1. The block model for the Griffiths phase 2.The renormalization group for the phase transition in disordered systems (a) Replica trick. (b) Replica symmetry breaking. 3.The solutions of the saddle point equation of Landau-Ginzburg Hamitonian with random temperature. (a)Block model for ising-type systems. (b)The method to find elementary blocks and the couplings between blocks. 4.The application of block model. (a)Universality violation in Blume-Capel model with random bonds. (b)The pseudogap in strongly disordered conventional superconductors. 1. Experimetal evidence of Griffiths phase. 5.Summary. 吴新天 北京师范大学物理系

2. Griffiths Phase "compact" cluster of occupied sites R. B. Griffiths, Phys. Rev. Lett. 23, 17 (1969). A. J. Bray, Phys. Rev. Lett. 59, 586 (1987). The probability of these rare regions are proportional to is singular at The Griffiths singularities are probably unobservable in experiments. ?

3. O. Plantevin, et al, Phys. Rev. B, 65 (2002) 224505. Dynamic structure factor of liquid 4He in Geltechl at the roton wave vector at temperatures as shown. Roton at the temperature above the superfluid transition Tc=0.725 K. Specifically, we observe a well- defined roton peak at temperature T>Tc in Geltech where as observed in a torsional oscillator experiment. Since the existence of well-defined excitations at higher wave vectors above the phonon (sound) region depends on the existence of a Bose– Einstein condensate (BEC), the observation of a well-defined roton peak above Tc suggests that there is Bose condensation above Tc in Geltech.

4. This phase is characterized by the coexistence of ferromagnetic entities within the globally paramagnetic phase far above the magnetic ordering temperature. Tc=175K, below which the long range order exists. FMR: Ferromagnetic Resonance. J. Deisenhofer, et. al, Phys. Rev. Lett. 95, 257202 (2005).

5. Griffiths singularities and magnetoresistive manganites Deviation from the usual Curie-Weiss law. Anomalous critical exponents. M B Salamon and S H Chun, Phys. Rev. B, 68, 014411 (2003).

6. ISSN 0021-3640, JETP Letters, 2007, Vol. 85, No. 1, pp. 51–54. Nature Physics, Vol. 3, p248 (2007)

7. Landau-Ginzburg Hamiltonian with random temperature In order to describe the phase transition in the quenched disordered systems, Landau-Ginzburg Hamiltonian with random temperature is usually used. where, and is the local critical temperature. Its saddle point equation is given by For the pure system, it is obvious that above Tc, the saddle point solution is zero. However in the early stage of studying the disordered systems, it is still assumed that the saddle point solution is zero above Tc.

8. Replica trick Consider the short-range correlated disorder A weinrib and B I Halperin, Phys. Rev. B, 27 (1983) 413. The probability of random temerperature is given by The averaged free energy is given by Using the identity equation We can get The effective Hamiltonian is given by

9. There should be many isolated islands. Consider such isolated islands, There should be solutions For, there are some regions, where the saddle point solution can be nonzero. If the size of the region is, the average of reduced temperature is. The solution is given by Replica symmetry breaking In 1995, Dotsenko et. al proposed that they should be nonzero saddle point solutions above the critical temeprature. Consider the saddle point equation V. S. Dotsenko, et al, J. Phys. A: Math. & Gen., 28 (1995) 3093

10. Considering the solution should have some deviation from They proposed that the solutions obey the following distribution Where So, with the replica trick and some theory of spin glass, they obtained an effective Hamitonian with replica symmetry breaking Parameterizing with Parisi ansartz, the renorlization group is applied to the above Hamiltonian. However it is shown that the replica symmetry breaking is irrelevant. Wu X T, 1998 Physica A 251 (1998) 309. A. A. Fedorenko, J. Phys. A : Math. & Gen., 36 (2003) 1239.

11. The numerical solutions on the saddle point equation of Landau-Ginzburg Hamiltonian with random temperature For the phase transition in the quenched disordered systems, we consider the Landau-Ginzburg Hamiltonian with random temperature. Its saddle point equation Is given by where, and is the local critical temperature In order to solve the saddle point equation numerically, we model the random temperature as follows: the system is divided into cells with the same size and each cell has reduced temperature, that is and the distribution of the random temperature is given by Obviously the temperatures in different blocks are not correlated, the correlation length of the disorder is.

12. The following figure shows a typical solution of the two-dimensional saddle point equation with random temperature for and. X. T. Wu and K. Yamada, J. Phys. A: Math. & Gen., 37 (2004) 3363 X. T. Wu, Phys. Rev. B 71, 174204 (2005) Before our work, the saddle point equation has not been studied in detail although there are some qualitative arguments on it.

13. An example of excited state solution in two dimension This is a ground state solution for certain realization of random temperature. At the green part the order parameter is very small and near zero. At the blue part, the order parameter is nonzero and positive. The solution has the same sign over the whole system. This is an excited state solution for the same realization of random temperature as the above ground state solution. At the green part the order parameter is very small. At the blue part the order parameter is positive, ant at the red part it is negative. The curves separating the negative region and the positive region are domain wall, at which the order parameter changes sign.

