Non-local effects in the fermion Dynamical mean field framework. Application to the 2D Falicov-Kimball model

4 Pages·0.15 MB·English

by  Mukul S. Laad

Non-local effects in the fermion Dynamical mean field framework.
Application to the 2D Falicov-Kimball model.
Mukul S. Laad1 and Mathias van den Bossche2
1 Institut fu¨r Theoretische Physik Universit¨at zu K¨oln, D-50937 K¨oln, Germany
2 Labortoire de Physique quantique, Universit´e Paul Sabatier, 31062 Toulouse cedex, France
We propose a new, controlled approximation scheme that explicitly includes the effects of non-
9
local correlations on the D=∞ solution. In contrast to usual D=∞, the selfenergy is selfconsis-
9
9 tentlycoupledtotwo-particlecorrelation functions. Theformalism isgeneral, andisappliedtothe
1 two-dimensional Falicov-Kimball model. Our approach possesses all the strengths of the large-D
solution, and allows one to undertake a systematic study of the effects of inclusion of k-dependent
n
effectsontheD=∞picture. Resultsforthedensityofstatesρ(ω),andthesingleparticlespectral
a
J densityforthe2D Falicov-Kimballmodelalwaysyieldpositivedefiniteρ(ω),andthespectralfunc-
tionshowsstrikingnewfeaturesinaccessibleinD=∞. Ourresultsareingoodagreementwiththe
3
exact results known on the2D Falikov-Kimball model.
1

PACS numbers: 71.27+a,71.28+d
]
l
e
-
r The effects of strong correlations on the properties of the one particle Green’s function on this site, Gimp(ω),
t
s low-dimensional lattice fermion systems is still an open whichisafunctionalofG0(ω).Theselfconsistencycondi-
t. problem, inspite of many efforts spanning over thirty tionthatlinksGimp backtoG0 issolvediteratively. The
a years. Recently, the development of a dynamical mean- maindrawbackofthismethodisthatΣisk-independent.
m
fieldtheory(DMFT),exactinD =∞hasledtoamajor In this letter, we present an extension of the DMFT
- advancement in our understanding of the physics in the that allows one to treat effects of non-local spatial cor-
d
n local limit [1,2]. DMFT is a nonperturbative scheme. relations. To do so, we exploit the freedom involved in
o It has provided a detailed picture of the Mott transi- choosingtheinputbathpropagator. Weaccountfornon-
c tion in this limit, and has been fruitfully applied to a local correlations via the self consistent inclusion of two
[
large class of models. Moreover, it has proved to be a particle correlation function in the bath Green’s func-

3 rather good approximation to actual three-dimensional tion (GF) G (ω). This is achieved by using the Spectral
0
v transition-metal and rare-earth compounds [2]. Inspite Density Approximation (SDA) [3]. Thus, our aim is to
6
of its successes, DMFT has its shortcomings; the single showthatapropercombinationoftwomethods–DMFT
7
particle selfenergy is k-independent, and so is not cou- and SDA– is eminently suited to describe 1/D effects in
1
2 pledtok-dependentcollectiveexcitations. Thisisclearly lattice correlated fermionic systems. As far as applica-
0 serious, e.g. near a magnetic phase transition where it tion is concerned, we deal with the FKM as it was the
8 cannot show any precursor effects. A consequence of the first model to be solved exactly in D = ∞. Attempts
9
above is that the single particle spectral function can- at a 1/D expansion for the FKM have also been made.
/
t not access such effects, and so the approach cannot be Finally, there exist some exact results on the 2D FKM
a
m employed to describe e.g. angle-resolved photoemission whichareofgreathelpto evaluatethe qualityofour ap-
experiments, or to study the changes in Fermi surface proximation,making it a firstchoice for testing methods
-
d topologies driven by interactions in finite-D. The above in the context of extensions of the DMFT.
