Transcript Renormalization of a 2d Hubbard Model in a two

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
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)
La
O
• Antiferromagnetically long-range ordered;
• SU(2) symmetry spontaneously broken;
• Gapless for spin excitations (magnons).
Cu
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.
Modeling the System  The 2D Hubbard Model
Electrons on a 2D square lattice
Hubbard Hamiltonian (U > 0)
H
 t c  c     n   U  n
i , j 
†
ij i
j
i
t , if i and j are n.n. sites
tij  
0, otherwise
The noninteracting Hamiltonian can be diagonalized
H 0  k ck† ck   2t  cos(k x a)  cos(k y a)     ck† ck
k
k
n
i i
i
i
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 fieldtheoretical framework.
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;
The bandwidth W = 8t
• Density of states logarithmically
divergent (van Hove singularities).
Starting Point  The Lightly Doped Scenario
Removing electrons  doping with holes
e.g.  = - 0.15 t (x = 0.09)
Important features:
• Fermi surface approximately nested
for energies E > ||;
Umklapp Surface
• Density of states is not divergent at
the FS.
Adding the Hubbard Interaction Term
In momentum space, the Hubbard interaction reads
 U 
†
†
H int  

(
p
+p
-p
)

   1 2 3  ( p3 )  ( p2 )  ( p1 )
 N sites  p1 , p2 , p3
Continuous Symmetries
• Global U(1)  Charge Conservation;
• SU(2)  Spin Conservation.
The RG transformation must
respect these symmetries.
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.
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.
The 2D Hubbard Model Case
The full Lagrangian of the Hubbard model reads
Linearized energy dispersion
L

p, , a 

1
V
i t  vF ( p  k F )  (Ba†), ( p, t ) (Ba ), ( p, t )
 
p1 , p2 , p3  ,  , ,
 g 2 B     g 1B     (B†), ( p1 +p2 -p3 , t ) (B†), ( p3 , t ) (B ),  ( p2 , t ) (B ), ( p1 , t )
SU(2) invariant form
where
 V 
U
N
 sites 
g1B  g 2B  
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!!!
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).
RG step l
The floating scale at which the
renormalized parameters are to
be defined.

l  ln  
 
l 
The infrared (IR) fixed point
behavior.
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
Microscopic Model
Hubbard Model
Local interaction
(g1B=g2B  U)
Quantum
Fluctuations
Low-Energy Dynamics
Effective theory with
nonlocal interaction
g1R=g1R(p1//,p2//,p3//)
g2R=g2R(p1//,p2//,p3//)
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
Renormalization of (4) and (2) 1PI Vertices
Rewrite (‘renormalize’) the couplings and the fermionic fields
Counterterms
 (Ba ), ( p, t )   (Ra ), ( p, t; )   (Ra ), ( p, t; )  Z 1/ 2 ( p; ) (Ra ), ( p, t; )
4
g iB  U   Z 1/ 2 ( pi // ;  )  g iR ( p1// , p2 // , p3// ;  )  g iB ( p1// , p2 // , p3// ;  ) 
(i  1, 2)
i 1
The renormalized Lagrangian (i.e free of divergences) now reads
L

p, , a 


1
V
1
V
Z ( p) i t  vF ( p  k F )  (Ra†), ( p, t ) (Ra ), ( p, t )
 g
 
  
p1 , p2 , p3
, , ,
 g
 
  
p1 , p2 , p3
, , ,
    g1R    (R†), ( p1 +p2 -p3 , t ) (R†), ( p3 , t ) (R ),  ( p2 , t ) (R ), ( p1 , t )
2 R 
    g1R    (R†), ( p1 +p2 -p3 , t ) (R†), ( p3 , t ) (R ),  ( p2 , t ) (R ), ( p1 , t )
2 R 
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, 165113 (2005))
What is the nature of this resulting state?
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
Lext  hexternal
Rewrite




p , , a 
 B ( p) (Ba†) ( p, t ') (Ba ) ( p, t ')


 B ( p)  Z ( p)  R ( p)   R ( p) 
1
Counterterm
Charge (CS)
 R,CS ( p)   R ( p)   R ( p)
Spin (SS)
 R,SS ( p)   R ( p)   R ( p)
Symmetrization 
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.
Not IR divergent
Calculating them, we get

i(2,1)
R , ( p, q  0)  i R ( p// ) 




dk
g
(
p
,
k
,
k
;

)

(
k
)

g
(
p
,
k
,
k
;

)

(
k
)

//  2 R
//
//
//
R
//
1R
//
//
//
R
// 
4 2vF 



i
 i R ( p// )
(2,1)

We must now make a prescription iR, ( p// , p  kF , p0   , q  0)  i R ( p// ; ) .
Therefore
 R ( p// ;  ) 




dk
g
(
p
,
k
,
k
;

)

(
k
;

)

g
(
p
,
k
,
k
;

)

(
k
;

)

