Analytic impurity solver with the Kondo strong-coupling asymptotics

0.27 MB·English

by  V. Janis

Analytic impurity solver with the Kondo strong-coupling asymptotics
V. Janiˇs and P. Augustinsky´
Institute of Physics, Academy of Sciences of the Czech Republic,
Na Slovance 2, CZ-18221 Praha 8, Czech Republic∗
(Dated: February 4, 2008)
7 Wepresentananalyticuniversalimpuritysolverforstronglycorrelated electrons. Weextendthe
0 many-bodyperturbationexpansionviasuitabletwo-particlerenormalizationsfromtheFermi-liquid
0 regime to the critical region of the metal-insulator transition. The reliability of the approximation
2 in thestrong-coupling limit isdemonstrated byreproducing theKondoscale in thesingle-impurity
Andersonmodel. WedisclosetheoriginoftheKondoresonanceintermsofFeynmandiagramsand
n
find criteria for the existenceof theproper Kondoasymptotic behavior in approximate theories.
a
J
PACSnumbers: 72.15.Qm,75.20.Hr
8

]
I. INTRODUCTION Kondo resonance The standard weak-coupling, Fermi-
l
e liquid-based perturbation expansion renormalized with
-
r One must have at hand a reliable theory capturing various self-energy insertions becomes unstable for in-
st thesalientfeaturesofthestrong-couplingasymptoticsto termediate and strong electron interaction.8 Attempts
. to go beyond one-particle renormalizations by introduc-
t comprehend fully the effects of electron correlations in
a ing explicit vertex corrections and a two-particle self-
metals. Most theoretical approaches are based on the
m
consistency either do not lead to analytically solvable
well established weak-coupling expansion and we have
- equations or have not yet produced the desired Kondo
d only a few opportunities to test its extensions to the
asymptotics in SIAM.9,10
n strong-couplingregime. Themostreliablecheckofcredi-
o bilityofatheoryinthestrong-couplingregimeistocom- The aim of this paper is to construct an analytically
c pare its results with one of the existing exact solutions controllable approximation that would reliably interpo-
[ in this limit. There are two principal exact results for late between the weak- and strong-coupling regimes in
2 strongly correlated electrons. It is the single-impurity models with a screened electron repulsion. To this pur-
v Anderson model (SIAM) and the 1d Hubbard model. poseweuseanappropriatelyrenormalizeddiagrammatic
2 The ground state of both models at half filling was con- many-body expansion. We demonstrate that it is not
5 structed with the aid of the Bethe ansatz. The former theone-particleself-consistencybutratheratwo-particle
0 is a heavy-fermion liquid in the strong-coupling limit,1 one that must be introduced in the critical region of the
7
while the latter is an electron-hole liquid for arbitrary metal-insulator transition. We further show that only
0
6 interaction strength.2 a symmetric inclusion of electron-electron and electron-
hole scatterings leads to the proper critical behavior.
0 Recently SIAM has won on importance, since its so-
/ lutions form a fundamental building block in the con- The two-particle self-consistency will be achieved within
t
a struction of a dynamical mean-field theory (DMFT) of the parquet approximation in which we separate singu-
m lar and regular functions. Not to lose analytic control
strong electron correlations via the limit to infinite-
- dimensional lattices.3 Unfortunately, there are only ap- only the potentially divergent functions with long-range
d fluctuations are kept dynamical while the regular ones
proximateimpuritysolversinthestrong-couplingregime.
n with short-range fluctuations are replaced by constants.
The only analytic theory,non-crossingapproximation,is
o
Such a simplification does not affect the universal fea-
c not a Fermi liquid and displays low-energy pathological
: features.4 NumericalquantumMonte-Carlosimulations5 turesofthecriticalbehavior. Weexplicitlyconstructthe
v approximation for SIAM so that its reliability becomes
are restrictedto rather high temperatures. The most re-
i
X liable impurity solver proved to be the numerical renor- transparent by qualitatively correctly reproducing the
exact weak-coupling (Fermi-liquid) and strong-coupling
r malization group. It produces the expected three-peak
a (Kondo) regimes. We identify with our construction the
spectral function with the central Kondo resonance for
intermediate coupling.6,7 Due to limitations of the nu- minimalnecessaryconditionsonapproximateschemesto
cover the Kondo strong-coupling behavior.