n makesitimperativetodevelopsuchcontrolledextensions We now present this new method. We begin with
o
of DMFT as are able to rectify its unphysical aspects, the formalism for the generic Hubbard model at zero-
c
while preserving its strengths. temperature. We then go to the FK to solve the full set
:
v
of selfconsistent equations. We will focus on half-filling,
i
X The DMFA maps the lattice system onto a selfconsis- the extension away from n = 1 is straightforward. The
r tently embedded “impurity in a bath” problem that de- Hamiltonian is
a scribestheeffectsofthecouplingtootherlatticesites. As
ina Weiss meanfieldtheory,a selfconsistency condition H =− X tσ(c†iσcjσ+h.c)+UXni↑ni↓−µXniσ
isobtainedbyrequiringtheaveragedGreen’sfunctionof <ij>,σ i iσ
the bath G to coincide with the local G at this impu- on a D−dimensional lattice. The Hubbard model is ob-
0
rity site. DMFA becomes exact in D = ∞ because one tained when t = t = t, and the FKM is given by
↑ ↓
can show that the spatial correlations fall off at least as t = t and t = 0. In the case of half filling µ = U/2
↑ ↓
1/D|i−j|/2withdistance|i−j|[4]. This“Weissfield”still by particle-hole symmetry. These models have been ex-
has a nontrivial dynamics described by an effective local tensively studied using the DMFT.
actionS obtainedbyintegratingoutallsitesexclud- We get a bath propagator which includes 1/D effects
DMF
ing the impurity site. Given S , one can compute using the SDA, first pioneered by Roth [3] to describe
DMF
1
magnetism in narrow bands. Its basic idea is to com- SDAG astheinputoneparticlebathGreen’sfunction
0σ
pute exactly the first few moments of the spectral den- inanewDMFTschemeallowsonetointroduceinasim-
sity to reconstruct the spectral function A(k,ω). With ple way the non-local spatial fluctuations at order 1/D
Hasabove,itis easyto compute the firstfour moments. in the FKM.
They are given by the expectation value of the following We now come to the computation of the dynamical
n-fold commutator correlations. This results in a dynamically corrected self
energy. To do this beyond the single-site level is a prob-
M(n) =<[...[[c† ,H],H],...H]> (1)
kσ k,σ lem that has not been attempted, and here we use the
standardDMFT as anapproximationto the 2D case. In
Using usual scaling arguments [4], one sees that the
the case of the half filled FKM, the method drawn from
higher n, the higher will be the powers of 1/D involved
the lines sketched above can be developed the following
in the n-th moment. In the case of a generic Hub-
way.
bard model, it turns out that the M(3) contains all
kσ As is known [6,7], the equation-of-motion method
the o(1/D) contributions. It can be writen Mk(3σ) = (EOMM)givestheexactsolutiontotheFKMinD =∞.
o(D10)+U2n↓(1−n↓)Bkσ with We will use the EOMM to get the full (local) impurity
Green’s function which we denote by G (ω).
1 imp,↑
n↓(1−n↓)Bkσ = N X{t−σ <c†i−σcj−σ(1−2niσ)> Theclosedsetofequationsfortheon-siteGreen’sfunc-
<ij> tionwithabathhavingΣ0,↑(k,ω)asaselfenergyisgiven
−t [<n n >−n2 −<c† c† c c > in ref. [7], replacing ǫk by ǫk+Σ0,↑(k,ω).
σ i−σ j−σ −σ iσ i−σ j−σ jσ
ThelocalG iscomputedtohavethesameformasthat
−<c†iσc†j−σcjσci−σ >]e−ik·(Ri−Rj)} (2) found in [2], a↑nd the local self-energy is computed from
Dyson’s equation, Σ (ω) = G−1 (ω)−G−1 (ω)
The SDA then permits one to write an explicit closed imp,↑ 0,imp,↑ imp,↑
where G (ω)=[ω+µ−∆(ω)]−1, with
formexpressionforA↑(k,ω)=δ[ω−ǫk+µ−Σ0,↑(k,ω)], 0,imp,↑
whereΣ (k,ω)isgiven–forthegenericHubbardmodel
0,↑ t2
– by ∆(ω)=X k (4)
k ω+µ−ǫk−Σ0,↑(k,ω)
Σ0,↑(k,ω)=Un↓+U2n↓(1−n↓)[ω+µ−U(1−n↓)−Bk↓]−1
One then gets
This is the self energy in the SDA. It is interesting to
notice that all the order 1/D effects enter only through U2n (1−n )
↓ ↓
Σ (ω)=Un + . (5)
Bkσ, whereby the self energy acquires a non-trivial k- imp,↑ ↓ ω+µ−U(1−n↓)−∆(ω)
dependence. Atthisstage,thisquantityisundetermined,
and will have to be obtained self consistently. The SDA This local dynamical self energy is used to correct the
one particle Green’s function can then be computed as localpartofthebathselfenergy,tointroducedynamical
G0↑(k,ω)=[ω−ǫk−Σ0,↑(k,ω)]−1. effects in the k-dependent bath GF. This is done by
We now consider the FKM, (t =0). In this case, the
↓
model has a local U(1) symmetry, [ni↓,H] = 0 ∀i. It Σ↑(k,ω)=Σimp,↑(ω)+Σ0,↑(k,ω)−XΣ0,↑(k,ω) (6)
thus follows from Elitzur’s theoremthat only the second k
correlator in Bkσ is non zero. This leads to
and provides us with a dynamically corrected non-local
n↓(1−n↓)Bk↓=−Nt X e−ik·(Ri−Rj)[<ni↓nj↓ >−n2↓] Gwhreicehn’sisfeuxnaccttioinnGb↑o(tkh,tωh)e=ato[ωm+icµan−dǫtkhe−bΣa↑n(dk,liωm)]i−ts1.
