1. 1 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 VI. INTRODUCTION to THERMAL FLUCTUATIONS in TYPE II SC. BKT TRANSITION in 2D A. Two scales of thermal fluctuations 1. The microscopic thermal fluctuations’ place in the GL description

2. 2 One loosely describes the effective mesoscopic order parameter as a classical field of Cooper pairs with center of mass at x. More mathematically, the remaining mesoscopic part of the statistical sum obeys the “scale matching”:

3. 3 Quantum effects on the mesoscopic level are usually small (only when temperature is very close to T=0 they might be of importance) and will be neglected. In this case the mesoscopic classical field is independent of time. Dynamical generalization of the GL approach will be introduced later.

4. 4 are dependent on temperature expressing these “microscopic” thermal fluctuations. The dependence can, in principle, be derived from a microscopic theory (example: Gor’kov’s derivation from the BCS theory of “conventional superconductors). The coefficients also depend on UV cutoff  or a, but we will see that this dependence can be “renormalized away”. In practice the “constants” are also weakly (typically logarithmically) temperature dependent The coefficients of the GL energy

5. 5 Normal Mixed state Meissner leading for example to “curving down” of the Hc2(T) line:

6. 6 2. Two kinds of mesoscopic thermal fluctuations: “perturbative” and “topological” Since under magnetic field the order parameter takes a form of vortices, the mesoscopic fluctuations can be qualitatively viewed as distortions of a system of vortices or “thermal motion”: vibrations, rotations, waves. The mesoscopic fluctuations qualitatively are of two sorts: “perturbative” small ones and “topologically nontrivial” or vortex ones.

7. 7 Broadening of the resistivity Major thermal fluctuations effects include broadening of the resistivity drop (paraconductivity) and diamagnetism in the normal phase and melting of the vortex solid into a homogeneous vortex liquid state Magnetization or conductivity are greatly enhanced in the “normal” region due to thermally generated “virtual” Cooper pairs. TTCTC Resistance Nonfluctuating SC Normal Metal Fluctuating SC

8. 8 Influence of the fluctuations on the vortex matter phase diagram Due to the thermally induced vibrations lattice can melt into “vortex liquid” and the vortex matter phase diagram becomes more complicated.

9. 9 Since symmetry of the normal and liquid phases are same, the normal – liquid line becomes just a crossover. Just crossover Normal Vortex liquid Meissner FLL H T U(1) breaking E2 breaking Two different symmetries are at play: the geometrical E(2) (including translations and rotations) and the electric charge U(1). Symmetries broken at the two transitons

10. 10 1. Gaussian fluctuations around the Meissner state B. Nontopological excitations. The Ginzburg criterion. We ignore the thermal fluctuations of the magnetic field which turn out to be very small even for high Tc SC in magnetic field. The effect of thermal fluctuations on the mesoscopic scale is determined by

11. 11 Ginzburg number In the new units the Boltzmann factor becomes: It is convenient to use units of coherence length (which might be different in different directions),  in units of   and energy scale in units of critical temperature

12. 12 Here an important dimensionless parameter characterizing strength of thermal fluctuations is with the anisotropy parameter Where the temperature independent Ginzburg number characterizing he material was introduced:

13. 13 To calculate the thermal effects via a somewhat bulky functional integral, the simplest method is the saddle point evaluation assuming that  is small. One minimized the free energy around the “classical” or nonfluctuating solution (which we found in the preceding parts also for the SC phase) and then expands in “small” fluctuations” Perturbation theory

14. 14 Now we consider the normal phase t>1 in which the saddle point value of the order parameter is trivial: However the superfluid density The thermal average of the order parameter is still zero due to the U(1) symmetry: due to fluctuations. First we calculate the fluctuation correction to free energy and thereby to specific heat which is impacted the most.

15. 15 From now on fl is dropped. The expansion of the partition function results in gaussian integrals: The fluctuation contribution to free energy

16. 16 It is more convenient to perform the calculation in momentum space which is presented graphically as “diagrams”:

17. 17 since the basic gaussian integral in momentum space becomes a product Therefore the fluctuations contribution to the free energy density is: The leading order

18. 18 The fact that free energy depends on the UV cutoff a means that it is not a directly measurable mesoscopic quantity: energy differences or derivatives are. The mesoscopic Cooper pairs contribution to the entropy density is less dependent on the division of degrees of freedom into micro and meso: just a constant Entropy and specific heat

19. 19 The second derivative, the fluctuation contribution to specific heat, is already finite

20. 20 Closer than that to Tc the perturbation theory cannot be used due to IR divergencies. Physics in critical region is therefore dominated by fluctuations. A nonperturbative method like RG required. The fluctuation contribution outside the critical region is called gaussian since fluctuations were considered to be non-interacting. Corrections already do depend on interactions. They are smaller by a factor at least. It is more instructive to see this on example of the correlator. 2. Interactions of the excitations and “critical” fluctuations. Ginzburg criterion.

