Insulating Spin Liquid in the 2D Lightly Doped Hubbard Model

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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
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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
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Effect of Doping ? The Phase Diagram
Hole Doped Compound ? La(2-x)SrxCuO4
At T0 several ground states emerge as we vary x

• 0 lt x lt 0.02 ? AF Mott insulator
• 0.02 lt x lt 0.1 ? Pseudogap, spin glass, stripes,
  ISL... ???
• 0.1lt x lt 0.15 ? Superconductor with d-wave
  pairing.

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Modeling the System ? The 2D Hubbard Model
Electrons on a 2D square lattice
Hubbard Hamiltonian (U gt 0)
The noninteracting Hamiltonian can be diagonalized
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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.
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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
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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 gt ?
• Density of states is not divergent at the FS.

Umklapp Surface
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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.
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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.
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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!!!
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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.
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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
Effective theory with nonlocal interaction g1Rg1R
(p1//,p2//,p3//)
g2Rg2R(p1//,p2//,p3//)
Hubbard Model Local interaction (g1Bg2B ? U)
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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
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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
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A Novel RG Fixed Point for Moderate U / W
(H. Freire, E. Corrêa and A. Ferraz, Phys. Rev. B
71, 165113 (2005))
Results for a Discretized FS (4X33 points)
What is the nature of this resulting state?
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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 ?
Symmetrization ?
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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.

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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 Z1.
As a result
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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.

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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) ?
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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)!!!

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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.

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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.

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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
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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
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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 1230-1300)
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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.
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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) ?
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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/0506682)
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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.