1. FIELD THEORETICAL RG FOR A 2D FERMI SURFACE
Alvaro Ferraz
Internacional Centre for Condensed Matter Physics
University of Brasilia
Brasilia-Brazil

2. General Scope
1-Introduction
2-Lagrangian Model & its 2D Fermi Surface
3-Coupling Function Renormalization at One-Loop Order
4-Self-Energy Corrections
5-RG at Higher Orders
6-Renormalized Coupling Flows
7- Density Wave & Pairing Susceptibilities
8-Conclusion

3. 1-Introduction
RG proved to be very useful in probing strongly interacting systems.
This is even more so in 2D.
We will describe a field theoretical RG calculation of a 2D electron gas in a presence of nearly flat Fermi surface (FS).
Our results can be related to the cuprate high-Tc superconductors.

4. A FS identical to ours was observed recently in La2-xSrxCuO4 thin epitaxial film under strain (Abrecht et al, PRL 91,57002( 2003)).
Cuprates are Mott insulators at ½-filling which turn into a spin liquid at very low dopings ( x~0.02! ).

5. At higher temperatures and concentrations there appears an exotic pseudogap phase and finally at even higher concentration a D-wave high-Tc superconductor.

6. 2-Lagrangian Model and its 2D Fermi Surface
To describe 2D electrons consider the renormalized lagrangian (A.F.,EPL 61,228(2003)).
Here a= refers to the upper (lower) or right (left) flat patch of our FS.

7. Thanks to the flat FS the single-particle dispersion is simply
with and
In general
The bare couplings represent backward and forward scatterings

8. Since the non-interacting propagators are
both part-part and part-hole diagrams are IR log divergent
with being a fixed upper energy cut-off

9. 3 – Coupling Function Renormalization at One-Loop Order
We will now proceed with the calculations of the one-particle irreducible functions
within the framework of the FTRG.

10. Up to one-loop order the calculation is simple. We set
Using appropriate Feynman rules, we find

11. This gives
Similarly, using a similar prescription we define

12. Using, again, appropriate Feynman rules it turns out that
Or, equivalently

13. The counterterms are in this way continuous functions of the external momenta and the RG scale parameter
Since the bare parameters don’t depend on the RG scale we are naturally led to the RG equations

14. It follows immediately that the one-loop renormalized coupling flows are then

15. 4 – Self-Energy Correction
To calculate self-energy corrections we need to estimate at least two-loop contributions

16. The first four diagrams produce constant shifts in and renormalize .
Since in our calculation both and SF are kept fixed we may ignore those contributions altogether.
In contrast, the remaining diagrams give us

17. Defining the renormalized one-particle irreducible function such that
where Z is the quasiparticle weight which relates the bare and renormalized fields:
Defining the renormalized one-particle irreducible function
such that

18. it follows immediately that
where the anomalous dimension is given by

19. It is now straightforward to estabilish the RG equation for the quasiparticle weight Z:
We point out that our renormalized Lagrangian can now be put in a more convenient form:

20. where

21. 5 – RG at Higher Orders
To calculate corrections for and we need to take into account the higher order diagrams which are also
These non-parquet diagrams in two-loop order for both backscattering and forward scattering channels are

22. Taking again into account the RG condition
the RG flow equations for the renormalized coupling functions in two-loops become

23. 6 – Renormalized Coupling Flows
It is impossible to solve our RG equations analytically.
We need therefore numerical methods to estabilish the flow of the renormalized coupling functions (H. Freire, E. Corrêa, A. F., PRB 71, (2005)).
To do this we discretize the FS replacing by a finite set of points.
For convenience we take where is our fixed upper energy cutoff and l our RG step.
Notice that max l is limited by the fact that cannot be shorter than the distance between neighboring points in our discretization procedure.

24. Initially we depict the one-loop results for different choices of external momenta.
They reproduce previous one-loop results.

25. We show next the quasiparticle weight Z

26. If we take initially , Z is mildly reduced from unity.

27. We show next the RG flows for and
in two-loop order
In contrast with one-loop results the renormalized couplings approach plateau values in a fixed point like regime.

28. Many renormalized couplings now approach zero continuously as a result of the suppression of Z.
This is a strong indicative that there is no symmetry breaking and no onset of long range order in the physical system.
To test the leading instabilities in this new regime we need to calculate the charge and spin susceptibilities.

29. 7 – Density Wave and Pairing Susceptibilities
Since the susceptibilities are essentially mean values of composite operators we add to our original Lagrangian the contributions (E. Corrêa, H. Freire and A. Ferraz (2005)).
and

30. The addition of composite operators generate new divergencies which must be regularized in their own right.
As a result we must have
and

31. The density wave renormalized vertex should be symmetrized with respect to the spin to give
and
Similarly, associated with the ’s we define the singlet and triplet pairing vertices
and

32. Diagrammatically the ’s and ’s are directly related with the one-particle irreducible vertex function
In one-loop order we get

33. For the DW channel we use the prescription
We use a similar condition for the SC channels

34. Taking into account the RG condition for the bare vertices we arrive at the RG equations
with or

35. Due to the particular shape of our flat FS the renormalized couplings must be symmetrical with respect to the exchange of + and – particles and change of sign of the external ’s :

36. In view of that it turns out that the RG equations for the renormalized vertices are symmetrical with respect to the sign reversal of in for a fixed (A. Zheleznyak et al PRB 55, 3200 (1997)).
We therefore define two irreducible representations of this symmetry which never mix
with a = S,T.
is associated with s-wave symmetry whereas shows d-wave character.

37. With b = S,C. Instead, for density wave symmetries
Here, the antisymmetrical ones are associated with the so-called flux phases.
Once the are found the related susceptibilities ,
associated with the related , follow immediately.

38. or, equivalently
and

39. Following the same numerical procedure as before we can estimate if there is any symmetry breaking and of what kind.
Here we take a Hubbard like initial condition together with

40. We show the corresponding one-loop and two-loops
contributions for the various symm and
antisymm renormalized

41. 8 - Conclusions
Although the one-loop c’s seem to announce symmetry breaking and, in particular, the predominance of the SDW+ instability the two-loop c’s seem to approach plateau values characteristic of short-range ordered states only.
Among them the SSC- (d-wave like) & SDW- (spin flux phase) appear closely together and slightly above the remaining susceptibilities.
The exception to that is the SDW+ which appears to grow indefinitely. However as opposed to the one-loop result, this growth seems spurious since it is slowered down considerably when we consider high-order effects.

42. To check this conclusion we calculated the associated spin and charge uniform susceptibilities (H. Freire, E. Corrêa and A. Ferraz, cond-mat/ ) and we find indeed no sign of long-range order since both
and