21. 21 A measure of the SC correlation (or “virtul density of Cooper pairs”) in the normal phase is Fourier transform of correlator and small wave vector: The leading correction is: A measure of coherence in the Meissner phase (density of Cooper pairs) is Correlator and its divergence at criticality

22. 22 This expression is a small perturbation only when and therefore cannot be applied close to Tc. Perturbation theory seems to be useless due to UV divergencies. However it is natural to assume that quantities measurable on the mesoscopiclevel should not depend on cutoff a. It is reasonable to assume that when the mesoscopic fluctuations are “switched on” the superconducting correlations are destroyed at below which takes into account only the microscopic ones. How to quantify it? Renormalized perturbation theory

23. 23 One can try to improve on it by resumming some diagrams At correlator decays slower:

24. 24 To first order in fluctuations criticality occurs at Since usually is not known theoretically and is not a quantity of interest, one expresses it via measured critical temperature used as an “input parameter”. Ginzburg criterion After renormalization we return to the perturbation theory applicability test:

25. 25 This might be compared to the jump between Meissner and normal phases before mesoscopic fluctuations are taken into account: The condition that fluctuations do not become “stronger” than the mean field effect is This is known as Ginzburg criterion. Plugging correct constants one obtains:

26. 26 F leads to a nontrivial minimum The negative coefficient of the quadratic term 3. Fluctuations in the Meissner phase. Goldstone modes.

27. 27 Since one of many possible “shifts” was chosen it is convenient to present the real and imaginary parts of the fluctuation separately: The energy in terms of two real fields O and A becomes: -v (+) +1/2 ( + 2+ ) Where the “masses” of excitations are Feynman rules with shift and the Goldstone mode.

28. 28 Goldstone mode, “acoustic” Dispersion relation of the A mode is that of acoustic phonons: Regular massive mode, “optic” And in the confuguration space the correlator is: in 2D in 3D

29. 29 The field A itself is not the order parameter since it does not transforms linearly under the symmetry transformation. The order parameter is Where the phase of is related to the fields by: in 2D in 3D Its correlator is:

30. 30 Fluctuations due to Goldstone bosons in 2D “destroy” perfect order. Such a phase is called quasi – long range order phase (or Berezinski-Kosterlitz-Thouless phase). We started from the assumption of nonzero VEV. It seems that fluctuations destroy this assumption! The 2D correlator in the “ordered” phase decays albeit slowly (as a power rather than exponential in the disordered phase

31. 31 4. Destructions caused by IR divergencies in D b 2 and the MWC theorem. The energy to the one loop level is: The corrected value of v is found by minimizing it perturbatively in “loops”: Noting that

32. 32 one obtains a logarithmically divergent correction to VEV: To higher orders the logs can be resummed: The VEV decays – do not diverges, indicating that order is “slowly” restored. This is Hohenberg-Mermin-Wagner theorem: in 2D continuous symmetry is not broken. More importantly this does not mean the perturbation theory is useless.

33. 33 For such quantities the “collective coordinates” method simplifies into perturbation theory around “broken” vacuum. All the IR divergencies cancel. Let us see this for the energy to two loops order. Jevicki, (1987) O(2) invariant quantities There also is correction due to change in v:

34. 34 The leading IR divergencies are easy to evaluate: Subleading divergencies also cancel although it is much less obvious. Cancellations occur to all orders F.David (1990) in loop expansion. What is the mechanism behind this cancellation of “spurious divergencies? It is hard to say generally, but at least in extreme case of 1D the answer is clear.

35. 35 For D=1 the model is equivalent to QM of particle on a plane with Mexican hat potential Physics below the lower critical dimension Ground state is O(2) invariant but is very far from the origin (0,0): pert. Ground state is bad, but theory “corrects” it using IR divergent matrix elements Kao,B.R.,Lee PRB61, 12652 (2000) O QM ground state Pert. ground state A

36. 36 5. Heuristic argument about destruction of order by Goldstone bosons. For the XY model (same universality class as GL) a Typical excitation is a wave of phase L

37. 37 Its energy in various dimensions is for no order order D=2 is a border case in which there exists “almost long range order”

38. 38 C. Topological excitations. The dual picture It is therefore advantageous in such a case to reformulate the theory in terms of vortex degrees of freedom only: 1. An extreme point of view: topological fluctuations dominate thermodynamics. Vortices are the most important degrees of freedom in an extremely type II SC (even in the absence of magnetic field ):

39. 39 Minimal excitation (Cooper pair) = Smallest vortex ring The Feynman-Onsager excitation and the “spaghetti” vacuum. The normal phase is reinterpreted as a proliferation od loops: SC Normal

40. 40 1. In 3D “non ligh Tc” materials k is not very large. 2. Vortex loop in 3D is a complicated object: infinity of degrees of freedom (unlike in 2D) This point of view is not commonly accepted or used with one notable exception: the “BKT” transition in thin films. Reasons: The dual picture was nevertheless advanced after the discovery of high Tc. Its extentions to include external magnetic field were unsuccessful so far.

