issp workshop/symposium: masp 2012

ISSP Workshop/Symposium: MASP 2012ISSP Workshop/Symposium: MASP 2012

Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 1

Yasutami TakadaYasutami TakadaInstitute for Solid State Physics, University of Tokyo

5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan

Seminar Room A615@ISSP, University of Tokyo10:00-11:30, Monday 25 June 2012

◎ Collaborators:

Drs.Drs. Hideaki Maebashi Hideaki Maebashi and Masahiro SakuraiMasahiro Sakurai

Many-Body Many-Body NonNon-Perturbative Approach -Perturbative Approach to the Electron Self-Energyto the Electron Self-Energy

Many-Body Many-Body NonNon-Perturbative Approach -Perturbative Approach to the Electron Self-Energyto the Electron Self-Energy

OutlineOutline


1. 1. Many-Body Perturbation TheoryMany-Body Perturbation Theory ○ Luttinger-Ward theory ○ Baym-Kadanoff conserving approximation ○ GW approximation

2. Self-Energy Revision Operator Theory2. Self-Energy Revision Operator Theory ○ Route to the exact electron self-energy ○ Relation with the Hedin’s theory ○ Good functional form for the vertex function ○ The GWGWscheme

3. Application3. Application ○ Electron liquids at metallic densities: Typical Fermi

liquid○ Relation with the G0W0 approximation

○ One-dimensional Hubbard model: Typical Luttinger liquid

4. Singularities at Low-Density Electron Liquids 4. Singularities at Low-Density Electron Liquids ○ Dielectric anomaly ○ Spontaneous Electron-hole Pair Formation?

5. Conclusion5. Conclusion

IntroductionIntroduction


Our ultimate goal is to obtain accurate, if not rigorous, solutions for both ground and excited states in this system with an infinite number of electrons. But how? Let us go with the Green’s-function formalismthe Green’s-function formalism. This is not necessarily meant to perform the many-body perturbation calculationthe many-body perturbation calculation.

The interaction part in The interaction part in HH is exactly the same as that in is exactly the same as that in the the electron-gas modelelectron-gas model::

H: H: ab initio ab initio Hamiltonian Hamiltonian in condensed matter physicsin condensed matter physics

Many-Body Perturbation TheoryMany-Body Perturbation Theory


Usual Perturbation-Expansion Theory Usual Perturbation-Expansion Theory Choose an appropriate nonperturbed one-electron Hamiltonian H0, together with its complete eigenstates {|n>: H0|n>=En

(0)|n>}

But the problem is that we need to sum up to infinite order, at least in some set of terms like the ring terms. Required to construct a formally rigorous framework to perform this kind of infinite sum. Luttinger-Ward theory Luttinger-Ward theory (1960)

Luttinger-WardLuttinger-Ward


Thermodynamic potential Thermodynamic potential is given by

where GG is the one-electron Green’s function, is the electron self-energy, and [[GG]] is the Luttinger-Ward energy the Luttinger-Ward energy functionalfunctional, given grammatically as

The problem here is that the number of terms in increases exponentially with the increase of the order.

Conserving ApproximationConserving Approximation


Luttinger-Ward is formally exact, but we have to give terms in by hand. Since we cannot give all these infinite number of terms in , it is practically impossible to get exact results from this theory.

Can we consider a general approximation algorithm to obtain physically appropriate thermodynamic quantities as well as correlation functions in which various conservation laws are satisfied automatically?

By exploiting the theoretical framework of Luttinger and Ward, Baym and Kadanoff Baym and Kadanoff proposed a good conserving approximationconserving approximation algorithmalgorithm (1961,1962).

Baym-KadanoffBaym-Kadanoff


Procedure of the Baym-Kadanoff algorithmProcedure of the Baym-Kadanoff algorithm 1) Choose your favorite functional form for [[GG]]. 2) Calculate the self-energy through ((pp:[:[GG])])==[[GG]/]/GG((pp).). 3) Obtain GG((pp)) self-consistently: GG((pp))-1-1==GG00((pp))-1-1((pp;[;[GG])]) 4) Solve the Bethe-Salpeter equation of the integral kernel defined in terms of the irreducible electron-hole effective interaction I I ((pp;;p’p’))==((pp;[;[GG])/])/GG((p’p’)=)=22[[GG]/]/GG((pp))GG((p’p’)) to determine various correlation functions.Examples: Examples: (1) Hartree-Fock approximation:

Ladder approximation in the Bethe-Salpeter equation

(2) Hedin’s GW approximation (1965) :

~~

GW ApproximationGW Approximation


◎ Not 0 but is a physical polarization function.◎ G may be regarded as not a physical quantity but just a building block to construct a physically correct , like the Kohn-Sham states in DFT.

