Universal contact for a strongly
interacting 1D Bose gas
Anna Minguzzi
Laboratoire de Physique et Modélisation des Milieux Condensés, Grenoble
in collaboration with Patrizia Vignolo (INLN, Nice), arXiv:1209.1545
PEPS-PTI 2012 – p.1/35
Universality
Universal properties: do not depend on microscopic
details (eg in proximity of a phase transition)
In quantum gases: possibility to tune the interactions to
very large values (eg with Feschbach resonances)
−1
s-wave scattering amplitude fk = a−1 +ik−k
2 r /2
s
e
if as → ∞, re ≪ as , n−1/3 ⇒ spatial scale invariance, no
length scale associated to interactions
A largely studied case: 2-component Fermi gas in the
unitary limit 1/kF as = 0
In this work: universal aspects of a 1D Bose gas with
strong repulsive interactions
PEPS-PTI 2012 – p.2/35
Plan
exact solutions for strongly interacting 1D gases:
external confinement and effects of temperature
Overview on 1D gases
theoretical approaches: Luttinger
liquid, Bethe Ansatz, TonksGirardeau
Psi_rel(x) a_ho
0.2
0.1
2
1
0
-4
-2
n(k)/aho (Log scale)
0
n(k)/aho
One-body coherences for a TG
gas at finite temperature:
momentum distribution
large momentum tails
0.3
0
x/a_ho
4
0.1
0.01
0.001
0.0001
0
2
5
6
10
kaho (Log scale)
kaho
10
15
15
PEPS-PTI 2012 – p.3/35
Many-body properties of strongly
interacting 1D Bose fluids
PEPS-PTI 2012 – p.4/35
1D quantum gases
Quasi-1D geometry:
ultracold atoms in tight transverse confinement
µ, kB T ≪ ~ω⊥
2D deep optical lattices, chip traps
z
x
y
PEPS-PTI 2012 – p.5/35
Some experimental results
1D bosons in the strongly interacting regime
density profiles, momentum distribution, correlation functions,
collective modes, transport, number fluctuations...
[T Kinoshita et al, (2005)]
[B. Paredes et al, (2004)]
(a)
1.52
0.51
0.30
#$
0.22
0.15
16
(b)
40
t = 5.4
N
[T Kinoshita et al (2004)]
20
N2
12
0
-30
8
0 z /! 30
(c)
t
4
0
0
10
20
30
N
40
50
7
6
5
0.7 0.77 " 0.85
[T. Jacqmin et al, (2011)]
[E Haller et al, (2009)]
[S. Palzer et al, (2009)]
PEPS-PTI 2012 – p.6/35
The model
ultracold dilute bosonic gases in 3D: binary interactions
through s-wave scattering length as
for atoms in a tight waveguide [Olshanii, 1998]
v(x) = gδ(x) with g = 2as ~ω⊥ (1 − 0.4602 as /a⊥ )−1
model Hamiltonian [Lieb and Liniger, 1963]
H=
X
i
X
~2 ∂ 2
+ V (xi ) + g
δ(xi − xj )
−
2
2m ∂xi
i<j
Lieb-Liniger model with external potential
coupling strength:
γ = gn/(~2 n2 /m)
note: strong coupling at weak densities
PEPS-PTI 2012 – p.7/35
Peculiar properties in 1D
No BEC for a homogeneous 1D Bose gas
ideal gas:
n=
Z
1
1
g1/2 (eβµ )
=
dk
exp[β(εk − µ)] − 1
λdB
always invertible, no macroscopic occupation of nk=0
interacting gas:
Bogoliubov-Hohenberg-Mermin inequality at T > 0
mkB T
1
nk ≥ 2 2 n0 −
~k n
2
PEPS-PTI 2012 – p.8/35
Peculiar properties in 1D
No BEC for a homogeneous 1D Bose gas
ideal gas:
n=
Z
1
1
g1/2 (eβµ )
=
dk
exp[β(εk − µ)] − 1
λdB
always invertible, no macroscopic occupation of nk=0
interacting gas:
Pitaevskii-Stringari inequality at T = 0 S(k): structure factor
n0
1
nk ≥
−
