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Dipolar and Coulomb one-dimensional gases

Dipolar and Coulomb one-dimensional gases. Departament de F ì sica i Enginyeria Nuclear, Campus Nord B4-B5, Universitat Politècnica de Catalunya, Barcelona, Spain. G.E. Astrakharchik Summer school, Trier , August 16 (2012). CONTENTS:. • Dipole-dipole interactions

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Dipolar and Coulomb one-dimensional gases

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  1. Dipolar and Coulomb one-dimensional gases Departament de Fìsica i Enginyeria Nuclear, Campus Nord B4-B5, Universitat Politècnica de Catalunya, Barcelona, Spain G.E. Astrakharchik Summer school, Trier, August 16 (2012)

  2. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  3. OVERVIEW: SHORT-RANGE INTERACTION IN 1D • Short-range interaction can be modeled by a δ-function interaction potential (Lieb-Liniger model) •high density (small γ) – mean-field regime, properties are described by Gross-Pitaevskii equation • low density (large γ) – Tonks-Girardeau regime, diagonal properties and energy same as in ideal Fermi gas • low density (large and negative γ) – super-Tonks-Girardeauregime, properties of in the gas-like metastable state are similar to that of a gas of hard-rods •is it possible to realize a stable gas in super-Tonks-Girardeau regime with long-range potentials?

  4. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  5. DIPOLE-DIPOLE INTERACTION

  6. MODEL HAMILTONIAN

  7. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  8. ARE DIPOLAR INTERACTIONS LONG RANGE?

  9. DESCRIBE DIPOLAR INTERACTIONS AS δ-POTENTIAL

  10. MONTE CARLO METHODS

  11. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  12. GROUND STATE ENERGY Energy per particle as a function of n1Dr0 (red solid line), energy of the Tonks-Girardeau gasETG (dashed line), energy of a classical crystal Ecr(dot-dashed line). A. S. Arkhipov, G. E. A., A. V. Belikov, and Yu. E. Lozovik, JETP Letters, 82, 39 (2005)

  13. CLASSICAL CRYSTAL LIMIT • In the limit of large density (n1Dr0 1) the potential energy dominates, in 2D geometry a crystal gets formed • Phonons in one-dimensional geometry destroy crystalline order • Still the energy can be calculated assuming a crystalline order • Energy per particle • i.e. energy of corresponding classical crystal - is cubic in the density.

  14. GENERIC BOSE-FERMI MAPPING

  15. TONKS-GIRARDEAU LIMIT Following quantities are known exactly: 1) Energy per particle: i.e. energy of corresponding fermigas - is quadratic in the density.

  16. TONKS-GIRARDEAU LIMIT II 2) Pair distribution function (gives the possibility to find a particle at a distance xfrom another particle)In the Tonks-Girardeau regime is the same as in the corresponding Fermi system and experience Friedel-like oscillations:3) Static structure factor (correlation function of the momentum distribution between elements –k and k)In the Tonks-Girardeau regime is the same as in the corresponding fermi system, is linear up to 2kf, with kf= πnbeing the fermi momentum.

  17. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  18. PAIR DISTRIBUTION FUNCTION Pair distribution function g2 (z) obtained from a DMCcalculation for densities n r0=10-2; 0.1; 1; 10 (larger amplitude of oscillationscorrespondto higher peaks)

  19. PAIR DISTRIBUTION FUNCTION Pair distribution function g2 (z) obtained from a DMCcalculation for densities n r0=10-2; 0.1; 1; 10 (larger amplitude of oscillationscorrespondto higher peaks)

  20. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  21. LUTTINGER LIQUID

  22. LUTTINGER PARAMETER

  23. STATIC STRUCTURE FACTOR Static structure factor S(k)obtained from a DMC. A higher peakcorresponds to a higherdensity. The red line corresponds to the static structure factor S(k)in the Tonks-Girardeau regime.

  24. LUTTINGER PARAMETER IN 1D DIPOLAR GAS Luttinger exponentK. Classical crystal and Tonks-Girardeau limits are shown with dashed lines. Inset: small-momentum part of the static structure factor. Fig. from R. Citro, E. Orignac, S. De Palo, M.-L. ChiofaloPRA 75, 051602 (2007)

  25. LUTTINGER PARAMETER IN LIEB-LINIGER GAS Luttinger exponentK versus γ. The dashed lines are the small γ approximations obtained from Bogoliubovtheory whereas the dotted-dashed lines correspond to the asymptotic expressions for large γ. Fig. from M. A. CazalillaJ. Phys. B 37, S1 (2004)

  26. STATIC STRUCTURE FACTOR Static structure factor S(k)for obtained in Reptation Quantum Monte Carlo calculation with N=40particles and different values of n1Dr0=0.01; 50; 100; 1000 . Decreasing slopes for small momentum and the emergence of additional peaks correspond increasing n1Dr0. Fig. from R. Citro, et al. PRA 75, 051602 (2007)

  27. FREQUENCIES OF COLLECTIVE OSCILLATIONS Square of the breathing mode frequencyω2(in units of trap frequency) as a function of characteristic parameter Nr02/ az2. The external potential is taken to be harmonic:

  28. COMPARISON WITH SHORT-RANGE POTENTIAL We make comparison of the properties of a long-range dipole potential with the properties of a short-range potential Vint(z)=g1Dδ(z) , where the coupling constant is related to 1D scattering length g1D = -22/(m a1D2). Properties of the system are governed by a one-dimensional gas parameter na1D. Two situations should be considered separately: 1) Repulsive interaction (Lieb-Liniger gas): - coupling constant g1D>0 - s-wave scattering length a1D<0 The gas state is always stable. 2) Attractive interaction - coupling constant g1D - - s-wave scattering length a1D  +0 The ground state has large negative energy and corresponds to soliton-like solution. The gas like state is stable in the regime of small densities n a1D <0.3 (Super-Tonks gas) and has analogies with a gas of hard-rods

  29. GROUND STATE ENERGY: COMPARISON Energy in units of ћ2/mr02 (or ћ2/ma1D2). Solid lines- green: system of dipoles, blue: Lieb-Liniger gas, red:Super-Tonks gas. Dashed lines- green ~n1,dark green: ~n2, blue: ~n3.At small n LL and ST energy correction to TG gas has same absolute value.

  30. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  31. sTG GAS OF DIPOLES: IDEA

  32. sTG GAS OF DIPOLES: WAVE FUNCTION - - - + + + + + + - - -

  33. STABILITY OF DIPOLAR sTG GAS

  34. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  35. COULOMB INTERACTION POTENTIAL

  36. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  37. FERMIONS: SIGN PROBLEM

  38. MODEL HAMILTONIAN

  39. ONE COMPONENT SYSTEM:IDEAL FERMI GAS

  40. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  41. GROUND STATE ENERGY

  42. ONE COMPONENT SYSTEM:WIGNER CRYSTAL

  43. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  44. PLASMON DISPERSION RELATION

  45. STATIC STRUCTURE FACTOR

  46. LUTTINGER LIQUID

  47. PAIR DISTRIBUTION FUNCTION

  48. CONTENTS: •Dipole-dipole interactions •Are dipolar interactions long-range? •Ground-state energy: crystal, Tonks-Girardeau regimes •Correlation functions •Luttinger liquid • Attractive dipoles: dipolar analog of sTG gas •Coulomb gas •Fermionic sign problem •Energy •Excitation spectrum: plasmons •Momentum distribution: bosons vs fermions •Trapped gases •Local density approximation • Frequency of collective oscillations

  49. MOMENTUM DISTRIBUTION

  50. MOMENTUM DISTRIBUTION

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