1. Self-consistent Calculations for Inhomogeneous Correlated Systems Amit Ghosal IISER, Kolkata HRI, 12 Nov, 2010

2. Mean-Field decoupling of four-fermionic interaction term Diagonalize full H using guess for these expectation values, calculate them in terms of the eigenfunctions (and eigenvalues) of H. Modify guess and keep iterating until guess and calculated values are equal. Then calculate expectation values self-consistently: Works well for disordered/inhomogeneous situations (disorder breaks all symmetries)‏ Scheme for self-consistent calculation for interacting system : Justification comes in comparison with other methods, e.g. QMC, ED etc. See, for example, Chen et al. J. Phys.: Cond. Matter, 20, 345211

3. Outline: Results from Anderson-Hubbard model → metallic phase in a 2D interacting disordered system? Results from Attractive Hubbard model with disorder → Superconductor-Insulator transition driven by impurities Structure of a d-wave vortex lattice → Charged vortices? Outlook and Conclusions

4. Part I: Anderson-Hubbard model in 2D

5. Anderson-Hubbard Model MF decoupling: with: Effective Hamiltonian contains expectation values of operators with respect to its own eigenstates. → Must be calculated in a self-consistent manner For a system of size N = L X L, there are (3N+1) SC parameters: 2N values of for σ = ↑ and ↓ iσ N values of h One value of μ to fix =1 (half-filling)‏ i (Heiderian & Trivedi, PRL, 2004) Model parameters: U = 4t, N = 28 X 28, =1 P(V )‏ -VV V i i

6. Motivation: 2D Metal-Insulator Transition (2D MIT): real metals, 2D or not 2D??  For dimension d ≤ 2, ALL non-interacting (single particle) states are localized for arbitrarily small disorder!! Scaling theory of localization (1979)‏ Kravchenko, Sarachik, Rep. Prog. Phys. 67, 1 (2004)‏ What’s “NEW” ??  strong correlation  disorder? Si-MOSFET

7.  Clean system (V=0) @ =1 is a Mott insulator (U >> t)‏ Hallmark of Mott insulator: (a) Mott gap (b) AFM spin correlation U = 4t, N = 24 X 24, =1 Anderson-Hubbard Model (contd.)‏

8. How do Mott-gap and AFM order behave as a function of V? Energy scale for charge fluctuation (gap) U Energy scale for AFM coupling t²/U Expect AFM order to vanish with V before the gap does! Surprise!! Insight comes from spatial correlations...

9. Look at the spatial structure / distributions: Staggered magnetization Correlation of AFM sites (m > 0.3)‏ paramagnetic (PM) sites (m < 0.1)‏ with site disorder AFM in WD regions, singly occupied sites and large Mott gap SD regions have low lying excitations, 0/2 occupations, AFM vanishes locally. SD regions grow with V V=3 t

10. Why does AFM order survives up to large V? Percolation-based model is at work for U >> t sites with V U/2 → empty sites with |V| < U/2 are singly occupied, and have free spins with t → 0 Turning on t up to 2 nd order leads to AFM order by standard mechanism of exchange (J modified by V)‏ → Fraction of singly occupied (magnetic) sites x = U/2V AFM vanishes at the critical V when doubly and unoccupied sites percolate! V = 3.4t consistent with classical percolation of vacancies on a 2D square lattice C2 Condition for AFLRO: x > x 0.59 (percolation threshold)‏ c

11. Localization properties of the Wave-function at ɛ F Metallic phase in 2D in Anderson-Hubbard model at intermediate disorder!! What is the phase between V and V ? C1C2

12. Screening of strong disorder by interaction: Why metal in 2D for effective non-interacting disorderd system? PM sites (SD region) suffer significant screening Screening of AFM sites (WD region) negligible  Screened potential correlated  Inhomogeneous magnetic field (h)‏ correlated with disorder Herbut, PRB, 63, 113102 Note: direct evidence for a metallic phase from conductivity calculation Kobayashi et al. arXiv:08073372

13. Proposed Phase Diagram: 0 < V < U/2 → Mott Insulator (brown)‏ U/2 < V < 5U/6 → Insulator A (pink)‏ V > 5U/6 → Insulator B (blue)‏ metal in between (gray)‏

