Insulating Spin Liquid in the 2D Insulating Spin Liquid in the 2D Lightly Doped Hubbard ModelLightly Doped Hubbard Model

International Centre of Condensed Matter Physics

University of Brasília, Brazil

Hermann FreireHermann Freire

Renormalization Group 2005Helsinki, Finland 30 August - 3 September 2005

Motivation Motivation The High-Tc Cuprates The High-Tc Cuprates

Parent Compound La2CuO4

Cu

O

La

• 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).

Effect of Doping Effect of Doping The Phase Diagram The Phase Diagram

• 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.

Hole Doped Compound La(2-x)SrxCuO4

At T=0 several ground states emerge as we vary “x”

Modeling the System Modeling the System The 2D Hubbard Model The 2D Hubbard Model

Electrons on a 2D square lattice

†

,ij i j i i i

i j i i

H t c c n U n n

,

0,

if and are n.n. sites

otherwiseij

t i jt

Hubbard Hamiltonian (U > 0)

The noninteracting Hamiltonian can be diagonalized

† †0 2 cos( ) cos( )k k k k k

k kx yH c c t k a k a c c

• 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?

What are the fundamental questions?What are the fundamental questions?

We will show our conclusions regarding these questions based on a complete Two-Loop Renormalization Group calculation within a field-theoretical framework.

The Noninteracting BandThe Noninteracting Band

The electron density can be adjusted by tuning the chemical potential

• = 0 (half-filled case)

Important features:

• Fermi surface perfectly nested;

• Density of states logarithmically divergent (van Hove singularities).

W

The bandwidth W = 8t

Starting Point Starting Point The Lightly Doped Scenario The Lightly Doped Scenario

e.g. = - 0.15 t (x = 0.09)

Umklapp Surface

Important features:

• Fermi surface approximately nested

for energies E > ||;

• Density of states is not divergent at the FS.

Removing electrons doping with holes

Adding the Hubbard Interaction TermAdding the Hubbard Interaction Term

In momentum space, the Hubbard interaction reads

† †int

, ,

( ) ( ) ( ) ( )1 2 3

1 2 3 3 2 1p p p

p +p -p p p psites

UH

N

Continuous 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.

The RG transformation must respect these symmetries.

Patching the FS Patching the FS The 2D g-ology notation 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 CaseThe 2D Hubbard Model Case

The full Lagrangian of the Hubbard model reads

†( ), ( ),

,

† †2 1 ( ), ( ), ( ), ( ),

, , , , ,

( ) ( , ) ( , )

1( , ) ( , ) ( , ) ( , )

t F F a aa

B B

L i v p k t t

t t t tV

1 2 3

B B

p,

B B B B1 2 3 3 2 1

p p p

p p

p +p -p p p pg g

Linearized energy dispersion

SU(2) invariant form

wheresites

VU

N

1B 2Bg g

Naive perturbation theory Lots of infrared (IR) divergent Feynman diagrams!!!

The model is defined at a scale of a few lattice spacings (microscopic scale) Bare (B) theory

Field Theory RG PhilosophyField Theory RG Philosophy

The microscopic Hubbard model (bare theory).

The floating scale at which the renormalized parameters are to be defined.

The infrared (IR) fixed point behavior.

Rewrite the bare theory in terms of renormalized parameters plus appropriate counterterms Reorganization of the perturbation series and cancellation of the infrared divergences.

ln

l

l

RG step

l

Renormalizing the Theory Towards the FSRenormalizing 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.

Hubbard Model

Local interaction

(g1B=g2B U)

Effective theory with nonlocal interaction

g1R=g1R(p1//,p2//,p3//)

g2R=g2R(p1//,p2//,p3//)

Quantum

Fluctuations

Schematically, we will obtain for instance

Microscopic Model Low-Energy Dynamics

Where should we look for divergences?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,q0) function Uniform response functions

(0,2)(q) function All kinds of susceptibilities

Renormalization of Renormalization of (4)(4) and and (2)(2) 1PI Vertices 1PI Vertices

Rewrite (‘renormalize’) the couplings and the fermionic fields

1/ 2( ), ( ), ( ), ( ),( , ) ( , ; ) ( , ; ) ( ; ) ( , ; )B R R Rp p p p pa a a at t t Z t

4

1/ 2// 1// 2 // 3// 1// 2 // 3//

1

( ; ) ( , , ; ) ( , , ; ) ( 1,2) iB i iR iBi

U Z p p p p p p p i

g g g

The renormalized Lagrangian (i.e free of divergences) now reads

†( ), ( ),

,

† †2 1 ( ), ( ), ( ), ( ),

, , , , ,

†2 1 ( ),

( ) ( ) ( , ) ( , )

1( , ) ( , ) ( , ) ( , )

1(

1 2 3

R R

p,

R R R R1 2 3 3 2 1

p p p

R1

p p p

p +p -p p p p

p +p

t F F a aa

R R

R R

L Z i v p k t t

t t t tV

V

g g

g g †( ), ( ), ( ),

, , , , ,

, ) ( , ) ( , ) ( , )1 2 3

R R R2 3 3 2 1

p p p

-p p p pt t t t

Counterterms

A Novel RG “Fixed Point” for Moderate U / WA Novel RG “Fixed Point” for Moderate U / W

What is the nature of this resulting state?

(H. Freire, E. Corrêa and A. Ferraz, Phys. Rev. B 71, 165113 (2005))

Results for a Discretized FS (4X33 points)

Uniform Response Functions Uniform Response Functions (2,1)(2,1)(p,q(p,q0) 0)

†( ) ( )

, ,

( ) ( , ') ( , ')p

p p pB Bext external B a a

a

L h t t

For the uniform susceptibilities, the infinitesimal field couples with both charge and spin number operators

Rewrite 1( ) ( ) ( ) ( )p p p pB R RZ

Symmetrization

• The Uniform Charge and Spin Functions

Charge (CS)

Spin (SS)

, ( ) ( ) ( )p p pR CS R R

, ( ) ( ) ( )p p pR SS R R

Counterterm

Earlier Methods Encountered in the LiteratureEarlier 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.

