1. Vortices as line - like objects. The small tilt appr.

1. 1. Vortices as line - like objects. The small tilt appr. VII. MELTING of VORTEX SOLID into LIQUID. A. The path integral theory of thermal fluctuations of flux lines in London appr. Abrikosov vortices are approximated by infinitely thin elastic lines with line energy Interaction is assumed mainly magnetic, thereby pair - wise (superposition principle)

2. Normal H Hm(T) Liquid Solid Meissner T Lindemann Melting criterion:

3. Brandt JLTP4 (1991) Thermodynamics is defined by a statistical sum This statistical sum is the “line representation of “dual” relativistic scalar QED in 3D Kovner, B.R. IJMPA7, 7419 (1992), MPLA8, 1343 (1993); Sudbo et al (1999)

4. The small tilt approximation This theory is too complicated. Generally all possible configurations including vortex loops contribute. However when magnetic field is not too weak, vortices are nearly aligned with magnetic field. In this case loops overhangs… can be ignored and one develops a small tilt approximation.

5. We can regards z as “time” and then the approximation is just a nonrelativistic approximation of the dual theory. Interaction also simplifies and becomes “instantaneous” Abrikosov repulsion: In this approximation the statistical sum becomes that of many -body problem of interacting bosons: With T analogous to h, to mass Nelson, PRL (1988)

6. 2. Example of the calculation with QM analogy: average displacement in vortex lattice The standard way of calculating displacement due to thermal fluctuations or disorder in lattice is via elasticity theory Brandt, FSS (1970s) Here I will employ the less rigorous but simpler Einstein approximation: a vortex is considered in a field of all the other vortices as if they are not vibrating: the cage model. This will be enough to qualitatively map the melting line via Lindemann criterium

7. 3. Lindemann crirerium for melting where a is a distance between vortices: Typical values of the Lindemann constant range between

8. The cage model To simplify the calculations only harmonic part of this potential will be used. For small fields , distances are large and theinteraction is exponentially weak

9. Therefore only 6 nearest neighbours are important: For larger inductions the interaction is logarithmic and more neighbours should be summed up Now we consider the thermal motion of a single vortex in harmonic potential: quantum mechanics of harmonic oscillator.

10. The quantum mechanical analogy Within the quantum mechanical analogy thermal average of a physical quantity is equivalent to VEV Therefore for square of displacement in harmonic oscillator ground state one obtains:

11. Lindemann criterium therefore takes a form: For weak fields one obtains For larger fields one gets

12. Bm(75K) =92G Bm(70K) =140G Bm(60K) =250G 3. Melting of the Vortex-Lattice 3.3. Experimental Results T = 60, 70, 75, 80 K Bcenter – Bext [G] Bext [G]

13. Conclusion • Within the range of applicability of the London approximation (not very close to Hc2) one can qualitatively and sometimes quantitatively calculate properties of the vortex solid. • Lindemann criterium gives a reasonable location of the melting line. • The theory can be extended to include disorder with similar results/

14. VIII. THERMAL FLUCTUATIONS in SOLID and LIQUID within LLL. A. Supersoft phonons in solid near Hc2 1. Exxitations in vortex lattice On the microscopic level temperature modifies properties of the electron gas and the pairing interaction responsible for the creation of Cooper pairs. When we “integrated out” the microscopic (electronic) degrees of freedom to obtain the effective mesoscopic GL theory in terms of the distributions of the order parameter with ultraviolet (UV) cutoff a

15. Ginzburg – Landau energy its simplification From very general arguments of criticality (second order transition) and U(1) gauge invariance one writes “near Tc” Gibbs free energy in terms of order parameter and vector potential: Near Hc2 it is convenient to use coherence length as unit of lengthand magnetic field in units of Hc2 and rescaled by

16. Magnetic translations and basis of Landau levels and quasimomenta To leading order in the solution belongs to LLL and should have hexagonal symmetry. Because of gauge invariance translations are represented nontrivially. Naive translations generators do not commute with H . However there are “magnetic translation” generators which do. Here In our gauge

17. From the generators one constructs magnetic translations However they do not commute among themselves Except when the phase factor disappears: Hexagonal symmetry of the Abrikosov lattice is imposed by:

18. The hexagonal state, normalized to is: The most convenient basis on Landau levels is constructed applying magnetic translations to this state The quasi momentum states are eigenfunctions of magnetic translations:

19. With following parameters:

20. Thermal fluctuations are taken into account via statistical sum Since the problem with thermal fluctuations is much more complicated we consider LLL only in which case the energy simplifies With only one dimensionless parameter: LLL scaled temperature

21. Supersoft phonons in vortex solid There are two types of the fluctuation modes in expansion around Abrikosov solution: Substituting this into energy and diagonalizing quadratic part of free energy one obtains: Eilenberger, PR (1967) The cubic and quartic parts define vertices.