<ij>
Thedensityofstatesisthenobtainedbytakingtheimag-
=−tχ (k) (3)
↓ inary part of the k-summed G(k,ω).
where χ (k) is the order 1/D static structure factor of Tocompletetheprocedure,oneshouldselfconsistently
↓
compute the order 1/D correlatorwhich enters the bath
the down spin electrons.
It is interesting to notice thatthe only order 1/D con- Green’s function as anounced earlier. In the case of the
tributiontothesingleparticleselfenergyisinfactχ (q). FKM, this means one has to compute the down spin
↓
static susceptibility. To begin with, we compute the
The above equations are exact in the band as well as in
vertex function exactly in D = ∞. The full χ (q) is
the atomic limit, as can be easily checked. This has im- ↓
then obtained from the Bethe-Salpeter equation in the
portant consequences, for it points to a way for further
framework of the EOMM [6]. The result is χ (q) =
improvementoftheformalism. Itallowsonetoformulate ↓
n (1−n )/D(q), where
aDMFT forthe localpart,moreorlessalongsamelines ↓ ↓
as in D = ∞ [2]. However, in contrast to the regular
dn ∂Σ G2 +χ0(q)
DnoMn-FtTrivwiahlekre-dΣep0eσn=den0t, fGea0tucorenst.ainTshienrfeoformrea,tuiosningabtohuist D(q)=1−Xν dG−ν↓1 ∂n↓ν(cid:12)(cid:12)(cid:12)(cid:12)Gν G2ν[χν0↑(q)dd↑GΣνν +1] (7)
2
and χ0(q) = (−1/N) G (k+q)G (k), where the ν- be satisfied to high accuracy.
↑ Pk ↑ ↑
sum is on the Matsubara indices (we take the zero tem- We now describe the results of our computation.
perature limit). In the above, the vertex corrections in Working with a lattice size L = 64, and a threshold
the equation for the charge susceptibility of the down- ε=10−9, we observea veryfast convergenceof the dou-
spinselectronsareapproximatedbytheirD =∞values. bleselfconsistentprocedure,thatneedslessthan10steps
To compute the vertex function exactly to order 1/D to reach the convergence threshold for G (ω). The Lut-
↑
would be a formidable task, which has not been success- tingersumruleissatisfiedwithanerroroflessthan10−3.
fully attempted even for the electron gas.
0.40
0.1
U/t=0
U/t=2
U/t=8
G (ω) G (ω )
O O 0.30
χ(q)
0.0
−0.3 0.0 0.3
G(ω) G(ω) ω)
( 0.20
ρ
t
(a) (b)
0.10
FIG. 1. (a) The selfconsistency loop in usual DMFT.
