Download Quantum Antiferromagnetism and high TC Superconductivity

Quantum Antiferromagnetism and
High TC Superconductivity
A close connection between the t-J model and the
projected BCS Hamiltonian
Kwon Park
References
• K. Park, Phys. Rev. Lett. 95, 027001 (2005)
• K. Park, preprint, cond-mat/0508357 (2005)
High TC superconductivity
• The energy scale of TC is very suggestive
of a new pairing mechanism!
• In contrast to low TC superconductors
which are metallic, cuprates are insulators
at low doping.
Non-Fermi liquid behaviors :
pseudogap and stripes
• Superconductivity is destroyed when
even a small amount of Cu is replaced
by non-magnetic impurities such as Zn.
• Pairing symmetry is d-wave.
Magnetic origin
Time line
Figure courtesy of H. R. Ott
Setting up the model
2D copper oxide
La
weak interlayer coupling
O
Cu
La2CuO4
2D copper oxide
1. Strong Coulomb repulsion: good insulator
2. Upon doping, high TC superconductor
Minimal Model
• 2D square lattice system
• electron-electron interaction alone
• strong repulsive Coulomb interaction
}
antiferromagnetism at half filling
(half filling = one electron per site = zero doping)
Hubbard model
Heisenberg model
(t-J model)
superconductivity upon doping: d-wave pairing
this talk
Why antiferromagnetism?
Hubbard model
In the limit of large U, the Hubbard model at half filling
reduces to the antiferromagnetic Heisenberg model.
Perturbative expansion of t/U
M. Takahashi (77), C. Gros et al.(87), A.H. MacDonald et al.(88)
0th order :
Degenerate low-energy Hilbert space
1st order :
High energy excitation by creating doubly occupied sites
2nd order :
The Heisenberg model
Derivation of the Heisenberg model
super-exchange
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
Si  S0 (1)
ix i y
zˆ
superconductivity upon doping: d-wave pairing this talk
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)
Anderson’s conjecture (87): if electrons are already paired
at half filling, they will become superconducting when
mobile charge carriers (holes) are added.
Goal
t-J model
Gutzwiller-projected
BCS Hamiltonian
where
: Gutzwiller projection (no double occupancy)
• Numerical evidence for a close connection between the t-J model and
the Gutzwiller-projected BCS Hamiltonian
K. Park, Phys. Rev. Lett. 95, 027001 (2005)
• Analytic proof for the equivalence between the two Hamiltonians
at half filling
K. Park, preprint, submitted to PRL
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).
 RVB  PNPG BCS
• 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)
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.
t-J model
Gutzwiller-projected
BCS Hamiltonian
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
The largest system accessible via exact diagonalization is very small in spatial
dimension (4-6 lattice spacing), but has a huge Hilbert space (103-105 basis states).
The significance of correlation function is ambiguous in finite-size systems unless
its long-distance limit is well-defined (we are interested in the long-range order).
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)
• The main reason for unequivocal trust in the CF theory is the
amazing agreement between the CF wavefunction and the exact
ground state.
The overlap is practically unity for the Coulomb interaction in all
available finite-system studies (typically much higher than 99 ).
For example, Perspectives in Quantum Hall Effects, S. Das Sarma and A. Pinczuk
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
P N h : number projection operator
PN h
G
 t J ( Nh | N ) P N h  BCS
(Nh , Nh  2 | N )
: wavefunction overlap
Undoped regime (half filling)
in the 4×4 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.
Optimally doped regime
2 holes in the 4×4 square lattice system
• Two distinctive regions of high overlap:
 J/t  0.1 and /t < 0.1 : trivial equivalence
 J/t > 0.1 and /t > 0.1 (physically relevant parameter range) :
High overlaps in this region are adiabatically connected to
the unity overlap in the strong coupling limit.
• Superconductivity
}•
in the t-J model !
J
Overdoped regime
4 holes in the 4×4 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.
Analytic derivation of the equivalence
at half filling
• While the numerical evidence is quite convincing, questions
regarding the validity of finite-system studies linger:
Q
1. Is the overlap exactly equal to unity, or just very close to it ?
2. Is there a fundamental reason why the overlap is so good ?
A
The overlap is exactly equal to unity at half filling.
The antiferromagnetic Heisenberg model is equivalent to the strong-pairing
Gutzwiller-projected BCS Hamiltonian at half filling.
Analytic derivation of the equivalence

U
The Hubbard model

The Heisenberg model
Are these two Hamiltonians identical in the asymptotic limit of
large U ? Note that U= is trivial. We are interested in the limit U  .

U
Strong-pairing BCS Hamiltonian
with finite on-site interaction U
Strong-pairing Gutzwiller-projected
BCS Hamiltonian
Outline for the derivation
1. HBCS+U and HHub are separated into two parts: the saddle-point Hamiltonian,
HBCS+U and HHub, and the remaining Hamiltonian, HBCS+U and HHub, describing
quantum fluctuations over the saddle-point solution.
2. The ground states of HBCS+U and HHub become identical in the large-U limit. Let us
denote this state as gr. Excitation spectra of HBCS+U and HHub have an energy gap
proportional to U so that the low-energy Hilbert space is composed only of states
connected to gr via rigid spin rotation.
3. All matrix elements of HBCS+U 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.]
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.
• Re-write the on-site repulsion term:
• Decompose the spin operator into the stationary and fluctuation parts:
,where
Step (1) for the derivation (continued)
,where
• Similarly, one can decompose HHub into HHub and HHub.
Step (2) for the derivation
• Saddle-point Hamiltonian in momentum space:
where
• Energy spectrum:
,
and
Ek 
2k  (0 / 2) 2
• Minimizing the ground state energy with respect to 0:
0  U in the limit of large U
The ground state is completely separated
from other excitations of HBCS+U.
.
Step (2) for the derivation (continued)
• Ground state:
where
and
• The ground state of HBCS+U becomes identical to the ground state of HHub in the limit
of infinite U.
Step (3) for the derivation
• The true low-energy excitation must be massless, as required by Goldstone’s
theorem (the spin rotation symmetry is broken).
• Low-energy fluctuations come from HBCS+U and HHub.
 Eventually, it boils down to the question whether the two stationary spin
expectation values,  and , are the same.
Saddle-point equation for HBCS+U
Saddle-point equation for HHub
Step (3) for the derivation (continued)
Saddle-point equation for HBCS+U
ky
Saddle-point equation for HHub
ky
k
Constant shift
by (,0)
kx

The integral is identical if t= !
k
kx
Step (4) for the derivation
1.
The ground states of the two saddle-point Hamiltonians, HBCS+U 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.
2.
Fluctuation Hamiltonians, HBCS+U 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.]
Conclusion
Real copper oxides
Minimal model
Hubbard model
Perturbative expansion
Analytic derivation
Heisenberg model
(the t-J model)
Equivalence at half filling
(strong-pairing limit)
Exact diagonalization
High overlaps at moderate
doping
Gutzwiller-projected
BCS Hamiltonian
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.
 RVB  PNPG BCS
Jastrow factor
concerning the short-range correlation
due to strong on-site repulsion
Quasi-particle wave function
concerning the long-range correlation
due to the BCS Hamiltonian
(e.g.) (1) the Bijl-Jastrow wave function for liquid Helium
(2) the composite fermion wave function for the FQHE
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.
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.
Acknowledgements
• S. Das Sarma (University of Maryland)
• A. Chubukov
• V. Yakovenko
• V. W. Scarola
• J. K. Jain (Penn State University)
• S. Sachdev (Yale University)