Mean-field approximation for thermodynamic and spectral functions of correlated electrons: Strong-coupling and arbitrary band filling

1.1 MB·

by  Václav Janiš

Mean-field approximation for thermodynamic and spectral functions of correlated
electrons: Strong-coupling and arbitrary band filling
V´aclav Janiˇs and Vladislav Pokorny´
Institute of Physics, The Czech Academy of Sciences,
Na Slovance 2, CZ-18221 Praha 8, Czech Republic
Anna Kauch
Institute of Physics, The Czech Academy of Sciences,
Na Slovance 2, CZ-18221 Praha 8, Czech Republic and
7 Institute of Solid State Physics, TU Wien, Wiedner Hauptstr. 8-10/E138, 1040 Wien, Austria
1 (Dated: March 20, 2017)
0
We present a construction of a mean-field theory for thermodynamic and spectral properties
2
of correlated electrons reliable in the strong-coupling limit. We introduce an effective interaction

r determinedself-consistentlyfromthereducedparquetequations. Itisastaticlocalapproximationof
a
thetwo-particleirreduciblevertex,thekernelofapotentiallysingularBethe-Salpeterequation. The
M
effectiveinteractionenterstheWardidentityfromwhichathermodynamicself-energy,renormalizing
the one-electron propagators, is determined. The dynamical Schwinger-Dyson equation with the
7
thermodynamic propagators is then used to calculate the spectral properties. The thermodynamic
1
and spectral properties of correlated electrons are in this way determined on the same footing and

in a consistent manner. Such a mean-field approximation is analytically controllable and free of
]
l unphysical behavior and spurious phase transitions. We apply the construction to the asymmetric
e
Anderson impurity and the Hubbard models in the strong-coupling regime.
-
r
t PACSnumbers: 72.15.Qm,75.20.Hr
s
.
t
a
m I. INTRODUCTION doesnotallowustogaincontroloverthecriticalbehavior
(e.g. Mott-Hubbard metal-insulator transition) via ana-
-
d lyticfunctions. Moreover,DMFTaddressesdirectlyonly
Tangible electron correlations cause phenomena that
n one-electronfunctions,wherebythetwo-particleonescan
remainfarfrombeingsatisfactorilyandfullyunderstood.
o beobtainedonlyindirectlywithlittlecontroloftheircrit-
c Only non-perturbative techniques can reliably describe
ical behavior. Extensions of DMFT such as dynamical
[ intermediate and strong coupling regimes of correlated
vertex11 or dual fermions12 partly amend this drawback
electrons. With the increasing computer power vari-
2 by addressing also two-particle vertex functions respon-
v ous numerically exact methods such as quantum Monte sible for nonlocal fluctuations. They, however, do not
2 Carlo,1,2 numerical,3,4 density matrix5,6 or functional7,8
guaranteeconsistencybetweenthelocalone-electronand
0 renormalizationgroup, arewidelyusedtoobtainquanti-
non-local two-electron functions.
4 tative non-perturbative results in the whole range of the
0
interaction strength. Numerical solutions are, however,
There are two fundamental relations between the one
0
restrictedtofinite-sizeclustersandrelativelysmallsetsof
. andtwo-particlemany-bodyGreenfunctions. TheWard
1 Matsubara frequencies or low-lying excited states. Prox-
identity between the self-energy and the two-particle ir-
0 imity of the critical points in intermediate and strong
reducible vertex is necessary for making the solution
7
coupling demands controlling two-particle and response conservingandthermodynamicallyconsistent.13,14While
1
functions that may diverge in the thermodynamic limit.
: theSchwinger-Dysonequationconnectingtheself-energy
v The singularities in two-particle functions and the crit-
with the full two-particle vertex determines the spectral
Xi ical behavior can be identified and fully controlled only and dynamical properties of the microscopic model.14–16
analytically. The singularities and divergencies must be
r The general unsurmountable problem is that the two
a treated separately from the non-divergent quantities so equations cannot be simultaneously obeyed in the ap-
that to reach stable solutions and to avoid instabilities proximate treatments.17,18 We suggested in Ref. 18 to
and spurious behavior in the numerical calculations and
resolve this inconsistency by resigning on the full one-
iterations.
particle self-consistency in the Schwinger-Dyson equa-
The simplest analytic static mean-field approxima- tion used in the standard Baym-Kadanoff construction
tions work only in the weak coupling and semi-analytic with the generating Luttinger-Ward functional. We pro-
methods must be used to include dynamical fluctuations posed to use the two-particle irreducible vertex from a
in the intermediate and strong coupling. Dynamical Bethe-Salpeter equation as the generating function from
mean-fieldtheory9 (DMFT),basedonthelimitoftheex- which a thermodynamic self-energy is determined via
act solution in infinite spatial dimensions10 offers an ap- a (linearized) Ward identity. In this way thermody-
propriateframework. Unfortunately,afullsolutionofthe namicconsistencybetweentheoneandtwo-particlefunc-
DMFT equations is available only numerically. It hence tions is qualitatively achieved. That is, a singularity in
2
theBethe-Salpeterequationgeneratesthecorresponding II. THERMODYNAMIC AND SPECTRAL
symmetry breaking in the self-energy. The thermody- PROPERTIES IN MEAN-FIELD
namicself-energydoesnotgenerallyobeytheSchwinger- APPROXIMATIONS
Dyson equation and hence it does not reflect adequately
the critical behavior in the spectral function. We thus We demonstrate the construction of mean-field ap-
introduced a spectral self-energy from the Schwinger- proximationsforthermodynamicandspectralproperties
Dyson equation in which we used the thermodynamic of strongly correlated electrons on the lattice Hubbard
self-energy to renormalize the one-electron Green func- model with a local interaction
tion and the two-particle vertex in it.18 Although the
(cid:88) (cid:88)
thermodynamic and spectral self-energies are equal only H(cid:98)H = ((cid:15)(k)+µ+σh)c†kσckσ+U n(cid:98)i↑n(cid:98)i↓ (1)
intheexactsolutiontheygeneratequalitativelythesame kσ i
critical behavior in thermodynamic and spectral func-
tions. The approximate solutions in this two-particle wherehisanexternalmagneticfield. Operatorsc†kσ,ckσ
construction match the emergence of the order param- createanddestroyelectronwithquasi-momentumkand
eter in both self-energies with the unique singularity in n(cid:98)iσ is the operator of the electron density on site Ri.
