1. Mott –Hubbard Transition & Thermodynamic Properties in Nanoscale Clusters. Armen Kocharian (California State University, Northridge, CA) Gayanath Fernando (University of Connecticut, Storrs, CT) Jim Davenport (Brookhaven National Laboratories, Upton, NY) Kalum Palandage (University of Connecticut, Storrs, CT)

2. Outline Motivation Small Hubbard clusters (2-site, 4-site) Ground state properties Exact Thermodynamics –Charge dos and Mott-Hubbard crossover –Spin dos and AF Nee l crossover –Phase diagrams QMC calculations in small clusters Conclusions

3. Quantum Monte Carlo Exact analytical results and QMC h=0

4. Motivation Electron Correlations - Large Thermodynamic System: Interplay between charge and spin degrees Mott-Hubbard Transition AFM-PM (Nee l Transition) Magnetic and transport properties -Nanoscale Clusters: Mott-Hubbard crossover? Charge and spin degrees? AFM-PM crossover?

5. Finite size Hubbard model Simplest lattice model to include correlations :  Tight binding with one orbital per site  Repulsion: on-site only  Nearest neighbor hopping only  Magnetic field Bethe ansatz solution [ Lieb & Wu. (’67) ]  Ground state but not correlation functions

6. Finite size Hubbard cluster Lieb & Wu. (’67) Thermodynamics (T≠0) Ground state (T=0) Weak correlations in 1d systems :  power law decay (Schulz ’91, Korepin & Frahm ’90) Long range order in finite clusters :  saturated ferromagnetism (Nagaoka’65) Signature of short range correlations :  weak magnetization (Aizenman & Lieb’90)  correlations decay faster than power law like (Koma & Tasaki ’92) No long range correlations :  no magnetic order in 1d (Mermin & Wagner. ’66, Ghosh ’71)

7. Large Clusters : Bethe-ansatz calculations Lanczos Monte Carlo Numerical diagonalization DMFT Small Clusters : Exact analytical diagonalization Charge and spin gaps (T=0) Pseudogaps (T≠0) Lieb & Wu. (’67) Dagotto et al. (’ 84) Canio et al. (’96) Jarrell et al. (’70) Kotliar. et al. (’97)

8. Mott-Hubbard transition: Temperature Magnetic field AF-PM Transition: Exchange Susceptibility HTSC superconductivity: Pseudogap formation Chemical potential (n≠1) Kotliar (’67) Schrieffer et al. (’ 90) Canio et al. (’96)

9. Neel Magnetic Phase T N Mott-Hubbard Phase T MH Two phase transitions in Hubbard Model From D. Mattis et al. (’69) M. Cyrot et al. (’70) J. R. Schrieffer et al. (’70)

10. Approaching to T MH from metallic state: U↑, T↓ T N consequence of Mott- Hubbard phase Mott Hubbard and AF transitions Brinkman et al. (’70) Anderson (’97) Slater (’51) T MH consequence of Neel anti-ferromagnetism Approaching to MH phase from insulator: T↑,U↓ Hubbard (’64) Evolution of dos and pseudogaps, T MH and T N for 2 and 4 site clusters at arbitrary U, T and h

11. Thermodynamics of small clusters From Shiba et al., (‘72) Specific heat of finite chains N=2, 3, 4, 5 Low temperature peak – AFM-PM High temperature peak – MH transition

12. Focus on 2 and 4-site clusters Mott-Hubbard Transition AFM-PM Transition Driven by h and T A single hydrogen molecule acting as a nanowire Shumann (’02)Shiba et al. (’70) Harris et al. (’72)Kocharian et al. (’ 96)

13. Exact ground state properties Exact mapping of 2-site Hubbard and Heisenberg ground states at half filling (A. Kocharian et al. ’91, 96): e.g., h C =J(U) h C - critical field of ferromagnetic saturation

14. Ground state charge gap (N=2) e.g., h<h C e.g., h≥h C Half filling  Gap is monotonic versus U and non monotonic versus h

15. Ground state charge gap (N=1, 3) Quarter and three quarter fillings  e.g., h<h C e.g., h≥h C Charge gap versus h and U is monotonic everywhere

16. Exact thermodynamics (T≠0) Number of particles N at h=0 versus µ and T Sharp step like behavior only in the limit T  0 h=0 2 sites: n  2 4 4 sites: n  4 4

17. h=0 N versus chemical potential (T/t=0.01) Real plateaus exist only T=0 (not shown) h=0

18. Chemical potential in magnetic field Number of particles N at h/t=2 versus µ and T Sharp step like behavior only in limit T  0 h/t=2.0, U/t=5.0 Plateaus at N=1 and N=3 increases with h Plateau at N=2 decreases with h

19. Magnetic susceptibility χ at half filling Susceptibility versus h at T=.05  As temperature T  0 peaks of χ closely tracks U dependence of h C (U) h c (U)/t U/t 4

20. Number of electrons vs. μ clusters h=0 Plateaus at integer N exist only at T=0 (not shown in figure)

21. Charge pseudogap at infinitesimal T≠0 h=0

22. Charge and spin dos in 2-site cluster Charge dos for general N has four peaks Spin dos at half filling has two peaks U=6 and h=2

