1. Ab initio simulations of symmetry- determined nonlinear dynamical regimes in single-layer graphene S. Shcherbinin G. Chechin, D. Ryabov Russia, Rostov-on-Don 2016

2. 1 Nonlinear normal modes (NNMs) We consider nonlinear vibrations in Hamiltonian systems with N degrees of freedom x i (t)=a i *f(t), i=1..N a i – constant coefficients f(t) – time-periodic function NNM by Rosenberg System of N differential equations System of N-1 algebraic nonlinear equations + 1 differential “governing” equation Rosenberg NNMs can exist only in some very specific classes of dynamical systems, particularly, in systems whose potential energy is a homogeneous function of all its arguments.

3. 2 Symmetry determined nonlinear normal modes (SD NNMs) Existence of NNMs in systems of general type may be provided by discrete symmetry groups General case: - V.P. Sakhnenko, G.M. Chechin. Phys. Dokl. 38, 219 (1993). - G.M. Chechin, V.P. Sakhnenko. Physica D 117, 43 (1998). FPU-chains: - G.M. Chechin, N.V. Novikova, A.A. Abramenko. Physica D 166, 208 (2002). - G.M. Chechin, D.S. Ryabov, K.G. Zhukov. Physica D 203, 121 (2005)

4. 3 Bushes of nonlinear normal modes The concept of bushes of nonlinear normal modes for dynamical systems with discrete symmetry groups was introduced in -V.P. Sakhnenko, G.M. Chechin. Phys. Dokl. 38, 219 (1993) - G.M. Chechin, V.P. Sakhnenko. Physica D 117, 43 (1998) m-dimensional bush consists of m nonlinear normal modes (hereafter, we deal only with vibrational modes), which are connected with each other by force interactions. The set of bush modes is conserved during the time-evolution, while amplitudes of these modes change in time. Bush represents an exact solution to nonlinear dynamical equations. The energy of initial excitation of a given bush turns out to be trapped in this dynamical object until it loses stability (then the bush transforms into another bush of lower symmetry and larger dimension).

5. 4 Symmetry properties of the bushes of NNMs Existence of bushes is due to a certain symmetry selection rules for excitation transfer between NNMs of different symmetry. In particular, the excitation from the mode with lower symmetry can transfer to modes with higher symmetry, but not vice versa! Each bush possesses its own symmetry group G which is a subgroup of the dynamical system symmetry group G 0. The group-theoretical methods for construction of bushes were developed in the above cited papers. They don’t require any information about interparticle interactions in the physical system. We must know only its symmetry!

6. 5 Properties of the bushes of NNMs As a rule, in physical systems, there can be excited bushes of modes of different dimensions: 1D, 2D, 3D, 4D, etc. One-dimensional bush represents a Rosenberg mode. Bush with m > 1 describes a quasiperiodic dynamical regime determined by m governing differential equations. The bush stability loss at large amplitudes of its modes can occur due to the “parametric interactions” with other modes of the physical system, which we call “sleeping” modes. This phenomenon is similar to the parametric resonance.

7. 6 Bushes of NNMs: root mode and secondary modes Different modes of a given bush possess different symmetries and, as a consequence, they have “different rights”. The mode (modes) with the lowest symmetry is the root mode of the bush, because its exciting leads to excitation of some other modes with a higher symmetry (we call them secondary modes). The full collection of all root and secondary modes form the given bush. Its total symmetry determined by the symmetry of the root mode.

8. 7 Linear normal modes and the theory of bushes of NNMs Conventional or linear normal modes (LNMs) represent exact solutions for Hamiltonian system in harmonic approximation. They cease to be exact, when we take into account some anharmonic terms. If anharmonism is weak, one can speak about interactions between LNMs and then construct some approximate solutions. Fundamental question: “Are there any exact solutions for the Hamiltonian systems with discrete symmetries beyond the harmonic approximation?” The theory of bushes of NNMs gives the answer to this question.

9. 8 Bush theory: decomposition of the configuration vector According to the Wigner theorem, LNMs can be classified by irreducible representations (irreps) of the system symmetry group G 0. According to the bush theory, the configuration vector X(t) can be decomposed into contributions from basis vectors of irreps of the group G 0 : For a given m-dimensional bush with symmetry group G<G 0, this decomposition contains only m terms! X ̅ (t)=∑c j (t)ξ ̅ j

10. 9 Ab initio calculations For checking validity of the bush theory, we use ab initio calculations based on the density functional theory (DFT), which proved to be very effective and rather correct for studying molecules and crystals We used DFT methods realized in Quantum espresso package http://www.quantum-espresso.org

11. 10 Single-layer graphene Symmetry group G 0 =P6mm=C6v

12. 11 One-dimensional bushes in graphene 1D bush with space group Cmm21D bush with space group P31m 1D bush with space group P6mm

13. One-dimensional bush with space group Cmm2 in 5% uniformly stretched graphene f,THz A, angstrom Initial displacement = 5% of lattice parameterInitial displacement =10% of lattice parameter 12

14. 13 One-dimensional bushes in graphene Complete condensates of order parameters corresponding to two-dimensional vibrational bushes. Root modes are highlighted by gray color

15. Thank for your attention

16. Displacement patterns corresponding to the basis vectors of irreducible representations of the graphene symmetry group Symmetry group G 0 =P6mm=C6v