Departament de Fìsica i Enginyeria Nuclear,
Campus Nord B4-B5, Universitat Politècnica de Catalunya,
Barcelona, Spain
Dipolar and Coulomb
one-dimensional gases
G.E. Astrakharchik
Summer school, Trier, August 16 (2012)
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
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-Girardeau
regime, 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?
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
DIPOLE-DIPOLE INTERACTION
MODEL HAMILTONIAN
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
ARE DIPOLAR INTERACTIONS LONG RANGE?
DESCRIBE DIPOLAR INTERACTIONS AS δ-POTENTIAL
MONTE CARLO METHODS
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
E/N
GROUND STATE ENERGY
10
6
10
5
10
4
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
dipoles
Tonks-Girardeau gas
crystal
-4
10
0.01
0.1
1
10
n1D r0
100
Energy per particle as a function of n1D r0 (red solid line), energy of the TonksGirardeau gas ETG (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)
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
ECC
2
  (3) 2 ( nr0 )3
N
mr0
i.e. energy of corresponding
classical crystal - is
cubic in the density.
E/N
Energy per particle
10
6
10
5
10
4
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
dipoles
Tonks-Girardeau gas
crystal
-4
10
0.01
0.1
1
10
n1D r0
100
GENERIC BOSE-FERMI MAPPING
Following quantities are
known exactly:
1) Energy per particle:
ETG  2 2n 2

N
6m
i.e. energy of corresponding
fermi gas - is quadratic in the
density.
E/N
TONKS-GIRARDEAU LIMIT
10
6
10
5
10
4
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
dipoles
Tonks-Girardeau gas
crystal
-4
10
0.01
0.1
1
10
n1D r0
100
TONKS-GIRARDEAU LIMIT II
2) Pair distribution function (gives the possibility to find a particle at a
distance x from 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
 | k | /(2n ), | k | 2n
as in the corresponding fermi system, is S ( k )  
| k | 2n
1,
linear up to 2kf, with kf = π n being the
fermi momentum.
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
PAIR DISTRIBUTION FUNCTION
2
n1Dr0 = 0.01
n1Dr0 = 0.1
n1Dr0 = 1
g2(x)
n1Dr0 = 10
1
0
0
1
2
n1D x
3
Pair distribution function g2 (z) obtained from a DMC calculation for densities
n r0 =10-2; 0.1; 1; 10 (larger amplitude of oscillations correspond to higher peaks)
PAIR DISTRIBUTION FUNCTION
2
n1Dr0 = 0.01
n1Dr0 = 1
n1Dr0 = 0.1
n1Dr0 = 10
g2(x)
1
0
0
1
2
3
4
5
6
7
8
9
10
n1D x
Pair distribution function g2 (z) obtained from a DMC calculation for densities
n r0 =10-2; 0.1; 1; 10 (larger amplitude of oscillations correspond to higher peaks)
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
LUTTINGER LIQUID
LUTTINGER PARAMETER
STATIC STRUCTURE FACTOR
3
2
n1Dr0 = 0 (TG gas)
Sk
2
n1Dr0 = 0.1
2
2
n1Dr0 = 1
1
0
0
5
10
k / n1D
15
Static structure factor S(k) obtained from a DMC. A higher peak corresponds to a
higher density. The red line corresponds to the static structure factor S(k) in the
Tonks-Girardeau regime.
LUTTINGER PARAMETER IN 1D DIPOLAR GAS
Luttinger exponent K. 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. Chiofalo PRA 75, 051602 (2007)
LUTTINGER PARAMETER IN LIEB-LINIGER GAS
Luttinger exponent K versus γ. The dashed lines are the small γ approximations
obtained from Bogoliubov theory whereas the dotted-dashed lines correspond to
the asymptotic expressions for large γ.
Fig. from M. A. Cazalilla J. Phys. B 37, S1 (2004)
STATIC STRUCTURE FACTOR
Static structure factor S(k) for obtained in Reptation Quantum Monte Carlo
calculation with N=40 particles and different values of n1D r0 =0.01; 50; 100; 1000
. Decreasing slopes for small momentum and the emergence of additional peaks
correspond increasing n1D r0. Fig. from R. Citro, et al. PRA 75, 051602 (2007)
FREQUENCIES OF COLLECTIVE OSCILLATIONS
Square of the breathing mode frequency ω2 (in units of trap frequency) as a function
of characteristic parameter N r02/ az2. The external potential is taken to be harmonic:
Vext   1m z2 zi2
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 solitonlike 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
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
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
6
4
0
MF
-GP
a1D
2
resonance
TG,
g1D /(aoscosc); a1D / aosc
8
g1D +
sTG GAS OF DIPOLES: IDEA
g1D
-2
sTG,
ne
collapse / solito
g 1D
-
-4
-6
-3
-2
-1
0
1
2
3
a3D / aosc
0.4
(x)
Repulsion, g1D > 0
Attraction, g1D < 0
0.3
0.2
0.1
0.0
-0.1
-10
x, [a.u.]
a1D < 0 a1D > 0
-5
0
5
10
sTG GAS OF DIPOLES: WAVE FUNCTION
3
Repulsion, +1/|x|
0.003
(x)
0.002
0.001
x, [a.u.]
