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Model
Bosonic two-leg ladder under flux:commensurate-incommensurate transition
and second incommensuration
E. Orignac
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique,F-69342 Lyon, France
May 10, 2016
Edmond Orignac Boson ladder in flux
Model
Collaboration & Publications
Coworkers
Roberta Citro (Universita degli Studi di Salerno)
Maria Luisa Chiofalo (Universita di Pisa)
Stefania De Palo, Mario Di Dio (Universita degli Studi diTrieste)
References
1 Phys. Rev. B 92, 060506(R) (2015)
2 arXiv:1601.06573 to appear in New J. Phys. (2016)
Edmond Orignac Boson ladder in flux
Model
Outline
1 Meissner-Ochsenfeld effect in superfluids
2 Non-interacting boson ladder in flux
3 Interacting boson ladder in a flux : C-IC transition and firstincommensuration
4 Second incommensuration
5 conclusions
Edmond Orignac Boson ladder in flux
Model
The Meissner-Ochsenfeld effect
B<Bc1
<B<Bc2
Bc1
For B < Bc1, supercurrents fully screen the external field.
For Bc1 < B < Bc2 , vortices are formed around quantized fluxtubes.
Edmond Orignac Boson ladder in flux
Model
Vortex phase with bosonic cold atoms
s-wave superconductor superfluid bosons
order parameter 〈ψ†↑ψ†↓〉 = ∆e iθ 〈b〉 = ∆e iθ
generalized rigidity j = e|∆|22m∗
(~i ∇θ − eA
)j = |∆|2
2m~i ∇θ
Analogue of the vector potential
Rotation: Lorentz force e~v × ~B ⇔ Coriolis Force ~Ω× ~vArtificial gauge fields
Edmond Orignac Boson ladder in flux
Model
Rotating Superfluid Helium 4
E. J. Yarmchuk et al. Phys. Rev. Lett. 43, 214 (1979)
Edmond Orignac Boson ladder in flux
Model
Rotating ultracold Rubidium 87 condensate
From K. W. Madison et al. Phys. Rev. Lett. 84, 806 (2000)
Vortex position signaled by low boson densityVortex number increases with angular velocity
Edmond Orignac Boson ladder in flux
Model
Artificial gauge fields from geometric phase
Two-level system and adiabatic approximation
|χ〉 state vector with two components.
|Ψ〉 = ψ(r)|χ(r)〉
~∇i|Ψ〉 = ~
∇iψ(r)|χ(r)〉+ ~ψ(r)
∇i|χ(r)〉
~〈χ|∇i|Ψ〉 = ~
∇iψ(r) + ~ψ(r)〈χ(r)|∇
i|χ(r)〉
A = ~〈χ(r)|∇i|χ(r)〉
J. Dalibard et al. Rev. Mod. Phys. 83, 1523 (2011).
N. Goldman et al. Rep. Prog. Physics 77, 126401 (2014).
Edmond Orignac Boson ladder in flux
Model
Two-leg boson ladder in a flux
Spinless bosons on a ladder with flux ϕ
A
A
=+ϕ/2
=−ϕ/2
ϕ
Landau gauge: ~A ‖ chains
Edmond Orignac Boson ladder in flux
Model
Two-leg boson ladder in a flux
Hamiltonian and dispersion
H = −J‖∑j ,σ
(b†j ,σeiσϕ/2bj+1,σ + H.c.)− J⊥
∑j
(b†j ,↑bj ,↓ + H.c.)
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
-3 -2 -1 0 1 2 3
E-(
k)
k
φ=3π/4φ=0
Edmond Orignac Boson ladder in flux
Model
Equivalences
Definition of the currents in the ladder
j‖ = −iJ‖∑j ,σ
σ(b†j ,σeiσϕ/2bj+1,σ −H.c.)
j⊥ = −iJ⊥∑j ,σ
σb†j ,σbj ,−σ
Mapping to spin-1/2 bosons with spin-orbit coupling
j‖ → spin current
j⊥ → local magnetization along y
Edmond Orignac Boson ladder in flux
Model
Zero temperature phases of the ladder
Meissner phase: single minimum in dispersion
j‖(j) = J‖ sin(ϕ/2) ; j⊥(j) = 0.
