Quantum Antiferromagnetism and High TC Superconductivity

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Quantum Antiferromagnetism and High TC
Superconductivity

• A close connection between the t-J model and the
  projected BCS Hamiltonian

Kwon Park
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References

• K. Park, Phys. Rev. Lett. 95, 027001 (2005)

• K. Park, preprint, cond-mat/0508357 (2005)

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High TC superconductivity

• The energy scale of TC is very suggestive
• of a new pairing mechanism!

Time line
Figure courtesy of H. R. Ott
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Setting up the model
2D copper oxide
La
O
Cu
2D copper oxide
La2CuO4
1. Strong Coulomb repulsion good insulator
2. Upon doping, high TC superconductor
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Minimal Model

• 2D square lattice system
• electron-electron interaction alone
• strong repulsive Coulomb interaction

Hubbard model
Heisenberg model (t-J model)
superconductivity upon doping d-wave pairing
this talk
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Why antiferromagnetism?
Hubbard model
In the limit of large U, the Hubbard model at
half filling reduces to the antiferromagnetic
Heisenberg model.
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Derivation of the Heisenberg model
super-exchange
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Minimal Model

• 2D square lattice system
• electron-electron interaction alone
• strong repulsive Coulomb interaction

Hubbard model
Heisenberg model (t-J model)
antiferromagnetism at half filling
Néel order
superconductivity upon doping d-wave pairing
this talk
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Why superconductivity (pairing)?
Both the pairing Hamiltonian and the
antiferromagnetic Heisenberg model prefer the
formation of singlet pairs of electrons in the
nearest neighboring sites.
antiferromagnetism
pairing (BCS Hamiltonian)
Andersons conjecture (87) if electrons are
already paired at half filling, they will become
superconducting when mobile charge carriers
(holes) are added.
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Goal
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A short historic overview of ansatz wavefunction
approaches

• Anderson proposed an ansatz wavefunction for
  antiferromagnetic
• models the Gutzwiller-projected BCS
  wavefunction, i.e., the RVB
• state (1987).

• It was realized that the RVB state could not
• be the ground state of the Heisenberg model
• on square lattice because it did not have
• Néel order (long-range antiferromagnetic
  order).

• Is it a good ansatz function for the ground
  state at non-zero doping?

C. Gros (88), Y. Hasegawa et al.(89), E. Dagotto
(94), A. Paramekanti et al. (01), S. Sorella et
al. (02)
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A new approach

• We study the Gutzwiller-projected BCS
  Hamiltonian instead of the Gutzwiller-projected
  BCS state.
• The ground state of the Gutzwiller-projected BCS
  Hamiltonian is different from the
  Gutzwiller-projected BCS state the former has
  Néel order at half filling, while the latter does
  not.

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Numerical evidence

• Exact diagonalization (via modified Lanczos
  method) of
• finite-size systems an unbiased study

It is compared with uncontrolled analytic
approximations (such as large-N expansion) and
variational Monte Carlo simulations (which assume
trial wavefunctions to be the ground state)

• Wavefunction overlap between the ground states
  of the t-J model
• and the Gutzwiller-projected BCS Hamiltonian an
  unambiguous study

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Digression to the FQHE

• The fractional quantum Hall effect (FQHE) is a
  prime example of
• highly successful ansatz wavefunction approach
  the Laughlin
• wavefunction the composite fermion (CF)
  theory, in general.

R. B. Laughlin (83), J. K. Jain (89)
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A new numerical technique
Applying exact diagonalization to the BCS
Hamiltonian is not straightforward.
Why?

• Particle-number fluctuations are coherent in
  the BCS theory, which is essential
• for superconductivity.

• How do we deal with number fluctuations in
  finite systems?
• ? combining the Hilbert spaces with different
  particle numbers
• ? adjusting the chemical potential to eliminate
  spurious finite-size effects

wavefunction overlap
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Undoped regime (half filling)
in the 44 square lattice system with periodic
boundary condition

• The overlap approaches unity in the limit of
  strong pairing, i.e., ?/t??.

• It can be shown analytically that the overlap is
  actually unity in the strong-pairing
• limit the Heisenberg model is identical to the
  strong-pairing Gutzwiller-projected
• BCS Hamiltonian.

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Optimally doped regime
2 holes in the 44 square lattice system

• Two distinctive regions of high overlap

? J/t ? 0.1 and ?/t lt 0.1 trivial equivalence

• J/t gt 0.1 and ?/t gt 0.1 (physically relevant
  parameter range)
• High overlaps in this region are
  adiabatically connected to
• the unity overlap in the strong coupling
  limit.

