U
QEINO 2013, Poznan October 16, 2013
Universality of
extended Bose-Hubbard models
with local three-body interactions
Tomasz Sowinski
Institute of Physics of the Polish Academy of Sciences
Center for Theoretical Physics of the Polish Academy of Sciencesand
U
QEINO 2013, Poznan October 16, 2013
Universality of
extended Bose-Hubbard models
with local three-body interactions
Tomasz Sowinski
Institute of Physics of the Polish Academy of Sciences
Center for Theoretical Physics of the Polish Academy of Sciencesand
in
U
• Hamiltonian of the one-dimensional chain
ˆ
n
i= ˆ a
†iˆ a
i• ultra-cold atoms (bosons) in optical lattice
D. Jaksch et al.: Phys. Rev. Lett. 81, 3108 (1998)
J U
H = J ˆ
X L
i=1
ˆ
a † i (ˆ a i 1 + ˆ a i+1 ) + U
2
X L
i=1
ˆ
n i (ˆ n i 1)
- the tunneling amplitude J is determined
by the shape of the lattice potential
- the interaction energy U is determined
by the shape of the lattice and details
of the interaction potential
standard Bose-Hubbard model
U
H = J ˆ
X L
i=1
ˆ
a † i (ˆ a i 1 + ˆ a i+1 ) + U
2
X L
i=1
ˆ
n i (ˆ n i 1)
• grand canonical ensemble
µ N ˆ
J / U Jc
Jc
S. Ejima et al.: EPL 93, 3002 (2011)
N ˆ =
X
Ni=1
ˆ
n
iphase diagram
U
J / U Jc
Jc
S. Ejima et al.: EPL 93, 3002 (2011)
H = J ˆ
X L
i=1
ˆ
a † i (ˆ a i 1 + ˆ a i+1 ) + U
2
X L
i=1
ˆ
n i (ˆ n i 1) µ N ˆ
• grand canonical ensemble
how the properties of the studied model
will change when local three-body
interactions are taken into account
????
+ W
6
X L
i=1
ˆ
n i (ˆ n i 1) (ˆ n i 2)
phase diagram
U
J. Silva-Valencia, A. Souza: Phys. Rev. A 84, 065601 (2011)
in the presence of three-body
interactions the first insulating lobe
remains almost unchanged
DMRG with L up to 512
first insulating lobe
U
J. Silva-Valencia, A. Souza: Phys. Rev. A 84, 065601 (2011)
DMRG with L up to 512
first insulating lobe
in the presence of three-body
interactions the first insulating lobe
remains almost unchanged
T. Sowiński, R. W. Chhajlany: Phys. Scripta (in press) (2013)
on mean-field level
J c (µ) = µ(U µ)
z(U + µ)
U
T. Sowiński: Phys. Rev. A 85, 065601 (2012)
Exact Diagonalization with L up to 16
phase diagram
shape of the second insulating lobe in
contrast to the first one, depends crucially
on the three-body interactions
U
universality class
• Berenzinskii-Kosterlitz-Thouless transition
- one-dimensional Bose-Hubbard model belongs to
the universality class of the two-dimensional XY spin model
- the transition from the MI to the SF phase is of
the Berenzinskii-Kosterlitz-Thouless type
- the correlation length diverges as
• question
do the local three-body interactions
change the critical behavior of the system?
1 ⇠ ⇠ ⇠ exp
✓ const
p J c J
◆
U
universality class
local three-body interactions do not change
the characteristic properties of the phase
transition (K-T universality class)
T. Sowiński: Phys. Rev. A 85, 065601 (2012)U
pure three-body limit
in the limit of infinite local three-body
interactions the quantum phase transition
remains in the BKT class
T. Sowiński: ArXiv:1307.6852 (2013)U/W ➜ 0
DMRG with L up to 256
U
attractive two-body forces
U < 0
H = J ˆ
X
Li=1
ˆ
a
†i(ˆ a
i 1+ ˆ a
i+1) + U
2
X
L i=1ˆ
n
i(ˆ n
i1) + W
6
X
L i=1ˆ
n
i(ˆ n
i1) (ˆ n
i2) µ ˆ N
U
H = J ˆ
X
L i=1ˆ
a
†i(ˆ a
i 1+ ˆ a
i+1) + U
2
X
L i=1ˆ
n
i(ˆ n
i1) + W
6
X
L i=1ˆ
n
i(ˆ n
i1) (ˆ n
i2) µ ˆ N
infinite three-body repulsion limit (W ➜ ∞)
attractive two-body forces
DENSITY
SF PSF
● Mott Insulator (MI)
hˆai 6= 0, hˆa
2i 6= 0
hˆai = 0, hˆa
2i 6= 0
hˆai = 0, hˆa
2i = 0
● pair-superfluid (PSF)
● superfluid (SF)
S. Diehl et al., Phys. Rev. Lett. 104, 165301 (2010) Y. Lee, M. Yang, Phys. Rev. A 81, 061604(R) (2010)
Wha t will ha
ppe n if
W is la rge b ut finite
?