Vortex phase: two minimas at ±kc in dispersion
j‖(j) =J2⊥ cos(ϕ/2)
2 sin2(ϕ/2)√
J2⊥ + 4J2
‖ sin2(ϕ/2)+ C1(ϕ) cos(2kc j)
j⊥(j) = C2(ϕ) sin(2kc j)
Edmond Orignac Boson ladder in flux
Model
Zero temperature phase diagram
From Atala et al. Nat. Phys. (2014)
Edmond Orignac Boson ladder in flux
Model
Experimental measurements
From Atala et al. Nat. Phys. (2014)
Edmond Orignac Boson ladder in flux
Model
What is the effect of interaction ?
Bose-Hubbard Hamiltonian
H = −J‖∑j ,σ
(b†j ,σeiσϕ/2bj+1,σ + H.c.)− J⊥
∑j
(b†j ,↑bj ,↓ + H.c.)
+U∑j ,σ
nj ,σ(nj ,σ − 1)
Edmond Orignac Boson ladder in flux
Model
Limit J⊥ J‖,U
Bosonization approach [EO, T. Giamarchi PRB 64, 144515 (2001)]
H = H0 + V
H0 =∑σ
∫dx
2π
[uK (πΠσ − σϕ/(2a))2 +
u
K(∂xφσ)2
]∂xθσ = πΠσ ; [φσ(x),Πσ′(x
′)] = iδσσ′δ(x − x ′)
bj ,σ = e iθσ (ja)∑m
Amcos(2mφσ − 2mπρσx)
nj ,σ/a = ρσ −1
π∂xφσ +
∑m
Bmcos(2mφσ − 2mπρσx)
V = 2J⊥
∫dxA2
0 cos(θ↑ − θ↓) + . . .
Edmond Orignac Boson ladder in flux
Model
Spin Charge separation
Change of variables
Πc =1√2
(Π↑ + Π↓) ; Πs =1√2
(Π↑ − Π↓)
φc =1√2
(φ↑ + φ↓) ; φs =1√2
(φ↑ − φ↓)
H = Hc + Hs
Hc =
∫dx
2π
[ucKc(πΠc)2 +
uc
Kc(∂xφc)2
]Hs =
∫dx
2π
[usKs(πΠs − ϕ/(
√8a))2 +
us
Ks(∂xφs)2
]−2J⊥A2
0
∫dx cos
√2θs
Edmond Orignac Boson ladder in flux
Model
Commensurate-Incommensurate transition
Semiclassical picture
cos√
2θs is imposing 〈θs〉 = nπ√
2
ϕ is imposing 〈θs〉 = ϕx/(√
8a)
low ϕ : stay with 〈θs〉 = 0 (Meissner)
large ϕ : form solitons in θs (Vortex)
x
θ−(x)
cϕ<ϕ
Meissner/Commensurate
θ−(x)
x
ϕ>ϕc
Incommensurate/
Vortex
Edmond Orignac Boson ladder in flux
Model
In the Meissner phase: commensurate phase
Observables
〈j‖〉 =uK
4πϕ
〈(nj↑ − nj↓)(n0↑ − n0↓)〉 = O(e−|j |/ξ)
〈j⊥(ja)j⊥(0)〉 = O(e−|j |/ξ)
〈bj ,σb†0,σ′〉 ∼
1
j1/(2Kc )
Mermin-Wagner ⇒ no U(1) breaking long-range order
Edmond Orignac Boson ladder in flux
Model
In the vortex phase: first incommensuration
Observables
〈j‖〉 =uK
4π(ϕ−
√ϕ2 − ϕ2
c)
〈(nj↑ − nj↓)(n0↑ − n0↓)〉 = − K ∗s4π2j2
+ B20
cos(2πρ0j)
|j |K∗s /2+Kc/2
Q ∝√ϕ2 − ϕ2
c
〈j⊥(ja)j⊥(0)〉 ∼ cos(2Qj)
|j |1/K∗s
〈bj ,σb†j ,σ′〉 ∼ δσσ′
cos(Qj/2)
|j |1/(2Kc )+2/(2K∗s )
Edmond Orignac Boson ladder in flux
Model
Limit U J⊥, J‖: Bogoliubov approximation
[A. Tokuno, A. Georges New J. Phys. 16, 073005 (2014)]
For ϕ < ϕc : reproduce Hc and Meissner state.