14. Part II: Disordered s-wave superconductors

15. Dirorder driven SIT: Motivation (1) Cooper Pairing: (2) Phase Coherence: Pair size ξ phase Pairing amplitude externally applied phase twist Φ Clean system What happens when we add another axis -- disorder? clear separation of SC and I phase for T → 0 V Haviland et al. PRL, 62, 2180

16. Model and Method: Attractive Hubbard model with impurities minimal model to study the interplay of SC and disorder Mean-Field decomposition in HF-Bogoliubov channel attractiondisorder Self-consistently determine: Local pairing amplitude Local density μ that fixes average density

17. Standard way to diagonalize effective H using Bogoliubov-de Gennes transformation: with: → so that: Recalculate local pairing amplitude and local density using u's and v's (T = 0)‏

18. Distribution of pairing amplitude: ξ(V = 0) 10a V 1.75 t C Δ(V=0)‏ How is inhomogeneous Δ distributed spatially? AG, Randeria & Trivedi, PRL, 81, 3940; PRB, 65, 14501

19. Spatial profile of Δ: SC “islands” (of size ξ)‏

20. Where in space do SC-islands form? SC-islands support low-lying excitations! “Islands” form where |V – μ| 0 μ Δ 0 Δ finite

21. Evolution of Energy Gap and superfluid stiffness with V Gap robust to V Stiffness decreases obtained from Kubo Formula

22. Once the SC-islands are formed, system susceptible to phase fluctuations Quantum Phase Fluctuations: → Renormalization of D due to quantum phase fluctuations S In clean system any applied phase twist distributed uniformly Inhomogeneous system gain energy by distributing phase twist non-uniformly Most of the twist lives in the sea, keeping the phase on SC-islands almost uniform

23. Evolution of D renormalized by phase fluctuations: S Good agreement with QMC QMC by Trivedi et al, PRB, 54, R3756

24. Proposed phase diagram on U-V plane based on BdG results: Of the 3 possibilities for U → 0, (a) represents the correct result

25. Part II: d-wave vortex lattice with competing AFM order

26. d-wave vortex lattice: Model for a d-wave SC (t-J type)‏ Orbital magnetic field introduced through the Peierl's factor: → results into Abrikosov vortex lattice Study the interplay of d-SC and AFM at the vortex cores

27. Mean-Field decoupling: Self-consistent variables Working parameters: J = 1.15 t, = 0.875 Minimum energy configuration has m = 0 at all sites, in the absence of orbital field. Allows AFM + d-SC ordering In HTSC cuprates, AFM stabilized at = 1, and suppressed quickly away from half-filling. AG, Kallin & Berlinsky, PRB, 66, 214502 where,

28. Spatial distributions on a cell containing one vortex: d-SC pairing amplitude AFM order parameter Charge density AFM develops only @ vortex cores, where d-SC vanishes due to orbital field In the absence of AFM, spatial charge density structureless When AFM is allowed to develop self-consistently, reorganizes itself to accommodate AFM. → 1 locally near core, a filling favorable for AFM order to stabilize! Coulomb repulsion, at the HF level, does not wash out such charge accumulation in AFM vortex core Knapp et al. PRB,71, 64504 AG, Kallin & Berlinsky, PRB, 66, 214502

29. Other self-consistent calculations oninhomogeneous correlated system: (1) Superfluid-Bose glass transition in disordered Bose-Hubbard model Sheshadri et al. PRL, 75, 4075 (2) Electrodynamics of s-wave vortex lattice Atkinson & MacDonald, PRB, 60, 9295 (3) Impurity effects on d-wave superconductors AG, Randeria & Trivedi, PRB, 63, R20505; Hirschfeld Group (4) Metal Insulator transition in 3D Anderson-Hubbard model Chen & Gooding, PRB, 80, 115125 And many others...

30. Self-consistent calculations are relatively simple, and are capable of exploring interesting physics in inhomogeneous correlated systems Conclusions:

31. Quantum Phase Fluctuations: Phenomenology: Quantum XY-Model Variational Method: self-consistent harmonic approximation Wood & Stroud, PRB, 25, 1600 Estimate D by finding out the best harmonic H that describes H S θ0 → Renormalization of D due to quantum phase fluctuations S Note: disorder enters only through TVR, Phys. Scripta, T27, 24