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

Time line

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

Figure courtesy of H. R. Ott

• 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.

Magnetic origin

• Pairing symmetry is d-wave.

Setting up the model

La2CuO4

La

Cu

O

2D copper oxide

1. Strong Coulomb repulsion: good insulator

2. Upon doping, high TC superconductor

2D copper oxide

weak interlayer coupling

Minimal Model

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

superconductivity upon doping: d-wave pairing

} Hubbard model

Heisenberg model (t-J model)

this talk

antiferromagnetism at half filling(half filling = one electron per site = zero doping)

Why antiferromagnetism?

Hubbard model

In the limit of large U, the Hubbard model at half filling reduces to the antiferromagnetic Heisenberg model.

0th order : Degenerate low-energy Hilbert space

2nd order : The Heisenberg model

1st order : High energy excitation by creating doubly occupied sites

Perturbative expansion of t/U M. Takahashi (77), C. Gros et al.(87), A.H. MacDonald et al.(88)

Derivation of the Heisenberg model

super-exchange

Minimal Model

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

superconductivity upon doping: d-wave pairing

antiferromagnetism at half filling

} Hubbard model

Heisenberg model (t-J model)

this talk

zS yx iii ˆ)1(0

S

Néel order

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 pairedat half filling, they will become superconducting whenmobile charge carriers (holes) are added.

Goal

: Gutzwiller projection (no double occupancy)

t-J model Gutzwiller-projectedBCS Hamiltonian

where

• Numerical evidence for a close connection between the t-J model and

the Gutzwiller-projected BCS Hamiltonian

• Analytic proof for the equivalence between the two Hamiltonians at half filling

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

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).

BCSGNRVB PP

• 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-projectedBCS Hamiltonian

Numerical evidence

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

• 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 tobe the ground state)

The significance of correlation function is ambiguous in finite-size systems unlessits long-distance limit is well-defined (we are interested in the long-range order).

The largest system accessible via exact diagonalization is very small in spatialdimension (4-6 lattice spacing), but has a huge Hilbert space (103-105 basis states).

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

)|2,( hhGBCS NNN

hNP hNP

: number projection operator

)|( h NNJt hNP : wavefunction overlap

• 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

Applying exact diagonalization to the BCS Hamiltonian is not straightforward. Why?

Undoped regime (half filling)

• 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.

in the 4×4 square lattice system with periodic boundary condition

Optimally doped regime2 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 regime4 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:

Q1. 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

Strong-pairing BCS Hamiltonian with finite on-site interaction U

Strong-pairing Gutzwiller-projected BCS Hamiltonian

The Hubbard model

U

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 .

Outline for the derivation1. 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.

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.]

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.

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:

20

2 )2/( kkE• Energy spectrum:

• Minimizing the ground state energy with respect to 0:

where and, .

U0 in the limit of large U

The ground state is completely separatedfrom other excitations of HBCS+U.

Step (2) for the derivation (continued)

where

• Ground state:

• The ground state of HBCS+U becomes identical to the ground state of HHub in the limit of infinite U.

and

Step (3) for the derivation

• Low-energy fluctuations come from HBCS+U and HHub.

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

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

Saddle-point equation for HHubSaddle-point equation for HBCS+U

Step (3) for the derivation (continued)

ky

kx kx

ky

Constant shift by (,0)

k k

The integral is identical if t= !

Saddle-point equation for HHubSaddle-point equation for HBCS+U

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

Hubbard model

Minimal model

Heisenberg model (the t-J model)

Perturbative expansion

Equivalence at half filling (strong-pairing limit)

Analytic derivation

Gutzwiller-projected BCS Hamiltonian

High overlaps at moderate doping

Exact diagonalization

Physical reason for the validity of the RVB state

BCSGNRVB PP

The RVB state can be viewed as a trial wave function for the Gutzwiller-projected BCS Hamiltonian

with the Jastrow-factor type correlation.

Jastrow factor concerning the short-range correlation

due to strong on-site repulsion

Quasi-particle wave functionconcerning the long-range correlationdue 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 tothe ground state of the projected BCS Hamiltonian, G

BCS .

• 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)