Critical Phenomena in Disordered Membranes

1. Critical phenomena in disordered membranes V.Yu.Kachorovskii IoffePhysico-Technical Institute, St.Petersburg, Russia (also Landau/KIT) Co-authors: I.S. Burmistrov (Landau/KIT) I.V. Gornyi(KIT/Ioffe/Landau) M.I. Katsnelson(Radboud Un) J.H. Los(Radboud Un) A.D. Mirlin(KIT/PNPI/Landau) D. R. Saykin(Skoltech/Landau) Petersburg Nuclear Physics Institute Euler Symposium on Theoretical and Mathematical Physics, June 15-20, 2019, St. Petersburg, Russia

2. The talk is based on: Gornyi, Kachorovskii, Mirlin Conductivity of suspended graphene at the Dirac point,PRB (2012) Rippling and crumpling in disordered free-standing graphene, PRB (2015) Anomalous Hooke's law in disordered graphene,2D Mater. (2017) Burmistrov, Gornyi, Kachorovskii, Katsnelson, Mirlin, Quantum elasticity of graphene: Thermal expansion coefficient and specific heat, PRB (2016) Burmistrov, Gornyi, Kachorovskii, Katsnelson, Los, Mirlin, Stress-controlled Poisson ratio of a crystalline membrane: Application to graphene,PRB (2018) Burmistrov, Kachorovskii, Gornyi, Mirlin, Differential Poisson’s ratio of a crystalline two-dimensional membrane, Annals of Physics (2018) Saykin, Burmistrov ,Gornyi, Kachorovskii,Mirlin, Crumpling of disordered membranes, unpublished (2019)

3. Introduction. Isolated crystalline membrane. Flexural • phonons and ripples • Phase diagram of the crystalline membrane. Crumpling • and buckling transitions • Anomalous Hooke’s law (AHL). Nonlinear scaling of • deformation with applied stress • Disordered membrane. Crumpling transition and AHL in the • membrane with random curvature • Experiment and numerical simulations. Observation of AHL. • Geometry of disordered membrane. • Other anomalous properties of membranes. Negative thermal • expansion coefficient and negative Poisson ratio Outline

4. Suspended graphene dynamic out-of-plane deformations(flexural phonons) T + static frozen-out deformations (ripples)  b

5. Flexural phonons (FP) out-of-plane flexural mode bending rigidity soft dispersion of FP in-plane elastic coefficients in-plane phonons

6. Global shrinking of membrane induced by FP or ripples “Membrane effect”: thermal fluctuations in y-direction lead to shrinking in x direction I.M. Lifshitz, JETP (1952) hidden area stretching parameter global deformation in-plane andout-of-plane fluctuations

7. Geometry of the membrane, effect of the external tension fractalgeometry x = xL External tension s“irons” thermal or static fluctuations

8. Crumpling transition Paczuski, Kardar, Nelson , PRL (1988) ; David and Guitter, Europhys. Lett. (1988); Nelson, Piran, Weinberg , “Statistical Mechanics of Membranes and Surfaces “ (1989); Crumpled phase Flat phase Analogy with ferromagnetic transition T<Tc spontaneous symmetry breaking !!!

9. Physics behind crumpling transition Crumpled Flat CT Competitionbetweentwoeffects: 1) “membrane effect” → shrinking of membranedue to FP or ripples 2) anharmoniccoupling between out-of-plane deformations and in-plane modes → infrared divergence of bending rigidity tendency to crumpling stabilization of the flat phase • η≈ 0.7- 0.8 • critical exponent

10. Buckling transition (BT) spontaneous symmetry breaking T = 0 Membrane with fluctuations, T ≠0 ???

11. Manifestation of BT: anomalous Hooke’s law Hooke’s law (1678):α=1 α- critical index of BT Graphene : 0 1) Guitter, David, Leibler, Peliti, PRL (1988); Aronovitz, Colubovic, LubenskyJ.Phys. France(1989) Gornyi, Mirlin, Kachorovskii., 2D Mater. (2017) anomalous Hooke’s law at SMALL (!) tension

12. Phase diagram of clean crystalline membrane Guitter,David,Leibler,Peliti, PRL (1988) ; Aronovitz,Colubovic,Lubensky, J.Phys.France(1989) FLAT, 1 STRETCHED, BUCKLED quantum effects CRUMPLED,

13. Crumpling and buckling in disordered membrane STRETCHED, non-zero tension,s> 0 , FLAT BUCKLED ξ= 0, CRUMPLED Gornyi, Kachorovskii, Mirlin , 2D Mater. (2017)

14. Experimental evidence of anharmonicity • huge (unrealistic) theoretical prediction • for out-of- plane fluctuations calculated in • harmonic approximation • several order of magnitude discrepancy • between theoretical and experimental • values of mobility limited by FP • experimental measurements of anomalous • Hooke’s law (AHL) in suspended graphene. • Numerical study of AHL and fractal • dimension of membranes

15. Huge out-of-plane fluctuations Random classical field: correlation function of FP graphene at T=300 K: 0 • unrealistic (proportional to the system size !!!) thermal out-of-plane fluctuations

16. Scattering off FP in graphene FP contribution to the deformation potential Theory: Golden-rule calculation simple theory yields unrealistic (too small) values of conductivity at the Dirac point Experiment: K. Bolotinet al ., PRL (2008) N = 4 spinvalleys, dimensionless e-ph coupling constant 0

17. Theory of crumpling transition Paczuski, Kardar, Nelson , PRL (1988) ; David and Guitter, Europhys. Lett. (1988); Nelson, Piran, Weinberg , “Statistical Mechanics of Membranes and Surfaces “ (1989); E= For physical membranes d=3, D=2 in-plane and out-of-plane fluctuations global deformation