22. 2. IR divergencies. Dos vortex lattice exists? Naively higher order contributions to energy are hopelessly divergent: Since experimentally the corrections are small one can speculate that it is not analytic. The perturbation theory was abandoned. Is this correct? A question of principle Is there a thermodynamic solid state for T>0 or experimentally observed vortex lattice is just a finite size effect or a quasi-long range solid?

23. The first surprise The most divergent two loop diagram, the “setting sun” is in fact convergent! Careful evaluation shows that vertices are supersoft and all the divergencies cancel. B.R. PRB60, 4268 (1999), Other two loop diagrams are IR divergent, but only logarithmically divergent. Moreover the divergencies look similar to “spurious divergencies” in critical phenomena of models with broken continuous symmetry . It turns out indeed that all the divergencies cancel. It is more instructive to consider a simple model. I will show in some detail what happens in D=2 O(2) symmetric model (which we completely understand) and then return to GL.

24. Back to Ginzburg – Landau theory Analogous events take place in GL up to two loops. Since the correction to the order parameter VEV is divergent, which means after resummation that it slowly vanishes. Therefore translation noninvariant solid is “destroyed” by thermal fluctuations and becomes a “quasi – solid” with quasi long rage order. However the perturbation theory for translation invariant quantities remains valid: no nonanalyticity. B.R. PRB60, 4268 (1999)I believe cancellations occur beyond two loops, but mathematical proof is not available now.

25. Free energy For energy, which is invariant under translation, we get to the two loop order: B.R. PRB60,4268 (1999), D.P. Li and B.R. PRB65,024514(2001) Even at melt ( ) the precision is 0.1%. From this one calculates magnetization, specific heat. Structure function and other physical quantities are also calculated perturbatively.

26. Bragg peaks and Lindemann criterium Kim et.al. PRB60, R12589 (1999) Sasik, Stroud PRL75,2582 (1995) DP Li, B.R. PRB60,9704 (1999) Generalized Lindemann criterium successfully determines melting temperature when

27. Theory of vortex liquid and quantitative description of melting Perturbation theory at high temperatures and Gaussian resummation Standard high temperature perturbation theory is defined only only for with the excitation energy and is therefore useless for the experimentally interesting region around melting. To get to negative one can perform “bubble resummation” also called Hartree – Fock, gaussian, renormalized…

28. The gap equation For the excitation energy has solution for any temperature. It shows that overcooled liquid is metastable all the way down to T=0 with excitation energy vanishing as Perturbations around Gaussian state were pushed to 9th order. Brezin et al PRL65,1949 (1990) Unfortunately the series are asymptotic and can be used only for long before melting occurs.

29. Liquid: quest for a nonperturbative result

30. Recent improvements • We constructed the Optimized gaussian series which are convergent rather than asymptotic. DP Li, B.R. PRL86,3618 (2001) Radius of convergence was found to be still a bit short. 2. However it allowed us to check the validity of Borel-Pade method which provided a convergent scheme everywhere down to T=0. Precision was finally good enough (0.1%) to study melting quantitatively

31. Melting line and discontinuities at melting The melting point is: The magnetization jump: Specific heat jump:

32. Comparison with experiments on fully oxidized YBCO Melting line Nishizaki et al, Physica C341,957 (2000) The magnetization jump Welp et al, PRL76,4809 (1996) Specific heat jump Willemin et al, PRL81,4236 (1998)

33. Conclusions • Ginzburg –Landau theory of vortex matter near Hc2 is qualitatively and even quantitatively well understood: vortex liquid first order melts into solid although liquid survives till zero temperature. • The theory agrees well with experiments on vortex melting. • Overcooled liquid has Madelung constant smaller by latent heat of the melting from that of solid .

34. Effect of disorder Point disorder in YBCO is mainly due to oxygen deficiencies. In the framework of GL it leads to local random temperature (potential): With variance The relation to the conventional disorder parameter is

35. Methods and results briefly • We used the replica method to tackle this nonperturbative problem. Results are following: • There is no broken replica symmetry homogeneous state. • This means that in statics there is no homogeneous glass although it might have dynamic glassy properties. • 2. There is a replica symmetry broken state in the lattice phase (Bragg glass). • 3. There are only two phases. Liquid gains more than solid from pinning.

