1. ISSP Workshop/Symposium: MASP 2012 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 1 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 Masahiro Sakurai Drs. Hideaki Maebashi and Masahiro Sakurai Many-Body Non-Perturbative Approach to the Electron Self-Energy

2. Outline Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 2 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  ○ Relation with the Hedin’s theory  ○ Good functional form for the vertex function  GW  ○ The GW  scheme 3. Application ○ Electron liquids at metallic densities: Typical Fermi liquid ○ Relation with the G 0 W 0 approximation ○ One-dimensional Hubbard model: Typical Luttinger liquid 4. Singularities at Low-Density Electron Liquids ○ Dielectric anomaly ○ Spontaneous Electron-hole Pair Formation? 5. Conclusion

3. Introduction Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 3 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? the Green’s-function formalism  Let us go with the Green’s-function formalism.  This is not necessarily meant to perform the many-body perturbation calculation the many-body perturbation calculation. The interaction part in H is exactly the same as that in the electron- gas model: H: ab initio Hamiltonian in condensed matter physics H: ab initio Hamiltonian in condensed matter physics

4. Many-Body Perturbation Theory Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 4 Usual Perturbation-Expansion Theory Usual Perturbation-Expansion Theory Choose an appropriate nonperturbed one-electron Hamiltonian H 0, together with its complete eigenstates {|n>: H 0 |n>=E n (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)

5. Luttinger-Ward Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 5 Thermodynamic potential  Thermodynamic potential  is given by G   [G] the Luttinger-Ward energy functional where G is the one-electron Green’s function,  is the electron self-energy, and  [G] is the Luttinger-Ward energy functional, given grammatically as The problem here is that the number of terms in  increases exponentially with the increase of the order.

6. Conserving Approximation Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 6 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 Baym and Kadanoff Luttinger and Ward, Baym and Kadanoff conserving approximation proposed a good conserving approximation algorithm algorithm (1961,1962).

7. Baym-Kadanoff Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 7 Procedure of the Baym-Kadanoff algorithm  [G] 1) Choose your favorite functional form for  [G].  (p:[G])=  [G]/  G(p). 2) Calculate the self-energy through  (p:[G])=  [G]/  G(p). G(p)G(p) -1 =G 0 (p) -1  (p;[G]) 3) Obtain G(p) self-consistently: G(p) -1 =G 0 (p) -1  (p;[G]) 4) Solve the Bethe-Salpeter equation of the integral kernel defined in terms of the irreducible electron-hole effective I (p;p’)=  (p;[G])/  G(p’)=  2  [G]/  G(p)  G(p’) interaction I (p;p’)=  (p;[G])/  G(p’)=  2  [G]/  G(p)  G(p’) to determine various correlation functions.Examples: (1) Hartree-Fock approximation:  Ladder approximation in the Bethe-Salpeter equation (2) Hedin’s GW approximation (1965) : ~

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

9. Improvement on Baym-Kadanoff Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 9 In principle, the Baym-Kadanoff algorithm never give the exact solution, because the exact  [G] is never known. the exact result can be obtained making the loop to determine  and  fully self-consistent!! I find, however, that the exact result can be obtained without explicitly giving  [G] by making the loop to determine  and  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 

10. Self-Energy Revision Operator Theory Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 10 Key idea: Determine I (p;p’) during the iteration loop rather than give it a priori  Map in {  (p;[G])} rather than give it a priori, but how?  Map in {  (p;[G])} Procedure to define the map:  input (p;[G] 1) Choose your favorite self-energy  input (p;[G]). G(p)G(p) -1 = G 0 (p) -1  input (p;[G]). 2) G(p) is given by G(p) -1 = G 0 (p) -1  input (p;[G]). I input (p;p’) =  input (p;[G])/  G(p’). 3) Determine I input (p;p’) =  input (p;[G])/  G(p’).  (p,p’) 4) Determine  (p,p’) by the solution of the Bethe-Salpeter I input (p;p’). equation with the integral kernel I input (p;p’).  (q) =   p  G(p)G(p+q)  (p+q,p). 5) Calculate  (q) =   p  G(p)G(p+q)  (p+q,p). W(q) = V(q)/[1+V(q)  (q)]. 6) Determine W(q) = V(q)/[1+V(q)  (q)].  input (p;[G])  output (p;[G]) 7) Revise the self-energy from  input (p;[G]) to  output (p;[G])  output (p;[G]) =   p’ W(p  p’)G(p’)  (p,p’). by  output (p;[G]) =   p’ W(p  p’)G(p’)  (p,p’). Mapping F in the function space {  (p;[G])} Mapping F in the function space {  (p;[G])} F:  input (p;[G])   output (p;[G]) F:  input (p;[G])   output (p;[G])  input (p;[G]) =  output (p;[G]). 8) Iterate 2)-7) until we obtain  input (p;[G]) =  output (p;[G]). ~ ~ ~