the two-particle vertex. We also use the single-impurity Anderson model
(SIAM) with the free conduction electrons hybridized
with the interacting local ones d†,d
σ σ
The existence of two self-energies is a rather com-
mon phenomenon of approximate treatments of corre- (cid:88) (cid:88)
lated systems. Apart from approaches that use auxil- H(cid:98)I = (cid:15)(k)c†kσckσ+Ed d†σdσ
iary self-energies19 all non-self-consistent or partly self- kσ σ
(cid:88)(cid:16) (cid:17)
consistent expansions around mean-field-type solutions + V d†c +V∗c† d +Undnd (2)
k σ kσ k kσ σ (cid:98)↑(cid:98)↓
introduce two self-energies, the perturbed and unper-
kσ
turbed ones. It is the case when we expand around
thestaticHartreeapproximation20–22 oraroundthelocal todemonstratereliabilityofthelocalmean-fieldapprox-
dynamical mean-field solution.11,23–25 The perturbed or imation as an impurity solver in the strong-coupling,
Kondo regime.
corrected self-energy does not change the critical behav-
ioroftheunperturbed,mean-fieldself-energy,butitmay
leadtoanunphysicalbehaviorsuchasnegativecompress-
A. Thermodynamic and spectral self-energies:
ibility. One has to determine the static, spin symmetric
Hartree approximation with vertex correction
part of the physical self-energy via the self-consistency
withfullpropagatorsandusetheSchwinger-Dysonequa-
Critical behavior of correlated electrons is induced
tion with the thermodynamic propagators to determine
by divergences in response functions and in a Bethe-
its dynamical correction.
Salpeter equation in the appropriate two-particle scat-
teringchannel. Athermodynamicallyconsistentdescrip-
The aim of this paper is to extend the two-particle tion demands that the irreducible vertex Λ, generating a
approach introduced in Refs. 26 and 18 to a full- singularityinaBethe-Salpeterequation,isproperlycon-
fledged mean-field theory of correlated electrons at low- nected with the self-energy so that the latter breaks the
temperatures,arbitrarycouplingandbandfillingsothat symmetry of the order parameter at the critical point.
the critical behavior of Bethe-Salpeter equations is re- It is the case when the two functions obey a functional
flected in thermodynamic and spectral functions consis- Ward identity.14 In the simplest, Hartree approximation
tently and on the same footing. The resulting approx- the two-particle irreducible vertex reduces to the bare
imation is analytically controllable, free of spurious in- interaction and the Ward identity
stabilities and qualitatively correctly describes quantum
criticality in the strong-coupling limit. δΣ[U;G]
Λ[U;G]= =U (3)
δG
The layout of the paper is as follows. We outline is exactly solvable. Here G is the propagator renormal-
the concept of two self-energies for thermodynamic and ized with the self-energy from the Ward identity. A so-
spectral properties in Sec. II. A general mean-field ap- lution of Eq. (3) is the Hartree self-energy
proximation with a static local irreducible vertex from
the reduced parquet equations is presented in Sec. III. U (cid:88)
ΣT(k )= G (k(cid:48) )≡UnT (4)
Weapplytheapproximationonthestrong-couplinglimit σ n βN −σ n(cid:48) −σ
k(cid:48),n(cid:48)
ofthesingle-impurityAndersonmodelwithgenerallocal
density of states in Sec. IV. The numerical solution of with the Hartree renormalized propagator
the mean-field approximation for the impurity and lat-
tice models is presented in Sec. V. The last section VI 1
G (k,iω )= . (5)
brings conclusions. σ n iω +µ+σh−(cid:15)(k)−UnT
n −σ
3
We sometimes use a four-vector notation k = (k,iω ) equations. To improve the Hartree approximation by re-
n n
for fermionic frequency-momenta and q = (q,iν ) for placing the static Hartree self-energy in the Schwinger-
m m
the bosonic ones. Dysonequationbyadynamicallyrenormalizedoneinthe
The Hartree self-energy leads to nontrivial response so-called FLEX approximation,27 is a step in the wrong
functions and to critical points. It does not, however, direction. The new renormalized self-energy breaks the
obeytheSchwinger-Dysonequationwiththetwo-particle Ward identity, shifts the critical point and disconnects
vertex derived from the Hartree response functions. The thesymmetrybreakingintheself-energyfromthecritical
staticself-energyserveswellforassessingthermodynamic point of the two-particle vertex. The correct procedure
quantities of the equilibrium state but completely fails to go beyond the Hartree approximation is to improve
to account for the dynamical spectral properties. The upon the irreducible vertex Λ in the Ward identity and
dynamics of the interacting system is contained in the to keep two self-energies, thermodynamic and spectral.
Schwinger-Dyson equation and hence the latter must be
reinstalled in the thermodynamic mean-field theory.