36. Phase diagram for various disorder strength

37. Discontinuities and comparison with optimally doped YBCO 4- A. Schilling et. al., Nature 382, 791 (1996) 6- K. Shibataet .al., unpublished (2002). 8- F. Bouquet , Nature 411, 448 (2001). 28- K. Deligiannis et. al, Physica C341, 1329 (2000).

38. Conclusions 1. In the London limit only a qualitative picture of thermal fluctuations and melting exist. 2. Ginzburg –Landau theory of vortex matter near Hc2 is qualitatively and even quantitatively well understood: vortex liquid first order melts into solid although liquid survives till zero temperature. 3. The theory agrees well with experiments on vortex melting. 4. In the presence of disorder there are two phases separated by a wiggling line. 5. Overcooled liquid has Madelung constant smaller by latent heat of the melting from that of solid.

39. A sample of open theoreticalquestions 1. Quantitative theory of melting in London approximation. 2. Theory of dynamics near Hc2 in the presence of pointlike disorder. In particular the glass transitions. 3. Layered structure (Prof. Yeshurun’s lectures). A sample of open experimental questions 1. Is the transition line in optimally doped YBCO a single continuous line? 2. What is the microscopic mechanism of pinning? What is correct temperature dependence of pinning strength? 3. Measurement of discontinuities along Bragg glass – “vortex glass” line.

40. B. Fluctuations effects in magnetic field. 1. Gaussian and “critical” fluctuations in the formerly normal state With following parameters:

41. Perturbation theory near Hc2 The expansion in respecting the hexagonal symmetry is therefore: Inserting it to the equation one obtains in the order

42. Making the scalar product with one obtains: The correction is obtained in the similar way. It has all the Landau levels in it: To find one makes products with HLL functions

43. The LLL component is found from the order etc. All the corrections are very small numerically partially due to to factor 1/n with multiples of 6 contributing due to symmetry

44. Conclusion • The perturbation theory in • converges well up to surprisingly low fields and temperatures, roughly above the line • 2. LLL is by far the leading contribution

45. Thermal fluctuations in the flux lattice • Thermal fluctuations are taken into account via statistical sum Since the problem with thermal fluctuations is much more complicated we consider LLL only in which case the energy simplifies With only one dimensionless parameter: LLL scaled temperature

46. Supersoft phonons in vortex solid There are two types of the fluctuation modes in expansion around Abrikosov solution: Substituting this into energy and diagonalizing quadratic part of free energy one obtains: Eilenberger, PR (1967) The cubic and quartic parts define vertices.

47. IR divergencies Naively higher order contributions to energy are hopelessly divergent: Since experimentally the corrections are small one can speculate that it is not analytic. The perturbation theory was abandoned. Is this correct? A question of principle Is there a thermodynamic solid state for T>0 or experimentally observed vortex lattice is just a finite size effect or a quasi-long range solid?

48. The first surprise The most divergent two loop diagram, the “setting sun” is in fact convergent! Careful evaluation shows that vertices are supersoft and all the divergencies cancel. B.R. PRB60, 4268 (1999), Other two loop diagrams are IR divergent, but only logarithmically divergent. Moreover the divergencies look similar to “spurious divergencies” in critical phenomena of models with broken continuous symmetry . It turns out indeed that all the divergencies cancel. It is more instructive to consider a simple model. I will show in some detail what happens in D=2 O(2) symmetric model (which we completely understand) and then return to GL.

49. Back to Ginzburg – Landau theory Analogous events take place in GL up to two loops. Since the correction to the order parameter VEV is divergent, which means after resummation that it slowly vanishes. Therefore translation noninvariant solid is “destroyed” by thermal fluctuations and becomes a “quasi – solid” with quasi long rage order. However the perturbation theory for translation invariant quantities remains valid: no nonanalyticity. B.R. PRB60, 4268 (1999)I believe cancellations occur beyond two loops, but mathematical proof is not available now.

50. Free energy For energy, which is invariant under translation, we get to the two loop order: B.R. PRB60,4268 (1999), D.P. Li and B.R. PRB65,024514(2001) Even at melt ( ) the precision is 0.1%. From this one calculates magnetization, specific heat. Structure function and other physical quantities are also calculated perturbatively.