(b) Our method: k-dependent features are incorporated via
aisnceoxmtepruntaeldinsepluftcoχn↓sitsotetnhtelybaasthshGorwenen.’s function G0, which 0.00−8.5 ω0./0t 8.5
FIG. 2. DOS for the 2D Falicov-Kimball model for sev-
eralvaluesofU/tathalf-filling,withourmethod. Itisalways
To sum up, the equations (3)-(7) above form a com-
positivedefinite. TheinsetshowsthebehavioroftheU/t=2
plete set of selfconsistent equations for the FKM. These
gapatω=0whenchangingtheartificialbroadeningη ofthe
include explicit1/D effects throughthe k-dependence of δ functions. η/t = 5.10−2 thick line, η/t = 5.10−3 thin line
Σ0,↑(k,ω). Tosolvetheseequations,westartwithanar- and η/t=2.10−4 dashed line.
bitraryχ (q) asaninput in ourinitialG . This is used
↓ 0↑
to compute Σ (k,ω), and the full G (ω) from the
0,↑ imp,↑ Fig.2showsthe density ofstates(DOS) ρ(ω)obtained
large-D solution along the lines of [7]. At this step, we
by our method for several values of U/t. Firstly, notice
have a DMF approximation to the FKM that is equiva-
that this function is always positive definite, in contrast
lenttoaselfconsistentloop,withanexternalinputquan-
to results of ref. [8]. This is a positive feature of the
tity,thestructurefactor. Thesecondstepofourmethod
presentmethod. We should stress that we are interested
is to compute χ (q). For this purpose, we use the DMF
↓ in the effect of coupling of self-energy to 1/D fluctua-
approximationtothestructurefactorequation(7),plug-
tions, which, for the FKM, are completely static. The
ging the obtained G (k,ω) along with the full local ver-
↑ effect of the interaction is very clear. Starting from the
tex function in the Bethe-Salpeter equation to compute
2D free fermions value at U/t = 0, it opens a gap at
χ (q). This is fed back into the equation for Σ and
↓ 0,↑ ω = 0. The critical (U/t) for opening up a gap is very
c
the whole process is iterated to convergence. The differ-
small. Asamatteroffact,workingwithaL×Llatticewe
ence between the regular self consistent DMFT and our
cannotresolveenergydetails smaller than2πv /L≃0.2
F
method is illustrated in fig.1.
here. However,playing with the parametersof the prob-
Weimplementednumericallythisnewmethodtostudy lem,weseearealgapwithU/taslowas0.1. Thisresult
the Falicov-Kimball model on a 2D square lattice at agreeswiththeexactresultonthe2DFKMatweakcou-
T = 0. The up spin electron dispersion relation is pling [11], where the gap and the related checkerboard
thus ǫk = −2t(coskx + cosky), and the corresponding order arises from the nested FS at n = 1. The inset of
density of states has a van Hove singularity at ω = 0. fig.2 shows the behavior of the DOS at low frequency
The k sums are performed on a L × L mesh, while for U/t=2, obtained by changing the broadening of the
the ω-integrals are computed using standard integra- delta functions to get a better precision. The features
tion procedures, using a small broadening η to repre- observed at higher frequencies are related to the unper-
sent δ functions. Convergence is obtained when χ2 = turbed 2D band structure and are absent in infinite D.
|G (ω)−G (ω)|2 <ε where ε is a small thresh- The computed χ (q), which contains information
Pω new old ↓
old. At each step of the iterative process, a check on about long-range order (LRO), shows interesting behav-
the accuracy of the computation is provided by the Lut- ior;alongthezone-diagonal,atthepoint(k=[π,π]),itis
tinger sum-rule µ dωA (k,ω) = n , which must a constant, scaling approximately with lattice size. This
PkR−∞ σ σ
3
is the expected behavior in the broken-symmetry phase that the coupling to 1/D fluctuations induces new fea-
[12], and implies that the n = 1 ground state is charac- tures in A(k,ω) compared to those observed in DMFT.
terized by checkerboardLRO of the down-spin electrons In D = ∞, A(k,ω) has a two-peak structure with the
[9]. Inref.[12],thebehaviorofχ (q)canbestudiedonly dispersioncontrolledsolelybythefreebandstructure. In
↓
in1D, becauseoffinite size restrictions. Incontrast,our our approach, this is modified because of the extra k-
approachyields unambiguousconclusionsin the thermo- dependence coming through the Bk↓. This shows that
dynamic limit in 2D. Again, this is in clear agreement a formalism capable of explicitly treating intersite cor-
with the exact result [11]. relations permits one to access k-dependent features in
A(k,ω), so wesuggestthatthis methodcanbe fruitfully
appliedto compute ARPESlineshapes in correlatedsys-
tems. Amoredetailedapplicationtospectroscopywould
tAHk,ΩL require the use of the actual bandstructure DOS, and is
left for future work.