(2,1), // // 2 // // // // 1 // // // //2

//

( , 0) ( ) ( , , ; ) ( ) ( , , ; ) ( )4

( )

p q

R R R R R RF

R

ii i p dk p k k k p k k k

v

i p

g g

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

(2,1), // 0 //( , , , 0) ( ; )qR F Ri p p k p i p

// // 2 // // // // 1 // // // //2

1( ; ) ( , , ; ) ( ; ) ( , , ; ) ( ; )

4

R R R R RF

p dk p k k k p k k kv

g g

Not IR divergent

// // // 2 // // // // 1 // // // //2

1( ; ) ( ) ( , , ; ) ( ; ) ( , , ; ) ( ; )

4

R B R R R RF

p p dk p k k k p k k kv

g g

Symmetrizing, we get for the charge response function

, // , // // 1 // // // 2 // // // , //2

1( ; ) ( ) ( , , ; ) 2 ( , , ; ) ( ; )

4

R CS B CS R R R CSF

p p dk p k k p k k kv

g g

And, similarly for the uniform spin response function

, // , // // 1 // // // , //2

1( ; ) ( ) ( , , ; ) ( ; )

4

R SS B SS R R SSF

p p dk p k k kv

g

These equations are then calculated self-consistently.

• This is indeed a Random-Phase-Approximation (RPA);

• Not consistent with the RG philosophy.

, ( ) ( ) ( )p p pR CS R R

, ( ) ( ) ( )p p pR SS R R

Uniform Susceptibilities in this RPA ApproximationUniform 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)

2

// , //2

1( ) ( ; )

4RCS R CS

F

dp pv

2

// , //2

1( ) ( ; )

4RSS R SS

F

dp pv

Numerical ResultsNumerical 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 FunctionsFull 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.

The Feynman Diagrams up to Two-Loop OrderThe 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.

Calculating these Feynman diagrams, we get

(2,1), // // // 1 2 2 1 1 24 2

2 1 1 1 2 2 // 1 1 2 2 1 1 2 2

// //

( , 0) ( ) {[32

2 2 ] ( ) [ ]

( )}ln ( )

p qR R R R R R R RF

R R R R R R R R R R R R R R R

R R

ii i p dk dq g g g g g g

v

g g g g g g q g g g g g g g g

q i p

where the dots mean that we are omitting the parallel momenta dependence in the coupling functions.

(2,1), // 0 //( , , , 0) ( ; )qR F Ri p p k p i p

We now establish the following renormalization condition

IR divergent

Therefore, we have

// // // 1 2 2 1 1 2 2 14 2

1 1 2 2 // 1 1 2 2 1 1 2 2

//

1( ; ) {[

32

2 2 ] ( ; ) [ ]

( ; )}ln

R R R R R R R R RF

R R R R R R R R R R R R R

R

p dk dq g g g g g g g gv

g g g g q g g g g g g g g

q

In this way, the bare and renormalized parameters are related by

1// // // //( ) ( ; ) ( ; ) ( ; )B R Rp Z p p p

Since the bare parameter (i.e. the quantity at the microscopic scale) does not know anything about the scale , we have

//( ) 0B

dp

d

As a result, we obtain the RG equations

// // // //( ; ) ( ; ) ( ; ) ( ; )R R R

d dp p p p

d d

where is the anomalous dimension of the theory and it is

given by

// //( ; ) ln ( ; )d

p Z pd

// // // 1 1 2 1 1 2 2 14 2

1( ; ) [2 2 ]

32 R R R R R R R RF

p dk dq g g g g g g g gv

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)

Similarly, we get for the uniform spin response function , ( ) ( ) ( )p p pR SS R R

, // // // 1 2 2 1 1 2 2 14 2

1 1 2 2 , // // , //

1( ; ) {[

32

2 2 ] ( ; ) ( ; ) ( ; )

R SS R R R R R R R RF

R R R R R SS R SS

dp dk dq g g g g g g g g

d v

g g g g q p 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.

Symmetrizing, we get for the charge response function , ( ) ( ) ( )p p pR CS R R

, // // // 1 2 2 1 1 2 2 14 2

1 1 2 2 1 1 2 2 1 1 2 2 , //

// , //

1( ; ) {[

32

2 2 2 2 2 2 ] ( ; )

( ; ) ( ; )

R CS R R R R R R R RF

R R R R R R R R R R R R R CS

R CS

dp dk dq g g g g g g g g

d v

g g g g g g g g g g g g q

p p

The Uniform Susceptibilities up to Two-Loop OrderThe 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.

Therefore, the corresponding analytical expressions are also the same

Charge Compressibility (CS)

Uniform Spin Susceptibility (SS)

2

// , //2

1( ) ( ; )

4RCS R CS

F

dp pv

2

// , //2

1( ) ( ; )

4RSS R SS

F

dp pv

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!!!

The Insulating Spin Liquid StateThe Insulating Spin Liquid State

• 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.

Starting Point (bare theory) Metallic State

Initial DOS for both charge and spin finite

(H. Freire, E. Corrêa and A. Ferraz, cond-mat/0506682)

Conclusions and OutlookConclusions and Outlook

• 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;

Within a complete Two-Loop RG calculation, and taking into account strong quantum fluctuations, we find for a 2D lightly-doped Hubbard model that

These results may be of direct relevance for the understanding of the underlying mechanism of high-Tc superconductivity.