Extremely correlated Fermi liquid theory meets Dynamical mean-field theory:Analytical insights into the doping-driven Mott transition

R. Zitko,1, 2 D. Hansen,3 E. Perepelitsky,3 J. Mravlje,1 A. Georges,4, 5, 6 and B. S. Shastry3

1Jozef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia2Faculty for Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia

3Physics Department, University of California Santa Cruz, CA 95064, USA4Centre de Physique Theorique, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France

5College de France, 11 place Marcelin Berthelot, 75005 Paris, France6DPMC-MaNEP, Universite de Geneve, CH-1211 Geneve, Switzerland

(Dated: September 16, 2018)

We consider a doped Mott insulator in the large dimensionality limit within both the recentlydeveloped Extremely Correlated Fermi Liquid (ECFL) theory and the Dynamical Mean-Field The-ory (DMFT). We show that the general structure of the ECFL sheds light on the rich frequency-dependence of the DMFT self-energy. Using the leading Fermi-liquid form of the two key auxiliaryfunctions introduced in the ECFL theory, we obtain an analytical ansatz which provides a goodquantitative description of the DMFT self-energy down to hole doping level δ ' 0.2. In particular,the deviation from Fermi-liquid behavior and the corresponding particle-hole asymmetry developingat a low energy scale are well reproduced by this ansatz. The DMFT being exact at large dimen-sionality, our study also provides a benchmark of the ECFL in this limit. We find that the mainfeatures of the self-energy and spectral line-shape are well reproduced by the ECFL calculations inthe O(λ2) ‘minimal scheme’, for not too low doping level δ & 0.3. The DMFT calculations reportedhere are performed using a state-of-the-art numerical renormalization-group impurity solver, whichyields accurate results down to an unprecedentedly small doping level δ . 0.001.

PACS numbers: 71.10.Ay, 71.10.Fd, 71.30.+h

I. INTRODUCTION

Strong electronic correlations constitute one of the ma-jor challenges in condensed-matter physics and continueto inspire new theoretical approaches. In search for novelfunctionalities, new materials are being synthesized on aregular basis, giving the field a steady impetus. Signifi-cant progress in the understanding of electronic correla-tions has been achieved from the Dynamical Mean-FieldTheory (DMFT), in which the self-energy is assumedto be momentum-independent (see Ref. 1 for a review).This theory becomes exact in the limit of infinite dimen-sionality.

The situation in low dimensions has further challengesrelating to the k dependence of the self energy, and thusnew methods for strongly correlated electrons continueto be developed. One promising approach is Shastry’sExtremely Correlated Fermi Liquid theory (ECFL), de-veloped in a recent series of papers2–5. This theory startsfrom the infinite-U limit and is based on the Schwingerequation of motion for Gutzwiller projected electrons,these non-canonical objects requiring special attention.The theory leads to a set of analytical expressions thatare in principle exact. So far, solutions of the secondorder expansion of these expressions in a partial projec-tion parameter λ are available. They can be obtainedfor any lattice by an iterative process analogous to theskeleton diagram method. The ECFL theory expres-sions have been successfully applied to account for theAngle Resolved Photo-Emission Spectroscopy (ARPES)line-shapes of cuprate superconductors6,7 in the normal

state.

In this work, we perform a comparative study of thesetwo methods. We use as a test-bed the single-band dopedHubbard model at strong coupling U , in the limit of largedimensionality. This limit leads to simplifications in theECFL theory which we introduce here (the details of theformalism are provided elsewhere8). The comparison fo-cuses on the frequency-dependence of the self-energy andsingle-particle spectral line-shapes, and their evolution asthe Mott insulator is approached by reducing the dopinglevel δ (defined in Eq. (5)).

The first outcome of the present work is that, by look-ing at the DMFT results within an ECFL perspective,we are able to obtain new analytical insights into theDMFT description of the doping-driven Mott transition.Within the DMFT, the single-particle self-energy Σ(ω)displays a rich and complex frequency dependence. Thishas been known for some time (see e.g. Ref. 9 for a recentstudy), but is further investigated in the present workdown to unprecedentedly low doping levels δ . 0.001using a state-of-the-art numerical renormalization group(NRG) solution of the DMFT equations. Local Fermi-liquid behavior ImΣ ∝ ω2 +(πT )2 is obeyed only below avery low energy-scale. Above this energy scale, a markedparticle-hole asymmetry develops, a feature which is be-yond the Fermi liquid theory. Furthermore, the strongsuppression of spectral weight in the intermediate rangeof energies separating the quasiparticle peak from thelower Hubbard band corresponds to a marked quasi-polein the self-energy.

We show that all of these features can be well re-

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produced by constructing an analytical ansatz for theone-particle self-energy which is directly motivated bythe ECFL construction. The ECFL introduces two keyquantities, Ψ and χ, which play the role of auxiliary self-energies in the Schwinger construction. The proposedanalytical ansatz is obtained by retaining only the dom-inant Fermi-liquid terms in the low-frequency expansionof these auxiliary quantities. This is found to providea satisfactory fit of the DMFT results for doping levelsδ & 0.2. Hence, quite remarkably, the marked deviationsfrom Fermi liquid behavior, and the particle-hole asym-metry in the physical single-particle self-energy, can beaccounted for by an underlying Fermi-liquid form of theECFL auxiliary quantities. For very large U , and espe-cially for very small doping levels, additional structuresappear in the DMFT results which are not present inthis simplest ECFL ansatz, and presumably require addi-tional terms beyond the Fermi liquid ones in the auxiliaryfunctions Ψ and χ.

Another synergistic outcome of our study is that, be-cause the DMFT provides an exact solution in the limitof large dimensionality, it can be used to benchmark theECFL in this limit. We present here quantitative resultsobtained within the ‘minimal scheme’ implementation ofthe ECFL in high dimensions8, giving rise to an expan-sion to order λ2 in the projection parameter λ. We findthat the main features of the self-energy and the spec-tral line-shape are well reproduced by the O(λ2) ECFLcalculations, on a semi-quantitative level, for not too lowdoping δ & 0.3. Improvement will require further de-velopments of the ECFL approach. Since the DMFT isable to handle any finite U , while the ECFL constructionis motivated by the very large U limit, this comparisonalso sheds light on the adiabatic connection between theregime of moderate and extreme correlations.

We emphasize that ECFL can be used on two differ-ent levels. On one level, it provides a functional form forthe physical Green’s function and the corresponding self-energy in terms of the auxiliary ECFL self-energies Ψ(ω)and χ(ω). By assuming the simplest Fermi-liquid formfor these two self-energies over a certain frequency rangecentered around ω = 0, we successfully fit the physi-cal self-energy obtained through DMFT in this frequencyrange for δ & 0.2. This is remarkable since the frequencyrange used is substantially larger than the characteristicfrequency at which the physical self-energy begins to de-viate from Fermi-liquid behavior, and even encompassesthe quasi-pole in the physical self-energy at negative fre-quencies. This phenomenological approach to ECFL isthe one used in the first five sections of the paper, andthe results of this fit are displayed in Figs. 8, 9, and 10.On the second level, ECFL provides a microscopic the-ory by which one can obtain concrete results for Ψ andχ via an expansion in the projection parameter λ. Inthe remainder of the paper, the results obtained fromthe O(λ2) theory are benchmarked against the resultsobtained from DMFT, which are exact in the limit of in-finite dimensions. In the long run, further combined use

of the ECFL and the DMFT approaches could lead toa better understanding of the momentum-dependence ofthe self-energy that becomes important in lower dimen-sions.

The paper is organized as follows. After defining themodel in Sec. II, the general structure of the ECFL for-malism is reviewed in Sec. III. In Sec. IV, we present de-tailed DMFT results for the hole-doped Hubbard modelusing high-precision Wilson’s NRG as a solver. In Sec. V,the DMFT self energies are interpreted in light of theECFL-motivated analytical expressions. The second partof the paper is devoted to the O(λ2) ECFL minimal im-plementation. The basic equations and their simplifica-tion in infinite dimensions are established in Sec. VI, andin Sec. VII, a quantitative comparison is made to theDMFT results.

II. MODEL

We study the Hubbard model defined by the Hamilto-nian

H =∑kσ

εkc†kσckσ + U

∑i

ni↑ni↓, (1)

where εk is the bare band dispersion relation obtainedby Fourier transforming the hopping matrix. In thisstudy we consider a doped Hubbard model with nearest-neighbor hopping on a Bethe lattice, with semicirculardensity of states:

ρ0(ε) =2

πD2

√D2 − ε2, (2)

where D is the half-bandwidth, and thus any sum overthe band energy can be converted to an integral as:

1

N

∑k

A(εk) →∫ D

−Ddε ρ0(ε)A(ε). (3)

We note that the Fermi energy εF satisfies

sin−1(εFD

)+(εFD

)√1−

(εFD

)2

= −π2

(1− n), (4)

and vanishes near n ∼ 1 as εF = −π4 (1 − n) D. Thehole doping level δ is related to the particle density n(n = N/Nsites) as:

δ = 1− n. (5)

We will use ρQ(ω) as a short-hand notation for thespectral function associated with any relevant quantityQ(iωn) (Green’s function, self-energy):

ρQ(ω) = − 1

πIm(Q(ω + i0+)

). (6)

3

III. ECFL: GENERAL FRAMEWORK

A. ECFL formalism

The ECFL methodology has been discussed exten-sively in recent literature2,3; here we highlight only theaspects that are of relevance to this work. ECFL dealswith Gutzwiller-projected states obtained in the limit ofU → ∞, with the no-double-occupancy constraint builtinto the electron operators, leading to the well known t-Jmodel. This results in a non-canonical theory, where fa-miliar Feynman diagram methods fail due to the absenceof Wick’s theorem. The ECFL formalism is an exactalternative to the Feynman diagram technique. Insteadit works with the Schwinger equations of motion for theprojected electrons. It provides results for the electronicGreen’s functions that describe the physics of the low-energy sector in the problem, namely the dynamics ofthe quasiparticle (QP) states near the Fermi energy andof the lower Hubbard band (LHB).

