1. Quantum Antiferromagnetism and High T C Superconductivity A close connection between the t-J model and the projected BCS Hamiltonian Kwon Park

2. References K. Park, Phys. Rev. Lett. 95, 027001 (2005) K. Park, preprint, cond-mat/0508357 (2005)

3. High T C superconductivity Time line The energy scale of T C is very suggestive of a new pairing mechanism! Figure courtesy of H. R. Ott In contrast to low T C 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.

4. Setting up the model La 2 CuO 4 La Cu O 2D copper oxide 1. Strong Coulomb repulsion: good insulator 2. Upon doping, high T C superconductor 2D copper oxide weak interlayer coupling

5. 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)

6. Why antiferromagnetism? Hubbard model In the limit of large U, the Hubbard model at half filling reduces to the antiferromagnetic Heisenberg model.  0 th order : Degenerate low-energy Hilbert space  2 nd order : The Heisenberg model  1 st 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)

7. Derivation of the Heisenberg model super-exchange

8. 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 Néel order

9. 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. antiferromagnetismpairing (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.

10. Goal : Gutzwiller projection (no double occupancy) t-J model Gutzwiller-projected BCS 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

11. 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)

12. 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

13. 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 to be the ground state) 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). The largest system accessible via exact diagonalization is very small in spatial dimension (4-6 lattice spacing), but has a huge Hilbert space (10 3 -10 5 basis states).

14. 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

15. A new numerical technique : number projection operator : 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?

16. 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

17. 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 }

18. 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.

19. 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.

20. 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  .

21. Outline for the derivation 1. H BCS+U and H Hub are separated into two parts: the saddle-point Hamiltonian, H BCS+U and H Hub, and the remaining Hamiltonian,  H BCS+U and  H Hub, describing quantum fluctuations over the saddle-point solution. 3. All matrix elements of  H BCS+U and  H hub, 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 H BCS+U and H Hub become identical in the large-U limit. Let us denote this state as  gr. Excitation spectra of H BCS+U and H Hub 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.

22. 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

23. Step (1) for the derivation (continued),where Similarly, one can decompose H Hub into H Hub and  H Hub.

24. Step (2) for the derivation Saddle-point Hamiltonian in momentum space: Energy spectrum: Minimizing the ground state energy with respect to  0 : whereand,. in the limit of large U The ground state is completely separated from other excitations of H BCS+U.

25. Step (2) for the derivation (continued) where Ground state: The ground state of H BCS+U becomes identical to the ground state of H Hub in the limit of infinite U. and

26. Step (3) for the derivation Low-energy fluctuations come from  H BCS+U and  H Hub. 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 H Hub Saddle-point equation for H BCS+U

27. Step (3) for the derivation (continued) kyky kxkx kxkx kyky  Constant shift by ( ,0) kk kk The integral is identical if t=  ! Saddle-point equation for H Hub Saddle-point equation for H BCS+U

28. Step (4) for the derivation 1.The ground states of the two saddle-point Hamiltonians, H BCS+U and H Hub, 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,  H BCS+U and  H Hub, 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.]

29. 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

30. 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. 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

31. Connection between  RVB and  G BCS The projected BCS wave function,  RVB, is a good approximation to the 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.

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

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