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Dynamic Correlation Functions of 1D Quantum Liquids

PRL 96 , 196405 (2006);. PRL 99 , 110405 (2007); PRB (2007), PRA (2008). Dynamic Correlation Functions of 1D Quantum Liquids. Alex Kamenev. in collaboration with. Leonid Glazman, Yale Maxim Khodas, BNL. Michael Pustilnik, Georgia Tech. Moscow, SC4 , May 2009. Models.

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Dynamic Correlation Functions of 1D Quantum Liquids

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  1. PRL 96, 196405 (2006); PRL 99, 110405 (2007); PRB (2007), PRA (2008). Dynamic Correlation Functions of 1D Quantum Liquids Alex Kamenev in collaboration with Leonid Glazman, Yale Maxim Khodas, BNL Michael Pustilnik, Georgia Tech Moscow, SC4 , May 2009

  2. Models N –interacting quantum particles on a ring • Thermodynamic limit • Bosons or spinless fermions • Translationally invariant • Bosons with Lieb-Liniger, integrable • (1+1)D complex field theory, ``critical’’

  3. Dynamic Structure Factor (DSF) at T=0 Observables Density

  4. I. Bloch, et al 2004 Cold Atoms in Optical Lattices T. Kinoshita, et al 2004 H. Moritz, et al 2003

  5. Bragg Scattering W. Ketterle, et al 2000-…

  6. 3D Condensates Belayev 1958 Bogoliubov mode • What about 1D?

  7. Dimensionless coupling constant: Lieb-Liniger (1963) model • N bosons with delta-functional interactions on a 1D ring • Two characteristic momenta: mc and n =N/L

  8. Bethe Ansatz integers

  9. Ground state: Lieb’s I mode “particles”: Lieb’s II mode “holes”: Lieb’s Modes Bogoliubov forg  0 Lower bound of the spectral continuum

  10. Strongly Interacting Bosons = Free Fermions

  11. Linear dispersion • Exact result within the Luttinger approximation. Structure Factor (free fermions) How does the dispersion curvature and interactions affect the structure factor ?

  12. Algebraic BA exact numerics J-S. Caux, P. Calabrese, 2006 N. Slavnov, 1989

  13. DSF singularities at Lieb’s modes

  14. Single deep hole + low-energy excitations Fermi-edge singularity problem. Power-law edge singularities. Effective model states, contributing to the leading logarithm corrections in each order of the perturbation theory. the idea: project all other states out; linearize remaining spectum. Mahan 67, Nozieres-DeDominicis 69

  15. Why Power-Law ? Deep hole creation operator (instantaneous shift of density and current): power-law of Dynamic structure factor Band of low energy excitations: Popov, 1973 Efetov, Larkin, 1975 Haldane, 1981

  16. Exactly solvable models can be fixed by: (i) comparing finite size spectrum of the effective model and Bethe Ansatz spectrum with fixed total momentum q Pereira, White, Affleck, 2008, 2009 Cheianov, Pustilnik, 2008 Imambekov, Glazman, 2008 Khodas, 2009 (ii) Galilean invariance + dispersion relation of the mode Kamenev, Glazman, 2008 Lamacraft, 2009

  17. Numerics (preliminary) Courtesy of J-S. Caux

  18. Singularities at Lieb’s modes • For integrable models: positions and exponents are known from TBA • Scaling functions ??? • Singularities are NOT smeared by temperature • Non-integrable models: three-body scattering smears singularity in the bulk, but NOT at the edge

  19. Weakly Interacting Bosons Phonon modes shake-up  power-law singularity

  20. Weakly Interacting Bosons What are the excitations below line? For infinite number of quasiparticles is excited.

  21. Dark Solitons Gross-Pitaevskii equation

  22. Observations of Dark Solitons Cornell, et al. 2001 Sengstock, et al. 1999 Phillips, et al. 2000

  23. Dark Solitons as Lieb II excitations P.P. Kulish, S.V. Manakov and L.D. Faddeev 1976 Dark soliton Lieb II mode

  24. Probability to excite a soliton is suppressed by orthogonality Photo-Solitonic Effect

  25. Wake up! • Power-law singularities at Lieb modes, where exponents are functions of q • Single energetic particle + low-energy excitations • Single particle = quasiparticle or soliton • Photo-Solitonic Effect

  26. 1D Spinor Condensates Hyperfine states, e.g. F=1. Ferromagnetic ground state J.M. Higbie, et al 2008 M. Vingalatore, et al 2008

  27. 1D Spinor Condensates Ferromagnetic magnon

  28. Atomic number Gravitational Fall in Spinor Condensate M. Kohl mg detector

  29. No tail ! l>1 repulsive fermions Pustilnik, 2006 Calogero-Sutherland model S =k +n dim integral Haldane 1994, Ha 1995

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