1. Insulating Spin Liquid in the 2D Lightly Doped Hubbard Model
Hermann Freire
International Centre of Condensed Matter Physics
University of Brasília, Brazil
Renormalization Group 2005
Helsinki, Finland 30 August - 3 September 2005

2. Motivation  The High-Tc Cuprates
Parent Compound  La2CuO4
Planes of Cu and O (2D system);
1 electron per site from the 3d shell of the Cu atoms (half-filled band);
Coupling between electrons rather strong;
Mott insulator (charge gap ~ 2ev)
Antiferromagnetically long-range ordered;
SU(2) symmetry spontaneously broken;
Gapless for spin excitations (magnons). La O Cu

3. Effect of Doping  The Phase Diagram
Hole Doped Compound  La(2-x)SrxCuO4
At T=0 several ground states emerge as we vary “x”
0 < x < 0.02  AF Mott insulator;
0.02 < x < 0.1  Pseudogap, spin glass, stripes, ISL... ???
0.1< x < 0.15  Superconductor with d-wave pairing.

4. Modeling the System  The 2D Hubbard Model
Electrons on a 2D square lattice
Hubbard Hamiltonian (U > 0)
The noninteracting Hamiltonian can be diagonalized

5. What are the fundamental questions?
What is the nature of the ground state of this model for electron densities slightly away from a half-filling condition?
Is this state long-range ordered or short-range ordered? Is there a spontaneous symmetry breaking associated?
The elementary excitations associated with the charge degrees of freedom are gapped or not?
The elementary excitations associated with the spin degrees of freedom are gapped or not?
We will show our conclusions regarding these questions based on a complete Two-Loop Renormalization Group calculation within a field-theoretical framework.

6. The Noninteracting Band
The electron density can be adjusted by tuning the chemical potential  W
 = 0 (half-filled case)
Important features:
Fermi surface perfectly nested;
Density of states logarithmically divergent (van Hove singularities).
The bandwidth W = 8t

7. Starting Point  The Lightly Doped Scenario
Removing electrons  doping with holes
e.g.  = t (x = 0.09)
Important features:
Fermi surface approximately nested for energies E > ||;
Density of states is not divergent at the FS.
Umklapp Surface

8. Adding the Hubbard Interaction Term
In momentum space, the Hubbard interaction reads
Continuous Symmetries
The RG transformation must respect these symmetries.
Global U(1)  Charge Conservation;
SU(2)  Spin Conservation.
The interesting regime happens when U ~ W, which will be the case considered here, since we are mostly interested in getting a qualitative idea of what should happen in the lightly-doped cuprates.

9. Patching the FS  The 2D g-ology notation
By dimensional analysis, the marginally relevant interaction processes are
Backscattering processes
Forward scattering processes
Here we neglect Umklapp processes since we are not at half-filling condition.

10. The 2D Hubbard Model Case
The full Lagrangian of the Hubbard model reads
Linearized energy dispersion
SU(2) invariant form
where
The model is defined at a scale of a few lattice spacings (microscopic scale)  Bare (B) theory
Naive perturbation theory  Lots of infrared (IR) divergent Feynman diagrams!!!

11. Field Theory RG Philosophy
Rewrite the bare theory in terms of renormalized parameters plus appropriate counterterms  Reorganization of the perturbation series and cancellation of the infrared divergences.
The microscopic Hubbard model (bare theory).
The floating scale at which the renormalized parameters are to be defined.
The infrared (IR) fixed point behavior.

12. Renormalizing the Theory Towards the FS
The renormalization procedure implies in approaching the low-energy limit of the theory  Only the normal direction to the FS is reduced.
The normal direction to the FS is irrelevant in the RG sense  It can be neglected;
The parallel direction to the Fermi surface is unaffected by the RG transformation  All vertices acquire a strong dependence on the parallel momenta.
Schematically, we will obtain for instance
Low-Energy Dynamics
Microscopic Model
Quantum
Fluctuations
Hubbard Model
Local interaction
(g1B=g2B  U)
Effective theory with nonlocal interaction
g1R=g1R(p1//,p2//,p3//)
g2R=g2R(p1//,p2//,p3//)

13. Where should we look for divergences?
Elementary Dimensional Analysis for the 1PI Vertices
(4)(p1,p2,p3) function  Effective two-particle interaction
(2)(p) function  Self-energy effects
(2,1)(p,q) function  Linear response w.r.t. various perturbations
(2,1)(p,q0) function  Uniform response functions
(0,2)(q) function  All kinds of susceptibilities

14. Renormalization of (4) and (2) 1PI Vertices
Rewrite (‘renormalize’) the couplings and the fermionic fields
Counterterms
The renormalized Lagrangian (i.e free of divergences) now reads

15. A Novel RG “Fixed Point” for Moderate U / W
Results for a Discretized FS (4X33 points)
(H. Freire, E. Corrêa and A. Ferraz, Phys. Rev. B 71, (2005))
What is the nature of this resulting state?