merical procedure it cannot be extended to arbitrarily
largeinteractions. Neither ofthese solutions,however,is
capabletoproduce adetailedultimate picture ofthe the
strong-couplinglimitandtodecidewhetherandinwhich II. MODEL AND BASIC EQUATIONS
form the Kondo resonance survives or whether the sys-
tem undergoes a metal-insulator transition. This ques- Weformulatetheconstructionofanapproximatethe-
tion can only be answered by an analytically controlled ory interpolating between the weak and strong coupling
theory. regimesindependentlyoftheunderlyingmodelviarenor-
There is presently no global analytic theory capable malized Feynman diagramsand equations ofmotion. To
of tracing down the genesis of the correlation-induced demonstrate reliability of our approachwe, however,use
2
in our explicit calculations the single impurity Anderson (cid:27)!n (cid:27)!n0
model the Hamiltonian of which reads
H = ǫ(k)c† c +E d†d (cid:0)(cid:27)(cid:27)0
kσ kσ d σ σ
kσ σ (cid:1)
b X + Vkd†σckσX+Vk∗c†kσdσ +Und↑nd↓ . (1) (cid:27)0!n+(cid:23)m (cid:27)0!n0+(cid:23)m
Xkσ (cid:16) (cid:17) FIG. 1: Diagrammatic assigning of dynamical variables, fre-
quencies and spins, to the vertexΓσσ′(iωn,iωn′;iνm).
b b
We denoted nd = d†d . When calculating the grand
σ σ σ
potential and thermodynamic properties of this impu-
Wefurtherintroducetwo-particleirreduciblevertices,
rity model we can explicitly integrate over the degrees
b
that is, vertices that cannot be diagrammatically dis-
of freedom of the delocalized electrons. To this pur-
connected by cutting specific pairs of one-particle lines
pose we standardly replace the local part of the prop-
(propagators). Thechoiceofthetwo-particleirreducibil-
agator of the mobile electrons by a constant ∆(ǫ) =
π k|Vk|2δ(ǫ−ǫ(k)) =. ∆ the value of which we set as ity is ambiguous.12 Here we choose only the singlet
electron-hole and electron-electron scattering channels.
theenergyunit. Forsimplicityweassumehalf-filledcase
P We denote the respective irreducible vertices Λeh and
µ=E +U/2. Theimpuritygrandpartitionfunctioncan
d
thenbe representedviaalocalGrassmannfunctionalin- Λee. With their aid we represent the full two-particle
vertex via two nonequivalent Bethe-Salpeter (BS) equa-
tegral
tions. The Bethe-Salpeter equation in the eh channel
reads
Z = DψDψ∗exp ψ∗(iω +µ¯+isign(ω )∆)ψ
Z ( n n n n n Γ↑↓(iωn,iωn′,iνm)=Λe↑h↓(iωn,iωn′,iνm)
X
−U βdτ nd↑(τ)nd↓(τ) . (2) − β1 Λe↑h↓(iωn,iωn′;iνm)G↑(iωn′′)G↓(iωn′′+m)
Z0 ) Xn′′
×Γ↑↓(iωn”,iωn′;iνm) . (4a)
b b
For this integral we can straightforwardly build up the
standardweak-coupling(diagrammatic)perturbationex- In the ee channel we obtain
pansion in powers of the interaction strength U (dia-
grammatically represented as a vertex) and the bare Γ↑↓(iωn,iωn′;iνm)=Λe↑e↓(iωn,iωn′;iνm)
propagator, being in the impurity case G (x + iy) = 1
1/(x+i sign(y)(∆+|y|)) (diagrammatically0represented − β Λe↑e↓(iωn,iωn′′;iνm+n′−n′′)G↑(iωn′′)
as an oriented line). Xn′′
We know that in the particular case of SIAM the ×G↓(iωn+n′+m−n′′)Γ↑↓(iωn′′,iωn′;iνm+n−n′′) . (4b)
non-renormalized weak-coupling expansion is preferred
Equations(3)and(4)reducethesolutionoftheinves-
andconvergesforarbitraryinteractionstrength.11Toen-
tigatedmodeltothedeterminationoftheirreduciblever-
able a treatment of phase transitions and singularities
tices Λeh and Λee. The principal idea of the parquet ap-
in more general extended lattice models we have, how-
proach is to use the topological nonequivalence of differ-
ever,to renormalizethe bare expansion. The fundamen-
entBethe-Salpeterrepresentationsofthefulltwo-particle