It is easy to derive the full two-particle vertex from B. General concept of thermodynamic and spectral
the Hartree magnetic susceptibility. It reads self-energy
U Thermodynamic consistency of quantum critical be-
Γ (q )= , (6)
↑↓ m 1−U2φ (q )φ (q ) havior demands unambiguous meaning of two-particle
↑ m ↓ m
Green functions and their singularities. The standard
where we denoted the electron-hole bubble φσ(qm) = Baym-Kadanoff construction with the Luttinger-Ward
(βN)−1(cid:80)k,nGσ(kn)G−σ(kn + qm). We use this ver- generating functional leads to ambiguous definitions of
tex to introduce another, spectral self-energy from the two-particle functions.18 We invert the procedure and
Schwinger-Dysonequationwiththisvertex. Wetakeonly take the irreducible vertex Λ[U;G], the kernel of the
its dynamical part so that not to break thermodynamic singular Bethe-Salpeter equation, as a generating func-
consistency of the static self-energy. That is tion of thermodynamically consistent theories. The one-
electron propagators in the representation of the irre-
Σsp(k )=−U2 (cid:88)φσ(qm)G−σ(qm+kn). (7) duciblevertexarerenormalizedbyathermodynamicself-
σ n βN 1−U2φ↑(qm)φ↓(qm) energyobeying,tocertaindegree,theWardidentity. The
q,m
Ward identity, being a functional differential equation,
We then define the total (physical) self-energy as cannot be resolved and used to obtain the self-energy
for a given irreducible vertex. The full thermodynamic
Σ (k )= U (cid:0)n−σmT(cid:1)+Σsp(k ). (8a) correspondence between the irreducible vertex and the
σ n 2 σ n self-energycan,nevertheless,bereplacedbyaqualitative
consistency reached by resolving the Ward identity only
The total particle density n = n +n in the full self-
↑ ↓ linearly with respect to the corresponding symmetry-
energy is determined from the one-electron propagator
breaking field. The latter is the Legendre conjugate to
renormalized with the full self-energy
the order parameter induced by the singularity in the
Bethe-Salpeter equation. If we assume that the spin-
1
G (k,iω )= , (8b) reflection symmetry gets broken at the critical point the
σ n iω +µ+σh−(cid:15)(k)−Σ (k,iω )
n σ n Ward identity will be linearized with respect to the ex-
ternal magnetic field h. The thermodynamic self-energy
whilethemagnetizationmT =nT−nT,Legendreconju-
↑ ↓ obeying the linearized Ward identity with the given ver-
gatetothesymmetry-breakingfieldh, iscalculatedwith
tex Λ, similarly to the Hartree approximation, is
the thermodynamic propagator, Eq. (5). This choice of
the static part of the physical self-energy is needed not ΣT =Λ[U;G]·G . (9)
σ −σ
to affect the magnetic instability of the thermodynamic
We must symmetrize the spin-dependent vertex
self-energy andto guaranteenon-negativityof thephysi-
Λ(k ,k ;q ) = [Λ (k ,k ;q )+Λ (k ,k ;q )]/2
cal compressibility calculated from the the full propaga- n n(cid:48) m ↑↓ n n(cid:48) m ↓↑ n n(cid:48) m
sothatitdependsonlyonevenpowersofthesymmetry-
tor G in Eq. (8). It means that only the spin-symmetric
breaking field. Only with this symmetrization the
static part of the physical self-energy is determined self-
thermodynamicself-energyΣT fromEq.(9)isasolution
consistently beyond the thermodynamic self-consistency σ
of a linearized Ward identity. The one-particle prop-
in propagators G (k,iω ) in approximate treatments.
σ n agators used in the determination of the two-particle
The above construction may serve as a prototype of
irreducible vertices Λ are renormalized with the
mean-field approximations for thermodynamic and spec- σσ(cid:48)
thermodynamic self-energy
tralproperties. TheHartreeapproximationlosesreliabil-
ity in intermediate and strong-coupling regimes. Cred- G (k,iω )
σ n
ible mean-field approximations in the strong coupling
1
must contain a two-particle self-consistency so that to = . (10)
iω +µ+σh−(cid:15)(k)−ΣT (iω )+ΣT
avoidunphysicalandspuriouspolesintheBethe-Salpeter n k,σ n 0
4
We introduced a correction ΣT to the thermodynamic it the same critical behavior. It hence extends the ther-
0
self-energysothattocomplywiththeexactelectron-hole modynamic approximation appropriately to the dynam-
symmetryandtokeepcompressibilitynon-negative. The ical and spectral functions without affecting the thermo-
vertex, the self-energy, and the one-electron propagator dynamic critical behavior. The full physical self-energy
must obey the following symmetry relations to comply is constructed from the spectral one as in the Hartree
with the electron-hole symmetry µ→U −µ approximation, Eq. (8). Only the Hartree propagator,
Eq. (5) is replaced by the appropriate thermodynamic
Gσ(kn)=−G(cid:101)−σ(−kn), (11a) one, Eq. (10).
ΣTσ(kn)=2ΣT0 −Σ(cid:101)T−σ(−kn), (11b)
Λσσ(cid:48)(kn,kn(cid:48)(cid:48);qm)=Λ(cid:101)−σ−σ(cid:48)(−kn,−kn(cid:48);−qm), (11c) III. THERMODYNAMICALLY CONSISTENT
MEAN-FIELD APPROXIMATION
where the hole functions X(cid:101)(iωn) are calculated for µ(cid:101) =
U − µ. The value of the shift in the self-energy is
The Hartree mean-field approximation suffers from
2ΣT = ΣT(∞)+Σ(cid:101)T(∞) to assure non-negativity of the
0 ↑ ↓ the deficit that it does not suppress unphysical spuri-
compressibility. HereΣT(∞)isthestaticpartoftheself-
σ ous poles in the two-particle vertex, which makes it falli-
energy at filling µ and magnetic field h. Its explicit form
ble in the intermediate coupling and inapplicable in the
depends on the specific approximation on vertex Λ .
σ−σ strong coupling. One must introduce a two-particle self-
Thefundamentalassumptionofthisapproximationis consistencytopreventtheexistenceofspurioussingular-
that the irreducible vertex Λ depends on even powers of ities. It is achieved by the parquet scheme. The full par-
the external magnetic field and the thermodynamic self- quet approximation16,31 is not analytically solvable and
energy complies with the Ward identity only in the limit moreover it does not reproduce the three-peak structure
of zero magnetic field, where the dependence of vertex of the DMFT local solution. It misses the the correct
Λ on the magnetic field can be neglected. The deriva- strong-coupling asymptotics and is not a suitable candi-
tive of the thermodynamic self-energy with respect to date for a mean-field approximation in the strong cou-
the magnetic field then diverges at the critical point of pling regime.32,33 We recently proposed a reduced set of
the Bethe-Salpeter equation with the irreducible vertex parquet equations that keep the structure of the singu-
Λ. Equation(9)determinesself-consistentlythethermo- larityintheelectron-holescatteringchannelandbalance
dynamicself-energyusedtorenormalizetheone-electron better multiple electron-hole and electron-electron scat-
propagators G in the perturbation expansion for vertex terings than the full set of the parquet equations.18 This
Λ. Notice that this construction of the thermodynami- reduction allows us to reach the critical region of the
cally consistent self-energy is similar to Hedin’s GW ex- Bethe-Salpeterequationintheelectron-holechanneland
tension of the Hartree-Fock approximation.28–30 the Kondo regime in the impurity model.