14. In the system shown in last page, there are 6 locally ordered regions (LOR). It can be shown that every LOR can take positive or negative sign arbitrarily for the saddle point solutions. Therefore there are states, two ground states and 62 excited states. We can assign every LOR a variable, and use the set of variables { } to label the states. On the saddle point level, in which the fluctuation around the saddle point solutions are neglected, the partition function is given by The free energy of each states is given by The free energy difference between the excited state and ground state is given by Block model

15. Consider two adjoining LORs labeled by 1 and 2. If they have the same signs, there is no domain wall between them. Otherwise there is a segment of domain wall between them. Assuming the free-energy increase due to this segment is. Then for these two LORs, whether there is a segment of domain wall between them or not, the free-energy increase due to these two wells can be given by. Therefore the free energy difference between an excited state and the ground state can be written Approximately The partition function becomes X. T. Wu, Phys. Rev. B 79, 184208 (2009). 12 12 This is also valid for multi-compenent spin systems, for example, XY-model. Xintian Wu, Physica A, 391 (2012) 6247.

16. From, and we get Domain walls around four elementary blocks obtained with opening windows. How to find the elementary block and calculate the coupling between blocks Homogeneous and inhomogeneous phase transitions in the Blume-Capel model with random bonds, X. T. Wu, PRE 82, 010101R (2010) The method of opening windows.

17. Universality violation in Blume-Capel model with random bond The Hamiltonian of Blume-Capel model with random bond is given by Where. The distribution of the bonds is In the MF approximation for the pure model, the tricritical point, For the phase transition is first order, otherwise is second order. In the recent MC simulation, for both and the phase transition are second order. Moreover the critical exponents are different. For the former, it has ; for the latter, it has. and A. Malakis, et. al, Phys. Rev. E 79, 011125 (2009).

18. X. T. Wu, PRE 82, 010101R (2010) Homogeneous and inhomogeneous phase transitions in the Blume-Capel model with random bonds Using the method mentioned above, we can get the coupling between blocks. The left figure is the distributions of the couplings between blocks. For, the distribution is exponential. The phase transition should be percolation like. Then. For, the distribution is Gaussian. The phase transition should be homogeneous. Then.

19. Disorder-Induced Inhomogeneities of the Superconducting State Close to the Superconductor-Insulator Transition, B. Sace´pe´, et. al., PRL 101, 157006 (2008) It is found that local superconductivity survives across the disorder-driven superconductorinsulator transition. Inhomogeneities of the Superconducting gap. Even in the insulating phase where Tc=0, the average superconducting gap still survives. Pseudogap in disordered conventional superconductors

20. Tunneling studies in a homogeneously disordered s-wave superconductor: NbN S. P. Chockalingam, et. al., PRB 79, 094509 (2009) Tunneling measurements reveal that for films with large disorder the superconducting transition temperature is not associated with a vanishing of the superconducting energy gap. Above Tc, the system is peudogapped?

21. We start from the negative-U hubbard model with site disorder The random potential is chosen independently at each site from a uniform distribution. The BdG Hamiltonian is gvien by Where Introducing Bogliubov transformation to diagonalize the Hamiltonian The free energy of each state is given by

22. The coupling distribution at different temperature for different disorder strength The distribution of couplings between neighbored elementary blocks and free energy of elementary blocks (insets). The disorder strength is V=0.5,1.0,1.5,2.0 for (a),(b),(c), and (d) respectively. Recall the formula, where labels the nearest neighbors of. According to Block model, the free energy of each state is approximately given by

23. One-particle density of states (DOS) defined in termsof the BdG eigenvalues T-evolutions of the local tunnelling conductance G characterized by the presence (a) or absence (b) of superconducting coherent peaks. Benjamin Sacépé, et. al, Nature Physics 7 (2011) 239. Weak disorderStrong disorder Weak disorder Strong disorder X T Wu and R Ikeda, PRB 83, 104517 (2011)

24. 1.At the temperature of superconductivity phase transition, the the energy gap is not zero. 2. The ratio increases at strong disorder. The phase diagram X T Wu and R Ikeda, PRB 83, 104517 (2011)

25. Summary The block model for the Griffiths phase is proposed at the saddle point level. It contains three basic points: 1.The systems is self-orgnized into blocks. The size of blocks depends on the the correlation and strength of disorder. 2. The blocks are coupled with each other and behave like superspins. The effective Hamiltonians are spin models with random bonds. 3. The couplings between elementary blocks can be obtained with the method of opening windows. These results are potentially valid for some systems, for example, conventional superconductors, of which the pure systems can be described by mean-field theory and their critical regimes are too small to be observed in experiments. Of course, the further study on how the fluctuations about these solutions renormalize them are worth. The fluctuation may change the block size and The coupling between blocks.