23. Thermodynamic charge dos and pseudogap Charge dos for general U≠0 has four peaks U=0 and h=0 Charge pseudogap disappears at T MH Two peaks merge in one peak saddle point Saddle point U=5 and h=0 T MH

24. Charge dos and pseudogap Charge dos for general N has four peaks Charge dos for general N has four peaks σ h=0 h=2t

25. Spin dos and pseudogap Spin dos at half filling has two peaks U=6 spin pseudogap at T N disappears (saddle point) Saddle point

26. Thermodynamic charge and spin dos Charge dos for general N has four peaks Spin dos at half filling has two peaks σ

27. Weak singularity in charge dos Infinitesimal temperature smears ρ(μ C )≠0 and results in pseudo gap At T MH, ρ(μ C )≠0 and ρ′(μ C )=0 ρ″(μ C )>0. It is a saddle point MH Transition at half-filling (N=2) True gap at μ C =U/2 exists only at T=0 n 1n 1 Forth order MH phase transition

28. Weak singularity in spin dos Infinitesimal temperature smears σ(0)≠0 at h=0 and results in pseudo gap At T N, σ(0)≠0 and σ′(0)=0 σ ″(0)>0.. It is a saddle point Neel Transition at N=2 True gap exists only at T=0 n 1n 1 Forth order Nee l phase transition

29. Weak singularity in charge dos Distance between charge peak positions versus temperature N=2 T MH versus μ MH crossover Bifurcations at μ=U/2 & μ≠U/2

30. Weak singularity in spin dos Distance between spin peak positions versus temperature N=2 T N versus h crossover

31. Spin magnetization Magnetization at quarter filling (no spin gap) Magnetization at half filling (spin gap)

32. Magnetization versus h h=0 No spin gap at N=1 and 3

33. Zero field spin susceptibility (N=2) T N from maximum susceptibility T N from peaks distance

34. T N temperature versus U T N versus U (AF gap) T N from maximum of spin susceptibility T N from spin dos peaks T F versus U (Ferro gap)

35. Zero field magnetic susceptibility χ h=0 At large U magnetic susceptibility ~T At U/t»1 χ increases linearly N=2

36. Spin susceptibility Susceptibility at quarter filling (no spin gap) Susceptibility at half filling (U C /t=6) h/t=2 N=1N=2

37. Phase diagram, T MH versus U At t=0 T MF MH =U/2 result at t=0. D.Mattis’69 At t=0 T MH = U/2ln2 and ρ(µ C )=2ln2/5U h=0

38. Phase diagram, T MH versus U h=0 Staggered magnetization  i (-1) x+y+z (spin at site i) T MH

39. Phase Diagram at half filling T MH & T N versus U, at which pseudogap disappears T/t U/t 4 MH + AF MH N T MH TNTN

40. Mott-Hubbard crossover T MH versus h and U σ N=2

41. 4-site clusters h=0

42. 4-site clusters

43. h=0

44. Plateaus at integer N exist only at T=0 (not shown in figure) N versus μ in 4 site cluster No gap at U=0 and N=4 T/t=0.01 U/t=4.0

45. h=0 Bifurcations at N = 2, 4 and 6 Weak singularity in charge dos

46. h=0 Bifurcations at N = 1, 2, 4, 6 and 7 Weak singularity in charge dos

47. Thermodynamic dos for 4-site cluster Analytical calculations h=0 DMFT calculations

48. Quantum Monte Carlo Exact analytical results and QMC h=0

49. Magnetism and MH crossovers in rings and pyramids h=0 Quantum Monte Carlo studies

50. QMC studies of small clusters h=0 Staggered magnetization  i (-1) x+y+z (spin at site i)

51. 5 sites pyramids h=0 Magnetization versus h

52. 5 sites pyramids h=0 Staggered magnetization versus n

53. 14 sites pyramids h=0 Staggered magnetization  i (-1) x+y+z (spin at site i)

54. 14 sites pyramids h=0 Staggered magnetization  i (-1) x+y+z (spin at site i)

55. Conclusions Exact mapping H ex ~H U in the ground state True spin and charge gaps exist only at T=0 E C Gap (U)≠E S Gap (U ) at U≠0 Pseudogaps appear at infinitesimal T Charge dos - MH crossover (T MH >T N ) Spin dos - AFM-PM crossover (T N ) Temperature driven bifurcation – generic feature 1d Hubbard model, U C =0 and true gap in ρ(μ C )=0 exists only at T=0 and n=1 2 and 4 site Hubbard clusters reproduces main features of small and large system Evolution of pseudogap versus μ in HTSC

56. Rigid spin dynamics h=0

57. QMC studies of small clusters h=0 Staggered magnetization  i (-1) x+y+z (spin at site i)

58. 14 sites pyrmids h=0 Staggered magnetization  i (-1) x+y+z (spin at site i)

59. QMC studies of small clusters h=0 Staggered magnetization  i (-1) x+y+z (spin at site i)

60. 14 sites pyrmids h=0 Staggered magnetization  i (-1) x+y+z (spin at site i)

61. 14 sites pyrmids h=0 Staggered magnetization  i (-1) x+y+z (spin at site i) Staggered magnetization (spin at site i)

62. 14 sites pyrmids h=0 Staggered magnetization  i (-1) x+y+z (spin at site i)

63. 4-site clusters h=0

64. Ground state charge gap Gap is monotonic versus U and non monotonic versus h