The thermodynamic self-energy ΣT and the ther-
Theideaofthereducedparquetequationsistoselect
modynamic propagators G generally do not guarantee
the singular Bethe-Salpeter equation determining quan-
that the critical behavior is correctly described also in
tum criticality we want to study and decouple the full
the spectral functions. The thermodynamic functions
two-particle vertex to a regular irreducible kernel of the
serve well for assessing the static thermodynamic quan-
Bethe-SalpeterequationΛandareducibledivergingpart
tities of the equilibrium state. They may, however, give
K. AnothernonsingularBethe-Salpeterequationisthen
poor results for dynamical spectral properties, in par-
used to derive a self-consistent, nonlinear equation for
ticular within simple (static) mean-field approximations.
the irreducible vertex Λ. It is important that the self-
The dynamical properties of the equilibrium state are
consistency contains only regular expressions and can be
extracted from the Schwinger-Dyson equation and the
solved numerically. The selection of the singular equa-
latter must be reinstalled in the approximate theories.
tions depends on the studied problem. It is the electron-
Hence,another,spectralself-energyobeyingthedynam-
hole scattering channel for the repulsive interaction and
ical part of the Schwinger-Dyson equation is introduced
the electron-electron channel for the attractive coupling.
Σsp =−UG G (cid:63)Γ [U;G]·G . (12)
σ σ −σ σ −σ
The two-particle singlet vertex Γ [U;G] there is a solu- A. Effective-interaction approximation
σ
tion of a Bethe-Salpeter equation
SincetheirreduciblevertexΛisregularandbounded
Γσ[U;G]=Λ[U;G] we can use a static approximation for it. It means that
−Λ[U;G]G G (cid:63)Γ [U;G] (13) we neglect all finite fluctuations and keep only large,
σ −σ σ
critical ones in the reducible vertex K. If we use the
with the same irreducible vertex Λ used in the Ward singlet electron-hole singular channel combined with the
identity. The spectral self-energy has a richer dynamical non-singular electron-electron channel we obtain a sim-
structure than the thermodynamic one, but shares with ple equation for the irreducible vertex from the reduced
5
parquet equations of Ref. 18 The thermodynamic occupation number then is
(cid:90) ∞
U
Λσ = 1+Ψ [Λ] (14) nTσ = d(cid:15)ρ((cid:15))f((cid:15)−µ¯σ). (21)
σ −∞
with Theeffectivechemicalpotentialµ¯ inthethermodynamic
Green function is determined from a doping parameter
Λ2 (cid:88) 1 (cid:88) x=µ−U/2 at zero magnetic field and temperature as
Ψ [Λ]=−
σ 2 β
α=±1 n (cid:90) |µ¯|
x=µ¯+Λ sign(µ¯) d(cid:15)ρ((cid:15)). (22)
φ (−iω −iαπT)G (iω )G (−iω )
σ n σ n −σ n (15) 0
1+Λφ (−iω −iαπT)
σ n The physical particle density is determined from the
propagatorrenormalizedwiththefullself-energy,Eq.(8),
and Λ = (Λ +Λ )/2. We extended here the effective-
↑ ↓
interaction approximation from Ref. 18 to non-zero tem-
(cid:90) ∞ dω (cid:90) ∞
peratures in that we added the average over the low- n = f(ω) d(cid:15)
est fermionic Matsubara frequencies to the vertex func- σ −∞ π −∞
tions and kept the irreducible vertex a real constant. ρ((cid:15))
. (23)
Such a straightforward extension allows us to represent ω+i0++µ+σh−(cid:15)− U (n−σmT)−Σsp(k )
the screening factor analytically with Fermi, f(ω) = 2 σ n
1/[exp(βω)+1], and Bose, b(ω) = 1/[exp(βω)−1], dis- The spectral self-energy of the effective-interaction ap-
tribution functions proximation has an explicit representation
Λ2 (cid:90) ∞ (cid:26) (cid:20) φ (−ω−iπT) (cid:21) Σsp(ω)
Ψ [Λ]= dω f(ω)(cid:60) σ σ
σ π −∞ 1+Λφσ(−ω−iπT) =−UΛ(cid:90) ∞ dx(cid:26)b(x)G (x+ω)(cid:61)(cid:20) φσ(x) (cid:21)
(cid:20) φ∗(−ω) (cid:21) π −σ 1+Λφ (x)
×(cid:61)[G (ω)G∗(−ω)]−b(ω)(cid:61) σ −∞ σ
σ 1+Λφ∗(−ω) φ∗(x−ω) (cid:27)
σ −f(x) σ (cid:61)G (x) . (24)
×(cid:60)[G (ω−iπT)G(−ω+iπT)]} (16) 1+Λφ∗(x−ω) −σ
σ σ
and ThethermodynamicandspectralGreenfunctionsare
not the same in this approximate treatment, hence they
(cid:90) ∞ dx leadtodifferentbandfillingsandbandedgesforthegiven
φ (ω)=− f(ω)[G (x+ω)(cid:61)G (x)
σ π −σ σ input parameters, density of states, chemical potential
−∞
and interaction strength. The band edges of the ther-
+ G∗(x−ω)(cid:61)G (x)] . (17)
σ −σ modynamic propagator are at |µ¯| = w, if the bare band
has support on interval (−w,w). Then (cid:61)φ(ω) = 0 and
The variables along the cuts are always taken as the
consequently also (cid:61)Σ(ω) = 0 for |µ¯| = w. The effective
limit from the upper complex half-plane, that is G(x)=
G(x+i0+) and G∗(x) = G(x−i0+) denotes complex interactionequalsthebareone,Λ=U atthebandedges.