Improvement on Baym-KadanoffImprovement on Baym-Kadanoff


In principle, the Baym-Kadanoff algorithm never give the exact solution, because the exact [G] is never known. I find, however, that the exact result can be obtainedthe exact result can be obtained without explicitly giving [G] by making the loop to determine making the loop to determine and and fully self-consistent!! fully self-consistent!! cf. YT, PRB52, 12708 (1995)

G in Baym-Kadanoff is not necessarily a physical quantity, because self-consistency is not imposed between and

Self-Energy Revision Operator TheorySelf-Energy Revision Operator Theory


Key idea: Key idea: Determine Determine I I ((pp;;p’p’) during the iteration loop ) during the iteration loop rather than give it rather than give it a prioria priori, but how? Map in {Map in { ((pp;[;[GG])}])}

Procedure to define the map: Procedure to define the map: 1) Choose your favorite self-energy inputinput((pp;[;[GG]]).2) GG((pp)) is given by GG((pp))-1 -1 = = GG00((pp))-1-1inputinput((pp;[;[GG]).]).3) Determine IIinputinput((pp;;p’p’) ) = = inputinput((pp;[;[GG])/])/GG((p’p’).).4) Determine ((pp,,p’p’) ) by the solution of the Bethe-Salpeter equation with the integral kernel IIinputinput((p;p’p;p’))..5) Calculate ((qq)) = = pp GG((pp))GG((pp++qq))((pp++qq,,pp).).6) Determine WW((qq)) = = VV((qq)/[1+)/[1+VV((qq))((qq)].)].7) Revise the self-energy from inputinput((pp;[;[GG]) ]) to outputoutput((pp;[;[GG]) ]) by outputoutput((pp;[;[GG]) ]) = = p’p’ WW((ppp’p’))GG((p’p’))((pp,,p’p’).).

Mapping Mapping FF in the function space { in the function space { ((pp;[;[GG])}])} F: F: inputinput((pp;[;[GG]) ]) outputoutput((pp;[;[GG]) ])

8) Iterate 2)-7) until we obtain inputinput((pp;[;[GG]) =]) =outputoutput((pp;[;[GG]).]).

~~

~~

~~

Fixed-Point PrincipleFixed-Point Principle


Key featuresKey features of this algorithm: of this algorithm: 1) If the iteration process converges, the converged (p;[G]) does not depend on input(p;[G]); or we can start from arbitrary input(p;[G]) to get converged.2) The converged (p) turns out to be the exact solution.

The exact self-energy appears The exact self-energy appears as as a fixed point of a fixed point of FF; ; ==F [F []]..

Thus the problem of obtaining the exact solution is reduced to considering the nature of F around its fixed point, which is nothing to do with the perturbation treatment. We may treat We may treat non-Fermi liquidsnon-Fermi liquids as well as well in this non-perturbative algorithmin this non-perturbative algorithm..

Because this is not a perturbation theory, there is no problem of no problem of double countingdouble counting, which is always troublesome in implementing the usual many-body perturbation theory, in particular, in using the Kohn-Sham basis.


Relation with the Hedin’s TheoryRelation with the Hedin’s Theory

Hedin has derived a closed set of rigorous relationsa closed set of rigorous relations among the exact values of G, W, , , and [PR139, A796(1965)].

In our algorithm, similar relations hold, but not quite the same, because Iinput is generally different from the exact I.