For our purposes, we need to express the ECFL theoryin the large-dimensionality limit. The related technicalproblems outlined in Ref. 10 (paragraph 3) have been re-cently solved8 by analyzing the infinite-dimensional limitof the Schwinger equations of motion in the ECFL11.The exact mapping of the momentum-independent self-energy of the infinite-dimensional Hubbard model ontothat of a self-consistent Anderson impurity model12 pro-vides a roadmap for a suitable formulation of the ECFLequations in this limit.

In the simplest version of the ECFL theory13, the phys-ical (i.e. projected) electronic Green’s function is ex-pressed as a product of an auxiliary Green’s functiong(k) and a caparison term denoted in the present workas µ(k). Thus,

G(k) = µ(k)× g(k), (7)

where k ≡ (~k, iωk) and ωk is a Fermionic Matsubara fre-quency. Here g(k) is a Fermi-liquid-like Green’s function

g−1(k) = iωk + µ−(

1− n

2

)εk − Φ(k), (8)

and µ is the chemical potential. The factor µ(k) (heredistinguished from the chemical potential µ by the tilde),plays the role of an adaptive spectral weight, and is givenby

µ(k) = 1− n

2+ Ψ(k), (9)

with Ψ(k) vanishing at infinite frequency. The func-tions Φ(k) and Ψ(k) are the twin self-energies in thetheory, and are exactly defined as the appropriate func-tional derivatives of g−1 and µ respectively2,3. Theterm µ(k) is termed the caparison (i.e., dressing) fac-tor, since it provides a second layer of renormalization tothe propagator g, which is already dressed by Φ. BothGreen’s functions satisfy an identical number sum rule

∑k,ωkG(k) = n

2 =∑k,ωk

g(k); this enables us to satisfythe Luttinger-Ward volume theorem.

In the large-d limit, a further simplification can be

established8: Ψ is independent of ~k and Φ is decompos-

able into two ~k independent functions,

Ψ(k) = Ψ(iωk), (10)

Φ(k) = χ(iωk) + εkΨ(iωk), (11)

i.e., the two frequency-dependent (but ~k independent)functions χ and Ψ determine the Green’s function.

The single-electron physical (Dyson) self-energy Σ isdefined from the single-electron Green’s function G inthe usual manner, as (using the analytic continuationiωk → ω + iη, η = 0+):

G(k, ω + iη) =1

ω + iη + µ− εk − Σ(ω + iη). (12)

Within the large-dimensional ECFL, the Dyson self-energy Σ can be related to Ψ and χ as follows:

Σ(ω + iη)− µ− ω =χ(ω + iη)− µ− ω1− n

2 + Ψ(ω + iη). (13)

We see that the Dyson self-energy is manifestlymomentum-independent in this limit. Note also that,as seen from (13), its real part grows linearly with ω asω → ∞. This is a consequence14 of the Gutzwiller pro-jection in the U → ∞ limit. At finite U , this behavioris regularized at high-enough frequency and Σ goes to aconstant.

For a concrete implementation, the ECFL formalismallows for a perturbative expansion in a projection pa-rameter λ ∈ [0, 1], ultimately identified with the doubleoccupancy density3. The theory to O(λ2) is expected tobe quantitatively accurate for densities up to n<∼ 0.715.We postpone the description of these equations to SectionVI, but note an important general insight gained from ex-amining and evaluating such an expansion2,3,15; the twoself-energies χ and Ψ have simple Fermi-liquid functionalforms, with a dissipative part that is quadratic in ω, atsufficiently low energies (see Figs. 11 in Sec. VI). Thisinsight is used in the following to obtain a low-energyexpansion for the Green’s function.

B. Low-frequency expansion of self-energies andGreen’s functions

In this section we derive the low-frequency behavior ofthe Green’s function and self-energy within the ECFL.We obtain an analytical expression which will be used tointerpret and fit the DMFT results in Sec. V. We show,in particular how a characteristic particle-hole asymme-try in the Dyson self-energy is generated even when theexpansion of Imχ and ImΨ is limited to the particle-holesymmetric lowest-order Fermi-liquid terms.

4

Indeed, as mentioned above, the first few terms of asystematic λ expansion of the ECFL equations indicatethat the self-energies Ψ and χ are very similar functionsand resemble the self-energy of a Fermi liquid at lowenough T, ω, with suitable scale constants. For low ωand low T , up to a low-frequency cutoff scale Ωc, so that|ω| ≤ Ωc D, we define (with kB = 1)

R(ω, T ) ≡ π[ω2 + (πT )2

], (14)

and write a Fermi-liquid-like expansion for the complexECFL self-energies:

Ψ(ω) ∼ Ψ0 + cΨ ω +i

ΓΨR(ω, T ) + Ψrem(ω), (15)

χ(ω) ∼ χ0 − cχ ω −i

ΩχR(ω, T ) + χrem(ω), (16)

where

cΨ = 2ΩcΓΨ

and cχ = 2ΩcΩχ

. (17)

Ωχ and ΓΨ are parameters that determine the curva-tures of the two imaginary parts, with Ωχ having thedimensions of energy, while ΓΨ has the dimensions ofthe square of energy. Consequently, cΨ has the dimen-sions of an inverse energy, while cχ is dimensionless. Theterms Ψrem(ω) and χrem(ω) in Eq. (15) represent the re-mainders containing the leading corrections to the Fermi-liquid behavior, of the type O(ω2) for the real part andO(ω3) for the imaginary parts of these functions. Inthe initial analysis below, we simply ignore these terms.They can be readily incorporated to find a systematicimprovement of the fits, and lead to further correctionsto the low-frequency behavior of the imaginary part ofthe Dyson self-energy beyond the terms considered here.Note that in Eq. (17) the constants cΨ and cχ also receivecontributions from higher terms beyond the quadratic.Hence these approximate relations become exact if weretain the imaginary terms only to quadratic order, i.e.,

assuming ρΨ ≡ −R(ω,T )πΓΨ

and ρχ ≡ R(ω,T )πΩχ

. In general,

however, cΨ and cχ can be considered as additional freeparameters.

Some further remarks about this expansion are calledfor:

1. Expressions (15) and (16) are of the standardFermi-liquid type (symmetric in ω for the imagi-nary parts). Nonetheless, when processed throughthe ECFL formalism, they lead to important con-tributions to the imaginary part of the Dysonianself-energy which are anti-symmetric in ω. Reveal-ing the origin of this important asymmetry is oneof the main strengths of the ECFL analysis.

2. We shall find that as we approach half filling, Ωχand ΓΨ turn out to be similar functions of the elec-tron density, in view of their parallel role in the twoself-energies within the λ expansion. In the analy-sis below, we will find that as n→ 1, it is consistentto choose Ωχ,ΓΨ ∼ δ.

3. The energy scale Ωc, which determines the range offrequencies where the quadratic behavior of Im Ψand Im χ applies, is itself a function of the density.It shrinks linearly with δ = 1 − n as n → 1, andtherefore cΨ and cχ are finite as n→ 1. We shouldnote that these are leading-order assumptions inδ = 1− n.

Thus we find at low T, ω:

G(k, ω + iη) (18)

∼α0 + cΨ ω + i

ΓΨR

ω(1 + cχ) + µ+ iΩχR− εkα0 + cΨ ω + i

ΓΨR

,

where we have introduced

α0 ≡ 1− n

2+ Ψ0, (19)

and χ0 has been absorbed into the chemical potential µ.The entire momentum-dependence is contained in εk. AtT = 0 and ω = 0 we must require G−1(kF , 0) = 0, so weneed to set

µ = α0 εF . (20)

At low ω + iη and a fixed ~k, we can write a usefulexpression

G−1 ∼ −εk +ω(1 + cχ) + α0 εF + i

ΩχR

α0 + cΨ ω + iΓΨR

(21)

and therefore

Σ(ω+iη) ∼ α0 εF+ω−ω(1 + cχ) + α0 εF + i

ΩχR

α0 + cΨ ω + iΓΨR

. (22)

Note that we adjusted the self-energy so that Re Σ(0) =µ−εF , thereby placing the zero-energy pole at the Fermimomentum. We now extract the wave function renormal-ization factor Z from Z−1 = ∂

∂ωG−1(k, ω)|ω=0 as

Z =α0

1 + cχ − εF cΨ. (23)

Using the above expansion we find the spectral function ρG(k, ω) (or equivalently A(k, ω), as denoted in the exper-

5

imental literature), at low ω and k ∼ kF

A(k, ω) ∼(α0

2

πΩΣ

) R(1− ω

∆

)(1 + cχ − cΨ εk)ω − α0(εk − εF )2 + α0

2R2/Ω2Σ, (24)

where

ΩΣ ≡ α0ΩχΓΨ

ΓΨ − εFΩχ, (25)

∆ ≡ α02 ΓΨΩχ

ΩΣ(1 + cχ)Ωχ − cΨΓΨ. (26)

In terms of the wave function renormalization factor Z,

A(k, ω)∼(Z2

πΩΣ

) R(1− ω

∆

)ω − Z(εk − εF )2 + Z2R2/Ω2

Σ.