The lower band edge is µ =−w+(U −Λ)/2, while the
conjugation. l
upper band edge is µ =w+(U −Λ)/2+U. The band
The full singlet local electron-hole vertex reads u
edges of the spectral function will have the same struc-
Λ ture, only the shift of the center of the band is different.
Γ (ω)= . (18)
σ 1+Λφσ(ω) Wefindwl =−w+U/2−Σ0 =−wandwu =w+U. The
bandwidthisthesame,onlythecenterofthebandofthe
thermodynamicGreenfunctionisshiftedwithrespectto
B. Thermodynamic and spectral self-energy and the exact value reproduced by the spectral self-energy.
band structure
The two self-energies used in this mean-field approxi- C. Magnetic susceptibility
mation lead to two particle occupations. The thermody-
namic self-energy renormalizes the bare propagator to a The critical behavior of the thermodynamic system
thermodynamic one the local element of which is is experimentally determined from response functions.
It is a transition to a magnetic order in the electron
(cid:90) ∞ d(cid:15)ρ((cid:15))
systems with the repulsive interaction. The thermody-
G (ω)= (19)
σ ω+i0++µ¯ −(cid:15) namic information about the magnetic critical behavior
−∞ σ
is contained in the (dynamical) magnetic susceptibility,
where
χ(q,iν ). The static susceptibility diverges at the tran-
m
U −Λ sition point at a specific transfer momentum q, depen-
µ¯σ =µ+σh− 2 −ΛnTσ . (20) dent on which type of magnetic order emerges at low
6
temperatures. Our approximate theory based on the re- Itisevidentthatboththethermodynamicandthephys-
duced parquet equations generates two self-energies and ical susceptibilities share the same critical behavior at
twoone-electronGreenfunctions. Theythenleadtotwo 1+Λφ(q,0) = 0. Since the spectral self-energy is fre-
susceptibilities but with qualitatively the same critical quency dependent we cannot sum over the Matsubara
behavior. It means that they both determine the same frequencies explicitly in the physical susceptibility. It is
critical point of the transition and the long-range order easytoverifythatathalffilling(µ¯=0)thesystemisan-
emergesatthesamepointofthephasediagram. Weeval- tiferromagneticallyorderedatzerotemperatureforarbi-
uate explicitly both susceptibilities and compare their traryinteractionstrengthU >0. Theelectron-holebub-
values in numerical calculations. ble logarithmically diverges φ (Q,0) = ρ(0)ln(|µ¯|/w)
AF
To determine the susceptibility from the Green func- for ρ((cid:15)) = ρ(−(cid:15)), T = 0 and for µ¯ → 0. As other static
tion using the thermodynamic self-energy, that is con- mean-fieldsolutions,theeffective-interactionapproxima-
stant and renormalizes the chemical potential to µ¯, we tion becomes unreliable in the description of the lattice
will need a momentum-dependent electron-hole bubble models in low-spatial dimensions, d=1,2. A dynamical
defined in the Matsubara formalism and non-local extension of the static effective interaction
is needed to improve the reliability to low-dimensional
φ(q,iν ) systems.
m
1 (cid:88) 1 (cid:88)
= G(k,iω )G(k+q,iω +iν ). (25a)
β N n n m
n k IV. SINGLE-IMPURITY ANDERSON MODEL:
The sum over Matsubara frequencies can easily be ana- STRONG-COUPLING LIMIT
lyticallycontinuedandweobtainananalyticrepresenta-
tion of the dynamical bubble The effective-interaction approximation was intro-
ducedtodescribequalitativelycorrectlyquantumcritical
1 (cid:88)f((cid:15)(k)−µ¯)−f((cid:15)(k+q)−µ¯)
φ(q,ω)= . (25b) behaviorandquantumphasetransitionsintheintermedi-
N ω−(cid:15)(k+q)+(cid:15)(k)
ateandstrongcoupling. Onecanusethesingle-impurity
k
Anderson model to demonstrate reliability of this mean-
In case of the ferromagnetic response with q=0 and
fieldapproximationinthestrong-couplingregime. There
(cid:15)(k+q)=(cid:15)(k) we have
isnocriticalpointinSIAMforfiniteinteractionstrength,
(cid:90) ∞ buttheKondoregimewithanexponentiallysmallKondo
φ (0,0)= d(cid:15)ρ((cid:15))f(cid:48)((cid:15)−µ¯). (26) temperature is a critical region of the singlet electron-
F
−∞ hole vertex. We demonstrated in Ref. 18 that both the
It reduces at zero temperature to φ(0,0) = −ρ(µ¯) and spectral function and the magnetic susceptibility at zero
we obtain a renormalized Stoner criterion for the ferro- temperature and in the charge and spin symmetric state
magnetic instability 1=Λρ(µ¯). In case of the antiferro- lead to a qualitatively correct Kondo temperature, the
magnetic response, q=Q, where vector Q is defined as logarithm of which linearly depends on the bare interac-
(cid:15)(k+Q)=−(cid:15)(k), we obtain tion strength. Here we apply the effective interaction to
describe the strong-coupling limit of the spin-symmetric
(cid:90) ∞ ρ((cid:15)) equilibrium state of SIAM away from half filling and for
φ (Q,0)=P d(cid:15)f((cid:15)−µ¯) . (27)
AF (cid:15) arbitrary density of states of the localized electrons.
−∞
The exact solution of the impurity model is known
The dynamical thermodynamic susceptibility then is only for the Lorentzian density of states (DOS),
2φ(q,ω)
χT(q,ω)=− . (28) 1 ∆
1+Λφ(q,ω) ρ(ω)= , (30)
πω2+∆2
The physical dynamical susceptibility has a more com-
plex expression and it mixes the thermodynamic Green where ∆ = π(cid:80)k|Vk|2δ(EF −(cid:15)(k)) is the effective hy-
function G(k,iω ) and the full one G(k,iω ) in a repre- bridizationbetweenthelocalandconductionelectronsat
n n
sentation the Fermi energy and sets the energy unit. Our mean-
field approximation allows us to formulate the strong-
2 1 (cid:88) 1 (cid:88) coupling limit for arbitrary form of the density of states
χ(q,iν )=− G(k,iω )
m 1+Λφ(q,iν )β N n of the local electrons. The only difference to the lat-
m
n k
ticemodelsisthattheimpuritymodeldoesnotallowfor
×G(k+q,iω +iν )[1+(Λ−U)φ(q,iν )
n m m a magnetic order and the equilibrium state remains the
+1 (cid:88) UΛφ(iνm(cid:48)) 1 (cid:88)G(k(cid:48),iω ) local Fermi liquid to infinite interaction strength. Our
β 1+Λφ(iν )N n+m(cid:48) approximate thermodynamic and spectral self-energies
m(cid:48)
m(cid:48) k(cid:48)
for the impurity model are the mean-field ones from
(cid:21)
×G(k(cid:48)+q,iω +iν ) . (29) Sec. IIIB. We apply the approximation to the zero-
n+m(cid:48) m
temperature spin-symmetric state, that is T =0,h=0.