41. 41 2. KT transition in thin films or layered SC Pearl (1964) may be very large. Interaction between 2D fluxons in logr up to scale SC - normal phase transition in thin films is of a novel type: the Berezinskii –Kosterlitz-Thouless continuous type (71).

42. 42 Dipole unbinding triggers the proliferation The basic picture is just 2D “slice” of the fluxon proliferation 3D picture : instead of vortex loop - dipoles. Dual dipoles: SC Free “dual charges”: normal - Kosterlitz, Thouless (72)

43. 43 Noeter theorem ensures that (if the symmetry is not spontaneously broken) any continuous symmetry has a corresponding conservation law. Examples: the electric charge global U(1) symmetry The magnetic flux symmetry Leads to charge conservation Other examples: rotations – angular momentum…

44. 44 Similarly one can interpret the Maxwell equation The symmetry is unbroken in the SC, while spontaneously breaks down (photon is a Golstone boson) in the normal phase As a conservation law of the magnetic flux, yet another global U(1) symmetry, sometimes calle “inverted” or “dual” U(1). The order parameter field was constructed and the GL theory in terms of it was established - Kovner, Rosenstein, PRL (92)

45. 45 The analogy of the charge U(1) and the magnetic flux U(1) is as follows: - Nelson, Halperin, PRB (81) The dual picture

46. 46 3. A brief history of phase transitions with continuous symmetry in 2D. The Hohenberg-Mermin-Wagner theorem demonstrates that fluctuations destroy long range order. According to the dual picture a continuous magnetic flux symmetry should be spontaneouly broken. This seems to be impossible in 2D, hence according to Landau’s “postulate” –no phase transition. a. They do not exist The correlator is a power decay at low temperatures - Berezinskii(71)

47. 47 b. The high temperature expansion: something happens in between. High temp. expansion gives an exponential Some qualitative change should happen in between! This “something” is unbinding of vortices.

48. 48 Number of states: c. The energy – entropy heuristic argument - Kosterlitz, Thouless (72) + - Energy of a pair of size r:

49. 49 Where a is the core size. This becomes negative for c. The energy – entropy argument - The free energy of a pair therefore is: which means that the pair proliferate and superconductivity is lost

50. 50 d. Systematic expansions and exact solution. Kosterlitz (74) and Young (79) developed a heuristic renormalization group (RG) approach to account for differences between pair’s sizes. We will follow this approach. The XY model or the Coulomb gas maps onto the sine- Gordon field theory for which perturbation theory exists Wiegman (78) Recent experiments on BSSCO and other layered high Tc superconductors found new area of applications for KT. Starting with Zamolodehikov’s (80) exact results were obtained that confirm approximate ones. Nowadays that KT theory is one of the most solid in theoretical physics.

51. 51 4. The RG theory of the BKT transition The energy of the KT pair neglecting interactions with other pairs is Let us assume that screening can be represented by the dielectric function e (r).

52. 52 It takes into account the polarization of the pairs due to smaller ones. The energy therefore gets reduced: r

53. 53 The dielectric function itself is created by the polarization due to dependence of the Boltzmann weight on the orientation of the thermally created KT dipoles: To calculate the polarization due consider constant electric field we first write the polarizability of a single dipole: E r q

54. 54 Therefore one gets since the number density of such pairs is

55. 55 Thereby we have derived an integral form of the RG eqs. for U and e. Differentiating it with respect to the pair size one obtains a differential form of the RG eqs. with initial conditions

56. 56 The equations can be turned into a set of autonomous ones by going to a log scale and rescaling the density variable

57. 57 Exercise 5: Plot the “vector field” of this autonomous system. Solve this system of differential equations either numerically or approximately analytically using the separatrix method. It is clear that the character of solution changes at

58. 58 Let us slightly redefine again new variables to make the solution evident With x small near the transition Exact solution near the KT temperature

59. 59 It is clear that parabolas are the flow lines The first equation now can be integrated With the result

60. 60 The critical value of dielectric constant is finite

61. 61 With criticality of the very weak KT type Density of bound pairs is With the result Singularities at transition

62. 62 It is clear that parabolas are the flow lines The first equation now can be integrated With the result

63. 63 5. Phenomenology of the BKT transition The energy of the KT pair neglecting interactions with other pairs is Let us assume that screening can be represented by the dielectric function e (r).