(27)

We thus obtain the following final form for the low-energyexpression of the Dysonian self-energy:

ImΣ(ω) ∼ − RΩΣ

1− ω∆

1 + ω cΨ/α02 +R2/(α0ΓΨ)2,

ReΣ(ω) ∼ α0εF + ω

−εF + ω(1 + cχ)/α0(1 + ωcΨ /α0) +R2/(α02ΩχΓΨ)

1 + ω cΨ/α02 +R2/(α0ΓΨ)2,

(28)

where we recall thatR(ω, T ) is defined in Eq. (14). Theseexpressions, and in particular that of ImΣ, are among thekey results of the present paper, and will be used belowin order to fit and interpret the DMFT data.

If we take the Ψrem and χrem terms in Eq. (15) andEq. (16) into account, then Eq. (28) receives higher-orderpolynomial corrections in ω in both the numerator anddenominator. Let us also note that Eq. (27) is of theform of a phenomenological version of the ECFL theorythat has been recently tested against experimental datawith considerable success, in some cases after adding aconstant times (εk − εF ) in the numerator2,6,7.

C. Low-doping limit n→ 1

At T = 0 and near half-filling we get εF = −π4 δDfrom Eq. (4). Further, from the single assumption thatΨ0 = −n2 near half-filling16, we find that the self-energyand wave function renormalization factor scale correctlywith δ as δ → 0. This assumption gives α0 = δ, anda scaling of various energy scales with δ. In particular,we find from the equations that Ωχ ∼ ΓΨ ∼ Ωc ∼ δ.This, together with Eq. (17), leads to cΨ ∼ O(1) andcχ ∼ O(1). This is consistent with the scaling behaviordescribed in remarks 2 and 3 in the previous section.Keeping the dominant terms in Eqs. (23), (25), and (26),

we find that

Z =α0

1 + cχ, (29)

ΩΣ = α0 Ωχ, (30)

∆ =α0 ΓΨ

(1 + cχ)Ωχ − cΨΓΨ. (31)

Near half-filling (δ → 0), we define

Z = δ Z, (32)

ΩΣ = δ2 ΩΣ, (33)

∆ ≡ δ ∆. (34)

All objects with an overline, such as Q, are determinedto be finite as δ → 0. Eq. (32) is expected on gen-eral grounds near the insulating limit: to leading orderG−1(k, ω) = εF − εk + ω

Z + O(ω2), and therefore thepropagating solutions correspond to quasiparticles withan energy dispersion Z(εk − εF ) that shrinks to zero atthe insulating point n = 1. We find here that this oc-curs as a linear function of δ. Eq. (33), together withEq. (28), implies that at small ω ∼ O(δ), the imaginarypart of the self-energy has a finite value. Further com-bined with a cutoff Ωc ∼ O(δ), it gives Re Σ ∼ ω−2ω Ωc

ΩΣ,

which is then consistent with the linear vanishing of Z inEq. (32). Eq. (34) shows that the particle-hole asym-metry in the spectral function increases as we approachhalf-filling. Finally, we see that the spectral density ImΣbecomes a scaling function of ω/δ at low doping levels.

IV. DOPED MOTT INSULATOR: SINGLE-SITEDMFT

The dynamical mean-field theory1 is based on thefact17 that in the limit of a large number of dimensionsd the self-energy becomes a momentum-independent lo-cal quantity, Σ(k, ω) → Σ(ω). This implies that thebulk problem for d → ∞ coincides with the problemof an interacting impurity embedded in an appropri-ate non-interacting bath12. DMFT formulates a prac-tical prescription for finding this effective impurity prob-lem and the self-consistency equation. For the Hub-bard model, the corresponding impurity problem is thesingle-impurity Anderson model, which can be efficientlysolved with the numerical-renormalization group (NRG)method18–22.

6

A. NRG method

The NRG calculations have been performed with thediscretization parameter Λ = 2 using the discretizationscheme with reduced systematic artifacts described inRef. 22. Furthermore, the twist averaging over Nz = 8different discretization meshes has been used to reducethe oscillatory NRG discretization artifacts23. The trun-cation cutoff in the NRG was set in the energy spaceat 10ωN (here ωN is the characteristic energy at the N -th NRG step); such results are well converged with re-spect to the truncation. The U(1) charge conservationand SU(2) spin rotational invariance symmetries havebeen used explicitly. The raw spectral data (weighteddelta peaks) were collected in bins on a logarithmic meshwith 1000 bins per frequency decade, then the broaden-ing scheme from Ref. 24 with α = 0.2 has been used toobtain the continuous representation of the spectral func-tions. To calculate the self-energy Σ, we have used theprocedure25 based on the following exact relation fromequations of motion:

Σσ(z) =〈〈[dσ, Hint]; d

†σ〉〉z

〈〈dσ; d†σ〉〉z=U〈〈nσdσ; d†σ〉〉z〈〈dσ; d†σ〉〉z

. (35)

Here dσ is the impurity annihilation operator, while Hint

is the interaction part of the Hamiltonian. The two cor-relators in this expression were computed using the full-density-matrix NRG algorithm24,26. To accelerate theconvergence of the DMFT self-consistency loop, Broy-den mixing algorithm has been used27. This techniqueis particularly important to ensure the convergence atsmall doping as the Mott transition is approached. TheBroyden solver has been used both to control the chem-ical potential to obtain the desired band filling and toapply the DMFT self-consistency equations27.

When performing the calculations in the large-U limit,it is important to note that the upper Hubbard band(UHB) is outside the NRG discretization energy window.The correlator Fσ(z) = 〈〈nσdσ; d†σ〉〉z receives a contribu-tion

FUHB(z) =wUHB

z − (εd + U)−→U→∞

wUHB

−U(36)

from the UHB, where wUHB is the total weight of theupper Hubbard band, which in the U →∞ limit is equalto n/2. The correlator F (z) in Eq. (35) is multipliedby a factor U , thus the UHB contribution to the nu-merator in the U → ∞ limit is equal to −wUHB. It iscrucial to correct the raw numerical results by makingthis subtraction when the UHB is outside the discretiza-tion window, otherwise the causality is very strongly vi-olated. (No such subtraction is necessary for the corre-lator Gσ(z) = 〈〈dσ; d†σ〉〉z, because the UHB only makesan O(1/U) contribution to the denominator.)

An analysis of the convergence of the NRG resultswith respect to the variation of various parameters inthe method is presented in Appendix B.

B. DMFT results

1. Scaling of quasiparticle weight Z vs. doping level δ

In this work, we only consider paramagnetic solutions.At low temperatures, the DMFT equations, dependingon the strength of the interaction U and the doping δ,give either an insulating or a metallic Fermi-liquid state.The key quantity characterizing the metallic state is thequasiparticle residue

Z =

(1− ∂ReΣ(ω)

∂ω

∣∣∣ω=0

)−1

. (37)

At δ = 0 (half-filled system), a metallic solution is foundfor U < Uc (this critical value of U is often denoted Uc2in the DMFT literature – the spinodal of the metallicsolution1 – and will be denoted Uc here for simplicity).For U > Uc, the DMFT equations only have a uniqueinsulating solution. At Uc/D = 2.918, a Mott metal-insulator transition takes place, with characteristics sim-ilar to the Brinkman-Rice picture28 in that the quasipar-ticle weight Z vanishes continuously and the quasiparticleeffective mass diverges.

Away from half filling, i.e., for any δ 6= 0, the solution isalways metallic (Z > 0); the Mott insulator (Z = 0) onlyexists exactly at half-filling (for U > Uc1, the spinodal ofthe insulator1). As δ → 0 for U > Uc, Z diminishes andvanishes at δ = 0. This doping-driven Mott transition isillustrated in Fig. 1. In panels a-d) we plot the resultsof Z vs. δ for a set of values of U . It is seen that, whenconsidered over a broad doping range δ . 0.5 (panelsa and b of Fig. 1), the overall doping-dependence of Zis fairly linear at intermediate values of U/D, while atstrong coupling (large U/D) a marked curvature is seen(approximately fit by a power-law with exponent close to1.4).

A plot of Z/δ vs. δ focusing on the low-doping region(panels c and d) reveals, however, that the asymptoticlow-δ behavior is actually linear, Z ∝ δ (except closeto the multicritical point U = Uc, δ = 0 where sizablecorrections are found). This is indeed the behavior ex-pected within the Gutzwiller approach28–30: Fig. 1c-dthus confirm that DMFT obeys this mean-field behavior.The prefactor of this linear dependence is also decreas-ing with U in reasonable agreement with the Gutzwillerestimate30 ∼ (1− Uc/U)−1/2. Note that the results dis-played here extend previous studies to much lower dopinglevels (δ . 0.001) than previously reported in the litera-ture, due to the improvements in the NRG methodology.

2. Self-energy and spectral function: overview of the mainstructures

We now address the properties of the self-energy Σ(ω)and one-particle spectral function in more detail.

7

0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5

Z

δ1.39

10-3

10-2

10-110

-4

10-3

10-2

10-1

100

Z

0.00 0.02 0.04 0.06 0.08 0.10

δ

0

1

2

3

4

Z/δ

10-3

10-2

10-1

100

δ

10-1

100

Z/δ

U=3U=4U=10U=∞

(a) (b)

(c) (d)

FIG. 1: (Color online) Doping-driven Mott transition within the DMFT: approach to the Mott insulating state for U > Uc =2.918D with decreasing doping δ. Top panels: doping dependence of Z for a range of δ on a lin-lin (a) and on a log-log (b)plot. The dashed line is a fit to a power-law function Z = δα with α = 1.39. Bottom panels: Z/δ vs. δ (c), and Z/δ vs. δ on alog-log plot (d).

