ISSP Workshop/Symposium: MASP 2012

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

2. 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 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)

3. Introduction 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)

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

5. Luttinger-Ward 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)

6. 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)

7. 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) : ~ Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

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

9. 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 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)

10. 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: • 1) Choose your favorite self-energy Sinput(p;[G]). • 2) G(p) is given by G(p)-1 = G0(p)-1- Sinput(p;[G]). • 3) Determine Iinput(p;p’) = dSinput(p;[G])/dG(p’). • 4) Determine G (p,p’) by the solution of the Bethe-Salpeter • equation with the integral kernel Iinput(p;p’). • 5) Calculate P (q) = -SpsG(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’). • 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]). ~ ~ Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

11. Fixed-Point Principle • Key featuresof 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 liquidsas 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. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

12. Relation with the Hedin’s Theory 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)

13. 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 formfor 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 andG ,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) Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

14. Scalar and Vector Vertex Functions: G and G 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)

15. Ratio Function & Exact Form for G ○ Definition: ○ Scalar vertex in terms of R: ○ Exact functional form for G, always satisfying WI -P(q) 15 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

16. 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 GWIis 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 GWIis essentially the same. • cf. H. Maebashi and YT, PRB84, 245134 (2011) 16 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

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

18. Improved GWG Scheme ~ ◎ 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” 18 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

19. Application to the Electron Gas 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)

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

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

22. 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 whereEp=66eV. (interacting • electron-gas model) • No abrupt changes in S (p,w). Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

23. Dynamical Structure Factor 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)

24. 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! 24 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

25. 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. 25 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

26. Prediction of n(p) for Lower Densities ○ 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. 26 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

27. 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. 27 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

28. A(p,w) at rs=8 ○ 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)

29. More Detailed Analysis at rs=8 　○ 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)

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

31. GWG for Insulators  =0   31 • In insulators and semiconductors: 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)

32. GWG for Luttinger Liquids  Maebashi  Application to the 1D Hubbard model Exact spectral function is obtained! • 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. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)

33. 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)-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! 33 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)