tal quantity for renormalization in the many-body per-
vertex via the irreducible ones. Due to the nonequiv-
turbation expansion is the self-energy. In our approach
alence of different types of two-particle scatterings the
we do not use the self-energy as the approximation-
reduciblevertexinonechannelisirreducibleintheother
generatingquantitydetermineddirectlyfromsumsofse-
one.12 If the vertex irreducible in both channels is the
lected (renormalized) diagrams. Instead, we represent it
bare interaction U (parquet approximation) we obtain
by means of the Schwinger-Dysonequation and the two-
the basic parquet equation in our two-channel scheme
particle vertex Γ as follows
Γ =Λeh+Λee −U . (5)
↑↓ ↑↓ ↑↓
U 1
Σσ(iωn)= G−σ(iωn′) 1− Gσ(iωn+m) We use this representationin Eqs.(4) to exclude the full
β β
Xn′ (cid:20) Xm vertex Γ from the BS equations that then become cou-
pledconvolutivenonlinearintegralequationsfortheirre-
×G−σ(iωn′+m)Γσ−σ(iωn+m;iωn,iνn′−n) (3) ducible verticesΛeh andΛee. These equationscannotbe
(cid:21)
solved analytically and their numerical solution is avail-
where ω = (2n+1)πT and ν = 2πmT are fermionic able only far from the Kondo regime.13 To gain an ana-
n m
and bosonic Matsubara frequencies in units k = 1, re- lytic assessmentof the low-temperature,strong-coupling
B
spectively. We assigned dynamical variables to the two- behavior of the parquet equations (4) using Eq. (5) we
particle vertex as shown in Fig. 1. have to resort to simplifications.
3
III. KONDO ASYMPTOTICS – VERTEX right-hand side of Eq. (6a). It reads
FUNCTIONS
2
U χ (iν )
Λsing(iν )=− eh m . (6b)
A. Simplified parquet equations ↑↓ m 1+Uχeh(iνm)
WenowuseEqs.(4b)and(5),andΛsingfromEq.(6b)
Equations (4) can be analytically solved (in an ap- to determine the effective interaction U. The right-hand
proximate way) individually for the full vertex Γ by ne- side. ofEq.(4b)remainsfrequency-dependentevenwith
glecting Eq. (5) and using the irreducible vertices as in- ouransatz. Sinceweneglectallfinite(short-time)fluctu-
put. Approximationsofthistypewithrenormalizedone- ationsinregularfunctions, wehavetoreplacethe vertex
particle propagatorsarenow called fluctuation-exchange Γ resulting from Eq. (4b) with a constant. This replace-
(FLEX). In the simplest case the irreducible vertices are mentisnotuniquelydefinedbuttheambiguityhasnoim-
replaced by the bare interaction. We find that vertex pactonthequalitative(universal)criticalbehaviorofthe
Γ tends to develop a pole in the BS equation with mul- solution. We found that the most appropriateway in re-
tiple electron-hole scatterings, Eq. (4a), caused by the placingthefullvertexwithaconstantistomultiplyboth
imminentmetal-insulatortransitionatzerotemperature. sides of Eq. (4b) with pairs of one-electron propagators
Whilethetwo-particlevertexΓremainsfinite(bounded) G(iωn)G(iωm−n)fromtheleftandbyG(iωn′)G(iωm−n′)
when calculated from Eq. (4b). We use the fact that fromthe right. We thenaverageoverthe fermionic Mat-
only equation (4a) develops a pole in simplifying the subara frequencies with a constraint m = 0, conserved
parquetapproximationinthecriticalregionofthemetal- during the electron-electron multiple scatterings. Doing
insulatortransitiontoananalyticallycontrollabletheory. soweobtainanexplicitequationfortheeffectiveinterac-
SincetheoutputoftheBSequationfromonechannelen- tion, the static part of the electron-hole irreducible ver-
ters the kernelof the other one, only Λee becomes singu- tex,
larinthecriticalregionofthemetal-insulatortransition.