An overview plot in Fig. 2a shows the main featuresin the local spectral function A(ω) and in the imagi-nary part of Σ(ω) in a broad frequency range. ImΣhas two very pronounced and sharp resonances (quasi-poles). These resonances are responsible for the suppres-sion of the spectral weight in A(ω) between the QP peakand the LHB and UHB, respectively. They are corre-spondingly positioned close to the minima of A(ω). Incontrast to the half-filled case, where these resonancesare symmetrically positioned on each side of ω = 0 at ascale31 ∝ ±

√Z, their locations in the doped case are no

longer symmetric and will be discussed below. In addi-tion, there are two broad humps in ImΣ in the frequencyranges associated with the two Hubbard bands. As Uincreases towards very large values at fixed doping, theUHB moves to higher frequencies, while the LHB andQP band gradually converge to their high-U asymptoticform. This convergence is, however, rather slow and thespectra start to very closely agree with the asymptoticones only for U on the order of 100D.

In Fig. 2b we plot a close-up on the low-energy struc-tures, i.e., the QP band and its vicinity. We notice thatthe Fermi-liquid quadratic behavior of ImΣ is limited toa very narrow frequency interval, much smaller than the

width of the QP peak itself. We also see (Fig. 2 andFig. 5) that at low doping level, ImΣ develops a markedparticle-hole asymmetry. These deviations to Fermi-liquid behavior are discussed below in a more quanti-tative manner.

One of the goals of this work is to provide an analyti-cal account of the complex frequency dependence of theself-energy that we just summarized. It should be kept inmind that the ECFL theory that we are going to use forthis purpose works with the U =∞ model, which beginsby throwing out the UHB altogether and deals only withthe LHB. Thus the comparison carried out later in thispaper refers only to the LHB and QP sector (no dou-ble occupancies), containing the interesting low-energyphysics of the problem.

3. Dynamical Particle-hole asymmetry

The local spectral function and self-energy are dis-played on Fig. 3 for U =∞, at two different doping levels(a small and large one, for comparison). An immediatelyapparent feature of these plots is the large asymmetrybetween hole-like (ω < 0) and particle-like (ω > 0) exci-

8

0

0.1

0.2

0.3

0.4

0.5

0.6

A(ω

)D

-3 -2 -1 0 1 2 3 4 5

1

2

-Im

Σ(ω

)/D

U/D=4, δ=0.02

max at 211max at 19

(a)

FIG. 2: (Color online) (a) Overview plot showing the fullstructure of spectra on high frequency scales (lower Hubbardband, quasiparticle band, upper Hubbard band) at finite U =4D. We show the DMFT local spectral function (top panel)and the imaginary part of the self-energy (bottom panel). (b)Close-up on the quasiparticle band at low frequencies.

tations.

Indeed, for ω < 0, |ImΣ| increases rapidly from ω = 0in order to connect with the negative-energy quasi-pole.The detailed form of this increase is somewhat differentdepending on the doping level. At large doping it is ap-proximately parabolic, in continuity with the low-ω FL∼ ω2 dependence. In contrast, at small doping, the low-ω parabolic dependence evolves into a more linear-likeincrease at higher frequency. The local spectral func-tion also displays an almost complete suppression of thespectral weight between the QP peak and the LHB atlow doping level, while this suppression is only modestat higher doping.

In contrast, for ω > 0, |ImΣ| rapidly flattens out afterits initial FL increase. It has a plateau-like behavior witha broad maximum at large and intermediate doping level(the maximum is sharper at smaller doping). Overall,|ImΣ| remains much smaller at ω > 0 than at ω < 0.

This asymmetry also reflects into the QP peak in thelocal spectral function A(ω), i.e., the local ρG(ω), whichhas a very asymmetric line-shape. The decrease from

its ω = 0 value A(0) is much faster on the ω < 0 side,in accordance with the large |ImΣ|. The detailed formof the line-shape on the more extended ω > 0 side isdifferent at lower and higher doping levels, with a convexand concave shape, respectively.

Finally, the particle-hole asymmetry has a very distinc-tive signature in the momentum-resolved spectra A(ε, ω),which are displayed in Fig. 4. It is seen there that thedispersion of the QP peak deviates from its low-energyform ωQP = Z(ε− εF ) much more rapidly on the ω > 0side, where a stronger dispersion closer to that of thebare band is rapidly found. This is mostly due to thedistinct behavior of the real part ReΣ for positive andnegative frequencies (shown later in Fig. 6). This find-ing, which is also supported by the ECFL results as dis-cussed below, is one of the main predictions of our work.It calls for the development of momentum-resolved spec-troscopies for unoccupied states (the ‘dark side’ that isnot directly accessible to ARPES). The physical signif-icance of this ‘dark side’ has also been recently pointedout in cluster-DMFT studies of the two-dimensional Hub-bard model32.

4. ω/Z scaling

Close to the Mott transition, all low-frequency prop-erties are expected to scale with Z, i.e., be described byscaling functions1,33,34 of ω/Z. This is indeed the case,as demonstrated in Fig. 5 in which good data collapse isobtained in the lowest frequency range when plotted vs.ω/Z. However, we also clearly observe that the scalingis limited to the asymptotic region of very small frequen-cies.

On panels b and c of Fig. 5, one can compare the evolu-tion of the shape of the local spectral function, discussedabove, as a function of the doping level. One sees that theQP peak becomes increasingly asymmetric at very lowdoping. We also observe that the LHB has some internalstructure, quite similar to that observed at half-fillingas the correlation-driven Mott transition is approachedfrom the metallic side22,35–37.

In Figs. 6, the real and imaginary part of the self-energy are plotted against ω/δD. The different panelscover different frequency ranges. While a better collapseof the different curves at very low frequency was obtainedabove when using ω/ZD as a scaling variable, it is seenfrom the plots in a broader frequency range that the over-all structures of the self-energy obey rather good scalingproperties with respect to ω/δD. For example, the sharppeak (quasi-pole) structure at ω < 0 in ImΣ is seen tobe located at a frequency proportional to doping level(ωpeak ' −0.7δD). This peak in ImΣ is associated withthe suppression of the spectral weight between the QPpeak and the LHB in the spectral function. Correspond-ingly, it is associated with a resonance-like structure inReΣ.

9

0.2

0.4

0.6

A(ω

)D

0.2

0.4

0.6

-1 -0.5 0 0.5 1-2

-1

0

ImΣ(

ω)/D

-0.1 -0.05 0 0.05 0.1-1

-0.5

0

-0.12 -0.08 -0.04 0 0.04 0.08 0.12ω/D

-1

-0.5

0

ImΣ(

ω)/D

-0.004 -0.002 0 0.002 0.004ω/D

-1

-0.5

0

δ=0.2 δ=0.005

FIG. 3: (Color online) Local spectral function and the imaginary part of the self-energy for large doping δ = 0.2 (left) andsmall doping δ = 0.005 (right), for U =∞.

5. Deviation from the low-frequency Fermi-liquid behavior

From panels b, c (for ReΣ) and e, f (for ImΣ) ofFig. 6, one can visualize the low-frequency deviationsfrom Fermi-liquid behavior. The latter is indicated bythe dashed straight and parabolic lines on this figure:ReΣ− ReΣ(0) = ω(1− 1/Z) and ImΣ ∝ −(ω/Z)2.

When visualized on an intermediate frequency scale(panels b and e) it is seen that deviation from the FL be-havior is more apparent on the ω > 0 side, in accordancewith the particle-hole asymmetry discussed above and aspointed out in previous studies9,38. ReΣ deviates fromlinearity and flattens upwards for ω > 0, resulting in thebending of the dispersion of ω > 0 quasiparticles towardsthe non-interacting bare dispersion, displayed above onFig. 4. Accordingly, the deviations from parabolic behav-ior in ImΣ(ω) are much more pronounced on the positivefrequency side.

Zooming further on the low-frequency range (panels cand f) allows one to locate more quantitatively the devia-tion from the FL behavior. At U =∞, it is seen to occurat ω?FL ' 0.1ZD, which is of order 0.025δD to 0.05δDdepending on δ. In agreement with previous studies9 atfinite U , the scale below which FL is found to apply isseen to be a very low one. It is one order of magnitudesmaller than the Brinkman-Rice scale ≈ δD which cor-responds to scaling the bare bandwidth by the (inverse)

of the effective mass. When converted to a temperaturescale, the Brinkman-Rice scale roughly corresponds tothe temperature at which QP excitations disappear al-together (and the resistivity approaches the Mott-Ioffe-Regel limit)9, but it should not be identified with themuch lower scale associated with deviations from FL be-havior.

The low-frequency zooms in panels c and f actuallyreveal that the deviations from FL behavior are seen bothon the ω < 0 and ω > 0 side, at similar scales ±ω?FL. Thisscale corresponds to a low-energy ‘kink’ in ReΣ. Thecorresponding low-energy kink in the QP dispersion39–41

is actually visible upon close examination of Fig. 4.