U <
0
U
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0 γ = 2.11
W = ∞
6 108 12 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
SF - PSF phase transition
• indicators of quantum phase transition
F (J, ) = |h G(J) | G(J + ) i|
- ground state fidelity
(J) = @ 2 F (J, )
@ 2 =0
- fidelity susceptibility QPT 1
QPT 0
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0 γ = 2.11
W = ∞
68 1012 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
# sites
T. Sowiński et al.: ArXiv:1304.4835 (2013)
W ➜ ∞
U
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0 γ = 2.11
W = ∞
68 1012 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
SF - PSF phase transition
T. Sowiński et al.: ArXiv:1304.4835 (2013)
• fidelity susceptibility near QPT
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0 γ = 2.11
W = ∞
68 1012 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
# sites
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0 γ = 2.11
W = ∞
6 108 12 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
(t, L) ⇠ L /⌫ (L 1/⌫ t)
- finite-size scaling
If all critical parameters were known then by plotting against for different system sizes all curves should collapse to the universal function.
L
/⌫L
1/t
W ➜ ∞
t = J J
cJ
c⇠ ⇠ |t|
⌫⇠ |t|
U
violation of the universality
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0
γ = 2.11
W = ∞
6 108 12 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0 γ = 2.11
W = ∞
6 108 12 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0 γ = 2.11
W = ∞
68 1012 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
# sites
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0 γ = 2.11
W = ∞
68 1012 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
0 20 40 60 80 100
0 0.1 0.2 0.3
χ
J/|U|
Jc = 0.19 ν = 1.0
γ = 2.11
W = ∞
6 108 12 0
0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
L 1/γ (J-J c )/J c
0 30 60 90 120
0 0.1 0.2 0.3
χ
Jc = 0.16ν = 1.0γ = 2.23
W = 4|U|
0 0.2 0.4 0.6
-8 -6 -4 -2 0 2 4 6
L -γ /ν χ
W ➜ ∞
the critical exponent γ varies with W and is thus
dependent on the MICROSCOPIC details of the model
T. Sowiński et al.: ArXiv:1304.4835 (2013)
U
Entanglement entropy
S(l) = c
6 log
sin
✓ ⇡l
L
◆
+ s(L) + O
✓ l
L
• entanglement entropy for block of size l ◆
hard constraint
model
DMRG with L up to 256
T. Sowiński et al.: ArXiv:1304.4835 (2013)
for one-dimensional systems the critical
exponents are determined by central charge c
U
conclusions
• repulsive two-body interactions (U>0)
• attractive two-body interactions (U<0)
H = J ˆ
X
L i=1ˆ
a
†i(ˆ a
i 1+ ˆ a
i+1) + U
2
X
L i=1ˆ
n
i(ˆ n
i1) + W
6
X
L i=1ˆ
n
i(ˆ n
i1) (ˆ n
i2)
- shape of the second insulating lobe in contrast to the first one, depends
crucially on the three-body interactions
- the universality class of MI-SF phase transition does not depend on the
strength of the three-body interactions (at least in 1D)
- critical behavior of the system undergoing a PSF-SF phase transition
depends on the value of the three-body repulsion
- critical exponent and the central charge governing the quantum phase
transitions are shown to have repulsion dependent features
T. Sowiński, R. W. Chhajlany, O. Dutta, L. Tagliacozzo, M. Lewenstein: ArXiv:1304.4835 (2013) T. Sowiński: Phys. Rev. A 85, 065601 (2012)
T. Sowiński: ArXiv:1307.6852 (2013)
T. Sowiński, R. W. Chhajlany: Phys. Scripta (in press) (2013)