For ϕ > ϕc : reproduces Hc + H∗s and the Vortex state.
Edmond Orignac Boson ladder in flux
Model
Density Matrix Renormalization Group approach
Momentum distribution [M. Di Dio et. al., Phys. Rev. B 92,060506(R) (2015)]
n(k) =∑
k,σ e−ikj〈bjσb†0,σ〉 Ω↔ J⊥ , λ↔ ϕ , U = +∞.
Edmond Orignac Boson ladder in flux
Model
Other observables
[M. Di Dio et. al., Phys. Rev. B 92, 060506(R) (2015)]
C (k) = F .T .[j⊥(j)j⊥(0)]; Ss(k) = F .T .[(n↑ − n↓)j(n↑ − n↓)0]
0
0.5
1
0 0.5 1 1.5 2
0
0.5
1
0 0.5 1 1.5 2
2 S
s(k
)
k(π)
b)
0
0.5
1
0 0.5 1 1.5 2
0
0.5
1
0 0.5 1 1.5 2
2 S
s(k
)
k(π)
b)
0
0.5
1
0 0.5 1 1.5 2
0
0.5
1
0 0.5 1 1.5 2
2 S
s(k
)
k(π)
b)
0
0.5
1
0 0.5 1 1.5 2
0
0.5
1
0 0.5 1 1.5 2
2 S
s(k
)
k(π)
b)
0
0.5
1
1.5
0 0.5 1 1.5 2
0
0.5
1
1.5
0 0.5 1 1.5 2
C(k
)
k(π)
a)
0
0.5
1
1.5
0 0.5 1 1.5 2
0
0.5
1
1.5
0 0.5 1 1.5 2
C(k
)
k(π)
a)
0
0.5
1
1.5
0 0.5 1 1.5 2
0
0.5
1
1.5
0 0.5 1 1.5 2
C(k
)
k(π)
a)
0
0.5
1
1.5
0 0.5 1 1.5 2
0
0.5
1
1.5
0 0.5 1 1.5 2
C(k
)
k(π)
a)
0
0.5
1
0 0.5 1 1.5 2
0
0.5
1
0 0.5 1 1.5 2
Sc(k
)/2
k(π)
c)
0
0.5
1
0 0.5 1 1.5 2
0
0.5
1
0 0.5 1 1.5 2
Sc(k
)/2
k(π)
c)
0
0.5
1
0 0.5 1 1.5 2
0
0.5
1
0 0.5 1 1.5 2
Sc(k
)/2
k(π)
c)
0
0.5
1
0 0.5 1 1.5 2
0
0.5
1
0 0.5 1 1.5 2
Sc(k
)/2
k(π)
c)
0
4
8
12
16
-1 -0.5 0 0.5 1
0
4
8
12
16
n(k
)k(π)
d)
0
4
8
12
16
-1 -0.5 0 0.5 1
0
4
8
12
16
n(k
)k(π)
d)
0
4
8
12
16
-1 -0.5 0 0.5 1
0
4
8
12
16
n(k
)k(π)
d)
0
4
8
12
16
-1 -0.5 0 0.5 1
0
4
8
12
16
n(k
)k(π)
d)
0
4
8
12
16
-1 -0.5 0 0.5 1
0
4
8
12
16
n(k
)k(π)
d)
Edmond Orignac Boson ladder in flux
Model
Flux-filling phase diagram
M. Piraud et al. Phys. Rev. B 91, 140406 (2015)
Edmond Orignac Boson ladder in flux
Model
flux-hopping phase diagram
M. Piraud et al. Phys. Rev. B 91, 140406 (2015)
Edmond Orignac Boson ladder in flux
Model
Second incommensuration at flux π
Observables [M. Di Dio et. al., Phys. Rev. B 92, 060506(R)(2015)]
At half-filling, for a flux λ = π and increasingJ⊥/J = 0.5(magenta), 1(green), 1.25(blue), 1.5(black), 1.75(red).