As seen on Figs. 5b and c, the full collapse of the data islimited to very low frequencies. Two kinds of deviationsfrom the universal behavior can be recognized. On neg-ative frequency side, at moderate doping the deviationsoccur at the onset of the Hubbard band as the quasipar-ticle peak is not clearly separated from the LHB. On thepositive frequency side, the different curves deviate fromeach other also in the small doping limit. Comparingthe two lowest dopings, for instance, reveals the excess ofthe spectral weight for the lower doping curve. This sug-gests that the quasiparticle peak is not fully characterizedby the renormalization factor Z alone: the quasiparticlepeak weight Wq.p. and Z are not necessarily simply pro-portional. This question is best addressed at very low

10

FIG. 4: (Color online) Intensity plots of the momentum (εk-) resolved spectral function A(ε, ω) for U = ∞ at four differentdoping levels, plotted as a function of ε/D and ω/ZD. The plain line locates the solution of the QP pole equation ω + µ −ReΣ(ω)− ε = 0 (neglecting ImΣ). By definition of the QP excitations, this line has slope unity (cf. dashed line) at low-ω whenplotted in this manner since ωQP = Z(ε− εF ) (i.e., v?F = ZvF ) within the DMFT.

doping, when the quasiparticle peak is well separatedfrom the LHB. We can then extract Wq.p. by integrat-ing the spectral function between the two local minimain A(ω). The results are plotted in Fig. 5d. We findthat at low δ, Wq.p. and Z are related by a power-lawWq.p. = Zγ , with γ close to one, but not exactly 1. Morespecifically, γ is found to be U -dependent: γ = 1.017 forU = 3, γ = 1.039 for U = 4, γ = 1.049 for U = 10 andγ = 1.067 for U =∞.

6. Charge compressibility: absence of phase separation

For some types of the (non-interacting) conduction-band density of states, there can be phase separation nearhalf filling42. We verify that this is not the case for theBethe lattice by plotting the band filling n as a functionof the chemical potential µ in Fig. 7, panel a) for U =∞. The dependence is monotonous, thus all solutionsare physically stable with positive charge compressibilityκ = ∂n/∂µ. We also plot the quasiparticle residue Z asa function of the chemical potential µ (panel b). Thecharge compressibility κ as a function of the band filling

(panel c) has a maximum near quarter filling. For smallern, the decrease is due to the particular form of the non-interacting DOS (semi-circular function). For larger n, κdrops to zero as the Mott transition is approached. Theasymptotic behavior is a power law δβ with β ≈ 1/5.

V. DOPED MOTT INSULATOR: AN ECFLPERSPECTIVE ON THE DMFT

In this section we make use of the general structure ofthe self-energy resulting from the ECFL in order to inter-pret, fit, and better understand the complex frequency-dependence of the DMFT self-energies. The emphasiswill be on the intermediate frequency range which en-compasses both the vicinity of ω = 0 and of the quasi-pole (sharp peak) in ImΣ on the negative frequency sideat ωpeak ' −0.7δD. We focus on intermediate dopinglevels which turns out to be the range where the ECFLapplies best, rather than on very low doping. For thesereasons, we can use as a scaling variable:

x ≡ ω/δ (38)

11

-2 -1 0 1 2-1

-0.8

-0.6

-0.4

-0.2

0

Im Σ

(ω)/D δ=0.005

δ=0.02δ=0.05δ=0.1δ=0.2δ=0.3

-20 -10 0 10 200

0.10.20.30.40.50.6

-2 -1 0 1 2ω/ZD

00.10.20.30.40.50.6

A(ω

)D

(a)

(b)

(c)

0.001 0.01 0.1Z

0.001

0.01

0.1

Wq.

p.

U=3U=4U=10U=∞

(d)

FIG. 5: (Color online) (a) Imaginary part of the self-energyΣ versus the rescaled frequency ω/Z for U = ∞. (b,c) Cor-responding local spectral functions. Note that when plottedvs. ω/Z, the peak related to the onset of the LHB moves tothe left with diminishing δ, as seen in panel b. The results forfinite U are qualitatively very similar. The arrows indicatethe direction of the increasing value of δ.

A. ECFL line-shapes: main features

The low frequency ECFL analysis from Sec. II gives asimple expression, Eq. (28), for ImΣ at T = 0. Using theoverline convention of Eqs. (32,33, 34) to denote variablesthat remain finite as δ → 0, e. g. P = P/δ, we rewrite

Eq. (28) as

ImΣ = − x2

Ω1

1− x/ ∆1

(1 + x/ ∆2)2 + x4/ Ω2

2

. (39)

This ansatz function is determined by two variables withthe meaning of curvature Ω1,2 that are simply related toΩΣ and ΓΨ, respectively, and by two parameters whichadjust the asymmetry ∆1,2 that are related to ∆ and cψ.

The numerator of the expression describes a parabolicdependence, with a cubic correction term. This ansatzfunction has a peak (quasi-pole) at frequency x = −∆2

for a finite Ω2, turning into a true pole when Ω2 → ∞.The low-frequency asymmetry of the self-energy, impor-tant for the low-temperature thermoelectric properties,arises through a combination of the terms present in thenumerator and denominator. Expanded to cubic orderin frequency, the ansatz gives

ImΣ = −x2/Ω1[1− (1/∆1 + 2/∆2)x]. (40)

The ansatz function and its evolution as the parametersare varied is illustrated in Fig. 8.

Summarizing, the ansatz function Eq. (39) contains aparabolic dependence, multiplied by a function with asharp peak at negative frequencies; therefore, it can beexpected to describes the coarse structure found in theDMFT very well already at this order including only theFermi-liquid structure of the underlying functions Ψ andχ.

B. ECFL fits of the DMFT self-energy

The DMFT results for U =∞ self-energy for a range ofdoping levels are presented in Fig. 9 together with fits toEq. (39). At large doping, the ansatz function describesthe DMFT data remarkably well: the low-frequency de-pendence and the main shape of the self-energy are fullyreproduced.

At smaller doping, the quasi-pole at negative frequen-cies becomes very sharp and ImΣ in DMFT develops a“nipple-like” structure at low-frequency, with semi-linearfrequency dependence at negative frequencies. These twofeatures (quasi-pole and nipple) cannot be simultane-ously well described by the simplest ECFL ansatz func-tion in a broad frequency range. Given that the ansatzhas a structure that already contains the pole, the fitsin a broad frequency window are more meaningful. TheDMFT data can still be described successfully, but termsbeyond the lowest-order Fermi-liquid form in Ψ and χneed to be retained. Work along these lines to reproducethe precise shape and to analyze its physical contentsshould be possible.

The evolution with doping of the fitting parameters isshown in Fig. 10. The first observation is that the fittingparameters (except for ∆1 to be discussed below) do notdepend substantially on the doping, hence validating ourassumptions stated above.

12

-2

-1

0

1

2

ReΣ

(ω)/

D

-0.5

0

0.5

1

1.5

-0.2

0

0.2

0.4

-1 -0.5 0 0.5 1

ω/δD

-2

-1.5

-1

-0.5

ImΣ

(ω)/

D

δ=0.004

δ=0.02

δ=0.1

δ=0.2

-0.4 -0.2 0 0.2 0.4

ω/δD

-0.2

-0.1

-0.4 -0.2 0 0.2 0.4ω/ZD

-0.2

-0.15

-0.1

-0.05

(a) (b) (c)

(d) (e) (f)

FIG. 6: (Color online) Real and imaginary parts of the DMFT self-energy Σ(ω) for a range of doping for U =∞, as a functionof ω/δD (left and centre) or as a function of ω/ZD (right). The arrows indicate the direction of the increasing value of δ. Thelinear (for real parts) and parabolic (for imaginary parts) are performed in the frequency range [−0.05 : 0.05]δD. In panel (f),the data is vertically offset for clarity.

-5 0 5 10μ/D

0

0.2

0.4

0.6

0.8

1

n

-8 -6 -4 -2 0 2 4 6 8 10 12μ/D

0

0.2

0.4

0.6

0.8

Z

8 9 100

0.01

0.02

0.03

0 0.2 0.4 0.6 0.8 1n

0

0.01

0.02

0.03

0.04

0.05

0.06

char

ge

com

pre

ssib

ilit

y, κ

D

NRG results

(1-δ)0.196U=∞

(a) (b)(c)

FIG. 7: a) Band filling n vs. chemical potential µ for U =∞. b) Quasiparticle residue Z vs. chemical potential µ for U =∞.c) Charge compressibility for U =∞.

The second observation is that Ω1 is usually found tobe very close to Ω2. This supports the conclusions of theλ2 analysis (to be discussed in the next section) whichalso finds that ΓΨ is close to Ωχ.

The third observation is that the bulk of the asym-metry does actually not come from the explicitly cubicterm (parametrized by 1/∆1), but rather from 1/∆2.Hence, it is the presence of the quasi-pole at negativefrequency which is responsible for the strong particle-hole asymmetry. This is seen most explicitly in thedoping range δ = 0.2 − 0.3, where 1/∆1 almost van-ishes and where, furthermore, the data is excellently de-scribed by the ansatz function from Eq. (39). The phys-ical content of this observation might be that the low-frequency particle-hole asymmetry is (at least at not toolow dopings) directly related to the presence of the LHBat much higher frequency scales, and thus ultimately tothe strong-correlation physics. This observation is con-sistent with the picture emerging from the ECFL, wherethe asymmetry is a consequence of the Gutzwiller pro-

jection.

C. Summary

To summarize, the ECFL-derived ansatz, Eq. (39),which retains only the lowest-order Fermi-liquid termsin Ψ and χ, describes the rather complex frequency de-pendence of the DMFT data remarkably well at lowto intermediate frequencies and for not too small dop-ing levels. Importantly, retaining only the Fermi-liquid(hence particle-hole symmetric) terms in these ECFLself-energies already yields a marked particle-hole asym-metry in the physical electron Dysonian self-energy. Theansatz also describes the pole at negative frequencies, as-sociated with the onset of the LHB. Whereas the Fermi-liquid behavior only applies at extremely low frequenciesin the Dysonian self-energy, the Fermi-liquid conceptscan still be used over a much broader frequency rangewhen proper auxiliary quantities are considered, within

13

decreasing Δ1

decreasing Ω2

decreasing Δ2

1.0 0.5 0.5 1.0x

1.5

1.0

0.5

1.0 0.5 0.5 1.0x

1.0

0.8

0.6

0.4

0.2

1.0 0.5 0.5 1.0x

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Im

FIG. 8: (Color online) Role of the parameters in the ECFLansatz function, Eq. (39), use for fitting in Figs. 9 and 10.

the broader framework provided, e.g., by the ECFL the-ory. At lower doping levels, however, the DMFT resultsdisplay structures (nipple) which signal the increasingimportance of corrections beyond the dominant Fermi-liquid terms in Ψ, χ.