11. Fixed-Point Principle Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 11 Key features 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 a fixed point of F ;  =  F [  ]. as a fixed point of F ;  =  F [  ]. We may treat non-Fermi liquids as well in this non-perturbative algorithm. 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. no problem of double counting 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.

12. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 12 Relation with the Hedin’s Theory a closed set of rigorous relations Hedin has derived a 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 I input 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. ~ ~ our algorithm provides an alternative route In this regard, 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.

13. 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. a good functional form for  (p,p’)  We need to find a good functional form for  (p,p’)  (p,p’;[  input ]). directly from  input (p;[G]) :  (p,p’;[  input ]).  Let us consider the electron-gas system to derive  (p,p’;[  ]). The Ward identity (1) The Ward identity: It relates the scalar and vector vertex functions,  and  directly with . The ratio function R (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)

14. Scalar and Vector Vertex Functions:  and  Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 14  : 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: Ward Identity (WI) Ward Identity (WI)

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

16. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) Approximate Form for  16 ○ Expansion in terms of “Landau parameters” for I: ○ Expansion in terms of “Landau parameters” for I: ● 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 p-wave part (related to m * /m) ● Inclusion of p-wave part (related to m * /m)  A more complex form for  (p+q,p) is derived, but  WI is essentially the same. cf. H. Maebashi and YT, PRB84, 245134 (2011) ~

17. Original GW  Scheme 17 ◎ GW  scheme in the original form [YT, PRL87, 226402 (2001)] 17 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 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 r s =5.25, where  diverges  Dielectric anomaly in the electron gas.  Dielectric anomaly YT, J. Superconductivity 18, 785 (2005).

18. Improved GW  Scheme 18 ◎ We need not go through  as long as I(q) depends only on q. Instead, let us define  WI ! 18 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)  WI (q)  “the modified Lindhard function”   Compressibility sum rule:

19. Application to the Electron Gas Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 19 Choose with use of the modified local field correction G + (q,i  q ), or G s (q,i  q ), in the Richardson-Ashcroft form [PRB50, 8170 (1994)]. G s (q,i  q ) This G s (q,i  q ) is not the usual G + (q,i  q ), but is defined for the true particle or in terms of  WI (q). G s (q,i  q ) Accuracy in using this G s (q,i  q ) was well assessed by Lein, Gross, and Perdew, PRB61, 13431 (2000). The peak height specified by  is further adjusted by us.

20. Typical Fermi Liquids Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 20 This  (p,  ) is shifted by  xc. Typical (textbook-type) Fermi liquid behavior ・ Typical (textbook-type) Fermi liquid behavior with clear quasiparticle spectra ・ m * /m ～ 1.0 and also E F * ～ E F Electron-hole symmetric excitations ・ Electron-hole symmetric excitations near the Fermi surface plasmaron ・ Broad plasmaron satellites are seen. ・ Nonmonotonic behavior of the life time of the the quasiparticle (related to the onset of the Landau damping Landau damping of plasmons) At usual metallic densities (r s ～  ） YT, Int. J. Mod. Phys. B15, 2595 (2001) Analytic continuation of  (p,i  ) into  (p,  ) by Pade approximant.

21. A (p,  ) at r s =4 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 21 At r s =4 At r s =4: Comparison of our results with those in G 0 W 0 (RPA) and GW In this case, m * /m (=0.89) < 1 at the Fermi level, but E F * ～ E F.

22. Quasiparticle Self-Energy Correction Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 22 Re  (p,E p ) and Im  (p,E p ) Re  increases monotonically.  Slight widening of the bandwidth Re  is fairly flat for p<1.5p F  reason for success of LDA Re  is in proportion to 1/p for p>2p F and it can never be neglected at p=4.5p F where E p =66eV. (  interacting electron-gas model) No abrupt changes in  (p,  ).

