Quantization of ϐ -Fermi-Pasta-Ulam lattice

1. Quantization of ϐ-Fermi-Pasta-Ulam lattice Bishwajyoti Dey Department of Physics SP Pune University, Pune, India with Aniruddha Kibey and Rupali Sonone, Dept of Phys., Pune University J.C. Eilbeck, Heriott-Watt University, Edinburgh, UK Physica D: Nonlinear Phenomena 294, 43 (2015). arXiv: 1408.0212

2. ϐ-FERMI-PASTA-ULAM (FPU) LATTICE • FPU model first introduced In 1953 to study heat conduction in solids and metals. • FPU Model – numerical simulation using first computer built. • Numerical simulation results were very surprising – beginning of • Nonlinear Dynamics. • Fermi-Pasta-Ulam problem: A status report. G. Gallavotti, Springer (2010).

3. Fermi-Pasta-Ulam models have been used for modeling various metamaterials • Discrete breathers in metamaterials. Tsironis et al. • Modeling of nonlinear metamaterials, P.L. Colestook et al, Proc. SPIE 8093 (2011). • Nonlinear phononics crystal or nonlinear acoustic metamaterials. Midtvedt et al, • Nature Comm. (2014). • Bose-Einstein condensate in optical lattice. Rev. Mod. Phys. Morsch and • Oberthaler (2006). Preparing and engineering quantum states in a controlled way. • Bose-Hubbard model. • Nonlinear dynamics of coupled granular networks: Granular acoustic metamaterials • Solitary waves in granular chain, S. Sen et al, Phys. Rep. 462, 21 (2008).

4. Outline of the talk 1. Nonlinear Lattice - beta-Fermi-Pasta-Ulam lattice - Classical Lattice vibration. Discrete breathers. 2. Quantization of the Classical Lattice vibration of beta-Fermi-Pasta-Ulam lattice. 3. Number conserving approximation. 4. Analytic mean field analysis. 5. Numerical Exact Diagonalization Method and energy spectrum. 6. Identifying quantum discrete breathers: Correlation functions. 7. FPU lattice with NN and NNN interactions. 8. Future scope.

5. Sievers and Takeno discovered spatially localized time-periodic • excitations in nonlinear Fermi-Pasta- Ulam lattice (1988). • These nonlinear localized excitations are termed as “Discrete Breathers” • or Intrinsic Localized Modes (ILM’s). Linearization around classical ground state give For Discrete Breather Xn(t+tb) = Xn (t) For we get Non resonance condition Linear lattice vibration- Extended state Nonlinear lattice vibration- Localized state

6. ENERGY LOCALIZATION : Local energy density If DBs are excited, initial local energy stay within the DB. Dashed line e5 vs time Solid line –Total energy of the chain e5

7. What is quantum counterpart of classical discrete breather or what is quantum discrete breather? Indications from early theories: Integrable classical field equation such as Sine-Gordon, nonlinear Schrodinger equation have breather solution which is a bound state of soliton and antisoliton. Bohr-Sommerfeld semi-classical quantization for classical breather shows discrete excitations which are bound states of small-amplitude wave excitations. Dashen, Hessler, Neveu, PRD (1974, 1975). Quantum field theory of solitons for integrable systems have shown breathers as multi-phonon bound states. Fadeev, Korepin, Phys. Rep. (1978). Similar results from numerical studies on Klein-Gordon type nonlinear lattice. Wang, Gammel , Bishop, PRL (1996). Indications from experiments: Quantum discrete breathers seen in expt. as bound states of up to seven phonons in PtCl (Swanson et al, PRL 1999) and as two-phonon bound states in CO2 crystals. Dows et al, JCP (1973).

8. Quantization of Nonlinear vibrational modes of FPU lattice. The quantum counterparts are expected to be bound states of arbitrary number of phonons. The Hamiltonian for beta-Fermi-Pasta-Ulam lattice is given by Define phonon creation and annihilation operators as Quantization Scheme: Number conserving approximation – disregard all number nonconserving terms.

9. By virtue of lattice periodicity and for lattice sites N →∞ the number conserving FPU Hamiltonian Analytic mean field study - Ivic and Tsironis (2006), Sarkar and Dey (2008) Mean field approximation: the last two terms describes effectively the single phonon tunneling. It can be treated in a mean field manner as follows: Independent of lattice sites, depends only on the number of quanta m in state

10. Two-phonon sector: Two-phonon state vector lead to the following equation for the biphonon amplitude Using lattice periodicity: Expanding in Fourier series:

11. The eigenvalue problem for the two-phonon sector reduces to an integral equation for the Fourier components where the energy of the two free phonon This integral equation have been solved analytically approximately to obtain the energy eigenvalue spectrum of the mean field 1D ϐ-Fermi-Pasta-Ulam chain (Ivic and Tsironis (2006)) . Similar equation for curved (wedge shaped) Fermi-Pasta-Ulam chain (Sarkar and Dey, 2008). Analytical mean field eigenvalue spectrum of the 1D ϐ-FPU chain for two values of the nonlinearity parameter. Ivic and Tsironis(2006). Band crossing?

12. Computational Method: The number state method (J.C. Eilbeck, 2002). Numerical exact diagonalization technique.

15. Allowed k-values for f lattice sites:

16. Results and Discussions Analytic mean field eigenvalue spectra Ivic and Tsironis (2006) Numerical mean field eigenvalue spectra (B. Dey et al, 2015)

17. Eigenvalue spectra for the full Hamiltonian (without mean field approx.) 2-quanta sector The overlap between the bands reduces with higher value 0f nonlinear interaction parameter 4-quanta sector Here we expect five bands as there are Five possibilities of distributing four quantas on f lattice sites. These are [4,0,0,….], [3,1,0,0…], [2,2,0,0…], [2,1,1,0,0…] and [1,1,1,1,0,0..]. The overlap between the bands reduces with increasing value of the nonlinearity parameter.

18. Identifying discrete breather bands

19. An arbitrary state vector of the system is written as For mean-field Hamiltonian Full Hamiltonian

20. Spatial correlation function Another way to identify discrete breathers is to calculate spatial correlation function to establish the main characteristic of spatial localization of breathers. The spatial correlation of the displacements at sites separated by n-lattice spacing Mean field Hamiltonian Full Hamiltonian

22. Future scopes Going beyond number conserving approximation – include number nonconserving terms in the Hamiltonian. Hamiltonian matrix is no more block diagonal. Have to deal with extended Hilbert space. Fix maximum number of quanta per site (truncation of Hilbert space). For maximum m=2 quanta per site and for f=3 sites, there are 27 states. [2 2 2], [2 2 1], [1 2 2], [2 1 2], [2 2 0], [0 2 2], [2 0 2], [2 1 1] ….. [0 0 0] etc This number of states runs into several thousands with higher m and f values. Numerically diagonalize the large matrix and obtain eigenvalue spectrum. Identification of breather bands not easy in this case as bands get mixed up. Usually breather bands are narrow . Bishop et al PRL (1996).

23. Quantization of more general Hamiltonian FPU case Compact-like discrete breather, Eleftheriou, Dey, Tsironis (2000) V(u) = V(u) = Morse potential

24. Conclusions: Classical lattice vibrations of Fermi-Pasta-Lattice are intrinsic localized modes or discrete breathers. These localized modes are quantized using numerical exact diagonalization method and number conserving approximation. From the calculations of the energy spectrum and the correlation functions it is shown that quantum breathers exists as bound states of multi-phonons on onsite and nearest sites . Inclusion of next-nearest interactions do not change the nature of eigenspectra and correlation functions except that the Brillouin zone is reduced to half.