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 [email protected], University of Tokyo
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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 [email protected], University of Tokyo
10:00-11:30, Monday 25 June 2012
◎ Collaborators:
Drs.Hideaki Maebashi andMasahiro Sakurai
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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 GWGscheme
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)
H: ab initio Hamiltonian in condensed matter physics
Our ultimate goal is to obtain accurate, if not rigorous, solutions for both groundandexcitedstates in this system with an infinitenumber 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 electron-gas model:
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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)
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
Thermodynamic potential Wis given by
whereGis the one-electron Green’s function,Sis 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 exponentiallywith the increase of the order.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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) :
~
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
◎ 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.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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 obtainedwithout explicitly giving F[G] bymaking 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)
~
~
~
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
Hedin has derived a closed set of rigorous relationsamong 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.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
Bethe-Salpeter equation:
: combined notation
g : bare vector vertex
◎ Gauge Invariance (Local electron-number conservation)
Ward Identity (WI)
◎ In the GW approximation, this basic law is not respected.
Ward Identity (WI)
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
○ Definition:
○ Scalar vertex in terms of R:
○ Exact functional form for G, always satisfying WI
-P(q)
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Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
~
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Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
◎ 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
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
YT, J. Superconductivity 18, 785 (2005).
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Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
~
◎ We need not go through Pas long as I(q) depends only on q.
Instead, let us define PWI!
Compressibility sum rule:
PWI(q) “the modified
Lindhard function”
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Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
Choosewith 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.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
ReS (p,Ep) and ImS (p,Ep)
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
Although it cannot be seen in
the RPA, the structure acan be
clearly seen, which represents
the electron-hole multiple
scattering (or excitonic) effect.
YT and H. Yasuhara,PRL89, 216402
(2002).
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
○ 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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Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
○ 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 rsless 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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Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
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Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
○ Anomalous behavior is already seen at rs=8!
・ Crossover effect:
m*/m> 1 for p << pFm*/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.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
○ On the imaginary-w axisS (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.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
◎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!
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
=0
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Quasiparticle energy same as in the G0W0 in the whole range of p.
cf.Ishii, Maebashi, & YT, arXiv: 1003.3342
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
Maebashi Application to the 1D Hubbard model
Exact spectral function is obtained!
long-range nature of interaction.
This is nothing but the Dzyaloshinskii-Larkin equation,
exactly describing the nature of the Luttinger liquid.
Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)
◎ 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)-1plays 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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Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)