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Overdoped regime
4 holes in the 44 square lattice system

• For general parameter range, the overlap is
  negligibly small.
• In the overdoped regime, the ground state of the
  projected BCS Hamiltonian is no longer a good
  representation of the ground state of the t-J
  model.

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Analytic derivation of the equivalence at half
filling

• While the numerical evidence is quite
  convincing, questions regarding the validity of
  finite-system studies linger

The antiferromagnetic Heisenberg model is
equivalent to the strong-pairing
Gutzwiller-projected BCS Hamiltonian at half
filling.
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Analytic derivation of the equivalence
Note that U? is trivial. We are interested in
the limit U? ? .
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Outline for the derivation
1. HBCSU and HHub are separated into two parts
the saddle-point Hamiltonian, HBCSU and HHub,
and the remaining Hamiltonian, ?HBCSU and
?HHub, describing quantum fluctuations over the
saddle-point solution.
3. All matrix elements of ?HBCSU and ?Hhub,
are precisely the same in the low-energy
Hilbert space with the same being true for those
of the saddle-point Hamiltonians.
4. Since the fluctuation as well as the
saddle-point solution is identical in the limit
of large U, the strong-pairing
Gutzwiller-projected BCS Hamiltonian and the
antiferromagnetic Heisenberg model have the
identical low-energy physics. Q.E.D.
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Step (1) for the derivation

• Effect of finite t the nesting property of the
  Fermi surface induces Néel order in
• the ground state of the Hubbard model at half
  filling.
• Effect of finite ? the strong-pairing BCS
  Hamiltonian with d-wave pairing
• symmetry also has a precisely analogous nesting
  property in the gap function.

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Step (1) for the derivation (continued)

• Similarly, one can decompose HHub into HHub and
  ?HHub.

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Step (2) for the derivation

• Saddle-point Hamiltonian in momentum space

where
and
,
.

• Energy spectrum

• Minimizing the ground state energy with respect
  to ?0

The ground state is completely separated from
other excitations of HBCSU.
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Step (2) for the derivation (continued)

• Ground state

where
and
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Step (3) for the derivation

• The true low-energy excitation must be massless,
  as required by Goldstones
• theorem (the spin rotation symmetry is broken).

• Low-energy fluctuations come from ?HBCSU and
  ?HHub.

• Eventually, it boils down to the question
  whether the two stationary spin
• expectation values, ? and ?, are the same.

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Step (3) for the derivation (continued)
ky
?k
?k
Constant shift by (?,0)
?
The integral is identical if t? !
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Step (4) for the derivation

• The ground states of the two saddle-point
  Hamiltonians, HBCSU and HHub, are
• identical in the limit of large U. The
  low-energy Hilbert space, which is
• composed of states connected to the
  saddle-point ground state via rigid spin
• rotations, is also identical.
• Fluctuation Hamiltonians, ?HBCSU and ?HHub,
  have identical matrix
• elements in the low-energy Hilbert space
  with the same being true for
• the saddle-point Hamiltonians.

• The antiferromagnetic Heisenberg model is
  equivalent to the strong-pairing
• Gutzwiller-projected BCS Hamiltonian.
  Q.E.D.

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Conclusion
Real copper oxides
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Physical reason for the validity of the RVB state
The RVB state can be viewed as a trial wave
function for the Gutzwiller-projected BCS
Hamiltonian with the Jastrow-factor type
correlation.
(e.g.) (1) the Bijl-Jastrow wave function for
liquid Helium (2) the composite fermion
wave function for the FQHE
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Connection between ?RVB and ?GBCS
The projected BCS wave function, ?RVB , is a good
approximation to the ground state of the
projected BCS Hamiltonian, ?GBCS .

• Hasegawa and Poilblanc (89) have shown that the
  RVB state has a good overlap ( 90)
• with the exact ground state of the t-J model for
  the case of 2 holes in the 10-site lattice
• system (i.e., for a moderately doped regime).

• The ground state of the projected BCS
  Hamiltonian is
• also very close to the exact ground state of the
  t-J model
• the optimal value of the overlap is roughly 98.

In other words, for a moderately doped regime,
the ground state of the t-J model, that of the
projected BCS Hamiltonian, and the RVB state are
very similar to each other.
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Future work

• Now, there is a reason to believe that the
  Gutzwiller-projected BCS
• Hamiltonian is closely connected to high TC
  superconductivity.
• So, it will be very interesting to investigate
  whether one can get
• quantitative agreements with experiment.

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Acknowledgements

• S. Das Sarma (University of Maryland)
• A. Chubukov
• V. Yakovenko
• V. W. Scarola
• J. K. Jain (Penn State University)
• S. Sachdev (Yale University)