The irreducible vertex Λeh remains bounded everywhere G G L2
↑ ↓ ↑↓
in the parquet approximation. Uχee(0)=Uχee− χ D+hG G LE i . (7)
ee ↑ ↓ ↑↓
Werestrictourinteresttotheleadingcriticalbehavior
ofthesolutionoftheparquetequationsintheasymptotic We denoted χee = β−1 nG↑(iωn)G↓(iω−n) the static
region of the metal-insulator transition. To this purpose part of the electron-electron bubble and
P
weusethereasoningoftherenormalizationgroupandne-
1
glect the short-range fluctuations. We take into account L↑↓(iωn)= β G↑(iωn′)G↓(iω−n′)Λs↑i↓ng(iν−n−n′) ,
explicitly only long-range (in time) fluctuations shaping n′.
X
the singular behavior. In this way we do not influence (8a)
the universal part of the critical asymptotics. We hence
1
keep only singular functions dynamical (dependent on hG G Xi= G (iω )G (iω )X(iω ) . (8b)
↑ ↓ ↑ n ↓ −n n
β
the relevant frequency only). Regular functions, where
n.
X
onlyshort-rangefluctuations contribute,canbe replaced
Equations (6b) - (8) form a closed set of relations deter-
by constants, suitably chosen averaged values. We must
mining the static effective interactionU and the dynam-
furtherguaranteethatonlytheleadinglow-frequencybe-
ical vertex Λsing(iν ) as functionals of the one-particle
haviorofsingularfunctionsdeterminesthecriticalbehav- m
propagators G and the bare interaction U. Together
ior in the Bethe-Salpeter equations. σ
with Eqs. (5) and (3) we have an analytic approxima-
As a first step we approximate Λeh with a static ef-
tion in closed form. The one-electron propagators may
fective interaction Λeh = U. Inserting this ansatz into
beeitherbareorrenormalizedwiththeself-energy,when
Eq. (4a) we obtain a FLEX-type equation for the irre-
one-particle self-consistency is used. The latter is not
ducible vertex from the electron-electron channel
mandatory and we show that it worsens the approxima-
tion with the Hartree one-electron propagators, in par-
2 ticular in the intermediate and high-frequency sectors.
U χ (iν )
Λe↑e↓(iωn,iωn′;iνm)=U − eh m (6a) The critical region of the metal-insulator transition
1+Uχ (iν )
eh m in SIAM corresponds to the strong-coupling regime,
U/∆ → ∞. It is reached when the denominator on
where we denoted the electron-hole bubble χ (iν ) = the r.h.s. of Eq. (6b) approaches zero. At zero tem-
eh m
β−1 G (iω )G (iω ). Since the static approxi- perature, half filling, and in the rotationally invariant
n ↑ n ↓ n+m
mation to the electron-hole irreducible vertex Λeh = U case, G↑ = G↓ we define a dimensionless Kondo scale
causPes a deviation from the exact high-frequency limit a = 1 + Uχeh(0) = 1 − U −0∞dωℑ[G+(ω)2]/π → 0.
Λeh →U,approximation(6a)holdsinthe low-frequency We used an abbreviation G (ω) ≡ G(ω ± i0+). The
±R
limit. We hence have to consider in our simplification denominator of the r.h.s. of Eq. (6b) can then be ex-
only the leading low-frequency contribution from the panded only to the lowest (linear) order in frequency if
4
we are interested in the leading singular behavior. We explicitlyevaluated,h(GG)n i=π2(n−1)ρ(2n−1)/(2n−1).