4nS(k) 2
⇒ from n =
R
dk nk then n0 = 0
PEPS-PTI 2012 – p.8/35
1D gases in harmonic trap
BEC possible at weak interactions and low temperature;
destroyed by thermal and quantum fluctuations
[Ketterle and Van Druten, 1996]
α=10, N *=100
10000
quasicondensate
true condensate
Thomas-Fermi profile
[Petrov and Shlyapnikov, 2000]
100
classical gas
gas of Tonks
10
degeneracy limit
10
T/ ω
100
1000
N
1000
PEPS-PTI 2012 – p.9/35
From quasicondensate to TG
Bose-Einstein condensation in 3D: off-diagonal long
range order for |x − x′ | → ∞ [Penrose and Onsager, 1965]
hΨ† (x)Ψ(x′ )i → n0
PEPS-PTI 2012 – p.10/35
From quasicondensate to TG
quantum fluctuations: important in one-dimension
in 1D quasi-long range order for |x − x′ | → ∞ [Haldane, 1981]
1
hΨ (x)Ψ(x )i →
|x − x′ |1/2K
K: Luttinger parameter
depends on interactions
′
5
Luttinger parameters
†
4
K
3
2
1
0
0.1
vs / v
1
F
g
10
100
Regimes of quantum degeneracy at T = 0:
γ ≪ 1 “quasicondensate”
condensate with fluctuating phase, K ≫ 1
γ ≫ 1 “Tonks-Girardeau” gas
impenetrable-boson limit, K = 1
PEPS-PTI 2012 – p.10/35
Several theory approaches
in addition to powerful numerical methods
homogeneus system, arbitrary interaction strengths:
exact solution with the Bethe Ansatz
(mainly) homogeneous system, arbitrary interactions, low
energy: the Luttinger-liquid approach
inhomogeneous system, infinite interactions: the
Tonks-Girardeau exact solution
PEPS-PTI 2012 – p.11/35
The Bethe-Ansatz solution
[E Lieb and W Liniger, Phys Rev 130, 1605 (1963)]
Many-body wavefunction
Ψ(x1 , ...xN ) =
X
a(P )e
i
P
j
kP (j) xj
P
with a(P ) amplitudes connecting different permutation
sectors, kj momentum rapidities defined from
kj L +
X
ℓ
2 arctan[(kj − kℓ )/c] = 2πIj
with c = mg/~2 , and Ij given integers (half integers) for N
odd (even), a set of quantum numbers
PEPS-PTI 2012 – p.12/35
The Bethe-Ansatz solution
Ground state energy E =
P
2
k
j j and excitation spectrum
Two excitation branches: the “Lieb-I” and “Lieb-II” modes
Several recent advances to compute correlation
functions: J.S. Caux, J.M. Maillet, ...
PEPS-PTI 2012 – p.13/35
The Luttinger liquid method
A quantum hydrodynamic approach (hyp: linear sound dispersion)
R dx 2
2
1
HLL = ~vs 2π K (∇φ(x)) + K (∇θ(x))
K: Luttinger parameter
θ(x) and φ(x) : fields for density and phase
Calculation of correlation functions: use the bosonic field
operator
P+∞
2miθ(x)+2miπρ0 x −iφ(x)
†
1/2
e
e
Ψ (x) = A [ρ0 + ∂x θ(x)/π]
m=−∞
P
and the mode expansion (HLL = ~vs j6=0 kj b†j bj )
−a|kj |/2
sign(kj )e
†
ik
x
−ik
x
j b
√
+ φ0 + πx
e j bj + e
J
j6=0
j
L
|kj |
−a|k |/2 P
e √ j
ikj x
−ikj x †
e
bj + e
bj + θ0 + πx
(N̂ − N )
j6=0
L
φ(x) =
q
π
2KL
θ(x) =
q
πK
2L
P
|kj |
PEPS-PTI 2012 – p.14/35
Impenetrable boson limit
Example: solution of the Schroedinger equation for two
particles in harmonic oscillator at increasing γ
Psi_rel(x) a_ho
wavefunction
0.3
0.2
0.1
0
-4
-2
0
2
x/a_ho
relative
coordinate
4
PEPS-PTI 2012 – p.15/35
Impenetrable boson limit
Example: solution of the Schroedinger equation for two