18. Energy of membrane stretching energy energy of fluctuations coupling between stretching and fluctuations In the absence of fluctuations ξ=1 Membrane effect: transverse fluctuations lead to decrease of membrane size in x-direction

19. Energy of fluctuations , clean membrane 0 strong anharmonicity strain tensor

20. Disorder Clean membrane 0 Random curvature 0 random vector In-plane disorder 0

21. Disordered membrane: Random curvature E random field Radzihovsky, Nelson, PRA (1991); Other models of disorder, see Le Doussal, Radzihovsky, PRB (1993) b0– strength of the disorder 0 Similar to dynamical fluctuations , 0 0

22. Scaling of x minimization of energy {db}

23. Renormalization of the stretching factor L –system size a - ultraviolet cutoff 0 0 dynamical fluctuations disorder-induced static fluctuations Flat phase is destroyed both by thermal fluctuations and by disorder x 0, for certain value of L

24. How to stabilize the flat phase? Physical mechanism: anharmonicity → interaction between h and u-modes

25. Renormalization of bending rigidity, clean case David, Guitter, Europhys. Lett. (1988), Le Doussal, Radzihovsky, PRL (1992) harmonic approximation: interaction between in-plane and out-of-plane modes is neglected However, such interaction dramatically change the small qbehavior of G(q) due to strong anharmonicity Anomalous scaling of bending rigidity

26. Integrate out the in-plane modes (D=2) 0 Interaction between out-of-plane modes Young modulus 0

27. Self-Consistent Screening Approximation Le Doussal , Radzihovsky, PRL (1992) Self-energy 0 Polarization operator

28. Perturbation theory: David, Guitter, Europhys. Lett. (1988) 0 → 0 0 ultraviolet cutoff : inverse Ginzburg length c

29. Anharmonicity-induced increase of the bending rigidity q ≪q*  universal scaling Ginzburg scale 0 Graphene: L* ~ 5 ÷10 nm • ≈ 0.7-0.8 - critical index 0 David ,Guitter, Europhys. Lett. (1988); Guitter, David, Leibler, Peliti, J. Phys. France (1989); Le Doussal ,Radzihovsky, PRL (1992) 0

30. Analytical and numerical results for h David, Guitter, Europhys. Lett. (1988) 1) 1/dc expansion for D=2 , dc >>1 Aranovitz, Lubensky, PRL (1988) 2) e– expansion for D=4-e , e<< 1 Aranovitz, Lubensky, PRL (1988) 3) self -consistent screening approximation Le Doussal, Radzihovsky, PRL (1992) for D=2, d=3 Bowick, Catterall, Falcioni, Thorleifsson, Anagnostopoulos, J. Phys. France (1996) 4) numerical simulations:h  0.7

31. Crumpling transition (clean case) critical temperature of CT for fixed bending rigidity critical bending rigidity of CT for fixed temperature

32. Rescaling of the bending rigidity rescaled bending rigidity unstable fixed point 0 crumpling transition

33. Fractalgeometryof the membrane Exactly at the transition point > 2 fractal (Hausdorff) dimension

34. Disordered graphene: Random curvature E random field Radzihovsky, Nelson, PRA (1991); Other models of disorder, see Le Doussal, Radzihovsky, PRB (1993) b – strength of the disorder Similar to dynamical fluctuations , Calculations: RPA + replica trick

35. Disordered case (random curvature) Replicas: interaction

36. Bending rigidity becomes a matrix in the replica space dimensionless disorder strength

37. Scaling in disordered graphene key parameter f ≫1 → ripples dominate f ≪ 1 → thermal fluctuations (FP) dominate f ≫1 strongly disordered case:

38. Crumpling transition in disordered membrane, s = 0 crumpled critical disorder strength flat bc ripples FP 1 f ~ 1 - critical bending rigidity for fixed temperature

39. RG flows non-monotonous scaling of rescaled bending rigidity and rescaled disorder

40. Higher order corrections in dc next order correction Saykin et al (unpublished, 2019) f unstable fixed point

41. Stabilization of the disordered phase new unstable point f = fc f = fc CT Saykin et al (unpublished, 2019)

42. Anomalous Hooke’s law External tension → new scale q = qs: scaling stops at q= qs

43. Finite tension 1) T=0, s≠0 → fluctuations are absent linear Hooke’s law 2) T≠0→fluctuations does not take into account suppression of fluctuations by s contribution of fluctuations at s = 0

44. 3) Effect of tension on fluctuations stress suppresses fluctuations !!! “hidden area”

45. Balance equation for membrane under isotropic tension normal anomalous global deformation for s = 0 сrossover tension critical indexof bucklingtransition

46. Anomalous Hooke’s law (experiment+simulation) Nicholl ,Conley, Lavrik, Vlassiouk , Puzyrev, Sreenivas, Pantelides , BolotinNat.Comm. 2015 Los, Fasolino,Katsnelson PRL (2016) Gornyi, Kachorovskii, Mirlin ,2D Mater. (2017)

47. Disorder-induced crumpling Giordanelli, Mendoza, Andrade, Gomes, Herrmann, Scientific Reports (2016) Pristine graphene membranes were damaged by adding random vacancies and carbon-hydrogen bonds.

48. Fractal dimension of crumpled graphene DH concentration of vacancies

49. Negative thermal expansion coefficient Agrees with experiment: Chen et al., 2009; Singh et al., 2010; Bao et al., 2009, Bolotin et al 2014 . T quantum effects ??? T→ 0: quantum fluctuations also renormalize (increase) bending rigidity

50. Quantum elasticity of graphene quantum coupling constant: Kats, Lebedev, PRB (2014) Burmistrov, Gornyi, Kachorovskii, Katsnelson, Mirlin, PRB (2016) →