16. Uniform Response Functions  (2,1)(p,q0)
The Uniform Charge and Spin Functions
For the uniform susceptibilities, the infinitesimal field couples with both charge and spin number operators
Counterterm
Rewrite 
Charge (CS)
Spin (SS)
Symmetrization 

17. Earlier Methods Encountered in the Literature
One-loop RG Calculation of the Uniform Response Functions
Feynman Diagrams 
Not a single IR divergent Feynman diagram;
Not possible to derive a RG flow equation for these quantities;
Very similar to a RPA approximation.

18. Not IR divergent
Calculating them, we get
We must now make a prescription
Therefore
Since in one-loop order there is no self-energy corrections Z=1. As a result

19. Symmetrizing, we get for the charge response function 
And, similarly for the uniform spin response function 
These equations are then calculated self-consistently.
This is indeed a Random-Phase-Approximation (RPA);
Not consistent with the RG philosophy.

20. Uniform Susceptibilities in this RPA Approximation
The Feynman diagram associated with both uniform susceptibilities is
The corresponding analytical expressions are the following
Charge Compressibility (CS) 
Uniform Spin Susceptibility (SS) 

21. Numerical Results
[C. Halboth and W. Metzner (Phys. Rev. B 61, 7364 (2000))]
AF dominating  Charge gap and no Spin gap (Mott insulator phase);
d-wave SC dominating  Spin gap and no Charge gap (Superconducting phase);
But they are not able to see anything in between (intermediate doping regime)!!!

22. Full RG Calculation of the Response Functions
A consistent RG calculation of the response function can only be achieved in two-loop order or beyond.
Two-Loop RG Calculation
At this order, it is possible to implement a full RG program in order to calculate the uniform response functions;
This is due to the fact that there are several IR divergent Feynman diagrams (the so-called nonparquet diagrams);
It has also the advantage of dealing properly with the strong self-energy feedback associated with our fixed point theory described earlier;
Physically speaking, it means including strong quantum fluctuations effects in the hope of understanding the highly nontrivial quantum state observed for the intermediate doping regime.

23. The Feynman Diagrams up to Two-Loop Order
Important Remarks
The two-loop diagrams are the so-called nonparquet diagrams.
We are neglecting the one-loop diagrams since they are not IR divergent and, therefore, they are unimportant from a RG point of view.

24. Calculating these Feynman diagrams, we get
IR divergent
where the dots mean that we are omitting the parallel momenta dependence in the coupling functions.
We now establish the following renormalization condition

25. Therefore, we have
In this way, the bare and renormalized parameters are related by
Since the bare parameter (i.e. the quantity at the microscopic scale) does not know anything about the scale , we have

26. As a result, we obtain the RG equations
where is the anomalous dimension of the theory and it is
given by
The anomalous dimension comes from the renormalization of the fields (self-energy effects) and it will be explained in more detail by A. Ferraz (Saturday 12:30-13:00)

27. Symmetrizing, we get for the charge response function 
Similarly, we get for the uniform spin response function 
Therefore, we see that now we do have a flow equation for the uniform response functions in contrast to the one-loop approach described earlier.

28. The Uniform Susceptibilities up to Two-Loop Order
The Feynman diagram associated with the uniform susceptibilities will be always the same regardless of the number of loops we go in our RG approach.
This is simply related to the fact that there is no way to find a logarithmic infrared divergence that is not generated by the other RG flow equations!!!
Therefore, the corresponding analytical expressions are also the same
Charge Compressibility (CS) 
Uniform Spin Susceptibility (SS) 

29. The Insulating Spin Liquid State
Starting Point (bare theory)  Metallic State
Initial DOS for both charge and spin finite
Strongly supressed charge compressibility and uniform spin susceptibility;
Absence of low-lying charged and/or magnetic excitations in the vicinity of the FS;
Charge gap (Insulating system) and spin gap;
No spontaneous symmetry breaking associated;
Short-range ordered state;
Insulating Spin Liquid behavior.
(H. Freire, E. Corrêa and A. Ferraz, cond-mat/ )

30. Conclusions and Outlook
Within a complete Two-Loop RG calculation, and taking into account strong quantum fluctuations, we find for a 2D lightly-doped Hubbard model that
The true strong-coupling ground state of this model has no low-lying charge and spin excitations;
Such a state is usually referred to as an Insulating Spin Liquid (ISL);
This state has short-range order and cannot be related to any symmetry broken phase;
These results may be of direct relevance for the understanding of the underlying mechanism of high-Tc superconductivity.