0.000
0.0
0.1
0.2
3
Attraction, -2/|x|
0.10
(x)
0.05
0.00
-0.05
x, [a.u.]
-0.10
0.0
0.1
0.2
STABILITY OF DIPOLAR sTG GAS
0.05
E/N
0.04
Tonks-Girardeau gas
repulsive dipoles
attractive dipoles
fit
0.03
0.02
0.01
0.00
0.0
0.1
0.2
n1D r0
0.3
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
COULOMB INTERACTION POTENTIAL
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
FERMIONS: SIGN PROBLEM
MODEL HAMILTONIAN
ONE COMPONENT SYSTEM:IDEAL FERMI GAS
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
GROUND STATE ENERGY
4
10
E/N
3
10
DMC
IFG / TG
E/N =  n /6m
Wigner cr.
E/N = e n ln N
2 2
2
2
2
10
N = 1000
N = 100
N = 10
1
10
0
10
-1
10
-2
10
-3
10
-3
10
-2
10
-1
10
0
10
1
10
2
n a0
10
ONE COMPONENT SYSTEM:WIGNER CRYSTAL
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
PLASMON DISPERSION RELATION
STATIC STRUCTURE FACTOR
2
10
Monte Carlo
phonons,  ~ |k|
1/2
plasmons,  ~ (3-2 ln |k|) |k|
harmonic theory (upper bound)
S(k)
1
10
0
10
-1
10
-2
10
-3
10
0.1
1
10
k/n
LUTTINGER LIQUID
PAIR DISTRIBUTION FUNCTION
g2(z)
DMC
bosonization
1/2
amplitude: 1 A1exp{-4 c2 ln z}
2
1
0
0
5
10
z / ao
15
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
MOMENTUM DISTRIBUTION
10
na=1
fermions
bosons
n(k)
5
-6
-3
0
3
k/n
6
MOMENTUM DISTRIBUTION
1.5
n a = 0.1
fermions
bosons
IFG
n(k)
1.0
0.5
0.0
-6
-3
0
3
k/n
6
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
TRAPPED SYSTEM
TRAPPED IDEAL FERMI / TONKS-GIRARDEAU GAS
1.5
n(x)
1.0
0.5
0.0
-5
-4
-3
-2
-1
0
1
2
3
4
x / aho
5
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
LOCAL DENSITY APPROXIMATION
LDA: POLYTROPIC EoS
LDA vs EXACT PROFILE or GPE
1.5
n(x)
1.0
0.5
0.0
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
x / aho
6
LDA: IFG / TG GAS / LIMIT OF SMALL CHARGE
LDA: LIMIT OF LARGE CHARGE
ENERGY OF A TRAPPED COULOMB GAS
4
10
Wigner crystal,
2
2/3
E/N = 3/10 (3q N ln N) 
E/N
3
10
IFG / TG,
E/N = N/2 
2
10
N=5
N = 10
N = 100
1
10
0
10 -2
10
-1
10
0
10
1
10
2
q
10
DENSITY PROFILE: q=0.1
0.15
n(r)
charge q = 0.1
LDA, IFG/TG
LDA, Wigner cr.
0.10
0.05
0.00
-10
-5
0
5
r / aho 10
DENSITY PROFILE: q=0.316
0.15
n(r)
charge q = 0.316
LDA, IFG/TG
LDA, Wigner cr.
0.10
0.05
0.00
-10
-5
0
5
r / aho 10
DENSITY PROFILE: q=1
0.15
n(r)
charge q = 1
LDA, IFG/TG
LDA, Wigner cr.
0.10
0.05
0.00
-10
-5
0
5
r / aho 10
DENSITY PROFILE: q=3.16
0.15
n(r)
charge q = 3.16
LDA, IFG/TG
LDA, Wigner cr.
0.10
0.05
0.00
-10
-5
0
5
r / aho
10
DENSITY PROFILE: q=10
0.15
n(r)
charge q = 10
LDA, IFG/TG
LDA, Wigner cr.
0.10
0.05
0.00
-20
-15
-10
-5
0
5
10
15
r / aho
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
FREQUENCIES OF COLLECTIVE OSCILLATIONS
Square of the breathing mode frequency ω2 (in units of trap frequency) as a function
of characteristic parameter N r02/ az2 (or N a1D2/ az2).
CONCLUSIONS 1/4
CONCLUSIONS 2/4
CONCLUSIONS 3/4
CONCLUSIONS 4/4
DANKE SCHÖN
FÜR IHRE
AUFMERKSAMKEIT!