ee 0
obtain 1+Uχ (ω +i0+) = a−iπUρ2ω. We denoted Using this resultwe obtainanexplicit representationfor
eh 0
the density of one-particle states at the Fermi energy the Kondo scale
ρ = −π−1ℑG (0). It does not depend at zero temper-
0 + 5
ature and half filling on the interaction strength. This a=exp − [Uρ −1] . (12)
0
singular low-frequency asymptotics of Λsing allows us 3
(cid:26) (cid:27)
to evaluate the leading contributions to the integrals in
The exact result is a=exp{−π2Uρ /8}. Determination
Eqs. (8). 0
of the exact prefactor at the linear dependence of the
Theleadingcontributionstotheintegralswithvertex
exponent of the Kondo scale on the interaction strength
Λsing in the asymptotic limit a→0 read
is beyond the reach of the present approximation. The
. G(z)G(−z) prefactor depends on the way in which we replace reg-
L↑↓(z)= π2ρ2 |lna| , (9a) ular frequency-dependent functions with constants. The
0 universalfeatureoftheKondoscaleislinearity inthein-
teraction strength of the exponent. This linearity is the
majorachievementofourconstruction. TheotherFermi-
hG G L i
↑ ↓ ↑↓
liquid-based approximations fail in the strong-coupling
=. − |lna| 0 dωℑ G (ω)2G (−ω)2 , (9b) regime of SIAM. The single-channelFLEX-type approx-
π2ρ2 π + − imations either do not see the metal-insulator transi-
(cid:18) 0(cid:19)Z−∞
(cid:2) (cid:3) tion at all (ee channel) or predict a quadratic exponent
and a∼exp{−U2ρ2} (eh channels).14
0
G G L2
↑ ↓ ↑↓
(cid:10) . |ln(cid:11)a| 2 0 dω IV. KONDO ASYMPTOTICS – ONE-PARTICLE
=− ℑ G (ω)3G (−ω)3 . (9c) FUNCTIONS
π2ρ2 π + −
(cid:18) 0(cid:19) Z−∞
(cid:2) (cid:3)
Thecoreoftheparquetapproachisaself-consistency
The effective interaction has an asymptotic solution for
determiningthetwo-particleirreducibleverticesfromthe
a→0
completelyirreduciblevertex(bareinteractioninthepar-
1 |lna| (GG)3 quet approximation) and the one-particle propagators.
U = χ Uχee− π2ρ2 h(GG)2eei (10) Both, the completely irreducible vertex and the one-
ee " 0 (cid:10) ee(cid:11)# particle propagators serve as input to the parquet equa-
where we denoted h(GG)n i = tions. Tocompletethe parquetapproximationandmake
ee it conservingand thermodynamically consistentwe have
−π−1 0 dωℑ[G (ω)nG (−ω)n].
−∞ + − to determine the physical one-particle functions, that is,
The dimensionless Kondo scale measuring the dis-
R the physical one-particle propagator and the self-energy.
tance from the metal-insulator transition is determined
All physical quantities in conserving theories are then
from an equation for the critical point of the metal-
generated from the full one-particle propagator or the
insulator transition. It reads 1 + Uχ (0) = 0. At
eh self-energyandtheirdependenceonappropriateexternal
half filling χ (0)=−χ and we obtain a leading-order
eh ee sources.15 The physicalone-particlepropagatorneednot
asymptotic solution for the distance to the critical point
be identicalwiththe ”auxiliary”oneusedintheparquet
equations for the two-particle vertices.
(GG)2
a=exp −π2ρ2 ee [Uχ −1] . (11)
0 h(GG)3 i ee
( (cid:10) ee(cid:11) )
A. Self-energy in the parquet approach
Solution (11) holds at zero temperature and half filling
when the argument in the exponential tends to infinity The fundamental quantity for any thermodynami-
and Uχee > 1. The latter condition can be viewed as callyconsistentandconservingapproximationistheself-
a characteristic of the strong-coupling regime, since the energy. Knowing the full two-particle vertex from the
FLEX, weak-coupling solutions are distinguished by the parquet equations we determine the self-energy fromthe
opposite inequality, Uχee <1. Schwinger-Dyson equation (3). It reduces in the criti-
cal region of the metal-insulator transition to a simple
FLEX-type expression
B. Hartree propagators
U G−σ(iωn′)
Σ (iω )= . (13)
σ n
TomanifestanalyticallythatEq.(11)reproducescor- β n′ 1+Uχeh(iνn−n′)
rectly the Kondo asymptotics we use the bare (Hartree) X
one-particlepropagatorsinEq.(11). Inthiscasetheinte- We can analytically continue the sum over Matsubara
grals with powers of the one-particle propagatorscan be frequencies to an integral with Fermi function. At zero
5

0,35 0,35
U = 3
0,30 0,30
U = 6
U = 12
0,25 0,25
U = 3
OS 0,20 OS 0,20 U = 6
D D U = 12
0,15 0,15
0,10 0,10
0,05 0,05
0,00 0,00
-8 -6 -4 -2 0 2 4 6 8 -0,10 -0,05 0,00 0,05 0,10
/ /
FIG. 2: (Color online) Density of states at zero tempera- FIG. 3: (Color online) Detailed dependence of the quasipar-
ture, half filling, and various interaction strengths without ticlepeakinthedensityofstatesontheinteractionstrength.
one-particle self-consistency.
cies, the parquet equations with the Hartree propaga-
temperature the real part of the self-energy reads
tors reproduce the high-frequency features of the spec-
tral function surprisingly well. The Hubbard bands are
ℜΣ+(ω) positioned very close to the exact values of the atomic
−ω 0 dx G (x+ω) levels of the impurity electrons at ±U/2.