VI. ECFL: EXPANSION TO O(λ2)

A. Summary of equations

We now summarize the results of the O(λ2) expan-sion of the ECFL equations in Ref. 8, which are thencomputed and compared with the DMFT results. Wenote that the ECFL reformulation of the Dyson self-energy Σ(ω) into the ECFL auxiliary self-energies Ψ(ω)and χ(ω) is exact. Therefore, if one could perform theλ expansion to infinite-order in λ, one would obtain theexact answer for these auxiliary self-energies, and conse-quently the Dyson self-energy. The resulting Dyson-selfenergy would then agree exactly with the one obtained

through DMFT for the case of infinite-U. Our aim hereis to benchmark the lowest non-trivial order of the λ ex-pansion against the exact DMFT results. Note that inthe d→∞ limit, in the paramagnetic phase, the single-particle properties of the t − J model are identical tothose of the U = ∞ Hubbard model. In other words, aslong as antiferromagnetic correlations are short-ranged,J does not enter single-particle properties in the d = ∞limit. Accordingly, the only coupling constant enteringthe simplified ECFL equations is the hopping (band dis-persion) itself, and not the superexchange.

In the O(λ2) scheme, the explicit density factors 1− n2

that occur in Eq. (8) and (9) are replaced by the rule

1− n

2→ aG ≡ 1− λn

2+ λ2n

2

4+O(λ3). (41)

The second rule is that the explicit self-energy expres-sions in these equations are multiplied by λ. As an illus-tration of these rules, we write Eq. (8) and (9) as

g−1(k) = iω + µ− aG εk − λΦ(k), (42)

µ(k) = aG + λ Ψ(k). (43)

The factor λ is set to 1 before actually computing withthese formulas. The two self-energy functions Φ(k) andΨ(k) satisfy the equations to second order in λ:

Ψ(iωk) = −λ∑pq

(εp + εq − u0)g(p)g(q)g(p+ q − k),

Φ(k) = (εk −u0

2)Ψ(iωk) + χ(iωk)− u0(λ

n2

8− n

2)

−∑p

εpg(p)

χ(iωk) = −λ∑pq

(εp+q−k −u0

2)(εp + εq − u0)×

g(p)g(q)g(p+ q − k). (44)

All equations in Eq. (44) are implicitly understood tohave O(λ3) corrections, so that the g and µ pieces of Gin Eq. (7) are correct to the stated order. As expected,the functions Ψ, χ depend on the frequency but not the

momentum ~k. Both Green’s functions satisfy an identi-cal number sum rule

∑k,ωnG(k) = n

2 =∑k,ωn

g(k), andthe theory has two chemical potentials necessary to im-pose these, namely µ and u0. As discussed in Ref. 3, thesecond chemical potential u0 arises from the requirementof satisfying a “shift invariance” in the theory. The shifttransformation in the present model acts as εp → εp + c.This transformation shifts the center of gravity of theband; it is absorbable in u0, and thus rendered incon-sequential. We can easily verify that χ and Ψ are inde-pendently shift-invariant. Combining the expressions, wewrite

µ′ ≡ µ+ u0

(λ2n

2

8− λn

2

)− u0

2aG + λ

∑p

εpg(p),

g−1(k) = iω + µ′ − (εk −u0

2) aG + λΨ(iωk) − λχ(iωk).

(45)

14

-2

-1.5

-1

-0.5

0

ImΣ

(ω)/

Dnumerical resultsfit to ECFL form

-1 -0.5 0 0.5 1

ω/δD

-3

-2

-1

ImΣ

(ω)/

D

-0.5 0 0.5 1

ω/δD

δ=0.3 δ=0.2

δ=0.1 δ=0.02

FIG. 9: (Color online) ImΣ(ω) vs. rescaled frequency ω/δD and fits to Eq. (39) for U =∞. The ansatz describes the DMFTself-energy at moderate dopings remarkably well. The nipple structure that becomes pronounced at small doping signals thatχ and Ψ develop non-Fermi-liquid corrections.

0 0.1 0.2 0.3 0.4 0.5 0.6δ

-1

-0.5

0

0.5

1

1.5

par

amet

ers

1/∆1

1/∆2

1/Ω1

1/Ω2

FIG. 10: (Color online) Parameters in the fit function,Eq. (39), for a range of doping δ at U = ∞. For a broadfitting energy range, the fitting parameters are smooth as afunction of δ, thus the fitting procedure is well defined. Atdopings where the fits work best (that is around δ = 0.25),Ω1 is found to be close to Ω2.

The Green’s function is then found by combiningEq. (45), (44) and (43) in the expression Eq. (7).

B. Setting up the computation

To set up the computation, we write a local Green’sfunction with weight m = 0, 1, . . . using Eq. (45) as

gloc,m(iωk) ≡∑~k

g(k) (ε~k)m

=

∫ D

−Ddε ρ0(ε)

εm

iω + µ′ − (aG + Ψ)(ε− u0

2 )− χ,

(46)

where χ and Ψ are functions of frequency iωk but not theenergy ε. We find that both gloc,0 and gloc,1 are needed tocompute the frequency-dependent self-energy. Similarly,a local G can be defined, and the number sum rules canbe written as n

2 =∑iω gloc,0(iω) and n

2 =∑iω Gloc,0(iω).

Where necessary, the usual convergence factor eiω0+

is

inserted. The two ~k independent functions Ψ and χ inEq. (44) can be written in a compact way if we first definea function with three indices (m1m2m3) from the weightfactors:

Im1m2m3(iω) = − 1

β2

∑ν1,ν2

gloc,m1(iν1)×

gloc,m2(iν2)gloc,m3

(iν1 + iν2 − iω). (47)

After continuation iω → ω + iη, and for all values ofthe indices, the low frequency and temperature I(iω) isa Fermi-liquid-like self-energy with an imaginary part ∝

15(ω2 + (πkBT )2

). We can now rewrite Eq. (44) as:

Ψ(iω) = −u0I000(iω) + 2I010(iω)

χ(iω) = −u0

2Ψ(iω)− u0I001 + 2I011(iω). (48)

Clearly, Eqs. (48) and (46) along with the definition (47)and the number sum rules form a self-consistent set ofequations that can be solved iteratively on a computer.The Dyson self-energy and the spectral function can becomputed in terms of these quantities using Eq. (13)

C. Auxiliary and Dyson self energies to O(λ2)

In Fig. 11 we present ρχ, ρψ, and ρΣ (from top to bot-tom). ρψ and ρχ have similar frequency-dependence anda Fermi-liquid form is obeyed much more accurately thanwhat is found for the Dyson self-energy ρΣ. This sup-ports the ansatz that we employed above. In particular,the auxiliary self energies are more particle-hole symmet-ric; most of the particle-hole asymmetry follows from thestructure of ECFL equations. This signals that the Fermiliquid concept has validity outside of the canonical Fermiliquid behavior.

VII. DETAILED COMPARISON OF O(λ2) ECFLRESULTS TO DMFT

A. The effective density of the ECFL spectralfunctions and its phenomenological adjustment

The O(λ2) equations of ECFL discussed here give ahigh-ω limiting behavior G ∼ aG

ω , differing from the exact

form G ∼ 1−n2ω due to the replacement 1 − n

2 → aG ≡1 − λn/2 + λ2n2/4 as per the rules of the calculation.This effect is due to the incomplete projection of theO(λ2) treatment of the ECFL equation of motion. Atn ∼ 0.75 the error in the high-frequency weight is 22.5%.

A phenomenological scheme for adjusting for this fea-ture defines an effective density neff , using the ratio ofparticle addition and removal states as the relevant met-ric, so that n

1−n+n2/4 = neff

1−neff, thus yielding

neff =n

1 + n2

4

. (49)

Clearly higher order calculations would have a corre-sponding mapping between the two densities. For severalof the comparisons below, agreement is greatly improvedby plotting the results of ECFL as a function of neff .

B. Comparison between O(λ2)-ECFL and DMFT

We find that the computed values of the quasiparticleweight Z from ECFL are close to the U/D = 4 DMFT

-1.0 -0.5 0.0 0.5 1.00.00

0.02

0.04

0.06

0.08

ΩD

ΡΧ D

Decreasing n

-1.0 -0.5 0.0 0.5 1.0

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

ΩD

ΡY

Decreasing n

-1.0 -0.5 0.0 0.5 1.00.00

0.05

0.10

0.15

0.20

0.25

ΩD

ΡS D

Decreasing n

FIG. 11: (Color online) Imaginary parts (spectral densi-ties) of the auxiliary self-energies χ and Ψ, and the Dysonself-energy Σ within the O(λ2) ECFL. The dotted lines areparabolic fits at the highest density. Recall that typicalFermi-liquid-type spectral functions exhibit a parabolic andtherefore particle-hole symmetric behavior over a large energyrange. From these fits one observes that the auxiliary func-tions Ψ and χ have a Fermi-liquid form over a wider energyrange than the Dyson self-energy Σ.

curve, we detail this in the Appendix A, where the mo-mentum distribution is also shown. This is suggestive ofan analogy between the two incompletely projected the-ories. In particular, making U finite, and truncating theλ expansion at second order, both introduce some doubleoccupancy into the system. It is therefore not surprisingthat the U/D = 4 DMFT results agree better with theO(λ2)-ECFL than with the U/D = ∞ DMFT results.