If is converged in our algorithm, however, our relations are reduced to those in the Hedin’s theory, because is the exact solution.

~~~~

In this regard, our algorithm provides an alternative route our algorithm provides an alternative route to solve the Hedin’s set of equations to solve the Hedin’s set of equations without resort to an perturbation expansion in terms of W.


Search for Approximation to Search for Approximation to FF

In actual calculation, it is better to avoid performing the functional derivative and solving the Bethe-Salpeter equation at each iteration step.

We need to find a good functional forma good functional form for for ( (p,p’p,p’)) directly from input(p;[G]) : ((pp,,p’p’;[;[inputinput])]). .

Let us consider the electron-gas system to derive (p,p’;[]). (1) The Ward identityThe Ward identity: It relates the scalar and vector vertex functions, anddirectly with .(2) The ratio function The ratio function RR, which is defined as the ratio of the scalar vertex to the longitudinal part of the vector vertex: If an approximation is made through R, the ward identity is always satisfied. cf. YT, PRL87, 226402 (2001)

Scalar and Vector Vertex Functions: Scalar and Vector Vertex Functions: and and


: bare vector vertex

◎ Gauge Invariance (Local electron-number conservation)

Ward Identity (WI)

◎ In the GW approximation, this basic law is not respected.

: combined notation

Bethe-Salpeter equation:Bethe-Salpeter equation:

Ward Identity (WI) Ward Identity (WI)

Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

Ratio Function & Exact Form for Ratio Function & Exact Form for

1515

　　○ Definition:

○ Scalar vertex in terms of R:

○ Exact functional form for , always satisfying WI

(q)


Approximate Form for Approximate Form for

1616

　○ 　○ Expansion in terms of “Landau parameters” for Expansion in terms of “Landau parameters” for II:: ●● s-wave approximation (related to s-wave approximation (related to ) ) “Exchange-correlation kernel” or “the local-field correction” (in the sense of Niklasson)

This WI is important in satisfying the Ward identity and also this is exactly the same function appearing in the Dzaloshinskii-Larkin theory for Luttinger liquids. This theory is seamlessly applicable This theory is seamlessly applicable to both Fermi and Luttinger liquids. to both Fermi and Luttinger liquids. 　　

● ● Inclusion of Inclusion of p-wave part (related to p-wave part (related to mm**//mm)) A more complex form for (p+q,p) is derived, but WI is essentially the same. cf. H. Maebashi and YT, PRB84, 245134 (2011) 　

~~

Original GWOriginal GW Scheme Scheme

17

◎ GW scheme in the original form [YT, PRL87, 226402 (2001)]

17Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

Difficulties in this scheme:Difficulties in this scheme: (1) Very much time consuming in calculating (2) Difficulty associated with the divergence of or the dielectric function (q,)=1+V(q)(q,) at rs=5.25, where diverges in the electron gas. Dielectric anomaly Dielectric anomaly

YT, J. Superconductivity 18, 785 (2005).

Improved GWImproved GW Scheme Scheme

18

◎ We need not go through as long as I(q) depends only on q. Instead, let us define WI!


WI(q) “the modified Lindhard function”

Compressibility sum rule:

Application to the Electron GasApplication to the Electron Gas


Choose with use of the modified local field correction G+(q,iq), or Gs(q,iq), in the Richardson-Ashcroft form [PRB50, 8170 (1994)].

This GGss((qq,i,iqq)) is not the usual G+(q,iq), but is defined for the true particle or in terms of WI(q). Accuracy in using this GGss((qq,i,iqq)) was well assessed by Lein, Gross, and Perdew, PRB61, 13431 (2000). The peak height specified by is further adjusted by us.

Typical Fermi LiquidsTypical Fermi Liquids


This (p,) is shifted by xc.

・ Typical (textbook-type) Fermi liquid Typical (textbook-type) Fermi liquid behaviorbehavior with clear quasiparticle spectra・ m*/m ～ 1.0 and also EF

* ～ EF

・ Electron-hole symmetric excitations Electron-hole symmetric excitations near the Fermi surface・ Broad plasmaronplasmaron satellites are seen.・ Nonmonotonic behavior of the life time of the quasiparticle (related to the onset of thethe Landau dampingLandau damping of plasmons)

At usual metallic densities (At usual metallic densities (rrss ～～）） YT, Int. J. Mod. Phys. B15, 2595 (2001)

Analytic continuation of (p,i) into (p,) by Pade approximant.

AA ((pp,,) at ) at rrss=4=4


At At rrss=4=4: Comparison of our results with those in G0W0(RPA) and GW

In this case, m*/m (=0.89) < 1 at the Fermi level, but EF* ～ EF.

Quasiparticle Self-Energy CorrectionQuasiparticle Self-Energy Correction


Re(p,Ep) and

Im(p,Ep)

• Re increases monotonically. Slight widening of the bandwidth•Re is fairly flat for p<1.5pF

reason for success of LDA•Re is in proportion to 1/p for p>2pF and it can never

be neglected at p=4.5pF where

Ep=66eV. (interacting

electron-gas model)

No abrupt changes in (p,).

Dynamical Structure FactorDynamical Structure Factor

Although it cannot be seen inthe RPA, the structure a can beclearly seen, which represents the electron-hole multiple scattering (or excitonic) effect.


YT and H. Yasuhara,PRL89, 216402

(2002).


Challenge to Low-Density Electron LiquidsChallenge to Low-Density Electron Liquids

2424

○ Dielectric anomaly Dielectric anomaly of (q0,0)=n2< 0 for rs>5.25

○ For long years, I could not obtain the convergent results for rs beyond this value, but I could not decide whether this is 　　　 1) due to intrinsic reason, related to new physicsnew physics? 2) due to inaccuracyinaccuracy in numerical multi-dimensional integral?

