Inotherwords,theactualvariationalwavefunction S = Sij = T˜ij−T˜i†j . (5) for the ground state of original Hamiltonian H is Xhiji Xhiji (cid:0) (cid:1) Ψ = (ǫ) ψ , (11) G The transformed Hamiltonian reads, | i U | i ǫ and depends both on θ and ǫ. The variational energy = (ǫ)†H (ǫ) H [S,H] H U U ≃ − √z functional per lattice site (where N is the total number ǫ2 ǫ3 (6) of sites), F(ǫ,θ) can be now obtained as the expectation + S,[S,H] S, S,[S,H] , value in the Gutzwiller wave function (9) of the Hamil- 2z − 6z3/2 (cid:2) (cid:3) h (cid:2) (cid:3)i tonian (7), which can be analytically computed in the whereǫis variationallydeterminedsoastominimise the infinitely coordinated Bethe lattice, energy. In the following, we shall assume that for any valueofU theoptimalǫissmallenoughtosafelyneglect 8T N (ǫ,θ)= Ψ H Ψ = ψ ψ , (12) 0 G G higher order terms, (ǫ4) in Eq. (6). A posteriori, we F h | | i h |H| i O shall check the validity of such assumption. We rewrite in reduced units of 8T . Here T is the hopping energy 0 0 − the transformed Hamiltonian in a more useful form, persiteof ψ ,whichinaBethelatticereadsT =8t/3π. 0 0 | i Already at this point, qualitative differences with re- t U H +ǫ [S ,T ]+ T spect to the standard Brinkman–Rice transition emerge ij kl ij H≃ z √z (cid:18) ijkl ij (cid:19) clearly. In our approach θ = 0 providing vanishing dou- X X − ǫ22 z3t/2 Sij,[Skl,Tmn] + Uz [Sij,Tkl] bacletuoaclcwupaavnefcuiensctiinon|ψ|GΨii.doEexspnlioctitlyyi,eltdhethdeensasimtyeofofrdothue- (cid:18) ijk ijkl (cid:19) bly occupied sites, d, can be calculated as lXmn(cid:2) (cid:3) X ǫ3 t (7) 1 + 6 z2 Sij, Skl,[Smn,Tpq] d≡N hψG |U(ǫ)† Pi(2) U(ǫ)|ψGi, (13) (cid:18) mXinjkplqh (cid:2) (cid:3)i (cid:16)Xi (cid:17) U + S ,[S ,T ] , which generically does not provide d=0 even if θ=0. z3/2 ij kl mn ijk (cid:19) In order to evaluate (12) as well as (13) we apply lXmn(cid:2) (cid:3) Wick’s theorem. Each expectation value resulting from this procedure can be conveniently visualised by a dia- where we made use of the following equality gramwithnodesdenotingsitesandedgesbeingaverages S ,H = U T . (8) oftheinter-sitesingleparticledensitymatrix. Sumofdi- ij int ij − agramswiththesamenumberxofnodesweshallshortly hXij i Xij denote as an x-vertex. We checked that a satisfying ac- The transformed low energy Hamiltonian (7) is then curacy is obtained by keeping all x-vertices up to x=4. analysed by a variational approach. Specifically, the The resulting energy functional reads 3 1 ǫu ǫu (ǫ,θ) 1 sin22θ+2u 1 cos2θ 8ǫτ 1 cos2θ F ≃ 8"−(cid:16) − 2τ(cid:17) (cid:16) − (cid:17)− (cid:18) − 2τ(cid:19) 3ǫ ǫu 9 ǫu 32ǫ3τ 1 sin22θ cos2θ+ǫ2 sin22θ+ cos2θ (14) − 8τ − 4τ! 4 − 2τ! 3 ǫ2 ǫu 7 155ǫ3 1 sin42θ+9 sin22θ cos22θ + sin22θ cos2θ , − 128τ2(cid:18) − 6τ(cid:19)(cid:18)− 2 (cid:19) 192τ # where interaction strength, U and hopping amplitude, t 0.06 MIT (a) are rescaled as 0.04 U t 3π u= , τ = = . (15) 0.02 8T 8T 64 0 0 0 The first line in Eq. (14) includes the 2-vertex contribu- tion, the second line is the 3-vertex one, and finally the E-0.02 third is the 4-vertex correction. Additionally, the expec- -0.04 tation value of the double occupancy d reads Energy GA+SW DMFT -0.06 total 1 kinetic d(ǫ,θ)= 2 2 4ǫ2 cos2θ -0.08 potential 8" −(cid:18) − (cid:19) E -0.1 BR ǫ 3ǫ2 + 2τ 1−ǫ2 sin22θ+ 32τ2 sin22θ cos2θ (16) 0.3 θ (b) ε + ǫ(cid:0)3 (cid:1)7sin42θ+9sin22θ cos22θ . ε, d 0.2 d 768τ3(cid:18)− 2 (cid:19)# θ, 0.1 uc1;DMFT uc2;DMFT uBR From(14)and(16)wecaneasilyrecovertheresultsofthe 0 standard Gutzwiller approximation applied to the Hub- 0.4 0.5 0.6 0.7 0.8 0.9 1 bardmodelby setting ǫ=0. Inthis casethe Brinkman– u Ricetransitiontakesplaceforu =1andtheinsulating BR state is characterisedby d=0. FIG.1: (a)Theequilibriumenergybalanceacrossthemetal toinsulatortransition (MIT).Foracomparison wehavepro- Wesearchforminimaofthefunctional withincreas- F vided data points of the real energies from DMFT calcula- ing u by standard methods. Namely, for each u we look tions [10]. Additionally for a reference we have also included for the pair of variables ǫ,θ satisfying { } theenergy correspondingtotheBrinkman-Riceresult (EBR) [2]. The transition takes place for quite similar critical in- ∂ /∂ǫ=∂ /∂θ=0, (17) F F teraction as for DMFT (uc2;DMFT). Also, alike