16

However the limitations of the O(λ2) calculation withinECFL preclude obtaining reliable results for doping levelssmaller than δ ≈ 0.25.

1. Spectral lineshapes

In Fig. 12 we compare the ECFL and the DMFT re-sults at U = 4D and U = ∞ for the ε-resolved spectralfunctions at two values of the band energy, εk = −Dand εk = εF . In general, the agreement is encourag-ing. At εk = −D (left panel), DMFT has a deeperminimum between the QP and the secondary feature athigh binding energy than is seen in ECFL, but the po-sition of the ECFL peaks agrees well with that of theDMFT peaks. At εF (right panel) the QP are of similarwidth but have different values of Z, as discussed above.The background of width ∼ D lies over essentially thesame frequency range for all three calculations, and hasa peak at ω = −0.5D, approximately the same positionfor each data set. However, the height of the peak is lesspronounced for the ECFL than the DMFT. At positivefrequencies the spectral functions are in excellent agree-ment. Plotting the spectral function as a function of thescaled frequency ω/ZD improves the agreement in theposition and width of the quasiparticle, as illustrated inthe more sensitive self-energy curves in Fig. 13. We notethat the scaled ECFL curves agree well with the DMFTcurves even for density n ∼ 0.8−0.9 for scaled frequency|ω| ≤ 0.5DZ. We find this agreement surprising in viewof our criterion discussed above, placing n ∼ 0.75 as thelimiting density.

The physical spectral function A(ω), when displayed asa color intensity plot using the scaled frequency ω/ZD asin Fig. 14, further emphasizes this similarity. At this levelof description, the U = 4D DMFT curve and the O(λ2)calculation look almost identical. In particular, as clearfrom this figure, both theories indicate that the quasipar-ticle peak becomes rapidly more dispersive as one movesto positive energies, corresponding to unoccupied states(i.e., the effective Fermi velocity increases as comparedto its low-energy value and becomes closer to the bandvalue). As discussed above (Sec. IV, Fig. 4), this is one ofthe primary common conclusions of both theories, whichcould be tested in future experiments able to probe theunoccupied states in a momentum-resolved manner.

In view of the remarkable similarity between the dif-ferent theories, as seen in Fig. 13 and Fig. 14, it appearsthat the O(λ2) version of ECFL has the correct shape ofthe spectra built into it, but requires a correction for atoo large value of the QP factor Z. This is the main con-clusion of this work regarding the benchmarking of theECFL.

VIII. CONCLUSION AND PROSPECTS

In this work we have presented a detailed comparisonbetween the DMFT and the ECFL theories, applied tothe doped Hubbard model at large as well as infinite U , inthe limit of infinite dimensions (Bethe lattice with infinitecoordination).

Our approach here is two-fold. On the one hand, wehave used the general structure of the Green’s functionand self-energy in the ECFL theory to obtain a usefulanalytical ansatz which reproduces quite well the richand complex frequency-dependence of the DMFT self-energy at not too low doping level. This ansatz relieson the lowest-order Fermi-liquid expansion of the twoauxiliary ECFL self-energies Ψ and χ. Quite remark-ably, the marked deviations from the Fermi-liquid formand the particle-hole asymmetry found in the physicalsingle-particle self-energy can be accounted for by thisunderlying Fermi-liquid form of auxiliary quantities. Inturn, the deviations observed between the DMFT resultsand this lowest-order ansatz at lower doping levels em-phasize the need for corrections to FL behavior in Ψ, χwithin the ECFL. This part of our study thus providesuseful analytical insights into the DMFT description ofthe doping-driven Mott transition.

On the other hand, we have used the DMFT results(obtained here with a high-accuracy NRG solver) as abenchmark of the ECFL theory. Specifically, we havesolved numerically the O(λ2) ECFL equations, appropri-ately simplified in the limit of large dimensions. For nottoo low doping levels, where this O(λ2) scheme is appli-cable, we found that the spectral properties agree wellprovided the comparison is made as a function of thescaled frequency ω/ZD, with Z the quasiparticle weight.A similar situation arises in comparing the ECFL methodfor the Anderson impurity model, where Z is rapidly sup-pressed as the Kondo limit is approached43. This ad-justment of the frequency scale compensates the knownweakness of the O(λ2) theory in obtaining Z quantita-tively, and enables, to some extent, a preview of theresults of the planned higher-order calculations in theECFL projection parameter λ.

From a physics point of view, we now summarize themost significant insights provided by our study.

Doped Mott insulators are found to be characterizedby a marked particle-hole dynamical asymmetry, as em-phasized in recent ECFL44 and DMFT9 studies. Inthe case of hole-doping, particle-like (ω > 0) excita-tions are longer lived than hole-like (ω < 0) ones, lead-ing to more ‘resilient’ electron-like quasiparticles9. Thisdynamical asymmetry has physical implications for thespectral lineshapes9,44 as well as thermopower9,45. Theasymmetric terms in the low-frequency expansion of theself-energy signal deviations from the Fermi-liquid the-ory which are usually ignored in weak-coupling studies.They become large at low hole doping and strong cou-pling, as demonstrated here in considerable detail, thusconfirming the proposal made originally in Ref. 44.

17

-2.0 -1.5 -1.0 -0.5 0.00.0

0.2

0.4

0.6

0.8

ΩD

DΡ G

HΕ,Ω

L

n=.7TD=.0025

¶D=-1

UD=4

UD=¥

ECFL

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

0.3

0.4

ΩD

DΡ G

HΕ,Ω

L

n=.7

TD=.0025

¶=¶F

UD=¥

UD=4

ECFL

FIG. 12: (Color online) Spectral functions within the DMFT (U =∞ and U/D = 4) and ECFL at two typical energies ε = −Dand ε = εF , with n = 0.7 and T = 0.0025D. The location of the quasiparticle peak near ω ∼ 0 and the broad secondary peakfor ω < 0 are common to both calculations. While there are subtle differences, especially in the magnitudes of the secondarypeaks, the main features of the three calculations match at high and low frequency.

Due to the importance of this asymmetry, we foundthat the energy vs. momentum dispersion of the quasi-particle state quickly deviates on the ω > 0 side from itslow-energy value (associated with the renormalized effec-tive Fermi velocity). The deviation is towards a weakerdispersion, closer to the bare band value. This is a pre-diction of both ECFL and DMFT which could be testedexperimentally once momentum-resolved spectroscopiesare developed in order to address unoccupied states (the‘dark side’ for photoemission).

Regarding ARPES lineshapes, we also emphasize thatthe recent successful comparison6,7 between the ECFLand the experimental ARPES lineshapes in the optimallydoped and overdoped cuprates along the nodal directioncan just as well be interpreted as the similar success ofthe DMFT interpretation of these lineshapes. The ad-justment of the momentum dependence of the caparisonfactor for different systems in Refs. 6,7 hints at the impor-tance of the momentum-dependence of the self-energy.This momentum-dependence is already present in theECFL in two dimensions, and also emerges from clusterDMFT calculations.

Further comparison between the nature of the momen-tum dependence in both theories is to be addressed in fu-ture work. More generally, we believe that this work laysthe foundation of a useful program where the momentum-dependent self-energies can be reliably computed and ex-pressed in simple analytic forms. While cluster DMFTmethods can already provide some answers to this im-portant problem, the ECFL theory readily treats low di-mensions and the momentum dependence. In order to getfurther solid results, the current limitation of the ECFLto the somewhat overdoped regime needs to be overcome.This limitation arises from the low order of the expan-sion in λ, and brute-force higher order calculations in λare planned. In this task, the insights gained from thepresent comparison with DMFT, are invaluable.

Acknowledgments

A.G., J.M. and B.S.S. acknowledge useful discussionswith Xiaoyu Deng at the initial stage of this project. Thework at UCSC was supported by DOE-BES under GrantNo. DE-FG02-06ER46319. A.G. and J.M. acknowl-edge the support of Ecole Polytechnique and College deFrance. R. Z. and J. M. acknowledge the support of theSlovenian Research Agency (ARRS) under Program P1-0044.

Appendix A: Quasiparticle occupation and Zk in theO(λ2) ECFL.

The momentum distribution function, Fig. 15, showsgood agreement with the DMFT at a density n = .7,and the large spillover for k > kF is of the same scalein both sets of calculation. Its importance in estimatingthe background spectrum in ARPES is well known, sothis is already a reasonably reliable common result. It isalso interesting that at the Fermi momentum, the mag-nitude of the distribution function is close to 1

2 in both

calculations, as argued in the literature15,46.

In Fig. 16 we compare the quasiparticle weight Z inECFL and DMFT. TheO(λ2) ECFL result has some sim-ilarity to the Gutzwiller approximation47 result 2δ/(1+δ)in the limited density range of validity. However it doesnot seem to vanish in any obvious way, if we extrap-olate by eye to higher density n, highlighting its mainweakness in the current state of development, but plot-ted against the effective density it becomes comparableto the U/D = 4 DMFT curve over a limited range.