+
=−U θ(ω) + θ(−ω) ℑ
π 1+Uχ (x)
(cid:20) Z−∞ Z−∞(cid:21) +
0 dx 1
−Uθ(ω) ℜG+(x+ω)ℑ B. One-particle self-consistency
π 1+Uχ (x)
Z−ω +
−ω dx 1
−Uθ(−ω) ℑG (x+ω)ℜ . (14a) Usage of the Hartree propagators may seem inferior
+
Z0 π 1+Uχ+(x) toatheorywithrenormalizedone-particlepropagatorsin
the parquetequations. Infact, the opposite is true,even
The imaginary part has a representation
though the one-particle self-consistency adds new self-
energy diagrams to those explicitly taken into account
0 −ω dx
in the parquet construction of the self-energy from the
ℑΣ (ω)=−U θ(ω) + θ(−ω)
+
π two-particlevertex. Thesolutionwithrenormalizedone-
(cid:20) Z−ω Z0 (cid:21)
1 particle propagatorsloses mostof attractivefeatures ob-
ℑ G (x+ω)ℑ . (14b)
+ tainedintheparquetapproachwiththeHartreepropaga-
1+Uχ (x)
+
tors. First, the asymptotic formula for the Kondo scale,
We distinguished with a subscript + the way the real Eq. (11) cannot be evaluated explicitly when renormal-
frequency axis is approached in functions with complex ized one-particle propagators are used. Hence, the exact
variables. That is, G (x) ≡ G(x + i0+), χ (x) ≡ strong-coupling asymptotics cannot be reached analyti-
+ +
χ (x+i0+). The one-particle propagators on the r.h.s. cally. Second, the satellite peaks are completely washed
eh
of Eqs. (14) may either be bare or full propagators with out andmerge with the centralquasiparticle peak of the
the self-energy from the left-hand side. Both construc- density of states as shown in Fig. 4. The central peak
tionsleadtoconservingapproximationswhenallthermo- broadens with respect to the solution with the Hartree
dynamic quantities are determined from the self-energy. propagators. Thesolutionwithrenormalizedone-particle
We first use the bare (Hartree) one-particle propaga- propagators shows overall worse agreement with the ex-
tors in Eqs. (14). The density of states calculated from act Kondo behavior derived from the Bethe ansatz. An
the one-particle propagator containing the self-energy analogousbehaviorwasobservedinastaticsimplification
fromEq. (14) is plotted for various interactionstrengths of the parquet equations.16 The failure of one-particle
inFig.2. The centralKondoquasiparticleiswellformed self-consistent theories to reproduce reliably the Kondo
anditswidthshrinkswithincreasinginteractionstrength strong-coupling regime in impurity models is in accord
in accord with Eq. (12). See Fig. 3 for a detailed de- with early analyses of SIAM using perturbation expan-
pendence of the quasiparticle peak on the interaction sion in the interaction strength.11
strength. Not only the central quasiparticle peak but The density of states resulting from the parquet
also the satellite Hubbard bands are well formed. Even approximation with renormalized one-particle propaga-
thoughourapproximationwasjustifiedforsmallfrequen- tors resembles the solution of the renormalized RPA of
6

0,35 3
NSC U=6 NSC U=6
0,30 SC U=6 2 SC U=6
0,25
1
DOS 0,20 Re 0
0,15
-1
0,10
0,05 -2
0,00 -3
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
/
/
FIG.4: (Color online) Density of states for one-particle non-
FIG. 5: (Color online) Real part of the self-energy for one-
self-consistent (NSC) and self-consistent (SC) solutions com-
particlenon-self-consistent(NSC)andself-consistent(SC)so-
pared.
lutions compared.