DMFT [3], potentialandkineticenergiesarecharacterized with thepro- under the condition that the Hessian is positive definite. nounced kinks while the total energy remains smooth. (b) We start observing that for u = 0, the minimum of the The equilibrium values of θ, ǫ and d vs u across metal to functional (14) is correctly determined by ǫ = 0 and insulator transition. We marked the critical values of inter- θ =π/4thatcorrespondtofullyuncorrelatedmetal. For action for aBrinkman–Rice transition (uBR) as well as those interactionstrengthroughlyuptou 0.4,theoptimised obtainedbyDMFT,inwhichcaseatuc1;DMFT both,metallic energy is almost coincident with tha≃t obtained either by and insulating solutions begin to coexist. Alike DMFT pre- dictions[3]weobtainnon-vanishingdoubleoccupancyalsoin Gutzwiller approximationor by DMFT. theinsulating phase. For stronger correlations, u & 0.4, the Gutzwiller ap- proximationstartstodeviateappreciablywithrespectto DMFT, while ourvariationalenergyremainsquite close. Following Brinkman and Rice [2], we associate the In Fig. 1(a) we plot the total energy, as well as sepa- Mottinsulatingstatewithθ =0,whichisalwaysasaddle rately kinetic and potential energies, of the minimum of point of the functional (14). However, this saddle point functional Eq. (14), as compared with DMFT [10], and becomes minimum only when metal becomes unstable; withthesoleGutzwillerapproximation,forwhichwejust the metal to insulator transition is thus continuous and show the total energy. occurs at a criticalinteraction, u 0.822,which is size- c ≃ 4 ably lower than the Brinkman-Rice value, u =1, and for the paramagnetic Mott transition in the half-filled BR quite close to DMFT, u 0.854. In Fig. 1(b) we single-band Hubbard model on a Bethe lattice with infi- c2;DMFT ≃ show the values of the variationalparameters ǫ and θ on nite coordination number. Although there are obviously the both sides of the transition. In the same figure, we differenceswithexactresults,neverthelessourvariational alsoplottheaveragedoubleoccupancyd(fromEq.(16)), wavefunction provides a description of the Mott transi- which is non-zero in the insulating phase and decreases tion much closer to reality than the Brinkman-Rice sce- almost linearly in the metallic state. nario. Moreimportantly,ourwavefunctionisabletopor- In the insulating phase, u& u when the optimal θ = tray a Mott insulator where charge fluctuations are not c 0, we can analytically calculate several quantities. For completelysuppressedasintheBrinkman-Ricescenario, instance, the saddle point value of ǫ can be obtained in and which therefore has a non-zero expectation value of power series of τ/u: the hopping. This variationaltechnique might open new possibilitiestoaccessMottphysicsorrelatedphenomena τ τ 3 τ5 in more realistic models with minimal computational ef- ǫ = 4 + , (18) ins u − u O u5 fort. (cid:16) (cid:17) (cid:18) (cid:19) Acknowledgements. WearegratefultoAdrianoAmar- whereas the energy per site is icci for providing us data by DMFT. MMW acknowl- edgessupportfromPolishMinistryofScienceandHigher τ2 4τ4 τ6 t2 4t4 Eins =−8T0 2u−3u3 +O u5 ≃−2U +3U3. (19) Education under the “Mobilno´s´c Plus” program, Agree- ment No. 1265/MOB/IV/2015/0. MF acknowledges (cid:16) (cid:17) (cid:16) (cid:17) Additionally, the averagedouble occupancy in powers of support from European Union under the H2020 Frame- τ/u reads work Programme, ERC Advanced Grant No. 692670 “FIRSTORM”. 1 τ 2 τ 4 τ6 d = 4 + , (20) ins 2 u − u O u6 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) which is indeed finite. Let us now compare more in detail the above results ∗ Electronic address: [email protected] with the exact DMFT ones [3]. Alike DMFT, we find † Electronic address: [email protected] continuous metal-insulator transition for quite similar [1] N. F. Mott, Phil. Mag. 6, 287 (1961). criticalinteractionU. However,inourcasethereisnoco- [2] W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970). existence region of the insulating and metallic solutions, [3] A.Georges,G.Kotliar,W.Krauth,andM.J.Rozenberg, which in DMFT spreads over significant region (u and c1 Rev. Mod. Phys. 68, 13 (1996). u obtainedbyDMFTaremarkedinFig.1(b)). 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