23. 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. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 23 YT and H. Yasuhara,PRL89, 216402 (2002).

24. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) Challenge to Low-Density Electron Liquids 24 Dielectric anomaly ○ Dielectric anomaly of  (q  0,0)=n 2  5.25 ○ For long years, I could not obtain the convergent results for r s beyond this value, but I could not decide whether this is new physics  1) due to intrinsic reason, related to new physics? inaccuracy 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. the convergent results up to r s =8, but never go beyond.  We obtain the convergent results up to r s =8, but never go beyond. ○ Last year, we developed the MPI code for about 100-core machine.  Seek convergent results for r s >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 r s ～ 8.6!

25. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) Momentum Distribution Function 25 n(p) ○ n(p) can be obtained without analytic continuation. This is a good index to check the accuracy of the results. sum rules ○ Check by sum rules: Our results satisfy these three sum rules at least up to three digits  Our results satisfy these three sum rules at least up to three digits, but those in recent QMC badly violates them except at r s =5.

26. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) Prediction of n(p) for Lower Densities 26 ○ 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 r s 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 r s ～ 10 and beyond.  Indication of some new phase for r s ～ 10 and beyond.

27. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) Effect of m * /m on the Functional Form 27 ○ Include the effect of m * /m (or the Landau parameter F 1 ) on the approximate functional form for the vertex function  cf. H. Maebashi and YT, PRB84, 245134 (2011) Determine m * /m self-consistently: ○ Determine m * /m self-consistently:  The results deviate from those  in the EPX [YT, PRB43, 5979 (1991)]  for r s > 4, as in the case of z F, indicating  that the perturbation approach does not  work well in that density region.

28. A(p,  ) at r s =8 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 28 ・ Crossover effect: m * /m > 1 for p p F ・ 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 r s =8!

29. More Detailed Analysis at r s =8 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 29 ○ On the imaginary-  axis  (p,i  )=i  [1-Z(p,i  )]+  x (p)+  c (p,i  ) Typical Fermi liquids Deviation from typical one With the increase of r s, the electron-hole excitations become asymmetric!  The concept of hole excitations  The concept of hole excitations should be examined. should be examined.

30. 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 , but because of the form of  WI, the equation for G it can be cast into the equation for G: a matrix equation ◎ Then, this can be solved by changing it into the form of a matrix equation of  p’ A(p,p’)G(p’) = 1. of  p’ A(p,p’)G(p’) = 1. ◎ The obtained results from this matrix equation turn out to be the same as those obtained previously for r s <8.6, but this matrix equation has no solution for r s beyond this value no solution for r s 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=G 0 /(1+G 0  ), there is a state at which the denominator becomes zero! From the very definition of the Green’s a one-electron wave-packet can be generated function, this implies that a one-electron wave-packet can be generated spontaneously! the spontaneous electron-hole excitation spontaneously! Or the spontaneous electron-hole excitation is indicated!

31. GW  for Insulators ◎ In insulators and semiconductors: Quasiparticle energy  same as in the G 0 W 0 in the whole range of p. cf. Ishii, Maebashi, & YT, arXiv: 1003.3342 31  =0 31 Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada)  

32. GW  for Luttinger Liquids Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) 32 ◎ 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 obtained! 

33. Many-Body Non-Perturbative Approach to the Electron Self-Energy (Takada) Summary 33 the self-energy revision operator theory ◎ Constructed “the self-energy revision operator theory”, a formally exact non-perturbative framework to calculate the electron self-energy . fixed point ◎ The exact  appears as a fixed point of the operator. ◎ An appropriate approximation form for the operator is proposed and GW  named the GW  method. The vertex function containing the factor G(p’) -1 - G(p) -1 ◎ 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 G 0 W 0 approximation works rather well in insulators, semiconductors, and clusters. open questions in the low-density ◎ There are still open questions in the electronic states in the low-density homogeneous electron liquids homogeneous 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. The GW  is useful in this respect! (TDDFT gives , not G!) Then a framework is needed to obtain G from the known correlation functions. The GW  is useful in this respect! 33