18

-1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

Ω

DZ

-Im

S

D

n=.6

UD=¥UD=4

ECFL

-1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

Ω

DZ

-Im

S

D

n=.7

UD=¥UD=4

ECFL

-1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

Ω

DZ

-Im

S

D

n=.8

UD=¥

UD=4

ECFL

-1.0 -0.5 0.0 0.5 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

w

DZ

-Im

S

D

n=.9

U êD=¶

U êD=4

ECFL

FIG. 13: (Color online) Spectral function (imaginary part) of the Dyson self-energy Σ versus the scaled variable ω/(DZ) inthe ECFL theory at order λ2, and the DMFT at two values of U . The ECFL predicts a value of Z which is too large at lowdoping, and significant U dependence creates differences between the U/D = 4 and U =∞ results of the DMFT. Nonetheless,all three cases overlap well at low frequencies when plotted against the scaled frequency. Surprisingly, this agreement survivesto densities far beyond the expected range of the current version of the ECFL.

Appendix B: NRG impurity solver convergence atsmall doping

In order to obtain well-converged spectral functionsusing the NRG impurity solver at low doping δ, sev-eral parameters in the method need to be apropriatelytuned. Their choice affects both low-frequency and high-frequency parts of the spectral functions. In addition,it also significantly affects the numerical requirements –both the duration of each NRG calculation and the num-ber of the DMFT cycles until self-consistency. Very closeto the Mott transition, obtaining fully converged resultsbecomes computationally very expensive (several hun-dreds of DMFT cycles) even with Broyden acceleration22.In this section, we explore the effects of different choiceson the quasiparticle residue Z (low-frequency property),Fig. 17, and on the shape of the LHB (high-frequencyproperty), Fig. 18.

We first explore the choice of the discretization scheme(i.e., how the coefficients of the Wilson chain are com-puted based from the input hybridization function). Wecompare the discretization scheme (denoted as Z) pro-posed by R. Zitko and Th. Pruschke in Ref. 22 which cor-rects the systematic discretization errors near band edges

present both in the conventional discretization scheme(Y), Ref. 48, and in the improved scheme by V. Campoand L. Oliveira (C), Ref. 49. At low frequencies, oneobserves excellent overlap of the results, as indicated inFig. 17a. This is in line with the common wisdom thatthe NRG is a reliable method for low-frequency prop-erties, having good spectral resolution in the vicinity ofthe Fermi level where the discretization grid is condensed.For this reason, the choice of the discretization schemehas little effect on Z. At high frequencies, however,one can clearly observe the systematic artifacts presentin schemes Y and C: the LHB presents spurious (non-physical) structure at the outer edge which is not presentin the results of scheme Z, see Fig. 18a. Recent com-parisons of the NRG (using scheme Z) and continuous-time quantum Monte Carlo at finite temperatures haveestablished that the NRG (using scheme Z) is, in fact,a rather reliable method also for high-frequency/finite-temperature properties. On the other hand, the NRGusing schemes Y or C is expected to exhibit more pro-nounced systematic errors at high-frequencies and at fi-nite temperatures.

The second important choice concerns the value of thediscretization parameter Λ, which controls the coarseness

19

FIG. 14: (Color online) Physical spectral function A(ε, ω). From left to right: U = 4 DMFT, U =∞ DMFT, and ECFL withn = 0.7 and T/D = 0.0025. Hot colors represent high intensity, while darker blue represents low intensity. Noting from theleft panel of Fig. 12 that the QP band has a slightly different width in each calculation, we plot the spectral function here as afunction of ω

DZ. This brings the low energy (QP) features of the spectral function into impressive agreement, indicating that

Z, rather than δ, is the fundamental energy scale of the extremely correlated state.

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

ΕD

mk

1-n2=.65

.5

n=.7

TD=.02,.01,.005,.0025

Lower Band weight

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

ΕD

mk

1-n2+n24=.7725

.5

n=.7

TD=0.0025, 0.0035, 0.0049, 0.0069,0.0096, 0.013, 0.019, 0.026, 0.037, 0.052

FIG. 15: (Color online) Momentum occupation versus εwithin the DMFT at U/D = 4 (top) and the ECFL (bot-tom).

of the logarithmic grid. The standard choice is Λ = 2,which is suitable to obtain well converged results at bothlow and high frequencies, see Figs. 17b and 18b. In fact,the results do not change much even when going to some-what higher Λ = 2.5, while for Λ = 3 we start to observesome systematic deviations at very low doping δ. Wehave also performed some test calculations for smallervalues Λ = 1.9, 1.8, 1.7; the results differ little from those

0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5

∆

Zk

ECFL

ECFL ∆eff

UD=4UD=20UD=40UD=¥

FIG. 16: (Color online) Quasiparticle weight Z as a functionof hole doping δ (or δeff = 1− neff , the effective hole doping)from the O(λ2) version of ECFL, and from DMFT for variousvalues of U. More detailed DMFT results are in Fig. 1. Theblue dashed line represents Z = δ, the simplest U =∞ slave-boson estimate, as a guide to the eye. The dotted red linerepresents the Gutzwiller approximation result Z = 2δ

1+δ.

for Λ = 2 while being significantly more computationallyexpensive to produce.

We now consider the broadening parameter α whichcontrols how the raw spectral function in the form ofa set of weighted delta peaks is processed to obtain asmooth continuous representation. Too small values leadto spurious oscillations, too high values to overbroaden-ing. These effects are nicely illustrated by the results forthe LHB part of the spectral funciton in Fig. 18c. Thelong high-frequency tail of the LHB for increasing α is aclear over-broadening effect, while the oscillatory featuresfor α = 0.1 are a discretization artifact. At low frequen-cies, the QP residue Z converges as α is decreased, seeFig. 17c. We find that for α ≤ 0.1, the results practi-cally overlap, while for α = 0.2 (the value used for most

20

0

0.5

1

Z

ZCY

0

0.5Z

Λ=2Λ=2.5Λ=3

0

0.5Z

α=0.1α=0.2α=0.3α=0.4α=0.5

0.001 0.01 0.1 1δ=1-n

0

0.5Z

10-7

10-6

10-5

10-4

3.16 10-4

discretization scheme

discretization coarseness

clipping cutoff value

broadening

(a)

(b)

(c)

(d)

FIG. 17: (Color online) Quasiparticle residue Z for the U =∞ Hubbard model as a function of doping δ. We comparethe DMFT results for different choices of the NRG impuritysolver parameters.

calculations in this work), the deviation from the asymp-totic value is of order one percent. For large values ofbroadening (as commonly done in NRG calculations), Zis underestimated. This is because the spectral weightis more spread around as α increases, thus less weightremains in the QP peak. Based on these results, we findthat α = 0.2 is a good compromise.

Finally, we discuss a subtle issue which becomes im-portant at very small dopings. The NRG discretizationhas difficulties if in the hybridization function there areextended regions of very low values. In particular, thisleads to very slow approach to the self-consistency. Forthis reason, it is convenient to use a small, but finite cut-off value for the hybridization function to clip the inputhybridization function to some minimum value at all fre-quencies. It is important, however, to choose this valueso that the results are not perturbed. We find that us-ing too high cut-off leads to incorrect Z vs. δ behaviorat low doping (a down-turn), see Fig. 17d. The effect isthus similar to overbroadening, since the spectral weightshifts from the quasiparticle peak to the region betweenthe LHB and the QP peak, where the clipping is applied(for small δ, where the LHB and the QP peak no longeroverlap, but rather the QP becomes an isolated specgtralpeak in the gap). There is also some effect of clipping onthe LHB itself, Fig. 18d. Again, this effect is analogousto overbroadening.

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noteworthy. In these works the momentum dependenceof the Dyson self energy in the limit of large d wasdiscussed, before the exact simplifications arising fromthat limit were known precisely. As shown in Ref. 8 andin the present work, ECFL does provide a momentum-independent Dyson self energy in this limit.

12 A. Georges and G. Kotliar, Phys. Rev. B 45, 6479 (1992).13 The minimal version of ECFL is obtained by setting to

zero all correlation functions satisfying the Gutzwillerconstraint- whereas the symmetrized version adds in someterms that are known to be zero, in order to make theeffective Hamiltonian manifestly Hermitean3 (p 11).

14 To avoid the linear growth with ω of the self energy, onemay define the Dyson Mori self energy, as in4 with a factor1− n

2in the numerator of the Greens function. However in

the present work, since DMFT directly yields the Dysonself energy, we simplify by quoting the results for this ob-ject directly.

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PhysRevB.87.245101.16 A more complete discussion of the approach to the insu-

lating limit within ECFL is not yet available. It is possibleto also achieve a vanishing quasiparticle weight near half-filling in other ways as well, such as letting cχ diverge as 1

δ.

We prefer the current scheme for its simplicity and econ-omy.

17 W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324(1989).

21

0

0.2

0.4

A(ω

)D

ZCY

0

0.2

A(ω

)D

Λ=2Λ=2.5Λ=3

0

0.2

A(ω

)D

α=0.1α=0.2α=0.3α=0.4α=0.5

-3 -2.5 -2 -1.5 -1 -0.5 0ω/D

0

0.2

A(ω

)D

10-7

10-6

10-5

10-4

3.16 10-4

discretization scheme

discretization coarseness

clipping cutoff value

broadening

(a)

(b)

(c)

(d)

FIG. 18: (Color online) Lower Hubbard band part of thelocal spectral function A(ω) for the U = ∞ Hubbard modelat low doping, δ = 0.01. In (a), the schemes C and Y producespurious features at the outer band edge and at ω = 0.3D(indicated by arrows). In addition, the (expected) feature atthe inner band edge at ω = 0.1D is overemphasized in C andY schemes. In (d), we note that the curves for clipping values10−7, 10−6 and 10−5 nearly overlap, while those for ≥ 10−4

exhibit some deviations.

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