Suhl.17 The centralquasiparticlepeak is, however,much
broader than in the FLEX-type approximations. It is
even broader than that from the parquet solution with Uπ2ρ2
the Hartree propagators. This indicates that the criti- x= 0 ω . (17a)
sU|lna|
cal region of the metal-insulator transition is reached, if
ever, for much stronger interactions in the solution with
Weresolvetheone-electrontildepropagatorbyusingthe
one-particle self-consistency than without it. The high-
Dyson equation of SIAM. We obtain an explicit solution
frequency behavior of the parquet approximation with
renormalizedone-particlepropagatorsandthe FLEXso-
Uπ2ρ2 x x2
lution seem to behave similarly. One can prove analyt- G (ω)= 0 −i 1− . (17b)
+
ically that the one-particle self-consistent theories with sU|lna|"2 r 4 #
simplifiedone-frequencytwo-particlevertexfunctionsbe- e
We see that one-particle self-consistent theories lead to
have in the high-frequency region universally and no
a universal high-frequency behavior with a semi-elliptic
Hubbard satellite bands emerge.
density of states spread over a large interval of order
To assess the high-frequency behavior of one-particle
propagatorsweevaluateexplicitly theleading-ordercon- U|lna|/Uπ2ρ2. TheFLEXsolutiondiffersonlyinthat
0
tribution to the self-energy in the critical region of the
qU =U. The semi-elliptic form of the density of states in
metal-insulator transition, that is, a → 0. We obtain
high frequencies is hence universal for one-particle self-
from Eqs. (14)
consistent theories. The Hubbard satellite bands are
completely smeared out. The reason for the nonexis-
. U
ℜΣ (ω)= |lna| ℜG (ω) tence of the satellite peaks lies in the behavior of the
+ Uπ2ρ20(cid:20) + real part of the self-energy in the low-frequency region.
Uπρ2 ω We compared the non-self-consistent and self-consistent
+arctan 0 ℑG (ω) , (15a)
+ solutions in Fig. 5. We can see that the sharp low-
a
(cid:18) (cid:19) (cid:21) frequency structure of the real part of the self-energy is
ℑΣ (ω)=. U ln 1+ U2π2ρ40 ω2 ℑG (ω) . significantly smoothed and the peaks are broadened by
+ 2Uπ2ρ20 " a2 # + the one-particle self-consistency. Most importantly, the
heightofthepeaksneartheFermienergyissomuchlow-
(15b)
eredthat|ω|>|ℜΣ(ω)|. Onlyifthisconditionisbroken,
If we now assume ω ≫ a∆ we can further simplify the as is the casein the non-self-consistentsolution, satellite
expression for the self-energy to peaks emerge.
U|lna| πsign(ω)
Σ (ω)= G (ω)+ ℑG (ω) . (16)
+ Uπ2ρ2 + 2|lna| + V. CONCLUSIONS
0 (cid:20) (cid:21)
e e e
We decorated the high-frequency one-particle quantities We presented a construction of a universal analytic
with tilde to distinguish them from the full ones. For impurity solver reliably interpolating between the weak
tilde functions we canintroducea dimensionlessvariable (Fermi liquid) and strong (Kondo) coupling metallic
7
regimes. The approximation is a simplification of two- handle singularities. Since singularities in models with
channelparquetequationswheretheboundedirreducible itinerant electrons emerge only at the two-particle level
vertex is replaced by a static effective interaction and in Bethe-Salpeter equations, we must introduce a two-
only the low-frequency singular vertex is kept dynam- particle self-consistency. The parquet construction of-
ical. Such a procedure is justified in the critical region fers a very natural way to reach this goal and to treat
ofthemetal-insulatortransitionwherethelow-frequency singularities in the Bethe-Salpeter equations appropri-
asymptotics of the singular two-particle vertex becomes ately. One-particleself-consistencyinfluencesthecritical
dominant. We set with our construction the minimal behavior of the two-particle vertex, but it does not offer
necessary conditions that must be fulfilled to reproduce any direct control of singularities in the Bethe-Salpeter
the Kondoexponentialscale. The Kondobehavioris ob- equations. Itcanneitherguaranteeintegrabilityofsingu-
served in an approximate solution of SIAM if the low- larverticesnoriscapabletoreproducethecorrectcritical
frequency singularity in the two-particle vertex domi- behavior in the Kondo regime. When used in the par-
natesthedynamicsandwhenelectron-holeandelectron- quetapproach,theone-particleself-consistencyinterferes
electron multiple scatterings are self-consistently mixed in the control of the two-particle singularities from the
in a balanced way. Bethe-Salpeter equations achieved by the two-particle
We analyzed two versions of the parquet approxima- self-consistency. The one-particle self-consistency can
tion. In the first one, we used the Hartree one-electron hence be used only when appropriately compensated so
propagatorsintheparquetequationsforthetwo-particle that it does not significantly affect the two-particle crit-
irreducible vertices and the Schwinger-Dyson equation icality. This is, however, not the case in the parquet
for the self-energy. In the second one we used the fully approximation and the non-self-consistent one-particle
renormalizedone-particlepropagators. Althoughthelat- propagatorsdeliverbetterresultsthantheself-consistent
terconstructioncontainsmoreFeynmandiagramsrenor- ones.
