Quantum Antiferromagnetism and High T C Superconductivity

1. Quantum Antiferromagnetism and High TC Superconductivity A close connection between the t-Jmodel 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. 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 High TC superconductivity • The energy scale of TC is very suggestive • of a new pairing mechanism! Time line Figure courtesy of H. R. Ott

4. weak interlayer coupling 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

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

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

7. Derivation of the Heisenberg model super-exchange

8. 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

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. 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.

10. Gutzwiller-projected BCS Hamiltonian t-J model 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 Goal

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. Gutzwiller-projected BCS Hamiltonian t-J model 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.

13. 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). 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

14. 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 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)

15. : number projection operator A new numerical technique Applyingexact 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

16. 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.

17. Superconductivity • in the t-J model ! } •   J 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.

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. Q • 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. 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.

20. The Hubbard model The Heisenberg model Are these two Hamiltonians identical in the asymptotic limit of large U ?   U U Strong-pairing BCS Hamiltonian with finite on-site interaction U Strong-pairing Gutzwiller-projected BCS Hamiltonian Analytic derivation of the equivalence Note that U= is trivial. We are interested in the limit U  .

21. 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. 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. 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.]

22. Re-write the on-site repulsion term: • Decompose the spin operator into the stationary and fluctuation parts: ,where 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.

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

24. in the limit of large U 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 HBCS+U.

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

26. Saddle-point equation for HBCS+U Saddle-point equation for HHub 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.

27. ky Saddle-point equation for HBCS+U Saddle-point equation for HHub kx kx Step (3) for the derivation (continued) ky k k Constant shift by (,0)  The integral is identical if t= !

28. Step (4) for the derivation • 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. • 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.]

29. Minimal model Hubbard model Perturbative expansion Heisenberg model (the t-J model) Analytic derivation Exact diagonalization Equivalence at half filling (strong-pairing limit) High overlaps at moderate doping Gutzwiller-projected BCS Hamiltonian Conclusion Real copper oxides

30. Quasi-particle wave function concerning the long-range correlation due to the BCS Hamiltonian Jastrow factor concerning the short-range correlation due to strong on-site repulsion 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

31. 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-Jmodel 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 TC 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)