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
S(k
)
k(π)
b)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
S(k
)
k(π)
b)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
S(k
)
k(π)
b)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
S(k
)
k(π)
b)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
S(k
)
k(π)
b)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
C(k
)
k(π)
a)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
C(k
)
k(π)
a)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
C(k
)
k(π)
a)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
C(k
)
k(π)
a)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
C(k
)
k(π)
a)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
ScB
OW
(k)/
2
k(π)
c)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
ScB
OW
(k)/
2
k(π)
c)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
ScB
OW
(k)/
2
k(π)
c)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
ScB
OW
(k)/
2
k(π)
c)
0
1
2
3
0 0.5 1 1.5 2
0
1
2
3
0 0.5 1 1.5 2
ScB
OW
(k)/
2
k(π)
c)
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
0
0.5
1
1.5
2
nσ(k
)k(π)
d)
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
0
0.5
1
1.5
2
nσ(k
)k(π)
d)
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
0
0.5
1
1.5
2
nσ(k
)k(π)
d)
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
0
0.5
1
1.5
2
nσ(k
)k(π)
d)
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
0
0.5
1
1.5
2
nσ(k
)k(π)
d)
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
0
0.5
1
1.5
2
nσ(k
)k(π)
d)
Edmond Orignac Boson ladder in flux
Model
Bosonized hopping for flux π
Gauge field along the rungs
Hhop. = J⊥∑
j
(−)jb†j ,σbj ,−σ,
bosonized form:
Hhop. =J⊥2πa
∫dx cos
√2φc
[cos√
2(θs + φs) + cos√
2(θs − φs)],
= J⊥
∫dx cos
√2φc(Jy
R + JyL ).
JyR/L are WZNW SU(2)1 currents.
Edmond Orignac Boson ladder in flux
Model
Mean-field treatment
Rotation and mean-field theory
rotation of π2 around x-axis
Jyν = Jz
ν , Jzν = −Jy
ν
represent Jzν with abelian bosonisation fields φs
Hhop.,MF =gc
πa
∫dx cos
√2φc +
hs
π√
2
∫∂x φs
hs ∼ J⊥〈cos√
2φc〉c,MF gc ∼ J⊥〈∂x φs〉s,MF
Edmond Orignac Boson ladder in flux
Model
Correlation functions from Mean-Field theory
Second incommensuration
hs ∼ J2⊥
〈j⊥(j)j⊥(j ′)〉 ∼ 1
2π2(j − j ′)2cos
(hs(j − j ′)
us
)+
(−1)j−j ′
|j − j ′|,
〈OπCDW (j)Oπ
CDW (j)〉 ∼ 1
2π2(j − j ′)2cos
(hs(j − j ′)
us
)+
(−1)j−j ′
|j − j ′|,
〈O0BOW (j)O0
BOW (j ′)〉 ∼ (−1)j−j ′
|j − j ′|cos
(hs(j − j ′)
us
)
Edmond Orignac Boson ladder in flux
Model
The second incommensuration disappears gradually
Flux ϕ slightly away from π
add to the Hamiltonian a term (ϕ− π)(JzR + Jz
L).
Rotation angle is now arctan[J⊥〈cos√
2φc〉/(ϕ− π)].
hs(ϕ) =√
hs(π)2 + J2‖ (ϕ− π)2
〈j⊥(j)j⊥(0)〉 ∼ (−)j
|j |hs(π)2 + J2
‖ (π − ϕ)2 cos(hs(ϕ)j/us)
hs(ϕ)2,
As φ− π increases, peaks at ϕ and 2π − ϕ are recovered.