7
The strong coupling limit U → ∞ generates an ex- A worse reliability of the effective interaction away from
ponentially small Kondo scale for the charge-symmetric half filling stems from the suppressed dynamics of mul-
situation, x = 0. The Kondo scale emerges there due to tiple scatterings from the electron-electron channel that
the sum rule making the density of states at the Fermi contribute to vertex Λ. This non-divergent vertex was
level independent of the interaction strength. It is no replaced by a constant. Fluctuations due to multiple
longer true if we move away from the charge-symmetric electron-electronscatterings,althoughnon-divergent,be-
case. Thestrong-couplinglimitthendependsontheway come more relevant the farther we are from half filling
we reach the limit of infinite interaction strength. and affect the way the critical, strong-coupling Kondo
In the first step we fix the effective chemical poten- regimeisreached. Amoreelaborateapproximationwith
tial µ¯ being equivalent to keeping the thermodynamic both two-particle vertices dynamical is needed to reach
occupation number nT constant. The strong-coupling better results in the strong-coupling regime away from
limit then is close in behavior to that from half-filled half filling.
case with a critical behavior in the two-particle vertex The two-particle vertices are further used to deter-
Γ(ω). Its critical behavior for a dimensionless Kondo mine the spin susceptibility and its dependence on the
scale a=1+Λφ(0)(cid:28)1 is dimensionless Kondo scale a. The physical susceptibil-
ity at zero magnetic field can be represented within the
φ(cid:48)(0)=−πρ(µ¯), (31a) effective-interaction approximation as18
. Λ
Γ(ω)= a−iπΛρ(µ¯)2ω , (31b) χ=−2 (cid:88)G(iω )2(cid:20)1− UX(iωn)(cid:21) , (35)
β n a
n
(cid:90) 0 where we introduced a two-particle function
φ(0)=−2 dxρ(µ¯+x)
×(cid:90)−∞∞d(cid:15)ρ(µ¯+x+(cid:15))−ρ(µ¯+x−(cid:15)). (31c) X(iωn)= β1 (cid:88)[G1+(iωΛnφ+(iνiνm)])2 {G(iωn+iνm)
(cid:15) m m
0
×[1+Λφ(iν )]+Λ[κ(iν )−κ(−iν )]} , (36)
m m m
The screening of the bare interaction and the Kondo
asymptotic scale read in which
1 (cid:88)
. |φ(0)||G(0)|2Λ κ(iν )= G(iω +iν )G(iω )2. (37)
ψ = |lna|, (32a) m β n m n
π2ρ(µ¯)2 n
(cid:26) (cid:27)
U|φ(0)| The Kondo scale is a two-particle quantity that regu-
a=exp − . (32b)
1+|(cid:60)G(0)|2/|(cid:61)G(0)|2 larizes the pole of the critical two-particle vertex. It en-
terstheone-particlespectralfunctionviatheSchwinger-
where
Dysonequationmatchingthetwo-particlevertexandthe
one-particle spectral self-energy, Eq. (24). The dynam-
(cid:61)G(ω)=−πρ(µ¯+ω), (32c)
ical part of the spectral self-energy in the Kondo limit
(cid:90) ∞ ρ(µ¯+ω+(cid:15))−ρ(µ¯+ω−(cid:15))
(cid:60)G(ω)=−2 d(cid:15) . a (cid:28) 1 with the dominant logarithmic singularity in the
0 (cid:15) effective-interaction approximation reads
(32d)
(cid:20)
. U
(cid:60)Σsp(ω)= |lna| (cid:60)G(ω)
For small dopings and.the Lorentzian density of states Λπ2ρ20
|µ¯|(cid:28)∆ we obtain 2µ¯ =x and
(cid:18)Λπρ2 ω(cid:19) (cid:21)
. (cid:26) U (cid:18) x2 (cid:19)(cid:27) +arctan a0 (cid:61)G(ω) , (38a)
a(x)=exp − 1− , (33)
π∆ 2∆2 . U (cid:20) Λ2π2ρ4 ω2(cid:21)
(cid:61)Σsp(ω)= ln 1+ 0 (cid:61)G(ω) . (38b)
while the exact Kondo scale determined from the mag- 2Λπ2ρ20 a2
netic susceptibility of the Bethe ansatz solution is a =
The spectral self-energy is significantly frequency mod-
exp{−π(U2/4−x2)/2∆U}.34,35 The approximate solu-
ulated only on a Kondo length scale of order a. Con-
tion correctly reproduces the quadratic increase of the
sequently the spectral function develops a quasiparticle
Kondo scale with parameter x (cid:28) ∆2 but misses the ex-
peak near the Fermi energy of the same width.
act prefactor. The Kondo regime in the approximate
Thestrong-couplinglimitforthefixeddopingparam-
solution sets in for a much stronger interaction strength.
