ISSP Workshop/Symposium: MASP 2012
Many-Body Non-Perturbative Approach
to the Electron Self-Energy
Yasutami Takada
Institute for Solid State Physics, University of Tokyo
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
Seminar Room A615@ISSP, University of Tokyo
10:00-11:30, Monday 25 June 2012
◎ Collaborators:
Drs. Hideaki Maebashi and Masahiro Sakurai
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Outline
1. Many-Body Perturbation Theory
○ Luttinger-Ward theory
○ Baym-Kadanoff conserving approximation
○ GW approximation
2. Self-Energy Revision Operator Theory
○ Route to the exact electron self-energy S
○ Relation with the Hedin’s theory
○ Good functional form for the vertex function G
○ The GWG scheme
3. 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
○ Dielectric anomaly
○ Spontaneous Electron-hole Pair Formation?
5. Conclusion
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Introduction
H: ab initio Hamiltonian in condensed matter physics
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 formalism.
 This is not necessarily meant to perform
the many-body perturbation calculation.
The interaction part in H is exactly the same as that in the electrongas model:
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Many-Body Perturbation 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 (1960)
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Luttinger-Ward
Thermodynamic potential W is given by
where G is the one-electron Green’s function, S is the electron
self-energy, and F[G] is the Luttinger-Ward energy
functional, given grammatically as
The problem here is that the number of terms in F increases
exponentially with the increase of the order.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Conserving Approximation
Luttinger-Ward is formally exact, but we have to give terms in
F by hand.  Since we cannot give all these infinite number of
terms in F, 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
proposed a good conserving approximation
algorithm (1961,1962).
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Baym-Kadanoff
Procedure of the Baym-Kadanoff algorithm
1) Choose your favorite functional form for F [G].
2) Calculate the self-energy through S (p:[G])=dF[G]/dG(p).
3) Obtain G(p) self-consistently: G(p)-1=G0(p)-1-S (p;[G])
4) Solve the Bethe-Salpeter equation of the integral kernel
defined in terms of the irreducible electron-hole effective
~
interaction I (p;p’)=dS(p;[G])/dG(p’)=d2F[G]/dG(p)dG(p’)
to determine various correlation functions.
Examples:
(1) Hartree-Fock approximation:
 Ladder approximation
in the Bethe-Salpeter equation
(2) Hedin’s GW approximation (1965) :
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GW Approximation
◎ Not P0 but P is a physical polarization function.
◎ G may be regarded as not a physical quantity but just
a building block to construct a physically correct P,
like the Kohn-Sham states in DFT.
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Improvement on Baym-Kadanoff
G in Baym-Kadanoff is not necessarily a physical quantity,
because self-consistency is not imposed between S and G.
In principle, the Baym-Kadanoff algorithm never give the exact
solution, because the exact F[G] is never known.
I find, however, that the exact result can be obtained without
explicitly giving F[G] by making the loop to determine S
and G fully self-consistent!! cf. YT, PRB52, 12708 (1995)
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Self-Energy Revision Operator Theory
~
Key idea: Determine I (p;p’) during the iteration loop
rather than give it a priori, but how?  Map in {S (p;[G])}
Procedure to define the map:
Choose your favorite self-energy Sinput(p;[G]).
G(p) is given by G(p)-1 = G0(p)-1- Sinput(p;[G]).
~
Determine Iinput(p;p’) = dSinput(p;[G])/dG(p’).
Determine G (p,p’) by the solution of the Bethe-Salpeter
~
equation with the integral kernel Iinput(p;p’).
5) Calculate P (q) = -Sps G(p)G(p+q)G (p+q,p).
6) Determine W(q) = V(q)/[1+V(q)P(q)].
7) Revise the self-energy from Sinput(p;[G]) to Soutput(p;[G])
by Soutput(p;[G]) = -Sp’ W(p-p’)G(p’)G (p,p’).
1)
2)
3)
4)
Mapping F in the function space {S (p;[G])}
F: Sinput(p;[G]) Soutput(p;[G])
8) Iterate 2)-7) until we obtain Sinput(p;[G]) =Soutput(p;[G]).
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Fixed-Point Principle
Key features of this algorithm:
1) If the iteration process converges, the converged S (p;[G])
does not depend on Sinput(p;[G]); or we can start from
arbitrary Sinput(p;[G]) to get converged.
2) The converged S (p) turns out to be the exact solution.
 The exact self-energy appears
as a fixed point of F; S = F [S].
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
non-Fermi liquids as well in this non-perturbative algorithm.
Because this is not a perturbation theory, there is no problem of
double counting, which is always troublesome in implementing
the usual many-body perturbation theory, in particular, in using
the Kohn-Sham basis.
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Relation with the Hedin’s Theory
Hedin has derived a closed set of rigorous relations among the
exact values of G, W, S, P, and G [PR139, A796(1965)].
In our algorithm, similar relations
hold, but not quite the same,
~
because Iinput is generally different
~
from the exact I.
If S is converged in our algorithm,
however, our relations are reduced
to those in the Hedin’s theory,
because S is the exact solution.
In this regard, our algorithm provides an alternative route
to solve the Hedin’s set of equations without resort to an
perturbation expansion in terms of W.
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Search for Approximation to F
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 form for G (p,p’)
directly from Sinput(p;[G]) : G (p,p’;[Sinput]).
 Let us consider the electron-gas system to derive G (p,p’;[S]).
(1) The Ward identity: It relates the scalar and vector vertex
functions, G and G , directly with S.
(2) The ratio function R, 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)
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Scalar and Vector Vertex Functions: G and G
Bethe-Salpeter equation:
: combined notation
g : bare vector vertex
◎ Gauge Invariance (Local electron-number conservation)
WardIdentity
Identity
(WI)
 Ward
(WI)
◎ In the GW approximation, this basic law is not respected.
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Ratio Function & Exact Form for G
○ Definition:
○ Scalar vertex in terms of R:
○ Exact functional form for G, always satisfying WI
 -P(q)
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Approximate Form for G
~
○ Expansion in terms of “Landau parameters” for I:
● s-wave approximation (related to k)  “Exchange-correlation
kernel” or “the local-field correction” (in the sense of Niklasson)
This GWI 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
to both Fermi and Luttinger liquids.
*
● Inclusion of p-wave part (related to m /m) A more complex form
for G(p+q,p) is derived, but GWI is essentially the same.
cf. H. Maebashi and YT, PRB84, 245134 (2011)
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Original GWG Scheme
◎ GWG scheme in the original form [YT, PRL87, 226402 (2001)]
Difficulties in this scheme:
(1) Very much time consuming
in calculating P
(2) Difficulty associated with the
YT, J. Superconductivity 18, 785 (2005).
divergence of P or the dielectric
function e(q,w)=1+V(q)P(q,w) at rs=5.25, where k diverges
in the electron gas. Dielectric anomaly
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Improved GWG Scheme
~
◎ We need not go through P as long as I(q) depends only on q.
Instead, let us define PWI!