particles in harmonic oscillator at increasing γ
Psi_rel(x) a_ho
wavefunction
0.3
0.2
0.1
0
-4
-2
0
2
x/a_ho
relative
coordinate
4
For γ → ∞ the many-body wavefunction vanishes at
contact
Ψ(...xj = xℓ ...) = 0
PEPS-PTI 2012 – p.15/35
Impenetrable boson limit
Example: solution of the Schroedinger equation for two
particles in harmonic oscillator at increasing γ
Psi_rel(x) a_ho
wavefunction
0.3
0.2
0.1
0
-4
-2
0
2
x/a_ho
relative
coordinate
4
For γ → ∞ the many-body wavefunction vanishes at
contact
Ψ(...xj = xℓ ...) = 0
“Fermionization”: interactions play the role of the Pauli
exclusion principle: the many-body wavefunction can be
constructed exactly
PEPS-PTI 2012 – p.15/35
Girardeau exact solution
Interactions are turned onto a cusp condition
Exact solution by mapping onto noninteracting fermions
[MD Girardeau, 1960]
1
Ψ(x1 ...xN ) = Π1≤j<ℓ≤N sign(xj − xℓ ) √ det[ul (xk )]
N!
with ul (x) single particle orbitals
satisfies the many-body Schroedinger equation and the
boundary conditions in each coordinate sector
valid for arbitrary external potential, also time dependent
Further progress in harmonic potential with ul (x) ∝ Hl (x)e−x2 /2 :
Ψ(x1 ...xN ) = Π1≤j<ℓ≤N |xj − xℓ |e
−
P
j
x2j /2
PEPS-PTI 2012 – p.16/35
Fermionic aspects of the TG gas
observables that do not depend on the sign of Ψ(x1 ...xN )
Density profiles of a TG gas in harmonic trap:
n(x) =
Z
dx2 ...dxN |Ψ(x, x2 ...xN )|2 =
N
X
j=1
|uj (x)|2
Green’s function method for
large N
[P Vignolo, AM, MP Tosi, (2000)]
n(x) a_ho profile
density
2
1.5
1
0.5
0
-5
0
x/a_ho in the trap
position
5
the density profile coincides with the one of a Fermi gas
PEPS-PTI 2012 – p.17/35
First-order coherences: the one-body
density matrix and momentum distribution
of a strongly interacting 1D Bose gas
PEPS-PTI 2012 – p.18/35
One-body density matrix
important for bosonic systems
A measure of first-order coherence and quasi long-range
order
ρ1 (x, y) = hΨ† (x)Ψ(y)i
associated to the momentum distribution
R
n(k) = dxdy eik(x−y) ρ1 (x, y)
a truly bosonic observable: very different from the one of the
mapped Fermi gas
PEPS-PTI 2012 – p.19/35
One-body density matrix
from Luttinger liquid approach
Large-distance behaviour from regularized LL model:
general structure at arbitrary interactions [N Didier, AM,
F Hekking, (2009)]
additional terms wrt [D Haldane,
1981] h ,P
P∞
∞
cos(2mx) P∞
b′n
a′n
1
ρ1 (x)∼|x|1/2K 1 + n=1 x2n + m=1 bm x2m2 K
+
n=0 x2n
P
i
P∞
′
∞
cn
sin(2mx)
c
,
n=0 x2n
m=1 m x2m2 K+1
the coefficients a′n , bm , b′n , cm and c′n are nonuniversal;
calculated by Bethe Ansatz [A Shashi et al, (2011)]
1.2
Popov Limit H
e 2-Γ
> 1.037L
4
1.0
B0
´
0.8
2Π2 A1
´
0.6
´
Tracy-Vaidya Limit H0.5214L
0.4
512Π10 A2
0.2
-8Π2B1
2
3
4
5
K
PEPS-PTI 2012 – p.20/35
One-body density matrix
of a Tonks-Girardeau gas, homogeneous case
in the TG limit: evaluation of a (N-1) dimensional integral
R
ρ1 (x, y) = N dx2 ..dxN ΨT G (x, x2 .., xN )Ψ∗T G (y, x2 , .., xN )
large-distance behaviour at large N : a mathematical
challenge [Lenard,
h Vadya and Tracy, Gangardt,..]