eterxisduetoEq.(22)qualitativelythesameasthatof
While the Kondo regime sets in the exact solution for
the fixed thermodynamic-occupation number nT. Quite
U (cid:29) 8(∆+x2/2U)/π in the approximate solution it is
a different strong-coupling limit is obtained if we fix the
for
charge density n. When U →∞ at a fixed n,
U (cid:18) µ¯2(cid:19)2 x2 Λ (cid:16)µ¯(cid:17) U
(cid:29) 1+ ≈1+ . (34) µ¯+ arctan ∝ (n−1) (39)
π∆ ∆2 2∆2 π ∆ 2
8
andtheeffectivechemicalpotentialµ¯goesto±∞linearly 0.3 n=1.0 0.3 n=0.99
withU accordingtowhethern>1orn<1. TheKondo
regime cannot then be reached, since Eq. (34) cannot be 0.2 0.2
scaotuipsfiliendg.lTimhietrefoirsthheencfiexnedo Kpaorntdicoleredgeinmsietyinnth(cid:54)=e 1st.roTnhge- nction0.1 0.1
u
tawfion-pitaertinictleeravcetritoenxsΓtr(eωn)grtehacUhmes(nm)a→xim∞umforfonr→ω =1. 0 at ectral f0.03 n=0.92 0.03 n=0.82
p
S0.2 0.2
0.1 0.1
V. NUMERICAL RESULTS
0 0
-10 -5 0 5 10 -10 -5 0 5 10
ω ω
Weusetheeffective-interactionapproximationforthe
irreducible vertex Λ from Eq. (14) at zero temperature
FIG. 1. (Color online) Spectral function of SIAM at zero
tocalculatemean-fieldspectralandresponsefunctionsin
temperature and U = 12. We used Lorentzian density of
thestrong-couplinglimitoftheFermi-liquidphaseofthe
states with ∆ taken as the energy unit. Doping the state
AndersonimpurityandHubbardmodelsatzerotemper-
with holes leads to merging of the central quasiparticle peak
ature and away from half filling. Numerical results for with the lower Hubbard satellite band.
non-zero temperatures will be presented elsewhere.
0.3
n=1, Parquet
0.2 NRG
A. Impurity model
0.1
We first evaluate the spectral function calculated 0
fEvnreqolm=.op(3snE0w)qh..+eFn(8nidgbuo)pr=ewin1igt1hsthhwteoihwteehlsecLhhtooorwroleensnt-.thhzoeiWlaesnpeseydcmcetarnmnasleitstfyerueincoctsfthoilasouttnatittodehnese-,, Spectral function000...231 nN=R0G.94, Parquet
↑ ↓ 0
height of the central quasiparticle peak decreases and 0.3
n=0.68, Parquet
the lower Hubbard satellite approaches the Fermi sur- 0.2 NRG
face. Finally, the lower Hubbard band merges with the
0.1
central one. We compared the spectral function calcu-
0
latedwithintheeffective-interactionapproximationwith -8 -4 0 4 8
ω
thespectralfunctionfromthenumericalrenormalization
group (NRG) for U = 8 in Fig. 2. The results were ob- FIG. 2. (Color online) Spectral function of SIAM for U = 8
tained by using the NRG Ljubljana Code.36 The central compared with NRG result.
quasiparticle peak of the reduced parquet equations is
narrower than that of NRG, since the logarithmic cor-
rections of the exact solution are missing in our approxi- the self-consistency with which they are determined.
mation. Themean-fieldapproximationreproduces, how- The mean-field approximation allows us to evaluate
ever, the correct strong-coupling asymptotics with the the strong-coupling limit and to check the behavior of
logarithm of the Kondo scale linearly dependent on the theKondoscalethereawayfromhalffilling. Thereduced
bare interaction for the fixed effective chemical poten- parquetequationsintheeffective-interactionapproxima-
tial µ¯. Moreover, the Hubbard satellites are better re- tionreproducetheKondoscaleonlyinthecriticalregion
produced in the high-energy region, where NRG gives ofthemetal-insulatortransitionwhereitisexponentially
less accurate results. The overall tendency of the spec- small. We can expect that the farther from half filling
tral function where the lower Hubbard band absorbs the the worse the Kondo scale from the Bethe ansatz will be
quasiparticle peak is in both solutions the same. reproduced. We compared in Fig. 4 the Kondo scale de-
We plotted in Fig. 3 the dependence of the auxiliary pendenceontheinteractionstrengthforthefixeddoping
thermodynamic occupation number nT from Eq. (21) parameter with the Bethe ansatz solution. Although the
andthereal(physical)occupationn,self-consistentlyde- numerical values of the Kondo scale differ, the linearity
termined from Eq. (8), on the doping parameter x = of the logarithm of the Kondo scale is correctly repro-
−E −U/2 and compared them with the charge density duced. The shift in the numerical values is caused by a
d
from the numerical renormalization group. The physical differenceinthenonuniversalprefactorsthattheparquet
occupation number quite well fits the NRG data for all approximation does not reproduce.
band fillings. The thermodynamic occupation number Similarly we calculated and compared the Kondo
nT is off in the similar way as the Hartree approxima- scale as a function of the doping parameter x for a few
tion. Both approximate solutions n(x) and nT(x) gener- fixed interaction strengths in Fig. 5. Kondo scale is no
ate nonnegative compressibility. It is the consequence of longer exponentially small, we are far from the critical
9
2 1
Parquet, n Parquet
Parquet, nT Bethe a.
NRG
1.5
e
al 0.1
g c
Fillin 1 ndo s U = 8
o
K
U = 12
0.5
0.01
U = 8 U = 16
0
-6 -4 -2 0 2 4 6 0 0.5 1 1.5 2
x x
FIG. 3. (Color online) Band fillings n and nT as a function FIG. 5. (Color online) Kondo scale calculated from the de-
ofthedopingparameterx=−E −U/2fromthestaticlocal nominator of the vertex function K(ω) at the Fermi energy
d
approximation in the reduced parquet equations compared ω=0asafunctionofthedopingparameterxcomparedfora
with the NRG result. few interaction strengths with the exact Bethe ansatz result.