PWI(q) “the modified
Lindhard function”
Compressibility sum rule:
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Application to the Electron Gas
Choose
with use of the modified local field
correction G+(q,iwq), or Gs(q,iwq), in the Richardson-Ashcroft
form [PRB50, 8170 (1994)].
This Gs(q,iwq) is not the usual G+(q,iwq), but is defined for
the true particle or in terms of PWI(q).
Accuracy in using this Gs(q,iwq) was well assessed by Lein,
Gross, and Perdew, PRB61, 13431 (2000). The peak height
specified by a is further adjusted by us.
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Typical Fermi Liquids
At usual metallic densities (rs～1-2）
YT, Int. J. Mod. Phys. B15, 2595 (2001)
Analytic continuation of S(p,iw) into S(p,w)
by Pade approximant.
・Typical (textbook-type) Fermi liquid behavior
with clear quasiparticle spectra
・m*/m ～1.0 and also EF*～EF
・Electron-hole symmetric excitations near the
Fermi surface
・Broad plasmaron satellites are seen.
・ Nonmonotonic behavior of the life time of
the quasiparticle (related to the onset of the
Landau damping of plasmons)
This S(p,w) is shifted by mxc.
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A (p,w) at rs=4
At rs=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.
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Quasiparticle Self-Energy Correction
ReS (p,Ep) and ImS (p,Ep)
• ReS increases monotonically.
 Slight widening of the
bandwidth
•ReS is fairly flat for p<1.5pF
 reason for success of LDA
•ReS 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 S (p,w).
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Dynamical Structure Factor
Although it cannot be seen in
the RPA, the structure a can be
clearly seen, which represents
the electron-hole multiple
scattering (or excitonic) effect.
YT and H. Yasuhara,PRL89, 216402
(2002).
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Challenge to Low-Density Electron Liquids
○ Dielectric anomaly of P(q0,0)=n2k < 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 physics?
2) due to inaccuracy 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 rs=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!
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Momentum Distribution Function
○ n(p) can be obtained without analytic
continuation. This is a good index to
check the accuracy of the results.
○ Check by sum rules:
 Our results satisfy these three sum
rules at least up to three digits, but
those in recent QMC badly violates
them except at rs=5.
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Prediction of n(p) for Lower Densities
○ We can also compare our results with those
of my old results in the EPX (effectivepotential 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 rs～10 and beyond.
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Effect of m*/m on the Functional Form
○ 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 m*/m self-consistently:
The results deviate from those
in the EPX [YT, PRB43, 5979 (1991)]
for rs > 4, as in the case of zF , indicating
that the perturbation approach does not
work well in that density region.
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A(p,w) at rs=8
○ Anomalous behavior is already seen at rs=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.
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More Detailed Analysis at rs=8
○ On the imaginary-w axis S (p,iw)=iw[1-Z(p,iw)]+cx(p)+cc(p,iw)
Typical Fermi liquids
Deviation from typical one
With the increase of rs, the electron-hole
excitations become asymmetric!
 The concept of hole excitations
should be examined.
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Change into the Self-Consistent Equation for G
◎ So far, the equation is written in terms of S, but because of the form of GWI,
it can be cast into the equation for G:
◎ Then, this can be solved by changing it into the form of a matrix equation
of Sp’ A(p,p’)G(p’) = 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 rs 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+G0S ), 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 generated
spontaneously! Or the spontaneous electron-hole excitation is indicated!
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GWG for Insulators
◎In insulators and semiconductors:

=0


Quasiparticle energy  same as in the G0W0 in the whole
range of p.
cf. Ishii, Maebashi, & YT, arXiv: 1003.3342
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GWG for Luttinger Liquids
◎ In 1D Tomonaga-Luttinger model,
long-range nature of interaction.
because of the

 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
obtained!
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Summary
◎ Constructed “the self-energy revision operator theory”, a formally exact
non-perturbative framework to calculate the electron self-energy S.
◎ The exact S appears as a fixed point of the operator.
◎ An appropriate approximation form for the operator is proposed and
named the GWG method.
◎ The vertex function containing the factor G(p’)-1-G(p)-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 in the electronic states in the low-density
homogeneous electron liquids.
◎ If we know P 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 P, not G!) Then a framework is needed to obtain G from the
known correlation functions. The GWG is useful in this respect!
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