//  2 R
//
//
//
R
//
1R
//
//
//
R
//

4 2 vF 



1
Since in one-loop order there is no self-energy corrections Z=1. As a result
 R ( p// ;  )   B ( p// ) 




dk
g
(
p
,
k
,
k
;

)

(
k
;

)

g
(
p
,
k
,
k
;

)

(
k
;

)

//  2 R
//
//
//
R
//
1R
//
//
//
R
//

4 2 vF 



1
Symmetrizing, we get for the charge response function   R,CS ( p)   R ( p)   R ( p)
 R ,CS ( p// ; )   B ,CS ( p// ) 
1
4 vF
2
 dk  g
//
1R
( p// , k// , k// ;  )  2g 2 R ( p// , k// , k// ;  )  R,CS ( k// ;  )
And, similarly for the uniform spin response function   R , SS ( p)   R ( p)   R ( p)
 R ,SS ( p// ; )   B ,SS ( p// ) 
1
4 2vF
 dk g
//
1R
( p// , k// , k// ;  ) R ,SS (k// ;  )
These equations are then calculated self-consistently.
• This is indeed a Random-Phase-Approximation (RPA);
• Not consistent with the RG philosophy.
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) 
 ( ) 
R
CS
 ( ) 
R
SS
1
4 vF
2
1
4 vF
2

 dp
//


 dp
//

 R ,CS ( p// ;  ) 
 R , SS ( p// ;  ) 
2
2
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)!!!
Full RG Calculation of the Response Functions
A consistent RG calculation of the response function can only be achieved in twoloop 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.
The Feynman Diagrams up to Two-Loop Order
Important Remarks
•The two-loop diagrams are the socalled 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.
Calculating these Feynman diagrams, we get

i (2,1)
R , ( p, q  0)  i R ( p// ) 
i
32 v
4 2
F
  dk dq {[ g
//
//
1R
 g 2 R  g 2 R  g1R  g1R  g 2 R
 g 2 R  g1R  2 g1R  g1R  2 g 2 R  g 2 R ] R (q// )  [ g1R  g1R  g 2 R  g 2 R  g1R  g1R  g 2 R  g 2 R ]

 R (q// )}ln    i R ( p// )
 

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

i(2,1)
(
p
,
p

k
,
p


,
q

0)


i

R,
//

F
0
R ( p// ;  )
Therefore, we have
 R ( p// ;  ) 
1
32 v
4 2
F
  dk dq {[ g
//
//
1R
 g 2 R  g 2 R  g1R  g1R  g 2 R  g 2 R  g1R
2 g1R  g1R  2 g 2 R  g 2 R ] R (q// ;  )  [ g1R  g1R  g 2 R  g 2 R  g1R  g1R  g 2 R  g 2 R ]

 R (q// ;  )}ln  
 

In this way, the bare and renormalized parameters are related by
 B ( p// )  Z 1 ( p// ;  )  R ( p// ;  )   R ( p// ;  ) 
Since the bare parameter (i.e. the quantity at the microscopic scale) does not know
anything about the scale , we have

d 
 B ( p// )  0
d
As a result, we obtain the RG equations

d 
d
 R ( p// ; )   ( p// ;  ) R ( p// ;  )  
 R ( p// ;  )
d
d
where  ( p// ; )  
d
ln Z ( p// ;  ) is the anomalous dimension of the theory and it is
d
given by
 ( p// ;  ) 
1
32 v
4 2
F
  dk dq [2 g
//
//
1R
 g1R  2 g 2 R  g1R  g1R  g 2 R  g 2 R  g1R ]
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)


Symmetrizing, we get for the charge response function   R,CS ( p)   R ( p)   R ( p)

d
1
 R ,CS ( p// ;  ) 
d
32 4 vF2
  dk dq {[ g
//
//
1R
 g 2 R  g 2 R  g1R  g1R  g 2 R  g 2 R  g1R
2 g1R  g1R  2 g 2 R  g 2 R  2 g1R  g1R  2 g 2 R  g 2 R  2 g1R  g1R  2 g 2 R  g 2 R ] R ,CS (q// ;  )
 ( p// ;  ) R ,CS ( p// ;  )
Similarly, we get for the uniform spin response function   R , SS ( p)   R ( p)   R ( p)

d
1
 R , SS ( p// ; ) 
d
32 4 vF2
  dk dq {[ g
//
//
1R
 g 2 R  g 2 R  g1R  g1R  g 2 R  g 2 R  g1R
2 g1R  g1R  2 g 2 R  g 2 R ] R , SS (q// ;  )   ( p// ;  ) R , SS ( p// ;  )
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.
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) 
R
 CS
( ) 
 SSR ( ) 
1
4 2vF
1
4 2vF






dp//  R ,CS ( p// ;  ) 
dp//  R , SS ( p// ;  ) 
2
2
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;
(H. Freire, E. Corrêa and A. Ferraz, condmat/0506682)
• Insulating Spin Liquid behavior.
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.