1.5
In conclusion, we have developed a simple, physically
appealing way to study the effects of non-local spatial
fluctuationsonthesingleparticlespectralproperties. We
1 n=8 have applied the formalism to compute the single parti-
n=7
n=6 cle spectral function and the DOS for the 2D FKM and
n=5 have obtained results in good accordance with what is
0.5 n=4 known from the exact solution. Extensions of the work
n=3
n=2 to look at the metallic phase in the FKM off n = 1, as
n=1 wellasfortheHubbardmodel,arebeingstudiedandwill
0 n=0 ω(cid:144)t be reported separately.
--66 --44 --22 0 Mukul S.Laad wishes to thank the Alexander von
FIG. 3. The spectral function A↑(k,ω) for the 2D Fali- Humboldt Stiftung for financial support during the time
cov-Kimball model at half-filling, for U/t = 4, and k along this project was started. We thank Prof. P.Fulde for
thezone diagonal, k=(nπ/8,nπ/8), n going from 0 to 8. advice and hospitality at the MPI, Dresden.
1 e-mail: [email protected]
2 e-mail: [email protected]
The greatest advantage of our method, however, is
that it allows a detailed study of k-dependent (at the
order of 1/D) effects in the one particle spectral func-
tion A(k,ω)=−ImG(k,ω)/π. In fig.3, we show A(k,ω)
for k along the zone-diagonal from k = 0 to k = (π,π)
for a representative value of U = 4t. We notice that
A(k,ω) satisfies all the symmetry properties consistent [1] J.Hubbard,Proc.Roy.Soc.(London)A281,401(1964).
with those expected frompure nearestneighbor hopping [2] A.Georges,G.Kotliar,W.KrauthandM.J.Rozenberg,
on bipartite lattices, and with particle-hole symmetry: Rev. Mod. Phys. 68, 13 (1996)
A(k ,ω) = A(k ,−ω) for k = [π/2,π/2], while along [3] L.M.Roth,Phys.Rev.184,461(1969).Formorerecent
0 0 0
the zonediagonalA(k +δ,ω)=A(k −δ,−ω),∀δ along work on SDA, see e.g. H. Beenen and D. M. Edwards,
the zone diagonal. We0show only the0A(k,ω) for ω < 0; Phys. Rev.B52, 13636 (1995).
[4] SeeD.Vollhardt,inCorrelated electrons, editedbyV.J.
theω >0partcanbeobtainedfromthesymmetryprop-
Emery, (World Scientific, Singapore, 1993)
ertymentionedabove. Wehavecheckedthatourcalcula-
[5] S. Elitzur, Phys.Rev. D12 3978 (1975).
tionsarefully consistentwiththe aboveproperty. More- [6] U. Brandt and C. Mielsch, Z. Phys B75 365 (1989)
over, we see that since Q = [π,π] is a nesting vector in [7] M. Laad, Phys. Rev. B49 2327 (1994).
2D, the above property of A(k,ω) with δ = [π/2,π/2] [8] A. Schiller and K. Ingersent, Phys. Rev. Lett. 75 113
leadstoanaturalexplanationofthe “shadow-band”fea- (1995).
tures observable in ARPES. It would be very interesting [9] See J. K. Freericks, Phys. Rev. B47, 9263 (1993), and
to see whether they survive away from n=1. ref. therein for numerical and analytical proofs of CDW
order at half-filling.
It is instructive to compare our results to those ob-
[10] B.Velicky´,S.KirkpatrickandH.Ehrenreich,Phys.Rev.
tained by Velicky´ et. al [10] in their pioneering paper on 175, 747 (1968).
CPA.CPAisequivalenttotheexactD =∞resultforthe [11] A.Su¨t¨o,Ch.GruberandP.Lemberger,J.Stat.Phys.56
FKM, and so a comparison with [10] allows us to study 261 (1989), Ch. Gruber, J. Jedrzejewski, J. Iwanski and
theeffectsof1/DeffectsonA(k,ω). Acomparisonofour P. Lemberger Phys. rev. B412198 (1990).
datawithresultsforidenticalparametersfrom[10]shows [12] P. Farkaˇsovsky´,Z. Phys. B104 553 (1997)
4