○ A few years ago, I could raise the accuracy by writing the openMP code applicable to about ten-core machine. We obtain the convergent results up to the convergent results up to rrss=8, but never go beyond.=8, but never go beyond.

○ Last year, we developed the MPI code for about 100-core machine. Seek convergent results for rs>8 Include the effect of m*/m in considering the approximate functional form for the vertex function, because m*/m seems to deviate much from unity in the low-density system.

It seems some anomaly exists at rs ～ 8.6!


Momentum Distribution FunctionMomentum Distribution Function

2525

○ nn((pp)) can be obtained without analytic continuation. This is a good index to check the accuracy of the results.○ Check by sum rulessum rules:

Our results satisfy these three sum Our results satisfy these three sum rules at least up to three digitsrules at least up to three digits, but those in recent QMC badly violates them except at rs=5.


Prediction of Prediction of nn((pp) for Lower Densities) for Lower Densities

2626

○ We can also compare our results with those of my old results in the EPX (effective- potential expansion) method. cf. YT & H. Yasuhara, PRB44, 7879 (1991) ○ From the results for rs less than 8, there is a method of extrapolation to predict n(p) for lower densities. cf. P. Gori-Giorgi & P. Ziesche, PRB66, 235116 (2002).

Indication of some new phase for Indication of some new phase for rrss ～～ 10 10

and beyond.and beyond.


Effect of Effect of mm**//mm on the Functional Form on the Functional Form

2727

○ Include the effect of m*/m (or the Landau parameter F1) on the approximate functional form for the vertex function

cf. H. Maebashi and YT, PRB84, 245134 (2011)

○ Determine Determine mm**//mm self-consistently: self-consistently:

The results deviate from those in the EPX [YT, PRB43, 5979 (1991)]for rs > 4, as in the case of zF , indicatingthat the perturbation approach does notwork well in that density region.

AA((pp,,) at ) at rrss=8=8


・ Crossover effect: m*/m > 1 for p << pF　 m*/m < 1 for p > pF

・ Quasiparticles are well defined only near the Fermi surface.

・ Average kinetic energy is about the same as its fluctuation in low density systems.

　○ ○ Anomalous behavior is already seen at Anomalous behavior is already seen at rrss=8!=8!

More Detailed Analysis at More Detailed Analysis at rrss=8=8


　○ On the imaginary- axis (p,i)=i[1-Z(p,i)]+x(p)+c(p,i) Typical Fermi liquids Deviation from typical oneTypical Fermi liquids Deviation from typical one

With the increase of rs, the electron-hole excitations become asymmetric! The concept of hole excitations The concept of hole excitations should be examined.should be examined.


Change into the Self-Consistent Equation for Change into the Self-Consistent Equation for GG

◎ So far, the equation is written in terms of , but because of the form of WI,

it can be cast into the equation for the equation for GG:

◎ Then, this can be solved by changing it into the form of a a matrix equation matrix equation of of

p’ p’ AA((pp,,p’p’))GG((p’p’) = 1.) = 1.

◎ The obtained results from this matrix equation turn out to be the same as those obtained previously for rs<8.6, but this matrix equation has no solution for no solution for rrss beyond this value beyond this value. ◎ The singular-value decomposition is made for the matrix A(p,p’) to find that one of the eigenvalue of this matrix becomes zero!◎ This means that if we write G=G0/(1+G0), there is a state at which the denominator becomes zero! From the very definition of the Green’s function, this implies that a one-electron wave-packet can be a one-electron wave-packet can be generated generated spontaneously! spontaneously! Or the spontaneous electron-hole excitation the spontaneous electron-hole excitation is indicated!

　

GWGWfor Insulators for Insulators ◎In insulators and semiconductors:

Quasiparticle energy same as in the G0W0 in the whole range of p.

cf. Ishii, Maebashi, & YT, arXiv: 1003.3342

31

=0


GWGWfor Luttinger Liquidsfor Luttinger Liquids


◎ In 1D Tomonaga-Luttinger model, because of the long-range nature of interaction.

This is nothing but the Dzyaloshinskii-Larkin equation, exactly describing the nature of the Luttinger liquid.

Maebashi Application to the 1D Hubbard model

Exact spectral function is Exact spectral function is obtained!obtained!


SummarySummary

33

◎ Constructed “the self-energy revision operator theorythe self-energy revision operator theory”, a formally exact non-perturbative framework to calculate the electron self-energy .◎ The exact appears as a fixed point fixed point of the operator.◎ An appropriate approximation form for the operator is proposed and named the GWGW method.◎ The vertex function containing the factor The vertex function containing the factor GG((p’p’))-1-1GG((pp))-1-1 plays a key role in satisfying the Ward identity, applicable to both Fermi and Luttinger

liquids on the same footing, and explaining the reason why the G0W0 approximation works rather well in insulators, semiconductors, and

clusters.◎ There are still open questions open questions in the electronic states in the low-densityin the low-density homogeneous electron liquidshomogeneous electron liquids.◎ If we know by other methods, we can include the information in constructing the vertex function. Note; so far we usually think to calculate G first and then the correlation functions, but there are so often the cases in which we can calculate the correlation functions much easier than G. (TDDFT gives , not G!) Then a framework is needed to obtain G from

the known correlation functions. The The GWGW is useful in this respect! is useful in this respect!33

issp workshop/symposium: masp 2012

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