i
ρTG
1 (z) =
ρ∞
|z|1/2
with z = kF (x − y)
1−
1 1
32 z 2
−
1 cos(2z)
8 z2
−
3 sin(2z)
16 z 3
+ .. ,
at short distance: cusp behaviour originating from the
effect of the interactions [Forrester et al (2003)]
ρ1 (z)/ρ(0) = 1 −
z2
6
+
|z|3
9π
Lenard’s important simplification: reduction to the
calculation of the determinant of single-variable integrals!
PEPS-PTI 2012 – p.21/35
One-body density matrix
in harmonic trap, not translationally invariant
generalization of the Lenard’s trick:
ρ1 (x, y) again reduced to the determinant of
single-variable integrals
2
2
ρ1 (x, y) ∝ e−(x +y )/2 det[bjk (x, y)]
[Forrester et al, 2003]
with an analytic expression for bj,k =
R
2
dte−t |x − t||y − t|tj+k−2
⇒ cusp |x − y|3 at short distances
Absence of Bose-Einstein condensation: from natural
orbitals φj (x)
R
dyρ1 (x, y)φj (y) = λj φj (x)
√
eigenvalue of the lowest natural orbital λ1 ∝ N
PEPS-PTI 2012 – p.22/35
Momentum distribution
results at arbitrary interactions
uniform case, at small k, n(k) ∝ k 1/2K−1
large-momentum tails [AM, P Vignolo, MP Tosi (2002),
M Olshanii, V Dunjko (2003)]
n(k) = Ck −4
for large
k
from a theorem on Fourier transforms,
R
dze−ik(z−z0 ) |z − z0 |α−1 F (z) → |k|2α F (z0 ) cos(πα/2)Γ(α)
tails fixed by the Tan’s contact [S Tan, (2008), M Barth,
W Zwerger, (2011)]
C=
4
a21D
hΨ† (x)Ψ† (x)Ψ(x)Ψ(x)i
measuring two-body correlations
PEPS-PTI 2012 – p.23/35
Momentum distribution
of a TG gas in harmonic trap at zero temperature
40
n(p) p_ho
30
20
10
0
-10
-5
0
p/p_ho
5
10
bosonic vs fermionic case
PEPS-PTI 2012 – p.24/35
Momentum distribution tails
of a homogeneous Bose gas at zero temperature: Bethe
Ansatz result
1
γ =0.5
N n(k)
0.1
γ =8
γ =1
γ =0.25
γ=64
γ =2
γ =4
0.1
0.01
0.01
0.001
0.001
0.0001
0.0001
2
0.125
4
0.25
1e-05
8
0.5
1
2
4
8
k
[JS Caux, P Calabrese, NA Slavnov (2007)]
the weight of the tails increases at increasing interaction
strength γ : focus on the Tonks-Girardeau limit
PEPS-PTI 2012 – p.25/35
Finite temperature effects
description of a thermal Tonks-Girardeau gas
The Bose-Fermi mapping holds at finite temperature
[M Girardeau and K Das, (2002)]: for any N-particle excited
state with quantum numbers α = {ν1 , ...νN }
F
i
=
Â|Ψ
|ΨB
N,α i
N,α
with  mapping operator
Statistical average of an observable Ô:
P
hÔi = N,α PN,α hΨFN,α |Â−1 Ô Â|ΨFN,α i
in the grand-canonical ensemble PN,α = e−β(EN −µN ) /Z,
PN
EN = j=1 ενj
PEPS-PTI 2012 – p.26/35
One-body density matrix
for a thermal Tonks-Girardeau gas
The fermionic observables are easy, eg density profile
PN
n(x) = j=1 fνj |uνj (x)|2
where fνj =
1
e
β(ενj −µ)
−1
Thermal one-body density matrix:
ρ1 (x, y) =
R
P
∗
dx
,
...dx
Ψ
(x,
x
..,
x
)Ψ
P
2
N
N
α
2
N
N,α
N α (y, x2 , .., xN )
N,α
...a formidable complexity?!