Edmond Orignac Boson ladder in flux
Model
Second incommensuration for ϕ = π〈n↑ + n↓〉 ?
Interchain coupling
Hhop. = J⊥
∫dx[e−i√
2φc (J+R + J+
L ) + e i√
2φc (J−R + J−L )]
A U(1) symmetry is present:
e i√
2φc → e iαe i√
2φc
J+R + J+
L → e iαJ+R + J+
L
Incommensuration but where ?
J±R/L have conformal spin 1, and are expected to give rise toincommensuration in correlation functions.How can we determine the incommensurate correlations ?
Edmond Orignac Boson ladder in flux
Model
Mean Field theory versus Mermin-Wagner theorem
Standard mean field
HMFhop. = J⊥
∫dx[〈e−i
√2φc 〉(J+
R + J+L ) + 〈e i
√2φc 〉(J−R + J−L )
]= J⊥
∫dx[e−i√
2φc 〈J+R + J+
L 〉+ e i√
2φc 〈J−R + J−L 〉]
〈e−i√
2φc 〉, 〈J+R + J+
L 〉 are U(1) breaking order parameters.Since the Mermin-Wagner prohibits a U(1) breaking ground state,strong fluctuations around that mean-field restore U(1).
Edmond Orignac Boson ladder in flux
Model
Partial inclusion of fluctuations
Modified mean-field theory
1 Choose a particular mean field state with〈e−i
√2φc 〉 = |〈e−i
√2φc 〉|e iα.
2 Calculate correlation functions for that mean-field groundstate.
3 average the correlations over α
Limitations:
Underestimates the fluctuations around the mean field.
in particular, incorrectly leaves a gap in φc .
Not a proof of incommensuration, only a hint
Edmond Orignac Boson ladder in flux
Model
Incommensurate rung current correlations
Modified mean field result
〈j⊥(j)j⊥(j ′)〉 ∼ cos 2πρ0(j − j ′)
(j − j ′)2
[A + B cos
(hs(j − j ′)
us
)]+
cosπρ0(j − j ′)
|j − j ′|
[C + D cos
(hs(j − j ′)
us
)]ρ0 = 〈n↑ + n↓〉 ∼ ϕ/π
Edmond Orignac Boson ladder in flux
Model
rung current correlations from DMRG
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25
JrJr
(k)
k
λ=0.75 πΩ=0.0625t
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25
JrJr
(k)
k
λ=0.75 πΩ=0.0625
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25
JrJr
(k)
k
λ=0.75 πΩ=0.0625
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25
JrJr
(k)
k
λ=0.75 πΩ=0.0625
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25
JrJr
(k)
k
λ=0.75 πΩ=0.0625
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25 0 2 4 6
k
λ=0.75 πΩ=0.75t
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25 0 2 4 6
k
λ=0.75 πΩ=0.75
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25 0 2 4 6
k
λ=0.75 πΩ=0.75
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25 0 2 4 6
k
λ=0.75 πΩ=0.75
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25 0 2 4 6
k
λ=0.75 πΩ=0.75
0
0.25
0.5
0.75
1
1.25
0 2 4 6 0
0.25
0.5
0.75
1
1.25 0 2 4 6
k
λ=0.75 πΩ=0.75
ρ 1.0 0.75 0.5 0.25 0.125color black red green blue magenta
Edmond Orignac Boson ladder in flux
Model
Conclusions
Summary
1 Spin-orbit coupling/artificial gauge field experimentallyaccessible with cold atoms.
2 Meissner/Vortex transition = C-IC transition
3 for flux commensurate with density: second incommensuration
Edmond Orignac Boson ladder in flux
Model
Perspectives
Many chain case
1 A few chains: Is there still a second incommensuration ?
2 2D array of chains and the Pfaffian at ν = 1 ?
Edmond Orignac Boson ladder in flux
