Title: Quantum Antiferromagnetism and High TC Superconductivity
1Quantum Antiferromagnetism and High TC
Superconductivity
- A close connection between the t-J model and the
projected BCS Hamiltonian
Kwon Park
2References
- K. Park, Phys. Rev. Lett. 95, 027001 (2005)
- K. Park, preprint, cond-mat/0508357 (2005)
3High TC superconductivity
- The energy scale of TC is very suggestive
- of a new pairing mechanism!
Time line
Figure courtesy of H. R. Ott
4Setting 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
5Minimal 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
6Why antiferromagnetism?
Hubbard model
In the limit of large U, the Hubbard model at
half filling reduces to the antiferromagnetic
Heisenberg model.
7Derivation of the Heisenberg model
super-exchange
8Minimal 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
9Why 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.
10Goal
11A 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)
12A 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.
13Numerical 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
14Digression 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)
15A 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
16Undoped 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.
17Optimally 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.
18Overdoped 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.
19Analytic 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.
20Analytic derivation of the equivalence
Note that U? is trivial. We are interested in
the limit U? ? .
21Outline 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.
22Step (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.
23Step (1) for the derivation (continued)
- Similarly, one can decompose HHub into HHub and
?HHub.
24Step (2) for the derivation
- Saddle-point Hamiltonian in momentum space
where
and
,
.
- Minimizing the ground state energy with respect
to ?0
The ground state is completely separated from
other excitations of HBCSU.
25Step (2) for the derivation (continued)
where
and
26Step (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.
27Step (3) for the derivation (continued)
ky
?k
?k
Constant shift by (?,0)
?
The integral is identical if t? !
28Step (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.
29Conclusion
Real copper oxides
30Physical 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
31Connection 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.
32Future 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.
33Acknowledgements
- S. Das Sarma (University of Maryland)
- A. Chubukov
- V. Yakovenko
- V. W. Scarola
- J. K. Jain (Penn State University)
- S. Sachdev (Yale University)