... how to sum on all the quantum states α and the
particle numbers N ?
PEPS-PTI 2012 – p.27/35
Lenard’s trick
(another one! [A. Lenard, J Math Phys 7, 1268 (1966)])
The bosonic one body density matrix as a series of
fermionic j−body density matrices
P∞ (−2)j
ρ1B (x, y) = j=0 j! (sign(x − y))j
Ry
× x dx2 ...dxj+1 ρj+1,F (x, x2 , ...xj+1 ; y, x2 , ...xj+1 )
PEPS-PTI 2012 – p.28/35
Lenard’s trick
(another one! [A. Lenard, J Math Phys 7, 1268 (1966)])
The bosonic one body density matrix as a series of
fermionic j−body density matrices
P∞ (−2)j
ρ1B (x, y) = j=0 j! (sign(x − y))j
Ry
× x dx2 ...dxj+1 ρj+1,F (x, x2 , ...xj+1 ; y, x2 , ...xj+1 )
Factorization of the fermionic density matrices
ρ1F (x1 , x2 , ...xn ; x′1 , x′2 , ...x′n ) = det[ρ1F (xi , x′ℓ )]i,ℓ=1,n
with fermionic one-body density matrix ρ1F (x, y) =
PN
∗
j=1 fνj uνj (x)uνj (y)
PEPS-PTI 2012 – p.28/35
Lenard’s trick
(another one! [A. Lenard, J Math Phys 7, 1268 (1966)])
The bosonic one body density matrix as a series of
fermionic j−body density matrices
P∞ (−2)j
ρ1B (x, y) = j=0 j! (sign(x − y))j
Ry
× x dx2 ...dxj+1 ρj+1,F (x, x2 , ...xj+1 ; y, x2 , ...xj+1 )
Factorization of the fermionic density matrices
ρ1F (x1 , x2 , ...xn ; x′1 , x′2 , ...x′n ) = det[ρ1F (xi , x′ℓ )]i,ℓ=1,n
with fermionic one-body density matrix ρ1F (x, y) =
PN
∗
j=1 fνj uνj (x)uνj (y)
The j-variables integration can be reduced as
combination of single-particle integrals
P
Pj+1
j
ρ1B (x, y) = ν1 ..νj+1 fν1 ...fνj+1 k=1 uν1 (x)Aν1 νk (x, y)u∗νk (y)
PEPS-PTI 2012 – p.28/35
n(k)/aho (Log scale)
n(k)/aho
Thermal momentum distribution
2
1
0
0.1
0.01
0.001
0.0001
0
5
6
10
kaho (Log scale)
kaho
10
15
15
Zoom on the tails: they increase with temperature !
n(k) ∼ C/k 4 from the j = 1 term (only) of the one-body
density matrix
R
j=1
3
3
C = π dR F (R)
ρ1b ∼ |x − y| F ((x + y)/2)
PEPS-PTI 2012 – p.29/35
Finite temperature contact
20
Ca3ho
the weight of the
momentum distribution tails increases
with temperature
10
5
0.1
1
kB T /h̄ω
10
GPF (NSR)
GG (Haussmann et al.)
3
G G
0
(Strinati et al.)
F
2
Virial3
/(
different from a unitary
Fermi gas: [Hui Hu et al, 2011]
Nk )
0
Virial2
1
0
0.0
0.2
0.4
0.6
T/T
0.8
1.0
F
PEPS-PTI 2012 – p.30/35
High-temperature contact
We use Tan’s sweep theorem (from Hellman-Feynman
relation)
∂H
~2
dE
=h
i=
C
da1D
∂a1D
2m
with a1D = −2~2 /mg
PEPS-PTI 2012 – p.31/35
High-temperature contact
We use Tan’s sweep theorem (from Hellman-Feynman
relation)
∂H
~2
dE
=h
i=
C
da1D
∂a1D
2m
with a1D = −2~2 /mg
in its thermodynamic form [Hui Hu et al, (2011)], with Ω the
grandthermodynamic potential
dΩ ~2
C
=
da1D µ,T
2m
(by changing the interactions changes only the internal
energy)
PEPS-PTI 2012 – p.31/35
Virial approach
to understand high temperature behaviour of the contact
Virial expansion :
Ω = −kB T Q1 (z + b2 z 2 + b3 z 3 + ....)