B. Lattice model
1 The effective interaction is used to generate a mean-
fieldapproximation,thatis,alocalapproximationtolat-
ticemodelsofinteractingelectrons. Inthemean-fieldap-
e
al 0.1 proximationweneedtoknowonlythedensityofstatesof
c
do s theunderlyingl√attice. Weusethesemi-ellipticdensityof
on states ρ((cid:15)) = 2 1−(cid:15)2/π corresponding to the infinite-
K
0.01 Parquet, x = 0.5 dimensional Bethe lattice. The halfwidth of the energy
x = 1.0 band is w = 1. Using the equations from Sec. III we
Bethe a., x = 0.5
calculate the spectral function obtained from the Green
x = 1.0
0.001 function renormalized with the spectral self-energy. The
0 5 10 15 20
spectral function at zero temperature and half filling,
U
µ = U/2, is plotted in Fig. 6 for several interaction
strengths covering weak and intermediate coupling. We
FIG. 4. (Color online) Kondo scale calculated from the de- can see that the three-peak structure with a quasiparti-
nominator of the vertex function K(ω) at the Fermi energy
cle peak and Hubbard satellites develops rather soon al-
ω=0asafunctionoftheinteractionstrengthU comparedfor
readyforamoderateinteractionU =2w. Thisstructure
two doping parameters x with the exact Bethe ansatz result.
is amplified with increasing the electron coupling. In the
strong-couplinglimitweseethesameKondobehavioras
in the impurity model and no metal-insulator transition
takes place, supposed that the low-frequency behavior
of vertex Γ(ω) remains decisive. We can also observe
point of the two-particle vertex and only a qualitative a non-analytic behavior in the region between the well
behavior of the Kondo scale as a function doping can formed central and satellite peaks due to the algebraic
be expected. The difference between our approximation band edges of the density of states. The non-analyticity
andtheexactsolutionincreaseswithdopingandstronger appears at the band edges of the noninteracting system.
interactionawayfromhalffilling. Thesimpleststaticap- It is more pronounced when the renormalized density of
proximationisinsufficienttoachieveagoodquantitative states is small.
agreement with the exact result there, since the farther WeplottedinFig.7developmentofthespectralfunc-
from half filling the more relevant the dynamical fluctu- tion whendoping theelectron-hole symmetricstatewith
ations due to multiple electron-electron scatterings be- holes x < 0 for two interaction strengths. The behavior
come. All fluctuations away from the Kondo regime are is very similar to the spectral function of the impurity
bounded, of the same order, and the electron-hole scat- model from Fig. 1. The lower satellite band moves to-
terings lose on dominance. A dynamical approximation wards the central one and finally absorbs it (for n < 1).
for the irreducible vertex Λ is needed to achieve more The satellite peaks are higher for stronger interaction
accurate quantitative results. strengths. Noticealsothatthecentralandsatellitepeaks
10
U=0 50
U=2 2
U=4
0.6 1.5
40
n 1
nction bility 30 0.5 nnn,T,, UUU === 1 828
pectral fu0.3 Suscepti 20 0-6 -4 -2 χ x0, UnT, =U 2 =8 12 4 6
S
χT, U = 8
10 χ, U = 12
χT, U = 12
0
0 1 2 3 4
0 x
-3 -2 -1 0 1 2 3
ω
FIG.8. (Coloronline)Dependenceofthelocalmagneticsus-
FIG.6. (Coloronline)Spectralfunctioninthemean-fieldap-
ceptibilitiesofSIAMχandχT onthedopingparameterxfor
proximationforthespectralself-energyoftheHubbardmodel
twointeractionstrengths. Theinsetshowsthecorresponding
atzerotemperaturewiththesemi-ellipticdensityofstatesat
occupation numbers.
half-filling.
0.6 n=0.98 0.6 n=0.98 Sec. VA and the same should be seen also on the local
magnetic susceptibility.
We plotted in Fig. 8 dependence of the zero-
0.3 0.3
temperature magnetic susceptibility of the impurity
model (Lorentzian density of states) on the doping pa-
0 0
ction0.6 n=0.88 0.6 n=0.88 craomupeltienrgxlimfoirt.twoWeintpelroatctteidonbsottrhentghtehrsmiondythneamstirconχgT-
n
u and physical χ magnetic susceptibilities. A fast decay
ctral f0.3 0.3 of their values with increasing doping is clearly demon-
pe strated. The dependence is symmetric with respect to
S
0 0 the electron-hole transformation x → −x. Both full
0.6 n=0.68 0.6 n=0.68 and thermodynamic susceptibilities behave qualitatively
inthesameway. Theinsetshowsthecorrespondingband
0.3 0.3 fillings.
If we perform the limit to strong coupling at a fixed
dopingparameterxthesystemtendstoreachthecritical
0 0
-3 -2 -1 0 1 2 3 -4 -2 0 2 4
ω ω Kondo asymptotics with an exponentially small Kondo
scale. Such a behavior is confirmed in the magnetic sus-
FIG.7. (Coloronline)Spectralfunctioninthemean-fieldap- ceptibility where it exponentially grows with increasing
proximationforthespectralself-energyoftheHubbardmodel theinteractionstrength,asdemonstratedonFig.9. Both
at zero temperature and off half-filling for U = 2 (left pane) susceptibilities coincide in the strong coupling regime,
and U =4 (right pane).
but they differ in the noncritical region. The full suscep-
tibilityslowsdownitsgrowthbeforeitreachesitsstrong-
couplingasymptotics. Thethermodynamicsusceptibility
are always connected and the density of states between
also bends its growth but less apparently.
them never vanishes.
The fixed doping parameter x does not mean fixed
occupation number n when going to the strong-coupling
regime. It is hence interesting to plot dependence of the
C. Magnetic susceptibilities susceptibility on the interaction strength at a fixed n.
Fig. 10 shows dependence of the magnetic susceptibility
Themajorassetofthetwo-particleapproachbasedon fortwofixedoccupationnumbers,oneclosetohalffilling
the reduced parquet equations with the self-consistency and another farther away. We can see a big difference in
via a linearized Ward identity is that the critical behav- shape. Whenclosetohalffilling,thesusceptibilitygrows
ior of the two-particle vertex is transferred simultane- exponentially in the strong coupling. While significantly
ously to the spectral and response functions. In the im- away from half filling we can see that the magnetic sus-
purity model it is the exponentially small Kondo scale ceptibility saturates at a finite interaction strength and
in the strong coupling. We discussed the dependence of starts do decrease, that is, the Kondo scale grows and
the Kondo scale in the spectral function on doping in the system moves away from the critical regime. It cor-