malizing the self-energy, its results are less reliable than To conclude, we derived a global approximation that
the results of the former approach. This conclusion is is analytically tractable, sufficiently simple, and univer-
not new and is also understandable.11,16 Renormaliza-
sal. It can be used in a number of physically interest-
tions due to the one-particle self-consistency unrealisti- ingsituations,includingrealistic(multiorbital),material
callysmearandunfoldthelow-frequencystructureofthe
specific models as kind ofa mean-fieldtheory for models
self-energy. Consequently,the high-frequency features of withstronglocalelectroninteraction. Itmaystandasan
the one-particle propagatorare washed out and no Hub-
alternativetoarecentlyproposedimpuritysolverforthe
bard satellite bands can be observed. strong-coupling regime.18 Unlike the solver from Ref. 18
One-particle self-consistency is standardly demanded our approximatescheme reproducescorrectlythe Kondo
in order to guarantee conservation laws and thermody- strong-coupling regime.
namic consistency. It is, however, not a necessary con-
dition for approximations to be conserving. Conserva-
tion laws are guaranteed in any theory with an approx-
imate self-energy functional containing physical external
Acknowledgments
sources. These sources are then used to generate the de-
sired physical quantities via linear response theory. On
the other hand, theories aiming at a description of crit- This research was carried out within a project
ical phenomena such as magnetic phase transitions or AVOZ10100520 of the Academy of Sciences of the
a metal-insulator transition must be self-consistent in Czech Republic and supported in part by Grant No.
someway. Self-consistencyisnecessaryforanytheoryto 202/04/1055oftheGrantAgencyoftheCzechRepublic.
∗ Electronic address: [email protected], [email protected] 10 N. E. Bickers and S. R. White, Phys. Rev. B43, 8044
1 A. M. Tsvelik and P. B. Wiegmann, Adv. Phys. 32, 453 (1991).
(1983). 11 V.Zlati´c and B. Horvati´c, Phys.Rev.B28, 6904 (1983).
2 E.H.LiebandF.Y.Wu,Phys.Rev.Lett.20,1445(1968). 12 V.Janiˇs, Phys. Rev.B60, 11345 (1999).
3 A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, 13 C. X. Chen and N. E. Bickers, Solid. State Commun. 82,
Rev.Mod. Phys.68, 13 (1996). 311 (1992).
4 T. Pruschkeand N.Grewe, , Z. Physik B74 , 439 (1989). 14 D.R. Hammann, Phys.Rev. 186, 549 (1969).
5 M. Jarrell, Phys. Rev.Lett. 69, 168 (1992). 15 G. Baym, Phys. Rev. 127, 1391 (1962).
6 A. C. Hewson, The Kondo Problem to Heavy Fermions 16 Y.M.VilkandA.M.S.Tremblay,J.Phys.(Paris)7,11309
(Cambridge University Press, Cambridge 1993). (1997).
7 R.Bulla, Phys.Rev. Lett. 83, 136 (1999). 17 H.Suhl, Phys.Rev. 171, 561 (1967).
8 N.E.BickersandD.J.Scalapino, Ann.Phys.(N.Y.)193, 18 X.Dai,K.Haule,andG.Kotliar, Phys.Rev.B72,045111
206 (19989). (2005).
9 R.A. Weiner, Phys. Rev.B4, 3165 (1971).