with z = eβµ ,
P −βǫrel
Q2
Q1
b2 = Q1 − 2 , Q2 = Q1 ν e ν
and Qn = Tre−βHn , n-body cluster
PEPS-PTI 2012 – p.32/35
Virial approach
to understand high temperature behaviour of the contact
Virial expansion :
Ω = −kB T Q1 (z + b2 z 2 + b3 z 3 + ....)
with z = eβµ ,
P −βǫrel
Q2
Q1
b2 = Q1 − 2 , Q2 = Q1 ν e ν
and Qn = Tre−βHn , n-body cluster
High-temperature virial expansion for Tan’s contact
C=
2m
2
k
T
Q
(z
c2
1
~2 λdB B
+ z 3 c3 + ...)
∂bn
with cn = − ∂(a1D
/λdB )
PEPS-PTI 2012 – p.32/35
Virial approach
to understand high temperature behaviour of the contact
Virial expansion :
Ω = −kB T Q1 (z + b2 z 2 + b3 z 3 + ....)
with z = eβµ ,
P −βǫrel
Q2
Q1
b2 = Q1 − 2 , Q2 = Q1 ν e ν
and Qn = Tre−βHn , n-body cluster
High-temperature virial expansion for Tan’s contact
C=
2m
2
k
T
Q
(z
c2
1
~2 λdB B
+ z 3 c3 + ...)
∂bn
with cn = − ∂(a1D
/λdB )
in the TG limit, consequence of scale invariance:
universality! cn =constant (ie independent on temperature)
as (a1D /λdB ) vanishes
PEPS-PTI 2012 – p.32/35
Universal contact coefficient
we need the eigenvalues ǫrel
ν solution of the interacting
two-body problem and ∂ǫrel
ν /∂a1D
in harmonic trap ǫrel
ν = ~ω(ν + 1/2)
with ν from the transcendental equation
Γ(−ν/2)
Γ(−ν/2+1/2)
=
√
1D
− a2a
HO
PEPS-PTI 2012 – p.33/35
Universal contact coefficient
we need the eigenvalues ǫrel
ν solution of the interacting
two-body problem and ∂ǫrel
ν /∂a1D
in harmonic trap ǫrel
ν = ~ω(ν + 1/2)
with ν from the transcendental equation
Γ(−ν/2)
Γ(−ν/2+1/2)
=
√
1D
− a2a
HO
analytic expression for the universal coefficient c2
in the TG limit, ν = 2n + 1:
2β~ωλdB P Γ(n+3/2) −β~ω(2n+3/2)
e
c2 =
n
π
n!
evaluating the sum, taking the large temperature limit
c2 = √12
universal contact coefficient for the Tonks-Girardeau gas
PEPS-PTI 2012 – p.33/35
Finite temperature contact
Ca3ho
20
10
5
0.1
1
kB T /h̄ω
10
2
3
k
T
Q
(z
c
+
z
c3 + ...)
C = ~22m
B
1
2
λdB
high-temperature leading behaviour
q
2
BT
C = πN3/2 k~ω
using Q1 = kB T /~ω, z = N ~Ω/kB T , λdB =
q
2π~2
mkB T
PEPS-PTI 2012 – p.34/35
Conclusions
n(k)/aho (Log scale)
Coherences at finite temperature: tails
of the momentum distribution and hightemperature contact coefficients
n(k)/aho
Unievrsal properties of a Tonks-Girardeau gas
2
1
0
0.1
0.01
0.001
0.0001
0
5
6
10
kaho (Log scale)
kaho
10
15
15
PEPS-PTI 2012 – p.35/35